Evolutionary Stability of Minimal Mutation Rates in an Evo-epidemiological Model

Abstract

We consider the evolution of mutation rate in a seasonally forced, deterministic, compartmental epidemiological model with a transmission–virulence trade-off. We model virulence as a quantitative genetic trait in a haploid population and mutation as continuous diffusion in the trait space. There is a mutation rate threshold above which the pathogen cannot invade a wholly susceptible population. The evolutionarily stable (ESS) mutation rate is the one which drives the lowest average density, over the course of one forcing period, of susceptible individuals at steady state. In contrast with earlier eco-evolutionary models in which higher mutation rates allow for better evolutionary tracking of a dynamic environment, numerical calculations suggest that in our model the minimum average susceptible population, and hence the ESS, is achieved by a pathogen strain with zero mutation. We discuss how this result arises within our model and how the model might be modified to obtain a nonzero optimum.

Appendices

Appendix 1: General Mutation Spectrum

Generally, one could write (1b) as

$$\begin{aligned} \frac{\partial i}{\partial t} = [S(t)\beta (\alpha , t) - (\alpha + \mu )] i(\alpha , t) - mi(\alpha ,t)\,+\,m\int _0^\infty K\left( \alpha ^\prime ,\alpha \right) i\left( \alpha ^\prime ,t\right) ~\hbox {d}\alpha ^\prime , \end{aligned}$$

(32)

where m is the mutation rate (although a different mutation rate than D defined below) and $K(\alpha ^\prime , \alpha )$ is the mutation spectrum (the rate of mutation from strain $\alpha ^\prime$ to $\alpha$). $K(\alpha ^\prime , \alpha )$ must have the following properties:

• (i) $K(\alpha ^\prime ,\alpha ) \ge 0$ for all $\alpha ^\prime ,\alpha \in [0,\infty )$

• (ii) $\int _0^\infty K(\alpha ^\prime ,\alpha )~d\alpha = 1$.

Property (i) follows from probabilities of mutation being non-negative and property (ii) ensures that mutation cannot change the total number of infectious individuals. This mutation spectrum can accommodate very general models of mutation (even among discrete phenotypes through the use of Dirac delta functions). Our diffusion model of mutation follows from this general spectrum under the assumptions that $K(\alpha ^\prime , \alpha )$ is symmetric in the sense that

$$\begin{aligned} K\left( \alpha ^\prime , \alpha \right) = K\left( \alpha , \alpha ^\prime \right) \end{aligned}$$

(33)

and for each $\alpha$, $K(\alpha ^\prime , \alpha )$ is narrowly peaked such that

$$\begin{aligned}&\int _0^\infty K\left( \alpha ^\prime ,\alpha \right) i\left( \alpha ^\prime ,t\right) ~d\alpha ^\prime \nonumber \\&\quad \approx \int _0^\infty K\left( \alpha ^\prime ,\alpha \right) \left[ i\left( \alpha , t\right) + \frac{\partial i}{\partial \alpha }\left( \alpha ^\prime - \alpha \right) + \frac{1}{2}\frac{\partial ^2 i}{\partial \alpha ^2}\left( \alpha ^\prime - \alpha \right) ^2 \right] ~\hbox {d}\alpha ^\prime . \end{aligned}$$

(34)

Using (33) and (34), as well as applying property (ii), in (32) we obtain

$$\begin{aligned} \frac{\partial i}{\partial t} = [S(t)\beta (\alpha , t) - (\alpha + \mu )] i(\alpha , t) + D(\alpha ) \frac{\partial ^2 i}{\partial \alpha ^2}, \end{aligned}$$

(35)

where $D(\alpha ) \equiv (m/2)\int _0^\infty (\alpha ^\prime - \alpha )^2 K(\alpha ^\prime , \alpha )~\hbox {d}\alpha ^\prime$. Finally, we obtain (1b) by assuming that $D(\alpha )$ is constant for all $\alpha$.

