Epidemiological SIR and SEIR ODE Models in Interdisciplinary Applications: Commonalities and Discipline-Specific Structural Differences

Abstract

Currently, epidemiological models can not only be found in epidemiology but also in other research disciplines. However, an interdisciplinary perspective that highlights the commonalities of epidemiological models across disciplines is missing. The goal of the current study is to foster such a perspective. To this end, a methodology is used that sets the current study apart from traditional review studies. Two benchmark epidemiological models formulated in terms of coupled ordinary differential equations, the susceptible–infected–recovered model and the susceptible–exposed–infected–recovered model, are followed through eight disciplines: epidemiology, virus dynamics within humans, computer viruses, drug addiction, voter dynamics, rumor spreading, sales dynamics, and viral marketing. Structural similarities and structural differences across these disciplines within the context of these two models are worked out. It is shown how the exact same mathematical structure can be applied for quite different interpretations across the selected disciplines. It is also shown that more complex model variants exhibit structural differences across research disciplines. In this way, this study helps researchers compare their own works on a structural level with related works in other disciplines. The particular importance of the current study is that it can boost progress in epidemiological modeling by making researchers aware of an interdisciplinary perspective.

1. Introduction

Epidemiological models are an indispensable tool to understand the spread of viral infections in human populations. However, time and again, they have popped up and have been studied in rather different disciplines. The popularity of epidemiological models in a variety of otherwise disconnected disciplines implies that scientists working in these disconnected disciplines run the risk of studying models that have been studied already in different contexts. Therefore, it is worthwhile to look at epidemiological models across disciplines, which is the objective of the current study.

The study of mathematical models describing the spreading dynamics of a virus in a human population has a long tradition [1,2]. As such, epidemiological models come in different types. Some frequently used epidemiological modeling approaches are listed in Figure 1: network models, ODE models, time-discrete models, stochastic models, time-delayed models, and models with nonlinear transition mechanisms. These characterizations do not necessarily describe mutually exclusive approaches. Rather, the approaches shown in Figure 1 are related to each other and may be combined. For example, ODE models can be derived from network models using mean field approximations [3,4,5,6]. ODE models are frequently simulated with the help of time-discrete models (e.g., [7,8,9]). Adding a noise term to an ODE model yields a model in terms of a stochastic differential equation [10,11]. Network models are typically stochastic models [3,4,5]. However, not every stochastic model is a network model (e.g., the aforementioned stochastic differential equation models are not network models). Time delays and nonlinear transition mechanisms can be incorporated into ODE models and stochastic models (e.g., [12,13,14], and see Section 3.9 and Section 5). In doing so, time-delayed models, nonlinear transition mechanism models, and ODE models or stochastic models may be combined with each other.

Addressing all approaches is a task beyond the scope of the current study. In the current study, the focus will be on the field of ODE models, and within that research field, two specific ODE models will be addressed in detail (see below). As such, all of the individual modeling approaches depicted in Figure 1 are important in their own merit. However, as indicated in Figure 1 and anticipated in part above, ODE models have a central position within that collection of theoretical advances. Accordingly, ODE models can be regarded as mean field approximations (i.e., simplifications) of network models. ODE models give rise to time-discrete models when discretizing time. ODE models can be generalized to take on the form of stochastic models and time-delayed models, and they can account for nonlinear transmission mechanisms (i.e., be combined with nonlinear transition mechanism models).

While over time epidemiological models have been improved and have increased in complexity, every now and then, a novel application of epidemiological models in a discipline different from epidemiology has been discovered. Currently, applications range from describing rumor spreading to the description of drug addiction and the explanation of sales curves of new market products. Reviews have typically taken either relatively narrow or relatively broad perspectives on this issue. For example, insightful (mini-)reviews focusing specifically on epidemiological models for infectious diseases, rumor spreading, drug addiction, and sales dynamics can be found in the literature [15,16,17,18,19,20]. That is, these reviews address a specific research field and review various different models or modeling approaches used in that field, as illustrated in panel (a) of Figure 2.

In contrast, interdisciplinary reviews typically take a broad perspective and address the modeling of complex network dynamics in general and in applications in epidemiology and other fields in particular [3,4]. That is, they focus on a general theoretical concept and work that concept out with the help of a variety of models and modeling approaches used in an array of research fields, as illustrated schematically in panel (b) of Figure 2. Both discipline-specific reviews on epidemiological models and interdisciplinary reviews on complex system dynamics are valuable sources of information. However, reviews of the first type, by definition, do not allow researchers to look across disciplines. Reviews of the second type tend to abound with a plenitude of diverse modeling approaches, which again makes it difficult for researchers to compare these approaches across disciplines. In particular, as indicated in panel (b), due to the relatively broad perspective of these reviews, there are naturally various empty cells (as indicated by dashes) in the matrix that plots the discussed models versus the addressed research fields. This indicates that the goal of such a review is typically not to demonstrate the general applicability of certain specific models or modeling approaches. These reviews are about the applicability of the general concept. At issue is the presentation of an interdisciplinary overview that focuses on some of the most fundamental epidemiological models, which is the objective of the current study. More explicitly, the goal of the present study is to present two fundamental epidemiological models, the SIR and SEIR models, and their applications in a variety of disciplines in order to work out explicit commonalities across disciplines and differences between disciplines. As mentioned above, to this end, we will focus on the ODE modeling approach (see Figure 1). That is, as such, in the current study, the modeling approach will be fixed and applications in various fields will be reviewed, as illustrated in panel (c) of Figure 2. However, in order to work out the general applicability of such ODE models in a more explicit, concrete way, the current review will narrow down the ODE modeling approach to the two aforementioned benchmark models: SIR and SEIR. In doing so, the current study will assume the structure shown in panel (d). In line with the schematic shown in panel (d), the first objective is to demonstrate that in a variety of disconnected disciplines, the exact same mathematical baseline models have indeed been studied and used, and will most likely be used in future studies. Making researchers aware of this issue allows them to take advantage of the findings obtained in unrelated disciplines. Roughly speaking, researchers do not have to reinvent the wheel again and again. The second objective is to discuss peculiarities and differences across disciplines. Looking across disciplines allows one to question an established approach in one discipline in view of an established approach in another discipline. For example, a particular mechanism may be well studied in discipline A but not in discipline B. There might be good reasons for that. If not, it might be worthwhile to address that mechanism in the context of discipline B. To be clear, the current study is not a review as such. As mentioned above, excellent discipline-specific reviews (i.e., type (a) reviews, see Figure 1) can be found in the literature. No attempt will be made to condense them to a single unified review. The current study is about two benchmark models (the SIR and SEIR models) that have been studied and applied in their own merit and are the basic building blocks of most more complex epidemiological models. The current study will address these models in the context of the following eight selected disciplines (see again panel (d) of Figure 2): epidemiology, virus dynamics, computer viruses, drug addiction, voter dynamics, rumor spreading, sales dynamics, and viral marketing/viral videos. In this way, this study attempts to deal with the trade-off of being specific and concrete on the one hand, and of illustrating generality on the other hand. This mixture of concreteness and generality should help and motivate researchers to look at models and disciplines not addressed in the current study in a similar way. Finally, it should be noted that this study is not about the solutions of models; it is about the (sub-)types of SIR and SEIR models used in epidemiology and other disciplines. It is about the structure of these models.

Before proceeding with the main findings, let us make a few comments on the methodological approach of the current study. First, in order to keep focused on an explicit set of mathematical models, in the current study, only SIR and SEIR models in the form of coupled ordinary differential equations will be considered. Second, in the current study, textbooks and (mini-)reviews have been evaluated in order to present the aforementioned SIR and SEIR models within coupled ordinary differential equation frameworks for the eight disciplines mentioned above. To this end, the following references have been used: Frank [1] and Rock et al. [2] for classical epidemiological models and virus dynamics models, Nwokoye and Madhussdanan [21] for modeling maleware spreading, van den Ende et al. [18] and Wang et al. [19] for epidemiological modeling of drug addiction, Turenne [17] for rumor spreading, Guidolin and Manfredi [20] for sales dynamics and its generalized research field, which is innovation diffusion, and Li et al. [16] for modeling viral marketing and the emergence and spread of viral videos. In addition, searches using Google Scholar and Scopus have been conducted to determine studies on SIR and SEIR models in all disciplines to supplement the aforementioned references if necessary. Note that in view of the study objectives listed above, the particular selection of references is not a crucial part of the current study. In particular, the first objective is to exemplify the utilization of the benchmark SIR and SEIR ordinary differential equation models in a variety of research fields. Any appropriate selection of references would fulfill that purpose. Likewise, note that the selection of the aforementioned eight disciplines or research fields is again not a crucial step of the current study. As will be shown in the results section below, the objective of the current study is to be as explicit as possible and to provide a back-to-back comparison of the epidemiological models across the selected disciplines. Such a enterprise requires a limitation of the number of disciplines. Any limited set of disciplines would serve the purpose of the current study. The selected disciplines should be considered as exemplary research fields in which the utilization of epidemiological models has attracted interest.

The remainder of this study is organized as follows. Section 2 discusses the use of SIR and SEIR (and their related SEI) models in the eight aforementioned disciplines. The focus is on mathematical equivalence or at least commonalities of the sub-types of SIR and SEIR models used in these fields. In Section 3, differences across disciplines are pointed out. That is, the focus is on discipline-specific peculiarities. For the sake of brevity, only the SIR model will be reviewed. In this context, a generalized SIR model will be introduced in Section 3.1. Section 4 continues the discussion about discipline-specific peculiarities. More precisely, in Section 4, different interpretations of the basic reproduction number, which is a key concept in epidemiology, will be briefly addressed. A brief discussion section including our conclusions will be presented in Section 5.

2. Commonalities: The SIR and SEI/SEIR Models Across Scientific Disciplines

As mentioned in the Introduction, in order to be explicit and focused, only models using coupled ordinary differential equations (ODEs) will be considered in what follows. There are various alternative mathematical descriptions that clearly have their own merit but are beyond the scope of the present study. Moreover, in order to point out the commonalities across scientific disciplines, only baseline versions of the SIR and SEI/SEIR models will be considered. Generalized versions will be addressed in Section 3.

2.1. The Baseline SIR Model in Epidemiology: The Susceptible–Infected–Recovered Model

The SIR model in epidemiology describes the spread of an infectious disease over time in a population with the help of three compartments [1,2,22]. The infectious disease is caused by a particular virus. The first compartment (or group) consists of susceptible individuals. They are not infected by the virus but can be infected by the virus under consideration. The second compartment (group) consists of infected individuals. In the context of the SIR model, these individuals are not just infected but are also infectious. That is, they can pass on the virus to susceptible individuals. In other words, they can infect susceptible individuals. Typically, the infected individuals show symptoms of the infectious disease caused by the virus. The third compartment (group) consists of recovered individuals. Typically, the word “recovered” refers to the notion that they have cleared the virus out of their bodies. Consequently, they cannot infect others. In addition, they have immunity against infection from the virus under consideration such that they cannot be infected again. While this is not crucial for the dynamics of the spread of the disease, it is typically assumed that individuals recover from the infectious disease as such at least to some extent. That is, they are in a healthier state as compared to infected individuals. Let S, I, and R denote the three aforementioned compartments. Then, $S\left(t\right)$, $I\left(t\right)$, and $R\left(t\right)$ are variables varying over time t. These variables correspond to real, positive numbers. They measure either densities or numbers of individuals. In the latter case, note that the variables $S,I,R$ should actually only assume integer values. However, for the sake of simplicity, real numbers are used. These variables will be referred to as state variables. The evolution equations for the state variables of the baseline SIR model read [1,2,22]

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\displaystyle \frac{\beta }{N}}SI-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& {\displaystyle \frac{\beta }{N}}SI-(\gamma +\mu )I,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}R& =& \gamma I-\mu R,\hfill \end{array}$$

(1)

with $B,\beta ,\mu ,\gamma \ge 0$, where B denotes the birth rate, $\beta$ denotes the effective contact rate (or transmission rate), $\mu$ denotes the natural death rate, and $\gamma$ denotes the recovery rate. In Equation (1), the variable N describes the total population size with $N=S+I+R$. The baseline model presented in Equation (1) does not account for disease-related deaths (for generalization, see Section 3). The stationary value $N_{st}$ of N is given by $N_{st}=B/\mu$ [1]. If the population is initially at its stationary value $N\left(0\right)=N_{st}$, then N is constant and the 3D model reduces to a 2D model involving the time-dependent state variables $S\left(t\right)$ and $I\left(t\right)$, defined by

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\displaystyle \frac{\beta }{N_{st}}}SI-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& {\displaystyle \frac{\beta }{N_{st}}}SI-(\gamma +\mu )I,\hfill \end{array}$$

(2)

and the auxiliary variable $R=N_{st}-S-I$. Using ${\beta }^{\prime }=\beta /N$ and ${\gamma }^{\prime }=\gamma +\mu$, the model may alternatively be expressed as

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\beta }^{\prime }SI-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& {\beta }^{\prime }SI-{\gamma }^{\prime }I.\hfill \end{array}$$

(3)

The model defined by Equation (1) accounts for vital dynamics (demographic effects) in terms of birth and death processes, as captured by the parameters B and $\mu$. If these vital dynamics are neglected, the 3D model reduces to the 2D model

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S=-{\displaystyle \frac{\beta }{N}}SI,{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I={\displaystyle \frac{\beta }{N}}SI-\gamma I$$

(4)

with $R=N-S-I$, where $N\ge S\left(0\right)+I\left(0\right)$ is the total population and constant. Again using ${\beta }^{\prime }=\beta /N$, the model can alternatively be expressed as

$$\begin{array}{c}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S={\beta }^{\prime }SI,{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I={\beta }^{\prime }SI-\gamma I.\end{array}$$

(5)

In the special case where $\gamma =0$, we consider a case in which infected individuals do not recover. That is, there is no compartment R. In this case, the SIR models (1)–(5) reduce to SI models. Most recently, in the context of the COVID-19 pandemic, SIR models (1)–(5) have been extensively used to describe the spread of COVID-19 infections (see, e.g., [1,7,15]).

2.2. The Baseline SI/SIR Model in Virus Dynamics: The Target Cells/Virus Load Model

The TV model of virus dynamics describes the spread of a virus in a human body in terms of the increase or decrease in the virus load in the body and the change in the number of target cells of the human being considered [1]. Target cells are cells that are attacked by the virus. Let T and V denote the target cells and the virus load, respectively. The variables $T\left(t\right)$ and $V\left(t\right)$ constitute the time-dependent state variables and are real, positive numbers. They are measured in suitable units and may reflect densities rather than cell counts or virus particle counts. Just as for epidemiological ODE models (see previous section), if T and V describe counts, then—in the context of the TV ODE modeling approach—the discrete nature of the cell or virus particle counts is neglected and the variables are considered as real numbers for the sake of simplicity. According to the TV model, the state variables evolve as follows [1]:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}T& =& B-{\beta }_{0}VT-\mu T,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}V& =& r{\beta }_{0}VT-k_{1}V\hfill \end{array}$$

(6)

with $B,{\beta }_0,\mu ,r,k_1\ge 0$, where B denotes the growth rate of the target cells, ${\beta }_0$ denotes the infection rate, $\mu$ denotes the natural death rate of target cells, r corresponds to a rescaled production rate parameter describing the production of virus particles by infected target cells, and $k_1$ is the effective decay rate of the virus load V (for details see [1]). An alternative interpretation of $k_1$ will be presented below in Section 2.10 on the TIV model. In general, T and V are measured in different units. The model can be rescaled by introducing the variable $I\left(t\right)=V\left(t\right)/r$ assuming $r>0$, which leads to

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}T& =& B-{\beta }^{\prime }IT-\mu T,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& {\beta }^{\prime }IT-k_{1}I\hfill \end{array}$$

(7)

with ${\beta }^{\prime }=r\beta$. It can be shown that due to this rescaling, the variable I exhibits the same units as T. A detailed discussion of this matter [1] shows that in fact, I describes the infected cells $I\left(t\right)$ associated with the virus load $V\left(t\right)$ at time t. Consequently, in the rescaled TV model defined by Equation (7), virus load is expressed in cell-like units, using the infected cells as an alternative measure. Importantly, for the purpose of the current study, Equation (7) is mathematically equivalent to the 2D SIR model defined by Equation (3). Whatever property the 2D SIR model (3) exhibits, the TV model (7) exhibits the same property, and after back-scaling, the original TV model defined by Equation (6) exhibits a similar, appropriately modified property. Whatever theorem is derived for the 3D SIR model (1), when the theorem is applied for the special case in which Equation (1) reduces to Equation (3), then the theorem holds as well for the rescaled TV model (7) and after appropriate back-scaling to the original TV model (6).

