A First Application of the Backward Technique in Social Sciences: Exploring Demographic Noise in a Model with Three Personality Types

Abstract

In the realm of dynamical systems described by deterministic differential equations used in biomathematical modeling, two types of random events influence the populations involved in the model: the first one is called environmental noise, due to factors external to the system; the second one is called demographic noise, deriving from the inherent randomness of the modeled phenomenon. When the populations are small, only space-discrete stochastic models are capable of describing demographic noise; when the populations are large, these discrete models converge to continuous models described by stochastic ordinary differential systems, maintaining the essence of intrinsic noise. Moving forward again from a continuous stochastic framework, we get to the continuous deterministic setting described by ordinary differential equations if we assume that noise can be neglected. The inverse process has recently been explored in the literature by means of the so-called “backward technique” in a biological context, starting from a system of continuous ordinary differential equations and going “backward” to the reconstruction and numerical simulation of the underlying discrete stochastic process, that models the demographic noise intrinsic to the biological phenomenon. In this study, starting from a predictable, deterministic system, we move beyond biology and explore the effects of demographic noise in a novel model arising from the social sciences. Our field will be psychosocial, that is, the connections and processes that support social relationships between individuals. We consider a group of individuals having three personality types: altruistic, selfish, and susceptible (neutral). Applying the backward technique to this model built on ordinary differential equations, we demonstrate how demographic noise can act as a switching factor, i.e., moving backward from the deterministic continuous model to the discrete stochastic process using the same parameter values, a given equilibrium switches to a different one. This highlights the importance of addressing demographic noise when studying complex social interactions. To our knowledge, this is also the first time that the backward technique has been applied in social contexts.

1. Introduction

The idealized scenario of a world entirely governed by deterministic mathematical equations, in which populations exhibit harmonious and predictable behavior, represents the purpose of deterministic models widely used in the field of biomathematics. However, the complexities of the real world introduce a disruptive element to these evolutions. Stochastic events, such as the onset of unpredictable external events, i.e., an epidemic in Biology or a cataclysm in Ecology, or decisions due to personal or family motivations in Social Sciences, bring a certain degree of uncertainty into the dynamics of the populations involved in the model. When randomness affects the evolution of a physical system, the model associated with the system is said to be affected by noise.

Noise can be distinguished into two main types: the first type is denoted as extrinsic or environmental noise and comes from outside influences affecting the populations, like external environmental changes or resource availability; the second type is called intrinsic or demographic noise, which stems from the inherent randomness in births, deaths, immigration, and emigration within the populations. In this case, fluctuations arise from the internal dynamics of the interacting populations.

We know that a deterministic system can be predictable or unpredictable in the case of the appearance of deterministic chaos. In this study, we will analyze cases in which the continuous system is described by differential equations and by several parameters and involves deterministic predictability. Starting from a model of deterministic predictability, we will analyze the effects of the addition of internal noise to the model. When dealing with small population units, it is necessary to manage complex models by means of stochastic processes with discrete state space that are able to capture the inherent randomness due to when an interaction (a “reaction” in chemical language) and what interaction, among the feasible ones, occurs. This approach arises in chemistry and biochemistry to describe the evolution of a set of chemical reactions in a homogenous chemical reaction system. In this setting, the populations are chemical molecule species [1]. The exact solution of these discrete stochastic systems is achieved by the Stochastic Simulation Algorithm (SSA) introduced by Gillespie in 1977 [2]. The problem with SSA is that it becomes computationally too demanding as the populations increase. For larger populations, i.e., when we can talk of densities or concentrations (number/volume), these discrete models turn into the more handling forms of continuous stochastic systems described by Stochastic Ordinary Differential Equations (SODEs), called the Chemical Langevin Equations (CLEs) [3]. These equations preserve the fundamental characteristics of intrinsic random fluctuations, although in a time-continuous setting. Finally, when the diffusion coefficient of the CLEs is put to zero, meaning that the term addressing demographic noise is totally neglected, the CLEs become the Reaction Rate Equations (RREs), consisting of a set of Ordinary Differential Equations (ODEs). In [4], Carletti and Banerjee introduced an innovative approach to reconstruct the discrete stochastic process modeling demographic noise from a continuous deterministic biological ODE system.

