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

Campillo-Funollet, Eduard; Wragg, Hayley; Yperen, James Van; Duong, Duc-Lam; Madzvamuse, Anotida

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.