Critical traveling wave solutions for a vaccination model with general incidence

Abstract

This paper is concerned with the existence of traveling wave solutions for a vaccination model with general incidence. The existence or nonexistence of traveling wave solutions for the model with specific incidence were proved recently when the wave speed is greater or smaller than a critical speed respectively. However, the existence of critical traveling wave solutions (with critical wave speed) was still open. In this paper, applying the Schauder's fixed point theorem via a pair of upper-and lower-solutions of the system, we show that the general vaccination model admits positive critical traveling wave solutions which connect the disease-free and endemic equilibria. Our result not only gives an affirmative answer to the open problem given in the previous specific work, but also to the model with general incidence. Furthermore, we extend our result to some nonlocal version of the considered model.

Propagation thresholds in a diffusive epidemic model with latency and vaccination

“…[ 13 ]. We also refer to some recent works [ 20 , 33 , 38 , 39 , 41 ] for the traveling wave solutions in epidemic models with the vaccinated class.…”