# American Institute of Mathematical Sciences

2004, 1(2): 267-288. doi: 10.3934/mbe.2004.1.267

## An Age-Structured Model of HIV Infection that Allows for Variations in the Production Rate of Viral Particles and the Death Rate of Productively Infected Cells

 1 Department of Mathematics, University of Michigan, 5860 E. Hall, Ann Arbor, MI 48109, United States 2 Department of Biology, University of New Mexico, Albuquerque, NM 87131, United States 3 Department of Mathematics, University of British Columbia, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada 4 Mathematical Modeling and Analysis, T-7, Los Alamos National Laboratory, Mail Stop B284, Los Alamos, NM 87545, United States 5 Theoretical Division T-10, Los Alamos National Laboratory, Los Alamos, NM 87545, United States

Received  April 2004 Revised  June 2004 Published  July 2004

Mathematical models of HIV-1 infection can help interpret drug treatment experiments and improve our understanding of the interplay between HIV-1 and the immune system. We develop and analyze an age-structured model of HIV-1 infection that allows for variations in the death rate of productively infected T cells and the production rate of viral particles as a function of the length of time a T cell has been infected. We show that this model is a generalization of the standard differential equation and of delay models previously used to describe HIV-1 infection, and provides a means for exploring fundamental issues of viral production and death. We show that the model has uninfected and infected steady states, linked by a transcritical bifurcation. We perform a local stability analysis of the nontrivial equilibrium solution and provide a general stability condition for models with age structure. We then use numerical methods to study solutions of our model focusing on the analysis of primary HIV infection. We show that the time to reach peak viral levels in the blood depends not only on initial conditions but also on the way in which viral production ramps up. If viral production ramps up slowly, we find that the time to peak viral load is delayed compared to results obtained using the standard (constant viral production) model of HIV infection. We find that data on viral load changing over time is insufficient to identify the functions specifying the dependence of the viral production rate or infected cell death rate on infected cell age. These functions must be determined through new quantitative experiments.
Citation: Patrick W. Nelson, Michael A. Gilchrist, Daniel Coombs, James M. Hyman, Alan S. Perelson. An Age-Structured Model of HIV Infection that Allows for Variations in the Production Rate of Viral Particles and the Death Rate of Productively Infected Cells. Mathematical Biosciences & Engineering, 2004, 1 (2) : 267-288. doi: 10.3934/mbe.2004.1.267
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