The Impact of Selective Predation on Host–Parasite SIS Dynamics

Abstract

While models of host–parasite interactions are widespread in the theoretical literature, we still have limited understanding of the impact of community dynamics on infectious disease dynamics. When the wider host ecology is taken into account, the underlying inter-species feedbacks can lead to counter-intuitive results. For example, the ‘healthy herd’ hypothesis posits that the removal of a predator species may not be beneficial for a prey population infected by an endemic disease. In this work, we focus on the effects of including a predator species in a susceptible–infected–susceptible model. Specifically, a key role is played by predator selectivity for either healthy or infected prey. We explored both cases and found important differences in the asymptotic behaviours of the system. Independently from selectivity, large portions of parameter space allow for the coexistence of the three species. However, when predators feed mainly on susceptible prey we find that a fold bifurcation can occur, leading to a region of bi-stability between coexistence and parasite extinction. Conversely, when predator selection is strongly towards infected prey, total prey population density can be maximal when the three species coexist, consistent with the ‘healthy herd’ hypothesis. Our work further highlights the importance of community interactions to infectious disease dynamics.

Appendices

Appendix A: Global Stability in the Host–Parasite System

It is possible to prove the global stability of $E_2$ by adapting the Lyapunov function commonly used for Lotka–Volterra systems (Takeuchi 1996). Explicitly,

$$\begin{aligned} V(X,Y)=X-{X_2}-{X_2}\ln \frac{X}{{X_2}}+\left( 1-\frac{\gamma }{\varGamma }+\frac{q}{\beta }\right) \left( Y-{Y_2}-{Y_2}\ln \frac{Y}{{Y_2}}\right) \end{aligned}$$

is well defined when $R_0>1$ and its derivative along the trajectories in the interior of ${\mathbb {R}}_2^+$ (Teschl 2012) is

$$\begin{aligned} \frac{\mathrm{d}V}{\mathrm{d}t}(X(t),Y(t))&=\left( X-{X_2}\right) \left[ a-b-q\left( X+Y\right) -\beta {Y}+\gamma \frac{Y}{X}\right] \\&\quad +\,\left( 1-\frac{\gamma }{\varGamma }+\frac{q}{\beta }\right) \left( Y-{Y_2}\right) (\beta {X}-\varGamma )\\&=-q\left( X-{X_2}\right) ^2+\frac{\gamma }{X{X_2}}\left( Y{X_2}-{Y_2}{X}+XY-XY\right) (X-{X_2})\\&\quad -\frac{\gamma }{{X_2}}\left( Y-{Y_2}\right) (X-{X_2})=-\left( q+\frac{\gamma {Y}}{X{X_2}}\right) \left( X-{X_2}\right) ^2\le 0. \end{aligned}$$

The only invariant set on the line $X={X_2}$ is $E_2$; therefore, by LaSalle’s invariant principle, all the trajectories in the interior of ${\mathbb {R}}_2^+$ converge to $E_2$.

Appendix B: Internal Equilibria and Stability Regions

Clearly, there is an evident connection between the regions created by the combinations of $R_{0I}$, $R_{0P}$, $R_{I}$ and $R_{P}$ and the number of solutions of (2). It is a bit less immediate to see where the relation lies analytically. To undercover it, we firstly name the left-hand side of (2) as $\pi (x)$. $\pi (x)$ is an downward-opening parabola when $\phi <1/2$ and an upward-opening one otherwise. Then, we notice that condition (4) corresponds to the requirement that the roots of $\pi (x)$ belong to the interval between ${X_2}$ and ${X_3}$. When $\pi (x)$ is evaluated at ${X_2}$, it can be re-written as

$$\begin{aligned} \pi ({X_2})={X_3}\left( {X_2}+\alpha +b\right) \left( R_P-1\right) , \end{aligned}$$

thus, it is positive if $R_P>1$, while

$$\begin{aligned} \pi ({X_3})&={X_3}\left[ \varPhi {q}\left( {X_1}-{X_3}\right) +\beta \left( {X_2}-{X_3}\right) \right] \\&={X_3}\left[ \varPhi {q}\left( {X_1}-{X_3}\right) +\beta {X_2}\right] \left( 1-R_I\right) . \end{aligned}$$

