Application and Extension of Mean-Field Theory such as SIR to Discuss the Non-Mean Field Problem of COVID-19

TL;DR

This paper extends mean-field SIR models for COVID-19 by introducing the effective infection opportunity population (EIOP), enabling better data fitting and prediction of infection waves despite data limitations.

Contribution

It proposes the effective SIQR model incorporating EIOP, bridging mean-field and non-mean field approaches, and develops methods for data-scarce scenarios.

Findings

EIOP curve fitting matches first and second wave data

Model predicts third wave using second wave data

Data fitting useful for qualitative infection analysis

Abstract

The concept of the effective infection opportunity population (EIOP) was incorporated into the SIQR model, and it was assumed that this EIOP would change with the spread of infection, and this was named as the effective SIQR model. When calculated with this model, the uninfected population S decreases with the passage of time. However, when the EIOP N increases because of any reason, the infection threshold becomes larger than 1. Even after the first wave seems to have subsided, the infection begins to spread again. Firstly, we find the curve of EIOP change so that the calculation result by this model matches the data of the first and second waves. Then, we use this curve to fit with only the data of the second wave alone, and the third wave is predicted. In the case of new coronavirus infection, there are various restrictions on data collection to identify individual coefficients of mathematical models, and the true value is almost unknown. Therefore, the discussion in this paper is only about data fitting for predictive calculation. Therefore, the simulation on the true value is not aimed. However, since the data of infected persons reflect the true values, the results of data fitting can be used for the prediction of infected persons, isolated care recipients, inpatients, and severely ill persons. They are useful for a qualitative understanding of infection. The idea of EIOP is important in the sense that it connects the mean-field and the non-mean field, but the existence of data is essential, and the theory alone cannot simulate the non-mean field. We have developed two methods for treating the non-mean field cases where we don't have enough data. We have briefly introduced them.