# American Institute of Mathematical Sciences

August 2018, 15(4): 1033-1054. doi: 10.3934/mbe.2018046

## 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

* Corresponding author: Xingfu Zou

Received  November 15, 2017 Accepted  November 15, 2017 Published  March 2018

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.

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. Mathematical Biosciences & Engineering, 2018, 15 (4) : 1033-1054. doi: 10.3934/mbe.2018046
##### References:
 [1] R. M. Bacon, K. J. Kugeler, K. S. Griffith and P. S. Mead, Lyme disease -United States, 2003-2005, Journal of the American Medical Association, 298 (2007), 278-279. [2] A. G. Barbour and D. Fish, The biological and social phenomenon of Lyme disease, Science, 260 (1993), 1610-1616. doi: 10.1126/science.8503006. [3] R. J. Brinkerhoff, C. M. Folsom-O'Keefe, K. Tsao and M. A. Diuk-Wasser, Do birds affect Lyme disease risk? Range expansion of the vector-borne pathogen Borrelia burgdorferi, Front. Ecol. Environ, 9 (2011), 103-110. doi: 10.1890/090062. [4] S. G. Caraco, S. Glavanakov, G. Chen, J. E. Flaherty, T. K. Ohsumi and B. K. Szymanski, Stage-structured infection transmission and a spatial epidemic: a model for Lyme disease, Am. Nat., 160 (2002), 348-359. [5] M. R. Cortinas and U. Kitron, County-level surveillance of white-tailed deer infestation by Ixodes scapularis and Dermacentor albipictus (Acari: Ixodidae) along the Illinois River, J. Med. Entomol., 43 (2006), 810-819. [6] D. T. Dennis, T. S. Nekomoto, J. C. Victor, W. S. Paul and J. Piesman, Reported distribution of Ixodes scapularis and Ixodes pacificus (Acari: Ixodidae) in the United States, J. Med. Entomol., 35 (1998), 629-638. doi: 10.1093/jmedent/35.5.629. [7] G. Fan, H. R. Thieme and H. Zhu, Delay differential systems for tick population dynamics, J. Math. Biol., 71 (2015), 1017-1048. doi: 10.1007/s00285-014-0845-0. [8] S. A. Gourley and S. Ruan, A delay equation model for oviposition habitat selection by mosquitoes, J. Math. Biol., 65 (2012), 1125-1148. doi: 10.1007/s00285-011-0491-8. [9] B. H. Hahn, C. S. Jarnevich, A. J. Monaghan and R. J. Eisen, Modeling the Geographic Distribution of Ixodes scapularis and Ixodes pacificus (Acari: Ixodidae) in the Contiguous United States, Journal of Medical Entomology, 53 (2016), 1176-1191. [10] S. Hamer, G. Hickling, E. Walker and J. I. Tsao, Invasion of the Lyme disease vector Ixodes scapularis: Implications for Borrelia burgdorferi endemicity, EcoHealth, 7 (2010), 47-63. doi: 10.1007/s10393-010-0287-0. [11] X. Lai and X. Zou, Spreading speed and minimal traveling wave speed in a spatially nonlocal model for the population of blacklegged tick Ixodes scapularis, in preparation. [12] J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079. doi: 10.1007/s11538-009-9457-z. [13] D. Liang, J. W.-H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded fields and their numeric computations, Diff. Eqns. Dynam. Syst., 11 (2003), 117-139. [14] K. Liu, Y. Lou and J. Wu, Analysis of an age structured model for tick populations subject to seasonal effects, J. Diff. Eqns., 263 (2017), 2078-2112. doi: 10.1016/j.jde.2017.03.038. [15] N. K. Madhav, J. S. Brownstein, J. I. Tsao and D. Fish, A dispersal model for the range expansion of blacklegged tick (Acari: Ixodidae), J. Med. Entomol., 41 (2004), 842-852. doi: 10.1603/0022-2585-41.5.842. [16] M. G. Neubert and I. M. Parker, Projecting rates of spread for invasive species, Risk Analysis, 24 (2004), 817-831. doi: 10.1111/j.0272-4332.2004.00481.x. [17] N. H. Ogden, M. Bigras-Poulin, C. J. O'Callaghan, I. K. Barker, L. R. Lindsay, A. Maarouf, K. E. Smoyer-Tomic, D. Waltner-Toews and D. Charron, A dynamic population model to investigate effects of climate on geographic range and seasonality of the tick Ixodes scapularis, Int J. Parasitol., 35 (2005), 375-389. doi: 10.1016/j.ijpara.2004.12.013. [18] N. H. Ogden, L. R. Lindsay, K. Hanincova, I. K. Barker, M. Bigras-Poulin, D. F. Charron, A. Heagy, C. M. Francis, C. J. O'Callaghan, I. Schwartz and R. A. Thompson, Role of migratory birds in introduction and range expansion of Ixodes scapularis ticks and of Borrelia burgdorferi and Anaplasma phagocytophilum in Canada, Applied and Environmental Microbiology, 74 (2008), 1780-1790. [19] J. W.-H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure —I. Traveling wave fronts on unbounded domains, Proc. Royal Soc. London. A, 457 (2001), 1841-1853. doi: 10.1098/rspa.2001.0789. [20] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870. [21] J. Van Buskirk and R. S. Ostfeld, Controlling Lyme disease by modifying the density and species composition of tick hosts, Ecological Applications, 5 (1995), 1133-1140. doi: 10.2307/2269360. [22] H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145. [23] X. Wu, G. Röst and X. Zou, Impact of spring bird migration on the range expansion of Ixodes scapularis tick population, Bull. Math. Biol., 78 (2016), 138-168. doi: 10.1007/s11538-015-0133-1.

