Abstract
In this paper, we consider a stochastic SIR epidemic model with regime switching. The Markov semigroup theory will be employed to obtain the existence of a unique stable stationary distribution. We prove that, if mathcal{R}^{s}<0, the disease becomes extinct exponentially; whereas if mathcal{R}^{s}>0 and beta(i)>alpha(i)(varepsilon(i)+gamma(i)), iinmathbb{S}, the densities of the distributions of the solution can converge in L^{1} to an invariant density.
Highlights
1 Introduction Infectious disease population dynamics is often influenced by different types of environmental noise, in which white noise is the most typical one
In reference [4], the authors assumed that the environmental white noises mainly influence the natural death rates μ of the populations
We prove that the disease becomes extinct exponentially if Rs < 0; whereas if Rs > 0 and β(i) > α(i)(ε(i) + γ (i)), i ∈ S, system (1.2) has a stable stationary distribution
Summary
Infectious disease population dynamics is often influenced by different types of environmental noise, in which white noise is the most typical one. The effects of white noise on epidemic models have already been considered by many authors By replacing μ dt by μ dt + σ dB(t) in the deterministic SIR model with saturated incidence, the authors obtained a stochastic SIR model, which takes the form of In this model, S(t) and I(t) denote the number of susceptible and infected individuals at time t, respectively. Due to the important effect of color noise on disease transmission, many author considered deterministic epidemic model with Markovian switching [9, 10]. Much literature considered asymptotic behavior of stochastic epidemic model under regime switching, e.g.
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