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Mathematical Biosciences xxx (2003) xxx–xxx
www.elsevier.com/locate/mbs
Analyzing bioterror response logistics: the case of smallpox
Edward H. Kaplan
a
a,*
, David L. Craft b, Lawrence M. Wein
c
Yale School of Management, and Department of Epidemiology and Public Health, Yale School of Medicine,
Box 208200, New Haven, CT 06520-8200, USA
b
Operations Research Center, MIT, Cambridge, MA 02139, USA
c
Graduate School of Business, Stanford University, Stanford, CA 94305, USA
Received 8 July 2002; received in revised form 16 April 2003; accepted 3 May 2003
Abstract
To evaluate existing and alternative proposals for emergency response to a deliberate smallpox attack, we
embed the key operational features of such interventions into a smallpox disease transmission model. We use
probabilistic reasoning within an otherwise deterministic epidemic framework to model the Ôrace to traceÕ,
i.e., attempting to trace (via the infector) and vaccinate an infected person while (s)he is still vaccine-sensitive.
Our model explicitly incorporates a tracing/vaccination queue, and hence can be used as a capacity planning
tool. An approximate analysis of this large (16 ODE) system yields closed-form estimates for the total
number of deaths and the maximum queue length. The former estimate delineates the efficacy (i.e., accuracy)
and efficiency (i.e., speed) of contact tracing, while the latter estimate reveals how congestion makes the race
to trace more difficult to win, thereby causing more deaths. A probabilistic analysis is also used to find an
approximate closed-form expression for the total number of deaths under mass vaccination, in terms of both
the basic reproductive ratio and the vaccination capacity. We also derive approximate thresholds for initially
controlling the epidemic for more general interventions that include imperfect vaccination and quarantine.
2003 Published by Elsevier Inc.
Keywords: Bioterror response logistics; Contact tracing; Queueing; Traced vaccination; Mass vaccination; Smallpox
1. Introduction
The threat of bioterrorism, that is, the deliberate use of viruses, bacteria, toxins, or even insects
to harm civilian populations, existed before the September 11 terrorist attacks in the United
*
Corresponding author. Tel.: +1-203 432 6031; fax: +1-203 432 9995.
E-mail addresses: edward.kaplan@yale.edu (E.H. Kaplan), dcraft@mit.edu (D.L. Craft), lwein@stanford.edu (L.M.
Wein).
0025-5564/$ - see front matter 2003 Published by Elsevier Inc.
doi:10.1016/S0025-5564(03)00090-7
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States. However, the September 11 attacks (which indicated the existence of terrorists willing to
murder thousands of civilians), together with the deliberate and fatal delivery of anthrax via the
US Mail (which provided a bioterror Ôproof of conceptÕ) comprised a bioterror wake up call in the
United States and around the globe. Consequently, government agencies, academics, and the
population at large have started in earnest to assess possible bioterror scenarios and countermeasures.
While there are many points of entry for the mathematical sciences in efforts to counter the
threat of bioterror, we focus in this paper on analyzing the response to a smallpox attack in a large
city, with an eye towards understanding the consequences (in terms of deaths and cases of illness)
of different proposed response policies. Smallpox is one of the most feared bioterrorist threats [1],
despite the fact that it was eradicated in 1979 by the World Health OrganizationÕs (WHO)
campaign in one of the twentieth centuryÕs major health achievements [2]. Although the US is
stockpiling 286 million doses of smallpox vaccine [3], the Centers for Disease Control and PreventionÕs (CDC) interim response plan [4] does not call for mass vaccination in the event of an
attack. Rather, the plan (along with an independent expert panel [5]) calls for a surveillancecontainment strategy that combines the isolation of symptomatic cases with the vaccination of
traced contacts from those cases. A critical reason for pursuing contact tracing is that unlike many
infectious diseases, a person infected with smallpox who is vaccinated shortly after infection can
avoid serious disease complications and infectiousness. To cite from the CDCÕs interim plan,
‘‘Contact identification is the most urgent task when investigating smallpox cases since vaccination of close contacts as soon as possible following exposure but preferably within 3–4 days may
prevent or modify disease. This was the successful strategy used for the global eradication of
smallpox’’. Recent editorials call for an analysis of the CDC plan [6] and express skepticism about
the planÕs efficacy [7].
