Epidemic Waves in a Discrete Diffusive Endemic Model with Treatment and External Supplies

In this paper, we study the traveling wave solutions of a discrete diffusive epidemic model with constant treatment and external supplies. In terms of the basic reproduction number R0 and the critical wave speed c∗ > 0, we establish the existence of traveling wave solutions for R0 > 1 and c ≥ c, which accounts for phase transitions between the disease-free equilibrium and the endemic steady state. In addition, we show the nonexistence of epidemic waves for R0 > 1 and c ≥ c (including negative values). Finally, numerical simulations and discussions are also provided to illustrate the theoretical results. We also investigate the effects of external supply, the proportion of treatment and the reduction factor in infectiousness due to the antiviral treatment on the basic reproduction number and critical wave speed. Our results generalize some known ones and indicate that the effective treatment can reduce the spread of the disease while the external supplies could encourage the spread of disease. The approach developed in this work might provide some insights into the dynamics (especially for the boundedness and asymptotic behavior of solutions) for the discrete diffusive system with nonmonotone structure.