As argued above, the TV model may be considered as a counterpart to the baseline 2D SIR model of epidemiology. Since the TV model involves only two state variables and the SIR model as such involves three state variables, the TV model may be alternatively considered as a counterpart to the two-variable SI model. If so, an SI model that accounts for vital dynamics by means of two different decay parameters that describe the death rates of susceptibles and infected individuals, respectively, should be used. In such a model, it is assumed that due to the disease, the death rate of the infected individuals differs from the death rate of the non-infected, susceptible individuals.

2.3. The Baseline SIR Model Describing the Spread of Computer Viruses: The Susceptible–Infected–Removed Model

Various studies have used the SIR model and SIR-like models to examine the spread of computer viruses in networks of computers (see, e.g., [23,24,25,26,27,28,29,30]). To this end, the collection of computers is divided into susceptible computers that are not affected by the virus but can be infected, infected computers, and removed computers that have been removed from the network due to infection. Just as for the SIR model in epidemiology, infected computers are machines that not only are infected but can also pass the computer virus on to other machines. That is, they are infectious. Let S, I, and R denote the classes of susceptible, infected, and removed computers such that $\left(t\right)$, $I\left(t\right)$, and $R\left(t\right)$ constitute the relevant time-dependent state variables under consideration. For the sake of simplicity, they, again, correspond to real numbers rather than integers. The SIR-like models mentioned above are understood as modified SIR models that may feature time delays and/or additional classes of computers. Regardless of their modifications, SIR-like models have in common that the susceptible entities described by these models are infected due to contact with infected entities, and when this happens, make a transition from the class S to the class I [1]. In order to present the baseline SIR model, all modifications will be neglected in what follows (some of them will be addressed below in Section 3). Accordingly, for the baseline SIR model describing computer virus spreading, the evolution equations of the state variables read

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\beta }_{0}SI-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& {\beta }_{0}SI-(\gamma +\mu )I,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}R& =& \gamma I-\mu R,\hfill \end{array}$$

(8)

with $B,{\beta }_0,\mu ,\gamma \ge 0$. In Equation (8), B denotes the installation rate at which new computers are added to the network and $\mu$ denotes the rate at which computers are removed out of the network because they have reached their lifespan. $\gamma$ denotes the removal rate of computers that have not yet reached their lifespan but have been infected. ${\beta }_0$ denotes the infection rate. As such, the class of removed computers in the 3D model defined by Equation (8) may be neglected because that class does not play a role in the spread of the computer virus under consideration. However, when modifying the baseline model by taking into account that removed computers can be cleaned from the virus, restored, and entered into the network again, then the class R becomes part of the infection dynamics [23,24,25,26,29]. SIR models, as described by Equation (8), have also been used to describe the spread of malware in wireless sensory networks [21].

2.4. The Baseline SIR Model for Drug Addiction: The White–Comiskey Susceptible–Addicted–Treated Model

White and Comiskey [31] introduced one of the earliest epidemiological ODE models describing the spread of drug addiction in a population. Their model involves compartments of susceptible individuals, drug users who are not under treatment, and drug users who are receiving treatment. Let us denote these compartments with S, $U_1$, and $U_2$, respectively. The group of susceptible individuals consists of individuals who are not addicted to the drug under consideration but who could potentially develop an addiction. Typically, non-addicted individuals at the age of 15 to 64 years constitute this group [31]. The White–Comiskey 3D model is a modified version of the epidemiological, baseline SIR model (1). When neglecting the modification, the evolution equations for the state variables S, $U_1$, and $U_2$ read

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\displaystyle \frac{{\beta }_1}{N}}{U}_{1}S-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{U}_1& =& {\displaystyle \frac{{\beta }_1}{N}}{U}_{1}S-(\gamma +\mu )U_1,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{U}_2& =& \gamma U_1-\mu U_2,\hfill \end{array}$$

(9)

with $B,{\beta }_1,\mu ,\gamma \ge 0$. In Equation (9), B denotes the rate at which individuals enter the susceptible group (also called the recruitment rate), whereas $\mu$ denotes the removal rate out of the susceptible compartment. According to the White–Comiskey approach, the parameter $\mu$ corresponds to the natural death rate. Accordingly, the terms $-\mu U_1$ and $-\mu U_2$ in the evolution equations of $U_1$ and $U_2$ describe the decay of their respective compartments due to natural deaths. Note that death due to drug addiction is not accounted for in Equation (9). Note also that the term $-\mu S$ should actually read $-(\mu +{\delta }_0)S$, where ${\delta }_0>0$ takes into account that individuals leave the compartment S due to aging (and not because they become deceased). Both issues can easily be accommodated by making some slight modifications to the model (see [31] and Section 3.5). As such, in Equation (9), we assume that ${\delta }_0$ can be neglected relative to $\mu$ and, likewise, drug-associated deaths can be neglected relative to natural deaths. The parameter ${\beta }_1$ denotes the probability that susceptible individuals will become drug users due to contact with drug users who are not under treatment. The parameter $\gamma$ describes the drug rehabilitation program enrollment rate. N is the population, made up by the three groups addressed by the model, t hat is, $N=S+U_1+U_2$. The simplified SUU model defined by Equation (9) is mathematically equivalent to the SIR model (1). We will briefly return to the full version of the SUU model in Section 3.

The baseline SIR model of epidemiology (1) and SIR-like models [1] have specifically been applied to describe alcohol addiction and smoking addiction. Let S, A, and R denote time-dependent state variables describing the number of susceptible individuals (i.e., non-addicted individuals), addicted individuals, and recovered individuals. In this context, recovered individuals are actually understood as individuals who are under treatment. That is, they may overcome their addiction until they pass away or they relapse. Ignoring relapse and other modifications, these SIR-like modeling approaches involve the simplified SUU model (9) as a core model, inputting $A=U_1$ and $R=U_2$. In this context, for the SIR-like modeling approach for alcohol addiction, see, for example, [32,33]; for SIR-like modeling approaches for smoking addiction see, for example, [34,35,36].

2.5. The Baseline SIR Model for Voter Dynamics: The Undecided–Decided–Nonvoter Model

In general, epidemiological ODE models have been used to describe voter dynamics [37,38,39] and party membership dynamics [40,41,42]. In this context, voter dynamics describe how the group of people who would vote for a particular party or candidate increases or decreases in size over time (e.g., prior to an upcoming election). Similarly, party membership dynamics describe how political parties increase or decrease over time in terms of their number of members. The main idea underlying such studies is that voters who are convinced of voting for a party or candidate can influence undecided voters to vote for that party or candidate as well. Likewise, party members can recruit others to become party members. That is, the idea to vote in a particular way or to enroll in a political party can be passed from an individual to another individual like a virus can be transmitted from an individual to another. The parameter $\beta$ in Equation (1) in this context is the persuasion rate at which voters of a particular party convince undecided voters to vote for their candidate or, likewise, the persuasion rate at which party members persuade non-party members to join their party.

The SIR model defined by Equation (1) was used by Yong and Samat [37] to describe the evolution of the group of voters who would vote in an election for a particular political candidate. $S\left(t\right)$, $I\left(t\right)$, and $R\left(t\right)$ correspond to the time-dependent state variables of the voting model and describe the number of undecided voters (S), voters who are decided to vote for the candidate of interest (I), and so-called recovered voters (R). The latter group of individuals consists of voters who at some point in time would had supported the candidate. However, at time t, these recovered individuals no longer have any interest in supporting or voting for the candidate. In this context, the parameter $\gamma$ in Equation (1) denotes the loss-of-interest rate, that is, the rate at which decided voters lose interest in their candidate and become recovered voters. Since the model focuses on a single candidate only, these individuals may be considered as non-voters (they may actually vote for another candidate—if so, it would be irrelevant to the model dynamics). According to the baseline SIR model (1), when applied to voting dynamics, a recovered voter stays in the recovered class for his or her remaining lifespan. Finally, $\mu$ in Equation (1) denotes the death rate just as in the case when Equation (1) is applied to describe the spread of an infectious diseases.

2.6. The Baseline SIR Model for Rumor Spreading: The Ignorant–Spreader–Stifler Model

Daley and Kendall [43,44] are two of the first authors to have suggested that the spread of a rumor in a population could be modeled in analogy to the spread of a virus in a population. Just as for the SIR model (1) of epidemiology, they distinguished three compartments: individuals who have not heard the rumor, individuals who are actively spreading the rumor, and individuals who are no longer spreading the rumor and, in doing so, stifle or suppress further spreading of the rumor. These compartments are now frequently referred to and denoted as ignorants (I), spreaders (S), and stiflers. In order to avoid confusion with the notation, stiflers are also called recovered individuals and are denoted by the symbol R. Using this notation, we are dealing with an ISR model rather than an SIR model. When an ignorant I comes into contact with a spreader S, then the ignorant I may learn about the rumor and may turn into a spreader S, such that an $I\to S$ transition occurs. A spreader S, in turn, may lose interest in spreading the rumor, and if so, turns into a recovered individual R. In this case, an $S\to R$ transition occurs. Note that the meaning of the symbols of I and S in the field of rumor spreading modeling is just opposite of their original meanings used in epidemiology. While in epidemiology S stands for the susceptible individual, when modeling rumor spreading, I stands for the susceptible individual. While in epidemiology I stands for the infectious person, in the context of rumor spreading, S stands for the infectious person. Using the ISR terminology, the evolution equations of the baseline rumor spreading model read [17,45,46,47,48]

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I=-\beta SI,{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S=\beta SI-\gamma S,{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}R=\gamma S$$

(10)

with $\beta ,\gamma \ge 0$. In Equation (10), the parameters $\beta ,\gamma$ describe the rates of the two aforementioned transitions: the $I\to S$ transition ($\beta$) and the $S\to R$ transition ($\gamma$). Rumors are typically short-lived, that is, the entire dynamics of a particular rumor, from its initial occurrence in a population to its disappearance, unfolds typically over a duration much shorter than the life of a human individual. Therefore, birth and death processes are frequently neglected when modeling rumors and are neglected in Equation (10). Consequently, the size of the total population of interest, where $N=I+S+R$, is constant. Equation (10) is equivalent to the 2D version (4) of the SIR model of epidemiology. Note that the original Daley–Kendall ODE model involves a term for $S\to R$ transitions that is not presented in Equation (10) and is specific for rumor spreading. We will briefly return to this peculiarity in Section 3.7. The baseline ISR rumor spreading model as presented in Equation (10) holds when this additional term can be neglected relative to the term $\gamma S$, which captures $S\to R$ transitions in Equation (10). Martins and Pinto [49] used a slightly different terminology to introduce their three-compartment rumor spreading model. They distinguished between ignorants (I), believers (B), and unbelievers (U), where believers and unbelievers are counterparts to the aforementioned discussed groups of spreaders and recovered individuals. Their model corresponds to Equation (10), setting $S=B$ and $R=U$ and scaling the total population to 1 for $N=1$, such that the state variables I,B, and U describe percentages of the total population. Interestingly, they take birth and death processes into account by adding appropriate terms into Equation (10) while, at the same time, the authors point out that these terms can typically be neglected.

2.7. The Baseline SIR Model of Sales Dynamics and Innovation Diffusion: The Non-Adopters–Adopters Model

In economics, epidemiological models in terms of ODEs have been used to describe the sales dynamics of new products on the one hand and innovation diffusion on the other hand [20]. The SIR model plays a crucial role in this context [20,50,51,52,53]. Epidemiological sales dynamics models typically describe the increase in the number of buyers of a new product over time (e.g., the number of buyers of black-and-white television sets when they came onto the market around 1950 [54]). The focus is on the market dynamics when a product is released for its first time and where it can be assumed that each buyer purchases only one product. In this scenario, the sales of the product are proportional to the number of buyers. As far as innovation diffusion is concerned, as such, it describes the spread of a new method, technology, or approach in appropriately defined groups or populations [20,53] (e.g., the use of electronic health records in hospitals [55]). The goal of epidemiological models for innovation diffusion is to describe the increase in users of such an innovative approach over time. Innovation diffusion is not necessarily linked to sales dynamics. However, the sales dynamics of new products may be considered as a special case of innovation diffusion: users get familiar with a new product, but in order to do so, they need to purchase it in the first place.

Irrespective of a possible connection between sales dynamics and innovation diffusion, buyers of a new product and entities (or individuals) who use a certain novel approach may be referred to as adopters [54]. In order to understand the adopter’s dynamics, the baseline SIR model of sales dynamics and innovation diffusion captures three types of individuals or entities [20]. First, there are the non-adopters, who have not yet bought the product of interest or have not yet adopted the new innovative approach under consideration. Second, there are the promoting adopters, who have bought the product or adopted the novel approach and convince other individuals or entities to buy or use it. Third, there are the non-promoting adopters, who previously promoted the product or approach but have lost interest in doing so. Consequently, only the second group can motivate non-adopters to become adopters. With this scenario in mind, the three groups can be seen as counterparts to the groups of susceptible, infected, and recovered individuals of the baseline SIR model presented in Section 2.1. In fact, in the literature, frequently, the SIR notation used in epidemiology is applied to sales dynamics and innovation diffusion. Accordingly, S denotes the compartment of non-adopters, I denotes the compartment of promoting adopters, and R denotes the compartment of non-promoting adopters. The baseline SIR model of sales dynamics and innovation diffusion is then defined by Equation (4) with $\beta ,\gamma \ge 0$ [20]. Birth and death processes are neglected since the focus is typically on sales or innovation dynamics that take place over a few years—a time span much shorter than the life expectancy of a human being. The term $\beta SI/N$, occurring in Equation (4), describes increases in the number of buyers of a new product or adopters of an innovative approach due to non-adopters who get into contact with promoting adopters. That is, it describes events in which non-adopters follow or imitate adopters. For this reason, the transition rate coefficient $\beta$ can be referred to as the imitation coefficient [54]. The coefficient $\gamma$ describes the rate at which promoting adopters turn into non-promoting adopters.

The baseline SIR model (4) captures sales or innovation dynamics due to contact between non-adopters and promoting adopters, that is, it accounts for word-of-mouth advertising. The famous Bass model of sales dynamics and innovation diffusion [54] takes into account that the number of adopters may increase independent of word-of-mouth advertising, which is an aspect that is neglected in the SIR model (4). We will return to the Bass model in Section 3. In summary, the baseline SIR model as defined by Equation (4) describes sales dynamics (in terms of buyer or adopter dynamics) and innovation diffusion when assuming that the word-of-mouth mechanism is the leading mechanism that derives these dynamics such that other mechanisms can be neglected.

2.8. The Baseline SIR Model for Viral Marketing and Viral Videos: The Unaware–Broadcaster–Inert Model

The research field of viral marketing and research centered around viral videos specifically focus on the spread of content in social media. Viral marketing takes place when individuals forward information about a product to others by using their social networks [56,57,58]. That is, they share marketing messages via emails, Facebook, Twitter, TikTok, etc. For companies, viral marketing is a word-of-mouth advertising method via the Internet that exploits the existing social contacts of potential customers [56,57]. Likewise, viral videos are videos that are shared and forwarded among friends on social platforms and, in doing so, rapidly increase in popularity [59,60,61]. While viral marketing is about selling a product, viral videos are, as such, free of a commercial aspect. Most frequently, viral videos are just entertaining [60]. The field of viral marketing may be considered as a special case in the broader field of the sales dynamics of novel products discussed in the previous section. Since spreading information on the Internet is, for modern societies, such an important factor, in what follows, viral marketing and its non-commercial counterpart, viral videos, will be discussed as a topic in its own merit.

