Traveling Wave Solutions for a Class of Discrete Diffusive SIR Epidemic Model

Abstract

This paper is concerned with the conditions of existence and nonexistence of traveling wave solutions (TWS) for a class of discrete diffusive epidemic model. We find that the existence of TWS is determined by the so-called basic reproduction number and the critical wave speed: When the basic reproduction number $$\\mathfrak {R}_0>1$$ R 0 > 1 , there exists a critical wave speed $$c^*>0$$ c ∗ > 0 , such that for each $$c \\ge c^*$$ c ≥ c ∗ the system admits a nontrivial TWS and for $$c<c^*$$ c < c ∗ there exists no nontrivial TWS for the system. In addition, the boundary asymptotic behavior of TWS is obtained by constructing a suitable Lyapunov functional and employing Lebesgue dominated convergence theorem. Finally, we apply our results to two discrete diffusive epidemic models to verify the existence and nonexistence of TWS.