Existence Of Traveling Wave Solutions For A Class Of Infectious Disease With Saturated Contact Rate

In this paper,we study the existence of traveling wave solutions for a class of infectious disease with saturated contact rate for the following infectious model(?)In the first part of this paper,we discuss the existence of traveling wave solution when the infected person is diffusive,that is d1=0,d2?0,and use the improved shooting method.By constructing Lyapunov function,we prove that the global positive solution of the system converges to the endemic equilibrium point when t?+?,that is the traveling wave solution exists,and the minimum wave velocity is obtained.Then we use numerical simulation to verify the results.In the second part,we discuss the existence of traveling wave solutions when both the susceptible and the infected are diffusive,that is d1?0,d2?0.The wave discussed here is not necessarily monotone,and has a velocity c*>0.For c>c*,the traveling wave moves at the velocity c.This proof uses the shooting method and constructs the Lyapunov function to prove the existence of traveling wave solutions.This traveling wave solution is equivalent to the heteroclinic orbit in the four-dimensional phase space.