Dynamics of an epidemic controlled by isolation and quarantine: A probability-based deterministic model

Abstract

Assuming a homogeneous population, we employ a deterministic model based on first principles of probability to explore dynamics of an epidemic controlled by isolation alone, quarantine alone, and the two together. We develop explicit closed-form equations for key metrics of control performance: cumulative fraction of population infected over the course of the epidemic (final size), maximum fraction infected at any one time, and epidemic duration. We derive an analytical solution for final size of an epidemic controlled by isolation, when final size is small, and develop empirical relations for the other cases. We frame equations in terms of reproduction numbers, measures of intervention effort and initial conditions. We model both strength and speed of interventions, assume second order gamma distributions for intervention waiting times and employ non-time-invariant equations for quarantine. We also account for quarantine of unexposed, susceptible individuals and for imperfect intervention.