In our study, we move beyond Biology and explore how demographic noise affects a novel model of social interaction. Our field will be psychosocial, that is, the connections and processes that support social relationships between individuals. We consider a group of individuals with three personality types: altruistic, selfish, and susceptible (neutral). Demographic noise can be thought of as the random variation of the populations due to birth, death, immigration, and emigration [5].

This allows us to show how the results and conclusions can differ dramatically between a world with perfect predictability and one filled with inherently occurring random events. For this model, demographic noise can act as a switching factor, where the stable scenario predicted by the deterministic model switches to different scenarios in the discrete stochastic system for the same parameter values. This highlights the importance of addressing randomness when studying complex social interactions. To sum up, the main idea of this article is a preliminary test of whether the addition of intrinsic noise to scenarios of deterministic social interactions can give rise to different or opposite landscapes from those predicted by the mathematical discrete model.

The paper is, thus, organized as follows: in Section 2, we introduce a novel ODE model describing social interactions among three different populations. In Section 3, we show the equilibria of the ODE model and discuss a preliminary study of the stability in the Appendix A of this article. In Section 4, we briefly recall the backward technique introduced in [4] and apply it to our ODE model, allowing us to reconstruct the discrete stochastic process modeling demographic noise associated with the original continuous deterministic system. In Section 5, we provide extensive numerical experiments representing crucial situations of both the continuous and discrete models and widely discuss the results. Finally, in Section 6, we offer some concluding remarks.

2. An ODE Model of Social Interaction

A focal point of research in both biology and sociology is comprehending how behavioral and cultural characteristics, emerging as reactions to alterations in socioeconomic or environmental circumstances, disseminate within populations [6]. Examples of behavioral traits being transmitted have been documented among various animal species, such as reef fish [7], birds [8], and between parents and their progeny [9]. These transmissions are facilitated by social interactions and can occur not only between parents and their direct offspring (vertical transmission) but also among peers (horizontal transmission) or between parents and the offspring of others (oblique transmission) [10,11].

In this paper, we focus on transmission between members of a cohort (horizontal transmission), which is different from what we recently did in [12], in which we examined the transmission from parents to their descendants (vertical transmission). We are interested in the evolutionary processes of behavioral traits spread by social interactions among individuals within a population. To examine this, we consider a specific cultural or behavioral trait, such as the propensity for altruism [13], that is horizontally culturally transmitted in its possible dichotomic values $A$ (altruistic) or $E$ (egoistic or selfish) in an individual. We do not consider altruism in economic or similar fields, but we idealize a test model of social interactions between susceptible, altruistic and selfish individuals.

The choice of altruism as a psychosocial characteristic in our model is indicative: A and E could be investigated as characteristics that could emerge when a dichotomous characteristic divides a group of individuals into two categories: for example, P(A) could be the proportion of individuals who have the characteristic A, while P(E) the proportions of who have the characteristic antagonistic to A, that is E.

Let $S\left(t\right)$ denote the proportion of the total population susceptible at time t, $A\left(t\right)$ the proportion of altruistic individuals at time t and $E\left(t\right)$ the proportion of selfish individuals at time t. When a susceptible individual meets an altruistic one, there is a probability that he becomes an altruist himself. However, there is a certain probability that he will not maintain this attitude forever, and he will lose this acquired character and become susceptible again. The same can occur in the case of a susceptible individual meeting a selfish one. When an altruist meets a selfish individual, he can become selfish himself with a certain probability, or vice versa. The test model is represented in a diagram figure (Figure 1).

In order to illustrate our technique for finding how behavioral and cultural traits are spread in populations by social interactions among individuals, we consider an ODE model describing the dynamics of interaction among susceptible, egoistic and altruistic individuals, introducing a number of parameters.

