Analysis of an epidemiological model driven by multiple noises: Ergodicity and convergence rate

Abstract

Environmental noise is unavoidable in the spread of infectious diseases. In this paper, we present a mathematical system to investigate the impact of environmental noise on disease transmission dynamics. The model incorporates Brownian noise, Markovian switching noise and nonlinear incidence. The results show that the long-term dynamics of the stochastic system is determined by a threshold parameter which is closely related to the stochastic noise. If the threshold is greater than zero all solutions converge exponentially to a unique invariant probability distribution, while if the threshold is less than zero the infectious diseases are extinct at an exponential rate and the level of susceptible individuals converges weakly to a unique invariant probability distribution. The threshold parameter also provides essential guidelines for accessing control of the diseases and implies that the environmental noise may be beneficial to contain the infectious diseases. The results extend and generalize previous work in understanding the dynamics of stochastic epidemic models with Markov switching. The theoretical approach can also be applied to the stochastic systems driven by white noise.