Studying the spatial temporal spread of the citrus tristeza virus through ODEs and Bernoulli trials.

The Citrus tristeza virus (CTV) is one of the most economically devastating citrus diseases worldwide. The spread of CTV in eastern Spain was studied by Gottwald et al. with the goal of determining the spatio-temporal mechanisms of spread. Since the subjects in this study are individual trees, it is natural to think of infections as Bernoulli trials. This approach is difficult however, due to the spatial and temporal dependence of the observations. Consequently, a system of ordinary differential equations (ODE) was used to model the probabilities of infection as well as the spatial and temporal dependence. Given the parameters in the ODE, the probabilities of infection are treated as conditionally independent. Using the conditional independence we then specify the joint likelihood function as a Poisson binomial distribution. For the purpose of model selection and hypothesis testing we, employed accumulated prediction error (APE) which has connections to both Bayesian and frequentist frameworks. We demonstrated the robustness of our method in accounting for spatio-temporal dependencies in the data by accurately predicting the spatial distribution of the disease through Join Counts.