  We introduce a two-strain model with asymmetric temporary immunity periods
and partial cross-immunity. We derive explicit conditions for competitive
exclusion and coexistence of the strains depending on the strain-specific basic
reproduction numbers, temporary immunity periods, and degree of cross-immunity.
The results of our bifurcation analysis suggest that, even when two strains
share similar basic reproduction numbers and other epidemiological parameters,
a disparity in temporary immunity periods and partial or complete
cross-immunity can provide a significant competitive advantage. To analyze the
dynamics, we introduce a quasi-steady state reduced model which assumes the
original strain remains at its endemic steady state. We completely analyze the
resulting reduced planar hybrid switching system using linear stability
analysis, planar phase-plane analysis, and the Bendixson-Dulac criterion. We
validate both the full and reduced models with COVID-19 incidence data,
focusing on the Delta (B.1.617.2), Omicron (B.1.1.529), and Kraken (XBB.1.5)
variants. These numerical studies suggest that, while early novel strains of
COVID-19 had a tendency toward dramatic takeovers and extinction of ancestral
strains, more recent strains have the capacity for co-existence.
