Interplays of a waterborne disease model linking within- and between- host dynamics with waning vaccine-induced immunity

In this paper, we propose a multi-scale waterborne disease model and are concerned with a heterogenous process of waning vaccine-induced immunity. A completely nested rule has been adopted to link the within- and between-host systems. We prove the existence, positivity and asymptotical smoothness of the between-host system. We derive the basic reproduction numbers associated with the two-scale system in explicit forms, which completely determine the behavior of each system. Uncertainty analysis reveals the trade-offs of the kinetics of the within-host system and the transmission of the between-host system. Numerical simulations suggest that the vaccine waning process plays a significant role in the estimation of the prevalence at population level. Furthermore, the environmental heterogeneity complicates the transmission patterns at the population level.