Dynamical modeling of viral spread in spatially distributed populations

Henry Tuckwell and Laurent Toubiana

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Submission date: 10. Nov. 2005
published in: Biosystems, 90 (2007) 2, p. 546-559 
DOI number (of the published article): 10.1016/j.biosystems.2006.12.006
Bibtex
with the following different title: Dynamical modeling of viral spread in spatially distributed populations : stochastic origins of oscillations and density dependence
Keywords and phrases: viral dynamics

Abstract:
In order to understand the structure of epidemiological data beyond that permitted with classical SIR 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 homogeneous 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 of each individual at any time are given by the solution of a pair of standard coupled ordinary differential equations for the virus and an immune system effector. Transmission is assumed to depend on the viral loads in donors and a temporal window which is disease specific. In simulated solutions we choose a constant temporal transmission factor. The implementation of the model is described in detail in Section 3. 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, and the latter may be in the form of single occurrences or multiple occurrences, sometimes in a roughly periodic pattern. The mechanisms of this oscillatory behaviour are analyzed in terms of three parameters, of the many dynamical and demographic parameters, in the first instance. These are , the probability that an encounter between an infected and another host, results in viral transmission; the population density N, and the quantity which is a threshold viral load required for viral growth in a newly infected host. A large number of trials is performed to examine the roles of these parameters in producing multiple outbreaks and these roles are analyzed in detail.