Traveling waves in a diffusive influenza epidemic model with vaccination

• We derive a mathematical model of diffusive influenza disease with vaccination.
• We obtain the explicit formula of the basic reproduction number R0R0 for the model.
• We establish the existence of the traveling wave solutions for the model if R0>1R0>1 and c > c*.
• We further study the non-existence of the traveling wave solutions for the model when R0<1R0<1 and c   ≥ 0 or R0>1R0>1 and cϵ(0, c*).

In this paper, a mathematical model of influenza disease with vaccination is formulated to incorporate a spatially homogeneous structure. The explicit formula of the basic reproduction number R0R0 for the model is obtained. It is shown that the existence and non-existence of the traveling wave solutions for the model are determined completely by the threshold value R0R0. Here, by introducing an auxiliary system and applying Schauder fixed point theorem, we show that the auxiliary system admits a nontrivial traveling wave solution. And then, by the limit arguments and Arzelà–Ascoli’s theorem, we establish the existence of the traveling wave solutions for the model if R0>1R0>1 and c > c*. Furthermore, by the two-sided Laplace transform, we show that the model has no nontrivial traveling wave solutions when R0>1R0>1 and c ∈ [0, c*). In addition, we further study the non-existence of the traveling wave solutions for the model when R0<1R0<1 and c ≥ 0.