Abstract
In recent years, the world knew many challenges concerning the propagation of infectious diseases such as avian influenza, Ebola, SARS-CoV-2, etc. These epidemics caused a change in the healthy balance of humanity. Also, the epidemics disrupt the economies and social activities of countries around the world. Mathematical modeling is a vital means to represent and control the propagation of infectious diseases. In this paper, we consider a stochastic epidemic model with a Markov process and delay, which generalizes many models existing in the literature. In addition, we show the stochastic threshold for the extinction of the disease. Furthermore, numerical examples are discussed to confirm the theoretical result.
1. Introduction and Preliminaries
For years, the development of more realistic mathematical models in epidemiology has attracted the attention of many researchers. Consequently, many deterministic models have been suggested to understand and control the propagation of infectious diseases.
The first mathematical model that describes the spread of an epidemic in a human population divided into three categories (susceptible (S), infected (I), and recovered (R)) was initially proposed by Kermack and Mackendrick [1]. Then, several generalizations of the SIR model of Kermack and Mackendrick were proposed. For example, in [2], a mathematical model called SEIR was proposed, which describes the propagation of an epidemic in a human population divided into four classes: susceptible, exposed, infected, and recovered. In [3], Cheng et al. investigated a network-based SIQS infectious disease model with a particular incidence rate. They proved that the disease-free equilibrium of the system is globally asymptotically stable, and the unique endemic equilibrium is globally attractive. Zhang et al. [4] constructed an epidemic model with a saturated treatment function described by the following ordinary differential equation system:
The parameters in system (1) have the following meaning: represent the recruitment rate of the susceptible class, specify the death rate of the population, and denotes the death rate due to the epidemic. Moreover, the parameter is the death rate due to the epidemic. is the saturated incidence rate of disease [5], is the contact rate, and is a parameter that measures the psychological or inhibitory effect of the population. There exist other types of incidence rates, and each one represents some advantages in modeling (for more detail, see Table 1). is saturated treatment function [4]; designates the cure rate and measure the extent of the effect of the infected being delayed for treatment (see, [4]).
However, the stochastic model represents some advantages compared with the deterministic one. In effect, the propagation of transmissible diseases is naturally randomized. Thus, it is necessary to introduce an environmental noise into the system to present the fluctuation of the environment. Generally, there exist many types of environmental noise: white noise [11], colored noise [12], Lévy noise [13], etc., and every one represents some advantages in modeling. Rajasekar et al. [14] studied a stochastic epidemic model with a saturated incidence rate and saturated treatment function. While El Koufi et al. [12] proposed an epidemic model incorporating with the Lévy process. Other works have used two types of noise in their models. For example, Li and Guo [15] presented a stochastic SIS epidemic model in which they mixed three types of noise, namely, white noise, colored noise, and Lévy noise. In the works [16, 17], the authors proposed models with delay and Markovian switching for ecological populations. We can use the telegraph (or colored) noise to describe the mutation of the system from one state to another. Indeed, the propagation of an epidemic is affected by external factors such as nutrition, pandemics, climate, etc. For example, the transmission rate of the influenza epidemic in the spring is not the same as in the summer. Thus, the system frequently transitions from one regime to another. Many scholars studied the effect of telegraphic noise on the transmission of an epidemic (see, for example, [18–23]).
To develop the model (1) and the stochastic model proposed by Rajasekar et al. [14], we suppose that the transmission rate of disease fluctuates randomly around an average value and that the system often switches between two or more environmental regimes.
We assume that the stochastic perturbations are of the white and telegraphic noises types, that is,where is a Brownian motion and is the intensity of noise, is a right-continuous Markov chain.
Then, we present a new stochastic epidemic model for a human population with a saturated incidence rate and saturated treatment function in regime-switching described by the following stochastic hybrid differential equations:with the initial valuewhere with is the Banach space of continuous functions, mapping the interval into , and .
The parameter stands for the successful vaccination rate of susceptible individuals. The quantity represents the case that an individual survives natural death before reverting to the susceptible class, is the length of immunity period of the recovered. The quantity indicates the period in which the vaccine has not yet given its effect, and is the validity period of the vaccination.
Throughout this paper, is a Brownian motion defined in a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right-continuous while contains all -null sets), is the intensity of noise. is a right-continuous Markov chain defined in the same space as the Brownian motion, and supposedly independent of the Brownian motion. The Markov chain takes values in a finite-state space with the generator defined for bywhere is the transition rate from to and if , while . To ensure that the system goes switching from one regime to another regime. In the present paper, we suppose that the Markov chain is irreducible. Then, there exists a unique stationary distribution of such that , with and , for each in . Therefore, for any sequence , let and .
The stochastic differential equations with Markovian switching [24] have the following form:where , and . We present the operator associated with equation (6) and for any function twice continuously differentiable for bywhere
Then the generalized Itô’s formula [25] is displayed by
Theorem 1. For any initial value defined in (4), the model (2) has a unique solution for all and the solution will remain in with probability one.
