A Delay Differential Equation approach to model the COVID-19 pandemic

SEIR (Susceptible - Exposed - Infected - Recovered) approach is a classic modeling method that has frequently been applied to the study of infectious disease epidemiology. However, in the vast majority of SEIR models and models derived from them transitions from one population group to another are described using the mass-action law which assumes population homogeneity. That causes some methodological limitations or even drawbacks, particularly inability to reproduce observable dynamics of key characteristics of infection such as, for example, the incubation period and progression of the disease's symptoms which require considering different time scales as well as probabilities of different disease trajectories. In this paper, we propose an alternative approach to simulate the epidemic dynamics that is based on a system of differential equations with time delays to precisely reproduce a duration of infectious processes (e.g. incubation period of the virus) and competing processes like transition from infected state to the hospitalization or recovery. The suggested modeling approach is fundamental and can be applied to the study of many infectious disease epidemiology. However, due to the urgency of the COVID-19 pandemic we have developed and calibrated the delay-based model of the epidemic in Germany and France using the BioUML platform. Additionally, the stringency index was used as a generalized characteristic of the non-pharmaceutical government interventions implemented in corresponding countries to contain the virus spread. The numerical analysis of the calibrated model demonstrates that adequate simulation of each new wave of the SARS-CoV-2 virus spread requires dynamic changes in the parameter values during the epidemic like reduction of the population adherence to non-pharmaceutical interventions or enhancement of the infectivity parameter caused by an emergence of novel virus strains with higher contagiousness than original one. Both models may be accessed and simulated at https://gitlab.sirius-web.org/covid-19/dde-epidemiology-model utilizing visual representation as well as Jupyter Notebook.