Abstract for: The Limits of Discrete Modeling of Continuous Individual Dynamics: A Cautionary Tale from Immuno-Epidemiogical Dynamics

The gradual waning of immunity can have a substantial impact on spread of many common infections, and is important to represent in epidemiological models for those infections. Current representations of the gradual waning of immunity diverge widely, with implementations ranging from very popular dichotomous representation in classic mathematical epidemiology compartment models to detailed descriptions characterizing immune system dynamics in individual-level models. We sought to compare two simple models, one based on the traditional SIRS discrete representation, and the other based on an exponential loss of immunity. We then analytically solved these models and analyzed how much they differed. We found that these representations differed significantly, in measures such as the mathematical structures and the limits of the closed forms of the probabilities. These divergences suggest that there are major, perhaps difficult to reconcile, discrepancies between different methods of describing the same phenomenon of waning of immunity. These differences offer cautions to those who attempt to use, for example, a dichotomous model to represent a continuous phenomenon such as the loss of immunity.