The Dynamics of a Stochastic SIR Epidemic Model with Nonlinear Incidence and Vertical Transmission

In this study, we build a stochastic SIR epidemic model with vertical infection and nonlinear incidence. The influence of the fluctuation of disease transmission parameters and state variables on the dynamic behaviors of the system is the focus of our study. Through the theoretical analysis, we obtain that there exists a unique global positive solution for any positive initial value. A threshold 
 
 
 R
 0
 s
 
 
 is given. When 
 
 
 R
 0
 s
 
 <
 1
 
 , the diseases can be extincted with probability one. When 
 
 
 R
 0
 s
 
 >
 1
 
 , we construct a stochastic Lyapunov function to prove that the system exists an ergodic stationary distribution, which means that the disease will persist. Then, we obtain the conditions that the solution of the stochastic model fluctuates widely near the equilibria of the corresponding deterministic model. Finally, the correctness of the results is verified by numerical simulation. It is further found that the fluctuation of disease transmission parameters and infected individuals with the environment can reduce the threshold of disease outbreak, while the fluctuation of susceptible and recovered individuals has a little effect on the dynamic behavior of the system. Therefore, we can make the disease extinct by adjusting the appropriate random disturbance.


Introduction
At the beginning of 2020, a sudden epidemic (COVID- 19) has disrupted people's normal life. In order to curb the spread of the epidemic, the state has taken measures such as closing cities and delaying the opening of schools to protect people's lives to the maximum extent. So far, the disease has not been completely controlled. Emerging infectious diseases have brought fear and inconvenience to people's lives and have a great impact on the global economy. It is very important to study the spread of infectious diseases.
Dynamic modeling is an important method to study the spread of infectious diseases. In the classical epidemic model, such as the SIR model, the total population is generally divided into three categories, the number of susceptible individuals by S(t), the number of infectious individuals by I(t), and the number of permanently immune individuals by R(t). e model is as follows.
where A is the recruitment rate of susceptible corresponding to immigration, b is the birth rate, d is the nature death rate, μ is the disease induced mortality rate, c is the rate of recovery from infection, a positive constant p (0 ≤ p ≤ 1) is the proportion of infection in the offspring of infected mothers, 1 − p is the proportion of susceptible individuals in the offspring of infected mothers, and β is the transmission coefficient between compartments S(t) and I(t). f(I) is continuously differentiable function and assume f(0) � 1 and f ′ (I) ≥ 0. For the theoretical research of this model (1), there are a lot of literatures; interested readers can refer to the literatures [1][2][3][4][5]. e classical ODE model can reflect the spread of disease to some extent, but in real life, the spread of infectious diseases is also affected by many random factors (as shown in [6]), for example, the unpredictability of person-to-person contact, which means that there is not necessarily uniform contact between individuals. In other words, the process of disease transmission is inevitably affected by random factors. In this way, the stochastic differential model is more suitable than the deterministic model.
ere are many literatures on the stochastic differential model of infectious diseases [7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Here, we mainly introduce the literature of stochastic differential models related to the SIR epidemic model in detail. In literature [8], Aadil and Omari built the SIRS stochastic differential model with parameter perturbation, vaccination of recruitment susceptible, and nonlinear incidence rate. ey found that large enough random disturbances can suppress outbreaks. In 2015, Liu and Chen [7] considered a model of literature [8] and observed that the solution of the system fluctuates near the disease-free equilibrium under suitable conditions. Zhou et al. [16] studied an SIR model with the bilinear infection rate and stochastic perturbation of parameter and state variables and got the conditions of survival and stationary distribution.
Since the contact between populations and the population itself are affected by many random factors in the environment, we will introduce stochastic white noise perturbations into system (1) by two different approaches.
First of all, inspired by the literature [21], we consider a discrete time Markov chain. For a fixed time increment Δt > 0, we define the process X Δt (t) � (X Δt 3) be three sequences of random variables. Suppose that these variables are jointly independent and that within each sequence the variables are identically distributed, such that where the parameter σ i ≥ 0 (i � 1, 2, 3) represents the intensity of the environmental white noise. We assume that X Δt grows within that time period according to the deterministic system (1) and the random amount en, we will show that X Δt (t) converges weakly to a diffusion process as Δt ⟶ 0. To determine the drift coefficients of the diffusion, first, let p Δt (x, dy) denote the transition probabilities of the homogeneous Markov chain X Δt (kΔt) ∞ k�0 , that is, (4) for all x � (s, i, r) ∈ R + 3 and any Borel set A ∈ R + 3 . Combining with (2), we have for all x � (s, i, r), 1 Δt 1 Δt To determine the diffusion coefficients, we consider the moments Using (2), we obtain erefore, for all 0 < K < ∞. Similarly, one can obtain that Finally, we will extend the definition of X Δt (t) to all t ≥ 0 by setting X Δt (t) � X Δt (kΔt) for t ∈ [kΔt, (k + 1)Δt). According to eorem 7.1 and Lemma 8.2 of [22], from (5)-(13), we can conclude that as Δt ⟶ 0, X Δt (t) converge weakly to the solution X(t) � (S(t), I(t), R(t)) of the following SDE: Discrete Dynamics in Nature and Society 3 where B i (t), i � 1, 2, 3 is a standard Brownian motion.
e stochastic SIR epidemic model with vertical transmission and nonlinear incidence is as follows.
where B 4 (t) is a standard Brownian motion and parameter σ 4 ≥ 0 represents the intensity of the environmental white noise. e structure of this study is as follows: In Section 2, we study the existence and uniqueness of the positive solution of system (15). In Section 3, we give sufficient conditions for the extinction of the disease. In Section 4, we prove the existence of the stationary distribution of system (15) under some conditions by using the appropriate Lyapunov function. In Section 5, we discuss the solution of system (15) spirals around the disease-free equilibrium and endemic equilibrium of deterministic system (15) under proper conditions. Finally, the correctness of the theoretical results is verified by numerical simulation.