Motivated by this sequence of events, we previously reported a numerical analysis comparing
the performance of the CDCÕs Ôtraced vaccinationÕ (TV) strategy and a mass vaccination (MV)
strategy, where the entire population is vaccinated as soon as possible after an attack is detected,
in the event of a bioterror attack in a major metropolitan area [8]. In the present paper, we
perform an approximate analysis that results in closed-form expressions for the number of deaths
under TV and MV in terms of the various problem parameters.
From a modeling viewpoint, it is crucial to balance the epidemiological details against response
logistics in modeling different response policies, and the analysis of this paper underscores this
point. TV and MV operate on different time scales: TV requires the identification of new cases to
trigger further vaccination, and thus proceeds at the pace of the epidemic, whereas the time required for MV is dictated by the number of available vaccinators and the speed with which they
work, independently of the state of the epidemic. Our analysis shows that as a result of these
differences, the number of deaths that will occur under TV will scale with the population size more
or less independently of the initial number infections, while for MV, the reverse is true: deaths
scale with the initial number of infections, independently of the population size.
Our TV model is novel in two respects. First, it incorporates scarce vaccination resources and
places people in a queue, where they wait to be traced and vaccinated. As such, our model cannot
only compare response alternatives such as TV or MV, but can also be used as a capacity planning
tool. Second, we include a detailed accounting of the Ôrace to traceÕ: To assess the efficacy of TV,
we compare the time from when someone is infected until (s)he is no longer vaccine-sensitive to
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the time from when the infection occurs until the infector becomes symptomatic, seeks medical
care, names this contact, and this contact is traced and vaccinated (perhaps after waiting in a
queue).
While we know of no other work that has incorporated queueing into an epidemic model or
modeled the race to trace, there are several related papers that deserve mention. M€
uller [9] derived
an elegant stochastic model of contact tracing as an STD intervention. While there are some
analogous results from their analysis and our own, their model is fundamentally different in that
contact tracing in their model is a means of identifying infected persons (by way of screening tests
or medical exam) who can be removed from the population and treated, whereas contact tracing
in our TV model is a means of finding and vaccinating susceptibles or infected persons still in the
vaccine-sensitive stage of disease. Also, in the M€
uller et al. paper, only persons actually infected
by index cases are traced, whereas in our model, all reported but as yet untraced contacts, whether
infected or not, are traced and vaccinated. Finally, as will become clear, the tracing and vaccination efforts in our TV model are constrained by the number of available personnel and the rates
with which they can operate; there are no resource constraints in the model of M€
uller et al. In
addition, several papers [10–12] analyze various physical ring vaccination strategies, whereby
persons (or animals) within a certain distance from a disease outbreak are vaccinated. By contrast
our model does not consider physical distance as a variable. Finally, Halloran et al. [13] constructed a microsimulation model for the spread of small outbreaks in a ÔcommunityÕ of 2000
persons; though their simulation methodology is completely different than our analytical approach, the results they obtained are almost identical to those in [8] when compared on the same
scale, as expected based on the analysis in the current paper.
There have also been some statistical analyses that have fit historical smallpox outbreaks to
disease transmission models. Most notably, Gani and Leach [14] used such models to estimate the
basic reproductive ratio R0 for numerous smallpox outbreaks. Some of their models do include
contact tracing; however they do not directly tie the progress of infection in a contact to the
progress of infection in the infecting index. Also, Meltzer et al. [15] have constructed a simple
smallpox model (essentially a geometric progression), and considered the impact of vaccination
and quarantine policies by modifications of the geometric growth rates and removal rates, respectively, within their model. They conclude that vaccination alone is unlikely to control a
smallpox outbreak, but that vaccination in concert with quarantine can eradicate disease within
one year. A valuable contribution of their work is the assembly of data describing the duration of
various stages of infection for smallpox and proposed probability distributions for these stage
durations.
Our paper is organized as follows. In the next section, we embed TV operations into a smallpox
transmission model. Following the derivation of our TV model, in Section 3 we develop analytical
approximations for the number of deaths and the queue length process. Section 4 is devoted to a
discussion of threshold conditions for various TV-based policies that are necessary for the initial
control of a smallpox outbreak. Examples include the tracing and vaccination of Ôcontacts of
contactsÕ, more aggressive quarantine policies, and the CDCÕs interim smallpox response plan for
an attack in the United States. Section 5 analyzes the special case of mass vaccination (MV), and
Section 6 assesses the accuracy of the analytical approximations in Sections 3–5, and in Section
6.5 also shows how the recent results reported by Halloran et al. [13] are consistent with our
analysis. Our concluding remarks appear in Section 7.
