Abstract

Diseases with waning immunity cause people with prior immunity (previous infection or vaccination) to be susceptible to infection again. This was evidenced in COVID -19. We formulate a model at the population level to understand how the number of new infections changes in the long run in a closed population.

In the same spirit as Kermack and McKendrick (1927, 1932 and 1933), we make use of two key features in our model: (i) time since infection and (ii) time since start of recovery. The former allows us to keep track of how the infectiousness of an infected individual changes with time, while the latter enables protection due to prior immunity to be included.

We perform stability and bifurcation analyses to obtain explicit conditions for which a disease goes extinct, stays at the endemic equilibrium, or causes an oscillatory number of new infections. The last regime (oscillatory solution) is a long-term dynamic that is not observed in the simple continuous-time geometric (exponential) distribution Susceptible-Infectious-Recovered-Susceptible (SIRS) model, which is commonly used by modellers. Finally, we show how our model can be applied to SARS -CoV-2 Omicron variant, making use of parameter values from the rich COVID -19 literature.