A reductive analysis of a compartmental model for COVID-19: data assimilation and forecasting for the United Kingdom

Abstract

We introduce a deterministic model that partitions the total population into the susceptible, infected, quarantined, and those traced after exposure, recovered and the deceased. We introduce the concept of 'accessible population for transmission of the disease', which can be a small fraction of the total population, for instance when interventions are in force. This assumption, together with the structure of the set of coupled nonlinear ordinary differential equations for the populations, allows us to decouple the equations into just two equations. This further reduces to a logistic type of equation for the total infected population. The equation can be solved analytically and therefore allows for a clear interpretation of the growth and inhibiting factors in terms of the parameters in the full model. The validity of the 'accessible population' assumption and the efficacy of the reduced logistic model is demonstrated by the ease of fitting the United Kingdom data for the total number of infected cases. The model can also be used to forecast further progression of the disease. The approach further helps us to analyze the original model equations. We show that the original model equations provide a very good fit with the United Kingdom data for the cumulative number of infections. The active infected population of the model is seen to exhibit a turning point around mid-May, suggesting the beginning of a slow-down in the spread of infections. However, the rate of slowing down beyond the turning point is small and therefore the cumulative number of infections is likely to saturate to about 3.8 × 105 only towards the end of July or beginning of August, provided the lock-down conditions continue to prevail. Noting that the fits obtained from the reduced logistic equation and the full model equations are equally good, the underlying causes for the limited forecasting ability of the reduced logistic equation is elucidated. The model and the procedure adopted here are expected to be useful in fitting the data for other countries and forecasting the progression of the disease.