Hybrid analysis with phylogeny and population modeling to estimate the recent founding date of a population: a case study in the origins of COVID-19 illustrates how a branching process approximation can simplify a hybrid analysis

Abstract

The exact date of the primary infection in COVID-19 remains unknown. One influential article (Pekar et al (2021)) estimated the date with a hybrid analysis combining epidemiological and phylogenetic methods. The phylogenetic methods analyzed 583 SARS-COV-2 complete genomes to estimate the sample tMRCA (time of the most recent common ancestor). Before igniting as an epidemic, however, COVID-19 may have had several population bottlenecks with only a single infected person, so the MRCA merely represents the last such bottleneck. Pekar et al (2021) therefore used epidemiological methods to estimate the time from the primary infection to the sample MRCA. The hybrid method involved several arbitrary decisions, however, reflecting the fact that the epidemiological and phylogenetic analyses overlap at the sample MRCA and are generally probabilistically dependent. Towards removing the dependence, note that the start of an epidemic has a branching process approximation. Let the branching process have a single ancestor. If the branching process does not go extinct, define skeleton particles (individuals) to be particles whose lineages do not go extinct, and define the long-time MRCA as the earliest skeleton particle with at least two skeleton offspring. A linear phylogeny of skeleton particles therefore separates the ancestor from the long-time MRCA. Probabilistically, the linear phylogeny is a defective renewal process of skeleton particles, making the generation count geometrically distributed. Moreover, the terminology “long-time MRCA” is apt, because as time becomes arbitrarily large, the MRCA of the corresponding extant population approaches the long-time MRCA. Effectively, the focus on the long-time MRCA makes the forward epidemiological and backward phylogenetic analyses probabilistically independent. The present article can therefore confirm most of the epidemiological conclusions of the hybrid analysis of Pekar et al (2021). Its use of branching process approximations also points the way to noticeable simplifications in the hybrid method.

1. Introduction

The exact date of the primary infection in COVID-19 remains unknown [1]. The epidemic origins involve two prominent alternative hypotheses [2]: (1) a zoonotic jump from an intermediate host to humans; and (2) a lab leak at the Wuhan Institute of Virology. Either origin remains possible [3] in the absence of definitive confirmation by either: (1) viral samples from the unknown intermediate host; or (2) detailed patient data pertinent to the lab leak.

The exact date of the primary infection can resolve some hypothetical ambiguities, as follows. On one hand, a zoonotic jump is not tied to a particular date. On the other hand, the extant lab leak hypothesis entailed three infected researchers at the Wuhan Institute of Virology who were hospitalized in November 2019 “with symptoms consistent with both Covid-19 and common seasonal illness” [4]. A primary infection in November 2019 therefore supports the lab leak hypothesis, although the resulting weight of evidence for it remains difficult to quantify, e.g., see the eLetter “Assessing Probabilities for the Two Leading SARS-CoV-2 Origin Narratives” following [5].

Several methods have been used to estimate the date of the primary infection, e.g., extreme-value distributions from conservation science [6]. Phylogenetic methods [7] can also estimate the time of the most recent common ancestor (tMRCA) from SARS-COV-2 sequence samples [8]. If all samples derived from a single primary infection, the primary infection precedes the sample MRCA. (Here and throughout, our usage of “precedes” permits the primary infection and the sample MRCA to be identical). As a superspreading epidemic [9], however, COVID-19 may have encountered several epidemic bottlenecks with only a single infected person. If sequence sampling is late but thorough, the sample tMRCA dates the last such bottleneck, so epidemiological modeling must supplement phylogenetic methods to date the primary infection before the tMRCA.

Of central interest here, one study [10] analyzed 583 SARS-COV-2 complete genomes collected before the end of March 2020. A phylogenetic analysis concluded that 95.1% of the posterior density for the tMRCA postdated 2019.11.17 (i.e., Nov 17, 2019, in our year.month.day notation). Epidemiological modeling then extrapolated back from the tMRCA to conclude that the primary infection had a median date of 2019.11.03; a 95% upper highest posterior density (HPD) back to 2019.10.14; and 99% upper HPD back to 2019.10.04. The epidemiological estimates were robust against the Bayesian priors chosen for the parameters.

A later, related study [5] of less interest here elaborated on these findings by inferring two separate zoonotic jumps using a hybrid of phylogenetic and epidemiological methods. In [5], the FAVITES package [11] modeled the COVID-19 epidemic as independent Poisson processes on a fixed network graph. The authors comment with due caution, however, “It is not always clear which method/setting combination performs best for a specific downstream use-case or for specific epidemiological conditions.” The comment partially motivates the present investigation.

The hybrid analysis combining phylogenetic and epidemiological analysis in [10] and [5] models considerable biological detail, making details correspondingly difficult to assess. An eLetter from S. Huang for [10], e.g., notes several problematic decisions during the hybrid analysis of the tMRCA, such as hidden probabilistic dependencies and biased rejection sampling. To clarify the implications of the decisions, we model the epidemic on a directed complete random graph whose nodes represent individuals undergoing non-Markovian transitions between disease states. Many random graph models have a branching process approximation [12; 13], a concept going back to Flory [14]. Mathematical rigor can justify the branching process approximation for some epidemic models [9], including the one here [15]. In present article, the theory of branching processes clarifies the consequences of some decisions in the hybrid analysis, with the aim of improving its statistical precision.

The Methods and Materials section presents an epidemiological model for COVID-19 superspreading and the corresponding branching process approximation. The Results section uses succinct, explicit simulation algorithms to estimate probability distributions derived from the branching process approximation. The Discussion considers some implications of the Results section and future directions.

Sections and tables within “Supplementary Information.docx” in the SI are preceded by “S”, e.g., Section S1 and Table S1. Below, sections, figures, and tables without the prefix “S” refer to the main article.

2. Methods and Materials

2.1. The epidemiological model

See Fig 1 and its caption for the epidemiological model underlying our results. Table 1 gives the parameters used in simulations, and the SI contains references for the parameters. If a parameter had broad range of potential values, the simulation uses more than one representative parameter value. Fig 1 and Table 1 assume that the parameters controlling spread of infection were similar for symptomatic and asymptomatic infection [16; 17; 18; 19; 20; 21; 22].

Fig 1. Model diagram for the duration of disease states.

