Do fatal infectious diseases eradicate host species?

Abstract

In simple SI epidemic and endemic models, three classes of incidence functions are identified for their potential to be associated with host extinction: weakly upper density-dependent incidences are never associated with host extinction. Power incidences that depend on the number of susceptibles and infectives by powers strictly between 0 and 1 are associated with initial-constellation-dependent host extinction for all parameter values. Homogeneous incidences, of which frequency-dependent incidence is a very particular case, and power incidences are associated with global host extinction for certain parameter constellations and with host survival for others. Laboratory infection experiments with salamander larvae are equally well fitted by power incidences and certain upper density-dependent incidences such as the negative binomial incidence and do not rule out homogeneous incidences such as an asymmetric frequency-dependent incidence either.

Appendix: persistence without uniqueness

Since the systems (1.1) and (7.7) may not have unique solutions if $I(0)=0$ or $r(0)=0$ respectively, we prove a persistence result which does not require uniqueness of solutions and is general enough to apply to both systems.

Let $\emptyset \ne D \subseteq {\mathbb {R}}^N$, $N \in {\mathbb {N}}$, and $F:D \rightarrow {\mathbb {R}}^N$ be continuous.

Assume that, for any $x^\circ \in D$, there exists a solution $x:{\mathbb {R}}_+ \rightarrow D$ of $x'=F(x)$ on ${\mathbb {R}}_+$, $x(0) = x^\circ$ (which does not need to be unique).

Let $\rho : D \rightarrow {\mathbb {R}}_+$ be continuous. The vector field F is called uniformly $\rho$- persistent if there exists some $\epsilon > 0$ such that $\liminf _{t\rightarrow \infty } \rho (x(t)) \ge \epsilon$ for all solutions $x:{\mathbb {R}}_+ \rightarrow D$ of $x'=F(x)$ on ${\mathbb {R}}_+$ with $\rho (x(0)) > 0$.

The vector field F is called uniformly weakly $\rho$-persistent if the same statement holds with the limit inferior being replaced by the limit superior.

For concrete ODE systems, it may be more convenient to say that the system $x'=F(x)$ rather than the vector field F is uniformly (weakly) $\rho$-persistent.

Obviously, uniform $\rho$-persistence implies uniform weak $\rho$-persistence. We explore conditions under which the converse holds. Since we do not assume that solutions to $x'=F(x)$ are uniquely determined by their initial conditions, the results in Smith and Thieme (2011) and Zhao (2003) cannot be directly applied though the proofs can be adapted.

(CA) There exists a closed subset B of D such that the following hold:

• (i) For any $0< \epsilon _2 < \infty$, the set $\{x \in B; \rho (x) \le \epsilon _2 \}$ is compact.

• (ii) For any solution $x:{\mathbb {R}}_+\rightarrow D$ to $x'=F(x)$ on ${\mathbb {R}}_+$ with $\rho (x(0))>0$, there exists some $r > 0$ such that $x(t) \in B$ for all $t\ge r$.

• (iii) There exists no $t \in (0,\infty )$, $s\in (0,t)$ and no solution $x:[0,t] \rightarrow B$ of $x'=F(x)$ on [0, t] such that $\rho (x(0)) > 0$, $\rho (x(s)) =0$ and $\rho (x(t))> 0$.

Theorem 9.1

Assume that F maps bounded subsets of D into bounded subsets of ${\mathbb {R}}^N$ and that the assumption (CA) is satisfied. Then F is uniformly $\rho$-persistent if it is weakly uniformly $\rho$-persistent.

Proof

Suppose that F is weakly uniformly $\rho$-persistent but not uniformly $\rho$-persistent. Then there exists some $\epsilon _0 > 0$ such that

$$\begin{aligned} \limsup _{t\rightarrow \infty } \rho (x(t)) > \epsilon _0 \end{aligned}$$

(9.1)

for all solutions $x: {\mathbb {R}}_+ \rightarrow D$ to $x'=F(x)$ on ${\mathbb {R}}_+$ with $\rho (x(0)) > 0$.

