The mathematics of contagious diseases and their limitations in forecasting

This article explores mathematical models for understanding the evolution of contagious diseases. The most widely known set of models are the compartmental ones, which are based on a set of differential equations. But these are not the only models. This review visits many different families of models. Additionally, we show these families, not as unrelated entities, but following a common thread in which the problems or assumptions of a model are solved or generalized by another model. In this way, we can understand their relationships, assumptions, simplifications, and, ultimately, limitations. Prompted by the current Covid19 pandemic, we have a special focus on spread forecasting. We illustrate the difficulties encountered to do realistic predictions. In general, they are only approximations to a reality whose biological and societal complexity is much larger. Particularly troublesome are the large underlying variability, the problem's time-varying nature, and the difficulty to estimate the required parameters for a faithful model. Additionally, we will also see that these models have a multiplicative nature implying that small errors in the system parameters cause a huge uncertainty in the prediction. Stochastic or agent-based models can overcome some of the modeling problems of systems based on differential or stochastic equations. Their main difficulty is that they are as accurate and realistic as the data available to estimate their detailed parametrization, and very often this detailed data is not at the modeller's disposal. Although the predictive power of mathematical models to forecast the evolution of a contagious disease is very limited, these models are still very useful to plan interventions as they can calculate their impact if all other parameters stay fixed. They are also very useful to understand the properties of disease propagation in complex systems.