Traveling Waves For Nonlocal Dispersal SIR And SIRI Epidemic Models

In this thesis, we study mainly the traveling waves for a nonlocal dispersal SIR epidemic model with nonlocal delay and a nonlocal dispersal SIRI epidemic model with constant external suply.First, we introduce the research background and process concerning the SIR epidemic model as well as some main results of this essay.Second, a nonlocal dispersal SIR model with nonlocal delay is established. In order to prove the existence and asymptotic behavior of non-trivial traveling wave solutions, we construct an invariant cone in a bounded domain and apply the fixed point theorem as well as limiting arguments. Furthermore, we show that there does not exist traveling wave solution if the speed is less than the critical velocity.Finally, we consider a nonlocal dispersal SIRI epidemic model with constant ex-ternal suply. This model can be used to describe some diseases with relapse. Firstly, the well posededness of the model is obtained via the Banach contracting theorem. Then the sufficient conditions for the existence of traveling waves are derived by the partial quasi-monotone condition, the super-lower solutions method and the fixed-point theorem. Finally the non-existence of traveling wave solutions is proved by the comparison principle.