Temporal evolution of immunity distributions in a population with waning and boosting

We investigate the temporal evolution of the distribution of immunities in a population, which is determined by various epidemiological, immunological and demographical phenomena: after a disease outbreak, recovered individuals constitute a large immune population, however their immunity is waning in the long term and they may become susceptible again. Meanwhile, their immunity can be boosted by repeated exposure to the pathogen, which is linked to the density of infected individuals present in the population. This prolongs the length of their immunity. We consider a mathematical model formulated as a coupled system of ordinary and partial differential equations, that connects all these processes, and systematically compare a number of boosting assumptions proposed in the literature, showing that different boosting mechanisms lead to very different stationary distributions of the immunity at the endemic steady state. In the situation of periodic disease outbreaks, the waveforms of immunity distributions are studied and visualized. Our results show that there is a possibility to infer the boosting mechanism from the population level immune-dynamics. AMS Classification 92D30, 34K60, 34K34, 37M05