The model is the following:

$$\begin{array}{c}{\displaystyle \frac{dS\left(t\right)}{dt}}=-\mathit{aS}\left(t\right)A\left(t\right)-\mathit{eS}\left(t\right)E\left(t\right)+\alpha A\left(t\right)+\epsilon E(t)\\ {\displaystyle \frac{dA\left(t\right)}{dt}}=\mathit{aS}\left(t\right)A\left(t\right)-\mathit{pA}\left(t\right)E\left(t\right)-\alpha A\left(t\right)+qA\left(t\right)E(t)\\ {\displaystyle \frac{dE\left(t\right)}{dt}}=\mathit{eS}\left(t\right)E\left(t\right)-qA\left(t\right)E\left(t\right)-\epsilon E\left(t\right)+pA\left(t\right)E(t)\end{array}$$

(1)

where $S\left(t\right)$ is the fraction of the total population susceptible of being in contact at time t with an altruistic individual belonging to $A\left(t\right),$ that is the fraction of the altruistic population at time t or with $E\left(t\right)$ the egoistic fraction at time t. $a>0$ is the altruistic transmission rate, i.e., the constant contact rate with altruistic individuals, $e>0$ is the egoistic transmission rate, i.e., the constant contact rate with egoistic individuals, $\alpha >0$ is the rate of “loss altruistic” diffusion, i.e., a constant that measures the force for an altruistic individual to return susceptible, $\epsilon >0$ is the rate of “loss egoistic” diffusion, i.e., a constant that measures the force for an egoistic individual to return susceptible, $p>0$ is the switch constant rate from altruistic to egoistic, and $q>0$ is the switch constant rate from egoistic to altruistic.

The initial condition for system (1) is any point in $R^3$ with $S\ge 0, A\ge 0, E\ge 0$ and $S+A+E=1$. For the sake of simplicity, this model is assumed to be in dimensionless form.

Parameters e, α, ε, p, and q vary in the interval [0, 1], representing different situations shown in the caption of Figure 2. Several relations among parameters, on which the dynamics depend yield different points of equilibrium in the social interactions. It can be proven that four different equilibria exist, ranging from the susceptible equilibrium to the positive (endemic) one.

3. Equilibria of the ODE Model

In this section, we present a preliminary study of the equilibrium points of the nonlinear system. Other considerations are shown in the Appendix A.

Firstly, we can easily see that the susceptible (boundary) equilibrium point $E_S=\left(1, 0, 0\right)$ exists.

Secondly, for $\alpha \le a$ the altruistic equilibrium $E_A=\left({\displaystyle \frac{\alpha }{a}}, 1-{\displaystyle \frac{\alpha }{a}}, 0\right)$ exists and it is achievable.

Thirdly, we can easily see that for $\epsilon \le e$ the egoistic equilibrium $E_E=\left({\displaystyle \frac{\epsilon }{e}}, 0, 1-{\displaystyle \frac{\epsilon }{e}}\right)$ exists and it is achievable. These equilibria do not depend on p and q.

Lastly, we can evaluate the endemic equilibrium resulting in $E_{+}=\left({\displaystyle \frac{A^{\mathrm{*}}\left(q-p\right)+\epsilon }{e}},A^{\mathrm{*}},{\displaystyle \frac{a{A}^{\mathrm{*}}\left(q-p\right)+\epsilon a-e\alpha }{e(p-q)}}\right)$, $A^{\mathrm{*}}$ is equal to

$$A^{*}={\displaystyle \frac{\left(q-p\right)\left(\epsilon -e\right)-\epsilon a+e\alpha }{(q-p)(a-e+p-q)}}.$$

(2)

