# MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint
14/2011

### Predicting the probability of persistence of HIV infection with the standard model

Henry Tuckwell and Patrick Shipman

#### Abstract

We consider the standard three-component differential equation model for the growth of an HIV virion population in an infected host in the absence of drug therapy. The dynamical properties of the model are determined by the set of values of six parameters which vary across host populations. There may be one or two critical points whose natures play a key role in determining the outcome of infection and in particular whether the HIV population will persist or become extinct.

There are two cases which may arise. In the first case, there is only one critical point $P_1$ at biological values and this is an asymptotically stable node. The system ends up with zero virions and so the host becomes HIV-free. In the second case, there are two critical points $P_1$ and $P_2$ at biological values. Here $P_1$ is an unstable saddle point and $P_2$ is an asymptotically stable spiral point with a non-zero virion level. In this case the HIV population persists unless parameters change. We let the parameter values take random values from distributions based on empirical data, but suitably truncated, and determine the probabilities of occurrence of the various combinations of critical points. From these simulations the probability that an HIV infection will persist, across a population, is estimated. It is found that with conservatively estimated distributions of parameters, within the framework of the standard 3-component model, the chances that a within host HIV population will become extinct is between 0.6% and 6.9%. With less conservative parameter estimates, the probability is estimated to be as high as 24%. The many factors related to the transmission and possible spontaneous elimination of the virus are discussed.