Abstract
In this paper, two stochastic SIRS epidemic models with standard incidence were proposed and investigated. For the non-autonomous periodic model, the sufficient criteria for extinction of the disease are obtained firstly. Then we show that the stochastic system has at least one nontrivial positive T-periodic solution under some conditions. For the model that are both disturbed by the white noise and telephone noise, we construct a suitable Lyapunov functions to verify the existence of a unique ergodic stationary distribution. Meanwhile, the sufficient condition for the extinction of the disease is also established. Finally, examples are introduced to illustrate the theoretical analysis.
Highlights
In the natural world, various systems are inevitably affected by the random factors [1,2,3]
For any k ∈ S, A(k), d(k), β(k), δ(k), γ (k), α(k) and σi(k) (i = 1, 2, 3) are positive constants. Another goal of this paper is to prove the existence of a unique ergodic stationary distribution of the positive solution to the system (3)
For the non-autonomous stochastic system (2), we investigate the extinction criteria of the disease, firstly
Summary
Various systems are inevitably affected by the random factors [1,2,3]. (2019) 2019:22 has a unique global positive solution for any initial value (S(0), I(0), R(0)) ∈ R3+ and has an ergodic stationary distribution under some conditions. In 2015, Lin et al [33] proposed a stochastic SIR epidemic model with seasonal variation and analyzed the existence of a nontrivial positive periodic solution.
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