Persistence of a pandemic in the presence of susceptibility and infectivity distributions in a population: Mathematical model

Abstract

The birth and death of a pandemic can be region specific. Pandemic seems to make repeated appearance in some places which is often attributed to human neglect and seasonal change. However, difference could arise from different distributions of inherent susceptibility (σinh) and external infectivity (ιext) from one population to another. These are often ignored in the theoretical treatments of an infectious disease progression. While the former is determined by the immunity of an individual towards a disease, the latter depends on the duration of exposure to the infection. Here we model the spatio-temporal propagation of a pandemic using a generalized SIR (Susceptible-Infected-Removed) model by introducing the susceptibility and infectivity distributions to comprehend their combined effects. These aspects have remained inadequately addressed till date. We consider the coupling between σinh and ιext through a new critical infection parameter (γc). We find that the neglect of these distributions, as in the naive SIR model, results in an overestimation in the estimate of the herd immunity threshold. That is, the presence of the distributions could dramatically reduce the rate of spread. Additionally, we include the effects of long-range migration by seeding new infections in a region. We solve the resulting master equations by performing Kinetic Monte Carlo Cellular Automata (KMC-CA) simulations. Importantly, our simulations can reproduce the multiple infection peak scenario of a pandemic. The latent interactions between disease migration and the distributions of susceptibility and infectivity can render the progression a character vastly different from the naive SIR model. In particular, inclusion of these additional features renders the problem a character of a living percolating system where the disease cluster can survive by spatial migration.