Abstract
In this paper, the dynamic behavior of a class of hybrid SIRS model with nonlinear incidence is studied. Firstly, we provide the condition under which the positive recurrence exists and then give the threshold for disease extinction, that is, when , the disease will die out. Finally, some examples are constructed to verify the conclusion.
1. Introduction
Disease is one of the most dangerous enemies of human beings. COVID-19 has spread throughout the world since the end of 2019 and brought great harm to people’s life and global economy. It is one of the important measures to understand the characteristics of disease transmission and control the disease to establish and study the disease model. Because of its importance and rich research content, different scholars have used various methods to study dynamical behavior and properties of epidemic models, such as persistence and extinction [1–7], stationary distribution and ergodicity [5, 8–12], stability [13, 14], bifurcation [15, 16], and optimal control of disease [17]. In [17], the optimal vaccination strategies were used to minimize the numbers of the susceptible and infected individuals as well as to maximize recovered individuals.
Incidence rate plays an important role in the spread of diseases. Liu et al. [18] abandoned the assumption that incidence rate is proportional to the quantity of infected and susceptible ones to study the influence of nonlinear incidence function on SIRS model. This is more reasonable and suitable for practical situations, because the premise of bilinear incidence rate is that the population is homogeneously mixed, and the probability of being infected for each person is the same. Meng et al. [3] discussed SIS model with the form and double epidemic hypothesis. In [6], Cai et al. studied the ratio-dependent transmission rate with the form in SIRS epidemic model. Recently, Wei-Xue studied the asymptotic stability of SEIR model with incidence rate [19]. In this paper, we concentrate on general incidence rate with this form.
Life is full of stochastic disturbances, which often have an important impact on the system and will essentially change the properties of the system. White noise depicted by Brownian motion and telegraph noise characterized by continuous time Markov chain are two important types of noise. Some scholars have studied the effect of noise on epidemic models [20, 21]. Although many scholars have studied the different properties of epidemic models with white noise and Markovian switching [22–26], there are few studies on the positive recurrence and extinction of SIRS models with the abovementioned general incidence rate. In this paper, we will study the abovementioned properties of stochastic hybrid SIRS model with the following form:where . , , and denote the numbers of the susceptible, infected, and recovered. is the recruitment rate, d represents the natural death rate, means valid contact rate, stands for the disease-caused death rate, signifies the recovery rate of infected individuals, and expresses the rate at which the recovered ones lose their immunity and go back to the susceptible class.
, are independent Brownian motions, , are the intensities of the disturbances and Markov chain will be introduced in detail in Section 2. Moreover, Section 2 will give an important lemma which will be used in this paper. Obviously, Markovian switching and nonlinear incidence rate make the model more applicable, but also increase the difficulty of research. Section 3 is devoted to prove the positive recurrence and stationary distribution of model (1) by constructing suitable Lyapunov functions. Section 4 gives the threshold for disease extinction. Some examples and their simulations will be presented in Section 5 to verify our results.
2. Preliminaries
Some background knowledge about differential equations and an important lemma of the paper will be introduced in this section.
Throughout this paper, denote by the complete probability space with a filtration which satisfies the usual conditions. Furthermore, let and assume that , are independent Brownian motions on the probability space.
Assume that is a right continuous Markov chain in a finite state space and its generator is , satisfyingfor . Here, if and , for all . In this paper, we assume that the Markov chain is irreducible; thus, it has a unique stationary distribution which can be obtained by solving the equation
Considering the -dimensional SDEs with Markovian switching,with the initial value and , where , , and . Let be the set of nonnegative functions defined on and the functions are continuously twice differential in . The differential operator on the function is defined by
Lemma 1. (see [27]). Supposing that the following three conditions hold true,(A1) if ;(A2)For every and the positive constant , satisfies (A3)There exists a bounded set with compact closure and a nonnegative function such that for any and where is a positive constant, and then is the positive recurrent. That is to say, there exists a unique stationary distribution such thatfor any integrable function with respect to .
For the model we discuss, our first concern is whether it has a unique positive solution. The following theorem will answer this question.