4. The Backward Technique and Its Application to the ODE Model

We briefly recall the backward technique introduced in [4] and apply it to our ODE model. Consider $S_1, S_2, \dots , S_N$ represent N interacting molecular species at constant temperature in a fixed volume Ω through a network of M chemical reaction channels. This system of interactions can be described mathematically by a discrete nonlinear jump Markov process in which a state vector $X\left(t\right)$ of dimension N, representing numbers of the N molecular species at time t, evolves through time. Capturing the full complexity of such a discrete system is a challenge. At the heart of this challenge lies the following time evolution equation, called the Chemical Master Equation (CME):

$${\displaystyle \frac{\partial }{\partial t}}P\left(x,\left.t\right|x_0,t_0\right)={\sum }_{J=1}^M{a}_J\left({x-v}_j\right)P\left(x{-v}_j,\left.t\right|x_0,t_0\right)-a_J\left(x\right)P\left(x,\left.t\right|x_0,t_0\right).$$

(3)

Solving the CME either analytically or numerically is often unfeasible and alternative methods are required to simulate its solution. A powerful tool is the Stochastic Simulation Algorithm (SSA), which treats each molecular species as a discrete stochastic process and randomly picks up one of the m reactions $R_1,\dots ,R_M$ to occur [2,3,14]. It is characterized by M stoichiometric vectors ${\nu }_1,\dots ,{\nu }_M$ which represents the update of the numbers of molecules in the system if reactions $R_1,\dots ,R_M$ occur, respectively, and by the set of M propensity functions $a_1\left(X\left(t\right)\right),\dots ,a_M\left(X\left(t\right)\right)$ such that $a_j\left(X\left(t\right)\right)dt$ is the probability that reaction $R_j$ will occur inside Ω in the next infinitesimal time interval $[t, t+dt)$ [1].

There are many biological examples successfully simulated by the SSA algorithm [15,16,17]. This method is mathematically rigorous but also computationally expensive, especially when the system is characterized by large amounts of molecules or slow reactions.

By applying the Central Limit Theorem and matching the first two moments of the Chemical Master Equation, we obtain the Chemical Langevin Equation (CLE) [3]. For large systems, the CLE provides a continuous approximation of the discrete dynamics captured by the SSA, in particular, intrinsic noise. CLE provides a good compromise between the accuracy of the solution and its computational cost. This equation is an Itȏ stochastic ordinary differential equation (SODE) and takes the form:

$$dX\left(t\right)={\sum }_{j=1}^M{v}_j{a}_j\left(X\left(t\right)\right)dt+B\left(X\left(t\right)\right)dW\left(t\right),\hspace{1em}\hspace{1em}\hspace{1em}X\left(0\right)=X_0,$$

(4)

where $X\left(0\right)=X0$ is the initial condition and ${{W\left(t\right)=(W}_1\left(t\right),\dots ,W_M\left(t\right))}^{\prime }$ is an N dimensional vector in which the components are independent Wiener processes, and B(X(t)) is an N × N matrix such that

$$B^2\left(X\left(t\right))=C\left(X(t\right)\right)=\nu Diag\left(a_1\left(X\left(t\right)\right),\dots ,a_M\left(X\left(t\right)\right)\right)v^T,$$

(5)

where $\nu =[{\nu }_1, . . . , {\nu }_M]$ is the N × M stoichiometric matrix [3]. $X\left(t\right)$ is now a continuous jump Markov process.

When the number of molecules is large [18] so that we can talk of population densities and noise can be neglected, the CLE becomes the Reaction Rate Equation (RRE)

$$dX\left(t\right)={\sum }_{j=1}^M{v}_j{a}_j\left(X\left(t\right)\right)dt,\hspace{1em}\hspace{1em}X\left(0\right)=X_0.$$

(6)

The RRE is an ODE that can be solved by usual non-stiff or stiff numerical methods, depending on the variation of the Lyapunov exponents [1].

The SSA, CLE and RRE can be seen as the three modeling regimes in discrete stochastic, continuous stochastic and continuous deterministic settings. Due to their computational costs, they are sometimes called slow, intermediate and fast regimes, and the process of moving through different scales in the direct sense from the slow to the fast regime can be interpreted as a “forward” technique to model intrinsic noise for an increasing number of molecules constituting biochemical systems.

