Studying and approximating spatio–temporal models for epidemic spread and control

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Abstract
A class of simple spatio–temporal stochastic models for the spread and control of plant disease is investigated. We consider a lattice–based susceptible–infected model in which the infection of a host occurs through two distinct processes: a background infective challenge representing primary infection from external sources, and a short–range interaction representing the secondary infection of susceptibles by infectives within the population. Recent data–modelling studies have suggested that the above model may describe the spread of aphid–borne virus diseases in orchards. In addition, we extend the model to represent the effects of different control strategies involving replantation (or recovery). The Contact Process is a particular case of this model. The behaviour of the model has been studied using Cellular–Automata simulations. An alternative approach is to formulate a set of deterministic differential equations that captures the essential dynamics of the stochastic system. Approximate solutions to this set of equations, describing the time evolution over the whole parameter range, have been obtained using the pairwise approximation (PA) as well as the most commonly used mean–field approximation (MF). Comparison with simulation results shows that PA is significantly superior to MF, predicting accurately both transient and long–run, stationary behaviour over relevant parts of the parameter space. The conditions for the validity of the approximations to the present model and extensions thereof are discussed.

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