University of Surrey

# Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : analysis of a spatially nonlocal model

Gourley, Stephen, Lai, X, Shi, J, Wang, W, Xiao, Y and Zou, X (2018) Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : analysis of a spatially nonlocal model Mathematical Biosciences and Engineering, 15 (4). pp. 1033-1054.

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## Abstract

Lyme disease is transmitted via blacklegged ticks, the spatial spread of which is believed to be primarily via transport on white-tailed deer. In this paper, we develop a mathematical model to describe the spatial spread of blacklegged ticks due to deer dispersal. The model turns out to be a system of differential equations with a spatially non-local term accounting for the phenomenon that a questing female adult tick that attaches to a deer at one location may later drop to the ground, fully fed, at another location having been transported by the deer. We first justify the well-posedness of the model and analyze the stability of its steady states. We then explore the existence of traveling wave fronts connecting the extinction equilibrium with the positive equilibrium for the system. We derive an algebraic equation that determines a critical value c* which is at least a lower bound for the wave speed in the sense that, if c < c*, there is no traveling wave front of speed c connecting the extinction steady state to the positive steady state. Numerical simulations of the wave equations suggest that c* is the minimum wave speed. We also carry out some numerical simulations for the original spatial model system and the results seem to confirm that the actual spread rate of the tick population coincides with c*. We also explore the dependence of c* on the dispersion rate of the white tailed deer, by which one may evaluate the role of the deer's dispersion in the geographical spread of the ticks.

Item Type: Article
Divisions : Faculty of Engineering and Physical Sciences > Mathematics
Authors :
NameEmailORCID
Gourley, StephenS.Gourley@surrey.ac.uk
Lai, X
Shi, J
Wang, W
Xiao, Y
Zou, X
Date : 22 February 2018
DOI : 10.3934/mbe.2018046
Copyright Disclaimer : Copyright 2017 American Institute of Mathematical Sciences. This is a pre-copy-editing, author-produced PDF of an article accepted for publication in Mathematical Biosciences and Engineering, following peer review. The definitive publisher-authenticated version is available online at: http://www.aimsciences.org/article/doi/10.3934/mbe.2018046
Uncontrolled Keywords : Dispersion, Ixodes scapularis, Lyme disease, Odocoilius virginianus, Spatial model, Ticks, Traveling wavefront, Spread rate.
Depositing User : Melanie Hughes
Date Deposited : 22 Nov 2017 11:48
URI: http://epubs.surrey.ac.uk/id/eprint/844987