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# Overdispersion decreases variant fixation rates

My first post on this blog was about fixation of a new beneficial mutation. I presented Haldane’s original 1927 result showing that with some reasonable assumptions, the likelihood that a new mutation with a small selective advantage $s$ fixed in the population was approximately $2s$. That derivation assumed a Poisson-distributed number of descendants. We’ll show some modeling with overdispersion, and see how that affects the result.

The specific probability distribution for the number of “children” (i.e., new infected hosts) of a particular organism is assumed to be Poisson by Haldane. If you recall, we were evaluating the fixed point of the generating function of the distribution. The generating function of the Poisson distribution is

$f(x) = e^{\lambda (x-1)}$.

We then solved the equation $f(x) = x$ for $\lambda = 1+s$ and $x = 1 - y$, where $y$ is the probability of extinction. We can naturally extend the Poisson distribution to a negative binomial distribution with mean $\mu$ and variance $\mu + \mu^2 / k$. $k$ is the overdispersion parameter, and we approach a Poisson distribution as it approaches infinity. The generating function of the negative binomial is instead

$f(x) = \frac{\mu^{k}}{(\mu + k - kx)^{k}}$

The goal is then to find a solution to the fixed point $f(x) = x$. Bartlett (1955) provides a useful approximation using the moments of the distribution:

$ln(x) = ln(f(1)) + ln(x) \mu + \frac{ln(x)^2}{2} \sigma^2 + \ldots$

This can be simplified to give an approximate solution valid for an arbitrary distribution:

$x \approx exp \left [ \frac{2(1 - \mu)}{\sigma^2} \right ]$

Applied to the Poisson distribution where the mean and sigma are both $\lambda = 1 + s$ we get:

$y = 1 - x \approx 1 - exp \left [\frac{-2s}{1 + s} \right ] = 2s - 4s^2 + \frac{22s^3}{3} \ldots$

And we see that we recapitulate Haldane’s result with the first term of the Taylor series approximation.

Using the negative binomial distribution, we get

$y = 1 - x \approx 1 - exp \left [\frac{-2s}{1 + 1/k + s + 2s/k + s^2/k}\right ]$

Assume $s$ is sufficiently small to neglect higher terms and we get:

$y \approx 1 - exp \left [\frac{-2s}{1 + 1/k + s + 2s/k}\right] = \frac{2s}{1 + 1/k} - \frac{4s^2}{1 + 1/k} \ldots$

And we see that the introduction of overdispersion by the negative binomial decreases the likelihood of fixation of the variant by approximately a factor of $1 + 1/k$. $k$ has been measured to be somewhere between 0.1 and 0.5 for the SARS2 virus, suggesting a 3-11-fold lower rate of fixation due to overdispersion of downstream infections than with a Poisson.