A nonstandard numerical scheme for a novel SECIR integro-differential equation-based model allowing nonexponentially distributed stay times

TL;DR

This paper introduces a novel integro-differential equation model for infectious disease spread that incorporates realistic stay time distributions and variable infectiousness, improving simulation accuracy especially during intervention changes.

Contribution

It develops a generalized IDE-based model with nonstandard numerical schemes, allowing for flexible stay time distributions and detailed clinical state tracking, surpassing traditional ODE models.

Findings

The model accurately captures disease dynamics with variable stay times.

The numerical scheme demonstrates convergence and stability.

The model better assesses intervention impacts.

Abstract

Ordinary differential equations (ODE) are a popular tool to model the spread of infectious diseases, yet they implicitly assume an exponential distribution to describe the flow from one infection state to another. However, scientific experience yields more plausible distributions where the likelihood of disease progression changes accordingly with the duration spent in a particular state of the disease. Furthermore, transmission dynamics depend heavily on the infectiousness of individuals. The corresponding nonlinear variation with the time individuals have already spent in an infectious state requires more realistic models. The previously mentioned items are particularly crucial when modeling dynamics at change points such as the implementation of nonpharmaceutical interventions. In order to capture these aspects and to enhance the accuracy of simulations, integro-differential equations (IDE) can be used. In this paper, we propose a generalized model based on IDEs with eight infection states. The model allows for variable stay time distributions and generalizes the concept of ODE-based models as well as IDE-based age-of-infection models. In this, we include particular infection states for severe and critical cases to allow for surveillance of the clinical sector, avoiding bottlenecks and overloads in critical epidemic situations. On the other hand, a drawback of IDE-based models is that efficient numerical solvers are not as widely available. We extend a recently introduced nonstandard numerical scheme. This scheme is adapted to our more advanced model and we prove important mathematical and biological properties. Furthermore, we validate our approach numerically by demonstrating the convergence rate. Eventually, we also show that our novel model is intrinsically capable of better assessing disease dynamics upon the introduction of nonpharmaceutical interventions.