The basic reproduction number in time-heterogeneous environments

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 $$ for structured population in general time-heterogeneous environments based on the generation evolution operator. Using the mathematical definition for cone spectral radius, we show that our $$ 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 $$ is a threshold value for population extinction and persistence in time-heterogeneous environments. Next we prove that even for nonlinear systems, our $$ plays a role of a threshold value for population extinction in time-heterogeneous environments. For periodic systems, we can show that supercritical condition $$ implies existence of positive periodic solution. Finally using the idea of $$ in time-heterogeneous environment, we examine existence and stability of periodic solution in the age-structured SIS epidemic model with time-periodic parameters.