Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

An age-structured model with immune response of HIV infection: Modeling and optimal control approach

Pages: 153 - 172, Volume 19, Issue 1, January 2014      doi:10.3934/dcdsb.2014.19.153

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Hee-Dae Kwon - Department of Mathematics, Inha University, 100 Inharo, Nam-gu, Incheon 402-751, South Korea (email)
Jeehyun Lee - Department of Computational Science and Engineering, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, South Korea (email)
Myoungho Yoon - Department of Mathematics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, South Korea (email)

Abstract: This paper develops and analyzes an age-structured model of HIV infection with various compartments, including target cells, infected cells, viral loads and immune effector cells, to provide a better understanding of the interaction between HIV and the immune system. We show that the proposed model has one uninfected steady state and several infected steady states. We conduct a local stability analysis of these steady states by using a generalized Jacobian matrix method in conjunction with the Laplace transform. In addition, we consider various techniques and ideas from optimal control theory to derive optimal therapy protocols by using two types of dynamic treatment methods representing reverse transcriptase inhibitors and protease inhibitors. We derive the necessary conditions (an optimality system) for optimal control functions by considering the first variations of the Lagrangian. Further, we obtain optimal therapy protocols by solving a large optimality system of equations through the use of a difference scheme based on the Runge-Kutta method. The results of numerical simulations indicate that the optimal therapy protocols can facilitate long-term control of HIV through a strong immune response after the discontinuation of the therapy.

Keywords:  HIV dynamics, age-structured model, immune response, optimal control, gradient method.
Mathematics Subject Classification:  Primary: 92B05; Secondary: 49K20.

Received: July 2012;      Revised: August 2013;      Available Online: December 2013.