Existence of traveling wave solutions of a deterministic vector-host\n epidemic model with direct transmission

We consider an epidemic model with direct transmission given by a system of\nnonlinear partial differential equations and study the existence of traveling\nwave solutions. When the basic reproductive number of the considered model is\nless than one, we show that there is no nontrivial traveling wave solution. On\nthe other hand, when the basic reproductive number is greater than one, we\nprove that there is a minimum wave speed $c^*$ such that the system has a\ntraveling wave solution with speed $c$ connecting both equilibrium points for\nany $c\\ge c^*$. Moreover, under suitable assumption on the diffusion rates, we\nshow that there is no traveling wave solution with speed less than $c^*$. We\nconclude with numerical simulations to illustrate our findings. The numerical\nexperiments supports the validity of our theoretical results.\n