Critical waves in a nonlocal dispersion delayed susceptible-infected-confined-quarantined-recovered outbreak model with general incidence function

Abstract

We study a delayed nonlocal epidemic model that includes the effects of confinement and a generalized incidence function. The model takes into account the spatial movements of the population through a nonlocal dispersal kernel and the behavioral control through a confinement parameter. First, we calculate the basic reproduction number $R_0$ and show its explicit dependence on the confinement rate. Second, we determine the minimal wave speed $\zeta^*$ of traveling wave solutions connecting the disease-free equilibrium to the endemic equilibrium. We show that $\zeta^*$ is given by the principal root of the corresponding characteristic equation and that (i) no traveling wave exists for $\zeta<\zeta^*$, and (ii) traveling waves exist for all $\zeta>\zeta^*$. Moreover, we show that the minimal wave speed is a monotone decreasing function of the confinement rate. Our results are obtained through a combination of spectral theory, upper-lower solution methods, and monotone iteration schemes that are modified to account for the joint effects of delay and nonlocal dispersal. Numerical simulations confirm the analytical prediction of the minimal wave speed and illustrate the quantitative slowing effect induced by confinement. These results provide a rigorous mathematical characterization of how mobility, delay, and confinement jointly determine epidemic invasion and spatial propagation.