Stability in a Ross epidemic model with road diffusion

Reaction-diffusion equations have been used to describe the dynamical behavior of epidemic models, where the spreading of infectious disease has the same speed in every direction. A natural question is how to describe the dynamical system when the spreading of infectious disease is directed diffusion. We introduce the road diffusion into a Ross epidemic model which describes the spread of infected Mosquitoes and humans. With the comparison principle the system is proved to have a unique global solution. By the approach of upper and lower solutions, we show that the disease-free equilibrium is asymptotically stable if the basic reproduction number is lower than 1 while the endemic equilibrium asymptotically stable if the basic reproduction number is greater than 1.