The basic reproduction number $R_0$ in time-heterogeneous environments

Abstract

In the previous paper (Inaba in J Math Biol 65:309–348, 2012), we proposed a new (most biologically natural) definition of the basic reproduction number $R_0$ for structured population in general time-heterogeneous environments based on the generation evolution operator. Using the mathematical definition for cone spectral radius, we show that our $R_0$ is given by the spectral radius of the generation evolution operator in the time-state space. Then as far as we consider linear population dynamics, our $R_0$ is a threshold value for population extinction and persistence in time-heterogeneous environments. Next we prove that even for nonlinear systems, our $R_0$ plays a role of a threshold value for population extinction in time-heterogeneous environments. For periodic systems, we can show that supercritical condition $R_0>1$ implies existence of positive periodic solution. Finally using the idea of $R_0$ in time-heterogeneous environment, we examine existence and stability of periodic solution in the age-structured SIS epidemic model with time-periodic parameters.