An SIS Reaction-diffusion-advection Model With Mass Action Infection Mechanism

The SIS ?susceptible-infected-susceptible? models provide essential frames in study-ing the dynamics of disease transmission in the field of theoretical epidemiology. To cap-ture the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease, we study an SIS epidemic reaction-diffusion-advection model with no-flux boundary conditions where the birth and death rate of susceptible individuals, and the death rate of infected individuals, are considered, thus the total population size may be not constant.Firstly, in this research, we will study the modified SIS epidemic model in which the birth rate and death rate of susceptible individuals are equal, and death rate of infected individuals is equal to zero. The basic reproduction number R₀ is defined which plays an essential role in determining whether the disease will extinct or persist. It is showed that the disease-free equilibrium is unique and globally asymptotic stable if R₀ is less than one, while the disease-free equilibrium is unstable when R₀ exceeds one. If the diffusion rate of the susceptible individuals is equal to the diffusion rate of the infected individuals,then the disease-free equilibrium is globally attractive if R₀ ? 1, whereas the endemic equilibrium is globally attractive if R₀> 1.Secondly, the model is particularly considered when the birth rate of susceptible individuals is greater than the death rate of susceptible individuals and the death rate of infected individuals is greater than zero. With the help of the bifurcation analysis, we prove the existence of the positive steady state. Furthermore, we obtain some a priori estimates of the the positive steady state by constructing upper and lower solutions in a special case. It is showed that when the initial data satisfy some conditions, the number of the susceptible individuals finally stays bounded while the infection level grows without bound.