Positive Recurrence and Extinction of Hybrid Stochastic SIRS Model with Nonlinear Incidence Rate

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 
 
 
 R
 0
 S
 
 
 for disease extinction, that is, when 
 
 
 R
 0
 S
 
 <
 1
 
 , the disease will die out. Finally, some examples are constructed to verify the conclusion.


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][2][3][4][5][6][7], stationary distribution and ergodicity [5,[8][9][10][11][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. is 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 (SI/(a + I)) and double epidemic hypothesis. In [6], Cai et al. studied the ratio-dependent transmission rate with the form g(S, I) � (S h I/(S h + αH h )) in SIRS epidemic model. Recently, Wei-Xue studied the asymptotic stability of SEIR model with incidence rate g(I)S � (SI/(1 + aI q )) [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 Le � vy noise on epidemic models [20,21]. Although many scholars have studied the different properties of epidemic models with white noise and Markovian switching [22][23][24][25][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 0 < q ≤ 1. S(t), I(t), and R(t) denote the numbers of the susceptible, infected, and recovered. Λ is the recruitment rate, d represents the natural death rate, β means valid contact rate, μ stands for the disease-caused death rate, c signifies the recovery rate of infected individuals, and δ expresses the rate at which the recovered ones lose their immunity and go back to the susceptible class. B i (t), i � 1, 2, 3 are independent Brownian motions, σ i (k), i � 1, 2, 3 are the intensities of the disturbances and Markov chain α t t≥0 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.

Preliminaries
Some background knowledge about differential equations and an important lemma of the paper will be introduced in this section. roughout this paper, denote by (Ω, F, F t t≥0 , P) the complete probability space with a filtration which satisfies the usual conditions. Furthermore, let R n + � x ∈ R n : x i > 0, 1 ≤ i ≤ n} and assume that B i (t), i � 1, 2, 3 are independent Brownian motions on the probability space.
Assume that α t t≥0 is a right continuous Markov chain in a finite state space N � 1, 2, . . . , N { } and its generator is for Δ↓0. Here, c ij > 0 if i ≠ j and N j�1 c ij � 0, for all i ∈ N. In this paper, we assume that the Markov chain is irreducible; thus, it has a unique stationary distribution π � (π 1 , π 1 , . . . , π N ) which can be obtained by solving the equation Considering the n-dimensional SDEs with Markovian switching, with the initial value be the set of nonnegative functions V(X, i) defined on R n × N and the functions are continuously twice differential in X. e differential operator L on the function Lemma 1 (see [27]). Supposing that the following three conditions hold true, (A2) For every i ∈ N and the positive constant λ, σ(x, i) satisfies (σ(x, i)ξ, ξ) ≥ λ|ξ| 2 ; (A3) ere exists a bounded set U with compact closure and a nonnegative function V(x, i) such that LV < − κ for any x ∈ U c and i ∈ N where κ is a positive constant, and then (X(t), α t ) is the positive recurrent. at is to say, there exists a unique stationary distribution Π(·, ·) such that for any integrable function h(x, i) with respect to Π.
For the model we discuss, our first concern is whether it has a unique positive solution. e following theorem will answer this question. (1) has a unique solution (S(t), I(t), R(t)) and the solution is still in R 3 + with probability one. e proof is common; we omit it here.

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 (S(t),
then the solution (S(t), (1) has a unique stationary distribution, which means that the disease will persist.
Proof. In order to prove eorem 2, we only need to check that the conditions in Lemma 1 are satisfied. First of all, we give some definitions of notations.
We verify that condition (A2) holds. Now, we move to prove condition (A3). Some functions are defined as follows: and ω k will be specified later. Using the continuity of the function V, we can take the minimum value of the function V(S * , I * , R * , ω * ) and let In this way, the nonnegativity of the selected function V can be guaranteed. By making use of the Ito formula, we get that Journal of Mathematics 2 (k))/2)), and select as well as Consider the system where A � (A(1), A(2), . . . , A(N)) T . Hence, it has the solution ω � (ω(1), ω(2), . . . , ω(N)) T such that N j�1 c kj ω j + A(k) � N k�1 π k A(k). erefore, we obtain that Here, λ ≔ R S 1 − 1 and R S 1 is defined in (7). Again, we apply the Ito formula to − lnS, − lnI, and V 3 to obtain Here, we choose sufficiently small constant p such that us, E < ∞ and Consequently, Select sufficiently large M and sufficiently small ϱ such that the following formulas hold: where the constants F, G, H, and K are all finite and their representations can been seen in (27)-(29) and (31). For the constant ϱ selected above, define a set Hence, we can divide the set R 3 ϱ , In what follows, we will prove that LV ≤ − 1 in R 3 + \D ϱ , that is, LV ≤ − 1 in the regions D i , i � 1, . . . , 6.

Journal of Mathematics
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.

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
By the nonnegative property of S(t), In addition, e proof is similar to that in [28]; we omit it here.
Assume further that en, the disease will extinct exponentially almost surely Proof. Applying Ito formula to lnI(t) yields Add the first two equations in (1) to get is, along with the third equation in (1), yields Integrating (24) and cd(S + I + mR) from 0 to t and dividing by t for both sides of formula, one has By virtue of Lemma 2 and ergodicity of Markov chain, (35) can be obtained. If R S 0 < 1, then ((lnI(t))/t) < 0, which means that the disease will extinct exponentially. e proof is completed.

Examples and Simulations
In this section, some examples are constructed and their simulations are presented to verify the abovementioned results.

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 R S 0 for disease extinction, that is, when R S 0 < 1, 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 Le � vy 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. ese are left to study in the future.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.