Dynamical modeling of viral spread in spatially distributed populations: stochastic origins of oscillations and density dependence

In order to understand the spatio-temporal structure of epidemics beyond that permitted with classical SIR (susceptible-infective-recovered)-type models, a new mathematical model for the spread of a viral disease in a population of spatially distributed hosts is described. The positions of the hosts are randomly generated in a rectangular habitat. Encounters between any pair of individuals are according to a Poisson process with a mean rate that declines exponentially as the distance between them increases. The contact rate allows the mean rates to be set at a certain number of encounters per day on average. The relevant state variables for each individual at any time are given by the solution of a pair of coupled differential equations for the viral load and the quantity of general immune system effectors which reduce the viral load. The parameters describing within-host viral-immune system dynamics are generated randomly to reflect variability across a population. Transmission is assumed to depend on the viral loads in donors and occurs with a probability ptransptrans. The initial conditions are such that one randomly chosen individual carries a randomly chosen amount of the virus, whereas the rest of the population is uninfected. Simulations reveal local or whole-population responses. Whole-population disease spread may be in the form of isolated or multiple occurrences, the latter often being approximately periodic. The mechanisms of this oscillatory behaviour are analyzed in terms of several parameters and the distribution of critical points in the host dynamical systems. Increased contact rate, increased probability of transmission and decreased threshold for viral transmission, decreased immune strength and increased viral growth rate all increase the probability of multiple outbreaks and the distribution of the critical points also plays a role.