Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model
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...
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2018-07-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2018046 |
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| author | Stephen A. Gourley Xiulan Lai Junping Shi Wendi Wang Yanyu Xiao Xingfu Zou |
| author_facet | Stephen A. Gourley Xiulan Lai Junping Shi Wendi Wang Yanyu Xiao Xingfu Zou |
| author_sort | Stephen A. Gourley |
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| description | 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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| institution | Kabale University |
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| language | English |
| publishDate | 2018-07-01 |
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| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-319f92806e654dd3a2d89ec8d9a6f7452025-01-24T02:40:57ZengAIMS PressMathematical Biosciences and Engineering1551-00182018-07-011541033105410.3934/mbe.2018046Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal modelStephen A. Gourley0Xiulan Lai1Junping Shi2Wendi Wang3Yanyu Xiao4Xingfu Zou5Department of Mathematics, University of Surrey, Guildford, Surrey GU2 7XH, UKInstitute for Mathematical Sciences, Renmin University of China, Beijing, ChinaDepartment of Mathematics, College of William and Mary, Williamsburg, VA, USAKey Laboratory of Eco-environments in Three Gorges Reservoir Region, and School of Mathematics and Statistics, Southwest University, Chongqing, ChinaDepartment of Mathematical Sciences, University of Cincinnati, Cincinnati, OH, USADepartment of Applied Mathematics, University of Western Ontario, London, ON, CanadaLyme 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.https://www.aimspress.com/article/doi/10.3934/mbe.2018046lyme diseaseblack-legged tickwhite-tailed deerspatial modeldelaydispersaltraveling wavefrontspread rate |
| spellingShingle | 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 Mathematical Biosciences and Engineering lyme disease black-legged tick white-tailed deer spatial model delay dispersal traveling wavefront spread rate |
| title | Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model |
| title_full | Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model |
| title_fullStr | Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model |
| title_full_unstemmed | Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model |
| title_short | Role of white-tailed deer in geographic spread of the black-legged tick Ixodes scapularis : Analysis of a spatially nonlocal model |
| title_sort | role of white tailed deer in geographic spread of the black legged tick ixodes scapularis analysis of a spatially nonlocal model |
| topic | lyme disease black-legged tick white-tailed deer spatial model delay dispersal traveling wavefront spread rate |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2018046 |
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