With the same step in [11], we can prove the above theorem, and the following region is almost surely positively invariant [12].
Our model represents many effects compatible with the natural word as(i)The memory effect represents an essential component of the natural population [26–28].(ii)The regime-switching. In fact, the epidemic disease often switches between two or more different regimes of the environment [18, 19]. So, integrating the colored noise into the epidemic system allows the latter to switch between different environmental regimes. Often, the switching between environmental regimes is usually memoryless. In addition, the waiting time for the ensuing switch follows the exponential distribution [25]. Then, we model the regime-switching by a Markov chain taking value in a finite-state space .(iii)The vaccination represents an intervention medical that plays a significant role in the reduction of the infected class [29].
This paper aims to study the extinction of disease. By proposing a threshold that includes the noise term of the stochastic system, we establish sufficient conditions for the extinction of disease. Moreover, our model (2) can be used to represent the switching between two or more regimes of environment, which differ by factors such as nutrition or socio-cultural factors. In addition, the model (2) represents a generalized form of many SIR epidemic models existing in the literature. The rest of this paper is as follows: in Section 2, we prove the sufficient conditions for the extinction of the disease. In Section 3, we make simulations to confirm our theoretical results.
2. Extinction
The main question in mathematical epidemiology is the determination of the conditions that ensure the disappearance of an epidemic in a population. For the deterministic system, the answer to this question is made when the value of (the basic reproduction number [30]) is less than or equal to one. In general, the value of depends on the parameters of the deterministic system. So, for our stochastic delayed system, we have defined a threshold value that depends on the delay and the random effects defined bywhere . The following theorem presents a condition for the extinction of the disease in model (2) in the function of the threshold value .
Theorem 2. Let be a positive solution of system (2) with the initial value given in (4), if , thenIn other words, will go to zero exponentially, namely, the disease dies out with probability one.
Proof. Applying Itô’s formula to yieldsBy (7), we getIntegrating both sides of the above inequality from 0 to and dividing by , we havewhere is a local martingale with quadratic variation expressed asBy the strong law of large numbers for local martingales (see, [31]), we resultOn the other hand, in view of the system (2), we deriveThus, we haveConsequently, we haveBy substituting the inequality (10) in (16), we findTaking the limit superior on both sides of (22) and combining with (18), using the ergodic theory of the Markov chain, we obtainIf , we obtain thatwhich implies thatHowever, the diseases in the system (2) die out exponentially with probability one.
Remark 1. Through Theorem 2, we find an impressive result, namely, when is less than one infective go to extinction. Thus, the evanishment of the disease depends on the delay and the noise value.
3. Numerical Simulations
In this part, we present some numerical simulations in order to support our analytical results. For this, we use the approximate numerical resolution method for the stochastic differential equations of Euler and Maruyama. Many authors have chosen this method to give a numerical approximation of solutions for some stochastic differential equations. Then, we consider the Markov chain taking values on the state space , with a generator defined byand the stationary distribution . Figure 1 demonstrates the path of the Markov chain .
Therefore, to simulate, we take the parameter values presented in Table 2. Next, a simple computation gives
Then by Theorem 2, for any initial (3), the solution of (3) satisfies
Thus, the disease in model (2) is extinct, Figure 2 supports this result.
By choosing different noise values as Case 1: , , Case 2: , , and Case 3: , , according to Figure 3, we see that when the value of noises increases, tends more rapidly to zero, then we conclude that a large value of noise leads to the disappearance of the epidemic from the population.
In order to examine the effect of vaccination immunity on the dynamics of an epidemic in the model (2), we keep the same parameter values in Table 1, and we vary the value of the period immunity . So, we see that when the period of validity of the vaccination is sufficiently large, the epidemic disease can be decreased in the population (see, Figure 4).
4. Conclusion
In this paper, we have proposed a stochastic epidemic model with Markov switching. Our model represents a generalization of many models existing in the literature. For example, the stochastic model proposed by Rajasekar et al. in [14], when , the regime-switching and delay are not considered. Also, using the stochastic Lyapunov approach, sufficient conditions are established to guarantee the extinction of epidemic disease by introducing a suitable stochastic threshold value , which depends on delay terms and stochastic noise. We have obtained exactly the following results:(i)Let be a positive solution of system (2) with the initial value given in (4), ifthen epidemic infection dies out with probability one.
Then, from the expression of , we remark that a large value of white noise is helpful to control the propagation of infectious diseases (see Figure 3). In addition, the long period of vaccination validation can reduce the number of infected people in the population (see Figure 4). In our future works, we can study the memory effect on the dynamics of model (2) by using the new generalized fractional derivative presented by Hattaf in [32].
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The author declares that there are no conflicts of interest.