Modeling and analysis of SIR epidemic dynamics in immunization and cross-infection environments: Insights from a stochastic model

. We propose a stochastic SIR model with two different diseases cross-infection and immunization. The model incorporates the effects of stochasticity, cross-infection rate and immunization. By using stochastic analysis and Khasminski ergodicity theory, the existence and boundedness of the global positive solution about the epidemic model are ﬁrstly proved. Subsequently, we theoretically carry out the sufﬁcient conditions of stochastic extinction and persistence of the diseases. Thirdly, the existence of ergodic stationary distribution is proved. The results reveal that white noise can affect the dynamics of the system signiﬁcantly. Finally, the numerical simulation is made and consistent with the theoretical results.


Introduction
Mathematical modeling is an important tool that can help us understand the transmission of an infectious disease. Many scholars [1-3, 6-11, 13, 15, 16, 18, 21, 24, 30] have put forward mathematical models and have made contributions to disease control.
As far as we know, a classic and important SIR model was early investigated by Kermazk and McKendrick [11] in 1927. In classical SIR models, the infected patients can recover health with treatment. Many scholars have also investigated the SIR model in different situations. Capasso et al. [3] summarized Kermazk-McKendrick model and took into account nonlinear incidence phenomena for large numbers of infectives. Meanwhile, Capasso et al. expanded the threshold theory and laid a foundation for solving the stochastic nonlinear threshold. Hethcote [10] gave a qualitative analysis of nonlinear incidence SIR model, which is appropriate for viral agent diseases and considered social impact. Liu [18] investigated a deterministic and modified nonlinear SIR model with periodic solutions. In [18], the author also explored the corresponding stochastic epidemic models and the asymptotic behavior of the solution. Ghosh et al. [7] gave an SIR model with nonmonotonic incidence and logistic growth. In [7], authors studied the condition for backward bifurcation and Hopf bifurcation and solved the optimal control problem. Dieu et al. [6] classified a stochastic SIR model and developed ergodicity of the underlying system.
In classical epidemiological models, bilinear and standard incidence rates [28] are suitable for a small number of people in a short time, so many scholars use nonlinear incidence [2,6,7,10,16,18,21]. Logistic model [7] is more in line with the law of social population growth. In references [16,21], the authors introduced the stochastic epidemic model with cross-infection of diseases. It has become a common phenomenon that people are infected with different diseases at the same time. On the basis of reference [2,6,7,10,16,18,21], a deterministic SIR model with cross-infection and permanent immunization is proposed in which the nonlinear incidence is used. The corresponding model is as follows: where S(t) and R(t) with the natural mortality rate µ 1 and µ 4 , respectively, are the susceptible class and the removed class, respectively. I 1 (t) and I 2 (t) are the individuals with cross-infection at time t. β 1 and β 2 are the contact rates. p is the proportion of patients infected by two diseases. µ 2 and µ 3 are the mortality of cross-infection diseases, which include natural mortality and mortality due to diseases. γ 1 and γ 2 are recovery rates of cross-infection diseases, respectively. δ is constant vaccination rate of susceptible class.
The susceptible class will have permanent immunity after vaccination. All parameters are positive. Functions β 1 SI 1 /(α 1 + S) and β 2 SI 2 /(α 2 + I 2 ) represent nonlinear incidence rates for cross-infection diseases. However, the spread of diseases is often affected by environmental noise [1,4,8,9,13,15,16,21,24,28,30], a lot of literatures add randomness to reflect real life more accurately. The properties for stationary distribution of random variables were proved in [8]. [15] gave an SIR epidemic model that a susceptible person is infected with a disease and temporarily immunized. All references [1,4,9,13,24,28,30] are stochastic epidemic models with nonlinear incidence that has been used in chemostat model [25]. We assume that the mortality rates of S(t), I 1 (t), I 2 (t), R(t) are disturbed by white noise in system (1), then we have Thus, we can consider the following system: In the next, we only consider the dynamic properties of system (2) through differential equation theories. Some notations of stochastic differential equations can be seen in [19]. Let (Ω, F, {F t } t 0 , P) be a complete probability space, which has a filtration {F t } t 0 and satisfies the usual conditions. The functions B 1 (t), B 2 (t), B 3 (t) are Brownian motion defined on this complete probability space. We define R m + = {y ∈ R m : y i > 0, 1 i m}. Let f (t) be an integrable function on [0, +∞) and define f (t) = (1/t) t 0 f (θ) dθ. This paper mainly studies mathematical modeling and theoretical proof as well as the influence of parameter changes on the model.
We arrange the article as follows: Section 2 proves that system (2) has a global positive solution, which is unique. In Section 3, stochastic boundedness of the solution is explored. Section 4 investigates the extinction and persistence conditions of the stochastic system (2). We show the maximum value at point S 0 and the existence of a unique ergodic stationary distribution in Section 5. At last, we give the numerical simulations and a brief conclusion.

