On expected durations of birth-death processes, with applications to branching processes and SIS epidemics

TLDR: An asymptotic expression is derived for the expected time to extinction of the SIS epidemic, but now starting at the endemic state, which is not independent of the distribution of Q.

Abstract: We study continuous-time birth–death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E[Q] = 1, and where the birth rate if the population is currently in state (has size) n is α(n). We focus on two important examples, namely α(n) = λn being a branching process, and α(n) = λn(N −n)/N which corresponds to an SIS (susceptible → infective → susceptible) epidemic model in a homogeneously mixing community of fixed size N. The processes are assumed to start with a single individual, i.e. in state 1. Let T , An, C, and S denote the (random) time to extinction, the total time spent in state n, the total number of individuals ever alive, and the sum of the lifetimes of all individuals in the birth–death process, respectively. We give expressions for the expectation of all these quantities and show that these expectations are insensitive to the distribution of Q. We also derive an asymptotic expression for the expected time to extinction of the SIS epidemic, but now starting at the endemic state, which is not independent of the distribution of Q. The results are also applied to the household SIS epidemic, showing that, in contrast to the household SIR (susceptible → infective → recovered) epidemic, its threshold parameter R∗ is insensitive to the distribution of Q.

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Applied Probability Trust (7 January 2015)
ON EXPECTED DURATIONS OF BIRTH-DEATH PROCESSES,
WITH APPLICATIONS TO BRANCHING PROCESSES AND SIS
EPIDEMICS
FRANK BALL,
∗
University of Nottingham
TOM BRITTON,
∗∗
Stockholm University
PETER NEAL,
∗∗∗
Lancaster University
∗
Postal address: The Mathematical Sciences Building, University Park, Nottingham, NG7 2RD, UK
∗∗
Postal address: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden.
∗∗∗
Postal address: Department of Mathematics and Statistics, Fylde College, Lancaster University,
LA1 4YF, United Kingdom
1

2 Frank Ball, Tom Britton and Peter Neal
Abstract
We study continuous-time birth-death type processes, where individuals have
indep endent and identically distributed lifetimes, according to a random
variable Q, with E[Q] = 1, and where the birth rate if the population is
currently in state (has size) n is α(n). We focus on two imp ortant examples,
namely α(n) = λn being a branching process, and α(n) = λn(N − n)/N which
corresp onds to an SIS (susceptible → infective → susceptible) epidemic model
in a homogeneously mixing community of fixed size N. The processes are
assumed to start with a single individual, i.e. in state 1. Let T , A
n
, C and
S denote the (random) time to extinction, the total time spent in state n,
the total number of individuals ever alive and the sum of the lifetimes of all
individuals in the birth-death process, respectively. We give expressions for
the expectation of all these quantities and show that these expectations are
insensitive to the distribution of Q. We also derive an asymptotic expression
for the expected time to extinction of the SIS epidemic, but now starting at the
endemic state, which is not independent of the distribution of Q. The results
are also applied to the household SIS epidemic, showing that, in contrast to the
household SIR (susceptible → infective → recovered) epidemic, its threshold
parameter R
∗
is insensitive to the distribution of Q.
Keywords: Birth-death process; branching processes; SIS epidemics; insensitiv-
ity results.
2010 Mathematics Subject Classification: Primary 60J80
Secondary 60G10;92D30
1. Introduction
A key question for population processes of a birth-death type, for example, branching
processes and epidemic processes (with infection and recovery corresponding to birth
and death, respectively), is what effect does the lifetime distribution have on key
quantities of scientific interest? For example, consider a single-type branching process,
where individuals have independent and identically distributed (iid) lifetimes according
to a random variable Q having an arbitrary, but specified, distribution and, whilst alive,
give birth at the points of a homogeneous Poisson point process with rate λ. The basic
reproduction number, R
0
= λE[Q], the mean number of offspring produced by an
individual during its lifetime, depends upon Q only through its mean E[Q]. The mean