In the context of viral marketing dynamics, a baseline ODE epidemiological model can be discussed that involves three types of individuals [57,58,62]: unaware individuals (U), broadcasting individuals (B), and inert individuals (I). The compartment of unaware individuals consists of people unaware of the marketing message. The broadcasting individuals are familiar with the marketing message and actively share it with their friends and contacts. The group of inert individuals consists of individuals that are, again, aware of the message of interest; however, they do not share it or no longer share it. The entire network under consideration consists of $N=U+B+I$ individuals. The simplest version of the UBI model assumes the standard form of the baseline SIR model (1) and reads

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}U& =& C-{\beta }_{0}UB-\mu U,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}B& =& {\beta }_{0}UB-(\gamma +\mu )B,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& \gamma B-\mu I,\hfill \end{array}$$

(11)

with $C,{\beta }_0,\mu ,\gamma \ge 0$. In Equation (11), C denotes the growth rate of the social network being considered. Similarly, $\mu$ describes the exit rate at which individuals leave the network. ${\beta }_0$ denotes the effective contact rate at which broadcasting individuals contact unaware individuals about the product of interest and lead them to share the information. That is, the ${\beta }_{0}UB$ term quantifies the transition rate of unaware individuals to broadcasting individuals. The parameter ${\beta }_0$ may be replaced by $\beta$ like ${\beta }_0=\beta /N$ in order to introduce an effective contact rate that is more or less independent of the size of the community of interest. $\gamma$ describes the rate at which broadcasting individuals lose interest in sharing information about the product under consideration. The original UBI model as proposed in [57,58,62] involves additional terms that will briefly be reviewed in Section 3.

Viral marketing models do not necessarily use the UBI notation. They may simply use the SIR notation of epidemiology [56]. Likewise, SIR models used to describe the occurrence dynamics of viral videos in certain populations frequently simply adopt susceptible–infected–recovered terminology [59,60,61]. Accordingly, susceptible individuals are individuals unaware of the video of interest. Infected individuals are those individuals who have watched the video and distribute it further. Recovered individuals are individuals who know the video but do not share it (or do not share it any more). In this context, the SIR model defined by Equation (1) is used as a model describing the spreading of viral videos. However, viral videos are typically relatively short-lived. Therefore, the size N of the community in which a video becomes viral is usually assumed to be constant during the period of interest. As a result, the parameters B and $\mu$ occurring in Equation (1) are ignored (i.e., Equation (1) is used with $B=\mu =0$). Accordingly, when using the SIR notation, the baseline SIR model for viral video dynamics is eventually given by Equation (4) with $R=N-S-I$.

2.9. The Baseline SEIR Model in Epidemiology: The Susceptible–Exposed–Infectious–Recovered Model

The SEIR model of epidemiology is a four-compartment model that extends the three-compartment SIR model by adding the compartment E of exposed individuals. The term “exposed” is somewhat misleading. In the context of the SEIR model, “exposed” means that the individuals have been exposed to the virus and due to this exposure, have been infected. That is, exposed individuals have been in contact with the virus and due to this contact, the virus has successfully entered the bodies of these individuals and is replicating. Exposed individuals are infected individuals. The key difference between exposed individuals and the group I of the SIR model is that—at least in the baseline SEIR model—exposed individuals are not infectious (for a relaxation of this property, see Section 2.17 below). In contrast, the compartment I of the SEIR model describes individuals who are infected and infectious (just like the compartment I of the SIR model, see Section 2.1 above). The SEIR model is a stage model that describes the episode of infection of an individual with the help of two infected stages: the exposed stage (infected but not infectious) and the infectious stage (infected and infectious). In the case of the baseline SEIR model, the state variables $S,E,I,R$ satisfy the evolution equations [1]

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\displaystyle \frac{\beta }{N}}SI-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}E& =& {\displaystyle \frac{\beta }{N}}SI-(\alpha +\mu )E,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& \alpha E-(\gamma +\mu )I,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}R& =& \gamma I-\mu R,\hfill \end{array}$$

(12)

with the total population size $N=S+E+I+R$ and parameters $B,\beta ,\mu ,\gamma \ge 0$ having the same meaning as for the SIR model defined by Equation (1). The parameter $\alpha >0$ describes the transition rate at which exposed individuals become infectious. The fixed-point value N is given by $N_{st}=B/\mu$. In analogy to Equations (2) and (3) for $N\left(0\right)=N_{st}$, the 4D model is reduced to a 3D model with

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\beta }^{\prime }SI-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}E& =& {\beta }^{\prime }SI-{\alpha }^{\prime }E,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& \alpha E-{\gamma }^{\prime }I,\hfill \end{array}$$

(13)

where ${\beta }^{\prime }=\beta /N_{st}$, ${\alpha }^{\prime }=\alpha +\mu$, ${\gamma }^{\prime }=\gamma +\mu$ and $R=N_{st}-S-E-I$. In analogy to Equation (4), the SEIR model without vital dynamics (describing epidemics evolving on relatively short time scales) reads

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& -{\beta }^{\prime }SI,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}E& =& {\beta }^{\prime }SI-\alpha E,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& \alpha E-\gamma I\hfill \end{array}$$

(14)

with ${\beta }^{\prime }=\beta /N$ and $R=N-S-E-I$, where $N=S\left(0\right)+E\left(0\right)+I\left(0\right)+R\left(0\right)$ is constant.

The SEIR model (12) includes the SIR model (1) as a special case in the limiting case $\alpha \to \infty$ [2]. In this case, the product $\alpha E$ can be considered a fast variable that assumes at every time point t the sliding fixed-point variable $\alpha E\left(t\right)=\beta S\left(t\right)I\left(t\right)/N\left(t\right)$. When substituting this expression into the evolution equation for I in Equation (12), Equation (12) is reduced to Equation (1). Likewise, Equations (13) and (14) include the SIR models (2) and (4) as special cases for $\alpha \to \infty$. In general, the SEIR models that will be reviewed below reduce to their corresponding SIR models when the respective decay rate parameter of $E\left(t\right)$ becomes large relative to the remaining model parameters. This implies that solutions and theorems that have been obtained for SEIR models also hold for the corresponding SIR models provided the aforementioned limiting case of infinitely large decay rates is considered (and can be carried out in the solutions or theorems of interest).

Just as for the SIR model (1) and its variants (2)–(5), in the special case of SEIR models with $\gamma =0$, we model infectious individuals who do not recover. That is, there is no compartment R. The SEIR models (12)–(14) are reduced to SEI models.

2.10. The Baseline SEI/SEIR Model in Virus Dynamics: The Target Cells–Infected Cells–Virus Load Model

A benchmark model in the field of virus dynamics is the three-compartment TIV model that describes the changes in non-infected target cells T, infected cells I, and virus load V [1]. The target cells–virus load model of Section 2.2 is a special case of the TIV model, which will be shown below.

2.10.1. The TIV Model

According to the TIV model, the virus infects non-infected target cells and turns them into infected cells. The infected cells, in turn, produce virus particles. When focusing on the baseline version of the TIV model, the evolution equations of the state variables T, I, and V read [1]

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}T& =& B-{\beta }_{0}VT-\mu T,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& {\beta }_{0}VT-k_{1}I,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}V& =& pI-k_{2}V\hfill \end{array}$$

(15)

with the parameters $B,{\beta }_0,\mu \ge 0$ having the same meaning as for the TV model discussed in Section 2.2. In the context of the TIV model, the parameters $k_1\ge 0$ and $p\ge 0$ denote the death rate of the infected cells and the production rate of virus particles, respectively. Furthermore, $k_2$ is the clearance rate at which the body removes virus particles.

2.10.2. $V\to V^{\prime }$ Rescaling and Equivalence to 3D SEIR Model

Rescaling the virus load variable as $V^{\prime }=k_{1}V/p$ with ${\beta }^{\prime }={\beta }_{0}p/k_1$ such that $\beta V={\beta }^{\prime }{V}^{\prime }$ holds, and Equation (15) becomes [1]

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}T& =& B-{\beta }^{\prime }VT-\mu T,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& {\beta }^{\prime }{V}^{\prime }T-k_{1}I,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{V}^{\prime }& =& k_{1}I-k_2{V}^{\prime },\hfill \end{array}$$

(16)

The rescaled TIV model (16) is mathematically equivalent to the 3D SEIR model (13) when ignoring the recovered compartment and considering $\mu ,\alpha$, and ${\alpha }^{\prime }$ in Equation (13) as being three independent parameters. In summary, the TIV models (15) and (16) can be considered as SEI models or, alternatively, as 3D SEIR models. Moreover, it can be shown [1] that the rescaled virus load $V^{\prime }$ is measured in the same units as the cell variables T and I. That is, the rescaling procedure yields a measure for virus load that measures virus load in cell equivalents.

2.10.3. Elimination of I and Derivation of 2D TV Models

Let us return to the original TIV model (15). As mentioned above, the TV model (6) can be considered as a special case of the TIV model (15) where either V or I can be considered as a fast variable. In the former case, we deal with systems for which $k_2$ is large relative to the remaining model parameters such that it can be assumed that V, at any time t, assumes the sliding fixed-point value $V\left(t\right)=pI\left(t\right)/k_2=rI\left(t\right)$ with $r=p/k_2$. When replacing I in the evolution equation for I in Equation (15) by $V/r$, Equation (15) becomes the 2D TV model (6), as also seen in ref. [1]. Consequently, an alternative interpretation of the parameter $k_1$ showing up in the TV model, as seen in Equation (5), is that $k_1$ actually describes the decay rate of infected cells that do not show up explicitly in the TV model. However, the this decay rate parameter $k_1$ turns up in the TV model as the decay constant of the virus load V.

Alternatively, let us assume that $k_1$ is large relative to the remaining model parameters such that I is assumed to be the fast variable. I assumes at any time t the sliding fixed-point value $I\left(t\right)={\beta }_{0}V\left(t\right)T\left(t\right)/k_1$. Substituting this solution into the evolution equation of V in Equation (15), the 3D TIV model (15) becomes the 2D TV model $\mathrm{d}T/\mathrm{d}t=B-{\beta }_{0}VT-\mu T$, $\mathrm{d}V/\mathrm{d}t=r{\beta }_{0}VT-k_{2}V$ with $r=p/k_1$, which has the same structure as Equation (6).

2.11. The Baseline SEIR Model Describing the Spread of Computer Viruses: The Susceptible–Exposed–Infectious–Recovered Model

As reviewed above, the motivation to introduce the SEIR model in epidemiology is to obtain a more fine-grained description of the infected phase of individuals by spitting the phase into two stages. Likewise, it has been suggested that when a computer is infected by a virus, the virus may stay for a certain period in an inactive state in which it does not make any attempt to spread to other computers [63,64]. In analogy to the SEIR model of epidemiology, this inactive infected state is then referred to as an exposed state E. In contrast, the infectious state I is characterized by attempts of the computer virus to infect other computers. The baseline SEIR model of computer virus spreading is then given as a generalization of the SIR model (8) as

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\beta }_{0}SI-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}E& =& {\beta }_{0}SI-(\alpha +\mu )E,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& \alpha E-(\gamma +\mu )I,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}R& =& \gamma I-\mu R,\hfill \end{array}$$

(17)

with $B,{\beta }_0,\mu ,\gamma \ge 0$ having the same meaning as in the SIR model defined by Equation (8). The additional parameter $\alpha \ge 0$ in the SEIR model (17) denotes the transition rate at which inactive viruses become active (i.e., computers transition from exposed to infectious).

In the literature on malware propagation in wireless sensory networks, the SEIR model of the form (17) has been used as well, again motivated by the notion that an infected node does not immediate start to infect other nodes. Rather, there is a temporary inactive state of the node [21,65].

2.12. The Baseline SEIR Models for Drug Addiction: Two-Stage Models

Drug addiction is assumed to develop over time in certain stages [66]. Therefore, in order to improve the realism of epidemiological models of drug addiction, stage models have been proposed that involve different stages of addiction. These stages differ in severity of the condition. The simplest, non-trivial stage models are two-stage models. Drug users start in stage 1 and then progress to stage 2. Two illustrative examples are models distinguishing between experimental drug users and addicted users [67] and models considering psychologically addicted users on the one hand, and physiologically addicted on the other hand [66,68].

2.12.1. Susceptible Users–Experimental Users–Addicted Users–Treated Users Model

The distinction between experimental drug users and addicted drug users has frequently been made in the literature on tobacco addiction. In this context, experimental smokers are occasional smokers who smoke on certain occasions when they are together with other smokers. That is, smoking is part of their social activities. Importantly, they could quit any time without experiencing withdrawal symptoms. Furthermore, they do not smoke alone. In contrast, addicted smokers are regular smokers who experience withdrawal symptoms when quitting smoking. Smoking is not necessarily associated with a social activity, and smoking alone is what they frequently practice. The four-compartment POSQ model involves non-smokers N, the aforementioned occasional smokers O, addicted smokers S, and finally, ex-smokers Q, who have quit smoking [11,69,70,71]. The baseline version of the POSQ (or SEIR) model in this context reads

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}P& =& B-{\beta }_{0}PS-\mu P,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}O& =& {\beta }_{0}PS-(\alpha +\mu )O,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& \alpha O-(\gamma +\mu )S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}Q& =& \gamma S-\mu Q,\hfill \end{array}$$

(18)

with $B,\mu \ge 0$ having the same meaning as in the baseline SIR model of drug addiction (9). ${\beta }_0$ is the contact-induced addiction rate at which non-smokers become occasional smokers when in contact with addicted, regular smokers. The parameters $\alpha$ and $\gamma$ in Equation (18) denote the progression rate from occasional smokers to addicted smokers and the rate of quitting smoking. In principle, occasional smokers may lure non-smokers into becoming occasional smokers just as regular smokers do. In the model (18), this contribution is assumed to be small and, consequently, is neglected. In order to account for this kind of addiction mechanism, the $2\beta$ SEIR model may be used, as seen in Section 2.17.

2.12.2. Susceptible–Psychologically Addicted–Physiologically Addicted–Treated Model

Addiction to a particular drug may begin in the form of a psychological dependency on that drug that over time turns into a physiological dependency. Following this line of thought, epidemiological ODE models for drug addiction have been formulated that involve the groups of psychologically addicted (P) and physiologically addicted (H) drug users [66,68]. Let S denote the susceptible individuals (non-addicted individuals) and T denote the addicted users under treatment. Then, in the case of the baseline SEIR model for drug addiction, the state variables S, P, H, and T satisfy the evolution equations

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\beta }_{0}SH-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}P& =& {\beta }_{0}SH-(\alpha +\mu )P,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}H& =& \alpha P-(\gamma +\mu )H,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}T& =& \gamma H-\mu T,\hfill \end{array}$$

(19)

where $B,{\beta }_0,\mu ,\alpha ,\gamma \ge 0$ should be interpreted in the same way as for the SEIR drug addiction model (18). In particular, $\alpha$ denotes the transition rate from psychological addiction to physiological addiction. The model (19) states that only physiologically addicted users inspire susceptible individuals to take drugs—which is an oversimplification. In principle, psychologically addicted drug users are also likely to lure non-addicted individuals into addiction just as physiologically addicted users. Consequently, in the baseline model formulated in Equation (19), this contribution is assumed to be small and is neglected. In general, a generalized version (a $2\beta$ version) should be used that captures the influences of both addicted groups on susceptible individuals, as seen in Section 2.17.