Extinction and persistence in mean
From system (2) we get We consider the stochastic equation From stochastic comparison theory we know that S(t) X(t). From Pasquali [22] we get Lemma 1.
where ω is an integrable function with measure ρ.
Proof. From system (4) we get the stationary Fokker-Plank equation with probability density π(X). Let h(X) = σ 2 1 X 2 π(X), then we can simplify the equation in the following form: Then we can get where H(X) = e X 1 g(τ )dτ , J is a constant. We can calculate where c 1 and c 2 are constants. We have From the conditions π(X) 0 and ∞ 0 π(X) dX = 1 we integrate the above formula, For the following proof, we define Lemma 2. (See [29].) If (S(t), I 1 (t), I 2 (t)) is the solution of system (2), then we have Theorem 3. The diseases I 1 (t) and I 2 (t) are said to be extinctive if R 1 < 1 and R 2 < 1, respectively.
Proof. We first prove the extinction of disease I 1 (t). Applying Itô's formula to system (2), one has From system (4) we learn S X. Then we have Then we get the following inequality by Lemmas 1 and 3 In the same way as in the proof of I 1 (t), we can get the following inequality from system (2): where S X. Then we get So we have Theorem 4. For any given initial value (S(0), I 1 (0), I 2 (0)) ∈ R 3 + , we have the following results of system (2): (i) If R 1 > 1 and R 2 < 1, then I 1 (t) will be persistent in mean, and I 2 (t) will be extinct. Besides, we have (ii) If R 1 < 1 and R 2 > 1, then I 1 (t) will be extinct, and I 2 (t) will be persistent in mean. Besides, we have (iii) If R 1 > 1 and R 2 > 1, then I 1 (t) and I 2 (t) will be persistent in mean and satisfy Proof. From the first equations of system (2) and system (4) one gets and https://www.journals.vu.lt/nonlinear-analysis From (5) and (6) one has that is, (i) By Theorem 3, since R 2 < 1, then lim t→+∞ I 2 (t) = 0 a.s. Since R 1 > 1, then 0 < I 2 (t) < and > 0 small enough such that Applying Itô's formula to the the Lyapunov function ln I 1 , it follows that Calculating (7) directly, we can obtain Simplifying the inequality above, we can obtain where ε → 0 as t → +∞. According to Lemma 4, one sees that lim t→+∞ I 1 (t)/t = 0 and lim t→+∞ ln I 1 (t)/t = 0 as I 1 (t) 1.

Stationary distribution
In this section, we use the Has'minskii theory [12] to prove the stationary distribution of system (2).

It is clear that for
Applying the Itô's formula yields where ζ ∈ (0, I 2 ) and From above equation we can get then we have Note that https://www.journals.vu.lt/nonlinear-analysis According to (10)-(11), we get G(S) G(S 0 ) = 0. Therefore, Next, Therefore, Construct a compact bounded subset U : and > 0 will be given in the later. In the set R 3 + \ U , choosing small enough such that Here B, C, D, E, F and J are positive constants defined in equations (20), (21), (22), (23), (24), (25), respectively. Next, six domains are given in the following: Together with (17), we have that LV −1.

