Spatial epidemic network models with viral dynamics

Abstract

A mathematical model is presented for the spread of viral diseases within human or other populations in which both the dynamics of viral growth within individuals and the interactions between individuals are taken into account. We thus bridge the classical macroscopic approach to the growth and population dynamics of disease at the microscopic level. Each member, $i,$ of the population of $n$ individuals is represented by a vector function of time whose components are antibody numbers ${a}_{i}(t),$ and the virion level ${v}_{i}(t).$ These quantities evolve according to $2n$ differential equations, which are coupled via a transmission matrix B with elements ${\ensuremath{\beta}}_{\mathrm{ij}},$ $i,j=1,\dots{},n,$ such that ${\ensuremath{\beta}}_{\mathrm{ij}}{v}_{i}$ is the expected rate of transmission of infectious particles from individual $i$ to individual $j.$ We study nearest-neighbor interaction and transmission which declines exponentially with distance between the individuals. Results are shown to be related to those of classical macroscopic (SIR) models. We find threshold effects in the occurrence of epidemics as the parameters of the viral and antibody dynamics change. The distribution of the final size of an epidemic is estimated, for various initial patterns of infection, at various values of the parameter which describes the mobility of the population. We also determine the final size in the cases of extreme clustering and dispersion of infected individuals.