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 $i\in\left\{1,...,n\right\}$ 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. $\mathcal{R}_i\neq \mathcal{R}_j \ \forall i\neq j$, 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.