Now, it is easier to see that in regions 2 and 5, ${E_4}$ is positive. More specifically, in region 2 the parabola is downward-opening since $\varPhi >1$. In fact,

$$\begin{aligned} R_P=\frac{{X_{2}}+\varPhi {Y}_I}{{X_{3}}}>1\Rightarrow \varPhi >\frac{{X_{3}}-{X_{2}}}{{Y_{2}}} \end{aligned}$$

and

$$\begin{aligned} \frac{{X_{3}}-{X_{2}}}{{Y_{2}}}&=\frac{{X_{3}}-{X_{2}}}{{X_1}-{X_{2}}}\frac{q+\beta \left( 1-\frac{\gamma }{\varGamma }\right) }{q}\\&\ge \frac{{X_{3}}-{X_{2}}}{{X_1}-{X_{2}}}=\frac{{X_{3}}-{X_1}+{X_1}-{X_{2}}}{{X_1}-{X_{2}}}\\&=\frac{{X_{3}}(1-R_{0P})}{{X_{2}}(R_{0I}-1)}+1\ge 1 \end{aligned}$$

because $R_{0P}<1$ and $R_{0I}>1$. In this region, $\pi ({X_2})$ is positive and $\pi ({X_3})$ is negative; thus, ${X_4}$ is in the interval of condition (4) (see Fig. 6a) and ${E_4}$ has all positive coordinates. Similarly, this happens in region 5 since $\pi (x)$ still assumes values of opposite sign at the border and ${X_4}$ remains in the interval. With the same reasoning, it can be proven the existence of ${E_5}$ in region 6 as shown in Fig. 6b. Notice that in region 3, $\pi (x)$ is upward-opening with positive values at the extremes; therefore, additional conditions are required to discern when (2) has two (Fig. 6c), one or zero solutions (Fig. 6d).

Fig. 6

Plots of $\pi (x)$ in the different stability regions. The grey regions mark the interval outside condition (4). In (a) $c=1$ and $\phi =0.2$, in (b) $c=1$ and $\phi =0.83$, in (c) $c=0.87$ and $\phi =0.7$, and in (d) $c=0.9$ and $\phi =0.7$

Appendix C: Direction of the Transcritical Bifurcations

We analyse first the transcritical bifurcation occurring between $E_2$ and an internal equilibrium when $R_P=1$ taking c as bifurcation parameter. The value of c is assumed fixed such that $R_P=1$. In order to distinguish between a forward or a backward bifurcation, we need to compute the quantities

$$\begin{aligned} {\tilde{a}}&=\sum _{k,i,j=1}^nv_kw_iw_j\frac{\partial ^2{f}_k}{\partial {x}_i\partial {x}_j}\left( {X_2},{Y_2},0\right) \\ {\tilde{b}}&=\sum _{k,i=1}^nv_kw_i\frac{\partial ^2{f}_k}{\partial {x}_i\partial {c}}\left( {X_2},{Y_2},0\right) \end{aligned}$$

where v and w are, respectively, the left and right eigenvectors corresponding to the zero eigenvalue of ${J\left( X_2,Y_2,0\right) }$, f represents the right-hand side of (1) and x are the system coordinates. We added a tilde to the original paper notation to not confuse ${\tilde{a}}$ and ${\tilde{b}}$ with the prey birth and death parameters. In our system,

$$\begin{aligned} v=\left[ \begin{array}{c} 0 \\ 0\\ \frac{1}{w_3} \end{array} \right] , \quad {w=} \left[ \begin{array}{c} \frac{c{(1-\phi })}{\beta }w_3\\ \frac{c}{\gamma -(q+\beta ){X_2}} \left[ \phi {X_2}+\left( \gamma \frac{{X_2}}{{Y_2}}+q{X_2}\right) \frac{{(1-\phi })}{\beta }\right] w_3\\ {w_3} \end{array} \right] \end{aligned}$$

and $w_3$ can be chosen arbitrarily, e.g., ${w_3=1}$. It follows that

$$\begin{aligned} {\tilde{b}}&=v_3w_3\theta {c}{Y_2}=\theta {\left[ \phi {X_I}+(1-\phi )Y_2\right] }>0\\ {\tilde{a}}&=2v_3\theta {c}[w_1w_3+\phi {w}_2w_3]=2\theta {c}(w_1+\phi {w}_2)\\&=\frac{c{(1-\phi })}{\beta {{X}_2}}\frac{{(1-\phi )(\gamma {Y_2}+qX_2^2)+\phi {X}_2(\gamma -qX_2)}}{\gamma -(q+\beta ){X_2}}. \end{aligned}$$

Thus, according with Castillo-Chávez and Song (2004) the transcritical bifurcation is forward (as in Fig. 2a–c) on the curve $R_P=1$, when ${\tilde{a}}<0$, i.e.,

$$\begin{aligned} \phi <{\frac{qX^2_2+\gamma {Y}_2}{2qX_2^2+\gamma ({Y}_2-X_2)}}={\bar{\phi }}, \end{aligned}$$

and backward (Fig. 2d) for $\phi >{\bar{\phi }}$.

Similarly, on the curve $R_I=1$, ${\tilde{b}}<0$ and the transcritical bifurcation is backward for decreasing c up to

$$\begin{aligned} {\bar{\varPhi }}={\frac{-(\varGamma +\gamma -qX_1)+\sqrt{(\varGamma +\gamma -qX_1)^2-4X_1(\gamma \beta -q\varGamma )}}{2qX_1}}, \end{aligned}$$

and forward for decreasing c for higher values of $\varPhi$ (lower values of $\phi$).