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We consider the dynamics of a virus spreading through a population thatproduces a mutant strain with the ability to infect individuals that wereinfected with the established strain. Temporary cross-immunity is includedusing a time delay, but is found to be a harmless delay. We provide somesufficient conditions that guarantee local and global asymptotic stability ofthe disease-free equilibrium and the two boundary equilibria when the twostrains outcompete one another. It is shown that, due to the immune evasion ofthe emerging strain, the reproduction number of the emerging strain must besignificantly lower than that of the established strain for the local stabilityof the established-strain-only boundary equilibrium. To analyze the uniquecoexistence equilibrium we apply a quasi steady-state argument to reduce thefull model to a two-dimensional one that exhibits a global asymptoticallystable established-strain-only equilibrium or global asymptotically stablecoexistence equilibrium. Our results indicate that the basic reproductionnumbers of both strains govern the overall dynamics, but in nontrivial ways dueto the inclusion of cross-immunity. The model is applied to study the emergenceof the SARS-CoV-2 Delta variant in the presence of the Alpha variant usingwastewater surveillance data from the Deer Island Treatment Plant inMassachusetts, USA.

The emergence of a virus variant: dynamics of a competition model with  cross-immunity time-delay validated by wastewater surveillance data for  COVID-19