Appendix 2: Derivation of Second-Order Moment Equations

Here we wish to approximate the full model, given by (1), by characterizing the infectious distribution, $i(\alpha , t)$, according to its first two moments. In addition, we shall assume that $i(\alpha , t)$ is normally distributed in $\alpha$ for every $t \ge 0$ and that $i(0, t) \approx 0$ (i.e. dynamics occur far from the boundary).

The mean virulence, $\left\langle \alpha \right\rangle$, is given by

$$\begin{aligned} \left\langle \alpha \right\rangle = \frac{\int _0^\infty \alpha i(\alpha , t)~\hbox {d}\alpha }{\int _0^\infty i(\alpha , t)~\hbox {d}\alpha }, \end{aligned}$$

and hence its time derivative is

$$\begin{aligned} \frac{\hbox {d}\left\langle \alpha \right\rangle }{\mathrm{d}t}&= \frac{\int _0^\infty \alpha \frac{\partial i}{\partial t}~\hbox {d}\alpha }{\int _0^\infty i(\alpha , t)~\mathrm{d}\alpha } - \frac{\int _0^\infty \alpha i(\alpha , t)~\hbox {d}\alpha }{\left[ \int _0^\infty i(\alpha , t)~\mathrm{d}\alpha \right] ^2} \int _0^\infty \frac{\partial i}{\partial t}~\hbox {d}\alpha \\&= \frac{\int _0^\infty \left( \alpha - \left\langle \alpha \right\rangle \right) \frac{\partial i}{\partial t}~\hbox {d}\alpha }{\int _0^\infty i(\alpha , t)~\mathrm{d}\alpha }. \end{aligned}$$

We now substitute $\partial i/\partial t$ for the expression in (1b) and simplify in order to obtain

$$\begin{aligned} \frac{\hbox {d}\left\langle \alpha \right\rangle }{\mathrm{d}t} = S(t)\left\langle (\alpha - \left\langle \alpha \right\rangle )\beta (\alpha , t) \right\rangle - {\sigma ^2_\alpha }(t), \end{aligned}$$

(36)

where ${\sigma ^2_\alpha }= \left\langle \alpha ^2 \right\rangle - \left\langle \alpha \right\rangle ^2$ is the variance of the virulence in the infectious population. We can now also take a Taylor expansion of $\beta (\alpha , t)$ about $\left\langle \alpha \right\rangle$ to second order in $\alpha$,

$$\begin{aligned} \left\langle (\alpha - \left\langle \alpha \right\rangle )\beta (\alpha , t) \right\rangle&= \Bigg \langle (\alpha - \left\langle \alpha \right\rangle ) \beta (\left\langle \alpha \right\rangle ,t) + (\alpha - \left\langle \alpha \right\rangle )^2\beta _\alpha (\left\langle \alpha \right\rangle ,t)\\&\qquad + \frac{1}{2}(\alpha - \left\langle \alpha \right\rangle )^3 \beta _{\alpha \alpha }(\left\langle \alpha \right\rangle , t) \Bigg \rangle = {\sigma ^2_\alpha }\beta _\alpha (\left\langle \alpha \right\rangle ,t), \end{aligned}$$

where the $(\alpha - \left\langle \alpha \right\rangle )^3$ term averages to zero since we have assumed normality (normal distributions have no skew). Therefore, the equation for the time evolution of $\left\langle \alpha \right\rangle$ is

$$\begin{aligned} \frac{\hbox {d}\left\langle \alpha \right\rangle }{\mathrm{d}t}&= \sigma ^2_\alpha \left[ S\beta _\alpha (\left\langle \alpha \right\rangle , t) - 1 \right] , \end{aligned}$$

(37)

as stated in (5c).

