Control and Simulation Center, Harbin Institute of Technology, Harbin, 15008, China.
Interdiscip Sci. 2018 Jun;10(2):375-389. doi: 10.1007/s12539-017-0275-y. Epub 2017 Dec 19.
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).
本文首次提出了一个具有潜伏期和延迟状态的乙型肝炎病毒（HBV）模型，并采用联合治疗方法以获得比单药治疗更长的效果。计算了平衡点和基本再生数，然后在此模型上分析了局部稳定性。我们随后基于 Pontryagin 最小原理提出了最优控制策略，该策略的目标函数不仅要使治疗结束时暴露细胞、感染细胞和游离病毒的水平几乎降至零，还要使药物副作用和治疗成本最小化。此外，我们还开发了一种基于前向和后向差分近似相结合的求解 HBV 模型的数值模拟算法。模拟了有无最优控制的未感染细胞、暴露细胞、感染细胞、游离病毒、CTL 和 ALT 的状态动态，结果表明，基于时变最优控制策略，HBV 几乎降至零，而没有控制则会导致疾病爆发。最后，通过模拟，我们证明了策略 A 是我们采用的三种策略中最好的，并进一步比较了模型（1）和模型（2）。