From this perspective, Carletti and Banerjee introduced the idea of inferring intrinsic noise, preferably denoted as demographic noise, from biological ODE models by means of a “backward” approach [4].

With the present paper, we turn our attention to a novel behavioral ODE model that arises in the social sciences. We apply the backward technique introduced in [4] and investigate if, and eventually how, it is possible to observe, for the same parameter values, different dynamics of the system when considered in a continuous deterministic setting and in a discrete stochastic setting where demographic noise plays a crucial role [19].

We can now interpret model (1) as an RRE of the kind (6). In the framework of the backward technique, we interpret the ODE model as the ∆t going to zero limits of the discrete jump Markov process $X\left(t\right)=\left(X_1\right(t), X_2(t), X_3(t\left)\right)=(S\left(t\right),A\left(t\right),E\left(t\right))$ describing the discrete population vector of susceptibles, altruistics, and egoistics. The initial number of units of each interacting species is assumed to be different from zero.

According to [4], we assume that six “reactions” among the three populations are taking place, namely:

$$\begin{array}{c} S+A\to 2A\\ S+E\to 2E\\ \begin{array}{c}\begin{array}{c} A\to S\\ E\to S\\ A+E\to 2A\end{array}\\ A+E\to 2E.\end{array}\end{array}$$

(7)

The associate propensity functions $a_j\left(X\right)$, the stoichiometric vectors ${\nu }_j$, and reaction rates $c_j$ are the following:

$$\begin{array}{l}{a}_1\left(X\right)=c_1{X}_1{X}_2, a_2\left(X\right)=c_2{X}_1{X}_3, a_3\left(X\right)=c_3{X}_2,\\ a_4\left(X\right)=c_4{X}_3, a_5\left(X\right)=c_5{X}_2{X}_3, a_6\left(X\right)=c_6{X}_2{X}_3;\\ v_1=[-\mathrm{1,1},0], v_2=[-\mathrm{1,0},1], v_3=[1,-\mathrm{1,0}],\\ v_4=[\mathrm{1,0},-1], v_5=[\mathrm{0,1},-1], v_6=[0,-\mathrm{1,1}];\\ c_1=a, c_2=e, c_3=\alpha , c_4=\epsilon , c_5=q, c_6=p.\end{array}$$

(8)

The CLE associated with the discrete system (7) and (8) takes the form of the following Ito SODE:

$$dX\left(t\right)={\sum }_{j=1}^6{v}_j{a}_j\left(X\left(t\right)\right)dt+B\left(X\left(t\right)\right)dW\left(t\right), X\left(0\right)=X_0$$

(9)

where $X_0$ is the initial vector, ${{W\left(t\right)=(W}_1\left(t\right), W_2\left(t\right),W_3\left(t\right))}^{\prime }$ is a 3-dimensional vector whose components are independent Wiener processes and B(X(t)) is the 3 × 3 diffusion matrix satisfying

$$B^2\left(X\left(t\right)\right)=C\left(X\left(t\right)\right)=\nu Diag\left(a_1\left(X\left(t\right)\right),\dots ,a_6\left(X\left(t\right)\right)\right)v^T,$$

(10)

where $\nu =[{\nu }_1, \dots , {\nu }_6]\prime$ is the 3 × 6 stoichiometric matrix. The computations show that the drift component is the column vectorR2022b

$$D=\left(\begin{array}{c}-a_1\left(X\right)-a_2\left(X\right)+a_3\left(X\right)+a_4\left(X\right)\\ a_1\left(X\right)-a_3\left(X\right)+a_5\left(X\right)-a_6\left(X\right)\\ a_2\left(X\right)-a_4\left(X\right)-a_5\left(X\right)+a_6\left(X\right)\end{array} \right)$$