Existence and Uniqueness of the Global Positive Solution
In order to analyze the dynamic behavior of the system, we first need to discuss whether the solution of the system is nonnegative and global existence? In stochastic differential equations, if their coefficients satisfy local Lipschitz conditions and linear growth conditions, system (15) has a global positive solution [24]. However, the coefficients of system (15) do not satisfy linear growth conditions, which may lead to diseases exploded at a finite time. Next, we will show that the solution of system (15) does not blow up in finite time, so that the solution of system (15) will be global.
Proof. Since the coefficients of system (15) are locally Lipschitz continuous, for any given initial value (S(0), I(0), where τ e is the explosion time. Next, we will verify that this solution is global, i.e., τ e � +∞ a.s. First, let ξ 0 be sufficiently large, such that (S(0), For each integer ξ ≥ ξ 0 , define the stopping time as [24] Without loss of generality, let inf ∅ � +∞ (∅ is the empty set); we have, τ ξ ≤ τ e . If we prove that τ ξ � +∞, a.s., then τ e � +∞ and (S(t), I(t), R(t)) ∈ R + 3 , a.s. for all t ≥ 0. If the statement is not true, then there is a positive constant T, such that Define a C 2 function V: For ω ∈ τ ξ < T and all t ∈ [0, τ ξ ), by Itô's formula, one can verify that 4 Discrete Dynamics in Nature and Society Using f(0) � 1 and f ′ (I) ≥ 0, the inequality (23) can be reduced to here, Integrating both sides of the inequality (24) from 0 to t yields Note that some components of S(τ ξ ), Letting t ⟶ τ ξ in (26), it leads to the contradiction: us, τ ξ � +∞ a.s. e proof is completed.