Fig 1 displays a superspreading SEIR model. According to context, the symbols ($S,E,I,R$) may denote: a compartment; an individual in a compartment; or the count of individuals in a compartment. Compartment counts have an implicit dependence on time, e.g., $S=S\left(t\right)$. In Fig 1, e.g., $S$ may denote a susceptible individual, the compartment of susceptibles, or the count of susceptibles, etc. Similarly, $E$ corresponds to exposed individuals (in their latent period); $I$, infectious individuals; and $R$, individuals after they have recovered from $I$. As the gray dashed arrow in Fig 1 suggests, compartment $I$ contributes to the force of infection $\lambda =\lambda \left(t\right)$ on each susceptible in $S$, which transits into the exposed compartment $E$. Each exposed individual $E$ waits a random exposed period of mean ${\mu }_E$ with gamma distribution $\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}\left({\mu }_E,k_E\right)$ before transiting to $I$. Each infectious individual $I$ waits a random infectious period of mean ${\mu }_I$ with gamma distribution $\mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}\left({\mu }_I,k_I\right)$ before transiting to recovery $R$. Here, the dispersions $k_E$ and $k_I$ are the shape parameters of their respective gamma distributions, consistent with usage in [9].

Table 1.

Representative COVID-19 epidemiological parameters.

Symbols	Parameter	Values Simulated
$R_0$	basic reproduction number	$R_0$=2.0, 3.0
E	latent (exposed) period Gamma $\left({\mu }_E,k_E\right)$	${\mu }_E$=3.5 days $k_E$=4
I	infectious period Gamma $\left({\mu }_I,k_I\right)$	${\mu }_I$=5.5 days $k_I$=0.3,0.7

Our model is unstratified, e.g., it does not specify age-specific contact rates with matrices [23]. The SI in [9] points out that individual variation in the number of secondary infections has many causes residing in the host, pathogen, and environment. Regardless of cause, however, given the basic reproduction number $R_0$, the individual reproduction number $Z\sim \mathrm{N}\mathrm{e}\mathrm{g}\mathrm{B}\mathrm{i}\mathrm{n}\left(R_0,k_I\right)$ pertains, if $Z\sim \mathrm{P}\mathrm{o}\mathrm{i}\mathrm{s}\mathrm{s}\mathrm{o}\mathrm{n}\left(v\right)$ and $v\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}\left(\mu ,k_I\right)$ [9]. Under our unstratified model, then, the shape parameter $k_I$ of $v$ equals the dispersion of secondary infections $Z\sim \mathrm{N}\mathrm{e}\mathrm{g}\mathrm{B}\mathrm{i}\mathrm{n}\left(R_0,k_I\right)$, so contact tracing studies can determine $k_I$ empirically.

Fig 1 and Table 1 display the Gamma and Negative Binomial distributions with the mean-dispersion parametrizations popular in epidemiology. Sections following rely heavily on Poisson processes, however, making the standard mathematical parametrizations of the Gamma distribution (shape-rate) and the Negative Binomial distribution preferable. We denote the mathematical parametrizations with underlines, e.g., $T_E\sim \underset{\_}{\mathrm{Gamma}}\left(k_E,k_E/{\mu }_E\right)$. Appendix A provides a convenient dictionary for translating between the notations.

Our epidemic model uses an unstratified population (all individuals probabilistically equivalent), so its approximation involves a single-type branching process. To avoid any unintended connotations from epidemiological terminology, we switch to branching process terminology, e.g., we use “particle” instead of “individual”.

A branching process approximation applies to the start of an epidemic, when the number of susceptible individuals is large in comparison the number of infective individuals. The epidemic branching process originates with a single ancestor (primary infection), involves only a single type of particle (infected individual), and each particle gives birth to offspring (secondary infections). The defining property of a branching process is that all particles are mutually independent, with the same probability law for lifetime and reproduction after birth.

The epidemic branching process is in fact a special case of the Crump-Mode-Jagers (CMJ) process. Let the particle $x$ be born at time ${\sigma }_x$. In the CMJ process, a random triple $\left(L_x,{\xi }_x,{\chi }_x\right)$ (whose elements may be dependent) characterizes the reproduction of $x$, as follows. The lifetime $L_x$ of the particle $x$ is a positive real random variate; and within the time-interval $\left[{\sigma }_x,t\right]$, the variate ${\xi }_x\left(t-{\sigma }_x\right)$ counts the offspring that $x$ bears, and ${\chi }_x\left(t-{\sigma }_x\right)$ indicates a random characteristic of $x$, where ${\chi }_x\left(t-{\sigma }_x\right)=0$ for $t<{\sigma }_x$. After a particle’s lifetime, further “births” are ignored, so ${\xi }_x\left(L\right)$ counts the total offspring of $x$ and the random birth measure ${\xi }_x$ effectively satisfies ${\xi }_x\left(t-{\sigma }_x\right)={\xi }_x\left(L\right)$ for ${\sigma }_x+L<t$. The triples $\left(L_x,{\xi }_x,{\chi }_x\right)$ for each particle $x$ are mutually independent, but the triples all have the same law as a generic triple $(L,\xi ,\chi )$ for the particle born at time $\sigma =0$ (the ancestor of the population). The branching process counted by random characteristics is the stochastic process

$$Z^{\chi }(t)=\sum _{(x)}{\chi }_x\left(t-{\sigma }_x\right),$$ (1)

where the summation is over all particles in the branching process (and therefore, over all particles born no later than $t$). For any statement $X$, define the Iverson bracket $\left[X\right]=1$ if $X$ is true and 0 otherwise. For example, the ancestor is alive at time $t$ iff (if and only if) $0\le t\le L$, so with ${\chi }_x\left(t\right)=[0\le t\le L],Z^{\chi }\left(t\right)$ in Eq (1) counts the particles alive at time $t$.