On expected durations of birth-death processes 3
total size of a subcritical branching process (R
0
< 1) with one ancestor is 1/(1 − R
0
),
which is again independent of the distribution of Q. However, other quantities of
interest, such as the probability of extinction and the Malthusian parameter of the
branching process, depend upon the distributional form of Q. Thus, in the language
of stochastic networks, R
0
can be viewed as an insensitivity result in that it depends
on Q only through its mean, see, for example, Zachary (2007).
Insensitivity results for stochastic networks are well known, see for example, Sev-
ast’yanov (1957), Whittle (1985) and Zachary (2007). In particular, in Zachary (2007),
Theorem 1, it is shown that for a wide class of queueing networks, where arrivals
(births) into the system are Poissonian with rate depending upon the total number
of individuals in the system and each arrival has an iid workload, the stationary
distribution of the total number of individuals in the system is insensitive to the
distribution of Q. It then follows automatically that, for example, the mean duration
of a busy period of the network (at least one individual in the system) is insensitive to
the distribution of Q.
Given the similarities between queueing networks and birth-death type models,
arrivals equating to births and workload equating to lifetime, we seek in this paper
to explore insensitivity results for birth-death type processes with particular emphasis
upon branching processes and SIS (susceptible → infective → susceptible) epidemic
models. In many cases, Zachary (2007), Theorem 1, cannot be applied directly to birth-
death processes, as many birth-death processes do not exhibit stationary behaviour.
For example, a branching process will either go extinct or grow exponentially. However,
we can exploit Zachary (2007), Theorem 1, for birth-death type processes whose mean
time to extinction is finite by introducing a regeneration step (cf. Hern´andez-Su´arez
and Castillo-Chavez (1999)) whenever the population goes extinct. That is, whenever
the population goes extinct, it spends an exponential length of time in state 0 (no
individuals) before a new individual is introduced into the population (regeneration).
The birth-death type process with regenerations then fits into the framework of Zachary
(2007), provided that the birth rate is Poissonian and depends upon the population
only through its size. Insensitivity results are then easy to obtain for the regenerative
process, and also for the original birth-death type process.
The paper is structured as follows. In Section 2, we formally introduce the generic

4 Frank Ball, Tom Britton and Peter Neal
birth-death type process with arbitrary birth rate α(n), where n denotes population
size, and introduce regeneration. We identify key insensitivity results for birth-death
type processes, including, that the mean duration, the mean time with n individuals
alive (n = 1, 2, . . .) and the mean total number of individuals ever alive in the process
are insensitive to the distribution of Q. In Section 3, we focus on three special
cases of the birth-death type process, namely, branching processes with constant birth
rate, and homogeneously mixing and household SIS epidemic models. In Section
3.1, we prove a conjecture of Neal (2014), that for a subcritical branching process,
the mean time with n (n = 1, 2, . . .) individuals alive is insensitive to Q and, using
Lambert (2011), Lemma 3.1, give a corresponding insensitivity result for critical and
supercritical branching processes. In Section 3.2, we apply the insensitivity results to
homogeneously mixing SIS epidemics and obtain a simple approximation for the mean
duration of the epidemic starting from a single infective. Moreover, we show that for
a supercritical epidemic (R
0
> 1), the mean duration of the epidemic starting from
the quasi-endemic equilibrium does depend upon the distribution of Q and we give
a simple asymptotic expression for this quantity. Finally, in Section 3.3 we exploit
the results obtained for the homogeneously mixing SIS epidemic to show that both
the threshold parameter R
∗
and the quasi-endemic equilibrium of the household SIS
epidemic are insensitive to the distribution of Q. These are interesting findings, as in
the household SIR (susceptible → infective → recovered) epidemic both R
∗
and the
fraction of the population ultimately recovered if the epidemic takes off do depend
upon the distribution of Q.
2. Generic model
The generic birth-death type process is defined as follows. The process is initiated
at time t = 0 with one individual. All individuals, including the initial individual,
have iid lifetimes according to an arbitrary, but specified, positive random variable Q
with finite mean. At the end of its lifetime an individual dies and is removed from
the population. New individuals are born and enter the population at the points of
an independent inhomogeneous Poisson point process with rate α(n) ≥ 0, where n
denotes the total number of individuals in the population. Without loss of generality,

On expected durations of birth-death processes 5
we assume that E[Q] = 1, since otherwise we can simply rescale time by dividing Q and
multiplying α(n) by E[Q]. The special cases of a branching process with individuals
giving birth at the points of independent homogeneous Poisson point processes with
rate λ and the homogeneously mixing SIS epidemic (see, for example, Kryscio and
Lef`evre (1989)) in a population of size N with infection rate λ, correspond to α(n) = nλ
and α(n) = nλ(N −n)/N , respectively.
The birth-death type process is similar to the single-class networks studied in Zachary
(2007), Section 2. In Zachary (2007), it is assumed that new individuals enter the
system (births) at the points of a Poisson process with state-dependent rate α(n),
where n is the total number of individuals currently in the system. Individuals have
iid workloads, according to a random variable Q with E[Q] = 1. While there are n
individuals in the system, the total workload is reduced at rate β(n) ≥ 0, with β(n) > 0
if and only if n > 0. In a biological setting, where the workload Q associated with an
individual is its lifetime, it only makes sense to take β(n) = n, so each individual’s
remaining lifetime decreases at constant rate 1.
In Zachary (2007), Theorem 1, it is shown that if the proper distribution π =
(π(0), π (1), . . .) satisfies the detailed balance equations
π(n + 1)β(n + 1) = π(n)α(n), n = 0, 1, . . . , (2.1)
and
∞

n=0
π(n)α(n) < ∞, (2.2)
then π is the stationary distribution of the size of the system, irrespective of the
distribution of Q.
For many biological systems, Zachary (2007) does not apply since a stationary
distribution for the total number of individuals alive does not exist. The solution
to make Zachary (2007), Theorem 1, relevant to birth-death type processes is to follow
Hern´andez-Su´arez and Castillo-Chavez (1999) and introduce regeneration by setting
α(0) = 1, leaving all other transition rates unchanged. Thus, if the process goes extinct,
it spends an exponentially distributed time, having mean 1, with no individual before
a new individual enters the p opulation leading to the process restarting (regeneration).