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##### References:
 [1] R. M. Bacon, K. J. Kugeler, K. S. Griffith and P. S. Mead, Lyme disease -United States, 2003-2005, Journal of the American Medical Association, 298 (2007), 278-279. [2] A. G. Barbour and D. Fish, The biological and social phenomenon of Lyme disease, Science, 260 (1993), 1610-1616. doi: 10.1126/science.8503006. [3] R. J. Brinkerhoff, C. M. Folsom-O'Keefe, K. Tsao and M. A. Diuk-Wasser, Do birds affect Lyme disease risk? Range expansion of the vector-borne pathogen Borrelia burgdorferi, Front. Ecol. Environ, 9 (2011), 103-110. doi: 10.1890/090062. [4] S. G. Caraco, S. Glavanakov, G. Chen, J. E. Flaherty, T. K. Ohsumi and B. K. Szymanski, Stage-structured infection transmission and a spatial epidemic: a model for Lyme disease, Am. Nat., 160 (2002), 348-359. [5] M. R. Cortinas and U. Kitron, County-level surveillance of white-tailed deer infestation by Ixodes scapularis and Dermacentor albipictus (Acari: Ixodidae) along the Illinois River, J. Med. Entomol., 43 (2006), 810-819. [6] D. T. Dennis, T. S. Nekomoto, J. C. Victor, W. S. Paul and J. Piesman, Reported distribution of Ixodes scapularis and Ixodes pacificus (Acari: Ixodidae) in the United States, J. Med. Entomol., 35 (1998), 629-638. doi: 10.1093/jmedent/35.5.629. [7] G. Fan, H. R. Thieme and H. Zhu, Delay differential systems for tick population dynamics, J. Math. Biol., 71 (2015), 1017-1048. doi: 10.1007/s00285-014-0845-0. [8] S. A. Gourley and S. Ruan, A delay equation model for oviposition habitat selection by mosquitoes, J. Math. Biol., 65 (2012), 1125-1148. doi: 10.1007/s00285-011-0491-8. [9] B. H. Hahn, C. S. Jarnevich, A. J. Monaghan and R. J. Eisen, Modeling the Geographic Distribution of Ixodes scapularis and Ixodes pacificus (Acari: Ixodidae) in the Contiguous United States, Journal of Medical Entomology, 53 (2016), 1176-1191. [10] S. Hamer, G. Hickling, E. Walker and J. I. Tsao, Invasion of the Lyme disease vector Ixodes scapularis: Implications for Borrelia burgdorferi endemicity, EcoHealth, 7 (2010), 47-63. doi: 10.1007/s10393-010-0287-0. [11] X. Lai and X. Zou, Spreading speed and minimal traveling wave speed in a spatially nonlocal model for the population of blacklegged tick Ixodes scapularis, in preparation. [12] J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latent period in a spatially continuous domain, Bull. Math. Biol., 71 (2009), 2048-2079. doi: 10.1007/s11538-009-9457-z. [13] D. Liang, J. W.-H. So, F. Zhang and X. Zou, Population dynamic models with nonlocal delay on bounded fields and their numeric computations, Diff. Eqns. Dynam. Syst., 11 (2003), 117-139. [14] K. Liu, Y. Lou and J. Wu, Analysis of an age structured model for tick populations subject to seasonal effects, J. Diff. Eqns., 263 (2017), 2078-2112. doi: 10.1016/j.jde.2017.03.038. [15] N. K. Madhav, J. S. Brownstein, J. I. Tsao and D. Fish, A dispersal model for the range expansion of blacklegged tick (Acari: Ixodidae), J. Med. Entomol., 41 (2004), 842-852. doi: 10.1603/0022-2585-41.5.842. [16] M. G. Neubert and I. M. Parker, Projecting rates of spread for invasive species, Risk Analysis, 24 (2004), 817-831. doi: 10.1111/j.0272-4332.2004.00481.x. [17] N. H. Ogden, M. Bigras-Poulin, C. J. O'Callaghan, I. K. Barker, L. R. Lindsay, A. Maarouf, K. E. Smoyer-Tomic, D. Waltner-Toews and D. Charron, A dynamic population model to investigate effects of climate on geographic range and seasonality of the tick Ixodes scapularis, Int J. Parasitol., 35 (2005), 375-389. doi: 10.1016/j.ijpara.2004.12.013. [18] N. H. Ogden, L. R. Lindsay, K. Hanincova, I. K. Barker, M. Bigras-Poulin, D. F. Charron, A. Heagy, C. M. Francis, C. J. O'Callaghan, I. Schwartz and R. A. Thompson, Role of migratory birds in introduction and range expansion of Ixodes scapularis ticks and of Borrelia burgdorferi and Anaplasma phagocytophilum in Canada, Applied and Environmental Microbiology, 74 (2008), 1780-1790. [19] J. W.