The epidemiological CMJ process starts with an ancestor at epidemic time $t=0$. In the shape-rate parametrization (see Appendix A), Fig 1 yields an exposed period $T_E\sim \underset{\_}{\mathrm{Gamma}}\left(k_E,k_E/{\mu }_E\right)$ and an infectious period $T_I\sim \underset{\_}{\mathrm{Gamma}}\left(k_I,k_I/{\mu }_I\right)$. The ancestor therefore has lifetime $L=T_E+T_I$ beyond which it is no longer fertile (able to infect). The following specifies the law of the random birth measure $\xi$. The particle is fertile during the infectious interval ($T_E,T_E+T_I$], during which its births (secondary infections) form a Poisson process of constant rate. The Poisson rate is $R_0/{\mu }_I$, so that over an mean infectious period ${\mu }_I$, the mean number of secondary infections is the basic reproduction number $R_0$, which completes the specification of $\xi$. In Eq (1), e.g., random characteristics ${\chi }_{\chi ,E}\left(t\right):=[0\le t<E]$ and ${\chi }_{\chi ,I}\left(t\right):=[E\le t<E+I]$ can count the exposed and infected particles, and other random characteristics can formalize other processes of interest. The law of the triple $(\lambda ,\xi ,\chi )$ for a generic particle born at time $\sigma =0$, given above, specifies the epidemiological CMJ process.

2.2. The Poisson process under a gamma-distributed time-constraint

Appendix B contains results for the epidemiological CMJ process, and we collect the results of immediate use here. Define the factorial $(k-1)!:=\mathrm{\Gamma }\left(k\right)$ for real $k\ne 0,-1,-2,\dots$, and the ascending factorial $k^{\bar{n}}:=k(k+1)\dots (k+n-1)$ [24] for real $k$ and integer $n\ge 0$.

Consider a Poisson process of constant rate $a\lambda$, with occurrences at random times $T_n$, so the Poisson process yields points $\mathbf{T}:=\left(T_1,T_2,\dots \right)$, where $T_1<T_2<\dots$. Let $T\sim \underset{\_}{\mathrm{Gamma}}(k,p\lambda )$ be the distribution of a random time $T$, i.e.,

$$P(t\le T<t+\mathit{dt})=e^{-p\lambda t}\frac{{(p\lambda t)}^{k-1}}{(k-1)!}p\lambda \mathit{dt}.$$ (2)

Let the integer random variate $Z$ count the Poisson points within the time constraint $[0,T]$, so $[Z=n]:=\left[T_n\le T<T_{n+1}\right]$. Let $t_1<\dots <t_n\le t,{\mathbf{t}}_{\left(n\right)}:=\left(t_1,\dots ,t_n\right)$, and ${\mathbf{T}}_{\left(n\right)}:=\left(T_1,\dots ,T_n\right)$, so ${\mathbf{T}}_{\left(n\right)}={\mathbf{t}}_{\left(n\right)}$ is a possible realization of the Poisson process up to time $t$. Then, $Z\sim \underset{\_}{\mathrm{NegBin}}(k,\rho )$, i.e.,

$$P(Z=n)=\frac{k^{\overline{n}}}{n!}{(1-\rho )}^n{\rho }^k,$$ (3)

where $\rho ≔p/(a+p)$.

Appendix B derives the joint distribution of $\left(Z,T,{\mathbf{T}}_{\left(Z\right)}\right)$ and shows how to simulate it efficiently.

Two different values of the free parameter $a$ are relevant to the present article. The second appears later, but the first corresponds directly to the epidemiological CMJ process: $a=1-p$ with $(1-p)\lambda =R_0/{\mu }_I$ for the rate of infection; and $p\lambda =k_I/{\mu }_I$ to match the mean of the gamma-distributed infectious period. Thus, the individual reproduction number, the number of offspring, $Z\sim \underset{\_}{\mathrm{NegBin}}\left(k_I,p\right)$, where $p=k_I/\left(R_0+k_I\right)$ [9].

We wish to condition the epidemic on non-extinction, however, so the following theory becomes relevant.

2.3. A Galton-Watson (GW) process and its Harris-Sevastyanov (HS) transformation

The GW process is the archetypal branching process, and the analysis of extinction in a CMJ branching process relies on its embedded GW process, as described below and in [25]. The reader should keep the epidemiological CMJ process in mind throughout the following.

A GW process starts with a single ancestral particle in Generation 0. The ancestor gives birth to a random number $Z:=Z_1$ of offspring according to a probability distribution $p_z=P(Z=z)$ ($z=\mathrm{0,1},\dots )$, e.g., $Z\sim \underset{\_}{\mathrm{NegBin}}\left(k_I,p\right)$ in the epidemiological CMJ process. To avoid trivialities, assume henceforth that $p_0,p_1<1$.

Given $Z_g$ offspring in Generation $g$, each offspring independently and simultaneously gives birth to offspring in Generation $g+1$ according to the distribution $\left(p_z\right)$. In a GW process, every particle occurs in a particular generation number ($g=\mathrm{0,1},\dots$). For a CMJ process, the induction from Generation $g$ to Generation $g+1$ defines a GW process embedded in the CMJ process.

The population goes extinct iff $Z_g=Z_{g+1}=\dots =0$ for $g$ large enough, i.e., population count 0 is an absorbing state. In the notation from [26], p.47, let $A$ be the event that the population survives forever; $B$, the complementary event of extinction. Let “≔” denote a definition and define the probability $q≔PB$ of extinction, so $PA=1-q$. Define the offspring probability generating function (pgf),

$$f(s)≔{\mathit{Es}}^Z=\sum _{z=0}^{\infty }{p}_z{s}^Z.$$ (4)

Here the dummy variable $s$ marks offspring, i.e., the exponent $z$ of the variable $s$ counts the offspring, e.g.,

$$f(s)=\sum _{n=0}^{\infty }\frac{k_I^{\overline{n}}}{n!}{(1-p)}^n{p}^{k_I}{s}^n={\left[\frac{p}{1-(1-p)s}\right]}^{k_I}.$$ (5)

in the epidemiological CMJ process [9].

Imagine now a complete realization with all generations of the GW process, so the $Z_g=A_g+B_g$ particles in each Generation $g$ can be partitioned into $A_g$ skeleton particles whose lineage survives forever and $B_g$ doomed particles whose lineage goes extinct. A particle is doomed iff all its offspring are doomed, so $q=f\left(q\right)=\sum _{z=0}^{\mathrm{\infty }} p_z{q}^z$.

