Evolutionary suicide through a non-catastrophic bifurcation: adaptive dynamics of pathogens with frequency-dependent transmission

Abstract

Evolutionary suicide is a riveting phenomenon in which adaptive evolution drives a viable population to extinction. Gyllenberg and Parvinen (Bull Math Biol 63(5):981–993, 2001) showed that, in a wide class of deterministic population models, a discontinuous transition to extinction is a necessary condition for evolutionary suicide. An implicit assumption of their proof is that the invasion fitness of a rare strategy is well-defined also in the extinction state of the population. Epidemic models with frequency-dependent incidence, which are often used to model the spread of sexually transmitted infections or the dynamics of infectious diseases within herds, violate this assumption. In these models, evolutionary suicide can occur through a non-catastrophic bifurcation whereby pathogen adaptation leads to a continuous decline of host (and consequently pathogen) population size to zero. Evolutionary suicide of pathogens with frequency-dependent transmission can occur in two ways, with pathogen strains evolving either higher or lower virulence.

Appendices

Appendix 1: General model

Here we consider a rather general family of SI models given by

$$\begin{aligned} \frac{dS}{dt}= & {} b_S({\tilde{\mathbf{E}}}(S,I,z),t)S + b_I({\tilde{\mathbf{E}}}(S,I,z),z,t)I -\mu ({\tilde{\mathbf{E}}}(S,I,z),t) S - \beta (z) \frac{SI}{N} \nonumber \\ \frac{dI}{dt}= & {} \beta (z) \frac{SI}{N}-\mu ({\tilde{\mathbf{E}}}(S,I,z),t)I-\alpha ({\tilde{\mathbf{E}}}(S,I,z),z,t)I. \end{aligned}$$

(25)

As in the main text, S and I are the density of susceptible and infected hosts, respectively, and $N=S+I$ is the total host population density. z is a trait that characterizes the infecting strain of the pathogen (e.g. its intra-host proliferation rate, see e.g. Boldin and Diekmann 2008). For the population dynamics of a given strain in (25), z is the parameter of interest for the bifurcation through which extinction happens. For the adaptive dynamics, z is the evolving trait. We assume that z determines the transmission rate $\beta (z)$ and also influences other demographic parameters (see below). In the models considered in the main text, we assumed that $\alpha$, the virulence, is a pathogen-specific constant (i.e., independent of ${\tilde{\mathbf{E}}}$ and time) and took $z=\alpha$.

The vector ${\tilde{\mathbf{E}}}$ contains environmental feedback variables, such as the densities of available resources, which depend on the number of hosts who exploit these resources. Susceptible and infected hosts may exploit the resources differently (for example, infected hosts may be less efficient foragers), and the exploitation of infected hosts may depend on the trait value z of the infecting strain (for example, the more the strain damages the hosts, the less efficiently they forage), so that ${\tilde{\mathbf{E}}}$ depends on S, I and z. The per capita birth rates of susceptibles and infecteds, $b_S$ and $b_I$, respectively, depend on the environmental feedback variables and, in the case of $b_I$, also on the infecting strain; with this latter assumption, we allow for a partially sterilizing pathogen whose degree of sterilization depends on its trait value z. Similarly, the background mortality rate ($\mu$) and the virulence ($\alpha$) depend on the environmental feedback variables, and, in the case of $\alpha$, on the infecting strain. Finally, all birth and death rates may depend explicitly on time, i.e., they may be affected by external factors.

This model subsumes a wide variety of ecological assumptions (how the demographic rates depend on population density via resources etc.) and also a variety of possible effects of the pathogen on its host’s demography, but retains the crucial assumption of frequency-dependent pathogen transmission. The model of Boots and Sasaki (2003) is a special case of (25) with $b_S({\tilde{\mathbf{E}}}(S,I,z),t)=b-h(S+I)$, $\mu ({\tilde{\mathbf{E}}}(S,I,z),t)=u$, $\alpha ({\tilde{\mathbf{E}}}(S,I,z),z,t)=\bar{\alpha }$ (where b, h, u and $\bar{\alpha }$ are positive numbers), and either $b_I({\tilde{\mathbf{E}}}(S,I,z),z,t)=0$ (the disease is fully sterilizing) or $b_I({\tilde{\mathbf{E}}}(S,I,z),z,t)=b-h(S+I)$ (the infected hosts have the same birth rate as the susceptibles).

