Critical waves in a nonlocal dispersion delayed SI1I2R model with generalized nonlinear incidence function

Critical waves in a nonlocal dispersion delayed SI1I2R model

Résumé

We investigate traveling wave solutions of a delayed nonlocal diffusion epidemic
model divided into four compartments: susceptible (S), two infectious classes (I1 and
I2), and recovered (R). The model integrates geographic dispersal using nonlocal integral
operators, as well as a temporal delay in the transmission process to account for
the latent period of infection. The incidence variables L1(S, I1) and L2(S, I2) allow for
broad nonlinear interactions between susceptibles and infectious classes. We demonstrate
the presence of traveling wave connecting the disease-free steady state to an
endemic equilibrium using appropriate kernel function and incidence rate assumptions.
The method is based on the development of higher and lower solutions. A detailed
mathematical examination reveals that the fundamental reproduction number R0 is
critical in recognising the presence of traveling waves. We present a threshold condition
for the least wave speed α∗ and analyse the impact of delay and nonlocal dispersal
on wave profile and propagation speed. The system permits nontrivial traveling wave
solutions at all speeds α ≥ α∗, but none exist for 0 < α < α∗. In contrast, if R0 ≤ 1,
no traveling wave solutions exist. Numerical simulations of sample nonlinear incidence
functions are presented to validate and illustrate the analytical conclusions.