Probabilistic Model for Control of an Epidemic by Isolation and Quarantine

Assuming a homogeneous population, we apply the mass action law for rate of new infections and a second-order gamma distribution for removal probability to model spread of an epidemic. In numerical examinations of higher-order gamma distributions for removal probability, we discover an unexpected pattern in maximum fraction of population infected. We develop from first principles of probability an eighth-order system of ordinary differential equations to model effects of isolation and quarantine. We derive analytical expressions for reproduction numbers modeling isolation and quarantine when applied separately and together and verify them numerically. We quantify strength and speed required of these interventions to contain epidemics of varying severity and examine how their effectiveness depends on when they begin. We find that effectiveness is highly sensitive to small changes of intervention strength in a critical region. Finally, adding two more differential equations to capture natural population dynamics, we calculate endemic disease equilibria when affected by isolation and examine dynamics of coming to an equilibrium state.