Dynamics Analysis for a Stochastic SIRI Model Incorporating Media Coverage

We investigate a stochastic SIRI model incorporating media coverage. Firstly, existence and uniqueness of global positive solution of the model are established. By using the Hasminskii theory, we study stationary distribution of the model. Then, we prove extinction of the disease. Moreover, the asymptotic properties of solutions are studied by using Lyapunov method. Finally, numerical simulations are presented. In addition, it shows that media coverage can strain the spread of infectious diseases.


Introduction
Although human society is constantly developing and progressing, various infectious diseases still plague humans (such as COVID-19 and mixed in uenza A and B). Governments around the world have taken a series of measures to prevent the spread of infectious disease and protect human life and health. Furthermore, with the acceleration of globalization, the convenient transportation and the frequent movement of population have accelerated the spread of infectious diseases, so as to study the infectious law of disease, predict its development trend, and seek for prevention and treatment strategies, scholars have established various epidemic models and studied their dynamic behaviors (see [1][2][3][4][5][6][7][8]).
As can be seen from the above discussion, in order to study the in uence of media coverage on the spread of infectious disease, previous models only considered the in uence of the number of infected individuals on media coverage. When infected individuals appear in an area, the public are conscious of the potential danger of infected individuals, they will voluntarily reduce their contact with the outside world for fear of infection. e more the infected individuals are, the less contact they have with the outside world.
at is, media coverage can minimize the contact between susceptible and infected individuals. In addition, the number of recovered individuals also partly re ects the severity of the epidemic. Hence, the number of infected and recovered individuals is larger, and the impact of media coverage is larger on the exposure rate. us, the reduction of contact rate is related to the number of infected and recovered individuals. Additionally, due to the in uence of psychological and social factors, susceptible individuals reduce their contact with infected individuals and strengthen protective measures, so that the effective contact rate between infected and susceptible individuals tends to saturation state. Based on this, a deterministic SIRI model with saturated incidence rate is established Here, S(t), I(t), R(t) are the susceptible, infected, and recovered individuals, respectively; Λ is the birth rate or migration rate of susceptible individuals; c is the recovery rate of infected individuals; μ is the natural mortality; δ denotes the recurrence rate of the recovered; β 1 is the maximum effective contact rate; and β 1 − β 2 (I + αR)/(m 1 + I + αR) denotes the contact rate under media coverage. m 1 > 0, m 2 > 0, 0 < α ≤ 1. Since media coverage will not completely prevent the spread of diseases, then β 1 > β 2 . For model (1), the basic reproduction number is easily obtained as . (2) In reality, infectious diseases are always influenced by environmental noise, which makes the related parameters (such as contact rate, mortality rate, and recovery rate) show random fluctuations. us, it is more realistic to research the dynamic behaviors of stochastic epidemic model (see [17][18][19][20][21][22][23]). At present, a lot of academics have studied stochastic epidemic models with media coverage and obtained some results (see [14,16,24]). In [24], a stochastic SIS model with media coverage on two patches was established, and almost sure exponential stability of disease-free equilibrium was obtained. In [14], asymptotic behaviors around the equilibria of a stochastic SIRI model were considered. In [16], ergodic stationary distribution of a stochastic SIRS model was investigated.
Considering the influence of environmental random factors, we establish the stochastic SIRI epidemic model where B i (t) is the Brownian motion on (Ω, F, P), which is a complete probability space; σ 2 i is the intensity of B i (t), i � 1, 2, 3.
In the following, we shall study existence of the stationary distribution and sufficient conditions for extinction of the disease. e dynamic properties of solutions near equilibria are to be considered. e results reveal the influence of noise intensity on disease transmission. In addition, the numerical simulation shows that increasing the media coverage can reduce the number of infected individuals, so media coverage can reduce the spread of infectious diseases. Besides, the less impact the number of recovered individuals has on media coverage, the lower the transmission rate is, which in turn leads to fewer infected individuals. us, it provides a scientific theoretical basis for the prevention and control of infectious diseases.

Stationary Distribution and Ergodicity
Let X(t) be a Markov process in R d and satisfy where σ k (X) � (σ 1 k (X), . . . , σ n k (X)) T . en, the diffusion matrix is Lemma 1 (see [25]). For a bounded domain U ⊂ R d with regular boundary zU, if is an integrable function with respect to the measure μ, then for any x ∈ R d , Denote Proof. For every (S(0), I(0), R(0)) ∈ (0, ∞) 3 , it follows from eorem 1 that there exists a unique global positive solution (S(t), I(t), R(t)). Let ε > 0 be sufficiently small. Denote and For every (S, us, where Here, Obviously, liminf l→∞,(S,I,R)∈(0,∞) 3 \U l V(S, I, R) � +∞, (18) where U l � (1/l, l) 3 . In addition, V(S, I, R) has a minimum point e It o formula shows that Hence, Denote Mathematical Problems in Engineering 5 Let ε > 0 be sufficiently small for Next, R 3 + \U ε is divided into six areas: We shall prove LV 2 (S, Case 2. Let (S, I, R) ∈ U 2 . en,

� −
en, e proof is complete. □ Remark 2. By eorem 5, if the noise intensity is small and R 0 > 1, the solution of (3) fluctuates around the endemic equilibrium. at is, if the intensity of random disturbance is small enough, the disease will persist.  . en, R s ≈ 0.6566 < 1. By eorem 3, the disease of (3) becomes extinct (see Figure 2). We

Numerical Simulations
It is known from eorem 4 that the solution of (3) fluctuates around the curve of (1). Figure 3 is consistent with this result.
is shows that the less impact the number of recovered individuals has on media coverage, the lower the transmission rate is, which in turn leads to fewer infected individuals.

Conclusions
We analyze a stochastic SIRI model with media coverage. To begin with, existence and uniqueness of global positive solution of the model are established. Using the Hasminskii theory, we prove existence of a stationary distribution. en, conditions of disease extinction are proved. By constructing the suitable Lyapunov functions, the dynamic properties of solutions near equilibria are studied. If R 0 < 1, when the intensity of random disturbance is sufficiently small, infectious diseases will be extinct; if R 0 > 1 and the random disturbance is small, infectious diseases will persist. In addition, the less impact the number of recovered individual has on media coverage, the lower the transmission rate is, which in turn leads to fewer infected individuals.

Data Availability
No data were used to support this study.

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

Authors' Contributions
All the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.