Role of white-tailed deer in geographic spread of the black-legged tick <i>Ixodes scapularis</i> : Analysis of a spatially nonlocal model

Stephen A. Gourley; Xiulan Lai; Junping Shi; Wendi Wang; Yanyu Xiao; Xingfu Zou; Stephen A. Gourley; Xiulan Lai; Junping Shi; Wendi Wang; Yanyu Xiao; Xingfu Zou

doi:10.3934/mbe.2018046

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 4: 1033-1054. doi: 10.3934/mbe.2018046

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

1.
Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UK
2.
Institute for Mathematical Sciences, Renmin University of China, Beijing, China
3.
Department of Mathematics, College of William and Mary, Williamsburg, VA, USA
4.
Key Laboratory of Eco-environments in Three Gorges Reservoir Region, and School of Mathematics and Statistics, Southwest University, Chongqing, China
5.
Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, USA
6.
Department of Applied Mathematics, University of Western Ontario, London, ON, Canada

Received: 15 November 2017 Accepted: 15 November 2017 Published: 01 August 2018
MSC : Primary: 92D20; Secondary: 34K20

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.
- Lyme disease,
- black-legged tick,
- white-tailed deer,
- spatial model,
- delay,
- dispersal,
- traveling wavefront,
- spread rate
Citation: Stephen A. Gourley, Xiulan Lai, Junping Shi, Wendi Wang, Yanyu Xiao, Xingfu Zou. Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model[J]. Mathematical Biosciences and Engineering, 2018, 15(4): 1033-1054. doi: 10.3934/mbe.2018046

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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.

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