Threshold dynamics and ergodicity of an SIRS epidemic model with Markovian switching

Abstract

This paper studies the spread dynamics of a stochastic SIRS epidemic model with nonlinear incidence and varying population size, which is formulated as a piecewise deterministic Markov process. A threshold dynamic determined by the basic reproduction number R 0 is established: the disease can be eradicated almost surely if R 0 < 1, while the disease persists almost surely if R 0 > 1. The existing method for analyzing ergodic behavior of population systems has been generalized. The modified method weakens the required conditions and has no limitations for both the number of environmental regimes and the dimension of the considered system. When R 0 > 1, the existence of a stationary probability measure is obtained. Furthermore, with the modified method, the global attractivity of the Ω-limit set of the system and the convergence in total variation to the stationary measure are both demonstrated under a mild extra condition.

A stochastic SIRS epidemic model with non-monotone incidence rate under regime-switching

“…• Compared with results in Liu et al [42] , Han et al [43] for a stochastic epidemic model with the fluctuation by both white and telegraph noises in the transmission rate of disease, the advantages in this paper lie in: (1) the existence of endemic stationary distribution is proved in the case of persistence; (2) the effect of the white and telegraph noises on the epidemiological consequences is studied based on realistic parameter values. • The effect of the white noise was not studied in Li et al [53] . The present paper shows that white noise play a very important role in determining persistence or extinction of diseases in the transmission of infectious disease.…”

“…The telegraph noise (namely, color noise) can be illustrated as a switching between two or more regimes of environment, which differ by factors such as nutrition or as rain falls [41] . Frequently, the switching among different environments is memoryless and the waiting time for the next switch is exponentially distributed [42,[44][45][46][47][48][49][50][51][52][53] . Thus we can model the regime switching by a continuous-time Markov chain with values in a finite state space, which drives the changes of the main parameters of epidemic models with state switchings of the Markov chain.…”

“…Thus we can model the regime switching by a continuous-time Markov chain with values in a finite state space, which drives the changes of the main parameters of epidemic models with state switchings of the Markov chain. In particular, Li et al [53] studied the spread dynamics of a stochastic SIRS epidemic model with nonlinear incidence and Markovian switching. They gave the threshold value of the disease persistence and extinction and then gave the ergodic analyzing of their model.…”

Random switching near bifurcations

“…The class of PDMPs considered in this article is also known under the names of hybrid systems [50] and random evolutions [28], [21,Chapter 12]. Randomly switched vector fields have applications to areas such as ecology [12], gene regulation [15], molecular motors [23], epidemiology [37], queueing theory [1], and climate science [38], to name just a few.…”

The Principle of Competitive Exclusion about a Stochastic Lotka-Volterra Model with Two Predators Competing for One Prey

“…In fact, there are some random perturbations, for example, the telephone noise. Recently, stochastic models with the telephone noise have been studied by many authors; see [19][20][21][22]. In the future study, we will consider a stochastic LotkaVolterra model with two predators competing for one prey perturbed by the telephone noise.…”