Nonlocal diffusion and pulse intervention in a faecal-oral model with moving infected fronts

How individual dispersal patterns and human intervention behaviours affect the spread of infectious diseases constitutes a central problem in epidemiological research. This paper develops an impulsive nonlocal faecal-oral model with free boundaries, where pulses are introduced to capture a periodic spraying of disinfectant, and nonlocal diffusion describes the long-range dispersal of individuals, and free boundaries represent moving infected fronts. We first check that the model has a unique nonnegative global classical solution. Then, the principal eigenvalue, which depends on the infected region, the impulse intensity, and the kernel functions for nonlocal diffusion, is examined by using the theory of resolvent positive operators and their perturbations. Based on this value, this paper obtains that the diseases are either vanishing or spreading, and provides criteria for determining when vanishing and spreading occur. At the end, a numerical example is presented in order to corroborate the theoretical findings and to gain further understanding of the effect of the pulse intervention. This work shows that the pulsed intervention is beneficial in combating the diseases, but the effect of the nonlocal diffusion depends on the choice of the kernel functions.