Assume that the GW process under scrutiny can go extinct, so $p_0>0$. Assume also that the mean number of offspring $m≔f^{\text{'}}\left(1\right)>1$, so it is supercritical, i.e., it does not go extinct with probability 1: $0<q<1$. Because $f\left(0\right)=p_0>0$ and $f\left(s\right)$ is a pgf, $f\left(s\right)$ is positive, increasing, and convex for $0\le s\le 1$, with $f\left(1\right)=1$. Consequently, (1) $q$ is the smallest positive root of $q=f\left(q\right)$; and (2) (draw a graph to see that in the notation of [26], p. 184) $\gamma ≔f^{\text{'}}\left(q\right)<1$.

The HS transformation assigns each particle randomly and independently at its birth a type, either “skeleton” or “doomed”, with probabilities $1-q$ and $q$, conditioning the particle’s offspring distribution on its type to conserve the probability law of the GW process, as follows. Let the variate $a$ mark skeleton particles; the variate $b$, doomed particles. The conservation equation

$$f((1-q)a+\mathit{qb})=(1-q)f_a(a,b)+{\mathit{qf}}_b(b).$$ (6)

combines the conditioned offspring pgfs for skeleton $f_a(a,b)$ and doomed $f_b\left(b\right)$ types, as follows. On the left, the particle has pgf $f\left(s\right)$ unconditioned on whether it has a skeleton or doomed type, and the argument $(1-q)a+qb$ assigns an independent random type to each offspring: skeleton with probability $1-q$ and doomed with probability $q$. The two terms on the right correspond to a particle itself having a skeleton or doomed type (i.e., the factors $1-q$ and q), and the factors $f_a(a,b)$ and doomed $f_b\left(b\right)$ give the corresponding conditional pgfs for offspring. A skeleton particle can have both skeleton and doomed offspring but must have at least one skeleton offspring, whereas a doomed particle can have only doomed offspring ($f_b\left(b\right)$). The HS transformation [27; 28] therefore maps a simple GW process into a two-type GW process by typing each offspring as either skeleton or doomed [29]: like the general treatment in [26], p. 184, we have

$$\begin{gathered} f_a(a,b)=\frac{f((1-q)a+\mathit{qb})-f(\mathit{qb})}{1-q} \\ {f}_b(b)=\frac{f(\mathit{qb})}{q}. \end{gathered}$$ (7)

explained as follows. The second equation is the dual reproduction law [29], reflecting the fact that a particle is doomed iff all its offspring are doomed. The first equation then follows from the dual reproduction law and Eq (6) for conservation of probability. Note $f_a(0,b)=0$, i.e., a skeleton particle must have at least one skeleton offspring.

2.4. The single-skeleton renewal in the epidemiological GW process.

The Introduction observes that a branching population may encounter several bottlenecks through a single particle, with the sample MRCA as the final bottleneck. The observation motivates an examination of the following single-skeleton renewal.

Consider a GW process that does not go extinct. The ancestor must be a skeleton particle, so $A_0=1$. Every skeleton particle must have at least one skeleton offspring, so $A_g>0$ for every $g=\mathrm{0,1},\dots$. Now, the sequence of events $\left[A_g=1\right](g=\mathrm{0,1},\dots )$ forms a renewal process because its probability law renews in every generation where $A_g=1$. The renewal is defective, because the number of skeleton particles is non-decreasing, and eventually the event $A_g>1$ occurs.

In the standard notation of [30], p. 1, let $\left[a^z\right]g\left(a\right)$ denote the coefficient of $a^z$ in the Maclaurin expansion of a function $g\left(a\right)$ analytic at $a=0$. The HS transformation of a GW process shows that the coefficient $\left[a^1\right]f_a(a,1)$ gives the probability that a skeleton particle has exactly one skeleton offspring. Thus, the probability that the renewal does not terminate in the transition from Generation $g-1$ to Generation $g$ is

$$\begin{gathered} P\left(A_g=1\mid A_{g-1}=1\right)=\left[a^1\right]f_a(a,1)=\left[a^1\right]\frac{f((1-q)a+q)-q}{1-q} \\ =\frac{\left[a^1\right]f((1-q)a+q)}{1-q}=\frac{\sum _{j=0}^{\infty }{p}_j\left[a^1\right]{((1-q)a+q)}^j}{1-q} \\ =\frac{\sum _{j=0}^{\infty }{p}_{j}j(1-q)q^{j-1}}{1-q}=\sum _{j=0}^{\infty }{p}_{j}j{q}^{j-1}=f^{\text{'}}(q)=\gamma \end{gathered}$$ (8)

Differentiating the dual reproduction law of Eq (7) at $b=1$ shows that $\gamma$ is the average number of (doomed) offspring per doomed particle. Because $\gamma <1$ (see Section 2.3), the expected total descendants in a doomed lineage is

$$1+\gamma +{\gamma }^2+\dots =1/(1-\gamma ).$$ (9)

Eq (8) provides an attractive alternative interpretation of $\gamma$ as the probability that the single-skeleton renewal continues into the next generation.

Let $G≔\mathrm{m}\mathrm{i}\mathrm{n}\left\{g>0:A_g>1\right\}$ be the first generation with more than one skeleton particle, i.e., if $g<G$, at long times the phylogeny of the population descends from a unique ancestor in Generation $g$, but after $g\ge G$, its phylogeny splits. The skeleton particle at Generation $G-1$ is therefore the MRCA of the long-time population. Accordingly, it is the long-time MRCA, and $G-1$ counts the generations from the primary infection to the long-time MRCA. Induction shows $P(G>g)={\gamma }^g$ for $g=\mathrm{1,2},\dots$, so

$$P(G=g)={\gamma }^{g-1}-{\gamma }^g={\gamma }^{g-1}(1-\gamma )$$ (10)

and $G\sim \mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}(1-\gamma )$. The mean $E[G-1]=\gamma /(1-\gamma )$ and its variance ${\sigma }^2(G-1)=\gamma /(1-\gamma {)}^2$, so unless $\gamma$ is close to 1, the distribution is tightly distributed on small counting numbers.

2.5. The epidemiological CMJ process and its HS transformation

The mean and variance at the end of Section 2.4 suggest that in many parameter ranges, only a few generations separate an ancestor and the long-time MRCA. The duration between them is often of greater interest, however. We therefore investigate the HS transformation of the epidemiological CMJ process.

