具有饱和感染率的随机SIR传染病模型的性质分析

Abstract

建立了一类具有饱和感染率的随机SIR模型，假设易感者、感染者及移出者三群体的自然死亡率和疾病感染率分别受到相互独立的高斯白噪声干扰。首先证明了在一定的条件下，感染者与移出者种群将依指数趋于灭绝。再就是相应确定性系统的地方病平衡点存在时，得到了该随机系统围绕该点具有稳定的分布且该分布是遍历的充分条件。A stochastically mathematical model of a stochastic SIR epidemic model with saturated incidence rates is proposed and analyzed, setting that all the death rate and incident rate are similarly per-turbed by an independent Gaussian white noise. First the paper shows that the infective population and recovered individuals will tend to zero exponentially almost surely under some additional condition. In addition, a sufficient condition for the stationary distribution around the endemic infection equilibrium state of the corresponding deterministic model is derived and the solution is ergodic.