Theorem 1. For the initial value , model (1) has a unique solution and the solution is still in with probability one.
The proof is common; we omit it here.
3. Positive Recurrence of Model (1)
In this section, positive recurrence of model (1) will be discussed by utilizing the theory put forward by Khasminskii under some conditions.
Theorem 2. For any initial value , ifthen the solution of model (1) has a unique stationary distribution, which means that the disease will persist.
Proof. In order to prove Theorem 2, we only need to check that the conditions in Lemma 1 are satisfied. First of all, we give some definitions of notations. Let , , and . The first condition (A1) holds obviously from the assumption of Markov chain.
Letwhere is a sufficiently small number. Thus, . From the expression of model (1), we can see that , , which implies that is positive definite and thusHence, for any , it has
We verify that condition (A2) holds. Now, we move to prove condition (A3). Some functions are defined as follows:and the function where the constants , , , , and will be specified later. Using the continuity of the function , we can take the minimum value of the function and letIn this way, the nonnegativity of the selected function can be guaranteed.
By making use of the formula, we get thatLet , and selectas well asConsider the systemwhere . Hence, it has the solution such that . Therefore, we obtain thatHere, and is defined in (7).
Again, we apply the formula to , , and to obtainHere, we choose sufficiently small constant such thatThus, andConsequently,Select sufficiently large and sufficiently small such that the following formulas hold:where the constants are all finite and their representations can been seen in (27)–(29) and (31).
For the constant selected above, define a setHence, we can divide the set into six regionsIn what follows, we will prove that in , that is, in the regions , . Case 1: when , it has where From the result of (21), one has . Case 2: when , it has where . can be obtained by (22). Case 3: when , we have Here, . can be obtained from the result of (23). Case 4: when , we havewhereThus, we can get that with the help of (24).
Similarly, and can be proved in a similar way. So far, we have proved that in region . By virtue of Lemma 1, the positive recurrence of model (1) can be obtained.
Remark 1. We can see from the proof that in order to obtain the positive recurrence of the model, different Lyapunov functions are constructed to reach the goal.
4. Extinction of Disease
In this section, we will discuss the conditions for extinction of disease in model (1). Let us start with a lemma as follows.
Lemma 2. Assume that there exists some constant such that and is the solution of model (1) with the initial value , then
By the nonnegative property of , it has
In addition,
The proof is similar to that in [28]; we omit it here.
Theorem 3. Let and . If for each , then
Assume further that
Then, the disease will extinct exponentially almost surely
Proof. Applying formula to yieldsAdd the first two equations in (1) to getThis, along with the third equation in (1), yieldsIntegrating (24) and from 0 to and dividing by for both sides of formula, one hasBy virtue of Lemma 2 and ergodicity of Markov chain, (35) can be obtained. If , then , which means that the disease will extinct exponentially. The proof is completed.
5. Examples and Simulations
In this section, some examples are constructed and their simulations are presented to verify the abovementioned results.
Let be the Markov chain with state space and its generator is Thus, its stationary distribution is .
Example 1. We will give the values of the abovementioned parameters in the two states in the form of vectors, that is, means , in two states. To verify the result of Theorem 2, we assume that , , , , , , , , , , , , and such that the threshold in Theorem 2, . According to the result of the Theorem 2, the model has unique stationary distribution and the disease will persist, see Figure 1.
Example 2. To verify the result of Theorem 3, let , , , , , , , , , , , , and such that the threshold in Theorem 3, . According to the result of Theorem 3, the disease will die out, see Figure 2.
6. Conclusions
In this paper, the dynamic behavior of a class of hybrid SIRS model with nonlinear incidence is studied. Firstly, the condition under which the positive recurrence exists is provided, and then we give the threshold for disease extinction, that is, when , the disease will die out. Finally, some examples are constructed to verify the conclusion.
Some other issues deserve further study. For example, the incidence function can be more generalized, or noise can be added to make the model more realistic. We can also investigate other properties of the model, such as stability and so on. These are left to study in the future.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (no. 11661065) and the Scientific Research Fund of the Jiangxi Provincial Education Department (no. GJJ191145).