2.13. The Baseline SEI Model for Voter Dynamics: Voter–Party Members Models and Voter–Cadres Models

The SIR model of voter dynamics discussed in Section 2.5 models decides voters who persuade undecided voters to vote for a candidate or party. The SEI model of voter dynamics distinguishes between voters and political influences. These influencers might be party members or high-ranking politicians (cadres) [41,72]. In the baseline SEI model, it is assumed that the political influencers convince undecided voters to vote in favor of a certain party or candidate. Let S denote the compartment of susceptible individuals or undecided voters. Let V denote the compartments of voters in favor of a party X and let M denote the compartment of party members of X. As usual, $S\left(t\right)$, $V\left(t\right)$, and $M\left(t\right)$ then describe the number of individuals in those compartments evolving over time t in terms of real numbers (ignoring the discrete nature of the problem at hand). In the case of the baseline SEI model of voter dynamics, the state variables S, V, and M satisfy the ODE model

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\displaystyle \frac{\beta }{N}}SM-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}V& =& {\displaystyle \frac{\beta }{N}}SM-(\alpha +\mu )V,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}M& =& \alpha V-\mu M\hfill \end{array}$$

(20)

with $B,\beta ,\mu ,\alpha \ge 0$. The parameters B and $\mu$ capture demographic effects and describe the rate at which individuals enter and leave the overall population of eligible voters. The size of this group is $N=S+V+M$. $\beta$ is the persuasion rate upon contact with a party member, that is, the rate at which undecided voters become voters of the party X or its candidate due to contact with party members. $\alpha$ is the recruitment rate at which voters of a party X become party members. The model (20) is mathematically equivalent to the baseline SEIR model (12) of epidemiology when putting $\gamma =0$ in Equation (12) and considering the corresponding SEI model. We will return to more sophisticated models of party membership dynamics in Section 3.

2.14. The Baseline SEIR Model for Rumor Spreading: The Ignorant–Exposed–Spreader–Stifler Model

Epidemiological ODE models for rumor spreading have also taken into consideration that rumor spreading might involve a latent, intermediate class E of individuals that are infected but not infectious [17,73,74,75,76,77]. In the context of rumor spreading, this means that there are people who have heard the rumor but do not yet share it with others. The idea is that individuals may need time to check the nature of the rumor. In the baseline SEIR model of rumor spreading, all exposed individuals eventually turn into spreaders. A more sophisticated model will be addressed in Section 3.

For the sake of simplicity, let us use the original SEIR notation in epidemiology (rather than the ISR notation used in the context of the SIR model, see Section 2.6). Accordingly, let S, E, I, and R denote the number of individuals in the compartments of ignorants, exposed, spreaders, and stiflers, respectively. Then, the baseline SEIR model of rumor spreading is given by Equation (14) with $R+N-S-E-I$, where N denotes the size of the total population of interest and is assumed to be constant. In this context, it is assumed that rumors are short-lived such that demographic effects can be neglected. Equation (14) illustrates the difference between exposed individuals E and stiflers R. Both kinds of individuals know the rumor and do not spread it. However, exposed individuals are individuals who will become spreaders and will spread the rumor. In contrast, stiflers are individuals who had spread the rumor previously and have stopped spreading it.

2.15. The Baseline SEI/SEIR Models of Sales Dynamics: The Buyers–Reviewers and Buyers–Reviews Models

The epidemiological SIR model of sales dynamics discussed in Section 2.7 was motivated by the idea that buyers of a product influence other individuals to buy the same product. The SEI/SEIR models of sales dynamics distinguish between buyers and their positive reviews of the purchased product [78,79]. Accordingly, positive reviews, rather than the buyers of the product, convince customers to buy a product. Let us illustrate this idea using two SEI/SEIR models of sales dynamics that have been suggested in the literature. In what follows, only simplified versions of these models that highlight the main SEI/SEIR structure will be presented (the original models will be briefly addressed in Section 3).

Let S denote the potential buyers that have not yet bought the product. Let I denote the number of buyers of the product who have not yet posted a review. Let P denote the number of buyers who have posted a positive review. Then, the baseline SEI model of sales dynamics reads [78]

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\displaystyle \frac{\beta }{N}}SP-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& {\displaystyle \frac{\beta }{N}}SP-(\alpha +\mu )I,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}P& =& \alpha I-\mu P\hfill \end{array}$$

(21)

with $B,\beta ,\mu ,\alpha \ge 0$. B and $\mu$ capture the usual demographic effects (which might be neglected, e.g., if the focus is on the sales of a novel product during its first few years). N denotes the total population size, given by $N=S+I+P$. $\beta$ denotes the imitation coefficients, just as in the SIR model of sales dynamics reviewed in Section 2.7. However, in the context of the SEI model, individuals imitate buyers of the product and purchase the product because of its positive reviews. $\alpha$ is the rate at which buyers write positive reviews. According to Equation (21), buyers start as buyers without reviews and eventually, at some point in time, write a positive review (provided they do not pass away, if death processes are taken into account). This is an oversimplification that can easily be relaxed. That is, models can be constructed such that buyers without reviews do not necessarily transition to the compartment of buyers with reviews, as seen in Section 3. Equation (21) assumes the structure of the baseline SEIR model (12) of epidemiology when putting $\gamma =0$ in Equation (12) and considering the corresponding SEI model.

The second variant of the buyer-and-reviews models for sale dynamics focuses on the number of reviews itself rather than the buyers who post reviews. The following model is tailored to address the sales dynamics of a particular online shop X. Let N denote the non-shoppers (individuals who have not yet used that shop X), S denote the shoppers of X, and P denote the positive reviews about X generated by the shoppers of X [79]. Then, when considering the baseline SEI model, the evolution equations of the state variables N, S, and P read [79]

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}N& =& B-{\beta }_{0}NP-\mu N,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& {\beta }_{0}NP-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}P& =& pS-k_{2}P\hfill \end{array}$$

(22)

with $B,{\beta }_0,\mu ,\alpha ,k_2\ge 0$. The total community T of interest consists of $T=N+S$ individuals. B and $\mu$ denote again parameters of demographic terms describing the growth (B) and decay ($\mu$) of the community size T. ${\beta }_0$ denotes the imitation coefficient (which, in Equation (22), is not scaled to T, compared to Equation (21)). p denotes the production rate of positive reviews produced by the online shoppers of X. The NSP model (22) is mathematically equivalent to the TIV model (15) of virus dynamics. Consequently, when Equation (22) is rescaled in the same way as the TIV model (see Section 2.10), then Equation (22) becomes mathematically equivalent to the baseline SEIR model (14) when ignoring the R compartment but keeping $\gamma >0$. That is, while we introduced Equation (22) above as an SEI model, Equation (22) may be alternatively interpreted as a 3D SEIR model, in particular, as a counterpart to Equation (14)).

2.16. The Baseline SEIR Model for Viral Marketing: The Exposed-Versus-Active-Sharing Models

Researchers studying viral marketing models have also taken advantage of the SEIR approach in order to increase the realism of their models [80,81,82]. Accordingly, individuals who are exposed to the market information of a particular product may not immediately share that information with others. If so, they belong to the compartment of exposed individuals. In the baseline SEIR model of viral marketing, all individuals first enter this exposed stage. Subsequently, after a certain period of reflection or checking, they start sharing the information of interest among their friends. Let T denote the target individuals unaware of the new product or marketing message, E denote the exposed individuals (who are not sharing), and A denote the actively sharing individuals. Moreover, let D denote individuals in the so-called dormant state. These correspond to the inert individuals discussed in Section 2.8 in the context of viral marketing and the SIR model. Dormant individuals come from the group of active individuals, but have lost interest in sharing the marketing information under consideration. Let $T\left(t\right)$, $E\left(t\right)$, $A\left(t\right)$, and $D\left(t\right)$ denote the time-dependent state variables describing the sizes of their respective compartments. Then, when focusing on the baseline SEIR model, the state variables evolve as [80,81,82]

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}T& =& -{\beta }_{0}TA,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}E& =& {\beta }_{0}TA-\alpha E,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}A& =& \alpha E-\gamma A,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}D& =& \gamma A\hfill \end{array}$$

(23)

with ${\beta }_0,\alpha ,\gamma \ge 0$. ${\beta }_0$ denotes the effective exposure rate at which unaware individuals are confronted with the marketing message due to contact with sharing individuals such that they eventually (after passing through E) spread the information by themselves. $\alpha$ denotes the transition rate at which exposed (non-sharing) individuals begin sharing the information of interest. $\gamma$ is the rate at which actively sharing individuals become dormant. It is assumed that the total community of interest $N=T+E+A+D$ does not vary in size during the period under consideration. Consequently entry and exit terms involving the parameters B and $\mu$ as shown in the SIR viral marketing model defined by Equation (11) are neglected in Equation (23).

Exposed individuals may decide against sharing the information and may transition immediately to the dormant state. This possibility is considered in the original study by Putra et al. [80,81] but is neglected in Equation (23). We will return to this issue and similar issues of generalization in Section 3.

The actively sharing individuals A may be considered as influencers. Using the influencer terminology, the variable A in Equation (24) may be replaced by I such that the TEAD model becomes a TEID model [82].

2.17. The 2$\beta$ SEIR Model

The baseline SIR and SEIR models are models involving a single infectious compartment. In epidemiology, there may be more than one infectious group. More precisely, in order to increase the robustness of an epidemiological model, the overall group of infectious individuals may be split into several specific groups. For example, one may distinguish between asymptomatically infected and symptomatically infected individuals who are both infectious [1]. The $2\beta$ SEIR model captures infection dynamics induced by two infectious groups, E and I, and reads [1]

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\displaystyle \frac{{\beta }_{1}E+{\beta }_{2}I}{N}}S-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}E& =& {\displaystyle \frac{{\beta }_{1}E+{\beta }_{2}I}{N}}S-(\alpha +\mu )E,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& \alpha E-(\gamma +\mu )I,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}R& =& \gamma I-\mu R\hfill \end{array}$$

(24)

with $B,{\beta }_1,{\beta }_2,\mu ,\alpha ,\gamma \ge 0$. In Equation (24), S and R denote the compartment sizes of susceptible and recovered individuals, respectively. Since the model (24) reduces for ${\beta }_1=0$ the baseline SEIR model (12), the two infectious compartments may be referred to as compartments of exposed (E) and infectious (I) individuals. If so, it must be made clear that the exposed group consists of infectious individuals as well. Exposed and infectious individuals differ in their degree of infectiousness. Typically, it is assumed that the first group (E) is the less infectious group, while the second group (I) is the more highly infectious group. In general, the two groups exhibit two different effective contact rate parameters, which are denoted in Equation (24) by ${\beta }_1$ and ${\beta }_2$. Equation (24) can be generalized to more than two infectious groups. In this case, each group exhibits its own effective contact parameter. For a more detailed discussion of the $2\beta$ SEIR model, see ref. [1]. Note that in the baseline $2\beta$ model defined by Equation (24), all individuals in stage E will eventually (i.e., if they do not pass away earlier) transition to the next infectious stage I. This fixed disease progression route can be relaxed by adding in Equation (24) an appropriate transition term for $E\to R$ transitions such that individuals of the stage-one infectious individuals E can recover without becoming stage-two infectious.

These $2\beta$ SEIR models have been used in the disciplines reviewed above.

Table 1. Some representative applications of the $2\beta$ SEIR model in the disciplines discussed in the current study.

Discipline	Study/Reference
Epidemiology	[1]
Virus dynamics	—
Computer viruses	[83]
Drug addiction	[66,67,68,84,85,86,87,88,89]
Voter dynamics	[41]
Rumor dynamics	[73,74,75]
Sales dynamics and innovation diffusion	generalized buyers-and-reviewers model (21)
Viral marketing and viral videos	generalized model (23), see text

lists some representative studies and references in which $2\beta$ SEIR models or generalizations of them have been used.

For example, in the field of drug addiction, Mubayi et al. [84] distinguished between two types of moderate drinkers who exhibited a moderate degree of alcohol addiction. According to the $2\beta$ SEIR model by Mubayi et al. [84], occasional drinkers with no alcohol addiction could become moderate drinkers due to contact with both types of moderate drinkers. Moreover, drug addiction models that distinguish between psychologically and physiologically addicted individuals (see Section 2.12.2) typically take into account that both groups of addicted individuals can lure non-addicted individuals into addiction. That is, $2\beta$ SEIR models are typically used [66,68,86,87,88,89] rather than the single-$\beta$ SEIR model (19). Likewise, when modeling opioid addiction, Battista et al. [85] noted that both addicted drug users who are addicted to prescribed medications and addicted drug users who are addicted to non-prescribed drugs can have a negative influence on non-drug users and lead them into addiction. Consequently, the authors used a $2\beta$ SEIR model [85]. The single-$\beta$ SEIR model (20) describing voter and party member dynamics is a simplification of the original model suggested by Romero et al. [41]. In the original study, it is taken into account that not only party members but also decided voters may convince undecided voters to vote for the party (or candidate) of interest, which means that we are dealing with a $2\beta$ model. In fact, Romero et al. split the class of voters into two sub-classes. Consequently, their epidemiological voter dynamics model is actually based on a $3\beta$ SEIR model. When modeling rumor spreading with the help of a compartment of exposed individuals, some authors define exposed individuals, as reviewed in Section 2.8, namely, as individuals who keep the rumor to themselves and do not attempt to persuade others. Other authors define an exposed individual as an individual who enters the first of two stages in which individuals spread the rumor [73,74,75]. In the first stage, individuals show reduced rumor-spreading activity because they by themselves are still somewhat undecided whether or not they should spread the rumor. In contrast, in the second stage, individuals spread the rumor with total conviction [73]. In both stages, individuals exhibit persuasive power. Therefore, these authors used $2\beta$ SEIR models.

The buyers-and-reviewers model (21) assumes that only buyers who post reviews inspire non-buyers to purchase the product under consideration. However, it is plausible to assume that buyers who do not post reviews also have some influential potential. When individuals who are not in possession of the product of interest come into contact with an individual of that group, they might be tempted to purchase the product. To account for such instances of persuasion, the $\beta SP/N$ term in Equation (21) can be replaced by a $({\beta }_{1}I+{\beta }_{2}P)S/N$ term. If so, a $2\beta$ SEIR for sales dynamics is obtained. This possibility is listed in Table 1.

The argument made above in the context of rumor spreading can also be applied to the viral marketing model (23). Accordingly, exposed individuals may be redefined as individuals who are sharing a particular marketing message with others while they are still in the process of contemplating whether or not they should share the message at all. As a result, they spend less effort in sharing than actively sharing individuals. In this case, we would replace the ${\beta }_{0}TA$ term in Equation (23) by a $({\beta }_{1}E+{\beta }_{2}A)T$ term and would be confronted with a $2\beta$ SEIR model. Again, this line of inquiry is listed in Table 1.

3. SIR Models with General Linear Transition Mechanisms and Discipline-Specific Peculiarities

3.1. SIR Models with General Linear Transition Mechanisms

The SIR model with general linear transition mechanisms can be formulated in a concise form using the state vector $\mathbf{X}=(S,I,R)$ and the transition matrix A, which is a $3\times 3$ matrix. Accordingly, the evolution equation of $\mathbf{X}$ reads

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}\mathbf{X}=B\left(\begin{array}{c}\hfill 1\\ \hfill 0\\ \hfill 0\end{array}\right)+{\displaystyle \frac{\beta }{N}}SI\left(\begin{array}{c}\hfill -1\\ \hfill 1\\ \hfill 0\end{array}\right)+A\mathbf{X}.$$

(25)

Equation (25) describes the most general case of possible linear transitions between the three compartments of the SIR model. As mentioned in the Introduction, there are endless possibilities to generalize Equation (25) even further (see Figure 1). Linear transition mechanisms may be replaced or supplemented with nonlinear mechanisms. The terms $a_{ik}{X}_k\left(t\right)$ on the right-hand side may be replaced by delayed state variables like $a_{ik}{X}_k(t-{\tau }_{ik})$, where ${\tau }_{ik}\ge 0$ denotes a time delay. Alternatively, memory effects may be introduced by using fractional time derivatives instead of ordinary ones. In line with the scope of the current study, Equation (25) will be used to compare SIR models across disciplines and will be referred to as an SIR model with general linear ordinary transition mechanisms. For the sake of brevity, the term “ordinary” will be dropped. In terms of individual components, Equation (25) reads

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\displaystyle \frac{\beta }{N}}SI+a_{11}S+a_{12}I+a_{13}R,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& {\displaystyle \frac{\beta }{N}}SI+a_{21}S+a_{22}I+a_{23}R,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}R& =& a_{31}S+a_{32}I+a_{33}R.\hfill \end{array}$$

(26)

Equation (26) includes the baseline SIR model defined by Equation (1) as a special case for $a_{11}=-\mu$; $a_{12}=a_{13}=0$; $a_{21}=0$; $a_{22}=-(\gamma +\mu )$; $a_{23}=0$; $a_{31}=0$; $a_{32}=\gamma$; and $a_{33}=-\mu$.

