A control theoretic approach to mitigate viral escape in HIV

A very important scientific advance was the identification of HIV as a causative agent for AIDS. HIV infection typically involves three main stages: a primary acute infection, a long asymptomatic period and a final increase in viral load with a simultaneous collapse in healthy CD4+T cell count during which AIDS appears. Motivated by the worldwide impact of HIV infection on health and the difficulties to test in vivo or in vitro the different hypothesis which help us to understand the infection, we study the problem from a control theoretic perspective. We present a deterministic ordinary differential equation model that is able to represent the three main stages in HIV infection. The mechanism behind this model suggests that macrophages could be long-term latent reservoirs for HIV and may be important in the progression to AIDS. To avoid or slow this progression to AIDS, antiretroviral drugs were introduce in the late eighties. However, these drugs are not always successful causing a viral rebound in the patient. This rebound is associated with the emergence of resistance mutations resulting in genotypes with reduced susceptibility to one or more of the drugs. To explore antiretroviral effects in HIV, we extend the mathematical model to include the impact of therapy and suggest different mutation models. Under some additional assumptions the model can be seen to be a positive switched dynamic system. Consequently we test clinical treatments and allow preliminary control analysis for switching treatments. After introducing the biological background and models, we formulate the problem of treatment scheduling to mitigate viral escape in HIV. The goal of this therapy schedule is to minimize the total viral load for the period of treatment. Using optimal control theory a general solution in continuous time is presented for a particular case of switched positive systems with a specific symmetry property. In this case the optimal switching rule is on a sliding surface. For the discrete-time version several algorithms based on linear programming are proposed to reduce the computational burden whilst still computing the optimal sequence. Relaxing the demand of optimality, we provide a result on state-feedback stabilization of autonomous positive switched systems through piecewise co-positive Lyapunov functions in continuous and discrete time. The performance might not be optimal but provides a tractable solution which guarantees some level of performance. Model predictive control (MPC) has been considered as an important suboptimal technique for biological applications, therefore we explore this technique to the viral escape mitigation problem