## 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 , risking
thousands of jobs. Ecological losses can extend beyond the a ected 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 deserti cation and permanent
habitat loss. Palm trees, which are threatened by Bayoud disease, in particular can help
to prevent deserti cation.
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 e ect of the parameters involved, in particular, the tree density, the
infection rate and e ective 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
con nement 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
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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 nd 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