Section 2.3 shows that every CMJ process has an embedded GW process, and the extinction of a lineage in the embedded GW process is equivalent to extinction of the lineage in the CMJ process [25]. Thus, the HS transformation of the embedded GW process endows every particle in the CMJ process with the skeleton or doomed type. As in Eqs (6) and (7), the randomization of particle types conditions the law of each triple $\left(L_x,{\xi }_x,{\chi }_x\right)$ accordingly. In general, the recipe determining the conditioned laws obtains the dual reproduction law first (as in the second Eq (7)) and then uses conservation (as in the conservation Eq (6)). The recipe can be carried out explicitly for the epidemiological CMJ process, as follows.

In the embedded GW process, the offspring distribution $Z\sim \underset{\_}{\mathrm{NegBin}}\left(k_1,p\right)$, so numerical solution of Eq (5) yields the extinction probability $q$ [9]. Thus,

$$q=f(q)={\left[\frac{p}{1-(1-p)q}\right]}^{k_I};$$ (11)

$$\gamma =f^{\text{'}}(q)=f(q)k_I\frac{1-p}{1-(1-p)q}=k_I\frac{(1-p)q}{1-(1-p)q}.$$ (12)

The defective single-skeleton renewal occurs between the births of the primary infection and the long-time MRCA. With the generation count $G\sim \mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}\mathrm{e}\mathrm{t}\mathrm{r}\mathrm{i}\mathrm{c}(1-\gamma )$ in hand, the distribution of the time between the successive skeleton births in the renewal determines the distribution of the total time from the ancestor to the long-time MRCA.

To derive the distribution of time-intervals between the successive skeleton births, note that the latent period $T_E$ is independent of offspring during the infectious period, so conditioning on a single skeleton daughter affects only the infectious period $T_I$. Now, infection occurs as a Poisson process under a gamma-distributed time constraint $T_I\sim \underset{\_}{\mathrm{Gamma}}\left(k_I,k_I/{\mu }_I\right)$, so we turn once again to the set-up in Section 2.2. Let $T=T_I$ be the time constraint on the Poisson process, where $p\lambda =k_I/{\mu }_I$ as above. Unconditioned, a particle bears offspring according to a Poisson process of rate $(1-p)\lambda$. As in the dual reproduction law in Eq (7), however, the doomed offspring form a Poisson process of the reduced rate $q(1-p)\lambda$. The complementary skeleton Poisson process therefore yields skeleton offspring at rate $a\lambda$, where $a=(1-q)(1-p)$, with truncation at 0 because every skeleton particle has at least one skeleton offspring. For the epidemiological CMJ process conditioned on a skeleton ancestor, the calculation in

Appendix B with values $p=k_I/\left(R_0+k_I\right),a=(1-q)R_0/\left(R_0+k_I\right)$, and $\rho =p/(a+p)=k_I/\left[(1-q)R_0+k_I\right]$ gives the following.

The number of skeleton daughters has distribution $A\sim \underset{\_}{\mathrm{NegBin}}\left(k_I,\rho \right)$ truncated at $A=0$, where $\rho ≔p/(a+p)$. Eq (11) yields $q={\rho }^{k_I}$ and direct substitution of $A\sim \underset{\_}{\mathrm{NegBin}}\left(k_I,\rho \right)$ truncated at $A=0$ yields $\gamma =P(A=1\mid A>0)$. The algebra in Appendix C shows that the new expression for $\gamma =P(A=1\mid A>0)$ accords with Eq (8).

During the single-skeleton renewal, the maternal age at the birth of the only skeleton offspring corresponds to the first Poisson time-point $T_1$ of a (0-truncated) skeleton Poisson process under time-constraint. The following therefore simulates the duration $S$ of the single-skeleton renewal.

Simulation.

(1) $G\sim \mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}(1-\gamma )$. If $G=1$, the ancestor and the long-time MRCA are identical and $S=0$. Otherwise, conditioned on $G>1$. (2) the sum of the latent periods in the single-skeleton renewal is ${\sum }_{g=1}^{G-1}{T}_E^{\left(g\right)}\sim \underset{\_}{\mathrm{Gamma}}\left(\left(G-1\right)k_{E,}{k}_E/{\mu }_E\right)$. For $g=\mathrm{1,2},\dots ,G-1:$ (3) the $g\text{-th}$ infectious period $T_I^{\left(g\right)}\sim \underset{\_}{\mathrm{Gamma}}\left(k_I+1,\left(a+p\right)\lambda \right)$; and (4) conditioned on $T_I^{\left(g\right)}$, the interval $\left[0,T_I^{\left(g\right)}\right]$ contributes $T_B^{\left(g\right)}\sim \mathrm{Uniform}\left[0,T_I^{\left(g\right)}\right]$ to the maternal age at the skeleton birth. Record the duration $S=\sum _{g=1}^{G-1} T_E^{\left(g\right)}+\sum _{g=1}^{G-1} T_B^{\left(g\right)}$ of the single-skeleton renewal.

2.6. Code and data availability

Our GitHub repository stores all relevant Python programs. In it, each directory contains a “requirements.txt” with the version numbers for package dependencies. All simulations used 1000 realizations.

3. Results

For the epidemiological parameter values considered in Fig 2 and Fig 3, the ancestor and the long-time MRCA are identical with probability about 0.5. Fig 2 shows that for the parameter values considered, the generations from the ancestor to the long-time MRCA rarely exceed 5; Fig 3, that the time between the birth of the ancestor and the long-time MRCA rarely exceeds 50 days.

Fig 2. Cumulative distribution functions (cdfs) for the generations in the single-skeleton renewal.

Fig 2 plots the cdfs $P(G-1\le g)=1-{\gamma }^{g+1}$ for the generations $G-1$ separating the ancestor from the long-time MRCA for $g=\mathrm{1,2},\dots$ The atoms at $g=1$ give the probability that the ancestor and the long-time MRCA are the same. From bottom to top: the purple circles correspond to $\left(R_0,k\right)=\left(\mathrm{2,\; 0.3}\right)$; the blue squares, to $\left(R_0,k\right)=\left(\mathrm{2,\; 0.7}\right)$; the orange diamonds, to $\left(R_0,k\right)=\left(\mathrm{3,\; 0.3}\right)$; and the red triangles, to $\left(R_0,k\right)=\left(\mathrm{3,\; 0.7}\right)$. For all sets of parameters examined, the single-skeleton renewal is short: the ancestor and the long-time MRCA are often the same, and effectively no single-skeleton renewal is longer than $G-1=6$ generations. The GitHub site contains the original data for the plot in the file Output/Generations/generations.csv.

