Asymptotic Dynamics of a Stochastic SIR Epidemic System Affected by Mixed Nonlinear Incidence Rates

This paper considers a stochastic SIR epidemic system affected by mixed nonlinear incidence rates. Using Markov semigroup theory and the Fokker–Planck equation, we explore the asymptotic dynamics of the stochastic system. We first investigate the existence of a positive solution and its uniqueness. Furthermore, we prove that the stochastic system has an asymptotically stable stationary distribution. In addition, the sufficient conditions for disease extinction are also obtained, which imply that the white noise can suppress and control the spread of infectious diseases. Finally, in order to illustrate the analytical results, we give some numerical simulations.


Introduction
In 1927, the threshold conditions for disease transmission were established by Kermack and McKendrick [1]. In recent decades, many studies have applied threshold theory to various epidemic systems, and there have been a large amount of results related to the dynamical behaviors for various models. In addition, nonlinear incidence rates are very significant and are used frequently in the dynamics of epidemic models [2][3][4][5][6][7][8][9][10][11]. Hethcote [12] introduced an SIR epidemic system affected by bilinear incidence rate λSI: where the parameters u, λ, and η are positive constants. S, I, and R represent the density of susceptible, infectious, and recovered individuals, respectively. Suppose that the recruitment rate is the same as the natural death rate, denoted by u. λ is the transmission coefficient, η represents the recovery rate, and λ 1 SI represents the bilinear incidence rate. Furthermore, some researchers [13] proposed SIR epidemic models with saturated incidence (λSI/(1 + mI)) similar to the following form: where m is a positive constant. (λ 2 SI/(1 + mI)) represents the saturated incidence rate. Particularly, Liu et al. [14] studied an SIS epidemic system with general nonlinear incidence rate effect by employing Markov semigroup theory. Inspired by previous works, we consider that the infected person may have immunity to return to recovery class after being cured or return to the susceptible class again. And the epidemic affected by mixed nonlinear incidence rates is more realistic compared to epidemic affected by a single nonlinear incidence rate. us, when two different incidence rates (bilinear and saturated incidence rates) are considered at the same time, the following deterministic SIR epidemic system affected by two different nonlinear incidence rates is studied: where the parameters α, λ 1 , λ 2 , and p are positive constants. α denotes the probability that the infected returns to recovery class, 1 − α denotes the probability that the infected returns to susceptible class correspondingly, λ 1 and λ 2 are the transmission coefficients, p represents the probability that bilinear incidence rate occurs, and 1 − p represents the probability that saturated incidence rate occurs correspondingly. e basic reproduction number R 0 � ((pλ 1 + (1 − p)λ 2 )/(u + η)) is the threshold of model (3). However, the population systems with stochastic effect present some complex dynamics, which attracts the attention of widespread researchers [15][16][17][18]. ere are many kinds of noise in the environment, among which the white noise is a relatively stable noise in the process of propagation. It has been widely used and studied in physics and has a relatively complete system in mathematical analysis and application. In order to consider a more realistic disease model and make it more practical to study the influence of environmental noise on infectious diseases, the white noise is used. ere is a long history of using the white noise to depict the influence of environmental randomness on the spread of disease; some researchers [19,20] put forward that the environmental white noise disturbs the system parameters stochastically, and parameter λ i (i � 1, 2) is an important parameter for the spread of disease. e researchers [21][22][23][24][25][26][27][28][29] investigated the effect of environment on the dynamic behaviors by introducing stochastic perturbation into deterministic models. Based on the discussion above, we assume that where B i (t) represents a standard Brownian motion with intensity σ i > 0 (i � 1, 2). Now, the stochastic SIR epidemic system corresponding to system (3) is as follows: Since d(S + I + R) � [u − u(S + I + R)]dt, for any initial value (S(0), I(0), R(0)) ∈ R 3 + and S(0) + I(0) + R(0) � 1, we always have S(t) + I(t) + R(t) � 1. erefore, (S(t), I(t), is a positive invariant set of system (4). For simplicity, we let R(t) � 1 − S(t) − I(t); then, the dynamics of system (4) is equivalent to the following two-dimensional system: Next, we study the dynamical behaviors of system (5) which are affected by mixed nonlinear incidence rates. Particularly, as the main purpose, we study the asymptotically stable stationary distribution and extinction of epidemic by establishing the corresponding sufficient conditions.
In this paper, (Ω, F, F { } t≥0 , P) represents a complete probability space with a filtration F t t ≥ 0 satisfying the
represents an integral Markov semigroup with a continuous kernel K(t; x; y) for t > 0, which satisfies X K(x, y)m(dx) � 1 for ∀y ∈ X. For the diffusion process (S(t), I(t)), i.e., the transition probability function is P(t, x, y, A) � Prob (S(t), en, this semigroup is asymptotically stable or sweeping with respect to compact sets [30,31]; this property is called the Foguel alternative [34].

