Entire Solution With Merging Fronts To A Diffusive Epidemic Model

With the development of society, the infectious disease has been the mainly reason which effect the people’s health. Therefore, many researchers established some mathematical models to illustrate the spread of infectious diseases and the prediction of the epidemic trend. We can also understand some global state of the epidemic course of the infectious disease. Moreover, epidemic spatial spread is an important topic in the research of epidemic model. In this paper, we studied the existence of three types of entire solution with merging fronts which means the connection of different traveling wave solutions to a diffusive epidemic model: we call by an entire solution a classical solution which is defined for all (x,t)∈R2.First of all, we illustrate the existence of traveling wave fronts in the monos-table and bistable cases respectively, then we show the exponential asymptotic behavior of traveling wave solution by eigenvalue method. In addition, we prove the existence of three types of entire solutions with merging fronts by construct-ing a pair of upper-lower solutions, which is motivated by the special properties of two functions, and by using the comparison principle. The system ignores the diffusion of susceptible population, namely, d2=0, because the diffusion rate of infectious bacteria is really higher, much more than the propagation speed of the susceptible. Finally, we claim the different course of construction of the entire solution with merging fronts in the susceptible population when d2>0.