# Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model

Roy, Priti Kumar and Chatterjee, Amar Nath and Greenhalgh, David and Khan, Qamar J A (2013) Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model. Nonlinear Analysis: Real World Applications, 14 (3). pp. 1621-1633. ISSN 1468-1218

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## Abstract

Infection with HIV-1, degrading the human immune system and recent advances of drug therapy to arrest HIV-1 infection, has generated considerable research interest in the area. Sebastian Bonhoeffer et al. [2], introduced a population model representing long term dynamics of HIV infection in response to available drug therapies. We consider a similar type of approximate model incorporating time delay in the process of infection on the healthy T cells which, in turn, implies inclusion of a similar delay in the process of viral replication. The model is studied both analytically and numerically. We also include a similar delay in the killing rate of infected CD4+ T cells by Cytotoxic TLymphocyte (CTL) and in the stimulation of CTL and analyze two resulting models numerically. The models with no time delay present have two equilibria: one where there is no infection and a non-trivial equilibrium where the infection can persist. If there is no time delay then the non-trivial equilibrium is locally asymptotically stable. Both our analytical results (for the first model) and our numerical results (for all three models) indicate that introduction of a time delay can destabilize the non-trivial equilibrium. The numerical results indicate that such destabilization occurs at realistic time delays and that there is a threshold time delay beneath which the equilibrium with infection present is locally asymptotically stable and above which this equilibrium is unstable and exhibits oscillatory solutions of increasing amplitude.

#### ORCID iDs

Roy, Priti Kumar, Chatterjee, Amar Nath, Greenhalgh, David ORCID: https://orcid.org/0000-0001-5380-3307 and Khan, Qamar J A;
• Item type: Article 49774 DateEventJune 2013Published31 October 2012Accepted HIV-1, time series solutions, cell lysis, time delay, asymptotic stability, reverse transcriptase inhibitor, cytotoxic T-lymphocyte, CD4+ T cells, Probabilities. Mathematical statistics, Economics, Econometrics and Finance(all), Computational Mathematics, Analysis, Applied Mathematics, Engineering(all), Medicine(all) Science > Mathematics > Probabilities. Mathematical statistics Faculty of Science > Mathematics and Statistics Pure Administrator 14 Oct 2014 08:30 26 Jun 2021 00:59 https://strathprints.strath.ac.uk/id/eprint/49774