Spatio-temporal models for plant epidemics : analytical and simulation studies

We formulate a stochastic spatio-temporal model for the spread of infectious diseases in plants. Studying the behaviour of a model which takes into account stochasticity and spatial extension usually involves intractable mathematics and requires the use of simulation. A challenging objective is to develop analytical methods for general application which provide predictions for the expected behaviour of the model. The individual-based model comprises primary and secondary infection and recovery processes. Using stochastic simulation we study the expected behaviour and variability of the epidemic size, and characterise the disease patterns through spatial correlation. Both stationary and transient behaviour are analysed over the parameter space. Simulation is also used to test empirical extensions of non-spatial models which attempt to account for heterogeneous mixing of susceptibles and infecteds. Analytical methods based on cluster approximations are commonly used for predicting the dynamics of stochastic models characterised by nearest neighbour (NN) interactions. On the other hand, for models with more general interactions, the rather simplistic and non-spatial Mean Field approximation has been extensively used. We propose an alternative general approach, built on individual-based ODEs and closure approximations, for predicting the behaviour of spatial models in which the individuals interact according to a generic function of their distance. The approximations, which take into account the development of correlations in the spatial distribution of the population, are tested against the simulation results showing excellent agreement in most of the parameter space. We also test the ability of cluster approximations to capture the effects of the anisotropic spread of the disease. To this end, we formulate a generalised NN model in which the dispersal of propagules depends on the direction of spread and use simulation to assess the performance of different approximations.