Qualitative analysis of stochastically perturbed SIRS epidemic model with two viruses

A stochastically perturbed SIRS epidemic model with two viruses is formulated to investigate the effect of intensities of white noise on each population. We prove that the stochastic model has a non-negative solution that belongs to a positively invariant set. By applying a novel combination of suitable Lyapunov functions, we obtain that the asymptotic behaviour of the stochastic model holds a disease-free equilibrium if R-0 <= 1 and investigate pth-moment exponential stability. Then we derive the sufficient conditions for the solution of the stochastic model that fluctuates around the endemic equilibrium if R-0 > 1. Furthermore, the suitable conditions for the extinction and persistence of the diseases are established due to intensities of white noise in large. Finally, the theoretical results are illustrated by numerical investigations. (C) 2018 Elsevier Ltd. All rights reserved.