Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis

Abstract

Please cite this article in press as: X. Meng et al., Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis, J. Math. Anal. Appl. (2015), http://dx. AbstractIn this paper, we propose new mathematical models with nonlinear incidence rate and double epidemic hypothesis. Then we dedicate to develop a method to obtain the threshold of the stochastic SIS epidemic model. To this end, first, we investigate the stability of the equilibria of the deterministic system and obtain the conditions for the extinction and the permanence of two epidemic diseases. Second, we explore and obtain the threshold of a stochastic SIS system for the extinction and the permanence in mean of two epidemic diseases. The results show that a large stochastic disturbance can cause infectious diseases to go to extinction, in other words, the persistent infectious disease of a deterministic system can become extinct due to the white noise stochastic disturbance. This implies that the stochastic disturbance is conducive to epidemic diseases control. To illustrate the performance of the theoretical results, we present a series of numerical simulations of these cases with respect to different noise disturbance coefficients.

Nonlinear stochastic analysis for a stochastic SIS epidemic model

“…Compare with previous study results [4,6,21], the additional condition of σ 2 βµ Λ is unnecessary. Therefore, we improve the main results of previous studies.…”

“…For another, a large number of literatures have used the Lyapunov functions to study the dynamics of infectious diseases and have given many good results [4,6,11,21]. However, this paper applies the Feller's test and canonical probability methods to study the dynamics of system (1.2) and gives some new results.…”

“…The parameters β, α, λ are all nonnegative constants and Λ, µ are positive constants. Recently, authors of [4,6,21] have done much research and have obtained plenty of significant conclusions for system (1.1). They set up the threshold condition R * , which determines disease extinction or permanence under stochastic perturbation, where…”

“…Recently, nonlinear analysis has been widely applied in mathematical modelling in biology [2,3,5,13,21,22,31]. Inspired by previous work for stochastic dynamics in [9,10,12,14,16,19,23,24,[26][27][28]32], this paper discusses a nonlinear SIS epidemic model with stochastic perturbation, which is described by the following stochastic system        dS(t) = Λ − µS(t) − βS(t)I(t) 1 + αI(t) + λI(t) dt − σS(t)I(t) 1 + αI(t) dB(t), dI(t) = βS(t)I(t) 1 + αI(t) − (µ + λ)I(t) dt + σS(t)I(t) 1 + αI(t) dB(t), (1.1) where S(t) and I(t), respectively, represent the number of susceptible and infectious individuals at time t, B(t)(B(0) = 0) is the standard Wiener process with intensity σ 2 > 0, Λ stands for the birth rate of susceptible individuals, µ denotes the natural mortality, λ is the recovery rate of disease.…”

Dynamics of a Nonautonomous Stochastic SIS Epidemic Model with Double Epidemic Hypothesis

“…Therefore, some deterministic SIS epidemic models have been studied by many authors [3][4][5][6][7][8][9][10]. Recently, the authors of [11][12][13] investigated the epidemic model with double epidemic hypothesis which has two epidemic diseases caused by two different viruses. For example, the deterministic SIS epidemic model with nonlinear saturated incidence rate and double epidemic hypothesis can be expressed as follows [11]:…”