Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination

Abstract

In this paper, a stochastic SIRS epidemic model with nonlinear incidence and vaccination is formulated to investigate the transmission dynamics of infectious diseases. The model not only incorporates the white noise but also the external environmental noise which is described by semi-Markov process. We first derive the explicit expression for the basic reproduction number of the model. Then the global dynamics of the system is studied in terms of the basic reproduction number and the intensity of white noise, and sufficient conditions for the extinction and persistence of the disease are both provided. Furthermore, we explore the sensitivity analysis of R s 0 with each semi-Markov switching under different distribution functions. The results show that the dynamics of the entire system is not related to its switching law, but has a positive correlation to its mean sojourn time in each subsystem. The basic reproduction number we obtained in the paper can be applied to all piecewisestochastic semi-Markov processes, and the results of the sensitivity analysis can be regarded as a prior work for optimal control.

Threshold dynamics of SAIRS epidemic model with Semi-Markov switching

“…Proof. Let us prove conditions ( 21) and (22). By using equation ( 14), we have that for all ω P Ω, and t ą T d lnpIptq `Aptqq dt ď q βprptqqpS 0 `εq ´pδ ´µq.…”

“…For overcoming this drawback, we use a semi-Markov process for describing environmental random changes. There are only very few semi-Markov switching epidemic models [13,22,3] and no one of these consider the role of the asymptomatic individuals in the disease dynamics. Thus, in this paper we want to fill this gap and to improve understanding of these type of hybrid systems.…”

Threshold dynamics of SAIRS epidemic model with semi‐Markov switching

“…Recently, a semi-Markov switching model has been used to analyze the coexistence and competitiveness of species in ecosystems [19]. In epidemiology, to the best of our knowledge, there are only very few semi-Markov switching models [17,20,21] and no one of these considers the role of the asymptomatic individuals in the disease dynamics. Thus, in this paper, we want to fill this gap and improve our understanding of these types of hybrid systems.…”