(11)

whereas the diffusion matrix B, such that $B^2=C$, is obtained as the matrix square root of C, where C is

$$C=\left(\begin{array}{ccc}{a}_1+a_2+a_3+a_4& -a_1-a_3& -a_2{-a}_4\\ -a_1-a_3& a_1+a_3{+a}_5+a_6& {-a}_5-a_6\\ -a_2{-a}_4& -a_5-a_6& {a_2+a_4+a}_5+a_6\end{array}\right)$$

(12)

5. Numerical Simulations and Discussion of Results

All the numerical experiments presented in this paper are performed on a last-generation PC and coded in a MATLAB R2022b environment run under Windows 11. We used standard ODE methods for the simulation of the solutions of the non-stiff differential system (1). We chose different parameter values representing an ordinary case, a case more inclined to altruism, a case more inclined to selfishness and finally, a balanced case among the proportions of the three different types.

We know that the numerical results from the discrete model obtained from a single trajectory cannot be used to demonstrate an equilibrium value when the model is stochastic, but our goal is only a preliminary illustration of an eventual discrepancy between the discrete social model and its counterpart with intrinsic noise.

The four situations are depicted in Figure 2. In all simulations, the initial state vector was (S₀, A₀, E₀) = (0.4, 0.3, 0.3).

The integration time is the interval [0, 500] days, although the significant results are found in the range [0, 100] days since for time greater than 100 days, the proportions of $S\left(t\right)$, $A\left(t\right)$, $E\left(t\right)$ are fixed and unmodified.

For the positive initial state vector and $a<\alpha$, $e<\epsilon$ (Figure 2a), the boundary susceptible equilibrium $E_S$ (1, 0, 0) is achievable. For several parameter values, we obtain three different equilibria: the altruistic equilibrium $E_A$ (Figure 2b), egoistic equilibrium $E_E$ (Figure 2c), and endemic equilibrium $E_{+}$ (Figure 2d).

If we apply the backward technique to the ODEs model (1), we obtain the discrete stochastic system (7) and (8) and SODE equation (9), whose exact solution can be computed via the SSA and the explicit Euler method to simulate the system [20]. We used a modified version of Gillespie’s SSA, working with a sorting algorithm to choose the firing reaction at each step. One would expect that the dynamics of the discrete stochastic system (8) and (9) have the same behavior as the continuous deterministic case. In Figure 3, we show these solutions associated with the continuous case through the backward technique. In a deterministic setting, we obtain the boundary equilibrium (1, 0, 0), the altruistic equilibrium, the egoistic equilibrium and the endemic (positive) equilibrium $E_{+}$ (where $S\left(t\right)>0, A\left(t\right)>0, E\left(t\right)>0$ are stable points for the choice of the parameter values (Appendix A). In a discrete stochastic setting, instead, for each kind of equilibrium, the numerical simulations do not show the existence of the same stable equilibrium as in the deterministic case. For example, the stochastic case for the scenario of the boundary susceptible equilibrium is not preserved: there are quite noticeable fluctuations for the selfish proportion, which often exceeds the susceptible proportion (Figure 3a). Moreover, the discrete stochastic case for the altruistic equilibrium scenario shows a kind of alternation of the altruistic and the selfish proportion (Figure 3b). Furthermore, the discrete stochastic case for the egoistic equilibrium scenario shows values for the proportion of selfish higher than altruists and susceptible, allowing the proportion of altruists to oscillate between non-negligible values (Figure 3c). Finally, the discrete stochastic case for the endemic equilibrium shows a kind of egoistic equilibrium (Figure 3d) or supremacy for the selfish proportion. We summarize the above results in

Table 1. Expected and observed stochastic scenarios in the discrete stochastic setting (8) and (9) for each of the equilibria of the deterministic model (1).

