Investigating Mechanisms of Spatiotemporal Epidemic Spread Using Stochastic Models

Abstract

To enable efficient control of a virus disease in an agricultural crop, it is desirable to understand the mechanisms that underlie its spread. Observations from small-scale field experiments, in which disease incidence is intensively mapped in space and at several time points, may shed some light on these mechanisms, and a range of analytic tools is available for their analysis and interpretation. For example, spatiotemporal distance class analysis (12,22) has been used to study the spatial distribution of newly detected infections in a population in relation to previous infections. Techniques based on spatiotemporal autocorrelation (1) such as STARIMA (spatiotemporal autoregressive integrated moving average) methods (13,23,24) have also been applied to model changes in disease incidence over time. A related approach is the use of ordinal regression (2,19,29), in which the probability of a change in the disease status of an individual at an observation time is modeled as a function of its status and the status of individuals in some neighborhood at a previous time. In common with STARIMA methods, this latter approach involves the fitting of a discretetime Markov model to the evolution of the epidemic. For discrete-time models to be applicable, frequent observation of a process may be required. If, however, the interval between observations is long in comparison with the time scales over which the disease spreads, then a continuous-time stochastic model may be more appropriate. Although, continuous-time stochastic models have been studied in the literature for many years (5,6,20,21), they have seen comparatively little use in the analysis of experimental data, with most emphasis placed on understanding their theoretical behavior and the macroscopic properties of the patterns they produce. However, recent developments in statistical methodology and computing power have led to a greater role for these models as analytic tools. Recently, Gibson and Austin (11) fitted continuous-time, spatiotemporal stochastic models to observations of a virus disease using stochastic integration methods. Their methods have been further refined using Markov chain Monte Carlo (MCMC) (3,9,14,15,28) by Gibson (10). The present paper illustrates and extends the techniques of Gibson (10) by applying them to observations of epidemics of citrus tristeza virus (CTV) in two experimental plots, each consisting of 216 trees of Washington navel orange (Citrus sinensis (L.) Osbeck) on Troyer citrange (Poncirus trifoliata (L.) Raf.), observed in Eastern Spain, where disease is spread by the vector Aphis gossypii. These data, illustrated in Figure 1, describe the distribution of CTV in the plots (which were originally disease free) as observed at three times, each 1 year apart. They were first reported and analyzed (along with several other CTV epidemics) by Gottwald et al. (12) using several methods mentioned above. The main goals of the analysis were to determine the relationship of diseased trees with each other and to detect directionality in the spread. Conclusions of Gottwald et al. (12) were that there was little evidence of anisotropy or aggregation in the patterns and that spread did not appear to occur preferentially to trees adjacent to those already diseased. The simultaneous presence of interand intraplot spread was suggested as the mechanism underlying the observations. The purpose of the new analysis presented here is to shed further light on the mechanisms that might underlie these data, by relating the disease maps directly to spatiotemporal stochastic models of spread. I believe the approach illustrated is a valuable addition to the methodology applicable to spatiotemporal data, which complements existing methods by allowing investigation of alternative transmission mechanisms using data in a direct, efficient manner. Spatiotemporal stochastic models. I consider a simple stochastic model for the spread of a virus disease through a discrete population. The model is more complex than related models considered by Gibson and Gibson and Austin (10,11) in that it represents virus transmission both from individuals in the population and from sources outside the population. The members of the population are represented by the vertices of a rectangular lattice. For ease of notation, I shall use x to denote both the spatial coordinates of a vertex and the individual located there. An individual x may be in one of two states: susceptible or diseased. I denote the set of diseased individuals at time t by D(t). The model considered is stochastic, meaning it represents the element of chance, or probability, in the acquisition of the disease by any susceptible individual. The probabilistic nature of disease transmission is incorporated as follows. If the individual x is susceptible at time t and, if dt is small, then x acquires the disease in the interval (t, t + dt) with some probability that is proportional to dt and is related to the locations of the diseased individuals in the population. Explicitly, the probability that x becomes diseased during (t, t + dt) is equal to ra(x)dt, in which