Evolution of acuteness in pathogen metapopulations: conflicts between “classical” and invasion-persistence trade-offs

Abstract

Classical life-history theory predicts that acute, immunizing pathogens should maximize between-host transmission. When such pathogens induce violent epidemic outbreaks, however, a pathogen’s short-term advantage at invasion may come at the expense of its ability to persist in the population over the long term. Here, we seek to understand how the classical and invasion-persistence trade-offs interact to shape pathogen life-history evolution as a function of the size and structure of the host population. We develop an individual-based infection model at three distinct levels of organization: within an individual host, among hosts within a local population, and among local populations within a metapopulation. We find a continuum of evolutionarily stable pathogen strategies. At one end of the spectrum—in large well-mixed populations—pathogens evolve to greater acuteness to maximize between-host transmission: the classical trade-off theory applies in this regime. At the other end of the spectrum—when the host population is broken into many small patches—selection favors less acute pathogens, which persist longer within a patch and thereby achieve enhanced between-patch transmission: the invasion-persistence trade-off dominates in this regime. Between these extremes, we explore the effects of the size and structure of the host population in determining pathogen strategy. In general, pathogen strategies respond to evolutionary pressures arising at both scales.

Appendix

McKendrick–von Foerster equations

We consider the spread of a disease in a well-mixed homogeneous population, where a transmission rate of a single host during an infection is varying, specified by the mechanistic model described earlier by Eq. 1. The transmission rate of a host infected a units of time ago is β(a). Let $\int _{a_1}^{a_2} \,i(t,a)\,\mathrm{d} {a}$ be the fraction of host at time t infected between times t − a ₁ and t − a ₂. Then, the fraction of infected host at time t that have progressed a units into their infection follows:

$$ \frac{{\partial}i}{{\partial}t}+\frac{{\partial}i}{{\partial}a}=-\mu(a)\,i, \qquad i(t,0) = \lambda(t)\, S(t), $$

(8)

where μ(a) is age-specific mortality, λ(t) is the force of infection, and S(t) is the fraction of the host population susceptible to infection at time t. The force of infection is

$$ \lambda(t) = \int_0^{a_c}\,\beta(a)\,i(t,a)\,\mathrm{d}{a} = \int_0^{a_c}\,\beta(a)\,\ell(a)\,i(t-a,0)\,\mathrm{d}{a}. $$

(9)

Here, a _c is the time when the infection is cleared in the host, and $\ell (a)=\exp \left ({-\int _0^a\,\mu (a')\,\mathrm{d} {a'}}\right )$ denotes the probability that an individual infected a time units ago has not yet died. We will assume that infections are nonlethal, and this amounts to assuming a constant death rate: ℓ(a)=e ^{− μa}. We assume that the total host population remains constant, and the fraction of susceptible S(t) obeys

$$ \frac{dS}{dt} = \mu\,(1-S)-\lambda(t)\,S. $$

(10)

Approximation of epidemic sizes

For small enough patches, we consider the pathogen extinct in a patch when the fraction of infected hosts

$$ H(t) = \int_{0}^{a_c}\,i(t,a)\,\mathrm{d}{a}, $$

reaches its first minimum value. We define the average duration of the epidemic $\bar {\delta }$ to be this duration by the time at which this occurs. Similarly, the average fraction of infected

$$ \bar{\iota} = \frac{1}{\bar{\delta}}\int_{0}^{\bar{\delta}}\,H(t)\,\mathrm{d}{t}. $$

The patch-level net reproductive number is $R_* = \bar {\iota }\,\bar {\delta }\,m\,n.$ The quantities $\bar {\iota }$ and $\bar {\delta }$ can be computed numerically either by simulation of the agent-based model or by numerical integration of Eqs. 8–10. Figure 7 shows estimates for R _∗ calculated both ways.

Fig. 7

Estimates of R _∗ for linear a and b saturating models, computed using both deterministic and stochastic frameworks. The curves are similar for both models, and each attains its maximum for r≈3.5

Sensitivity to birth and migration rates

To explore sensitivity to change in birth and migration rates, we take a metapopulation configuration (N _P = 20, and n = 100) and observe the change in the steady-state distributions of r. In Fig. 8, we see that increasing either migration or birth rates shifts the ESS r towards r ^∗. This is not surprising. Increasing either the birth rate or the migration rate sustains the epidemic locally for longer periods, allowing for within-patch competition to be more relevant.

Fig. 8

Sensitivities to birth and migration. The graphs show the ESS r, with the metapopulation configuration N _P = 20 and n = 100, as a birth rate μ and b migration rate m vary. For each parameter value, 1,000 replicate simulations were performed in the same fashion as for figures in the main text

Fig. 9

Sample trajectories of pathogen evolution through time in two multi-patch configurations. a N _P = 20 and n = 100, and b N _P = 300 and n = 20. The grayscale indicates the distribution of r(t) within the pathogen population; the dashed line plots the mean value of r(t)

Sample trajectories

We present sample trajectories for two chosen multi-patch configuration depicting the evolution of r over time. Compared to a single-patch dynamics that converges to ESS r that maximizes R ₀ (as seen in Fig. 4), the trajectories converges to different ESS rs depending on patch configuration.

Evolutionarily stable strategies in populations under pure birth–death process

Here, we relax the assumption of density-dependent mortality, by holding the host mortality constant and equal to the host birth rate, μ. In Fig. 10, comparable to Fig. 5 in the main text, we present the ESS as a function of host configuration. The notable difference is in transect C, in contrast to the observation in the main text; here, we see that changing the number of patches of size 100 decreases ESS of r.

Fig. 10

Evolutionarily stable strategy (ESS) in populations without density dependence. This figure is comparable to Fig. 5 in the main text, except that here, the host mortality rate is held constant and equal to the host birth rate μ, (i.e., it is no longer density dependent). a ESS r (mean r in the stable strain distribution, indicated by color) and survival fraction (chance of avoiding extinction by time 2,000, indicated by symbol size), at various metapopulation configurations. Metapopulations are characterized by size of patch, n (vertical axis), and number of patches, N _P (horizontal axis). Black crosses indicate extinction in all of 1,000 replicate runs by time 2,000. The bottom panels show steady-state strain distributions along three different transects across metapopulation structure. b The size and the number of patches vary, and the expected total regional population size remains constant at 2,000. c The number of patches varies, and the expected size of each patch remains constant at 100. d The size of patches varies, and the number of patches remains constant. Metapopulations with many large patches were not explored due to computational infeasibility