The effects of population dispersal and pulse vaccination on disease control

In this paper, we investigate a nonautonomous SIR type epidemic model with pulse vaccination in patchy environments. We obtain a threshold parameter which governs the extinction or the uniform persistence of the disease by applying Floquet theory and the comparison theorem of impulsive differential equations. Numerical results indicate that population dispersal has significant effects on disease transmission. Varying the population dispersal rates between two patches in which the disease would die out if they had remained isolated, could allow the disease to persist globally. Alternatively, similar variations between patches where the disease would be persistent if isolated can lead to extinction globally. Finally, the model was generalized and we extended the definition of the basic reproduction number in a continuous (autonomous or periodic) system to that for a hybrid system as the spectral radius of the next infection operator.