Modeling and Control of a Delayed Hepatitis B Virus Model with Incubation Period and Combination Treatment

In this paper, a hepatitis B virus (HBV) model with an incubation period and delayed state and control variables is firstly proposed. Furthermore, the combination treatment is adopted to have a longer-lasting effect than mono-therapy. The equilibrium points and basic reproduction number are calculated, and then the local stability is analyzed on this model. We then present optimal control strategies based on the Pontryagin's minimum principle with an objective function not only to reduce the levels of exposed cells, infected cells and free viruses nearly to zero at the end of therapy, but also to minimize the drug side-effect and the cost of treatment. What's more, we develop a numerical simulation algorithm for solving our HBV model based on the combination of forward and backward difference approximations. The state dynamics of uninfected cells, exposed cells, infected cells, free viruses, CTL and ALT are simulated with or without optimal control, which show that HBV is reduced nearly to zero based on the time-varying optimal control strategies whereas the disease would break out without control. At last, by the simulations, we prove that strategy A is the best among the three kinds of strategies we adopt and further comparisons have been done between model (1) and model (2).