Reformulating the SIR model in terms of the number of COVID-19 detected cases: well-posedness of the observational model

TL;DR

This paper reformulates the classical SIR epidemiological model using observed COVID-19 case data, proving well-posedness of the resulting boundary value problem and providing a numerical solution method.

Contribution

It introduces a new observational model for the SIR system based on detected cases, establishing its mathematical well-posedness and offering a numerical approximation approach.

Findings

Proved existence and uniqueness of the solution to the observational SIR model.

Developed a numerical algorithm for approximating the model's solution.

Enhanced the applicability of SIR models to real-world COVID-19 data.

Abstract

Compartmental models are popular in the mathematics of epidemiology for their simplicity and wide range of applications. Although they are typically solved as initial value problems for a system of ordinary differential equations, the observed data is typically akin of a boundary value type problem: we observe some of the dependent variables at given times, but we do not know the initial conditions. In this paper, we reformulate the classical Susceptible-Infectious-Recovered system in terms of the number of detected positive infected cases at different times, we then prove the existence and uniqueness of a solution to the derived boundary value problem and then present a numerical algorithm to approximate the solution.