# 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
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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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