The influence of host competition and predation on tick densities and management implications

Abstract

Host community composition and biodiversity can limit and regulate tick abundance which can have profound impacts on the incidence and severity of tick-borne diseases. Our understanding of the relationship between host community composition and tick abundance is still very limited. Here, we present a novel mathematical model of a stage-structured tick population to study the influence of host behaviour and competition in the presence of heterospecifics and the influence of host predation on tick densities. We examine the influence of specific changes in biodiversity that modify the competition among and the predation on small and large host populations. We find that increasing biodiversity will not always reduce tick populations, but depends on changes in species composition affecting the degree and type competition among hosts, and the host the predation is acting on. With indirect competition, tick densities are not regulated by increasing biodiversity; however, with direct competition, increased biodiversity will regulate tick densities. Generally, we find that biodiversity will regulate tick densities when it affects tick-host encounter rates. We also find that predation on small hosts have a limited influence on reducing tick populations, but when the predation was on large hosts this increased the magnitude of tick population oscillations. Our results have tick-management implications: while controlling large host populations (e.g. deer) and adult ticks will decrease tick densities, measures that directly control the nymph ticks could also be effective.

Appendices

Appendix A: Global stability of the extinction equilibrium and nonexistence of periodic orbits

Consider a system of differential equations d x/d t = f(x), where $x=(x_{1},x_{2},x_{3})\in \mathbb {R}^{3}$ and x(t, x ₀) is a solution of the equations which satisfies x(0, x ₀) = x ₀. We use a generalisation, to higher dimensions, of a criteria of Bendixson for the non-existence of invariant closed curves such as periodic or homoclinic orbits. The theory was developed by Li and Muldowney (1993, 1996) and shows that oriented infinitesimal line segments, y(t, y ₀), evolve as solutions of

$$ \frac{dy}{dt}=\frac{\partial f}{\partial x}(x(t,x_{0}))y $$

(13)

and oriented infinitesimal areas, z(t, z ₀) evolve as solutions of

$$ \frac{dz}{dt}=\frac{\partial f}{\partial x}^{[2]}(x(t,x_{0}))z $$

(14)

where $\frac {\partial f}{\partial x}^{[2]}$ is the second additive compound matrix. For a general matrix A, the corresponding second additive compound matrix is given by A ^[2] as follows,

$$\begin{array}{@{}rcl@{}} A&=&\left[\begin{array}{ccc} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \end{array}\right],\\ A^{[2]}&=&\left[\begin{array}{ccc} a_{11}+a_{22} & a_{23} & -a_{13} \\ a_{32} & a_{11}+a_{33} & a_{12} \\ -a_{31} & a_{21} & a_{22}+a_{33} \end{array}\right]\;. \end{array} $$

(15)

Thus, for Eqs. (5)–(7), the second additive compound matrix is given by Eq. (16).

$$\phantom{r}\left(\frac{\partial f}{\partial x}\right)^{[2]}=\left(\begin{array}{ccc} -\mu_{1}-\mu_{2}-\frac{\gamma_{1}a_{1}}{(a_{1}+x_{1})^{2}}-\frac{\gamma_{2}a_{2}}{(a_{2}+x_{2})^{2}} & 0& -\frac{\alpha_{1}a_{3}}{(a_{3}+x_{3})^{2}} \\ \\ -\frac{\alpha_{3}a_{2}}{(a_{2}+x_{2})^{2}}&-\mu_{1}-\mu_{3}-\frac{\gamma_{1}a_{1}}{(a_{1}+x_{1})^{2}}-\frac{\gamma_{3}a_{3}}{(a_{3}+x_{3})^{2}}&0\\ \\ 0&\frac{\alpha_{2}a_{1}}{(a_{1}+x_{1})^{2}}&-\mu_{2}-\mu_{3}-\frac{\gamma_{2}a_{2}}{(a_{2}+x_{2})^{2}}-\frac{\gamma_{3}a_{3}}{(a_{3}+x_{3})^{2}} \end{array}\right)\;. $$

(16)

By Theorem 3.3 of Li and Muldowney (1993) if for each $x_{0}\in \mathbb {R}^{3}_{+}$ (13) and (14) are uniformly asymptotically stable then all line segments collapse to the origin and we have global stability of (0, 0, 0) and there exists no invariant closed curves (periodic orbits, homoclinic or heteroclinic cycles) and the orbits converge to a single equilibrium.

Asymptotic stability of (13) and (14) is shown by constructing Lyapunov functions. Using the Lyapunov function V(x ₁, x ₂, x ₃) = |x ₁| + |x ₂| + |x ₃| and together with (13), we have

$$\begin{array}{@{}rcl@{}} \dot V(y)&=&(1,1,1)\cdot \frac{\partial f}{\partial x}=-\mu_{1}+\frac{a_{1}(\alpha_{2}-\gamma_{1})}{(a_{1}+x_{1})^{2}}\\ &-&\mu_{2}+\frac{a_{2} (\alpha_{3}-\gamma_{2})}{(a_{2}+x_{2})^{2}}-\mu_{3}+\frac{a_{3}(\alpha_{1}-\gamma_{3})}{(a_{3}+x_{3})^{2}} \end{array} $$

If $\dot V(y)<0$, we have global stability of the zero solution of (13). Since γ ₁ ≥ α ₂ and γ ₂ ≥ α ₃, then a sufficient condition for $\dot V(y)<0$ is μ ₃ > (α ₁ − γ ₃)/a ₃, condition (A) in Table 3. Showing that $\dot V(y) = (1,1,1)\cdot \left (\frac {\partial f}{\partial x}\right )^{[2]}<0$ guarantees asymptotic stability of (14) and gives condition (C).

