具有分布时滞离散SIR传染病模型的稳定性

传染病动力学是对传染病进行理论性定量研究的一种重要方法。用微分方程建立连续型传染病模型的研究较多,但是研究离散模型的较少。相对连续模型,离散模型能展示更丰富的动力学性态。许多无法求解或理论分析的连续模型往往需要化为离散模型进行数值模拟。因此,建立和分析离散传染病模型就更加实用。在连续SIR传染病模型的研究基础上,研究具有分布时滞,常数出生率、死亡率的离散SIR传染病模型,讨论模型在无病平衡点的稳定性。主要结论是当且仅当基本再生数小于等于时,系统存在唯一无病平衡点,且无病平衡点是全局渐近稳定。

The Stability of Discrete SIR Epidemic Model with Distributed Time Delay

ABSTRACT: Dynamics of Infectious Diseases is a significant method applied in the theoretical and qualitative research on infectious diseases. There have been many studies of continuous epidemic model through differential equations, but little research on discrete models. Compared with continuous model, discrete model is capable of expressing more abundant dynamic behaviors, where many continuous ones which cannot be solved or of theoretical analysis have to conduct numerical simulation through being converted into discrete model. Therefore, the construction and analysis of discrete epidemic model is more applicable. On the base of the studies of continuous SIR epidemic model, this paper studies the discrete SIR epidemic models with distributed time delay and constant birth rate and mortality rate, and discusses the stability of the model at the disease-free equilibrium. The main conclusion drawn from this study is that: if and only if the basic productive number , the only disease-free equilibrium exists in the system and it is of global asymptotic stability.