Abstract
Many methodologies in disease modeling are invaluable in the evaluation of health interventions. Of these methodologies, one of most fundamental is compartmental modeling. Compartmental models have many different forms with one of the most general characterizations occurring from the description of disease dynamics with nonlinear Volterra integral equations. Despite this generality, the vast majority of disease modellers prefer the special case where nonlinear Volterra integral equations reduce to systems of differential equations through the traditional assumptions that 1) the infectiousness of a disease corresponds to incidence, and 2) the duration of infection follows either an exponential or Erlang distribution. However, these assumptions are not the only ones that simplify nonlinear Volterra integral equations in such a way. In what follows, we illustrate a biologically more accurate description of the total infectivity of a disease that reduces systems of nonlinear Volterra integral equations to a class of novel compartmental models, as described by systems of differential equations. We demonstrate the consistency of these novel compartmental models to their traditional counterparts when the duration of infection follows either an exponential or Erlang distribution, and provide a novel compartmental model for a Pearson distributed duration of infection. Significant outcomes of our work include a compartmental model that captures any Erlang distributed duration of infection with only 3 differential equations, instead of the typical inflated model sizes, and a compartmental models that capture any mean, standard deviation, skewness, and kurtosis of the duration of infection distribution with only 4 differential equations.
Author summary Compartmental models are a powerful tool for predicting disease outbreaks, and evaluating public health policies and intervention effectiveness. However, such models typically have an inability to account for many of the biological features of a disease. For instance, the assumptions placed on the duration of infection required by most compartmental models are due to mathematical convenience, and are known to massively effect model behavior and quality of predictions. Our work illustrates a simple solution to these erroneous assumptions by proposing a new simplification of the general model proposed by Kermack and McKendrick. In doing so, we obtain a new class of compartmental models with many of the features that make traditional compartmental the go-to disease model for the vast majority the epidemiological modeling community, such as their formulations as systems of differential equations, while adding the ability to more accurately account for effects of variability in an individual’s duration of infection. As such, our work may be viewed as the starting point for multiple research avenues, as it opens up a new class of compartmental model for investigation under the contexts of mathematics, public health, and evolutionary biology.