Frequently asked questions

Q1. What are the contributions in "On expected durations of birth-death processes, with applications to branching processes and sis epidemics" ?

The authors study continuous-time birth-death type processes, where individuals have independent and identically distributed lifetimes, according to a random variable Q, with E [ Q ] = 1, and where the birth rate if the population is currently in state ( has size ) n is α ( n ). The authors give expressions for the expectation of all these quantities and show that these expectations are insensitive to the distribution of Q. 

Q2. What is the way to determine the extinction probability of an epidemic?

Note that, subject to E[Q] = 1, pQ is least when Q is constant (i.e. P(Q = 1) = 1), so the model with a constant infectious period has the shortest mean time to extinction starting from quasi-endemic equilibrium. 

Q3. What is the mean size of a subcritical branching process?

The meantotal size of a subcritical branching process (R0 < 1) with one ancestor is 1/(1−R0), which is again independent of the distribution of Q. 

Q4. What is the distribution of (n) in a supercritical epidemic?

The epidemic initiated from a single infective either goes extinct very quickly or takes off and reaches an endemic equilibrium of a proportion (λ − 1)/λ of the population infected, cf. Kryscio and Lefèvre (1989). 

Q5. What is the definition of the within-household epidemic?

The within-household epidemic without additional global infections is simply a homogeneously mixing SIS epidemic with N = h and λ/N = λL, so λ(n) = λLn(h− n), and (2.12) and (3.23) imply thatR∗ = λG h∑ n=1 (h− 1)! 

Q6. What is the basic reproduction number of the approximating branching process?

conditional upon S, the total number of global infectious contacts emanating from the household has a Poisson distribution with mean λGS, so the basic reproduction number of the approximating branching process is R∗ = λGE[S]. 

Q7. What is the extinction probability of the branching process?

For λ > 1 and var(Q) < ∞,E [ T(N) Q] ∼ 11− pQ µ(N),where pQ is the extinction probability of the branching process studied in Section 3.1. 

Q8. What is the simplest way to calculate the number of individuals in a system?

In Zachary (2007), Theorem 1, it is shown that if the proper distribution π =(π(0), π(1), . . .) satisfies the detailed balance equationsπ(n+ 1)β(n+ 1) = π(n)α(n), n = 0, 1, . . . , (2.1)and∞∑ n=0 π(n)α(n) < ∞, (2.2)then π is the stationary distribution of the size of the system, irrespective of the distribution of Q.For many biological systems, Zachary (2007) does not apply since a stationary distribution for the total number of individuals alive does not exist. 

Q9. What is the role of Q in E?

assuming that A (N) Q and B (N) Q are both o(µ (N)) for general Q, the authors have thatE [ T(N) Q] =11− P (N)Q{ 1π(N)(1) − P (N)Q A (N) Q − (1− P (N) Q )B (N) Q } ∼ 11− pQ × 1 π(N)(1) =11− pQ µ(N), (3.26)which highlights the role of Q in E [ T(N) Q] . 

Q10. what is the iid lifetime of the ancestor?

the mean number of births whilst the process is in state k is α(k)Ak, so,including the initial ancestor and noting from (2.9) that E[A1] = 1, the authors have thatE[C] = E [ 1 +∞∑ k=1 α(k)Ak] = 1 +∞∑ k=1 α(k) π(k) π(0) = E[A1] + ∞∑ k=1 (k + 1) π(k + 1) π(0)= E[A1] + ∞∑ k=2 kE[Ak] (2.11) = E[S]. (2.12)As mentioned above, the authors consider branching processes where individuals have iid lifetimes according to Q (with E[Q] = 1) and whilst alive give birth at the points of independent homogeneous Poisson point processes with rate λ. 

Q11. How can the authors know the time to extinction of a SIS epidemic?

By studying Gaussian approximations for the endemic equilibrium, qualitative results on the time to extinction have beenobtained, see N̊asell (1999) and Britton and Neal (2010). 

Q12. What are the key insensitivity results for the birth-death type process?

The authors identify key insensitivity results for birth-death type processes, including, that the mean duration, the mean time with n individuals alive (n = 1, 2, . . .) and the mean total number of individuals ever alive in the process are insensitive to the distribution of Q.