On a diffusive epidemic model with the tendency to move away from the infectious diseases

Abstract

This paper is purported to investigate an epidemic system with the tendency of the susceptible to move away from the infectious diseases in a bounded domain with no flux boundary condition. The local and global stabilities of positive equilibrium are investigated to this system without cross-diffusion. The sufficient conditions to nonexistence and existence of non-constant positive solution are considered for this epidemic system. The results show that there exists spatiotemporal pattern formation when the tendency of susceptible individuals is strong to move away from the infectious diseases with 1<R0<1+min⁡{dSd,βσ1Aσ2}, while there does not exist spatiotemporal pattern formation when the self pressure is big enough with a weak tendency of the susceptible individuals to keep away from the infectious individual. Moreover, there exists positive solution bifurcation from the semi-trivial solution when the basic reproduction number is equal to one, which implies that endemic disease exists locally.