Random periodic solution for a stochastic SIS epidemic model with constant population size

Dianli Zhao,Sanling Yuan,Haidong Liu

doi:10.1186/s13662-018-1511-4

Abstract

In this paper, a stochastic susceptible-infected-susceptible (SIS) epidemic model with periodic coefficients is formulated. Under the assumption that the total population is fixed by N, an analogue of the threshold {R_{0}^{T}} is identified. If {R_{0}^{T} > 1}, the model is proved to admit at least one random periodic solution which is nontrivial and located in (0,N)times(0,N). Further, the conditions for persistence and extinction of the disease are also established, where a threshold is given in the case that the noise is small. Comparing with the threshold of the autonomous SIS model, it is generalized to its averaged value in one period. The random periodic solution is illuminated by computer simulations.

Highlights

Mathematical epidemiology has made a significant progress in better understanding of the disease transmissions
Epidemic dynamics is always affected by the environmental noise, investigating the influence of the noises on dynamics of the epidemics is of interest to the researchers
We present a stochastic SIS epidemic model with periodic coefficients as follows:

Summary

Introduction

Mathematical epidemiology has made a significant progress in better understanding of the disease transmissions. For model (1), Lin and Jiang [2] studied the existence of a positive and global solution, they showed sufficient conditions for the survival and extinction of the disease. The authors proved that this model has a unique global positive solution and derived the existence of stationary distribution. We present a stochastic SIS epidemic model with periodic coefficients as follows:

Results

Conclusion