Extinction of the Disease
For system (1), using the notations in [25], we can deduce the basic reproductive number R 0 . First, we have two vectors F and V to represent the new infection term and remaining transfer terms, respectively: Discrete Dynamics in Nature and Society e infected compartment is I; then, Hence, the reproduction number is given by where d > 0 and 0 ≤ p ≤ 1. If R 0 ≤ 1, the disease will go to extinction. If R 0 > 0, the disease will persist. For system (15), we focus on whether diseases can be extincted by regulating the parameters of system (15). For the sake of simplicity, denote 〈x(t)〉 � (1/t) t 0 x(s)ds.

Lemma 1. Let (S(t), I(t), R(t))
be the solution of system (15) with initial value (S(0), Proof. Using the method of Lemma 1 in [26], it is easy to prove the conclusion (32) and (33). en, we prove that conclusion (34) holds. From system (15), we obtain Integrating both sides of (35) from 0 to t, then dividing by t yields Taking the limit of both sides of (36) and using (32) and (33), we have e proof of the lemma is completed. Define It is obvious that R s 0 ≤ R 0 .
Discrete Dynamics in Nature and Society Theorem 2. Let (S(t), I(t), R(t)) be the solution of system (15) with initial value (S(0), I(0), R(0)) ∈ R 3 + and suppose that one of the following conditions holds: then, I(t) will tend to zero exponentially with probability one, i.e., In addition, we also have where k is a positive constant.
Proof. Using Itô's formula, we have Taking integrate both sides of (43) from 0 to t, then dividing by t, we yield By the strong law of large numbers for martingales [24] and Lemma (32), we have If condition (A) holds, (44) becomes and using (45), we obtain If condition (B) holds, (44) becomes Furthermore, using (45), we can get So far, we proved (40) and (41). From these two limits, we can get that there is a constant k > 0, such that for almost all ω ∈ Ω, there exists a T 0 � T 0 (ω) > 0, when t > T 0 , Solving the third equation of system (15) and denoting It is obvious that erefore, using (51), we have Furthermore, from system (15), we can obtain Consequently, where us, we have the following corollary to supplement eorem 2. (15) with the initial value (S(0), I(0), R(0)) ∈ R 3 + . Assume that σ 4 � 0.

Existence of the Stationary Distribution
For the deterministic model (1), if R 0 > 1, there was an unique globally asymptotically stable endemic equilibrium which implies the disease will be persistent. But for system (15), although there is no endemic equilibrium, we also expect to know the trend of the positive solution of the system. e trend of positive solution can be explained by the stationary distribution. If we can prove that system (15) exists a stationary distribution, it can show that the disease persists for a long time. en, we will give a main theorem that there exists a stationary distribution for system (15) according to a well-known result from Khasminskii [27].
Let X(t) be a regular time-homogeneous Markov process in R d described by the stochastic differential equation: where h(x) � (h 1 (x), h 2 (x), . . . , h d (x)), g r (x) � (g 1 r (x), g 2 r (x), . . . , g d r (x)), and B r (t) (r � 1, 2 , . . . , k) are the standard Brownian motions defined on some probability space (Ω, F, F t t ≥ 0 , P). e diffusion matrix for equation (60) is defined as follows: where f(x) is a function integrable with respect to the measure π.
Remark 2. To validate (i), it suffices to prove that there is a positive number G, such that d i,j�1 a ij (x)ξ i ξ j ≥ G|ξ| 2 for all x ∈ U and ξ ∈ R d [28,29]. To validate (ii), it suffices to prove that there exists a nonnegative C 2 function V and a neighborhood U, such that for some c > 0, LV(x) < − c, x ∈ R d ∖U [30].
hold, then there exists a stationary distribution and ergodic property for system (15). Here, (S * , I * , R * ) is the unique endemic equilibrium of system (1), and (64)

Discrete Dynamics in Nature and Society
Proof. When R s 0 > 1, which means that R 0 > 1, there has an epidemic equilibrium E * (S * , I * , R * ) in system (1) and S * , I * , R * are satisfied with Next, build a function V: where (67) By computing and using (a + b) 2 ≤ 2(a 2 + b 2 ), f ′ (I) ≥ 0, and ab ≤ ((a 2 + b 2 )/2), we obtain Discrete Dynamics in Nature and Society (68) leads to where holds, then lies entirely in R + 3 . Namely, the compact set U ⊂ R + 3 . us, there is a constant c > 0, such that for any x � (S(t), I(t), R(t)) ∈ R + 3 ∖U, we have Finally, we obtain Discrete Dynamics in Nature and Society

Asymptotic Behavior around the Equilibria of System (1)
In the section, we will study the asymptotic behavior of the solution of system (15) around the disease-free equilibrium and endemic equilibrium under random perturbations.

