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

Authors: IPPOLITO, STEPHEN - American Seed Trade Association; LABORDE, JOSE - North Carolina State University; Gottwald, Timothy; IREY, MICHAEL - University Of Florida

Submitted to: Journal of Theoretical Biology
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 4/10/2020
Publication Date: 4/13/2020
Citation: Ippolito, S., Laborde, J., Gottwald, T.R., Irey, M.S. 2020. Studying the spatial temporal spread of the citrus tristeza virus through ODEs and Bernoulli trials. Journal of Theoretical Biology. 497:1-13. https://doi.org/10.1016/j.jtbi.2020.110279.
DOI: https://doi.org/10.1016/j.jtbi.2020.110279

Interpretive Summary: Past research by the third author and colleagues led to a mathematical model to describe the spread of plant pathogens through a crop. It allowed us to determine how much of the disease was due to disease spread within the crop and how much was due to more disease coming from outside that added to the epidemic. For insect vectored diseases this tool also helped us determine what insect vectors might be contributing the most to the disease. By looking at the progress of the disease in this way, it helped us to determine the underlying factors that influenced disease and thus how we might better control it. Several new and emerging mathematical techniques have evolved since that original model was developed that now help us add considerable depth and computing power to our original model. For example the original model was so computer computationally complex, that only about 500 plants could be considered without overloading the computer processor running the model. We have since developed new extensions that allow us to examine disease within a nearly unlimited crop size. The new model also allows us to more deeply examine the factors on internal versus external disease and its contribution to an epidemic. We tested the new model using citrus tristeza virus, a potentially severe disease of citrus, as we did for the first model. New model output suggests a mixture of both long and short range transmission for the citrus tristeza and allowed us to determine that initial long range contributions of disease are the most important driving factor of the disease, however, short range transmission becomes increasingly more important as the infection spreads. Once again this points to different insect vectors and that helps us determine the most efficacious control scenarios. Thus although this is a new extension of a complex mathematical model it has real world impact in aiding with disease control decisions.

Technical Abstract: The spread of the citrus tristeza virus (CTV) in Eastern Spain was studied by Gottwald et al., with the goal of determining the spatial mechanisms of spread. The results did not support patterns in which the spread necessarily followed from adjacent trees which were already infected. The suggested mechanism for the spread were both a mixture of long and short range spread. Consequently in 1997 Gibson at el. devised a Bayesian model for studying this problem with parameters corresponding to the suggested modes of transmission. The posterior distribution of these parameters was computed, using MCMC, followed by simulating from the posterior distribution in order to study the Spatio-Temporal characteristics of the disease. Though the contributions of this method are important we take a necessary departure staring with the same base model recast in a model selection or hypothesis testing framework. To achieve this we change the likelihood function based on order statistics making future predictions difficult, and instead consider a Poisson Binomial distribution where the probabilities of success are determined as the solution to a system of ordinary differential equations. Beyond this, while order statistics allowed temporal heterogeneity from external factors to be ignored, our model though not able to ignore it is able to recast it as a parameter, which in a Bayesian framework, could benefit the model though expert elicitation. To give proper credit, the system of ODEs is a natural extension of the model used by Gibson et al. Using ordinary differential equations, however, provides a natural framework for non-linear time series allowing us to make future predictions. Then since, as in the original model, the parameters have interpretations representing local and long range spread we may consider subsets of the full model with parameters removed. The models, representing different hypothesis, are then compared using Accumulated Prediction Error which is most relevant to the practitioner trying to predict next year’s incidence, and to the scientist as Accumulated Prediction Error is theoretically related to both Bayesian and Likelihood model selection frameworks. Although our goal is not to overlap with the analysis used by Gibson et al, we do preform simulations from the posterior distributions for the models in question as part of a post hoc analysis. Also as part of the post hoc analysis we consider prediction at the quadrat level, though quadrat specification is not necessary for model specification. Our results suggest a mixture of both long and short range transmission for the Citrus tristeza data where we see initial long range is the most important, however, short range transmission becomes increasingly more important as the infection spreads.