How does dispersal affect the infection size?

Human movement facilitates the spatial spread of infectious diseases and poses a serious threat to disease prevention and control. A large number of spatial epidemic models have been proposed and analyzed in the past few decades. The vast majority of these studies focus on establishing a threshold result between disease persistence and extinction in terms of the basic reproduction number. In reality, disease eradication is difficult and even impossible for many infectious diseases. Thus, it is crucial to understand how population dispersal affects the total infection size and its distribution across the environment. Based on a susceptible-infected-susceptible patch model with standard incidence, some general results on the number of infections over all patches and disease prevalence in each patch are obtained. For the two-patch submodel, we give a complete classification of the model parameter space as to whether dispersal is beneficial or detrimental to disease control. Particularly, fast diffusion decreases the basic reproduction number but may increase the total infection size, highlighting the necessity of evaluating control measures with other quantities besides the basic reproduction number. Higher infection risk means higher disease prevalence in the two-patch case. However, numerical simulations find that the patch with the highest risk of infection may not have the highest disease prevalence when three or more patches are concerned. Besides spatial heterogeneity and diffusion coefficient, the total infection size is also significantly affected by patch connectivity.