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2. Traced vaccination
Motivated by the CDC smallpox response policy, we turn our attention to a model of traced
vaccination. Before deriving the equations for this model in Sections 2.3–2.5, we construct in
Section 2.1 a basic smallpox transmission model in the absence of any intervention save the
isolation of symptomatically infected individuals. In Section 2.2, we outline the basic operations
entailed in such a policy, and then expand the state space (and notation) of the smallpox model
described in Section 2.1 to account for these operations. Values for the model parameters are
discussed in Section 2.6.
2.1. A model without intervention
In the absence of intervention beyond the isolation of symptomatic smallpox cases, we consider
a closed-population smallpox model with mass-action transmission (i.e., free mixing) and staged
disease progression dynamics. Since we will build on this model to incorporate vaccine-related
interventions, we include four stages of infection: asymptomatic, non-infectious and vaccinesensitive (stage 1); asymptomatic, non-infectious and vaccine-insensitive (stage 2); asymptomatic,
infectious and vaccine-insensitive (stage 3); and symptomatic and isolated (stage 4). We note that
the transition from stage 3 to stage 4 in our model occurs at the time the person is isolated, which
may be some time after symptoms appear (e.g., the person may observe symptoms, then seek
medical help, and finally begin isolation). The duration of infection is exponentially distributed
for all disease stages, with mean duration in stage j given by rj1 , j ¼ 1; 2; 3; 4. We further assume
that a fraction d of symptomatic cases die from disease, while the remaining cases recover and are
immune to reinfection, and we ignore non-smallpox sources of mortality.
We define SðtÞ as the number of susceptibles in the population at time t, and Ij ðtÞ as the number
of infected persons in stage j of infection at time t, j ¼ 1; 2; 3; 4. Since only those in stage 3 of
infection are infectious, mass-action sets the rate of new infections equal to bSðtÞI3 ðtÞ at time t
where b is the disease transmission parameter. This leads to the following system of ordinary
differential equations (ODEs):
dSðtÞ
¼ bSðtÞI3 ðtÞ;
dt
ð1Þ
dI1 ðtÞ
¼ bSðtÞI3 ðtÞ r1 I1 ðtÞ;
dt
ð2Þ
dIj ðtÞ
¼ rj1 Ij1 ðtÞ rj Ij ðtÞ; j ¼ 2; 3; 4:
dt
This SEIR-like system is governed by the basic reproductive ratio R0 , where
R0 ¼
bSð0Þ bN
;
r3
r3
ð3Þ
ð4Þ
where N is the population size. The approximation in (4) follows from our assumption that a very
small fraction of the population is exposed to the smallpox attack. Following the introduction of
infection into the population, the number of infected persons will only increase if R0 > 1, as is
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typically the case. Also, the fraction i, of the population that is ultimately infected is well approximated by the root of the equation
ð5Þ
i ¼ 1 eR0 i ;
as is well known for SEIR-like models, and thus the total number of cases and deaths in the
population essentially equal N i, and N id respectively.
2.2. Overview and notation
When a person infected with smallpox is identified as symptomatic, (s)he is immediately isolated
to prevent further disease transmission. We refer to a newly identified and isolated symptomatically
infected person as an index case. At the time of isolation, the index case is interviewed to obtain a
list of contacts who were potentially exposed to infection via the index. Imperfect recall and imperfect tracing are modeled by assuming that of all those truly exposed to infection via the index
case, only a proportion are both named and located. The names of all newly (i.e., first-time)
identified contacts are entered into a tracing/vaccination queue. Upon reaching the front of the
queue, the contact is located and vaccinated (unless the contact has already become symptomatic,
in which case (s)he becomes a new index case). Note that although all contacts are vaccinated, the
vaccine is only effective in contacts who are either susceptible or in the vaccine-sensitive stage 1 of
infection. Even for these contacts, the vaccine is not perfect, and in addition, a fixed fraction of all
those vaccinated will die from vaccine complications. The rate with which this queue is serviced
(that is, contacts are located and vaccinated) depends upon both the number of persons involved in
contact tracing and vaccination (ÔserversÕ in queueing theory parlance), and the rate with which
these servers can work (which depends upon the time required to find contacts and vaccinate them).
