  We consider the dynamics of a virus spreading through a population that
produces a mutant strain with the ability to infect individuals that were
infected with the established strain. Temporary cross-immunity is included
using a time delay, but is found to be a harmless delay. We provide some
sufficient conditions that guarantee local and global asymptotic stability of
the disease-free equilibrium and the two boundary equilibria when the two
strains outcompete one another. It is shown that, due to the immune evasion of
the emerging strain, the reproduction number of the emerging strain must be
significantly lower than that of the established strain for the local stability
of the established-strain-only boundary equilibrium. To analyze the unique
coexistence equilibrium we apply a quasi steady-state argument to reduce the
full model to a two-dimensional one that exhibits a global asymptotically
stable established-strain-only equilibrium or global asymptotically stable
coexistence equilibrium. Our results indicate that the basic reproduction
numbers of both strains govern the overall dynamics, but in nontrivial ways due
to the inclusion of cross-immunity. The model is applied to study the emergence
of the SARS-CoV-2 Delta variant in the presence of the Alpha variant using
wastewater surveillance data from the Deer Island Treatment Plant in
Massachusetts, USA.