Further, there exists a sequence $(\epsilon _n)$ in $(0,\epsilon _0/2)$, $\epsilon _n \rightarrow 0$, and a sequence $(x_n)$ of solutions $x_n: {\mathbb {R}}_+ \rightarrow D$ of $x_n'=F(x_n)$ on ${\mathbb {R}}_+$ such that, for all $n \in {\mathbb {N}}$, $\rho (x_n(0)) > 0$ and $\limsup _{t\rightarrow \infty } \rho ( x_n(t)) > \epsilon _0$ and $\liminf _{t\rightarrow \infty } \rho (x_n(t)) < \epsilon _n$.

Then there exist sequences $(r_n)$, $(s_n)$, $t_n)$, $(u_n)$ in $(0,\infty )$ such that, for all $n \in {\mathbb {N}}$, $r_n > n$ and

$$\begin{aligned} \begin{aligned}&\rho (x_n(r_n)) = \epsilon _0, \quad \rho (x_n(r_n+ s_n)) = \epsilon _n, \quad \rho (x_n(r_n + s_n + t_n)) =\epsilon _0/2, \\&\rho (x_n(r_n + s_n +t_n + u_n)) = \epsilon _0, \\&\rho (x_n(s)) < \epsilon _0, \quad s \in (r_n, r_n + s_n +t_n + u_n), \\&x_n(t) \in B, \quad t \ge r_n . \end{aligned} \end{aligned}$$

Set $y_n:{\mathbb {R}}_+ \rightarrow D$ by $y_n(t) = x_n(r_n + t)$ for $t \ge 0$. Then $y_n' = F(y_n)$ on ${\mathbb {R}}_+$,

$$\begin{aligned} \begin{aligned}&\rho (y_n(0)) = \epsilon _0, \quad \rho (y_n(s_n)) = \epsilon _n, \quad \rho (y_n(s_n+ t_n)) =\epsilon _0/2, \\&\rho (y_n(s_n +t_n + u_n)) = \epsilon _0, \\&\rho (y_n(s)) < \epsilon _0, \quad s \in (0, s_n +t_n + u_n), \\&y_n(t) \in B, \quad t \ge 0. \end{aligned} \end{aligned}$$

(9.2)

Since the set $B_0=\{x \in B; \rho (x) \le \epsilon _0\}$ is compact by (CA) (i), there is some $c > 0$ such that $\Vert y_n(t) \Vert \le c$ for all $n \in {\mathbb {N}}$ and $t \in [0,s_n +t_n + u_n]$. Since F maps bounded subsets of D into bounded subsets of ${\mathbb {R}}^N$, there is some $\tilde{c}>0$ such that $\Vert y_n'(t)\Vert = \Vert F(y_n(t))\Vert \le \tilde{c}$ for all $n \in {\mathbb {N}}$ and $t \in [0, s_n +t_n + u_n]$. Since $\rho$ is uniformly continuous on the compact set $B_0$, (9.2) implies that the sequence $(u_n)$ is bounded away from 0.

We claim that $s_n + t_n + u_n \rightarrow \infty$ as $n \rightarrow \infty$.

If not, after choosing subsequences, $s_n \rightarrow s$, $t_n \rightarrow t$ and $u_n \rightarrow u$ with $s , t, u \in (0,\infty )$, $u > 0$. By the Arzela-Ascoli theorem, $y_n \rightarrow y$ uniformly on $[0,s +t]$ with some continuous function $y:[0,s+t ] \rightarrow B$ which solves $y' =F (y)$ on $[0,s+t]$ and satisfies $\rho (y(0)) >0$, $\rho (y(s)) =0$, and $\rho (y(s+t)) >0$ by taking limits as $n\rightarrow \infty$ in (9.2).

This contradicts (CA).

So $s_n + t_n + u_n \rightarrow \infty$ as $n \rightarrow \infty$. By the Arzela-Ascoli theorem, $y_n \rightarrow y$ uniformly on all bounded subsets of ${\mathbb {R}}_+$ with some continuous function $y:{\mathbb {R}}_+ \rightarrow B$ which solves $y' =F (y)$ on ${\mathbb {R}}_+$ and satisfies $\rho (y(0)) >0$ and $\rho (y(t))\le \epsilon _0$ for all $t \ge 0$.

This is a contradiction to (9.1). $\square$