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Spread of SARS-CoV-2 in a SIS model with vaccination and breakthrough infection

Although previous infection and vaccination provide protection against SARS-CoV-2 infection, both reinfection and breakthrough infection are possible events whose occurrence would increase with time after first exposure to the antigen and with the emergence of new variants of the virus. Periodic vaccination could counteract this decline in protection. In the present work, our aim was to develop and explore a model of SARS-CoV-2 spread with vaccination, reinfection and breakthrough infection. A modified deterministic SIS (Susceptible-Infected-Susceptible) model represented by a system of differential equations was designed. As in any SIS model, the population was divided into susceptible and infected individuals. But in our design, susceptible individuals were, in turn, grouped into three consecutive categories whose susceptibility increases with time after infection or vaccination. The model was studied by means of computer simulations, which were analysed qualitatively. The results obtained show that the prevalence, after oscillating between peaks and valleys, reaches a plateau phase. Moreover, as might be expected, the magnitude of the peaks and plateaus increases as the infection rate rises, the vaccination rate decreases and the rate of decay of protection conferred by vaccination or previous infection increases. Therefore, the present study suggests that, at least under certain conditions, the spread of SARS-CoV-2, although it could experience fluctuations, would finally evolve into an endemic form, with a more or less stable prevalence that would depend on the levels of infection and vaccination, and on the kinetics of post-infection and post-vaccination protection. However, it should be kept in mind that our development is a theoretical scheme with many limitations. For this reason, its predictions should be considered with great care.

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