Initially, we will not consider quarantine beyond the isolation of symptomatic cases, but later we
will show how various quarantine policies can easily be incorporated into the model.
The model has the five basic disease states outlined earlier. These are replicated to account for
persons who have yet to be named as a contact by an index case, who have been named but have
not yet been located and vaccinated and hence remain in queue (though symptomatic cases do not
enter the queue), or who have been located but unsuccessfully vaccinated. In addition are two
states that account for individuals who are immune (via vaccination or recovery from disease) or
dead (from either disease or fatal vaccine complications).
The state notation is as follows (all variables are time-dependent):
S ‘ ¼ number susceptible in level ‘ (‘ ¼ 0 corresponds to untraced persons, while ‘ ¼ 1 corresponds to those unsuccessfully vaccinated), ‘ ¼ 0; 1;
Ij‘ ¼ number infected and in disease stage j of level ‘ (again ‘ ¼ 0 corresponds to untraced persons while ‘ ¼ 1 corresponds to unsuccessfully vaccinated persons), j ¼ 1; 2; 3; 4; ‘ ¼ 0; 1;
Qj ¼ number in disease stage j (j ¼ 0 refers to susceptibles) in the tracing/vaccination queue
(and hence unvaccinated), j ¼ 0; 1; 2; 3 (note that symptomatic cases never appear in queue);
Z ¼ number immune (whether from the vaccine or recovery from disease);
D ¼ number dead.
Fig. 1 depicts a flow diagram indicating the states and feasible population flows between states
in the model. In addition, Table 1 provides the notation for the model parameters.
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I 30
S0
I 10
I 20
Q0
Q1
Q2
Q3
S1
I11
I 21
I 31
Z
I 40
I 41
D
Fig. 1. Referring to the notation in Section 2.2, the first three rows of compartments represent people who are untraced,
in the tracing/vaccination queue, and unsuccessfully vaccinated, respectively. The five columns of compartments correspond to the five disease stages (susceptible; asymptomatic, non-infectious and vaccine-sensitive; asymptomatic, noninfectious and vaccine-insensitive; asymptomatic, infectious and vaccine-insensitive; and symptomatic and isolated). In
the last row of compartments, immunization ðZÞ results from successful vaccination and disease recovery, and death ðDÞ
results from vaccine complications and disease.
Table 1
Parameter values for the model
Parameter
b
c
p
N
r1
r2
r3
r4
n
l
m0
m1
d
f
I10 ðsÞ
s
Description
Infection rate: uncongested ðR0 ¼ 3Þ
congested ðR0 ¼ 6Þ
Names generated per index
Fraction of infectees named by index
Population size
Disease stage 1 rate
Disease stage 2 rate
Disease stage 3 rate
Disease stage 4 rate
Number of vaccinators
Service rate
Vaccine efficacy, stage 0
Vaccine efficacy, stage 1
Smallpox death rate
Vaccination fatality rate
Initial number infected
Detection delay
Value
7
Reference
1
1
10 person day
2 · 107 person1 day1
50
0.5
107
(3 days)1
(8 days)1
(3 days)1
(12 days)1
5000
50/day (TV), 200/day (MV)
1.0
1.0
0.3
106
103
5 days
[14]
[14]
[2]
Text
Text
[2]
[2]
[2]
[2]
[16]
Text
Text
Text
[2]
[2]
Text
Text
2.3. Contact tracing
2.3.1. Overview
In this section we describe our model of contact tracing. Note that while all contacts newlyidentified by an index will be traced, only some of these contacts will actually have been infected
by the index. This leads to two different classes of contacts: those who were not infected by the
referring index, and those who were. This distinction is important, for the disease status of a
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contact infected by an index depends on the time that has lapsed from infection through detection
of the index case. Of key interest is the probability that a contact referred and infected by the same
index is still in the vaccine-sensitive stage 1 of infection at the time the index is detected, for only
such contacts can be saved by vaccination. In contrast, contacts named but not infected by an
index could be susceptible or in any stage of infection. Consistent with our free-mixing transmission model, we assume that such contacts are distributed in proportion to the state populations in the model.
