一类随机SIR传染病模型的非标准数值离散化

Nonstandard numerical discretization of a stochastic SIR epidemic model

研究一个具有饱和发生率和疫苗接种率的随机离散SIR传染病模型的稳定性.首先,引入一个具有饱和发生率和疫苗接种率的确定性SIR模型,考虑到随机噪声对疾病传播有很大影响,应用非标准有限差分(NSFD)方法将模型离散化,最终得到一个随机离散的SIR传染病模型.这种离散方法是对系统的右侧进行局部离散,得出离散模型,然后系统左侧用广义前向差分法对一阶导数进行逼近,并且要选取恰当的分母函数.其次,应用Lyapunov函数和矩阵方法给出了系统平衡解稳定的充分条件,并且得到了非线性差分方程概率稳定的充分条件和线性差分方程渐近均方稳定的充分条件.最后,通过数值仿真对理论分析结论进行验证.

This paper studies the stability of a stochastic discrete SIR epidemic model with saturated incidence and vaccination rate.A deterministic SIR model with saturated incidence and vaccination rate is introduced.Considering the great influence of stochastic noise on disease transmission,the model is discretized by nonstandard finite difference(NSFD)method,and finally a stochastic discrete SIR epidemic model is obtained.This discretization method is to locally discretize the right side of the system to obtain the discrete model,and then use the generalized forward difference method to approximate the first derivative on the left side of the system,and select the appropriate denominator function.The sufficient conditions for the stability of the equilibrium solution of the system are given by using Lyapunov function method and matrix method,and the sufficient conditions for the probabilistic stability of nonlinear difference equations and the sufficient conditions for the asymptotic mean square stability of linear difference equations are also proposed.Finally,the conclusion is verified by numerical simulation.