As in the main text, we rewrite the system in terms of the total population density N and the fraction of susceptible hosts $x=\frac{S}{N}$ as

$$\begin{aligned} \frac{dN}{dt}= & {} N \left[ x b_S(\mathbf {E},t) + (1-x) b_I(\mathbf {E},z,t) -\mu (\mathbf {E},t) - (1-x) \alpha (\mathbf {E},z,t) \right] \nonumber \\ \frac{dx}{dt}= & {} (1-x) \left[ x b_S(\mathbf {E},t) + (1-x) b_I(\mathbf {E},z,t)-x \beta (z) + x \alpha (\mathbf {E},z,t) \right] \end{aligned}$$

(26)

where $\mathbf {E}=\mathbf {E}(N,x,z)={\tilde{\mathbf{E}}}(xN,(1-x)N,z)$. The invasion fitness of a mutant strain $z_\mathrm{mut}$ is given by

$$\begin{aligned} \rho (z_\mathrm{mut},z)=\beta (z_\mathrm{mut}) \langle x \rangle - \langle \mu (\mathbf {E}(N,x,z),t) \rangle - \langle \alpha (\mathbf {E}(N,x,z),z_\mathrm{mut},t) \rangle \end{aligned}$$

(27)

where the angle brackets $\langle \cdot \rangle$ denote the time-averages on the ecological timescale of Eq. (26) (Metz et al. 1992); we assume that these expectations exist. If the resident dynamics in (26) are autonomous (i.e., if the birth and death rates do not depend explicitly on time) and the system attains an equilibrium, then the time-averages reduce to the values at the resident equilibrium. In this case, the environmental feedback variables that determine the invasion fitness of a given mutant are the fraction of susceptibles, x, and the elements of $\mathbf {E}$ at the resident equilibrium.

The adaptive dynamics of the pathogen trait z is governed by the selection gradient

$$\begin{aligned} D(z)&= [\partial \rho (z_\mathrm{mut},z)/\partial z_\mathrm{mut}]_{z_\mathrm{mut}=z} \nonumber \\&=\beta '(z) \langle x \rangle - \left\langle \frac{\partial \alpha (\mathbf {E}(N,x,z),z_\mathrm{mut},t)}{\partial z_\mathrm{mut}} \bigg |_{z_\mathrm{mut}=z} \right\rangle . \end{aligned}$$

(28)

Following Boots and Sasaki (2003), assume first that the disease-induced death rate is independent of the strain infecting the host, i.e., that $\alpha (\mathbf {E}(N,x,z),z_\mathrm{mut})$ does not depend on $z_\mathrm{mut}$ (but it may still depend on z via the environmental feedbacks). In other words, different strains $z_\mathrm{mut}$ of the pathogen differ in their transmission rate $\beta (z_\mathrm{mut})$ and may also differ in their effect on the birth rate of an infected host, $b_I(\mathbf {E}(N,x,z),z_\mathrm{mut},t)$ so that a strain with a higher transmission rate may be more damaging to host fecundity. With this assumption, the selection gradient reduces to $D(z)=\beta '(z) \langle x \rangle$, which has the same sign as $\beta '(z)$. The pathogen therefore evolves to maximize its transmission rate. Assume further that the transmission rate can increase without bound, and $\beta (z) \rightarrow \infty$ as $z \rightarrow z_0$. Let the initial strain be such that the solution of $\tfrac{dz}{dt}=\beta '(z)$ tends to $z_0$ (if the function $z \mapsto \beta (z)$ does not have finite local maxima, then this is true for any initial value z). In this case, z evolves towards $z_0$.

If a strain z is viable, i.e., if in its resident population the density of infected hosts, I, is bounded and also bounded away from zero, then $\left\langle \frac{1}{I}\frac{dI}{dt} \right\rangle =0$ must hold (cf. Metz et al. 1992). By the second equation of (25), this is equivalent to

$$\begin{aligned} \beta (z) \langle x \rangle = \langle \mu (\mathbf {E}(N,x,z),t)+\alpha (\mathbf {E}(N,x,z),z,t) \rangle . \end{aligned}$$

(29)

Since $\mu$ and $\alpha$ are bounded, we have $\langle x \rangle \rightarrow 0$ as $z \rightarrow z_0$; a very highly transmissible disease infects all hosts. The dynamics of the total population density then converges to the orbit of

$$\begin{aligned} \frac{dN}{dt}= \left[ b_I(\mathbf {E}(N,0,z_0),z_0,t) -\mu (\mathbf {E}(N,0,z_0),t) - \alpha (\mathbf {E}(N,0,z_0),z_0,t) \right] N. \end{aligned}$$

(30)

If (30) has no other attractor than the trivial equilibrium $N=0$, then the entire host population goes extinct. In absence of Allee-effects, the birth and death rates are monotonic functions of the elements of $\mathbf {E}$, which, in turn, are monotonic in N. In this case, the trivial equilibrium is the only attractor of (30) if

$$\begin{aligned} \langle b_I(\mathbf {E}(0,0,z_0),z_0,t) -\mu (\mathbf {E}(0,0,z_0),t) - \alpha (\mathbf {E}(0,0,z_0),z_0,t) \rangle <0. \end{aligned}$$

(31)

As z evolves towards $z_0$, the entire host population goes extinct when (31) holds. The evolution of the pathogen thus results in its own extinction, i.e., in evolutionary suicide. In absence of Allee-effects, evolutionary suicide occurs as the host population density declines to zero continuously during the course of evolution. In autonomous systems, this happens via a local non-catastrophic bifurcation of population density.

