  The host immune response can often efficiently suppress a virus infection,
which may lead to selection for immune-resistant viral variants within the
host. For example, during HIV infection, an array of CTL immune response
populations recognize specific epitopes (viral proteins) presented on the
surface of infected cells to effectively mediate their killing. However HIV can
rapidly evolve resistance to CTL attack at different epitopes, inducing a
dynamic network of interacting viral and immune response variants. We consider
models for the network of virus and immune response populations, consisting of
Lotka-Volterra-like systems of ordinary differential equations. Stability of
feasible equilibria and corresponding uniform persistence of distinct variants
are characterized via a Lyapunov function. We specialize the model to a "binary
sequence" setting, where for $n$ epitopes there can be $2^n$ distinct viral
variants mapped on a hypercube graph. The dynamics in several cases are
analyzed and sharp polychotomies are derived characterizing persistent
variants. In particular, we prove that if the viral fitness costs for gaining
resistance to each epitope are equal, then the system of $2^n$ virus strains
converges to a "perfectly nested network" with less than or equal to $n+1$
persistent virus strains. Overall, our results suggest that immunodominance,
i.e. relative strength of immune response to an epitope, is the most important
factor determining the persistent network structure.
