Traveling waves in a Kermack–Mckendrick epidemic model with diffusion and latent period

• We present a diffusive Kermack–Mckendrick epidemic model to describe the dynamics of diseases.
• We consider the well-posedness of solutions of the model.
• We establish the existence and non-existence of traveling wave solutions of the model which is determined completely by the threshold value R0R0.
• We show the constant c∗c∗ is the minimum speed for the existence of traveling wave solutions of the model.

This paper is devoted to the study of a Kermack–Mckendrick epidemic model with diffusion and latent period. We first consider the well-posedness of solutions of the model. Furthermore, using the Schauder fixed point theorem and Laplace transform, we show that if the threshold value R0>1R0>1, then there exists c∗>0c∗>0 such that for every c>c∗c>c∗, the model admits a traveling wave solution, and if R0<1R0<1 and c≥0c≥0; or R0>1R0>1 and c∈(0,c∗)c∈(0,c∗), then the model admits no traveling wave solutions. Hence, the existence and non-existence of traveling wave solutions is determined completely by R0R0, and the constant c∗c∗ is the minimum speed for the existence of traveling wave solutions of the model.