Abstract
Abstract In this paper, we extend the deterministic single-group MSIRS epidemic model to a multi-group model, and we also extend the deterministic multi-group framework to a stochastic one and formulate it as a stochastic differential equation. In the deterministic multi-group model, the basic reproduction number R 0 is a threshold that completely determines the persistence or extinction of the disease. By using Lyapunov function techniques, we show that if R 0 > 1 , then the disease will prevail, the infective condition persists and the endemic state is asymptotically stable in a feasible region. If R 0 ⩽ 1 , then the infective condition disappears and the disease dies out. For the stochastic version, we perform a detailed analysis on the asymptotic behavior of the stochastic model, which also depends on the value of R 0 , when R 0 > 1 , we determine the asymptotic stability of the endemic equilibrium by measuring the difference between the solution and the endemic equilibrium of the deterministic model in time-averaged data. Numerical methods are used to illustrate the dynamic behavior of the model and to solve the systems.
Highlights
Many models of the outbreak and spread of disease have been analyzed mathematically and applied to specific diseases, and these models have provided some useful and valid reference data for the characteristics of disease transmission
Ji et al considered a multi-group SIR model with stochastic perturbation and deduced the globally asymptotic stability of the disease-free equilibrium when R ≤, which means the disease will die out; the determined that when R >, the disease will prevail, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model using time-averaged data [ ]
We considered an SEIR epidemic model with constant immigration and random fluctuation around the endemic equilibrium, and we performed a detailed analysis on the asymptotic behavior of the stochastic model [ ]; we investigated a two-group epidemic model with distributed delays and random perturbation [ ]
Summary
Many models of the outbreak and spread of disease have been analyzed mathematically and applied to specific diseases, and these models have provided some useful and valid reference data for the characteristics of disease transmission. Kuniya investigated the global stability of a multi-group SVIR epidemic model and considered the heterogeneity of the population and the effect of immunity induced by vaccination [ ]. Beretta et al proved the stability of the epidemic model using stochastic time delays influenced by the probability under certain conditions [ ] Such stochastic perturbations were first proposed in [ , ] and later were successfully used in many other papers for many different systems (see, e.g., [ – ]). Ji et al considered a multi-group SIR model with stochastic perturbation and deduced the globally asymptotic stability of the disease-free equilibrium when R ≤ , which means the disease will die out; the determined that when R > , the disease will prevail, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model using time-averaged data [ ].
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