Fig 3. Cumulative distribution functions for the durations of the single-skeleton renewal.

Fig 3 plots sample cdfs $P(S\le s)$ for the duration $S$ separating the ancestor from the long-time MRCA. The simulation used 1000 realizations. The atoms at $S=0$ give the probability that the ancestor and the long-time MRCA are the same. From bottom to top: the solid purple lines correspond to $\left(R_0,k\right)=\left(\mathrm{2,\; 0.3}\right)$; the long blue dashes, to $\left(R_0,k\right)=\left(\mathrm{2,\; 0.7}\right)$; the orange dashes, to $\left(R_0,k\right)=\left(\mathrm{3,0.3}\right)$; and the short red dashes, to $\left(R_0,k\right)=\left(\mathrm{3,0.7}\right)$. For all sets of parameters examined, the single-skeleton renewal is short: the ancestor and the long-time MRCA are often the same, and few single-skeleton renewals are longer than about 50 days. The GitHub site contains the original data for the plot in the file Output/Durations/durations.csv.

In the Introduction, Pekar et al [10] estimated that 95.1% of HPD of the tMRCA postdated 2019.11.17, and the dates for the ancestor had a median 2019.11.03; 95% HPD 2019.10.14; and 99% HPD 2019.10.04. An online service, Timeanddate.com [31], counted days before 2019.11.17 up to and including the ancestor’s birth to yield a median of 13 days; a 95% HPD of 30 days; and a 99% HPD of 44 days. Against Fig 3, the median in [10] is noticeably larger, but the estimates are otherwise comparable.

To help interpret Fig 3, Table 2 lists values derived from the parameters in Table 1. The parameters ${\mu }_E,k_E$, and ${\mu }_I$ are not in Table 2 but equal the unambiguous values in Table 1; the ambiguous parameters $R_0$ and $k_I$ in Table 1 yield the 4 combinations of ambiguous values in Table 2. The first 2 columns list in increasing order the basic reproduction number $R_0$ and the dispersion $k_I$ of the individual reproduction number. The last 4 columns list: the extinction probability $q$; the mean number $\gamma =f^{\text{'}}\left(q\right)$ of doomed offspring of a doomed particle; the mean number $\gamma /(1-\gamma )$ of the $G-1$ generations separating the ancestor and the long-time MRCA (add 1 to get $1/(1-\gamma )$, expected total descendants in a doomed lineage in Eq (9)); and the standard deviation $\sqrt{\gamma }/(1-\gamma )$ of the $G-1$ generations. The last 2 columns emphasize that in the parameter ranges examined, the generations from the ancestor to the long-time MRCA is usually a small non-negative integer. The atoms on the Y-axis of Fig 2 and Fig 3 are the probabilities $1-\gamma$ complementary to the values $\gamma$ in Column 3.

Table 2.

Summary statistics relevant to the single-skeleton renewal.

Basic reprod number	Dispersion	Extinction probability	Mean doomed offspring	Mean generations skeleton	St dev generations skeleton
$R_0$	$k_I$	q	γ	$\mathit{\gamma }/(1-\mathit{\gamma })$	$\sqrt{\gamma }/(1-\gamma )$
2	0.3	0.739	0.540	1.172	1.600
2	0.7	0.571	0.514	1.056	1.470
3	0.3	0.628	0.399	0.663	1.050
3	0.7	0.416	0.356	0.553	0.930

Fig 2 and Fig 3 suggest robustness of our results. Pekar et al [10] considered more extreme possibilities, that sampled sequences of SARS-CoV-2 evolved from a less infectious progenitor. If viral infectiousness evolves, present results provide bounds on duration of the single-skeleton renewal. In the present paradigm, one extreme possibility in [10] resembles a decrease in $R_0$, shifting the curves in Fig 2 and Fig 3 further to right than shown. Regardless, the sharp cut-off induced by the geometric tail of $G\sim \mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}(1-\gamma )$ promotes robustness of results, while the model simplicity promotes intuitive understanding. The GitHub site contains code to compute parameters in specific cases of interest.

Numerical evidence mentioned in Appendix D suggests that for a $\underset{\_}{\mathrm{NegBin}}(k,p)$ offspring distribution, $q$ and $\gamma$ increase (or decrease) together as the parameters ($k,p$) vary. As $\gamma$ increases, the geometric variate $G\sim \mathrm{G}\mathrm{e}\mathrm{o}\mathrm{m}(1-\gamma )$ increases (stochastically). Eq (18) therefore downweights the corresponding event $F$ through the non-extinction probability $1-q$, effectively downweighting ancestors of extremely low infectiousness.

4. Discussion

The present article emphasizes the role of the long-time MRCA, the MRCA of a population extant a long time after the population origins. In practice and at a minimum, a “long time” exceeds extinction times for doomed lineages (see Eq (9) and Table 2).

Although the present article analyzed the origins of COVID-19, it applies to epidemics in general, and even to any process or population whose early evolution is closely approximated by a branching process. In fact, many phylogenies implicitly use long-time MRCAs, because they sample a population long after its origins, e.g., many lineages have gone extinct since the origins of the human species, so the MRCA of the present human population is a long-time MRCA. In such cases, the relative temporal difference between the population ancestor and the long-time MRCA is often negligible.

Such is not the case when estimating recent epidemic origins, however, because small temporal differences may have an impact on scientific conclusions, as in COVID-19. The long-time MRCA then provides a useful reference for comparing MRCAs from alternative samples. On one hand, if a virus preceding the long-time MRCA enters a sample, the sample MRCA precedes the long-time MRCA. If the corresponding viral lineage goes extinct, the virus does not contribute to an epidemic at long times, increasing sampling variability [10]. On the other hand, if samples exclude lineages that survive at long times, the long-time MRCA can precede the sample MRCA. In both cases, when bootstrapping the tMRCA, the MRCA may vary during resampling, increasing the variance of the estimated tMRCA.

Towards reducing variance, the viral sample can easily be restricted to a long-time sample. Call the sample a full long-time sample if no larger long-time sample set has a different MRCA. If a full long-time sample is sufficiently dense, resampling with the bootstrap will usually select another full long-time sample, repeatedly estimating the long-time MRCA. In fact, the stable coalescence defined in [10] can be viewed as a sample estimator for the long-time MRCA. Other estimators are also possible.

