Traveling waves in a delayed reaction–diffusion SIR epidemic model with a generalized incidence function

This work examines the existence (resp. nonexistence) of traveling waves in a delayed SIR epidemic model with a generalized incidence function. It is demonstrated that the basic reproduction number, \(R_0\) $R_0$ , determines whether traveling waves occur or not. In particular, we demonstrate that if the basic reproduction number \(R_0\) $R_0$ exceeds the threshold 1, there exists a minimal wave speed \(c^* > 0\) $c^{\ast }>0$ such that: (i) the system admits a nontrivial traveling wave solution with wave speed c for every \(c \ge c^*\) $c\ge c^{\ast }$ ; (ii) for any \(c < c^*\) $c<c^{\ast }$ , there is no nontrivial traveling wave satisfying the system. In conclusion, we showcase a few numerical simulations that demonstrate the traveling wave existence of the system in some particular cases of some well-known nonlinear incidence functions.