9 A within-host HIV-I model
Case study: HIV-I
In this chapter we will look at another example of a within-host disease model, and in fact one of the first such models to be developed. Human immunodeficiency virus (HIV) infects immune cells called ‘CD4+ T-cells’, which form an important part of the human immune system. HIV enters these T-cells, replicates within the cell and then releases new virus particles in to the bloodstream. Immediately after first infection, the virus grows rapidly and produces common infection symptoms in the patient. After a few months, these symptoms disappear and the virus concentration reduces to a lower, but steady, level. This ‘asymptomatic period’ can last for years, with the virus density staying roughly constant, and the concentration of T-cells very slowly dropping. Eventually, the T-cell density becomes so low that the patient’s immune system is no longer effective, a condition called aquired immunodeficiency syndrome (AIDS). The patient is then at risk from life-threatening opportunistic infections. From the 1990s, much work has been done to explore the dynamics of HIV and to produce potential treatments, including the development of mathematical models (see chapter references).
A (pre-treatment) model
Let’s start by establishing our variables and drawing a schematic of what is happening in this system. The variables that we want to keep track of (before treatment, at least), are:
- Healthy T-cells ([latex]T[/latex])
- Infected T-cells ([latex]T^*[/latex])
- Virus particles ([latex]V[/latex])
We will only keep track of the concentration of virus particles in the bloodstream, not the ‘intracellular’ concentrations. Our schematic should look like this:
Healthy T-cells are produced at some constant rate [latex]s[/latex] by the body (not a per-capita rate), but also decay at some rate [latex]d[/latex]. These healthy cells become infected through contact with virus, assuming a mass-action process, with coefficient [latex]k[/latex]. These infected T-cells then decay at some rate [latex]\delta[/latex]. New virus particles are produced when infected T-cells die, with [latex]N[/latex] giving the average number of virus particles produced by each cell. The virus then also decays at some rate [latex]c[/latex]. The system can therefore be expressed with the following set of ODEs:
[latex]\begin{align} \frac{dT}{dt} &= s-dT-kVT\\ \frac{dT^*}{dt} &= kVT-\delta T^*\\ \frac{dV}{dt} &= N\delta T^*-cV \end{align}[/latex]
Looking at the equations, we can see that there is an HIV-free equilibrium for [latex](T,T^*,V)=(s/d,0,0)[/latex]. We have another example here of a 3-dimensional system. As we saw in the free-living parasite model, assessing stability here can be a bit more complicated, often reying on using the Routh-Hurwitz criteria. As it turns out, for the HIV-free equilibrium things simplify fairly nicely to the point that we can actually directly calculate one of the eigenvalues, and then just use the trace-determinant con