Remarks on basic reproduction ratios for periodic abstract functional differential equations

In this paper, we extend the theory of basic reproduction ratios $ \mathcal{R}_0 $ in [Liang, Zhang, Zhao, JDDE], which concerns with abstract functional differential systems in a time-periodic environment. We prove the threshold dynamics, that is, the sign of $ \mathcal{R}_0-1 $ determines the dynamics of the associated linear system. We also propose a direct and efficient numerical method to calculate $ \mathcal{R}_0 $.