-H. So, J. Wu and X. Zou, A reaction diffusion model for a single species with age structure —I. Traveling wave fronts on unbounded domains, Proc. Royal Soc. London. A, 457 (2001), 1841-1853. doi: 10.1098/rspa.2001.0789. [20] H. R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity, SIAM J. Appl. Math., 70 (2009), 188-211. doi: 10.1137/080732870. [21] J. Van Buskirk and R. S. Ostfeld, Controlling Lyme disease by modifying the density and species composition of tick hosts, Ecological Applications, 5 (1995), 1133-1140. doi: 10.2307/2269360. [22] H. F. Weinberger, M. A. Lewis and B. Li, Analysis of linear determinacy for spread in cooperative models, J. Math. Biol., 45 (2002), 183-218. doi: 10.1007/s002850200145. [23] X. Wu, G. Röst and X. Zou, Impact of spring bird migration on the range expansion of Ixodes scapularis tick population, Bull. Math. Biol., 78 (2016), 138-168. doi: 10.1007/s11538-015-0133-1.
The life-stage components of the model: questing larvae ($L$) find a host, feed and moult into questing nymphs ($N$), which then find a new host, feed and moult into questing adults ($A_q$). Adult females that find a deer host ($A_f$) feed, drop to the forest floor, lay $2000$ eggs and then die. Hatching eggs create the next generation of questing larvae. The $r$ parameters are the per-capita transition rates between each compartment
$H_1(\lambda, c)$ and $H_2(\lambda, c)$ for different $c$. (a) $c = 0.55$; (b) $c = c^* = 0.6176844021$, ($\lambda = 0.7081234538$); (c) $c = 0.7$. Here, the model parameters are taken as $b = 3000$, $r_1 = 0.13$, $r_2 = 0.13$, $r_3 = 0.03$, $r_4 = 0.03$, $d_1 = 0.3$, $d_2 = 0.3$, $d_3 = 0.1$, $d_4 = 0.1$, $\tau_1 = 20$, $\tau_2 = 10$, $N_{cap} = 5000$, $h = 100$ and $D = 1$
Dependence of $c^*$ on $b$ and $D$ respectively: (a) with $D = 1$; (b) with $b = 3000$. Other parameters are taken as: $r_1 = 0.13$, $r_2 = 0.13$, $r_3 = 0.03$, $r_4 = 0.03$, $d_1 = 0.3$, $d_2 = 0.3$, $d_3 = 0.1$, $d_4 = 0.1$, $\tau_1 = 20$, $\tau_2 = 10$, $N_{cap} = 5000$ and $h = 100$
There is no biologically relevant traveling wave front solution with speed $c = 0.1<c^* = 0.24$: $\phi_1$ may take negative values
There is a non-negative traveling wave front solution with speed $c = 0.4>c^* = 0.24$
(a): time evolution of $L(x, t)$; (b): time evolution of $N(x, t)$; (c): contours of (a) with region where $L(x, t)>0.1$ shown in grey; (d): contours of (b) with region where $N(x, t)>0.1$ shown in grey
(a): time evolution of $A_q(x, t)$; (b): time evolution of $A_f(x, t)$; (c): contours of (a) with region where $A_q(x, t)>0.1$ shown in grey; (d): contours of (b) with region where $A_f(x, t)>0.1$ shown in grey
Explanation of parameters
 Parameters Meaning Value $b$ Birth rate of tick $3000$ $1/r_1$ average time that a questing larvae needs to feed and moult $1/0.13$ $1/r_2$ average time that a questing nymph needs to feed and moult $1/0.13$ $1/r_3$ average time that a questing adult needs to successfully attach to a deer $1/0.03$ $r_4$ Proportion of fed adults that can lay eggs 0.03 $d_1$ per-capita death rate of larvae 0.3 $d_2$ per-capita death rate of nymphs 0.3 $d_3$ per-capita death rate of questing adults 0.1 $d_4$ per-capita death rate of fed adults 0.1 $\tau_1$ average time between last blood feeding and hatch of laid eggs 20 days $\tau_2$ average time tick is attached to a deer $10$ days
 Parameters Meaning Value $b$ Birth rate of tick $3000$ $1/r_1$ average time that a questing larvae needs to feed and moult $1/0.13$ $1/r_2$ average time that a questing nymph needs to feed and moult $1/0.13$ $1/r_3$ average time that a questing adult needs to successfully attach to a deer $1/0.03$ $r_4$ Proportion of fed adults that can lay eggs 0.03 $d_1$ per-capita death rate of larvae 0.3 $d_2$ per-capita death rate of nymphs 0.3 $d_3$ per-capita death rate of questing adults 0.1 $d_4$ per-capita death rate of fed adults 0.1 $\tau_1$ average time between last blood feeding and hatch of laid eggs 20 days $\tau_2$ average time tick is attached to a deer $10$ days
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