The time evolution for the variance is derived in a similar fashion. First note that, similar to the time derivative of $\left\langle \alpha \right\rangle$,

$$\begin{aligned} \frac{\hbox {d}\left\langle \alpha ^2 \right\rangle }{\mathrm{d}t} = \frac{\int _0^\infty \left( \alpha ^2 - \left\langle \alpha ^2 \right\rangle \right) \frac{\partial i}{\partial t}~\hbox {d}\alpha }{\int _0^\infty i(\alpha , t)~\mathrm{d}\alpha }, \end{aligned}$$

which, upon substitution of (1b), simplifies to

$$\begin{aligned} \frac{\hbox {d}\left\langle \alpha ^2 \right\rangle }{\mathrm{d}t} = S(t)\left\langle \left( \alpha ^2 - \left\langle \alpha ^2 \right\rangle \right) \beta (\alpha , t) \right\rangle - \left\langle \alpha ^3 \right\rangle + \left\langle \alpha \right\rangle \left\langle \alpha ^2 \right\rangle + 2D. \end{aligned}$$

This result allows us to easily compute the time derivative of ${\sigma ^2_\alpha }$,

$$\begin{aligned} \frac{\hbox {d}{\sigma ^2_\alpha }}{\mathrm{d}t}&= \frac{\hbox {d}\left\langle \alpha ^2 \right\rangle }{\mathrm{d}t} - 2\left\langle \alpha \right\rangle \frac{\hbox {d}\left\langle \alpha \right\rangle }{\mathrm{d}t} \\&= S(t)\left\langle \left( \alpha ^2 - \left\langle \alpha ^2 \right\rangle \right) \beta (\alpha , t) \right\rangle - \left\langle \alpha ^3 \right\rangle + \left\langle \alpha \right\rangle \left\langle \alpha ^2 \right\rangle \\&\quad + 2D - 2\left\langle \alpha \right\rangle S(t)\left\langle (\alpha - \left\langle \alpha \right\rangle )\beta (\alpha , t) \right\rangle + 2\left\langle \alpha \right\rangle {\sigma ^2_\alpha }(t) \\&= S(t)\left\langle \left( \alpha ^2 - \left\langle \alpha ^2 \right\rangle - 2\left\langle \alpha \right\rangle \alpha + 2\left\langle \alpha \right\rangle ^2 \right) \beta (\alpha , t) \right\rangle \\&\quad - \left( \left\langle \alpha ^3 \right\rangle - 3\left\langle \alpha \right\rangle \left\langle \alpha ^2 \right\rangle + 2\left\langle \alpha \right\rangle ^3 \right) + 2D \\&= S(t)\left\langle \left( \alpha - \left\langle \alpha \right\rangle \right) ^2\beta (\alpha , t) \right\rangle - S(t){\sigma ^2_\alpha }\left\langle \beta (\alpha , t) \right\rangle \\&\quad - \left( \left\langle \alpha ^3 \right\rangle - 3\left\langle \alpha \right\rangle \left\langle \alpha ^2 \right\rangle + 2\left\langle \alpha \right\rangle ^3 \right) + 2D. \end{aligned}$$

The term in brackets is the third cumulant of the infectious distribution and therefore is zero under our assumption of normality. We again expand $\beta (\alpha , t)$ to second order in $\alpha$. From this we obtain

$$\begin{aligned} \frac{\hbox {d}{\sigma ^2_\alpha }}{\mathrm{d}t}&= \frac{1}{2}S(t)\beta _{\alpha \alpha }(\left\langle \alpha \right\rangle , t)\left( \left\langle (\alpha - \left\langle \alpha \right\rangle )^4 \right\rangle - \left( {\sigma ^2_\alpha }\right) ^2 \right) + 2D. \end{aligned}$$

Finally, we use the fact that the fourth central moment is equal to the sum of the fourth cumulant (which again is zero by the assumption of normality) and three times the variance squared. This gives the result quoted in (5d),

$$\begin{aligned} \frac{\hbox {d}\sigma ^2_\alpha }{\mathrm{d}t}&= \left[ \sigma ^2_\alpha \right] ^2 S \beta _{\alpha \alpha }(\left\langle \alpha \right\rangle , t) + 2D. \end{aligned}$$

(38)

Equations (5a) and (5b) are obtained directly from the full model defined in (1) by simply expanding $\beta (\alpha , t)$ to second order as in the above two cases.