The following sections will focus on the transition matrix A. In this context, in these sections, vital dynamics will be taken into account. More explicitly, the death rate parameter $\mu$ will be listed in the to-be-discussed matrices A as one possible contribution to the diagonal elements $a_{11},\dots ,a_{33}$. If these vital dynamics are ignored (which is plausible in several applications, as discussed previously), then this particular contribution drops out. Depending on the discipline being considered, this may imply that certain diagonal coefficients vanish.

3.2. Peculiarities of Epidemiological SIR Models in Epidemiology

ODE models in epidemiology that assume the general form (26) or generalize Equation (26) in the sense discussed in Section 3.1 have been studied.

Table 2. Typically used coefficients and related mechanisms in epidemiological SIR models (where $\mu$ and $\gamma$ stand for the death rate and recovery rate, respectively, as in the baseline SIR model (1)). Parentheses “()” indicate that it is open for debate whether the $I\to S$ pathway in the context of the SIR framework is biologically plausible or not (see text). Triple stars “***” indicate effects that have been studied in the literature but typically in the context of models that are more complex than the SIR model.

Coefficient	Mechanism(s)
$a_{11}$	$-\mu$
	*** environmental contamination (spontaneous $S\to I$ transitions) ***
	*** vaccination where R includes vaccinated cases ($S\to R$ transitions) ***
$a_{12}$	($I\to S$ transitions in SIS model [2,90]
$a_{13}$	waning immunity ($R\to S$ transitions) [22]
$a_{21}$	*** environmental contamination ($S\to I$) ***
$a_{22}$	$-(\gamma +\mu )$
	$I\to S$ transitions as in SIS models [2,90]
$a_{23}$	n.a.
$a_{31}$	*** vaccination where R includes vaccinated cases ($S\to R$) ***
$a_{32}$	$\gamma$
$a_{33}$	$-\mu$
	waning immunity ($R\to S$ transitions) [22]

summarizes the coefficients that have typically been addressed in such studies. Table 2 also presents some of the mechanisms captured by these coefficients.

The model parameters $\mu$ and $\gamma$ and their mechanistic interpretations have already been addressed in the context of the baseline SIR model (1). The coefficient $a_{12}$ (in combination with $a_{11}$) may be used to describe $I\to S$ transitions as in SIS models [2,90]. If $a_{12}\ne 0$ is used in the context of an SIR model, then this means that a virus can affect individuals differently: after recovery, some of them may acquire at least some temporary immunity ($I\to R$), while others do not $(I\to S)$. In Table 2, this infection dynamics route is put in parentheses. The reason for this is that if a virus has a relatively defined disease progression, then recovered humans either have a noteworthy period of immunity or do not. In the former case, $a_{21}=0$. In the latter case, an SIS model is used [2,90] rather than an SIR model. That is, the entire compartment R is dropped. Having said this, in the wake of the COVID-19 pandemic [1] and still ongoing COVID-19 infections, and in view of the plenitude of SARS-CoV-2 mutations and variants circulating worldwide, it is in fact possible that individuals who have just recovered from COVID-19 can be re-infected with COVID-19 after a relatively short period. Therefore, the plausibility of SIR models featuring $S\to I$ transitions via $a_{12}\ne 0$ is at least open for debate.

As indicated in Table 2, the coefficient $a_{13}$ (in combination with $a_{33}$) can be used to describe waning immunity [22]. Moreover, the coefficient $a_{21}$ (in combination with $a_{11}$) has been used in the SEIR modeling literature to describe the effect of virus-contaminated environments on infection dynamics [91,92,93,94,95]. The stars “***” indicate that in the context of SIR modeling studies, the effect of a contaminated environment is typically not a study objective and, consequently, the effect is ignored such that $a_{21}$ is set to zero. Having said that, as such, the effect could be addressed within an SIR framework. That is, a non-vanishing coefficient $a_{21}$ is not implausible. Likewise, the coefficient $a_{31}$ could be used to describe the effect of vaccination when defining the class R as recovered individuals or individuals who have been vaccinated against the virus of interest. The stars “***” indicate that in most SIR model-based studies that consider vaccination, this coefficient is actually not used. Rather, such studies introduce a new compartment of vaccinated individuals. That is, the coefficient $a_{31}$ is typically set as equal to zero in SIR model-based studies. Having said this, as will be shown below, in other disciplines, the coefficient $a_{31}$ of the SIR model is indeed used in the aforementioned sense of a vaccination parameter.

As pointed out in Table 2, SIR modeling studies in epidemiology typically assume that $a_{23}=0$ holds. That is, in such studies, $R\to I$ transitions are considered either as biologically implausible or are not part of the study objectives.

3.3. Peculiarities of Epidemiological SIR Models in Virus Dynamics

In the current study, the TV model (6) is considered as an SIR model since the baseline SIR model (1) and the TV model (6) exhibit the same key mathematical structure (see Section 2.2). However, the TV model does not exhibit a third compartment R. Therefore, only four out of the nine matrix elements of A are relevant.

Table 3. Coefficients and related mechanisms in the SIR model of virus dynamics as they show up in the TV model (6) (where $\mu$ and $k_1$ stand for the death rate of cells and the clearance rate of virus particles, respectively, as in Equation (6)).

Coefficient	Mechanism(s)
$a_{11}$	$-\mu$
$a_{12}$	n.a.
$a_{21}$	n.a.
$a_{22}$	$-k_1$

shows these matrix coefficients and which mechanisms they represent.

The fact that in the model $a_{12}=0$ holds means that $V\to T$ transitions (like in an SIS model) are biologically implausible in the context of virus dynamics. Furthermore, the coefficient $a_{21}$ in SIR modeling approaches is set to zero in order to indicate that in typical cell biological systems, the spontaneous production of virus particles V by non-infected target cells T does not happen. The induced production of virus particles only takes place via infected cells I, which in the TV model are described by the nonlinear term $r{\beta }_{0}VT$ (see Equation (6) and see the derivation of the TV model from the TIV model in Section 2.10).

3.4. Peculiarities of Epidemiological SIR Models for Computer Virus Spreading

Table 4. Coefficients and some of their mechanistic interpretations for SIR models describing computer virus spreading (where $\mu$ and $\gamma$ denote the age-related removal rate and the removal rate due to infection, respectively, as in the baseline SIR model (8) for computer virus spreading).

Coefficient	Mechanism(s)
$a_{11}$	$-\mu$
	anti-virus software implementation ($S\to R$) [29,96,97]
$a_{12}$	cleaning with immediate reactivation ($I\to S$) [96,97]
$a_{13}$	reactivation of cleaned, removed ($R\to S$) [96,97]
	waning anti-virus protection ($R\to S$) [24,25,29]
$a_{21}$	n.a.
$a_{22}$	$-(\gamma +\mu )$
	cleaning with immediate reactivation ($I\to S$) [96,97]
$a_{23}$	reactivation of infected, removed ($R\to I$) [96,97]
$a_{31}$	anti-virus software implementation $S\to R$ [29,96,97]
$a_{32}$	$\gamma$
$a_{33}$	$-\mu$
	reactivation of cleaned, removed ($R\to S$) [96,97]
	waning anti-virus protection ($R\to S$) [24,25,29]
	reactivation of infected, removed ($R\to I$) [96,97]

shows the non-vanishing coefficients of the transition matrix A and possible mechanistic interpretations of them for studies using the general SIR model (26) or generalization of it to describe the spreading of computer viruses.

The coefficient $a_{12}$ (in combination with $a_{22}$) can be used to describe virus cleaning of infected computers with immediate reactivation of the machines [96,97]. Note that cleaning does not provide these computers any protection against future virus attacks. In a similar vein, the matrix coefficient $a_{13}$ (in combination with $a_{33}$) may describe that removed computers that have been infected are reactivated after clearing them of computer viruses [96,97]. Alternatively, the matrix coefficient $a_{13}$ (in combination with $a_{33}$) may describe that anti-virus protection becomes outdated such that computers with anti-virus protection become vulnerable to virus attacks again [24,25,29]. In this context, note that the compartment R is not interpreted as computers removed from the network but as active computers protected against virus attacks due to anti-virus programs. The coefficient $a_{23}$ (in combination with $a_{33}$) may be used to capture when infected computers removed from the network are (by mistake) reactivated and connected to the network [96,97]. In such an event, $R\to I$ transitions take place. This interpretation requires R to be carefully defined as a compartment that includes infected computers that have been removed from the network and have not been cleaned of the virus. For example, when modeling both $R\to I$ and $R\to S$ transitions, the compartment R must be defined as computers that have been removed from the network, where some have been cleaned of the virus of interest and others have not.

The coefficient $a_{31}$ (in combination with $a_{11}$) can be used to model the implementation of anti-virus programs on computers such that these machines are protect against virus attacks [29,96,97]. In the context of the SIR model (26), this mechanism corresponds to $S\to R$ transitions. That is, in this case, the compartment R is interpreted as active computers having anti-virus protection. Note that alternatively, the SIR model may be supplemented with an additional compartment A that describes computers exhibiting anti-virus protection [23]. These two possibilities to model the roll-out of anti-virus protection should be considered as the two modeling possibilities of vaccination discussed above in Section 3.1 in the context of SIR models for infectious diseases.

The previous discussion of coefficients $a_{ik}$ with either $i=3$ or $k=3$ related to the compartment R illustrates that when applying the SIR model (26) to computer virus spreading, it is crucial to give a clear definition of the compartment R. This definition determines which kind of phenomena (or mechanisms) can be modeled within the framework of the three-variable SIR model.

Finally, computer virus spreading studies based on the SIR model (26) usually involve a vanishing transition coefficient $a_{21}$. The assumption that $a_{21}=0$ holds states that computers cannot be infected spontaneously. They can only be infected due to contacts with other computers. In other words, it is assumed that there are only induced infections as captured by the nonlinear $\beta SI/N$ term occurring in Equation (26).

3.5. Peculiarities of Epidemiological SIR Models for Drug Addiction

When applied to model the spread of drug addiction in communities, the general SIR model (26) may be used with the coefficients listed in

Table 5. Coefficients and mechanistic interpretations of the SIR model (26) when applied to describe drug addiction epidemics (where $\mu$ denotes the death rate, ${\delta }_0$ denotes the exit rate at which susceptible individuals age out of the age groups of interest, and $\gamma$ denotes the drug rehabilitation enrollment rate as discussed in Section 2.4).

Coefficient	Mechanism(s)
$a_{11}$	$-(\mu +{\delta }_0)$
	prevention programs (drug education) effects $S\to R$ [35]
$a_{12}$	n.a.
$a_{13}$	waning therapy/education protection ($R\to S$) [33,35,36]
$a_{21}$	n.a.
$a_{22}$	$-(\gamma +\mu )$
$a_{23}$	spontaneous relapse ($R\to I$) [33,34,69,98]
$a_{31}$	prevention programs (drug education) effects $S\to R$ [35]
$a_{32}$	$\gamma$
$a_{33}$	$-\mu$
	waning therapy/education protection ($R\to S$) [33,35,36]

. As in the previous sections, possible mechanistic interpretations of these coefficients are summarized in Table 5 as well.

The coefficient $a_{13}$ (in combination with $a_{33}$) can describe $R\to S$ transitions that in the context of drug addiction may be interpreted in two alternative ways. First, in line with the White–Comiskey SUU model (9), when interpreting R in the SIR model (26) as individuals who receive treatment and have stopped taking drugs, we may consider the possibility that such individuals over time can become susceptible again. That is, therapy may lose over time its protective effect against relapse [33,36] (note: as such, we are not talking about relapse to addiction, just about transitions to S). Alternatively, when interpreting R as a compartment that contains (among other kind of individuals) individuals who are protected from becoming addicted because they have received drug education via prevention programs, then such a protection again may lose its effectiveness over time [35]. If so, $R\to S$ transitions of protected (or addiction-resistant) individuals R take place. The coefficient $a_{23}$ (in combination with $a_{33}$) has been used to describe spontaneous relapse of treated individuals [33,34,69,98]. Spontaneous refers to the notion that individuals become addicted again without the influence of others. The coefficient $a_{31}$ (in combination with $a_{11}$) may be used to describe the aforementioned effect of prevention programs aiming at drug use education [35]. To this end, as mentioned above, the compartment R must either be defined to consist of individuals who are protected from becoming addicted because of such education or be defined to include such individuals. For example, when modeling waning protection against addiction of treated individuals and waning protection against addiction of drug-educated individuals with the help of the three-variable SIR model (26), the compartment R must be defined as containing both types of protected (or addiction-resistant) groups.

As indicated in Table 5, the SIR model (26), when applied to drug addiction, typically features two zero-value coefficients. The value $a_{12}=0$ reflects that it is typically assumed that drug addiction dynamics do not feature $I\to S$ transitions (like in SIS models; see the discussion in Section 3.1). That is, individuals can return to the susceptible state S only via state R (like $I\to R\to S$). The value $a_{21}=0$ means that some kind of spontaneous addiction does not exist (or if it exists can be neglected). Within the SIR framework (26), addiction can only be induced by other addicted people, as described by the nonlinear $\beta SI/N$ term.

In the latter context of transitions towards the compartment I, note that the original White–Comiskey study [31] as well as other studies (e.g., [32]) have examined induced relapses from $R\to I$. In order to accommodate such transitions, the SUU model (9) can be generalized as

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\displaystyle \frac{{\beta }_1}{N}}{U}_{1}S-(\mu +{\delta }_0)S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{U}_1& =& {\displaystyle \frac{{\beta }_1}{N}}{U}_{1}I-(\gamma +\mu )U_1+a_{23}{U}_2+{\displaystyle \frac{{\beta }_2}{N}}{U}_1{U}_2,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}{U}_2& =& \gamma U_1-(a_{23}+\mu )U_2-{\displaystyle \frac{{\beta }_2}{N}}{U}_1{U}_2\hfill \end{array}$$

(27)

with $a_{33}=-(a_{23}+\mu )$. In Equation (27), the adjustment of $\mu$ in the dynamics of S via the ${\delta }_0$ coefficient mentioned in Section 2.4 is also taken into account, as well as the aforementioned possibility of spontaneous relapses. The ${\beta }_2{U}_1{U}_2/N$ term in Equation (27) describes the induced transitions of treated individuals who become addicted again due to contact with addicted (non-treated) individuals and describes a nonlinear transition mechanism such that Equation (27) exemplifies a nonlinear transition mechanism model (see Figure 1). In summary, Equation (27) captures both spontaneous relapses (if $a_{23}>0$ holds) and induced relapses to addiction (if ${\beta }_2>0$ holds). For $a_{23}=0$ and ${\delta }_0=0$, Equation (27) corresponds to the original White–Comiskey SUU model presented in ref. [31].

3.6. Peculiarities of Epidemiological SIR Models for Voter Dynamics

Table 6. Coefficients and mechanistic interpretations of the SIR model (26) applied to voter dynamics (where $\mu$ and $\gamma$ denote the death rate and loss-of-interest rate, respectively, as discussed in Section 2.5).

