Non-local 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. By incorporating these two critical factors, this article proposes an impulsive non-local faecal-oral model with free boundaries. In this framework, non-local diffusion captures the long-range movement of individuals, impulsive effects model periodic disinfection measures, and free boundaries delineate the advancing fronts of infection. We first establish the existence and uniqueness of a non-negative global classical solution to the system. Subsequently, by employing the theory of resolvent positive operators and their perturbations, we analyze the principal eigenvalue, which depends on the infected domain, impulse intensity, and non-local diffusion kernel functions. On the basis of this eigenvalue, we derive sharp threshold criteria distinguishing between disease extinction and propagation. The incorporation of non-local diffusion and impulsive interventions introduces significant mathematical complexities, which are addressed through novel analytical techniques. Finally, numerical simulations are conducted to validate the theoretical results and explore the impact of pulsed interventions. Our findings demonstrate that periodic disinfection effectively mitigates disease transmission, whereas the influence of non-local diffusion varies with the selection of kernel functions.