Dynamics of a seasonal brucellosis disease model with nonlocal transmission and spatial diffusion

Abstract Brucellosis has been increasingly concerned with the animal health and a huge loss of economics. Multiple transmission routes, seasonal pattern and spatial diffusion have been identified in brucellosis propagations. Meanwhile, infectious ability strongly depends on the distance of each adjacent infectious individual. Such property can be captured by a nonlocal infection with some integrable functions. In this paper, we first propose a seasonal brucellosis epidemic model with nonlocal transmissions and spatial diffusions. We calculate the next generation operator R ( x ) by a renewal process. The basic reproduction number R 0 is defined as the spectral radius of R ( x ) , which plays an equivalent role in a principal eigenvalue of a linear operator. It is shown that the proposed model exhibits the threshold dynamics in terms of R 0 , i.e, if R 0 1 , then the brucella-free steady state is globally asymptotically stable; otherwise, the model admits at least one positively periodic steady state. Numerical simulations elucidate that time heterogeneity enhances magnitudes of oscillations of animal population and brucella. Moreover, enlarging effective infection radii increases the risk of brucellosis propagations. As a result, we suggest that herdsmen should improve the sanitation of animals’ environment and isolate infected animals instantly to control brucellosis prevalence.