Coefficient	Mechanism(s)
$a_{11}$	$-\mu$
	positive-media-induced transitions $S\to I$ [38]
	negative-media-induced transitions $S\to R$ [38]
$a_{12}$	n.a.
$a_{13}$	re-establishing of undecided state ($R\to S$) [38]
$a_{21}$	positive-media-induced transitions $S\to I$ [38]
$a_{22}$	$-(\gamma +\mu )$
$a_{23}$	n.a.
$a_{31}$	negative-media-induced transitions $S\to R$ [38]
$a_{32}$	$\gamma$
$a_{33}$	$-\mu$
	re-establishing of undecided state ($R\to S$) [38]

lists the coefficients of the SIR model (26) when it is applied to voter dynamics. Some interpretations of these coefficients are listed there as well.

In the context of voter dynamics, the coefficient $a_{13}$ (in combination with $a_{33}$) has been used to describe that non-voters (in the sense of voters who decided not to vote for a particular candidate, see Section 2.5) re-enter a neutral, undecided state. That is, these individuals revoke their decision not to vote for the candidate under consideration [38]. The coefficients $a_{21}$ and $a_{31}$ (in combination with $a_{11}$) have been used to model the impact of media on voter dynamics [38]. Accordingly, positive and negative media reports can lead undecided voters to decide to voter for or against the candidate under consideration. In the former case, we deal with $S\to I$ transitions modeled by $a_{21}>0$, and in the latter case with $S\to R$ transitions modeled by $a_{31}>0$. These transitions may be considered as counterparts to the spontaneous transitions discussed in other disciplines in the sense that they are not induced by other voters.

The SIR model (26), when applied to voter dynamics, is characterized by vanishing coefficients $a_{12}$ and $a_{23}$. Fixing $a_{12}=0$ implies it is assumed that there are no $I\to S$ transitions. As discussed above, decided individuals I may return to the compartment S. However, if they do so, then they make this transition only via the $I\to R\to S$ route. Fixing $a_{23}=0$ implies that it is assumed that there are no $R\to I$ transitions describing individuals who revoke their decisions not to vote for the candidate being considered by deciding to give the candidate their vote. This transition can be considered as a shortcut to the transition route $R\to S\to I$, where this is possible when allowing for $R\to S$ transitions by selecting $a_{13}>0$ (and adjusting $a_{33}$ appropriately). In summary, it would be open for debate whether SIR models (26) with $a_{12}>0$ and/or $a_{23}>0$ describing the direct transition routes $I\to S$ and $R\to I$ should be used when discussing voter dynamics since the corresponding indirect transition routes $I\to R\to S$ and $R\to S\to I$ have been discussed and considered as plausible pathways in the literature.

3.7. Peculiarities of Epidemiological SIR Models of Rumor Spreading

Table 7. Coefficients and mechanistic interpretations of SIR rumor spreading models satisfying Equation (26) or a generalized version of Equation (26) (where $\mu$ and $\gamma$ denote the death rate and the loss-of-interest rate, respectively).

Coefficient	Mechanism(s)
$a_{11}$	$-\mu$  [49]
	education $S\to R$ [49]
$a_{12}$	n.a.
$a_{13}$	re-establishing indifferent state ($R\to S$) [17,48,99,100]
$a_{21}$	n.a.
$a_{22}$	$-(\gamma +\mu )$
$a_{23}$	n.a.
$a_{31}$	education $S\to R$ [49]
$a_{32}$	$\gamma$
$a_{33}$	$-\mu$
	re-establishing indifferent state ($R\to S$) [17,48,99,100]

presents the coefficients typically used in rumor spreading models of the SIR type, as defined by Equation (26), as well as a generalization of it. Mechanistic interpretations of these coefficients are presented in Table 7 as well.

The coefficient $a_{13}$ (in combination with $a_{33}$) describes $R\to S$ transitions, that is, transitions of stiflers who return to some kind of pseudo-ignorant state. More precisely, it is assumed that individuals who lost interest in the rumor and joined the group of stiflers can return to an ”indifferent” state in which they are undecided whether or not they want to spread the rumor [17,48,99,100]. In this context, the compartment S contains individuals that have not yet learned about the rumor and individuals who have heard the rumor but are indifferent in the sense mentioned above. In contrast, R denotes stiflers who have decided not to participate anymore in spreading the rumor in their social networks. Having S and R defined in this way, the $R\to S$ transitions can be regarded as counterparts to the waning immunity transitions in SIR models of infectious diseases (see Section 3.1).

The coefficient $a_{31}$ (in combination with $a_{11}$) reflects the effects of education, making ignorant individuals unresponsive to rumors [49]. In this context, the compartment R needs to be re-defined appropriately. In order to model such effects of education within an SIR modeling framework, R should be defined as a compartment that includes (among other types of individuals) individuals who are no longer susceptible to rumors because they were educated to ignore them.

As pointed out in Table 7, SIR models of rumor spreading typically feature several vanishing matrix coefficients: $a_{12}$, $a_{21}$, and $a_{23}$. The assumption $a_{12}=0$ means that it is assumed that rumor spreading does not exhibit $I\to S$ transitions. However, individuals may take the $I\to R\to S$ route in order to transition from spreaders I to pseudo-ignorants S. The assumption $a_{21}=0$ implies that $S\to I$ transitions are not considered. Ignorants (or pseudo-ignorants) only become spreaders by means of contact with other spreaders (as captured by the $\beta SI/N$ term in Equation (26)). That is, only induced transitions to I are considered. Finally, the assumption $a_{23}=0$ reflects that it is assumed that direct $R\to I$ transitions do not take place. However, individuals may take the $R\to S\to I$ route to make such transitions in an indirect way. Just as in the previous section, we are inclined to say that it would be open for debate whether or not it would be plausible to capture direct transitions $I\to S$ and $R\to I$ with the help of non-vanishing coefficients $a_{12}$ and $a_{23}$ in rumor spreading models. As pointed out in Section 2.6, the lifetimes of rumors are relatively short. For this reason, in the field of rumor modeling, birth and death processes can usually be neglected, which implies that the coefficient $a_{11}$ typically does not involve a death rate parameter. Nevertheless, for the sake of completeness, a few studies have considered birth and death terms. If so, the death rate parameter shows up in the coefficient $a_{11}$, as indicated in Table 7.

A peculiarity of rumor spreading modeling is the assumption that spreaders do not only spontaneously lose interest in spreading a rumor. Rather, it is assumed that there are induced transitions to the group of stiflers when (i) a spreader meets another spreader or (ii) a spreader meets a stifler [43,44]. In both cases, the spreader learns that the rumor is known already by other individuals, which may result in a loss of interest. These induced transitions can be modeled with nonlinear terms [3,43,44,48,101,102,103]. Using the ISR terminology of Equation (10), we arrive at the following generalized model:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& -\beta SI,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& \beta SI-\gamma S-({\alpha }_{1}S+{\alpha }_{2}R)S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}R& =& \gamma S+({\alpha }_{1}S+{\alpha }_{2}R)S\hfill \end{array}$$

(28)

The terms ${\alpha }_1{S}^2$ and ${\alpha }_{2}SR$ describe induced transitions to the state of stiflers when a spreader comes into contact with another spreader or when a spreader comes into contact with a stifler, respectively. In contrast, the $\gamma S$ term describes the spontaneous transition of spreaders where they stop spreading the rumor and become stiflers. The original Daley–Kendall benchmark model of rumor spreading [43,44,48,103] corresponds to Equation (28) with $\gamma =0$. The Daley–Kendall terms ${\alpha }_1{S}^2$ and ${\alpha }_{2}SR$ nicely illustrate how nonlinear transition mechanisms can be used in interdisciplinary applications of epidemiological models. Model (28) belongs to the class of nonlinear transition models (see Figure 1).

3.8. Peculiarities of Epidemiological SIR Models of Sales Dynamics and Innovation Diffusion

Table 8. Coefficients and mechanistic interpretations of sales dynamics and innovation diffusion models based on the SIR model (26) (where $\gamma$ denotes the loss-of-interest rate of promoting adopters discussed in Section 2.7 and $\mu$ denotes the death rate parameter).

Coefficient	Mechanism(s)
$a_{11}$	$\mu$, pure innovation dynamics [50,51]
$a_{12}$	n.a.
$a_{13}$	n.a.
$a_{21}$	pure innovation dynamics [50,51]
$a_{22}$	$-(\gamma +\mu )$
$a_{23}$	n.a.
$a_{31}$	n.a.
$a_{32}$	$\gamma$
$a_{33}$	$-\mu$

presents transition coefficients of sales dynamics and innovation diffusion models that take an SIR modeling perspective (26). Interpretations of relevant coefficients are given there as well.

The coefficient $a_{21}$ (in combination with $a_{11}$) describes pure innovation dynamics in the sense that non-adopters become adopters without being in contact with other adopters or without feeling social pressure from other adopters to imitate them [54]. These adopters learn about the product of interest from media or simply during shopping. They are individuals who have a strong interest in innovation (new products) and, for this reason, can be referred to as innovators [54]. The model described in Table 8 reads explicitly

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-{\beta }_{0}SI-(k_1+\mu )S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& {\beta }_{0}SI+k_{1}S-(\gamma +\mu )I,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}R& =& \gamma I-\mu R,\hfill \end{array}$$

(29)

where ${\beta }_0$ denotes the imitation coefficient (see Section 2.7). The parameter $k_1$ is called the innovation coefficient and determines the speed of the pure innovation dynamics. In the special case where $B=\mu =0$ (no vital dynamics) and $\gamma =0$, Equation (29) becomes the famous Bass model [54] of sales dynamics and innovation diffusion and features only two compartments. By adding the compartment R and the $\gamma I$ term, describing that promoting adopters lose interest in promoting a product, the Bass model is merged with the SIR model of epidemiology. Accordingly, Equation (29) has been referred to as the Bass–SIR model [50,51]. While the original Bass model (with $\gamma =0$) alone is a powerful tool to describe the sales dynamics of new products [20], it oversimplifies market dynamics by assuming that all buyers become promoting adopters [52]. The Bass–SIR model addresses a more realistic situation [52] by assuming that promoting adopters switch over time to non-promoting adopters.

As pointed out in Table 8, the Bass–SIR model is characterized by a variety of vanishing transition coefficients. We have $a_{12}=0$ (no $I\to S$ transitions); $a_{13}=0$ (no $R\to S$ transitions, meaning non-promoting buyers do not return to the group of potential buyers); $a_{23}=0$ (no $R\to I$ transitions, meaning non-promoting buyers do not become promoting buyers again); and $a_{31}=0$ (no $S\to R$ transitions, i.e., no “vaccination”). If vital dynamics are neglected, then $a_{33}=0$ holds, which means that non-promoting buyers do not change their state over time (i.e., no “waning immunity”).

3.9. Peculiarities of Epidemiological SIR Models of Viral Marketing and Viral Videos

When viral marketing or the spread of viral videos is studied with the help of the general SIR model (26) or generalization of it, models with matrix coefficients as shown in

Table 9. Coefficients and mechanistic interpretations of SIR models (26) of viral marketing and/or the spread of viral videos (where $\mu$ and $\gamma$ denote the death rate of individuals and the loss-of-interest rate for sharing marketing messages or videos, as discussed in Section 2.8).

Coefficient	Mechanism(s)
$a_{11}$	$-\mu$ [57,58]
$a_{12}$	mild loss of interest ($I\to S$) [61,79]
$a_{13}$	re-establishing of susceptible state ($R\to S$) [104]
$a_{21}$	n.a.
$a_{22}$	$-(\gamma +\mu )$
	mild loss of interest ($I\to S$) [61,79]
$a_{23}$	spontaneous relapse ($R\to I$) [58,61,62]
$a_{31}$	n.a.
$a_{32}$	$\gamma$
$a_{33}$	$-\mu$
	re-establishing of susceptible state ($R\to S$) [104]
	spontaneous relapse ($R\to I$) [58,61,62]

are typically considered.

The baseline SIR model of viral marketing and the spread of viral videos discussed in Section 2.7 state that when broadcasting individuals lose interest in sharing information or videos, they turn to a state of “intentional” inactivity. That is, they become inert individuals who do not share information. This $I\to R$ transition (using the SIR terminology) is determined by the parameter $\gamma$ (showing up in the matrix coefficients $a_{22}$ and $a_{32}$). In addition to this form of becoming inactive, it has been assumed that broadcasters may simply return to the compartment of susceptible individuals ($I\to S$ transitions) [61,79]. In this context, S has to be interpreted as a compartment involving two types of individuals: individuals unaware of the marketing message or viral video at hand and individuals who are aware of that kind of content and have shared that content for a while but turned into a state in which they are undecided whether or not they should continue sharing the content. The coefficient $a_{12}$ (in combination with $a_{22}$) describes transitions from I to S broadcasting individuals due to this second type of loss of interest, which is referred in Table 9 as mild loss of interest. The coefficient $a_{23}$ (in combination with $a_{33}$) describes the spontaneous relapse of inert individuals who left the group of broadcasting individuals to return back to that group of broadcasting individuals ($R\to I$ transitions) [58,61,62]. Finally, inert individuals may not become broadcasting individuals, but may give up their “intentional” inactivity. That is, inert individuals may return to the group of susceptibles in the generalized sense defined above. They become individuals who are undecided whether or not to share the viral content of interest. The coefficient $a_{13}$ (in combination with $a_{33}$) describes such $R\to S$ transitions [104]. In Table 9, the mechanism that moves inert individuals to the group of susceptibles is called the re-establishing of the susceptible state.

Table 9 illustrates that SIR models (26) of viral marketing and viral videos dynamics are typically characterized by two vanishing transition coefficients: $a_{21}$ and $a_{31}$. First, assuming that $a_{21}=0$ holds means (just as in the other disciplines reviewed above) that it is assumed that the dynamics do not exhibit spontaneous $S\to I$ transitions. Individuals can only become broadcasting individuals due to contact with other broadcasting individuals (i.e., there are only induced transitions, as described by the $\beta SI/N$ term). Second, the assumption that $a_{31}=0$ holds reflects that in the context of viral marketing and the spread of viral videos, some kind of “vaccination” mechanism does not exist that would lead to $S\to R$ transitions.

In concluding this section, let us return to the $R\to I$ relapse transitions described by the coefficient $a_{23}$. The transition matrix A in general describes linear transition mechanisms that can be regarded as “spontaneous” transitions (i.e., transitions not induced or affected by individuals of other compartments). Several authors have assumed that inert individuals may leave their condition when they come in contact with broadcasters. That is, these authors have entertained the possibility of induced transitions that lead inert individuals to start broadcasting again [57,58,62].

The SIR model (26) of viral marketing and viral videos that accounts for spontaneous and induced relapse of individuals reads

$$\begin{array}{ccc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}S& =& B-({\beta }_1+{\beta }_2)SI-\mu S,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}I& =& {\beta }_{1}SI-(\gamma +\mu )I+a_{23}R+{\beta }_{3}RI,\hfill \\ \hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}R& =& {\beta }_{2}SI+\gamma I-(\mu +a_{23})R-{\beta }_{3}RI.\hfill \end{array}$$

(30)

The linear term $a_{23}R$ describes spontaneous relapses of inert individuals. In contrast, the nonlinear term ${\beta }_{3}RI$ describes induced relapses of inert individuals. In Equation (30), another nonlinear term is added. The model also accounts for induced $S\to R$ transitions via the nonlinear term ${\beta }_{2}SI$. Such transitions describe individuals who come into contact with the content at hand and immediately reject sharing the content in their social networks. That is, just as the ${\beta }_{1}SI$ term describes a positive, supporting and promoting a reaction of susceptibles to the content presented by broadcasters, the ${\beta }_{2}SI$ term describes a negative, rejecting a reaction of susceptibles to the broadcasters’ content.

In the literature, a model with spontaneous relapse only ($a_{23}>0$, ${\beta }_2={\beta }_3=0$) was studied by Bauckhage [104]. A model with induced relapse only (${\beta }_3>0$, $a_{23}={\beta }_2=0$) was the original UBI model that was examined by Bhattacharya et al. [57] and that was presented in Equation (11) in Section 2.8 in a simplified form (using the UBI notation rather than the SIR notation). Modeling-induced $S\to R$ transitions via the term ${\beta }_{2}SI$ was proposed by Li et al. [61]. For ${\beta }_2>0$ or ${\beta }_3>0$, Equation (30) becomes a nonlinear-transition-mechanism–ODE model (i.e., merges the ODE modeling approach with the nonlinear transition modeling approach shown in Figure 1).