In this Appendix, we assumed that the transmission rate can evolve to arbitrarily high values. In reality, the transmission rate is bounded by the rate of contacts between host individuals, and increasing the transmission rate may be possible only at the cost of increasing virulence (Alizon et al. 2009). This model nevertheless shows, along the lines of Boots and Sasaki (2003) but in a much more general model, that with a sufficiently high contact rate and for some trade-off functions linking transmission and virulence, evolutionary suicide must be possible. In the main text of this article, we show that evolutionary suicide does happen through a non-catastrophic bifurcation also in models with bounded transmission traded off with virulence.

Appendix 2: (In)stability of the disease-free equilibrium of Model I

The aim of this Appendix is to verify that the disease-free equilibrium of (10) is unstable whenever ${\mathcal {R}}_0>1$ and locally stable when ${\mathcal {R}}_0$ is below 1. The linearization of (10) around $(N^*,1)$ gives

$$\begin{aligned} \begin{bmatrix} - N^* \mu '(N^*)&\quad * \\ 0&\quad -b_S(1) + \beta (\alpha ) - \alpha \end{bmatrix}. \end{aligned}$$

The upper left element of the Jacobian at $(N^*,1)$ is negative. Since $-b_S(1) + \beta (\alpha ) - \alpha >0 \iff {\mathcal {R}}_0 (\alpha )>1$, the disease-free equilibrium $(N^*,1)$ is unstable when ${\mathcal {R}}_0 (\alpha )>1$ and locally stable whenever ${\mathcal {R}}_0 (\alpha )<1$.

Appendix 3: Local stability of equilibria of Model II

In this Appendix we discuss stability of equilibria of Model II.

The linearization of (17) takes the form

$$\begin{aligned}&J(N,x) \\&\quad = \begin{bmatrix} (b_S(N)x + b_I(N)(1-x) - \mu (N) - \alpha (1-x))&\quad N(b_S(N)-b_I(N)+ \alpha ) \\ + N (b_S'(N)x + b_I'(N)(1-x) - \mu '(N))&\\ (1-x)(b'_S(N) + b_I'(N)(1-x))&\quad (1-x)(b_S(N)-b_I(N) -\beta (\alpha ) + \alpha ) \\&- (b_S(N)x + b_I(N)(1-x) - \beta (\alpha )x + \alpha x) \end{bmatrix}. \end{aligned}$$

The assumption $b_S(0)>\mu (0)$ implies that the equilibrium (0, 1) is unstable. If we further assume that ${\mathcal {R}}_0>1$, the disease-free steady state $(N^*,1)$ is unstable.

If an endemic equilibium $(\hat{N}, \hat{x})$ exists then the Jacobian evaluated in $(\hat{N}, \hat{x})$ has the form

$$\begin{aligned}&J((\hat{N}, \hat{x})) \\&\quad = \begin{bmatrix} \hat{N} (b_S'(\hat{N})\hat{x} + b_I'(\hat{N})(1-\hat{x})-\mu '(\hat{N}))&\quad \hat{N}(b_S(\hat{N}) - b_I(\hat{N}) + \alpha ) \\ (1-\hat{x})(b_S'(\hat{N})\hat{x} + b_I'(\hat{N})(1-\hat{x}))&\quad (1-\hat{x})(b_S(\hat{N})- b_I(\hat{N})- \beta (\alpha ) + \alpha ) \end{bmatrix} \end{aligned}$$

which has the sign structure

$$\begin{aligned} \begin{bmatrix} -&\quad + \\ -&\quad - \end{bmatrix}, \end{aligned}$$

implying that $(\hat{N}, \hat{x})$ is locally stable whenever it exists.

If the assumption of Case I holds, then there are no other equilibria of (17). The same conclusions holds in Case II (i). When the assumption of Case II (ii) holds, there is no endemic equilibrium of (17). There exists however an equilibrium $(0,x_0)$ with $0 < x_0 <1$. We have

$$\begin{aligned} J((0, x_0)) = \begin{bmatrix} \beta (\alpha ) x_0 - \mu (0) - \alpha&\quad 0 \\ *&\quad (1-x_0)(b_S(0)-b_I(0) - \beta (\alpha ) + \alpha ) \end{bmatrix}. \end{aligned}$$

Since both diagonal elements are negative, the equilibrium $(0,x_0)$ is locally stable.