Remark 2.
If the distribution of (S(t), I(t)) is absolutely continuous with respect to the Lebesgue measure with the density U(t, x, y), then U satisfies the Fokker-Planck equation [32]: are independent on R 2 + . us, according to Ho .. rmander theorem (see [35]), . e proof is completed. In addition, we can obtain that the semigroup + and a continuous function ϕ ∈ L 2 ([0, T]; R); the following system is first considered:
Step 2. Prove that the derivative D i x 0 ,y 0 ;ϕ has rank 2. Replace system (14) with en, we get and 4 Complexity en, where erefore, the rank of D i x 0 ,y 0 ;ϕ is 2.
Step 3. Verify that, for any (x 0 , y 0 ) ∈ E and (x, y) ∈ E, there exist control function ϕ and System (18) becomes Using the same proof method of Lemma 3.2 in [36], we can obtain that, for any ). Now, the positivity of K is proved by support theorems (see [37][38][39]).
for every density f and the semigroup P(t) { } t≥0 , the following conclusion holds: Proof. Substitute Z(t) � S(t) + I(t); we can obtain that Next, in the following three cases, for almost every ω ∈ Ω; this proof is completed.
Proof. Using the same proof method of Lemma 4.5 in [14], this means that our result holds. We omit it here.
} t≥0 satisfies the Foguel alternative according to Lemma 4. We can exclude sweeping by constructing a nonnegative C 2 -function V and a closed set U ∈ Σ satisfying sup (S,I)∈X\U

e Property of the Positive Solution.
e first result is the existence and uniqueness of the positive solution of system (5). Complexity Theorem 1. System (5) has a unique positive solution (S(t), I(t)) on t ≥ 0 with any initial value (S(0), I(0)) ∈ R 2 + almost surely. Proof. By system (5), we obtain that (65) Clearly, V(S, I) is nonnegative. Using Itô's formula and the fact S(t) + I(t) ≤ K, we get dV(S, I) � LV(S, I)dt + pσ 1 (I − S) dB 1 (t) Here, K 0 is a positive constant. en, refer to [42], and we complete the proof.

Asymptotically Stable Stationary Distribution and Extinction of System (5)
Theorem 2. Denote (S(t), I(t)) a solution of system (5) with any initial value (S(0), I(0)) ∈ R 2 + ; the distribution of (S(t), I(t)) has a density U(t, x, y) for every t > 0. If Proof. e proof of eorem 2 is the following steps: Step 1. According to Ho .. rmander theorem [35], we prove that the transition function of the process (S(t), I(t)) is absolutely continuous (see Lemma 1) Step 2. We verify the positivity of the density of the transition function on E by support theorems [37][38][39] (refer to Lemma 2) Step 3. We prove that the Markov semigroup satisfies the Foguel alternative (see Lemmas 3 and 4) Step 4. Excluding sweeping by verifying there is a Khasminski � i function (refer to Lemma 5) Notice that the above strategies can be proved by Lemmas 1-5; thus, this proof is complete. □ Theorem 3. For any initial value (S(0), I(0)) ∈ R 2 + , (S(t), I(t)) is a solution of system (5). If one of the following conditions holds, the epidemic I(t) becomes extinct with probability one.
By the conclusion of eorem 1, it is worthy to point out that two different incidence rates which are considered at the same time will not destroy a great property that existence and uniqueness of the positive solution. From eorems 2 and 3, we know that when two infection rates are considered at the same time, in addition to the values of σ 1 and σ 2 , the value of p also affects the threshold value R 0 − ((p 2 σ 2 1 +(1 − p) 2 σ 2 2 )/2(u + η)). In addition, from the sufficient conditions for disease extinction, parameter p is an important variable related to the extinction of the population. Especially, when the parameters p � 1 and α � 1, system (3) becomes (1) in [12], and system (3) is similar to (2) in [13] when the parameters p � 0 and α � 1. To some extent, our model is more realistic than considering the epidemic affected by a single nonlinear incidence rate.
Since the proposed system is degenerate, we use Markov semigroup theory to study the stationary distribution and ergodicity of the system, and we can also study color noise and other noises in the future. For other noises, this method is feasible as long as it conforms to the relevant properties of Gaussian white noise. In addition, two aspects can be used as a guide for further research. First of all, we can investigate some other systems, for example, one can consider the systems with the impulsive perturbation effects [43,44]. In addition, it is interesting to study the chemostat as well as population dynamics systems [45][46][47][48][49][50][51]. Notice that it is a meaningful question to investigate whether the way used in this article is applied to other epidemic systems. ese questions are worthy of further study.   16 Complexity