Reformulating the susceptible–infectious–removed model in terms of the number of detected cases: well-posedness of the observational model

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 are typically akin to 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 to yield what we term the observational model. We then prove the existence and uniqueness of a solution to the boundary value problem associated with the observational model and present a numerical algorithm to approximate the solution. This article is part of the theme issue ‘Technical challenges of modelling real-life epidemics and examples of overcoming these’.

A hospital demand and capacity intervention approach for COVID-19

“…We have so far demonstrated how the boundary value problem would be conceived for a simple SIR model and prove existence and uniqueness of the problem for fixed parameters. In the future we look at parameter identifiability, parameter estimation and efficient numerical algorithms using this method [ 56 ]. The standard methods of choice to solve these types of boundary value problems are the shooting method, nonlinear least squares of the equivalent initial value problem, and numerical continuation [ 81 – 84 ].…”

“…We consider the data starting from the first day of lockdown, and so we also need to infer the initial conditions. We have not yet conducted a formal investigation into the resulting log-likelihood, but it is clear that there is a continuous dependence between initial conditions and the parameters, see [ 56 ] for a comprehensive discussion. In practice, we see this manifest as an issue to calibrate p : if the first guess of initial conditions and parameters is not close to the “true” values, then it is p which changes in value the most.…”

Technical challenges of modelling real-life epidemics and examples of overcoming these

“…Panovska-Griffiths et al [5] and Hinch et al [6] are focused on the English COVID-19 epidemics and both estimate the transmissibility of variants and evaluate interventions, with the former combining the ABM with a statistical regression and making a policy contribution with the results, while the latter is taking a geospatial approach. improved methods [3][4][5][6][7][8] fitting data [3,[9][10][11] agent-based models [5,9,10,12,13] model relationships [5,6,14,15] estimating R (t) [8,11,13,16] methodology recommendations [17,18] variants […”

“…Li et al [8] use an ABM to estimate Rfalse(tfalse) and demonstrate improved statistical methods to do this in the face of noisy data, while Creswell et al [10] use classic modelling techniques to explore the role of imported COVID-19 cases on Rfalse(tfalse). Finally, Madzvamuse et al [11] present analytical results showing how to reformulate the standard differential equation epidemic model so that it is expressed in terms of observed information rather than the unobserved actual state of the population.…”