Computational modelling of the spread of tree disease through forests

Abstract

The impact of infectious tree diseases, such as Ash dieback, are wide-ranging: economic losses are estimated to be in the region of billions of pounds, for example with the cost of Ash dieback to the UK estimated to be £15 billion over the next century [1], risking thousands of jobs. Ecological losses can extend beyond the affected tree species to many dependent species, at a time when concern over the precariousness of many species is mounting; the loss of some tree species can also lead to desertification and permanent habitat loss. Palm trees, which are threatened by Bayoud disease, in particular can help to prevent desertification [2]. In order to control outbreaks of infections and mitigate this damage, it is desirable to develop reliable warning signals, which can indicate when a localised outbreak is likely to become an epidemic. Outbreak events can also be predicted or prevented by identifying characteristics, such as planting density or homogeneity, which may make a forest especially vulnerable or resilient. However, the possibility of identifying these through experiment is limited: tree diseases typically spread over long time scales, it is hard to control the relevant parameters, for example, maintaining a particular density of trees without continual human interference, and the areas of land required would be considerable. In this thesis, we will further develop existing mathematical models for disease spreading while making a broad examination of the effect of the parameters involved, in particular, the tree density, the infection rate and effective distance, and the uniformity of the forestscapes. Our approach uses a lattice-based model, which combines a compartmental model of infection with a spatial component. Each lattice point represents either a tree or an empty space. We start with an infection with local interactions, such that an infected tree can only transmit the disease to its nearest neighbours. We consider two types of neighbourhoods: von Neumann and Moore, and quantify the observed dynamics by measuring the spreading velocity of the disease. As the forestscape density increases, we see a transition from local confinement to widespread outbreak at a critical density. The critical density is consistent with the percolation threshold, that is, the point at which the domain is spanned by a single connected cluster. We also further investigate an established framework of early warning signals such as the standard deviation, skewness and kurtosis of the velocity to predict the occurrence of the transition to the outbreak regime. Further measures, such as the mortality, also indicate a critical shift at the same critical density. Previous work has focused on homogeneous forests, but this is not very representative of real forests, which frequently exhibit large scale features, such as man-made structures, or a natural clustering of trees. We develop an algorithm for generating forests with clustering, with a single parameter governing the degree of clustering. We characterise the observed distributions of trees in terms of Besag’s function, which captures how far a distribution is from random at particular length scales. We also extract data on the distribution of trees in real forests, using the Mahalanobis distance, a measure of how close the properties of an individual are to the properties of a set, to classify cells as either containing trees or not, and compare this to our synthetically generated data. Using our synthetically generated clustered forestscapes in our local model, we observe that the dynamics strongly depend on the degree of clustering. We find that an increase in the level of tree aggregation suppresses the infection propagation. The critical density for the transition to the epidemic is consequently shifted to higher values for clustered forestscapes. We then consider how the dynamics of an infection which spreads in a more realistic non local way differ from an infection which only spreads locally. We observe that, in addition to the tree density and the degree of clustering, the dynamics are strongly influenced by the characteristic spreading distance, with some interesting interplay between the parameters. Measuring the spreading velocity, we again see that clustering suppresses the infection, but as the spreading distance increases, this effect is diminished. For the non-local infection, the transition to the epidemic regime is not clear when considering the velocity, but can be seen in the mortality. However, it occurs at much smaller tree densities than for the local infection. Overall, the dynamics of the infection can be remarkably altered by the switch from local to non-local interactions, with several observed effects that are relevant to understanding and predicting the propagation of tree diseases.