Endemic states of integro-differential equation-based disease models

Abstract

As recently demonstrated by the SARS-CoV-2 pandemic, infectious diseases may have a huge impact on society. Mathematical models of infectious diseases allow to predict their behaviour. This process makes it easier to plan mitigation actions. Moreover, by studying the long-term behaviour of a model mathematically, one can see when disease dynamics start to stabilize around an equilibrium or under which circumstances the disease dies out.

Our contribution is the study of an IDE-based model with varying population size which does allow for endemic behaviour. In order to derive an endemic model based on integro-differential equations, we will include the possibility of natural birth and death in a model similar to the one presented in Wendler et al. (2026). Compared to other IDE-based models, this model is rather complex, also allowing for disease death. The fact that we have a varying population size influenced by both the natural birth and death rate, as well as the disease-induced mortality, make the model analysis more involved. Moreover, the definition of equilibria is unclear when considering non-constant population size. In order to study the model behaviour independently of the population size, we will introduce a normalized version of our model. While this technique was already applied to ODE-based models, this seems to be a novel approach to IDE-based models. As a main result, we show the stability of the disease-free equilibrium whenever the reproduction number is smaller than one. Moreover, we derive conditions under which the disease-free equilibrium becomes unstable for a reproduction number larger than one.