Numerical simulations
We construct the following equivalent model to facilitate computer simulation: where ξ, ζ, η are independent random variables, ∆t is the time taken divided by the step size.
In each figure below, every figure has two subfigures. The first subfigure represents the development trend of S(t), I 1 (t), I 2 (t), respectively. The second subfigure is the probability density of S(t), I 1 (t), I 2 (t). From Theorem 3 we know that the diseases I i (t) (i = 1, 2) will be extinct when R i < 1 (i = 1, 2). When R i > 1 (i = 1, 2), the diseases will be persistent.
In our simulations, we only consider the influence with the changes in white noise on the disease. When the values of white noises are large than a certain value, the disease will be extinct. When the white noise is less than a certain value, the disease will be persistent. The figures are consistent with the theorem in our paper. In Fig. 1, we let σ 2 = 0.8 and σ 3 = 0.75. By calculating we obtain that R 1 = 0.6899 < 1 and R 2 = 0.7334 < 1, then the conditions of Theorem 3 hold. So I 1 (t) and I 2 (t) are extinct.
In Fig. 4, we simulate the influence of different noise intensities on system (2). It is found that as the intensity of white noise increases, the number of infections will decrease.   In addition, as time goes on, it shows periodic outbreaks and the duration of the outbreak is shortened. Diseases are always affected by various noises in the environment, then the changes of environmental noise can lead to changes in diseases. According to Theorems 3 and 4, the conditions of extinction and persistence in mean about system (2) have been established. These theorems are in fact a development of the papers by Cai [2], Liu [16] and Meng [21]. Furthermore, we used a new stochastic method to investigate the extinction and persistence, which is different from the previous works [2,6,7,16,18,21]. The results that obtained in the present work can be applied to stochastic model of proportional disturbance. The obtained theory is a positive and effective guidance for cross infection. Many diseases, such as diphtheria, typhoid and influenza, are cured, the susceptible class can have permanent immunity. This feature can be well reflected in this model.
The model can introduce telephone noises such as continuous time Markov chain [5,17,27]. We only study one susceptible person, and we can study multiple susceptible persons. We also can investigate a susceptible person infected with more than three diseases [32]. We can explore the periodic solution of the epidemic model [23]. The impact of white noise on not only mortality rate but also infection rates also will be considered. The methods also can be used in stochastic food chain models [26,31]. In our future work, we will solve these problems.

Conclusion
This paper provides a modeling framework based on stochastic differential equations to explore the long-term dynamics of epidemic cross-infection with SIR epidemiological laws. Since the interaction between the disease and the environment is full of stochasticity, it is of great practical significance to explore the mechanism of environmental stochasticity on the dynamics of infection. For this reason, we assumed that each component of the population is subject to environmental stochasticity, which is positively correlated with the density of each component of the population. In addition, in view of the fact that immunization is widely used in the control of epidemic diseases and has achieved miraculous achievements repeatedly, we also considered the impact of immunization on the spread of diseases. With the help of stochastic analysis tools and auxiliary systems, we studied the properties of the global positive solution of the proposed system. Furthermore, we provided our main theoretical results including the extinction, persistence and the existence of a unique stationary distribution of the proposed model. The results show that: (i) When R j < 1, j = 1, 2, both types of diseases will be extinct with probability one. Since the intensities of environmental stochasticities σ i , i = 2, 3, are negatively correlated with the conditions for extinction R j , j = 1, 2, respectively, which indicates that environmental stochasticity is not conducive to the survival of the diseases. (ii) When R 1 > 1 and R 2 < 1, I 1 (t) will be persistent in the mean, while I 2 (t) will be extinct; when R 1 < 1 and R 2 > 1, I 2 (t) will be persistent in the mean, while I 1 (t) will be extinct; when R j > 1, j = 1, 2, both I 1 (t) and I 2 (t) will be persistent in the mean. Similar to the item (ii), this result suggests that small stochasticity is beneficial to the survival of the diseases. (iii) When R 1 > 1 and the parameters meet some other constraints (see Theorem 5 for detail), the stochastic system has a unique ergodic stationary distribution. Ergodicity means that the statistical properties of the stochastic system will not change over time, which allows us to estimate the contour of the stationary distribution by simulating a trajectory of the solution. In this scenario, the results also indicate that the small stochasticity is necessary for the existence of the stationary distribution, i.e., small stochasticity contributes to the survival of the diseases.
This work is just our preliminary exploration of how stochasticity affects the dynamics of disease transmission. In order to have a more comprehensive understanding of the interaction between environmental stochasticity and the diseases, the following explorations are needed: (i) The type of stochasticity considered in this article is white noise, and in complex actual environments, there are other types of stochasticity such as telegraph noise [5,17]. It has practical significance to investigate how these different types of noise synergistically affect the spread of diseases. (ii) This article only considers the situation of one type of susceptibles. Next, we can explore more complex scenarios such as by including more than three types of diseases [32]. (iii) This article assumes that stochasticity is positively correlated with the density of each component. Next, we can also consider the situation where stochasticity mainly perturbs the infection rate, which will induce a different stochastic system with degenerate diffusion terms, and its theoretical analysis is also more challenging.