Dynamics of a kind of stochastic SIRS models with two different nonlinear incidences

Abstract

A new kind of stochastic SIRS models with two different nonlinear incidences are extended. The obtained results can be expressed in two dimensions. In mathematics, the threshold values R s 1 and R s 2 which ensure permanent or extinct disease are presented, respectively. More concretely, when R s i .1(i = 1, 2), the two diseases are persistence in mean. When R s i \1 or R s i .1(i = 1, 2), the two diseases will either be extinct or be permanent, respectively. What's more interesting is the numerical results which show that the two diseases go to extinction at a large time, when R s 1 = 1 and R s 2 = 1. Furthermore, the sufficient conditions for the diseases that are extinct and permanent are given. In addition, the stochastic boundedness of the model is investigated. Epidemiologically, we can conclude that large noise fluctuations can effectively prevent the outbreak of disease, which can supply us beneficial methods to control the spreading of two diseases. Finally, a few numerical simulations are presented to demonstrate and verify the findings obtained.

Stochastic Extension of a Two-Strain SIR Model with Non-Monotone Incidence Functions: Analysis of Stochastic Thresholds and Disease Dynamics

“…In addition, hybrid stochastic SIS and SIR models using a nonlinear saturated incidence function alongside the dual epidemic hypothesis were further developed in studies by Boukanjime et al and Selvan et al [4,22]. Huang et al [12] presented an advanced stochastic SIRS model with two types of nonlinear incidence rates. Finally, Farah et al conducted a study on a stochastic model of two influenza strains with both bilinear and saturated incidence rates [8].…”

A Stochastic SIS Epidemic Model with Ornstein-Uhlenbeck Process and Brown Motion

“…In the past few decades, an increasing number of scholars have studied various forms of stochastic epidemic models and made many functional theoretical analyses and numerical discussions on the corresponding models. For instance, Krause [15] and Chen [16] have studied the nonlinear stochastic SIS epidemic system with white noise; Chang [17], Huang [18] and Gao [19] have proposed the nonlinear stochastic SIS epidemic system with multiple noises; Han et al [20] have studied the stochastic SIS epidemic models with saturated incidence rate; Wang et al [21] have proposed the stochastic SIS model with the mean regression Ornstein-Uhlenbeck process. In addition, one proved the stochastic persistence and the extinction of the disease with different conditions on the intensities of noises and different parameters of the model [24][25][26][27][28][29][30][31], and showed the stationary distribution of the existence of the disease and the expectation and the variance of the system [32][33][34].…”