When a new index is detected, (s)he identifies c contacts as having been potentially exposed to
infection. Of the indexÕs true contacts, however, only a proportion p are named and traced. In
particular, of the R0 ðtÞ persons an index detected at time t actually infected (R0 ðtÞ will be defined
mathematically below), only pR0 ðtÞ are named and traced. We refer to the tracing of contacts
infected by the index case as local tracing. The remaining c pR0 ðtÞ contacts named are persons
who were not actually infected by the index. We refer to the tracing of such contacts as random
tracing. The arrival rate from a level 0 (untraced) population compartment to the tracing/vaccination queue will thus be the sum of two terms: a random tracing term for contacts not infected by
the referring index case (which will be proportional to the population of the level 0 compartment)
and a local tracing term for contacts that are infected by the index case (which will not be proportional to the level 0 compartment size).
The rate at which new index cases are detected is given by r3 I3 ðtÞ where
I3 ðtÞ ¼ I30 ðtÞ þ Q3 ðtÞ þ I31 ðtÞ;
ð6Þ
the total number of stage 3 infectious persons in the population. For any level 0 (i.e., untraced)
population compartment, the random tracing term will always be of the form
½c pR0 ðtÞ compartment size
r3 I3 ðtÞ;
N
ð7Þ
while the local tracing term will always be of the form
E½# local contacts traced r3 I3 ðtÞ:
ð8Þ
The challenge is to model correctly the expected number of local contacts traced by disease stage
(clearly there are no susceptible local contacts traced), and approximate the resulting expression in
a way that allows us to continue with an ODE model.
2.3.2. Local tracing
For convenience, we assume that the initial bioterror attack occurs at time s, and that the TV
response begins at time 0 (so s is the detection delay). The expected number of persons infected by
an index case detected at time t is denoted by R0 ðtÞ, where
Z tþs
b½S 0 ðtÞ þ Q0 ðtÞ þ S 1 ðtÞ
er3 x b½S 0 ðt xÞ þ Q0 ðt xÞ þ S 1 ðt xÞ dx :
ð9Þ
R0 ðtÞ r3
0
The approximation assumes that the state populations are changing slowly relative to the relaxation time 1=r3 (3 days). To derive Eq. (9), we observe that a new index discovered at time t was
infectious x time units ago with probability er3 x , and was by free-mixing generating infections at
rate b½S 0 ðt xÞ þ Q0 ðt xÞ þ S 1 ðt xÞ. Note that early in the epidemic, one can neglect the Q0
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and S 1 terms and take S 0 ðt xÞ ¼ S 0 ð0Þ, which recovers the usual definition for R0 ¼ bS 0 ð0Þ=r3 bN =r3 at the start of the epidemic.
An index detected at time t will have generated R0 ðtÞ infections. However, only pR0 ðtÞ of these
will be detected. As stated earlier, this means two things: of the c contacts named, pR0 ðtÞ contacts
were infected by the index, while the remaining c pR0 ðtÞ contacts named were not infected by the
index case.
Suppose that an index case has just been detected at time t. What is the expected number of
previously untraced contacts currently in disease stage j that were infected by this index? Note that
while the index could have infected persons in the S 0 , Q0 or S 1 compartments, only those infected
who were in the S 0 compartment were untraced at the time of infection. Thus, we only have to
consider infections of persons in S 0 for purposes of local tracing. We also want to consider the
possibility that even if someone was in S 0 at the time the index infected him/her, there is a chance
that someone else could have randomly named this infectee before the index became symptomatic.
To proceed, we define
pj ðxÞ ¼ Prfperson in stage j at time x after infectiong:
ð10Þ
We need to determine the expected number of contacts infected by an index detected at time t who
are currently (i.e., at time t) in disease stage j; we refer to such contacts as local infectees, and
denote the expected number of local infectees in disease stage j by kj ðtÞ. We claim that
kj ðtÞ ¼ E½local infectees in level 0; disease stage j j index detected at time t
Z tþs
¼
er3 x bS 0 ðt xÞ Prfnot randomly named in ðt x; tÞgpj ðxÞ dx:
ð11Þ
0
To understand Eq. (11) above, note that the index was infectious x units prior to detection with
probability er3 x , was infecting level 0 susceptibles at rate bS 0 ðt xÞ at that time, that the new
infection at time ðt xÞ would now be in disease stage j with probability pj ðxÞ, but would only still
be untraced (in level 0) if (s)he was not randomly named in the time period ðt x; tÞ by someone
other than the index. Of the expected number computed above, a fraction p would be named by
the index case and have their names sent to the tracing/vaccination queue.
