Mutation and SARS-CoV-2 strain competition under vaccination in a modified SIR model

The crisis caused by the COVID-19 outbreak around the globe raised an increasing concern about the ongoing emergence of variants of the virus that may evade the immune response provided by vaccines. New variants appear due to mutation, and as the cases accumulate, the probability of the emergence of a variant of concern increases. In this article, we propose a modified susceptible, infected, and recovered (SIR) model with waning immunity that captures the competition of two strain classes of an infectious disease under the effect of vaccination with a highly contagious and deadlier strain class emerging from a prior strain due to mutation. When these strains compete for a limited supply of susceptible individuals, changes in the efficiency of vaccines may affect the behaviour of the disease in a non-trivial way, resulting in complex outcomes. We characterise the parameter space including intrinsic parameters of the disease, and using the vaccine efficiencies as control variables. We find different types of transcritical bifurcations between endemic fixed points and a disease-free equilibrium and identify a region of strain competition where the two strain classes coexist during a transient period. We show that a strain can be extinguished either due to strain competition or vaccination, and we obtain the critical values of the efficiency of vaccines to eradicate the disease. Numerical studies using parameters estimated from publicly reported data agree with our theoretical results. Our mathematical model could be a tool to assess quantitatively the vaccination policies of competing and emerging strains using the dynamics in epidemics of infectious diseases.