Optimal control therapy and vaccination for special HIV-1 model with delay

In this paper, we consider a four dimensional model of the human immunodeficiency virus-1 (HIV-1) with delay, which is an extension of some three dimensional models. We approach the treatment problem by adding two controllers to the system for inhibiting viral production. The optimal controller [Formula: see text] is considered for vaccine and [Formula: see text] for the drug. The Pontryagin maximum principle with delay is used to characterize these optimal controls. At the end, numerical results are presented to illustrate the optimal solutions. The validity of the model was confirmed by proper semi-quantitative simulation of some clinical data. The model was used to predict the possible beneficial effects of vaccine and anti-retroviral drug administration in HIV-1 disease.

Dynamics and control of a mathematical model for metronomic chemotherapy

A 3-compartment model for metronomic chemotherapy that takes into account cancerous cells, the tumor vasculature and tumor immune-system interactions is considered as an optimal control problem. Metronomic chemo-therapy is the regular, almost continuous administration of chemotherapeutic agents at low dose, possibly with small interruptions to increase the efficacy of the drugs. There exists medical evidence that such administrations of specific cytotoxic agents (e.g., cyclophosphamide) have both antiangiogenic and immune stimulatory effects. A mathematical model for angiogenic signaling formulated by Hahnfeldt et al. is combined with the classical equations for tumor immune system interactions by Stepanova to form a minimally parameterized model to capture these effects of low dose chemotherapy. The model exhibits bistable behavior with the existence of both benign and malignant locally asymptotically stable equilibrium points. In this paper, the transfer of states from the malignant into the benign regions is used as a motivation for the construction of an objective functional that induces this process and the analysis of the corresponding optimal control problem is initiated.