Disease-Induced Hydra Effect with Overcompensatory Recruitment

Abstract

In ecological systems, the hydra effect is an increase in population size caused by an increase in mortality. This seemingly counterintuitive effect has been observed in several populations, including fish, blowflies, snails and plants, and has been modeled in both continuous and discrete time. A similar effect induced by disease has recently been observed empirically. Here we present theoretical and simulation results for an infectious disease-induced hydra effect, namely conditions under which the total population size, composed of those that are infectious as well as those that are susceptible, at an endemic equilibrium is greater than the population size at the disease-free equilibrium. (For an endemic k-cycle, this can be similarly defined using the average population.) We find this disease-induced hydra effect occurs when the intra-specific competition is strong and disease infection sufficiently inhibits the reproductive output of infected individuals. For our continuous time model, we give a necessary and sufficient condition for a disease-induced hydra effect. This condition requires overcompensatory recruitment. With a discrete time model, we show there is no disease-induced hydra effect without overcompensatory recruitment. We illustrate by simulations that a disease-induced hydra effect may occur with Ricker recruitment when the endemic system converges to either a fixed equilibrium or a 2-cycle.

Appendix: Proof of Theorem 4

1.1 Existence of Endemic Equilibrium (EE)

From (11), the endemic equilibrium $(S^*, I^*)$ must satisfy

$$\begin{aligned} I^{*}=-\frac{1}{\beta }ln\frac{1-f(S^{*})}{1-d}>0. \end{aligned}$$

(14)

This requires that

$$\begin{aligned} S^*<f^{-1}(d)=S_{{\text {DFE}}}. \end{aligned}$$

(15)

Substituting (14) into (12) gives that $S^{*}$ must be a positive root of

$$\begin{aligned} \begin{aligned} L(S):=(d+\mu -d\mu )\frac{1}{\beta }ln\frac{1-f(S)}{1-d}-{\text{ d }}S+Sf(S)=0 \end{aligned} \end{aligned}$$

(16)

For the EE to exist, L(S) must be defined, and (14) must hold, and thus $d<f(S)<1$. In addition $S>0$. Let

$$\begin{aligned} \ell = {\left\{ \begin{array}{ll} 0, &{} r<1,\\ f^{-1}(1),&{} r\ge 1, \end{array}\right. } \end{aligned}$$

Then, the domain of L(S) is $[\ell , S_{{\text {DFE}}}]$.

To show the existence of an EE for ${\mathcal {R}}_0 >1$, we show that L(S) has opposite signs near the two endpoints of its domain. Firstly, note that $L(S_{{\text {DFE}}})=0$. Furthermore,

$$\begin{aligned} L'(S)=-(d+\mu -d\mu )\frac{1}{\beta }\frac{f'(S)}{(1-f(S))}-d+\frac{d}{{\text {d}}S}[Sf(S)]. \end{aligned}$$

Differentiating,

$$\begin{aligned} L'(S_{{\text {DFE}}})=-f'(S)f^{-1}(d)\left( \frac{1}{{\mathcal {R}}_0}-1\right) . \end{aligned}$$

Thus, if ${\mathcal {R}}_0> 1$, then $L'(S_{{\text {DFE}}})<0$. There then exists an $\varepsilon >0$ such that $L(S_{{\text {DFE}}}-\varepsilon )>0$. Secondly, $L(\ell )<0$. Thus, L(S) must have a positive root in the domain $(\ell , S_{{\text {DFE}}})$, i.e., there must exist an EE if ${\mathcal {R}}_0>1$.

1.2 Uniqueness of the Endemic Equilibrium

To show that the EE is unique, we show that $L''(S)<0$ for $S\le S_{{\text {DFE}}}$.

$$\begin{aligned} \begin{aligned} L''(S)=-(d+\mu -d\mu )\frac{1}{\beta }\frac{f''(S)(1-f(S))+[f'(S)]^{2}}{(1-f(S))^2}+[Sf(S)]'' .\end{aligned} \end{aligned}$$

(17)

Since $f''(S)>0$ and $f(S)<1$, the first term of $L''(S)$ is negative. The second term is $g''(S)<0$ where $g(S) = Sf(S)$. Thus, $L''(S)<0$, i.e., L(S) is concave down, and thus L(S) cannot have more than two roots. Since $S_{{\text {DFE}}}$ is already a root, this guarantees the uniqueness of the EE. $\square$