The likelihood that a person just infected is in disease stage j at time x after infection is just
(
)
j1
j
X
X
ð12Þ
Tk < x 6
Tk ;
pj ðxÞ ¼ Pr
k¼1
k¼1
where Tk represents the duration of time spent in disease stage k, and is assumed to be exponentially distributed with mean 1=rk . Eq. (12) can be evaluated directly from first principles for
each j, for example
p1 ðxÞ ¼ er1 x ;
Z x
p2 ðxÞ ¼
r1 er1 u er2 ðxuÞ du ¼
0
and so forth.
ð13Þ
r1
ðer2 x er1 x Þ;
r1 r2
ð14Þ
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We now address the probability that a contact is not randomly named over ðt x; tÞ. By Eq. (7),
regardless of which level 0 compartment the contact is in, at time u between t x and t, the
random tracing rate experienced by any individual is given by jðuÞ where
½c pR0 ðuÞ r3 I3 ðuÞ
:
ð15Þ
N
The probability that a contact would not have been randomly traced over ðt x; tÞ is thus given by
Rt
jðuÞ du
PrfNot randomly traced over ðt x; tÞg ¼ e tx
:
ð16Þ
jðuÞ Returning to Eq. (11) we have
Z tþs
Rt
jðuÞ du
r3 x
0
e bS ðt xÞ e tz
pj ðxÞ dx:
kj ðtÞ ¼
ð17Þ
0
Owing to imperfect recall, the expected number of local infectees actually named and traced from
disease stage j is given by pkj ðtÞ.
Eq. (17) is not terribly convenient for use in an ODE model, so we now approximate
it in a way
Rt
that depends only on time t. We accomplish this in three steps. First, we take tx jðuÞ du jðtÞx.
Second, we take S 0 ðt xÞ S 0 ðtÞ. The justification is that the random tracing rate jðtÞ (which
depends on the population compartment sizes) and the population of untraced susceptibles S 0 ðtÞ
are slowly varying relative to the relaxation time 1=r3 (3 days); this is the same assumption we
made in Eq. (9). This leads to the expression
Z tþs
eðr3 þjðtÞÞx bS 0 ðtÞpj ðxÞ dx
kj ðtÞ 0
j1
Y
k¼1
rk
bS 0 ðtÞ
;
rk þ r3 þ jðtÞ rj þ r3 þ jðtÞ
ð18Þ
where the third and final approximation follows from assuming that t þ s is large relative to 1=r3 .
2.3.3. Contact tracing ODEs
With R0 ðtÞ, kj ðtÞ, and I3 ðtÞ as defined in Eqs. (9), (18), and (6), we can now specify the ODEs
that govern the population flows through the level 0 (untraced) compartments in the model. These
are given by
dS 0 ðtÞ
S 0 ðtÞ
0
¼ bI3 ðtÞS ðtÞ ½c pR0 ðtÞ
ð19Þ
r3 I3 ðtÞ;
dt
N
dI10 ðtÞ
I10 ðtÞ
0
ð20Þ
¼ bI3 ðtÞS ðtÞ ½c pR0 ðtÞ
þ pk1 ðtÞ r3 I3 ðtÞ r1 I10 ðtÞ;
dt
N
(
)
0
dIj0 ðtÞ
I
ðtÞ
j
0
ðtÞ ½c pR0 ðtÞ
ð21Þ
¼ rj1 Ij1
þ pkj ðtÞ r3 I3 ðtÞ rj Ij0 ðtÞ for j ¼ 2; 3;
dt
N
dI40 ðtÞ
¼ r3 I30 ðtÞ r4 I40 ðtÞ:
dt
ð22Þ
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In addition to the usual disease transmission and progression terms, note how individuals leave
the untraced population via random and local tracing.
