Stochastic persistence and stationary distribution in an SIS epidemic model with media coverage

Abstract

h i g h l i g h t s• A stochastic epidemic model with media coverage is developed.• The global dynamics of the deterministic model is shown.• The stochastic dynamics of the SDE model is given.• The existence of a unique stationary distribution of the SDE model is displayed. a b s t r a c tThis paper aims to study an SIS epidemic model with media coverage from a general deterministic model to a stochastic differential equation with environment fluctuation. Mathematically, we use the Markov semigroup theory to prove that the basic reproduction number R s 0 can be used to control the dynamics of stochastic system. Epidemiologically, we show that environment fluctuation can inhibit the occurrence of the disease, namely, in the case of disease persistence for the deterministic model, the disease still dies out with probability one for the stochastic model. So to a great extent the stochastic perturbation under media coverage affects the outbreak of the disease.

The Positivity of Numerical Method for Susceptible-Infected-Recovered-Susceptible Epidemic Model

“…Cai et al [9] gave the conditions of persistence and extinction for the dynamic properties around the positive equilibrium of the SIRS model. Guo et al [10] used the Markov semigroup theory to prove the global behavior of a stochastic SIRS model with media convergence. An SIRS model with a nonlinear incidence rate and vaccination can be described by the following set of differential equations (see, e.g., [11]):…”

Dynamic behavior of a stochastic SIRS epidemic model with media coverage

“…Obviously, the more preventive knowledge that the people have, the more preparations people will make to prevent the spread of the disease. Thus, media coverage is of great importance, and people should pay more attention to this field …”

“…Thus, media coverage is of great importance, and people should pay more attention to this field. 7,8 To describe the dynamic of the infectious disease, this research firstly provides a model from Xiao and Ruan 5 :…”

The effect of media coverage on threshold dynamics for a stochastic SIS epidemic model

“…Thomas and Kevin [8] pointed out that global climate change may cause considerable uncertainty about the rates of change. Therefore, the environmental fluctuation (uncertainty change in temperature or humidity) may translate to affect the transmission rate of infectious diseases [15][16][17][18][19][20][21][22][23]. If we assume that the uncertainty change of transmission rate of infectious disease β 1 is subjected to a Gaussian white noise, with…”