4. The Basic Reproduction Number and Discipline-Specific Peculiarities

The basic reproduction number $R_0$ is a key concept in classical epidemiology [1,2,90]. It is defined as the average number of newly infected individuals produced by a single typically infected individual when assuming a completely susceptible population [1,2,90]. In the literature, there are several alternative ways to refer to $R_0$. Accordingly, the term “reproduction” may be replaced by “reproductive”. Likewise, the term “number” may be replaced by “ratio” [1]. Irrespective of the precise terminology, roughly speaking, $R_0$ quantifies how many individuals are infected by an infected person on average when a new virus invades a population. In addition to this quantitative aspect of $R_0$, $R_0$ can be used as a bifurcation parameter because if an infected person produces more than one infected case, then there will be a disease outbreak and the disease-free fixed point is unstable. In contrast, if an infected person produces (on average) less than one case, then the initial subpopulation of infected individuals will monotonically decay over time and the disease-free state is stable. For the baseline SIR model (1), $R_0$ reads [90]

$$R_0={\displaystyle \frac{\beta }{\gamma +\mu }}.$$

(31)

For the baseline SEIR model (12), $R_0$ reads [2]

$$R_0={\displaystyle \frac{\alpha \beta }{(\alpha +\mu )(\gamma +\mu )}}.$$

(32)

The question arises whether $R_0$ has been utilized and adopted in other disciplines such as those discussed in the current study. Let us dwell on this issue.

In the field of virus dynamics, the basic reproduction number $R_0$ has indeed been used in analogy to $R_0$, as defined above for classical epidemiology. Accordingly, the basic reproduction number $R_0$ corresponds to the average number of newly infected cells due to the presence of a single infected cell when assuming that the cell population under consideration consists entirely of non-infected target cells [105,106,107,108]. The quantitative aspect of $R_0$ may be exploited by estimating $R_0$ from virus load data. In doing so, a quantitative measure for the within-host infectiousness of a given virus can be obtained (e.g., see [108,109,110]). Just as in classical epidemiology, $R_0$ can be used a bifurcation parameter. For $R_0>1$, the virus will spread out in the human body. For $R_0<1$, the initial infection will decay, that is, the virus will be removed and cleared out faster than it can reproduce itself. Finally, note that in the field of virus dynamics, $R_0$ is a within-host reproduction number and describes the spread of a disease within an individual. In contrast, in classical epidemiology, $R_0$ denotes a reproduction number on the population level and describes the spread of an infectious disease across individuals.

Studies examining the spread of computer viruses have also adopted the concept of the basic reproduction number (see, for example, [23,26,27,29,30,63,64,65]). In this context, researchers frequently just point to the utilization of $R_0$ in classical epidemiology without presenting an explicit interpretation of $R_0$. Once $R_0$ is introduced in this heuristic way, it is used as a bifurcation parameter to examine the stability of fixed points of interest. Having said that, in the context of computer viruses, the basic reproduction number $R_0$ may be defined as the average number of newly infected computers produced by a single infected computer in a network of non-infected computers. In this definition, computers may be switched out by nodes if it is more appropriate to talk about the spread of malware across network nodes.

The basic reproduction number has been used in various studies on drug addiction [11,31,32,33,35,36,67,68,69,70]. Explicit definitions of $R_0$ have been given by some authors in the context of their respective studies. For example, White and Comiskey defined the basic reproduction number as the average “total number of people that each single drug user will initiate to drug use during the drug-using career” (see Section 3.1.1 in [31]). A similar definition can be found in Tang et al. [67]. In the context of studies focusing on tobacco addiction, it has been suggested to denote $R_0$ as the smoker’s generation number [35,36]. In line with ref. [36], when studying tobacco addiction at university campuses and how tobacco-addicted students lure fellow students into addiction, the smoker’s generation number may be defined as the average number of secondary cases of addicted smokers produced by a single addicted student smoker in a university population composed of non-addicted students. Note that in ref. [36], a slightly different definition is actually presented because the authors consider the so-called effective reproduction number, which applies to situations where an epidemic is ongoing [1]. Irrespective of the interpretation of $R_0$, studies concerned with drug epidemics frequently use $R_0$ as a bifurcation parameter. If $R_0>1$ holds, then there is an epidemic outbreak of a drug (e.g., a new synthetic drug [68]). In contrast, if $R_0<1$ holds, then the initial invasion of a population by a particular drug will fail to trigger a drug epidemic outbreak.

The basic reproduction number $R_0$ has been used as a bifurcation parameter in a few studies on voter dynamics that take advantage of epidemiological ODE modeling [37,39]. In addition, Romero et al. [41] defined specific reproduction numbers to capture the influence of specific groups (such as decided voters and party members) on undecided voters. Inspired by the terminology used by Romero et al., [41], in the context of the baseline SIR model (1) used in the study by Yong and Samat [37] to describe voter dynamics, the basic reproduction number $R_0$ may be explicitly defined as the average number of undecided voters that are influenced to vote for a particular candidate by a single voter (of that candidate) who is placed in an entire population of undecided voters.

Studies on rumor spreading that have used epidemiological ODE models have often adopted $R_0$ as a bifurcation parameter to discuss the stability of fixed points (see e.g., [47,73,74,75,76,77]). Accordingly, if $R_0>1$ holds for a particular rumor, then the rumor distributed by a single spreader generates more than one new spreader on average and the rumor spreads in the population. For $R_0<1$, a single spreader convinces on average less than one person to become a spreader and, consequently, the rumor will fade monotonically away. As such, $R_0$ may be explicitly defined as the average number of secondary cases of rumor spreaders caused by a single spreader in a population full of ignorant individuals [74,76].

In studies on sales dynamics and innovation diffusion, the basic reproduction number has rarely been used. For example, in the recent review by Svoboda et al. [53] on the utilization of the SIR model in the field of innovation diffusion, it is acknowledged that the basic reproduction number is a cornerstone concept of epidemiological modeling and that innovation diffusion and epidemiology exhibit strong similarities. However, the explicit utilization of $R_0$ in the field of innovation diffusion itself is not mentioned. In the review by Guidolin and Manfredi on innovation diffusion and the Bass sales model [20], the basic reproduction number is discussed in the context of the SIR model (1) for sales dynamics, as reviewed in Section 2.7. Applications of $R_0$ to more complex models such as the Bass–SIR model (29) or alternative models reviewed in ref. [20] are not given. As an exception to this seeming absence of $R_0$ in the sales dynamics and innovation literature, Sharma [79] used $R_0$ as a bifurcation parameter to analyze the buyers-and-reviews model (22) described in Section 2.15. In the context of the model (22), $R_0$ may be defined as the average number of new shoppers at the online shop X that are generated by a single shopper at X in an initial population composed of online non-shoppers. Recall that according to the model (22), these new shoppers emerge due to the positive reviews written by the shoppers at X. These reviews are not explicitly mentioned in the definition of $R_0$ — just as the virus is not explicitly mentioned in the definitions of $R_0$ given above for classical epidemiological systems and the spread of viruses in human hosts.

In closing this section, let us briefly address viral marketing modeling and the modeling of the emergence of viral videos. In the context of the unaware–broadcaster–inert model (11), the basic reproduction number $R_0$ has been defined as the average number of broadcasters that a single broadcaster produces when ignoring the inert class [57]. The requirement of ignoring the inert class can be replaced by requiring (in analogy to the various previous definitions of $R_0$ listed above) that a scenario is considered that involves a completely susceptible population [58]. That is, the population consists entirely of unaware individuals. Irrespective of the definition of $R_0$, in this research field, $R_0$ has typically been used as a bifurcation parameter for the purpose of stability analysis [56,57,58,82].

5. Discussion

It has been demonstrated that epidemiological models have found applications in a variety of disciplines outside of classical epidemiology. Importantly, it has been shown that the exact same mathematical ODE models utilized in epidemiology have been used in these alternative research fields. This has been exemplified for two benchmark epidemiological models: the SIR and SEIR models. The implication of this demonstration is far-reaching for researchers dealing in all kind of disciplines in which epidemiological models are currently applied: researchers studying a particular mathematical, epidemiological model in the context of a certain discipline should look beyond their specific discipline in order to find previous work relevant for their work. The reason for this is that theorems and solutions obtained in one discipline carry over to other disciplines as long as the mathematical models are identical across the disciplines. For example, in the wake of the COVID-19 pandemic, extensive work has been carried out to develop a perspective alternative to the state space perspective of epidemiological ODE models [1]. This alternative perspective uses amplitude equations that have certain benefits as compared to the original state space equations (for details, see ref. [1]). The amplitude equations for the baseline SIR models (1) and (4) with and without demographics are derived in chapter 4 in ref. [1]. Likewise, the amplitude equations for the baseline $1\beta$ and $2\beta$ SEIR models (17) and (24) in the absence of demographic terms are derived in chapter 5 in ref. [1]. The amplitude equation perspective obtained in this work in the context of classical virus spreading in human populations is ready for use in alternative disciplines. For example, it can be used to describe rumor spreading [48] or drug addiction dynamics [66]. In other words, once a problem associated with a particular epidemiological, mathematical model has been solved in a certain discipline, the same problem related to the same mathematical model does not need to be solved again in the context of another discipline. The demonstration provided in Section 2 of mathematical equivalent models used across disciplines comes with a plea to researchers not to reinvent the wheel again and again. More importantly, the demonstration in Section 2 comes with the invitation to researchers working in a particular research field to learn from and take advantage of results obtained in otherwise disconnected disciplines that just happen to use the same kind of mathematical, epidemiological model.

In particular, as mentioned earlier, analytical solutions of epidemiological models that have been developed in a particular field may be utilized by researchers working in alternative research fields despite characteristic differences across these fields. For example, characteristic time scales typically vary across the scientific disciplines that take advantage of epidemiological modeling. Computer viruses typically spread out in computer networks within hours or days [24,111]. Likewise, rumors may act on relatively short time scales. A rumor may become popular within a day. Subsequently, its popularity may decay dramatically [48,99]. Just like rumors, viral videos tend to reach their maximal popularity within a few days [60]. Viral infections within humans such as influenza and COVID-19 come with virus load dynamics that build up within a week, reach a peak level, and subsequently decay on a similarly fast time scale [1,109]. In contrast, virus infections on the population level typically take place on longer time scales of months [1,9,112], and the development of such epidemics may be studied over generations, that is, on even longer time scales [2]. Likewise, the sale of new products may reach a maximum only after several years [54]. Exact analytical solutions obtained for a particular model (e.g., the baseline SIR model), by definition, are not approximations and, consequently, hold for any parameter set of the model in question and for any time scale being considered. For this reason, exact solutions can conveniently be transferred across disciplines. An illustrative example in this regard is the equation for the maximum value of an infection wave described by the baseline SIR model without vital dynamics, as defined by Equation (4). Let $I_0$ and $S_0$ denote the initial values of I and S at the initial time $t=0$. Then, the maximal value of infected individuals, $I\left(\text{max}\right)$, can be computed from the following [1,90]:

$$I\left(\text{max}\right)=I_0+S_0(1-{\displaystyle \frac{1}{\xi }}[1+ln\left(\xi \right)],\xi ={\displaystyle \frac{\beta S_0}{\gamma N}}.$$

(33)

If influenza waves or COVID-19 waves are modeled by a SIR model (4) as in refs. [1,112,113], then Equation (33) can be applied. If the virus dynamics within humans infected by the influenza virus are modeled using the TV model as in ref. [109], then Equation (33) may be applied as well due to the equivalence of the TV model to the SIR model (see Section 2.2). In this context, it does not matter that in the former case, the infection waves evolve over several months, while in the latter case, the virus load trajectories evolve over a much shorter period of days. Having said that, the situation may be more complex when considering analytical approximative solutions of epidemiological ODE models. For example, in the field of virus dynamics, two-phase approximative solutions describing the increase in and decay of virus load have been proposed [1,114,115]. For illustration purposes, let us present here a slightly improved version of the two-phase approximative solution derived in ref. [1]. It reads

$$V_I\left(t\right)=b \text{exp}\left({\lambda }_{\text{max}}t\right),V_{II}\left(t\right)=V_r\text{exp}(-k_2(t-t_s)),t_r=t_p+\Delta .$$

(34)

In Equation (34), $V_I\left(t\right)$ describes the first phase, namely, the increase in the viral load towards the maximal (peak) value $V_{max}$, and $V_{II}\left(t\right)$ describes the second phase, namely, the decay of the virus load towards zero after the peak value has been reached. The approximations (34) can be derived from the TIV model (15) for $B=\mu =0$ [1]. Accordingly, ${\lambda }_{\text{max}}$ denotes the largest eigenvalue of the TIV model for which an analytical expression exists [1,115], and b can be computed from the so-called unstable eigenvector of the model [1]. $t_p$ is the time point at which the virus load becomes maximal (such that $V_{\text{max}}=V\left(t_p\right)$), and $k_2$ is the decay parameter occurring in Equation (15). $\Delta >0$ denotes a time shift that was neglected in ref. [1] and can be used improve the approximation. More precisely, $\Delta$ can be used to shift the reference time point $t_r$ away from $t_p$ into the second phase of exponential decay (see the example below). Finally, $V_r$ is the virus load at the reference time point $t_s$: $V_r=V\left(t_r\right)$. For $\Delta =0$, Equation (34) is reduced to the original two-phase approximation presented in ref. [1]. Figure 3 illustrates an application of Equation (34) for one of the COVID-19 patients discussed in ref. [1]. The virus load is shown as a function of time in a log-lin scale. The virus load data of the patient (blue circles) are fitted to the TIV model (15) [1]. The best-fit solution is shown as black line. From the best-fit model, parameters ${\lambda }_{max}$ and b are calculated. The dotted red line shows the first-phase approximation $V_I\left(t\right)$ thus obtained. According to the TIV model, the virus load peaks at 5.36 days. A shift $\Delta$ of 1 day moves the reference point into the second phase of exponential decay (which shows up as a linear decay in the log-lin plot). The dotted green line shows $V_{II}\left(t\right)$ with $t_p=5.36$d, $\Delta =1$d, and $k_2$, as obtained from the best-fit TIV model.

Due to the equivalence of the TIV model with the 3D SEIR model (13), as seen in Section 2.10, the two-phase approximation (34) can be used in different disciplines using the SEIR model (13). However, the accuracy of the approximation depends in general on the model parameters. Since the model parameters vary across disciplines just as the characteristic time scales mentioned above, the usefulness of the approximation must be checked for each discipline separately (and, in general, within disciplines for each application). Moreover, clinically relevant changes in virus load can be seen on log-scales. On the log-lin graph shown in Figure 3, the two-phase approximation based on the two exponential functions provides a reasonably good fit to the exact TIV model solution. In other disciplines, typically state variables are shown on linear scales. Consequently, the usefulness of the approximation (34) for state variables shown on linear scales would require additional testing. Overall, the two examples related to Equations (33) and (34) illustrate that transferring knowledge across different scientific disciplines that use the same kind of epidemiological models is possible and promising but should be done with caution.

In Section 3, it was also exemplified with the help of the eight research fields considered in the current study that there are differences across disciplines when using the same type of epidemiological ODE models. Naturally, the interpretation of the state variables and model coefficients differs across disciplines, as seen in Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9. However, when comparing Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9, it also becomes clear that in some disciplines, transition coefficients are relevant that are irrelevant (or have frequently been neglected) in other disciplines. To make this point more explicit,

Table 10. SIR model structures across disciplines (VM = viral marketing; “( )” and “***” see text).