Equilibria	Expected Stochastic Scenario	Observed Stochastic Scenario
$E_S$ Susceptible equilibrium	Case more inclined to be susceptible	Case with noticeable fluctuations for the selfish proportion, which often exceeds the susceptible proportion
$E_A$ Altruistic equilibrium	Case inclined to altruistic with non-negligible values of susceptible	Balanced case among the proportions of altruistic and selfish individuals
$E_E$ Egoistic equilibrium	Case more inclined to selfishness	The proportion of selfish individuals is higher than that of susceptible ones with non-negligible values for the proportion of altruistic individuals
$E_{+}$ Endemic equilibrium	Balanced case among the proportions of the three different types, the proportion of susceptible individuals slightly higher than selfish ones, non-negligible values for the proportion of altruistic individuals	Case more inclined to selfishness

. This different behavior of the system shows that intrinsic noise significantly modifies the dynamics of the system observed in the deterministic case. In the discrete stochastic case, due to intrinsic noise, not even the scenario of the positive equilibrium is preserved. Maybe in the discrete stochastic case, we can reasonably say that none of the three personality types is left to fade away. It is worth noticing that the discrete stochastic system (8) and (9) shows a much weaker dependence on the parameter values than the ODE system does. Different models other than (1) show a similar discrepancy effect, yielding different dynamics of the systems in stochastic discrete settings and in the deterministic continuous regime [19].

6. Discussion

This article explores the application of the backward technique introduced by Carletti and Banerjee in [4] to the novel test model for social interactions (1). As in chemical kinetics [21,22,23,24,25], models arising in the field of social sciences are inherently stochastic in the sense that they are influenced by intrinsic—or demographic—noise. Therefore, the comparison between deterministic and stochastic settings is likely to offer a deeper understanding of the complex population dynamics processes where demographic noise plays a crucial role. In [4], the authors hypothesized that the backward technique could be successfully applied to a broad range of ODEs from different areas of research. A point with it is that the technique shares the limitations that occur in the direct process (discrete stochastic, continuous stochastic and continuous deterministic), for which the Stochastic Simulation Algorithms work only for specific types of chemical reactions (first order, second and higher order heterodimeric, homodimeric, and Hill-type reactions) so that well-defined mathematical formulations exist [21,22]. With our paper, we presented an example of how the backward process (continuous deterministic, continuous stochastic (skipped from the treatment) and discrete stochastic), when going backward from the ODE system (1) to its associate discrete system (8) and (9), provides a switching effect, in the sense that, for the same parameter values and initial condition, an equilibrium point of (1) with its scenario switches, in (8) and (9), to a different scenario closer to a proper one of another equilibrium point of (1). This fact means that demographic noise can have implications on the nature and stability of the equilibrium solutions of the model (1). In Macrelli et al. [19], a comparable discrepancy effect is observed. Similar aspects about how noise can alter equilibrium dynamics or the convergence of demographic stochastic dynamics toward deterministic dynamics are shown in [26,27].

Substantially, our results suggest that ignoring demographic noise in ODE models could lead to inaccurate predictions, particularly in systems with multiple equilibrium points.

The effects investigated in this preliminary study can be taken into consideration when we want to investigate the outcomes that could emerge when a dichotomous decision is posed to a group of individuals, for example, to predict the outcomes of an electoral campaign between two contenders or on the decisions taken by a group of individuals to choose to purchase a certain product rather than another.

An issue that will be investigated more accurately in a future article is if the backward technique presents the same aspects illustrated by the Renormalization Group Theory [28,29]; that is, if the value of the parameters that affect where the system turns from one attractor to another, change going “backward” from a system of continuous ordinary differential equations to the reconstruction and numerical simulation of the underlying discrete stochastic process. If it is so, these changes in the scenarios analyzed in our article would be not so unexpected.

Further and more specific studies are required to investigate the advantages and limits of the applicability of the backward techniques to ODE models. In particular, advances in chemical kinetics could allow us to extend the use of the backward technique to a wider range of ODE models. A different issue that could be addressed is studying which types of ODEs are most susceptible to the switching effect and if there is a connection between the intensity of demographic noise, assuming we find a possible way to measure its grade in an ordinal scale, and equilibria in the switching effect.

Finally, the practical consequences of the switching effect in real-world systems and how our results compare to models incorporating different types of noise (e.g., environmental noise) are certainly important points to be investigated.