2.4. Queueing
Newly named contacts enter a tracing/vaccination queue, where they wait until they are located
and vaccinated. Disease progression among already infected contacts continues as before. Uninfected persons in queue remain susceptible to infection, while queued individuals in stage 3 of
infection continue to infect others. Queued infectious individuals who become symptomatic immediately exit the queue and are placed in isolation; in terms of the states in our model, such
individuals enter the Ôunsuccessfully vaccinatedÕ state I41 described in the next section. We let Qj ðtÞ
denote the number in queue in disease stage j (j ¼ 0 refers to susceptibles). Each tracer/vaccinator
can locate and vaccinate l persons per day. Since there are only n tracers/vaccinators available,
the population flow out of the queueing states can never exceed nl persons per day. If more than n
persons are in the queue, then queued individuals in disease stage j receive service at rate
nlQj ðtÞ=QðtÞ where
3
X
Qj ðtÞ:
ð23Þ
QðtÞ ¼
j¼0
Thus, queue departure rates are proportional to the relative numbers in queue when QðtÞ > n.
This explains the minð1; n=QðtÞÞ in the queueing state equations. The queueing ODEs are then
given by
dQ0 ðtÞ
S 0 ðtÞ
n
¼ ½c pR0 ðtÞ
r3 I3 ðtÞ bI3 ðtÞQ0 ðtÞ lQ0 ðtÞ min 1;
;
ð24Þ
dt
N
QðtÞ
dQ1 ðtÞ
I10 ðtÞ
n
¼ bI3 Q0 ðtÞ þ ½c pR0 ðtÞ
þ pk1 ðtÞ r3 I3 ðtÞ lQ1 ðtÞ min 1;
r1 Q1 ðtÞ;
dt
N
QðtÞ
ð25Þ
dQj ðtÞ
¼ rj1 Qj1 ðtÞ þ
dt
for j ¼ 2; 3:
(
)
Ij0 ðtÞ
n
þ pkj ðtÞ r3 I3 ðtÞ lQj ðtÞ min 1;
½c pR0 ðtÞ
rj Qj ðtÞ
QðtÞ
N
ð26Þ
2.5. Vaccination and death
The unsuccessfully vaccinated states (by stage of infection) constitute one set of destinations for
asymptomatic individuals departing the queue. Upon vaccination, those who do not die of vaccine
complications but are still unsuccessfully vaccinated (the vaccine only takes with probabilities m0
and m1 for susceptibles and those in disease stage 1 respectively) freely mix in the population,
contributing to disease transmission and progression in the usual way (except for symptomatically
infected individuals in isolation). We let S 1 ðtÞ and Ij1 ðtÞ denote unsuccessfully vaccinated susceptibles and disease stage j infecteds respectively; note that infectious individuals in queue
ARTICLE IN PRESS
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11
proceed directly to state I41 . The unsuccessfully vaccinated states are governed by the following
ODEs:
dS 1 ðtÞ
n
ð27Þ
¼ ð1 f Þð1 v0 ÞlQ0 ðtÞ min 1;
bS 1 ðtÞI3 ðtÞ;
dt
QðtÞ
dI11 ðtÞ
n
ð28Þ
¼ bI3 ðtÞS 1 ðtÞ þ ð1 f Þð1 v1 ÞlQ1 ðtÞ min 1;
r1 I11 ðtÞ;
dt
QðtÞ
dIj1 ðtÞ
n
1
¼ rj1 Ij1
ðtÞ þ ð1 f ÞlQj ðtÞ min 1;
rj Ij1 ðtÞ
QðtÞ
dt
for j ¼ 2; 3;
ð29Þ
dI41 ðtÞ
¼ r3 ðI31 ðtÞ þ Q3 ðtÞÞ r4 I41 ðtÞ:
ð30Þ
dt
Successful vaccinations occur with probability m0 and m1 for susceptibles and those in stage 1 of
disease respectively, while a fraction 1 d of those who progress to symptomatic smallpox
eventually recover (on average r41 days after development of symptoms) and remain immune. We
denote the number of immune individuals (whether by vaccination or recovery from disease) by
ZðtÞ, and obtain
dZðtÞ
n
¼ ð1 f Þðm0 Q0 ðtÞ þ m1 Q1 ðtÞÞl min 1;
ð31Þ
þ ð1 dÞr4 ðI40 ðtÞ þ I41 ðtÞÞ:
dt
QðtÞ
A fraction d of those who develop smallpox die of the disease, while a fraction f of all those
vaccinated die of vaccine-related complications. Letting DðtÞ denote the number of deaths in the
population at time t, we obtain
dDðtÞ
n
ð32Þ
¼ f lQðtÞ min 1;
þ dr4 ðI40 ðtÞ þ I41 ðtÞÞ:
dt
QðtÞ
2.6. Parameter values
The parameter values for our model, along with the associated references, appear in Table 1.
