On an impulsive faecal-oral model in a moving infected environment

This paper develops an impulsive faecal-oral model with a free boundary to investigate the combined effects of periodic disinfection and the spatial expansion of infected regions on disease transmission dynamics. We first establish the existence and uniqueness of a global classical solution for the impulsive model. Principal eigenvalues are then defined for the corresponding periodic eigenvalue problem at both initial and asymptotic time scales, with both eigenvalues shown to depend on impulse intensity and expansion rates. The long-term dynamics of the model are fully characterized using these principal eigenvalues, establishing a spreading-vanishing dichotomy. Numerical simulations support the theoretical findings and further investigate the impact of impulsive intervention and expansion capacity on disease transmission. Our results demonstrate that increasing impulse intensity and decreasing expansion rates both significantly contribute to effective disease control.