Bayesian Phylodynamics
Tim Vaughan, Jana Huisman, Nicola MÃ¼ller
cEvo group, DBSSE, The ETH Zurich
Taming the BEAST, 20^{th} Feb, 2019
Birthdeath 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.
Birthdeath branching process
Birthdeathsampling process
 The birthdeath sampling process extends the birthdeath 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: $X\overset{(1r)\psi}{\longrightarrow}X$
Additionally, the model allows each surviving lineage at the end of the process (present day) to be sampled with probability $\rho$.
Birthdeathsampling trees
The sampled tree probability
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+m1}\psi^{k+m}(4\rho)^n\prod_{i=0}^{n+m1}\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]
Birthdeath 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: timedependence
 The birthdeath skyline model allows piecewiseconstant 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, $R_e(t)=\lambda(t)/\mu(t)$.
Birthdeath model assumptions
 Samples are members of a stochastically varying population that is on average exponentially growing with rate $\lambda(\mu+r\psi)$.
 Sample fraction can be small, large or complete  tree prior is still valid.
 Sample number and times are assumed to be produced according to the model: deviation from this assumption results in biased inferences.
 Population is "well mixed", samples are drawn randomly.