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MiS Preprint

Dynamical modeling of viral spread in spatially distributed populations

Henry Tuckwell and Laurent Toubiana


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 $p_{trans}$, the probability that an encounter between an infected and another host, results in viral transmission; the population density $N$, and the quantity $v_{crit}$ 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.

viral dynamics

Related publications

2007 Repository Open Access
Henry C. Tuckwell and Laurent Toubiana

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

In: Biosystems, 90 (2007) 2, pp. 546-559