HYBRID ON-OFF CONTROLS FOR AN HIV MODEL BASED ON A LINEAR CONTROL PROBLEM

In this talk, we consider a model of HIV infection with various compartments, including target cells, infected cells, viral loads and immune effector cells, to describe the human immunodeficiency virus(HIV) type 1 infection. We show that the proposed model has one uninfected steady state and two infected steady states and investigate their local stabilities by using a Jacobian matrix method. In addition, we apply techniques and ideas from linear optimal control theory in conjunction with a structured treatment interruption(STI) control to derive optimal on-off HIV therapeutic strategies. We obtain the equations for the adjoint variables and the characterization of optimal control function by applying the Pontryagin’s Maximum Principle. The results of numerical simulations indicate that the optimal hybrid on-off therapy protocols can move the model system to a ”healthy” steady state in which the immune response is dominant in controlling HIV after discontinuation of therapy. HIV MODEL We introduce a nonlinear ordinary differential equation model that includes modeling for the immune response. The majority of the model is adapted from Callaway and Perelson[3]. In addition, the model contains an immune response component with Michaelis-Menten-type saturation nonlinearity [1,2]. The system of ODEs describing the compartmental infection dynamics is given by Target: Ṡ = λ− dS − (1 − α1e)kV S Infected: I = (1 − α1e)kV S − δI −mEI Virus: V = (1 − α2e)NTδI − cV − (1 − α1e)ρkV S Immune effectors: Ė = λE + bEI I + Kb E − dEI I + Kd E − δEE (1) together with an initial condition [S(0), I(0), V (0), E(0)]. This model includes the four key compartments: uninfected target cells (S, cells/mm) and infected cell (I , cells/mm), free virus (V , copies/mm), and immune response, CTL (E, EQ0 EQ1 EQ2 EQ3 S 1,000 162.571 580.770 968.863 I 0 11.958 1.675 0.075 V 0 64.389 9.023 0.402 E 0.01 0.031 180.178 347.356 local stability unstable stable unstable stable Table 1 Off treatment (e ≡ 0) steady states (cells/mm) for model (1). Non-physical steady states have been omitted. cells/mm). The constant α1 and α2 are the drug efficacy of reverse transcriptase inhibitors and protease inhibitors, respectively. Given the specified parameters, in the absence of therapy, the model exhibits several steady states as shown in Table 1. There is a locally unstable equilibrium EQ0 S = 1000, I = 0, V = 0, E = 0.01, which represents an unifected patient, as well as two locally stable equilibria for an infected patient in the absence of treatment. These stable steady states are as follows : “unhealthy” : S = 162.571, I = 11.958, V = 64.389, E = 0.031; “healthy” : S = 968.863, I = 0.075, V = 0.402, E = 347.356. LINEAR OPTIMAL CONTROL PROBLEM In this section, we formulate a linear optimal control problem in order to derive an optimal hybrid on and off therapy. We consider control of free HIV population in finite time intervals using e(t) as our control functions. The objective functional is defined as