Discipline	${\mathit{a}}_{11}$	${\mathit{a}}_{12}$	${\mathit{a}}_{13}$	${\mathit{a}}_{21}$	${\mathit{a}}_{22}$	${\mathit{a}}_{23}$	${\mathit{a}}_{31}$	${\mathit{a}}_{32}$	${\mathit{a}}_{33}$
Epidemiology	x	(x)	x	***	x		***	x	x
Virus dynamics	x		–		x	–	–	–	–
Computer viruses	x	x	x		x	x	x	x	x
Drug addiction	x		x		x	x	x	x	x
Voter dynamics	x		x	x	x		x	x	x
Rumor dynamics	x		x		x		x	x	x
(Sales dyn./innovation diffusion	x				x			x	x	)
(VM/viral videos	x	x	x		x	x		x	x	)
Sales dyn. and VM	x	x	x	x	x	x		x	x

provides an condensed overview of Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8 and Table 9. In Table 10, for each discipline, the non-vanishing transition coefficients of the SIR model with general linear transition mechanisms (26) are indicated. In doing so, Table 10 allows the identification of convenient structural similarities and differences in the SIR modeling approaches across the disciplines considered in the current study.

Note that in Table 10, the two research fields about sales dynamics/innovation diffusion and viral marketing/viral videos are listed separately, just like the other remaining six research fields. However, since these research fields are closely related to each other, we also condensed them into a single discipline called sales dynamics/viral marketing. Consequently, the following discussion will be centered around seven disciplines rather than eight. Note also that in the case of the SIR model for epidemiology, some coefficients are presented with parentheses “()” and stars “***”. The meanings of these quantifiers were discussed in Section 3.2. For the discussion about structural differences across disciplines, we just need to note that the relevance of these coefficients is open for debate. That is, SIR models in epidemiology in fact could exhibit more than just one irrelevant coefficient. For the sake of simplicity, let us count the transmission coefficients flagged by the aforementioned quantifiers “()” and “***” as relevant ones.

As can be seen in Table 10, there are three single-entry missing disciplines. That is, there are three disciplines featuring SIR models for which all transmission coefficients except for one are relevant. These disciplines are epidemiology, computer viruses, and the discipline centered around sales dynamics and viral marketing. These single-entry missing SIR models in the disciplines epidemiology, drug addiction, and sales dynamics/viral marketing feature different missing entries. For SIR models in epidemiology, the coefficient $a_{23}$, reflecting spontaneous $R\to I$ transitions, is typically ignored. In contrast, for SIR models describing computer virus spreading, it is the coefficient $a_{21}$, associated with spontaneous $S\to I$ transitions, that is typically ignored and set as equal to zero. Finally, for SIR models in the field of sales dynamics and viral marketing, the missing transition coefficient is the coefficient $a_{31}$, describing spontaneous $S\to R$ transitions. This implies that the maximally complex models (i.e., the models that exhibit all but one coefficient) in these three disciplines exhibit different mathematical structures. More precisely, they exhibit transition matrices with zero entries at different positions. Disciplines exhibiting (according to the discussion provided in Section 3) two missing entries are virus dynamics, drug addiction, and voter dynamics. Interestingly, virus dynamics SIR models and drug addiction SIR models exhibit the same vanishing coefficients: $a_{12}$ and $a_{21}$. Having said that, one should note that the virus dynamics SIR model does not exhibit an R compartment. Consequently, only the dynamics of the compartments S and I can be compared across the two disciplines when focusing on SIR modeling approaches. The virus dynamics model and the drug addiction model are mathematically equivalent (in the $S,I$ subspace) when the dynamics of R for the drug addiction SIR model do not affect the dynamics in the $S,I$ subspace. That is, we need to require that the coefficients $a_{13}$ and $a_{23}$ vanish in the drug addiction model. In summary, any solution or theorem that is about the $S,I$ subspace dynamics and holds either for the virus dynamics SIR model characterized in Table 10 or for the drug dynamics SIR model with a coefficient listed in Table 10 where $a_{13}=a_{23}=0$ can be carried over to the respective complementary discipline (i.e., from virus dynamics to drug addiction or vice versa from drug addiction to virus dynamics). As mentioned earlier, the third discipline featuring a maximally complex model with two vanishing transition coefficients is the research field for voter dynamics. The missing coefficients are $a_{12}$ and $a_{23}$. Consequently, the maximally complex SIR model in this research field differs from the maximally complex SIR models in the fields of virus dynamics and drug addiction. Finally, according to the review presented in Section 3, the maximally complex SIR model for rumor dynamics is characterized by a triple-entry-missing transition matrix. Overall, certain disciplines (those with single-entry-missing maximally complex models) exhibit a richer structure than other disciplines (those with double-entry- or triple-entry-missing maximally complex models). Moreover, the structures of the maximally complex models differ across all seven disciplines. That is, each discipline exhibits a maximally complex model with a unique structure.

As discussed in the context of Table 10, spontaneous $R\to I$ transitions (as described by $a_{23}>0$) are typically not considered in the disciplines of classical epidemiology, voter dynamics, and rumor dynamics. However, the argument made in the field of drug addiction modeling that spontaneous $R\to I$ transitions may occur could also be made for voter and rumor dynamics. As far as classical epidemiology is concerned, the class R typically describes recovered individuals that have cleared the virus under consideration out of their bodies. Consequently, in this context, spontaneous $R\to I$ transitions cannot occur.

When taking a data-driven approach, the maximally complex model of the respective discipline may be fitted to the data at hand. A purely data-driven approach avoids a detailed discussion about which transition coefficients should be included and which should not. In data-driven approaches, the data decides how the model at hand is parameterized without taking any a priori knowledge into account [117]. Only in a subsequent step would the estimated matrix coefficient $a_{ij}$ be interpreted in terms of the mechanisms listed in Table 2. In this step, the hypothesis about the different mechanisms underlying the coefficients can be checked. For example, Table 2 suggests that $a_{22}=-a_{32}$ should hold such that the coefficients can be interpreted as estimates for the decay parameter $\gamma$ like $a_{22}\approx -a_{32}\approx -\gamma$.

Note that in the context of such a data-driven approach, the coefficients are assumed to be independent from each other. In this context, the question arises if a data set at hand is sufficient to fit the parameters of a given model. According to the 1-to-10 rule of statistics for each to-be-fitted parameter, there should be at least 10 data points. The baseline SIR model (1) without vital dynamics comes with two parameters: $\beta$ and $\gamma$. Including vital dynamics, the model exhibits four parameters: $\beta ,\gamma ,B$, and $\mu$. The maximal complex models as listed in Table 10 can have up to eight non-vanishing matrix coefficients (which are considered to be independent when considering the data-driven approach). Consequently, when ignoring vital dynamics, a data set should have at least 20 data points to fit the two-parameter baseline SIR model and should contain at least 90 data points to fit a model featuring $\beta$ and eight independent matrix coefficients $a_{ij}$. A similar consideration can be made for the case when the vital dynamics are taken into account.

If a data set does not contain sufficient data points, a feasible solution is to fit mechanisms (i.e., factors) rather than coefficients. In this approach, the matrix coefficients are not independent and thus the number of independent parameters becomes relatively small. Moreover, one may fit just one additional mechanism (factor) at a time. In doing so, as is common practice in statistics, one may test models with different predictors against each other (see, for example, [118,119]). For example, let us assume the goal is to fit an SIR model to infectious disease data of a relatively short infection wave for which vital dynamics can be ignored. In this case, the two-parameter baseline model (26) with $\beta >0$ and $a_{22}=-a_{23}=-\gamma$, $\gamma >0$, and all other parameters set equal to zero, is fitted in a first step to the data. Subsequently, three-parameter models that exhibit one additional parameter (that will be denoted below by $\delta$) are considered, capturing one of the possible mechanisms listed in Table 2. For the four mechanisms listed in Table 2, we obtain the four models labeled 1–4 in

Table 11. Two-parameter and three-parameter SIR models derived from Table 2 and their mechanisms. Coefficients $\beta$ and $a_{ij}$ refer to Equation (26). All other parameters occurring in Equation (26) are set to zero. $\beta ,\gamma ,\delta >0$.

Model	Mechanisms	Parameters
0	Baseline	$\beta$, $a_{22}=-a_{23}=-\gamma$
1	Baseline and $S\to I$ transitions	$\beta$, $a_{22}=-a_{23}=-\gamma$, $a_{11}=-a_{21}=-\delta$
2	Baseline and $S\to R$ transitions	$\beta$, $a_{22}=-a_{23}=-\gamma$, $a_{11}=-a_{31}=-\delta$
3	Baseline and $I\to S$ transitions	$\beta$, $a_{22}=-a_{23}=-\gamma$, $a_{12}=-a_{22}=\delta$
4	Baseline and $R\to S$ transitions	$\beta$, $a_{22}=-a_{23}=-\gamma$, $a_{13}=-a_{32}=\delta$

. The goodness of fit of models 1–4 may be compared. If so, the mechanism (factor) that matters most to improve model fit can be identified. In addition, the performance of the three-parameter models 1–4 may also be compared with the two-parameter baseline model using the Akaike information criterion, which is a standard procedure when comparing the performance of models that vary in the number of their parameters (see, for example, [119]).

Having said that, in practice, a priori knowledge is used when it comes to model fitting (see, for example, [1] and the references therein). That is, usually not all parameter values are estimated; some are taken from the literature. For example, birth rate and death rate parameters can often be inferred on the basis of available demographic data. In doing so, the parameter-fitting step can be dramatically simplified. In closing these considerations, let us return to Figure 1. As it has been pointed out several times throughout this study, the SIR ODE model with linear transition mechanisms can be generalized in various ways. For example, adding the compartment E to the SIR model such that it turns into an SEIR model increases the number of potentially relevant elements in the transition matrix A from 9 to 16. Adding another compartment inflates the number of possibly relevant transition matrix elements to 25. Likewise, adding nonlinear transmission mechanisms or delays to an ODE model with linear transition mechanisms typically increases the number of to-be-fitted parameters. Consequently, overfitting a model that exhibits too many parameters relative to the length of a given data set can become a severe problem when dealing with complex, high-dimensional epidemiological models.

The current study focused on epidemiological models formulated in terms of ODEs. As discussed in the Introduction and illustrated in Figure 1, there are different types of epidemiological mathematical models that were not addressed in the current study and are beyond the scope of the current study. For example, delay differential equations rather than ordinary differential equations were used. Epidemiological delay differential equations models can be found in a variety of different disciplines such as classical epidemiology [12,13], virus dynamics [120], computer viruses [26,29,121], drug addiction dynamics [87,122], and rumor spreading [123,124]. In principle, the approach presented in the current study for epidemiological ODE models can be applied to any type of epidemiological mathematical models. That is, there is room to generalize the present study in future works. In this context, note that in Section 3, several examples of nonlinear transition mechanism models were presented, as seen in Equations (27), (28), and (30). We are inclined to believe that discussing nonlinear transition mechanism models using the approach of the present study would be a cumbersome and almost impossible enterprise due to the almost endless mathematical possibilities of formulating nonlinear transition terms. A feasible approach would be to focus on models exhibiting a particular type of nonlinearity. For example, the models (27), (28), and (30) all exhibit quadratic nonlinearities of the form $X_1^2$ or $X_1{X}_2$, where $X_1$ and $X_2$ are state variables.

Nonlinear functions may not only be implemented to generalize transition mechanisms as captured by the transition matrix A defined in Equation (26); they can also be used to generalize the standard infection term $\beta SI$ (see, for example, Equations (1) and (12)), describing the rate at which susceptible individuals become infected. Accordingly, $\beta SI$ is replaced by $\beta SI·h(X_1,\dots ,X_n)$, where $X_1,\dots ,X_n$ are state variables of the model under consideration and h is a function of $X_1,\dots ,X_n$. Saturation effects [29,64,125] or other effects slowing down the infection rate [6,8,126,127] are frequently accounted for by introducing this kind of nonlinearity.

Note that the bilinear term $\beta IS$ describes the infection of a susceptible individual due to their contact with an infected individual. As argued in the previous paragraph, under certain circumstances, infection processes should be described by expressions that go beyond this bilinear form. Another interesting mechanism that leads to additional nonlinearities $h(X_1,\dots ,X_n)$ in the infection term $\beta IS$ is to take multiple or higher-order interactions into account. For example, when describing the spread of infectious diseases, the infection of an individual due to two contacts with infectious individuals that happen over a relatively short period of time is better described by the term $\beta S{I}^2$, where the state variable I occurs in quadratic form [128]. Likewise, in the field of drug addiction, the impact of social pressure that lures non-addicted individuals into addiction may be better described by a quadratic function of the addicted individuals A like $\beta S{A}^2$ [98,122]. Again, the rationale here is that social pressure is not about a single individual but a group of individuals affecting a susceptible person. In the context of rumor spreading and information diffusion, it suggests itself that such higher-order interactions are likely to take place. For example, susceptible individuals may become aware of a new piece of information via the joint impact of two spreaders, that is, two individuals who tend to pass on the information under consideration [129].

Figure 1 not only points out the importance of nonlinear mechanisms but also indicates that network models outperform ODE models in terms of generality. While network models can capture effects related to the heterogeneity of networks that are neglected in ODE models, ODE models are a powerful tool to study a plenitude of mechanisms that hold irrespective of the explicit structure of networks. For example, ODE models provide a useful framework for studying the effect of vaccination on the spread of an infectious disease in a population [1,2]. Likewise, when considering, for example, the spread of computer viruses, various effects, as discussed in Section 3.4, can be adequately addressed with the help of ODE models, such as the roll-out of anti-virus software [29,96,97], the waning of such anti-virus software’s protection [24,25,29], the reactivation of cleaned computers [96,97], or the premature reactivation of computers that are still infected [96,97]. Such effects and many others can be studied without a specific network structure in mind. Of course, network structures may have an impact on the magnitude of an effect. Therefore, the results obtained from the ODE models establish baseline scenarios that may be re-investigated in follow-up studies involving network models.

In this study, eight research disciplines were considered in which epidemiological modeling makes an essential contribution. Other research disciplines could be considered and/or some of the disciplines mentioned in this study could be broadened to include a wider spectrum of applications. For example, not only could smoking and alcohol addiction be passed on from addicted individuals to non-addicted individuals, as discussed in Section 2 and Section 3, but also certain types of behaviors. In this context, Usaini et al. [130] pointed out that a negative attitude and behavior in some student populations against their instructors is known to spread during exam weeks. The authors used an epidemiological ODE model to describe this kind of transfer of attitude-guided behavior across students and the emerging behavioral epidemic. Likewise, related to the topic of innovation diffusion discussed in Section 2 and Section 3, one may consider the spread of knowledge in academia as a form of epidemic in which knowledge is passed on from scholars to scholars. The key idea in this context is to measure the amount of knowledge in terms of the number of appropriately selected publications [100,131,132]. Epidemiological ODE models can then be proposed and fitted to data, as in refs. [131,132]. In the wake of the COVID-19 pandemic, it became obvious how vulnerable supply chains are against disturbances and how easily they can break down. This observation fueled the interest to apply epidemiological models to other models and examine the spread of disturbances in supply chains [15]. In short, research fields and applications that have not been addressed in the current study may be evaluated in a similar way as in the present study either to advance the research in those fields or to learn from insights or peculiarities that have been obtained and addressed in those fields.

In certain cases, it may be useful to combine epidemiological models from two different research fields into a single comprehensive model. For example, the spreading of knowledge concerning an infectious disease and the spreading of the disease itself have been modeled by a single complex model that takes into account various interactions between these two layers [129]. Likewise, information diffusion as an epidemiological phenomenon and the aforementioned spread of supply chain disturbances has been studied by merging two epidemiological models [133]. Such approaches may be seen in analogy to well-established models of infectious zoonotic diseases, where the virus spreads both within animal and human populations and between human and animal populations [2,9]. While in zoonotic disease modeling two epidemiological models from the same kind of discipline (i.e., infectious disease epidemiology) are combined, the previous examples are about merging models from different disciplines. Being aware of this analogy can help researchers in the field of zoonotic diseases to take advantage of research on interdisciplinary two-layered epidemiological approaches and vice versa.