Table 3 Analytical criteria for tick eradication (A, B) and the absence of tick cycles (C, D) (see Appendix A for details of the derivations). Note that (A) and (B) are alternative criteria, only one of these needs to be satisfied and similarly for (C) and (D)

Alternatively, using the Lyapunov function V(x ₁, x ₂, x ₃) = sup{|x ₁|,|x ₂|, |x ₃|} gives stronger results (conditions B and D in Table 3).

Appendix B: $\mathcal {R}_{0}$ and tick-borne disease dynamics

While ticks can feed on a variety of hosts, it is commonly believed that pathogens are associated with a particular host that acts as a disease reservoir that maintains the pathogen in the environment (Randolph 2004). For instance, the spirochete Borrelia burgdorferi s.l. is maintained mainly in deer mice: the spirochete is transferred to the tick when it feeds on an infected deer mouse; after which, the infected tick can transfer the disease to a human, causing Lyme disease, or to another deer mouse—thus maintaining the disease in the environment. If the tick feeds on an alternate small or large host that is not a disease reservoir (e.g., pocket mice, rabbits, humans), the pathogen will either be eliminated by the immune system, or lead to the death of the host, or not be transferred to another host, in all cases effectively acting as a dead end that removes the pathogen from the environment.

Larval ticks typically hatch free from infection and can acquire infection through a blood meal with an infected small host, at which point they molt to become infected nymphs. So larval ticks cannot transmit the disease. Infected nymphs can transmit the infection to the hosts they feed upon and the infection remains in the ticks when they molt to the adult stage. Adopting the approach of Lou and Wu (2014) we can extend our model in a simple way to capture the disease dynamics of Lyme disease by describing the disease status of the individuals in our model. The rate of change of infected small H ₁ hosts ${H_{1}^{I}}(t)$, infected nymphs ${x_{2}^{I}}(t)$ and infected adult ticks ${x_{3}^{I}}(t)$ are given by Eqs. (19)–(20).

$$\begin{array}{@{}rcl@{}} \dot{{H_{1}^{I}}}&=&-\overbrace{\mu_{H_{1}} {H_{1}^{I}}}^{\text{\footnotesize{death}}}+\overbrace{\beta_{H}\frac{H_{1}-{H_{1}^{I}}}{H_{1}}\frac{\gamma_{2}^{\prime} {x_{2}^{I}}}{a_{2}+x_{2}}}^{\stackrel{\text{\footnotesize{infected nymphs transmitting}}}{\text{\footnotesize{disease to healthy hosts}}}}, \end{array} $$

(17)

$$\begin{array}{@{}rcl@{}} \dot{x}_{2}^{I}&=&-\overbrace{\mu_{2} {x_{2}^{I}}}^{\text{\footnotesize{death}}}+ \overbrace{\beta_{L}\frac{{H_{1}^{I}}}{H_{1}}\frac{\alpha_{2}^{\prime} x_{1}}{a_{1}+x_{1}}}^{\stackrel{\text{\footnotesize{larvae feeding on infected hosts}}}{\text{\footnotesize{molting to become infected nymph}}}} -\overbrace{\frac{\gamma_{2}^{\prime} {x_{2}^{I}}}{a_{2}+x_{2}}}^{\stackrel{\text{\footnotesize{infected nymphs molting}}}{\text{\footnotesize{to become infected adults}}}}, \end{array} $$

(18)

$$\begin{array}{@{}rcl@{}} \dot{{x_{3}^{I}}}&=&-\overbrace{\mu_{3} {x_{3}^{I}}}^{\text{\footnotesize{death}}}+\overbrace{\frac{\alpha_{3}^{\prime} (x_{2}-{x_{2}^{I}})}{a_{2}+x_{2}}}^{\stackrel{\text{\footnotesize{infected nymphs molting}}}{\text{\footnotesize{to become infected adults}}}} +\overbrace{\beta_{N}\frac{{H_{1}^{I}}}{H_{1}}\frac{\alpha_{3}^{\prime} (x_{2}-{x_{2}^{I}})}{a_{2}+x_{2}}}^{\stackrel{\text{\footnotesize{uninfected nymph feeding on infected hosts}}}{\text{\footnotesize{and molting to become infected adults}}}} -\overbrace{\frac{\gamma_{3} {x_{3}^{I}}}{a_{3}+x_{3}}}^{\text{\footnotesize{adults taking final blood meal}}}\;. \end{array} $$

(19)

