### Abstract

We study a mathematical model for the viral dynamics of HIV in an infected individual in the presence of HAART. The paper starts with a literature review and then formulates the basic mathematical model. An expression for R0, the basic reproduction number of the virus under steady state application of HAART, is derived followed by an equilibrium and stability analysis. There is always a disease-free equilibrium (DFE) which is globally asymptotically stable for R0 < 1. Deterministic simulations with realistic parameter values give additional insight into the model behaviour. We then look at a stochastic version of this model and calculate the probability of extinction of the virus near the DFE if initially there are only a small number of infected cells and infective virus particles. If R0 1 then the system will always approach the DFE, whereas if R0 > 1 then some simulations will die out whereas others will not. Stochastic simulations suggest that if R0 > 1 those which do not die out approach a stochastic quasi-equilibrium consisting of random uctuations about the non-trivial deterministic equilibrium levels, but the amplitude of these uctuations is so small that practically the system is at the non-trivial equilibrium. A brief discussion concludes the paper.

Original language | English |
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Pages (from-to) | 305-321 |

Number of pages | 16 |

Journal | Discrete and Continuous Dynamical Systems - Series B |

Volume | 12 |

Issue number | 2 |

DOIs | |

Publication status | Published - Sep 2009 |

### Keywords

- HIV
- AIDS
- HAART
- differential equations
- R0
- equilibrium and stability
- analysis
- deterministic model
- stochastic model
- probability of extinction
- simulation

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## Cite this

Dalal, N., Greenhalgh, D., Mao, X., & University of Strathclyde (Funder) (2009). Mathematical modelling of internal HIV dynamics.

*Discrete and Continuous Dynamical Systems - Series B*,*12*(2), 305-321. https://doi.org/10.3934/dcdsb.2009.12.305