1. Introduction
In ancient times, societies have grappled with numerous infectious diseases, posing significant threats to human health and the environment at large [
1,
2,
3]. According to the World Health Organization (WHO), infectious diseases account for more than 17 million deaths annually [
4]. Fortunately, modern health care systems, with the assistance of disease control methods, treatment, and spreading information to communities, continue making these endeavors more effective. Despite these strides, over the past three decades, urban and animal populations have witnessed the emergence of over 30 diseases, some exhibiting alarming characteristics, such as mutations facilitating transmission from animals to humans or between humans [
5]. Although modern medical innovation has made it possible to come up with cures and vaccines for some emerging diseases, the process of developing a vaccine that would be immune for all strains is arduous, as was the case with the COVID-19 pandemic that took a long time for an effective vaccine to be developed and rolled out to the world. Socioeconomic inequalities, particularly among impoverished communities unable to effectively isolate, contribute to disease spread. Understanding where diseases concentrate helps inform strategies to control outbreaks, with medical science focusing on vaccine and antibiotic development. Applied mathematicians and epidemiologists are pivotal in developing epidemiological models to analyze disease spread dynamics [
6,
7,
8,
9]. Kuniya et al. [
10] introduced various SIS epidemic models, while Mouaouine et al. [
11] explored diverse SIR epidemic models. Although SIS and SIR models are pivotal in epidemiology [
12], they overlook the latent period observed in certain infectious diseases like chickenpox, influenza, and tuberculosis, and vector-borne illnesses such as dengue hemorrhagic fever, west nile virus, and malaria. These diseases are often asymptomatic for a long incubation period, making controlling spread in the initial stages difficult. In other words, they are incubation period bites, during which non-infected individuals are carriers of the pathogen long before they become infectious. Recognizing this, SEIS epidemic models [
13,
14] and SEIR epidemic models [
15,
16] have been developed and analyzed.
Still, SEIR models do not represent well in practice because they presume that the individual remains immune to the disease for an infinite time, that is, until the person dies. However, this assumption does not hold for many diseases. For instance, in malaria and rotavirus, even after recovering from the disease, one will only be protected for a short duration, and even the time before this disease manifests is quite long. As a result, such people may be susceptible again with time once their temporary immunity wears off.
It is worth mentioning that some types of epidemic models include other forms of control such as, for instance, quarantine, which consists of confining persons who are suspected of being infected but not yet diagnosed with an infectious disease [
17,
18], or immunization [
19,
20,
21] as an efficient measure of controlling outbreaks. This approach is beneficial in curbing diseases such as polio and tuberculosis, and vaccination has primarily led to their destruction. The earlier models of epidemics concerning the power of immunization usually assumed total immunity after vaccination [
22]. However, this notion does not hold water in every situation. The disease can still be acquired by those vaccinated, although this occurs at a much lower probability than it does for those not vaccinated.
The traditional models have been highlighted as needing more improvement in the recent work conducted on COVID-19. A new “identifiability model” is developed to avoid the additional parameters in the standard Target Cell Limited model, which are rarely helpful in making predictions that “fit” the observations [
23]. Likewise, even simple quarantine or treatment compartmental models demonstrate that even minor alterations in the public health policy severely constrain the scope for infection [
24]. The discrete SIR models have also been well characterized in the propagation of disease and have aided in controlling the spread of epidemics remarkably [
25]. Relatively few such studies account for the combined effect of the two diseases. Co-infection dynamics studies have become predominating, for instance, the dynamics of interplays between dengue, chikungunya, and more complex HIV/HCV infections where a system of fractional derivatives is used to model the system [
26,
27]. These models help understand the spread strategy and available “best” treatment methods. A better understanding of disease dynamics has been achieved. Hence, some clinicians have advocated using fractional-order derivatives models due to their ability to include history in infectious transmission. Non-local fractional methods are used for a study of the transmission of SARS-CoV-2 [
28], and pathologies like multiple sclerosis have recently been modeled this way [
29] with an improvement in prediction and detection at an early stage. By integrating factors like co-infections and fractional calculus, these models provide deeper insights into disease spread and control, contributing to more effective public health strategies [
30,
31].
Delay differential equations (DDEs) can be classified under the dynamical system approach, and their behavior is more complex. The introduction of a suitable time lag in the model can, on the other hand, destabilize an equilibrium that formerly was stable, thereby resulting in population fluctuations. This article emphasizes the utility of time delay in the population models and population dynamics, especially models concerning disease outbreaks. Different infectious diseases have different patterns for their delay in population dissemination. To introduce the relevance of delay differential equations (DDEs) in the context of population dynamics and epidemics, a comprehensive overview delving into delayed logistic-type equations, which are commonly utilized in epidemic modeling, is provided in [
32,
33].
This study is motivated by recognizing the complexities in infectious disease dynamics, especially in vaccination programs. Traditional models often overlook the impact of time delays in disease modeling from significant factors such as incubation period, vaccination timing, and population response. By incorporating multiple delays, we aim to improve the model’s realistic nature and predictive accuracy, which enhances public health interventions and policy decisions.
