Asymptotic behavior of the basic reproduction number for periodic nonlocal dispersal operators and applications.

This paper is concerned with asymptotic behavior of the basic reproduction number defined by next generation nonlocal (convolution) dispersal operators in a time-periodic environment and applications. First we investigate the influence of the frequency and dispersal rate on the basic reproduction number, and we obtain that the basic reproduction number is monotone on the frequency. In the nonautonomous situation, the basic reproduction number is not a monotone function of dispersal rate in general. We derive the monotonicity for large frequency or dispersal rate. Then we apply the obtained results to a time-periodic SIS epidemic model and establish the existence and asymptotic profiles of the endemic periodic solution. Since solution maps of nonlocal system lack compactness, the standard uniform persistence theory and topological degree theory are unapplicable to obtain the existence of the endemic periodic solution. To overcome this difficulty, we apply the asymptotic fixed point theorem with the help of the Kuratowski measure of noncompactness.