Target reproduction numbers for time-delayed population systems

In the field of population dynamics, target reproduction number is a crucial metric that dictates the necessary control efforts for achieving specific prevention, intervention, or control goals. Recently, the concept of the target reproduction number has undergone significant extensions. Lewis et al. [1] presented a general framework of the target reproduction number for nonnegative matrices, and Wang and Zhao [2] further developed it to positive operators on an ordered Banach space. These extensions encompass fundamental metrics like basic reproduction number and type reproduction number, along with other threshold parameters from existing literature, elucidating their roles in population control. In the current paper, we establish the theory of target reproduction number for a large class of compartmental population models with time delay in the case where control is targeted toward either new infection/production or internal evolution/transition. It turns out that the target reproduction number of the original time-delayed population model can be viewed as a basic reproduction number of some modified system. At the end, we apply these analytic results to three epidemic models, which enhances our theoretical understanding and provides valuable insights for effective strategies in population-based interventions and control measures.