Numerous researchers have integrated delays into various scenarios following the initial exploration of time-delayed epidemic models. These include considerations such as the duration of vaccination periods [
34], delays in the asymptomatic carriage and infection periods [
35], delays in immunity periods, and delays in the incubation or latent periods [
35,
36]. Several studies have investigated disease transmission models with delays. For example, some studies incorporate delays in specific stages of the disease progression. In [
35], a model with delays in incubation and asymptomatic carriage periods is analyzed. Similarly, ref. [
37] explores an SEIRS model with constant delays in the latent and immune periods.
Other studies focus on models with general disease transmission dynamics and specific delay structures. In [
38], a general model with a latent period and relapse is studied, while [
36] investigates a time-delayed SIR model with a nonlinear incidence rate and a Holling type II treatment. Vaccination strategies within time-delayed models have also received attention. In [
39], a model with a delay representing the time for susceptible individuals to recognize their infection and seek vaccination is explored while [
40] studies different methods of introducing delays. Additionally, refs. [
41,
42] explore other time-delayed models incorporating vaccination. In [
43], the study investigates monkeypox by incorporating a quarantine compartment and optimal control strategies. Inspired by the above approaches, we propose an SVIR epidemic model with multiple delays to analyze the spread of infections in this work.
This paper is organized as follows: In
Section 2, we propose a mathematical model with multiple time delay, derive the basic reproduction number, and prove the positivity and boundedness of the solution.
Section 3 delves into the proposed models’ stability analysis and examines the sensitivity index of the basic reproduction number (BRN) concerning various parameters. In
Section 4, we present the formulation and solution of the associated optimal control system. In
Section 5, our theoretical results are affirmed through numerical simulations.
2. Model Formulation and Positivity of the Solution
Modeling the latent period plays a significant role in predicting the spread of infectious diseases. A recently studied model [
44,
45] has been developed by incorporating numerous improvements to enhance realism and accuracy in depicting the real-life spread of diseases.
This study extends the current framework by incorporating time delays, vaccine effectiveness, and the possibility of waning immunity. The model provides a more realistic representation of disease spread and vaccination efficacy by including these dynamics. The introduction of parameters for delayed transmission, vaccine effectiveness, and immunity loss significantly enhances the model’s applicability to real-world scenarios [
46].
This model below introduces an SVIR system for individuals exposed to infection and examines the latent period, which is the interval between an individual becoming infected with a virus and the appearance of disease symptoms. This period can vary significantly, depending on factors such as the specific disease, like polio or COVID-19, and whether individuals have been vaccinated. Vaccination may offer some degree of protection against the virus, potentially affecting the duration or progression of the latent period. However, the extent of this effect can vary based on factors such as the type of vaccine, the individual’s immune response, and the level of virus transmission within the community. Conversely, individuals who have not received any vaccine doses are generally more susceptible to infection and may experience a different latent period. Introducing a delay parameter into the model represents the length of the latent period, which can differ between individuals who have received partial vaccination and those who have not been vaccinated at all. In our analysis, we denote the latent period for susceptible individuals as
and the latent period for vaccinated individuals as
. This delay parameter represents the time interval between when an individual is infected and when they become infectious. The above epidemic scenario is modeled as
with the initial condition:
Here, with and
In the traditional infectious disease model, susceptible individuals (S) can contract the infection and transition to the infected (I) state. Infected individuals can then recover and become immune (R). With the rise of COVID-19 and polio vaccinations globally, vaccinated individuals (V) form a significant demographic that must be considered separately. In this context, vaccinated individuals have not received sufficient vaccine doses to confer lifelong immunity. Recent years have seen a resurgence of polio in particular African and Middle Eastern countries, as well as in conflict zones like Syria, as reported by the World Health Organization, see [
47]. Polio is a hazardous virus that necessitates multiple doses for lifelong immunity. Those who have only received a single dose remain at risk during periods of virus resurgence. Fully immunized individuals are categorized as recovered (R), indicating a reduced risk of reinfection. Our contribution includes delineating an SVIR epidemic model encompassing four key variables: the susceptible (S) population, the vaccinated (V) population, the infected (I) population, and the recovered (R) population. Additionally, our model accounts for time delays, specifically incorporating disease latent periods denoted by the parameters
, where
and
represent the respective latent periods. We consider the following assumptions to model the interactions between the four classes:
is between 0 and 1; when , the vaccines are invalid and, when , it shows that the vaccines are completely active.
Delayed effects: Since individuals who were infected at time (or ) become infectious at a later time t, we use to represent the contribution of infectious individuals from to the overall infectivity at time t. Similarly, represents the contribution of infectious vaccinated individuals from to the infectivity at time t.
We assume that vaccinated individuals are susceptible to infection before acquiring immunity, with a disease transmission rate
when in contact with infected individuals. It is reasonable to assume
because vaccinated individuals may have partial immunity during the vaccination process or may modify their behavior to reduce contact with infected individuals based on their understanding of disease transmission characteristics. The details of the population flux are shown in
Figure 1 and a description of the parameters is given in
Table 1.