In a hybrid analysis combining phylogenetic and epidemiological methods, therefore, a focus on the long-time MRCA can reduce variance, essentially by substituting analysis for a crude Monte Carlo method [32]. The phylogenetic methods work backward to the long-time MRCA; and the epidemiological methods can simulate the defective single-skeleton renewal to work forward to the long-time MRCA. Because both methods terminate in the long-time MRCA, their estimates are probabilistically independent, simplifying analysis enormously. The specification of the HS transformation of the epidemiological CMJ (Eq (7) and Section 2.5) provides the main obstacle to the scientific program.

Section 2.4 and the Results section achieved the immediate aim of verifying epidemiological estimates in [10]. They verified that the estimate the duration from the primary infection to the sample MRCA in COVID-19 is both short and relatively insensitive to the epidemiological parameters. Specifically, Section 2.4 shows that the number of generations is geometrically distributed, so the distribution is concentrated on small non-negative integers (Fig 2), with correspondingly small durations (Fig 3).

The present theory and the results in [10] are discrepant because the median duration in [10] is noticeably (but not exorbitantly) larger than the median duration here. Early viruses with doomed lineages might place the sample MRCA earlier than the long-time MRCA, but they would make the apparent duration smaller, not larger. Because the sample MRCA is not necessarily the long-time MRCA, the epidemiological and phylogenetic estimates may be probabilistically dependent, potentially complicating the practical analysis in [10] considerably.

Our analysis presents quantitative results for a range of epidemiological parameters derived from actual measurements (see Table 1). It also considers qualitative results for a more extreme possibility, that sampled sequences of SARS-CoV-2 evolved from a less infectious progenitor. Unsurprisingly, a less infectious progenitor increases the time of the single-skeleton renewal between the primary infection and the long-time MRCA. Appendix D shows, however, that a less infectious progenitor also increases the extinction probability of epidemic, so the more extreme the reduction in infectiousness, the smaller the progenitor’s probability.

The analysis in the present article generalizes naturally to multitype branching processes, to handle epidemiological models stratified by age, sex, etc. Extinction probabilities are readily calculated for multitype GW processes [33]. The HS transformation for multitype branching processes seems less explored (e.g., [34]), but the recipe at the end of Section 2.5 seems straightforward, even for multitype branching processes. The long-time MRCA makes the epidemiological and phylogenetic analyses probabilistically independent, simplifying a hybrid analysis enormously. The long-time MRCA generalizes to hybrid methods using multitype branching processes in a hybrid analysis of a population stratified by age, sex, etc. The advantages of using the long-time MRCA therefore seem accessible, even in models more intricate than the one used here.

Indeed, the transformation generalizes to many other branching processes besides GW processes. Simulations can implement the HS transformation to generate a skeleton branching process, often without excessive analysis or computation. Because many random graph models have an accurate branching process approximation [9; 12; 13; 15], with a corresponding HS transformation, instead of simulating the entire epidemic, simulations (e.g., in [10]) then can then focus on the single-skeleton renewal process and terminate at the long-time MRCA.

Highlights.

• Estimation of the founding date of a recent population can entail a hybrid analysis.

• Sequence phylogeny works backward to the MRCA of sequence samples.

• Population modeling works forward to the sample MRCA but depends on the sample.

• A “long-time” MRCA makes the forward and backward analysis independent.

7. Appendix A

Interconversion of distributional notations.

Gamma_distribution.

The shape-rate parametrization $T\sim \underset{\_}{\mathrm{Gamma}}(k,\lambda )$ of the gamma distribution denotes a distribution satisfying

$$P(t\le T<t+\mathit{dt})=e^{-\lambda t}\frac{{(\lambda t)}^{k-1}}{(k-1)!}\lambda \mathit{dt}.$$ (13)

The shape-rate parametrization is directly related a Poisson process occurring on a line (see Appendix B), suggesting the underline in our notation.

The mean-dispersion notation for the same variate $T\sim \mathrm{G}\mathrm{a}\mathrm{m}\mathrm{m}\mathrm{a}(\mu ,k)$ uses the mean $\mu =k/\lambda$ and characterizes the variance of the distribution with a dispersion, usually the shape parameter $k$. Sometime, however, the reciprocal of the shape parameter is also called the dispersion. The shape-rate parametrization is easily recovered from the mean-dispersion notation: $\lambda =k/\mu$.

Negative_binomial_distribution.

Let shape-rate parametrization of the gamma distribution is directly related to a parametrization $Z\sim \underset{\_}{\mathrm{NegBin}}(k,p)$ of the negative binomial distribution

$$P(Z=n)=\frac{k^{\overline{n}}}{n!}{(1-p)}^n{p}^k.$$ (14)

The mean-dispersion notation for the same variate $Z\sim \mathrm{N}\mathrm{e}\mathrm{g}\mathrm{B}\mathrm{i}\mathrm{n}(\mu ,k)$ uses the mean $\mu =k(1-p)/p$ and characterizes the variance of the distribution with a dispersion $k$. The original parametrization is easily recovered: $p=k/(k+\mu )$.

8. Appendix B

A Poisson process under a Gamma-distributed time-constraint.

Section 2.2 specifies the set-up for the Poisson process. Eq (15) below calculates the resulting point-distribution.

