SIR-based model with multiple imperfect vaccines

Since the introduction of vaccination in the current COVID-19 outbreak, many countries have approved and implemented vaccination campaigns to mitigate and ultimately curtail the pandemic. Several types of vaccines have been proposed and many of them have finally been approved and used in different countries. The different types of vaccines have different vaccine parameters, and therefore, this situation induces the necessity of modeling mathematically the scenario of multiple imperfect vaccines. In this paper, we introduce a SIR-based model considering different vaccines, and study the basic properties of the model, including the stability of the Disease-Free Equilibrium (DFE), which is locally asymptotically stable if the reproduction number is less than 1. A sequence of further results aims to enumerate the conditions where the reproduction number can be decreased (or increased). Two important mathematical propositions indicate that in general vaccination might not be enough to contain an outbreak and that the addition of new vaccines could be counterproductive if the leakiness parameter is greater than a threshold η . This model, despite its simplicity, was validated with data of the COVID-19 pandemic in five countries: Israel, Chile, Germany, Lithuania, and Czech Republic, observing that improvements for the vaccine campaigns can be suggested by the developed theory.