Theorem 4. Let (S(t), I(t), R(t))
be the solution of system (15) with initial value (S(0), Proof. Now, define a function V: where By calculating, we obtain  (1), respectively. R s 0 < 1 < R 0 shows that large white noise can lead to disease extinction, though the deterministic system is persistent.  Figure 3: e subgraphs (a) and (b) denote the numerical simulation of the stochastic system (15) with R s 0 > 1 and deterministic system (1) with R 0 > 1, respectively. is shows that the disease will be permanent in a long time. 16 Discrete Dynamics in Nature and Society where which is a continuous local martingale; then, taking expectation on both sides of (91) yields
Remark 4. From eorems 4 and 5, it can be seen that the solution of system (15) will oscillate around the disease-free equilibrium and endemic equilibrium of system (1) under some conditions, and the disturbance intensity is proportional to the white noise intensity. From a biological point of view, the solution of system (15) fluctuates most of the time around the disease-free equilibrium and endemic equilibrium of system (1) due to the small magnitude of the random disturbance.

Conclusions and Numerical Simulations
In this study, we are mainly concerned about the influence of stochastic factors on the behavior of the epidemic SIR model with a nonlinear incidence and vertical transmission. Here, the stochastic factors mainly include the disease transmission coefficient and state variables affected by white noise. System (15) has a unique global positive solution starting from the positive initial value. Furthermore, we give a threshold value R s 0 to distinguish the persistence and extinction of diseases, which differs from the basic reproductive number R 0 corresponding system (1) by a noise term. When R s 0 < 1, the conditions for extinction of disease are obtained in system (15). By constructing a stochastic Lyapunov function, we prove that an ergodic stationary distribution exists in system (15) when R s 0 > 1, which means that the disease will persist. Finally, we study the asymptotic properties of solution of system (15) around disease-free equilibrium and endemic equilibrium of the deterministic model (1) under certain conditions and under the existence of stationary distribution, respectively. From the biological point of view, the interference of environmental white noise may have certain influence on the stability of the biological system: the ability of population to adapt to the environment is limited. If the intensity of white noise in the environment is small enough, the stability of the population will not be damaged. If the intensity of white noise is large in the environment, it may lead to the extinction of species.
In the following part, we will give some examples to verify the theoretical results by numerical simulation. e method of numerical simulation is shown in reference [32]. For convenience, take f(I) � 1 + mI. Example 1. Fix the parameters A � 0.43, β � 0.9, d � 0.6, μ � 0.1, c � 0.2, b � 0.1, p � 0.2, σ 1 � 0.1, σ 2 � 0.5, σ 3 � 0.1, σ 4 � 1, m � 1 in system (15). It is easy to calculate Condition (B) of eorem 2 is satisfied. As shown in Figure 1(b), S(t), I(t), R(t) fluctuate around (A/(d − b)) � 0.86, 0, 0 with the increase of time, respectively. is proves that the conclusion of eorem 2 is correct. Furthermore, take σ 4 � 0 and other parameters keep the same with the first set of parameters. Under this set of parameters, R s 0 � 0.7375 < 1. From Figure 1(c), it can be seen that the result of Corollary 1 is true.
In the study, we only consider the white noise. In real life, the environmental impact may be not only white noise but also color noise, so can we consider both of them in the model? In addition, it may be more reasonable to consider the spatial factors and time delay when building the model. We leave these issues for further investigation and look forward to resolution in the future.

Data Availability
No data were used to support this study.

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