A Multi-Strain Virus Model with Infected Cell Age Structure: Application to HIV

We consider a general mathematical model of a within-host viral infection with $n$ virus strains and explicit age-since-infection structure for infected cells. In the model, multiple virus strains compete for a population of target cells. Cells infected with virus strain $i\in \{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ die at per-capita rate ${\delta }_i(a)$ and produce virions at per-capita rate $p_i(a)$, where ${\delta }_i(a)$ and $p_i(a)$ are functions of the age-since-infection of the cell. Viral strain $i$ has a basic reproduction number, ${\mathcal{R}}_i$, and a corresponding positive single strain equilibrium, $E_i$, when ${\mathcal{R}}_i>1$. If ${\mathcal{R}}_i<1$, then the total concentration of virus strain $i$ will converge to 0 asymptotically. The main result is that when ${\max }_i{\mathcal{R}}_i>1$ and all of the reproduction numbers are distinct, i.e. ${\mathcal{R}}_i\mathrm{\ne }{\mathcal{R}}_j\mathrm{\forall }i\mathrm{\ne }j$, the viral strain with the maximal basic reproduction number competitively excludes the other strains. As an application of the model, HIV evolution is considered and simulations are provided.