Within a probability, let a comma between events denote the corresponding intersection. Then, with $\rho ≔p/(a+p)$,

$$\begin{gathered} P\left(Z=n,T=dt,T_{(n)}=d{t}_{(n)}\right)=P\left(T=dt,T_{(n)}=d{t}_{(n)},T_n\le T<T_{n+1}\right) \\ ={\displaystyle \prod }_{j=1}^n\left\{e^{-a\lambda \left(t_j-t_{j-1}\right)}a\lambda d{t}_j{e}^{-a\lambda \left(t-t_j\right)}\right\}{e}^{-p\lambda t}\frac{{(p\lambda t)}^{k-1}}{(k-1)!}p\lambda dt \\ =\left\{n!{\displaystyle \prod }_{j=1}^{n}d{t}_j\right\}{e}^{-a\lambda t}\frac{{(a\lambda )}^n}{n!}{e}^{-p\lambda t}\frac{{(p\lambda t)}^{k-1}}{(k-1)!}p\lambda dt \\ =\frac{(k+n-1)!}{n!(k-1)!}{(1-\rho )}^n{\rho }^k\left\{\frac{n!}{t^n}{\displaystyle \prod }_{j=1}^{n}d{t}_j\right\}{e}^{-(a+p)\lambda t}\frac{{[(a+p)\lambda t]}^{k+n-1}}{(k+n-1)!}(a+p)\lambda dt \\ =\frac{k^{\overline{n}}}{n!}{(1-\rho )}^n{\rho }^k\left\{e^{-(a+p)\lambda t}\frac{{[(a+p)\lambda t]}^{k+n-1}}{(k+n-1)!}(a+p)\lambda dt\right\}\frac{n!}{t^n}{\displaystyle \prod }_{j=1}^{n}d{t}_j \\ =P(Z=n)P(T=dt\mid Z=n)dt\frac{n!}{t^n}{\displaystyle \prod }_{j=1}^{n}d{t}_j. \end{gathered}$$ (15)

The final factor in Eq (15) displays the density of the order statistics of a uniform distribution on $[0,T{]}^n$, because the order statistics are invariant under any of their $n!$ permutations.

Simulation of the time constrained Poisson process.

(1) $Z\sim \underset{\_}{\mathrm{NegBin}}(k,\rho )$; (2) conditioned on $Z,T\sim \underset{\_}{\mathrm{Gamma}}(k+Z,(a+p\left)\lambda \right)$ (I know of no reference for this useful fact); and (3) conditioned on $Z$ and $T$, generate independent variates $U_i\sim \mathrm{U}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}[0,T](i=\mathrm{1,2},\dots ,Z)$ and put them in ascending order to yield the coordinates of ${\mathbf{T}}_{\left(z\right)}$.

9. Appendix C

The probability γ that the single-skeleton renewal does not terminate.

Section 2.4 used the HS transformation of a GW process to calculate the probability $\gamma$ that the single-skeleton renewal continues for another generation. Section 2.5 gives an alternative calculation of $\gamma$. This Appendix verifies that the two calculations yield the same value.

In a Poisson process under time constraint without the 0-truncation, the number of skeleton daughters $A\sim \underset{\_}{\mathrm{NegBin}}\left(k_I,\rho \right)$, where $\rho ≔p/(a+p)$ and $a=(1-q)(1-p)$. Because $(1-q)(1-p)+p=1-(1-p)q$, so Eq (11) yields $q={\rho }^{k_I}$. The single-skeleton renewal focuses interest on $[A=1]$ within $[A>0]:P(A=0)={\rho }^{k_I}=q,P(A>0)=1-{\rho }^{k_I}=1-q$, and $P(A=1)=k_I(1-\rho ){\rho }^{k_I}=k_{I}aq/(a+p)$, again without 0-truncation. Even with 0-truncation accounted for,

$$P(A=1\mid A>0)=\frac{P(A=1)}{P(A>0)}=k_I\frac{a}{a+p}\frac{q}{1-q}$$ (16)

i.e., $P(A=1\mid A>0)=\gamma =P\left(A_g=1\mid A_{g-1}=1\right)$ from Eq (12), as it should.

10. Appendix D

Comparison of GW processes.

Given the distributions of the exposed and infectious periods, the GW parameters $q$ and $\gamma$ influence the robustness of the single-skeleton renewal in the Results. Consider the Negative Binomial pgf $f\left(s\right)$ given in Eq (5). For fixed $0<s<1$, the partial derivatives of $\mathrm{l}\mathrm{n}f\left(s\right)$ show that $f\left(s\right)$ is a decreasing function of $k_I$ and an increasing function of $p$, facilitating comparison.

The following places subscripts on $f\left(s\right)$ to compare GW processes. The subscripts will propagate to $q$ (the smallest non-negative root of $f\left(s\right)=s)$ or $\gamma ≔f^{\text{'}}\left(q\right)$, etc., without comment.

The extinction probability $q$ decreases with $f$.

Proposition 1. If $f_1\left(s\right)<f_0\left(s\right)$ for $0<s<\mathrm{1,0}<q_0$, and $q_1<1$, then $q_1<q_0$.

Proof: $0=f_1\left(q_1\right)-q_1<f_0\left(q_1\right)-q_1$. The graph of $f_0$ (because of its convexity properties) shows that $f_0\left(s\right)-s$ is strictly positive only for $0\le s<q_0$, so $0<f_0\left(q_1\right)-q_1$ implies $q_1<q_0$.

For a NegBinomial $(k,p)$ offspring distribution, Proposition 1 implies that $q$ decreases as $k$ increases or $p$ decreases. The behavior of $\gamma$ is less amenable to analysis than $q$, but the GitHub site code computing numerical values of $\gamma$ over a grid of $k$ and $p$. Over the grid, $\gamma$ decreases as $k$ increases or $p$ decreases, like $q$.

Bayesian weighting of Epidemiological Parameters.

Let $A$ be the non-extinction event, with probability $1-q_x≔P(A\mid X=x)$ given the parameter $X=x$, e.g., $x=(k,p)$ for the Negative Binomial offspring distribution. Let $F$ be any event occurring during the GW process. Let the parameter(s) $X=x$ have a known discrete unconditional probability distribution, e.g., a Bayesian prior, so

$$P(F)={\sum }_{(x)}P(F\mid X=x)P(X=x\mid A).$$ (17)

with obvious modification if $X$ has a continuous distribution. Conditional on the non-extinction event $A$,

$$\begin{gathered} P(F\mid A)=\sum _{(x)}P(F\mid X=x;A)P(X=x\mid A) \\ =\frac{{\sum }_{(x)}P(F\mid X=x;A)P(A;X=x)}{P(A)} \\ =\frac{{\sum }_{(x)}P(F\mid X=x;A)\left(1-q_x\right)P(X=x)}{P(A)}. \end{gathered}$$ (18)

Eq (18) shows that conditioning on non-extinction (unsurprisingly) reweights the parametrization from its original probability distribution $P(X=x)$ in Eq (17), multiplying it by the non-extinction probabilities $1-q_x$ (and then normalizing with $P\left(A\right)$).

5.2. Data Availability.

https://github.com/johnlspouge/2024-05-27_single_skeleton_renewal