Mathematical analysis of global dynamics and optimal control of treatment for an age-structured HBV infection model

Evidences show that cell-to-cell infection occurs in HBV infection and is a very efficient way which may lead to more than half of the viral infection. Hence we formulate an HBV infection model with cell-to-cell infection, meanwhile, we incorporate the infection age of infected cells. For dynamic behaviors, we firstly establish the well-posedness and asymptotic smoothness of the model; then show the global threshold dynamical behaviors, which are determined by the reproduction number; and finally perform numerical simulations not only to verify the theoretical results but also to explore effects of cell-to-cell infection on viral load and infection concentration. Our finding is that ignoring the cell-to-cell infection will underestimate the actual HBV infection status. For optimal control of treatment for HBV, we introduce two time-varying control measures and investigate the corresponding optimal control problem. We derive the characteristic expression of the optimal control and discuss effects of control costs and integrated control. The results show that high control cost will not result in better control effectiveness, while integrated optimal control can reach the best control objectives compared with the two single ones. Moreover, infection age can greatly influence HBV infection at the initial stage.