Dynamical analysis and optimal control of an age-since infection HIV model at individuals and population levels

To study the impact of HIV infection at the individual level on the transmission of HIV/AIDS at the population level, in this paper, we formulate a multi-scale HIV model coupled with a two stages viral model (lymphoid tissue and blood) and an infection-age structured HIV epidemic model. Then we establish the well-posedness problem, derive the basic reproduction numbers and analyze the stability of equilibria for the coupled model. In order to investigate the effect of giving treatments to individuals on the disease, we further consider an optimal control problem subject to multiple drug treatments for the within-host model. For such control problem, we can obtain the optimal drug administration strategies to minimize the total number of transmissions from an infected individual over the course of infection. Some numerical simulations are presented to investigate the impact of the optimal control quantitatively on the viral load within the host and the between-host transmission dynamics. Our results suggest that controlling the progression of HIV infections at the individuals level is an effective measure to decrease the HIV transmission at the population level.