Abstract
In this paper, a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis is proposed and analysed. We explain the effects of stochastic disturbance on disease transmission. To this end, firstly, we investigated the dynamic properties of the system neglecting stochastic disturbance and obtained the threshold and the conditions for the extinction and the permanence of two kinds of epidemic diseases by considering the stability of the equilibria of the deterministic system. Secondly, we paid prime attention on the threshold dynamics of the stochastic system and established the conditions for the extinction and the permanence of two kinds of epidemic diseases. We found that there exists a significant difference between the threshold of the deterministic system and that of the stochastic system. Moreover, it has been established that the persistent of infectious disease analysed by use of deterministic system becomes extinct under the same conditions due to the stochastic disturbance. This implies that a stochastic disturbance has significant impact on the spread of infectious diseases and the larger stochastic disturbance leads to control the epidemic diseases. In order to illustrate the dynamic difference between the deterministic system and the stochastic system, there have been given a series of numerical simulations by using different noise disturbance coefficients.
Highlights
Infectious disease is generally considered as the enemy of human health; in history, the epidemic of infectious diseases such as smallpox, cholera, AIDS and so on have brought great disaster to the national economy of a country and people’s livelihood [ ]
From Theorems . and . , it can be seen that there is a significant difference between the thresholds of the stochastic system and the deterministic system, from which it can be concluded that the conditions for two epidemic diseases to go to extinction in the stochastic system are weaker than those of the deterministic system
To illustrate the dynamic difference between the deterministic system and the stochastic system, we carry out some numerical simulations of these cases with respect to different noise disturbance intensity using the Euler Maruyama (EM) method [, ]
Summary
Infectious disease is generally considered as the enemy of human health; in history, the epidemic of infectious diseases such as smallpox, cholera, AIDS and so on have brought great disaster to the national economy of a country and people’s livelihood [ ]. The susceptible person get infected and becomes an infected person making contact with an infected person, and the infected person can be recovered taking treatments, the individuals who reach this class have permanent immunity for the relevant disease This type of model is called the SIR (susceptible-infected-removed) model, which can be mathematically expressed as Some diseases do not conform to the SIR system, such as influenza, infected individuals do not get permanent immunity for the disease they take proper treatments, in this case, there exists a high possibility of recovered individuals to be re-infected This type of model is called an SIS model, the mathematical system can be expressed in the following form The main aims of this paper are (a) to establish a set of most suitable conditions such that diseases to be died out or to be persistent, and (b) to obtain the thresholds (based on the basic reproductive number) of the above two SIS epidemic models
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