We assume that
. The disease-free equilibrium (DFE) point of model (
1) is given as
, where
Stability characteristics of the equilibrium points within the system are often influenced by a crucial parameter known as the basic reproduction number (BRN), denoted as
. This vital parameter represents the number of secondary infections from one infected individual throughout the entire infectious period in an entirely susceptible population. In the context of model (
1), the computation of the BRN can be found in references [
48,
49]:
This vital parameter helps us to identify the secondary infections resulting from the primary infected individual throughout the entire infectious period. It helps identify which model parameters are crucial in controlling the disease spread. The same will also be discussed in the sensitivity analysis of this paper.
Next, the endemic equilibrium point of model (
1) is denoted as
, where
where
Examining reveals its positivity, evident from the condition when . Thus, from the preceding analysis, we observe that the endemic point is positive under the condition .
Positivity and Boundedness of Solutions
In order to prove the positiveness and eventual boundedness of solutions
, and
of Equation (
1) we offer the following lemma on the related area.
Lemma 1. Let be a solution of system (1) with the initial conditions (2). Then, Moreover, for , if holds, where , then Proof. From initial conditions (
2), we obtain
Continuity of
ensures through our assumption that
represents the solution of system (
1). Therefore, it is clear that there exists a
neighborhood around “0” such that
According to the first equation in (
1), we obtain
, contradicting the assumption that
for
and
. Therefore, if there exists a positive
where
,
,
, and
for any
, then
.
Moreover, by the second, third, and fourth equations of (
1), we have
Consequently, if a positive
exists where
,
,
, and
for all
, then it follows that
and
. Hence, we derive Equation (
6). Consider the delayed system in (
1) as
From (
1) and (
2), we have
with initial conditions
We proceed by considering the following auxiliary equation (as discussed in [
33,
50]):
The above equation exhibits a unique globally positive equilibrium that is
. We define the following functional:
We proceed by analyzing the time derivative of
evaluated along the solution trajectory
of (
7),
From the equilibrium condition
, we have
We establish that the functional
is non-negative and monotonically decreasing. Furthermore, there exists a non-negative constant
such that
converges to
as
t approaches positive infinity:
. Equation (
8) is satisfied exclusively when
; we have
, which yields
. Thus, based on the results of the foregoing discussion and employing the comparison principle, we can conclude that
. □
3. Stability Analysis
The present section is dedicated to studying the stability characteristics of the equilibrium points of system (
1). Initially, we present the outcomes related to the local asymptotic stability of
[
51]. As stated in the upcoming theorem, the following is asserted:
Theorem 1. The DFE is locally asymptotically stable if .
Proof. At the equilibrium point
, the Jacobian matrix is
where
and
The characteristic equation adopts the following form for the system
To study the impact of the time delay on the stability of the equilibrium, model (
1) is analyzed in four cases:
Theorem 2. If and the following conditions are satisfied then the endemic equilibrium of (1) is locally asymptotically stable: Proof. The detailed proof of the theorem has been given as
Appendix A. □
Next, we study the sensitivity index of the BRN concerning the parameter values defined in model (
1). As in [
4], first, we define the following:
Definition 1. The normalized forward sensitivity index of a function for , is denoted by , and is defined as The following are the results calculated for the model parameters, which determine the BRN:
where
If
, or
values increase/decrease, the BRN also increases/decreases. Suppose
, or
values increase/decrease; then, the BRN value decreases/increases. By assuming the model parameter values to be as in
Table 2, the sensitivity index of the BRN over the model parameter is obtained, and those values are shown in
Table 3. Further, the same is depicted in
Figure 2.
Table 3 shows that parameters
,
and
exhibit negative indices, while
, and
display positive indices, as illustrated in
Figure 2. Negative (positive) indices indicate that
decreases (increases) with the parameter. For instance,
signifies that, if
increased by
, then
decreased by
. Similarly,
suggests that a
increase in
r results in a
increase in
. Furthermore, it is notable that
, the disease transmission rate, and
, the new recruitment, exhibit the highest positive sensitivity indices. Consequently, reducing both the new recruitment
and the disease transmission rate
effectively lowers the value of the basic reproduction number.
Figure 3 demonstrates that an increase in the spread of infection rate results in an increase in the basic reproduction number. Similarly,
Figure 4 shows the impact of the inflow population increasing also making the
value greater. These parameters are the key parameters to control the spread of infections. Further, we demonstrate for two different forms, reduction in population (death rate due to infection
) and the infection spread rate of the vaccinated
, and the results are shown in
Figure 5. This clearly represents an increase in the infection spread and a decrease in population due to infection death, make the spread of infection greater. Next, we discuss for the impact of recovery rate and loss of immunity in
Figure 6; here, it is evident that the lose of immunity plays a vital role in the infection spread, which is easily visible from
Figure 6. Finally, the impacts of inflow of the population and vaccine efficiency are considered; this shows that the loss of vaccine efficiency plays a vital role as compared with the inflow of the population to the model. This is clearly shown in
Figure 7.