We do not track infected large hosts as they can only transmit the infection to adult ticks which cannot pass the infection onto their offspring, so the large hosts are not acting as a reservoir for the disease the way that the small hosts are. We assume only the H ₁ small hosts (e.g. deer mice) are a competent reservoir for the disease and that the H ₂ small hosts are not (Ostfeld and Keesing 2000). β _H , β _L and β _N are the transmission coefficients of the infection to H ₁ hosts, larval ticks and nymphal ticks, respectively. $\gamma _{i}^{\prime }$ is the contribution to γ _i that comes from feeding on H ₁ hosts only similarly for $\alpha _{i}^{\prime }$. For example, $\gamma _{2}^{\prime }=\sigma _{1}(p_{s})H_{1}\lambda _{2,1}$. Assuming the tick population are at equilibrium then, we can study the disease dynamics in isolation replacing x ₁(t) and x ₂(t) by their equilibrium values $x_{1}^{*}$ and $x_{2}^{*}$ and noting that the equation for infected adult ticks decouples. Hence, two equations form the epidemiological model,

$$\begin{array}{@{}rcl@{}} \dot{{H_{1}^{I}}}=&-\mu_{H_{1}} {H_{1}^{I}}+\beta_{H}\frac{H_{1}-{H_{1}^{I}}}{H_{1}}\frac{\gamma_{2}^{\prime} {x_{2}^{I}}}{a_{2}+x_{2}^{*}}, \end{array} $$

(20)

$$\begin{array}{@{}rcl@{}} \dot{x}_{2}^{I}=&-\mu_{2} {x_{2}^{I}}+ \beta_{L}\frac{{H_{1}^{I}}}{H_{1}}\frac{\alpha_{2}^{\prime} x_{1}^{*}}{a_{1}+x_{1}^{*}} -\frac{\gamma_{2}^{\prime} {x_{2}^{I}}}{a_{2}+x_{2}^{*}}. \end{array} $$

(21)

We can calculate the basic reproduction number for the disease using the next generation matrix method (see Van den Driessche and Watmough 2002). The transmission matrix and transition matrix are given by

$$ F= \left(\begin{array}{cc} 0 & \frac{\beta_{H}\gamma_{2}^{\prime}}{a_{2}+x_{2}^{*}} \\ \frac{\beta_{L}\alpha_{2}^{\prime}x_{1}^{*}}{H_{1}(a_{1}+x_{1}^{*})} & 0 \end{array}\right) \;\; \text{and}\;\; V =\left(\begin{array}{cc} \mu_{H_{1}} & 0 \\ 0 & \mu_{2}+\frac{\gamma_{2}^{\prime}}{a_{2}+x^{*}_{2}} \end{array}\right) $$

(22)

respectively. Together these yield the next generation matrix

$$ FV^{-1}= \left(\begin{array}{cc} 0 & \frac{\beta_{H}\gamma_{2}^{\prime}}{\mu_{2}(a_{2}+x_{2}^{*})+\gamma_{2}^{\prime}} \\ \frac{\beta_{L}\alpha_{2}^{\prime}x_{1}^{*}}{H_{1}(a_{1}+x_{1}^{*})\mu_{H_{1}}} & 0 \end{array}\right), $$

(23)

the dominant eigenvalue of which gives the basic reproduction number $\mathcal {R}_{0}$ for the disease.

$$\begin{array}{@{}rcl@{}} \mathcal{R}_{0}&=&\sqrt{\frac{\beta_{H}\gamma_{2}^{\prime}}{\mu_{2}(a_{2}+x_{2}^{*})+\gamma_{2}^{\prime}} \frac{\beta_{L}\alpha_{2}^{\prime}x_{1}^{*}}{H_{1}(a_{1}+x_{1}^{*})\mu_{H_{1}}}}\\ &=&\sqrt{\frac{\beta_{H}\gamma_{2}^{\prime}}{\mu_{2}(a_{2}+x_{2}^{*})+\gamma_{2}^{\prime}} \frac{\beta_{L}\alpha_{2}^{\prime}}{H_{1}\alpha_{2}\mu_{H_{1}}}\left(\mu_{2}+\frac{\gamma_{2}}{a_{2}+x_{2}^{*}}\right)x_{2}^{*}} \end{array} $$

(24)

The unique endemic equilibrium is

$$\begin{array}{@{}rcl@{}} H_{1}^{I*}=H_{1}\left(1-\frac{1}{\mathcal{R}_{0}^{2}}\right) \end{array} $$

(25)

$$\begin{array}{@{}rcl@{}} x_{2}^{I*}= \beta_{L}\frac{\alpha_{2}^{\prime}}{\alpha_{2}}x_{2}^{*}\left(1+\frac{\gamma_{2}-\gamma_{2}^{\prime}}{\gamma_{2}^{\prime}+(a_{2}+x_{2}^{*})\mu_{2}}\right) \left(1-\frac{1}{\mathcal{R}_{0}^{2}}\right) \end{array} $$

(26)

Applying Theorem 2.1 from Lou and Jianhong (2014) shows that $\mathcal {R}_{0}$ determines the global stability of the endemic equilibrium. Specifically, if $\mathcal {R}_{0}>1$, the endemic equilibrium is globally asymptotically stable.