Theoretical Analysis Of An SIS Reaction Diffusion System With Standard Infection Mechanism

In this paper,we consider a susceptible-infected-susceptible(SIS)epidemic reaction-diffusion model under heterogeneous environment.We first prove the global existence and uniform bounds of solutions.Then,in accordance with the basic reproduction number,we establish the stability of the disease free equilibrium and the uniform persistence of the endemic equilibrium.By constructing appropriate Lyapunov functions,we prove that the global attractivity of both the disease free equilibrium and the endemic equilibrium under different environment conditions.Lastly,we investigate the asymptotic profile of the endemic equilibrium as the migration rate of the susceptible or infected individuals is small.Compared to the theoretical analysis results of the total population invariance model proposed by Allen et al,we find that control of the disease becomes more difficult when considering varying total population and natural mortality.More discussion on the implication of our results will be given in the last section.