Bayesian Phylodynamics
Tim Vaughan, Jana Huisman, Nicola Müller
cEvo group, D-BSSE, The ETH Zurich
Taming the BEAST, 20th Feb, 2019

# What is Phylodynamics?

• What does a phylogenetic tree tell us about a population?
• Places the emphasis on the population itself instead of the subset of individuals involved in a particular phylogeny.
• Term introduced by Grenfell et al. (2004) in the context of epidemiology, but modern phylodynamic methods are much more generally applied.

# Bayesian phylodynamics

• Generative models connect the population dynamics model to the observed sequence alignment.
• Use Bayesian inference to work backwards:

# Why use sequence data?

Sequence data can allow migration/transmission patterns (i.e. who infected whom) to be uncovered.

# Why use sequence data?

Time $\longrightarrow$

Genetic samples yield trees: information about events ancestral to samples.

# Classes of phylodynamic models

• Coalescent models
• Originally developed by John Kingman in a series of papers in 1982.
• Extensions for dynamics and structured populations introduced by several authors.
• Birth-death models
• Theory for completely sampled contemporaneous trees developed by Elizabeth Thompson in her 1975 book "Human Evolutionary Trees".
• Theory for trees conditioned on number of samples and extensions to incomplete sampling, sampling through time, structured populations, etc. developed by Tanja Stadler from 2008.
E.A. Thompson, "Human Evolutionary Trees", C.U.P. 1975

# Birth-death population dynamics

• Population size described by a positive integer.
• Changes through time according to a continuous time Markov process, similar to the substitution models we've already seen.

Can use a chemical reaction notation to describe rates and effects of possible events:

Birth:
$$X\overset{\lambda}{\longrightarrow} 2X$$
Death:
$$X\overset{\mu}{\longrightarrow} 0$$

The parameters $\lambda$ and $\mu$ are probabilities per [time unit] that any given individual experiences a birth or a death.

# Birth-death-sampling process

• The birth-death sampling process extends the birth-death process by incorporating a model for sample generation.
• This can be described using the following reactions:
• Birth: $X\overset{\lambda}{\longrightarrow}2X$
• Death: $X\overset{\mu}{\longrightarrow}0$
• Sampling (with removal): $X\overset{r\psi}{\longrightarrow}0$
• Sampling without removal: $X\overset{(1-r)\psi}{\longrightarrow}X$

Additionally, the model allows each surviving lineage at the end of the process (present day) to be sampled with probability $\rho$.

# The sampled tree probability

\begin{align*} g(t+\Delta t) =& (1 - \Delta t (\lambda + \mu + \psi)) g(t) + \Delta t\lambda g(t) p_0(t) \times 2 \end{align*}

Gives rise to differential equations which can be solved to obtain the following tree probability: \begin{equation*} P(T|\lambda,\mu,\psi,r,t_0) = g(t_0) =\lambda^{n+m-1}\psi^{k+m}(4\rho)^n\prod_{i=0}^{n+m-1}\frac{1}{q(x_i)}\prod_{i=1}^{m}p_0(y_i)q(y_i) \end{equation*} where $q(t)=4\rho/g(t)$. [Stadler, J. Theor. Biol., 2010]

# Birth-death parameterizations

There are several distinct parameterizations besides the basic $\lambda,\mu,\psi$ parameterization, including:

Epidemiological parameterization:
• Effective reproductive number $R_e$ ($=\lambda/(\mu + r\psi$)
• Becoming uninfectious rate $\mu+r\psi$
• Sampling proportion $\psi/(\psi+\mu)$
Macroevolutionary parameterization:
• Species diversification rate $\lambda - \mu$
• Turnover rate $\mu/\lambda$
• Fossilization rate $\psi$

# Extension: time-dependence

• The birth-death skyline model allows piecewise-constant variation in rate parameters.

# Example: Hepatitis C in Egypt

• Effective reproductive number $R_e(t)$ generalizes basic reproductive number $R_0=R_e(0)$.
• Analysis of 63 sequences, $R_e(t)=\lambda(t)/\mu(t)$.

# Birth-death model assumptions

1. Samples are members of a stochastically varying population that is on average exponentially growing with rate $\lambda-(\mu+r\psi)$.
2. Sample fraction can be small, large or complete - tree prior is still valid.
3. Sample number and times are assumed to be produced according to the model: deviation from this assumption results in biased inferences.
4. Population is "well mixed", samples are drawn randomly.

# Wright-Fisher Model Assumptions

• Discrete generations.
• Individuals identical. (No structure.)
• Fixed offspring distribution (mean of 1)
• Constant + finite population size.
• An interpretation: population at equilibrium (carrying capacity).

# Sampled WF phylogeny

• Generations between internal nodes related to population size.
• Can we quantify this relationship?

# Sampled 2-individual phylogeny

Probability of coalescence in generation $i-m$: $P(m)=(1-p_{\textrm{coal}})^{m-1}p_{\textrm{coal}}$

Continuous time limit (large $N$, small $g$): $P(t)=e^{-\frac{1}{Ng}t}\frac{1}{Ng}$

# Sampled k-individual phylogeny

Question: How can this be generalized to $k$ samples?

Answer: $p_{\text{coal}}=\frac{k(k-1)}{2}\frac{1}{N}=\binom{k}{2}\frac{1}{N}$

# Kingman's coalescent

• Backward-time Markov process which produces sampled time trees.
• Equivalent to sampled trees produced by WF model when $N\gg k$ and $g\ll 1$.
• Times between coalescence events drawn from exponential distributions with rate $\binom{k}{2}\frac{1}{Ng}$.
• Interesting: expected time to MRCA is always $\leq \frac{2}{Ng}$, regardless of how many samples there are.

# Effective population size

• If a real population were described precisely by a Wright-Fisher model with generation time $g$, the $N$ parameter in the coalescent would correspond to the real population size.
• However, real populations are more complicated:
• Random mating assumption is often violated,
• Offspring distribution may differ from that assumed by WF,
• Generations may be overlapping,
• ...
• We therefore define the effective population size $N_e$ to be the size of a statistically equivalent WF population. (We often also absorb other parameters such as $g$, the clock rate and ploidy into this number.)

# Probability of a coalescent tree

$$P(T|N_e)=\prod_{i=1}^{n-1}\left(\exp\left[-\Delta t_i\binom{k_i}{2}\frac{1}{N_e}\right]\frac{1}{N_e}\right)$$

# Coalescent-based inference

• We can therefore infer demographic parameters like $N_e$ given a known phylogeny!
$L(N_e|T)=P(T|N_e)$

# Bayesian skyline

• Piecewise-constant population model: population size can only change at a fixed number of break points. Each time interval can contain several coalescent intervals.
• Uses MCMC to integrate over all possible break point positions in addition to $N_e$ within each time interval.
• Graphs of marginal probability of population size vs time appear smoothed due to averaging over possible histories.

# General assumptions of the coalescent

1. Samples are members of a population that is at demographic equilibrium.
2. Number of samples is small compared to the total population size.
• Necessary for WF-based derivation, not all derivations.
3. Population is "well mixed", samples are drawn randomly.

## Birth-death vs coalescent HCV phylodynamics

Birth-death model can infer effective reproductive number dynamics:

Coalescent model can infer effective population size dynamics:

# Conclusions

• Both birth-death and coalescent models probabilistically relate a population's demography to its phylogenetic history.
• Both allow for model-based inference of demographic and epidemiological parameters but may differ in their parameterization.
• Which model to use depends on what assumptions you are willing to make and what you want to infer.