Figure 1.
Comparison of the birth-death model and the coalescent model in estimating epidemic growth rate.
For each plot, 100 trees simulated under the constant rate birth-death (BD) model with incomplete sampling (subfigure A) or coalescent (CE) model with exponential growth of the infected population (subfigure B) were analyzed assuming a birth-death model (blue bars) or a coalescent model with deterministic exponential population growth (red bars). 95% highest posterior density (HPD) intervals of the growth rate parameter are shown (y-axis). The trees are ordered (x-axis) by the median value of the posterior distribution of the growth rate parameter estimated by the coalescent (orange dot within the red bar) from the birth-death trees. Median of the posterior estimates for the growth rate parameter estimated by birth-death model is indicated as light blue dot within each blue interval. The true value of the growth rate parameter, i.e. the value under which the trees were simulated, is displayed as black horizontal bar. Here, we used and
(
). See Figure S1 for the plots of other parameter settings.
Table 1.
Summary of growth rate parameter estimation statistics.
Figure 2.
Influence of branch length extension in various parts of the tree on the growth rate parameter estimation.
For setting and
(
), we modified all 100 birth-death trees (A) and all 100 coalescent trees (B) by branch extension. The unchanged tree is denoted as “orig” on x-axis. We added 48 units of time, roughly corresponding to the full length of the longest trees, to the branches. We extended the branches that were present in the tree at 10% of the tree (going from the root), at 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90% (see x-axis from left to right). We then re-estimated the growth rate parameter for each such tree. Unlike in previous plot, here we display a summary in form of the median values of the start and the end of the 95% HPD intervals, and the median of the medians of the posterior estimates for all 100 trees per setting.
Table 2.
Summary of growth rate parameter estimation statistics in SIS/SIR trees.
Table 3.
Summary of growth rate parameter estimation statistics at various .
Figure 3.
Influence of sampling scheme on the growth rate parameter estimation.
For setting (
), we modified the birth-death tree simulations to include periods of higher (
) and lower sampling (either
, subfigures A and B, or
, subfigures C and D). We simulated 100 birth-death trees (A and C) and corresponding coalescent trees (B and D) under various sampling schemes (see x-axis annotation). We display a summary in form of the median values of the start and the end of the 95% HPD intervals, and the median of the medians of the posterior estimates for all 100 trees per setting. For the settings where the constant rate birth-death method produced very severe biases, we also analysed the trees with the birth-death skyline model with 10 intervals for the sampling probability (BDSKY, light-blue lines). The summary for trees simulated under constant sampling
throughout, is represented on the very left of each figure (
on the x-axis). Next, we varied the sampling as to e.g. sample no tips (
) in the early phases (
until
) when going forward in time and then sampling all the tips that die (
) from
onward (corresponding to the setting denoted as “p = 0 from t = 0 to t = 9”).
Figure 4.
Error on as a function of sampling probability
for fixed
and
.
In (A) the relationship between the error on , i.e. estimated
/true
, and the sampling probability
is plotted. The values
and
are fixed. For different
,
, and
and
, we calculate
and
, and plot the impact on
error when changing
during inference using Equation (3) in the Supplementary Material S1. In (B) we display how error on
depends on different assumptions of
during inference for
, and
and an array of true sampling probability
used for calculating
and
.
Figure 5.
Effect of different information used in the parameter inference.
For setting and
(
), we estimated the
parameter from the birth-death trees (A) and the coalescent trees (B) using four methods. First, using the coalescent posterior estimates of the growth rate
and the true
, we obtained
(red bars). Second, we used the birth-death posterior estimates of
(trees analysed under uniform priors for
,
, and
), and the true
in the post-processing (blue bars), similar to the procedure used for the coalescent. Third, we also analyzed the trees by fixing the prior on the death rate
to the true value,
(green bars) or by fixing the prior on the sampling probability
to the true value,
(purple bars) during the MCMC analysis. Note that y-axis now displays 95% HPD of the
parameter, and within each figure, the trees (simulations) are ordered (x-axis) by the median estimate of growth rate
parameter estimated by the coalescent on the birth-death trees.
Figure 6.
Comparison of the birth-death model and the coalescent model in estimating epidemic growth rate from trees with tips sampled at one point in time.
For simulated trees where all 100 tips are sampled at one point in time, we estimated the growth rate parameter assuming a birth-death model with fixed sampling probability (blue bars) and the coalescent model with a deterministic exponentially growing population (red bars). Here we used
and sampling probability
(
). See Figure S15 for the plots of other parameter settings.
Figure 7.
Comparison of growth rate point estimates of the birth-death model and the coalescent model.
For setting and
(
), we display the ML and MAP estimates for the birth-death trees (A) and the coalescent trees (B). As a comparison, the median values of the start and the end of the 95% HPD intervals, and the median of the medians of the posterior estimates for all 100 trees per setting are also displayed. The true value of the growth rate parameter, i.e. the value under which the trees were simulated, is displayed as a black horizontal bar. See Figures S17 and S18 for the plots of other parameter settings.