  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.