Many of the epidemiological parameters are taken from the classic literature, and require no
further comment. The infection rate b is chosen so that R0 ¼ 3 in the uncongested base case and
R0 ¼ 6 in the congested base case (these terms are defined in the next paragraph); these two values
correspond to the lower and upper estimates provided in [14]. Our base case considers an initial
attack size of 1000 in a large urban population of 10 million. We assume that the disease-detection
infrastructure would work as expected, so that the time of detection would correspond to a point on
the left tail of the probability density function (pdf) that convolves the first three disease-stage pdfs
(i.e., the time from the attack until several people are symptomatic). It then may take several days to
mobilize the intervention program. We chose s ¼ 5 days, at which point about 10% of the initial
infecteds are symptomatic. We assume for simplicity that no one exits disease stage 4 before time 0.
Turning to the logistical parameters, Ref. [16] reports that 0.78% of the US population is
employed in nursing, while 18.3% of employed nurses work in public or community health.
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Applying these percentages to N ¼ 10 million and dividing by 3 to produce round-the-clock 8-h
shifts yields 4758 which we have rounded to 5000. Vaccinators trace and vaccinate the contacts
named by index cases. In our model, it takes ST and SV time units, respectively, to trace and
vaccinate a named contact. Hence, l1 ¼ ST þ SV for TV and l1 ¼ SV for MV. We assume
ST ¼ 3SV , recognizing that the time required to locate contacts is greater than the time required
for vaccination [17]. Finally, while vaccine efficacy has been estimated to be 0.975 [18,19], our
analysis in Section 3 assumes 100% efficacy of susceptibles and people in disease stage 1; we
consider imperfect vaccination in Section 4.
3. TV analysis
This section contains an approximate analysis of TV. After investigating the pre-intervention
dynamics in Section 3.1, we transform the problem in Section 3.2, and estimate in Section 3.3 the
total number of deaths and the maximum queue length in the uncongested case, where QðtÞ 6 n
for all t. We consider the congested case in Section 3.4, where the maximum queue length exceeds
the number of vaccinators. Our approach is to find approximate closed-form expressions that
reveal dependencies of outcomes on model parameters.
3.1. Pre-intervention
In this subsection, we estimate the state vector at the time that intervention begins. Let us
suppose the attack occurs at time s, and the size of the initial attack is I10 ðsÞ. Intervention starts
s time units later, at time 0. Since I10 ðsÞ is much smaller than N , and the detection delay is
relatively short, we assume that S 0 ðtÞ N for t 2 ½s; 0, so that Eqs. (20) and (21) can be expressed as
dI10 ðtÞ
¼ r3 R0 I30 ðtÞ r1 I10 ðtÞ;
dt
ð33Þ
dI20 ðtÞ
¼ r1 I10 ðtÞ r2 I20 ðtÞ;
dt
ð34Þ
dI30 ðtÞ
ð35Þ
¼ r2 I20 ðtÞ r3 I30 ðtÞ:
dt
Because the matrix exponential solution to (33)–(35) is quite tedious, we pursue a simple linear
approximation that is used again in Section 5.1: I30 ðtÞ gðt þ sÞ for t 2 ½s; 0. To estimate the
growth rate g, we note that the vast majority of people in disease stage 3 at time 0 are people who
were exposed in the initial attack. Therefore, we ignore the infection term on the right side of (33),
and solve (33)–(35) sequentially to get
I30 ð0Þ ¼
r1 r2 eðr1 þr2 þr3 Þs ½ðr1 r2 Þ eðr1 þr2 Þs þ ðr2 r3 Þ eðr2 þr3 Þs þ ðr3 r1 Þ eðr1 þr3 Þs 0
I1 ðsÞ;
ðr1 r2 Þðr1 r3 Þðr2 r3 Þ
which provides a closed-form expression for g via g ¼
and (33), and solving (22), (33) and (34) yields
I30 ð0Þ
.
s
ð36Þ
Substituting gðt þ sÞ for I30 ðtÞ in (22)
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13
I10 ðtÞ ¼
er1 ðtþsÞ ½I10 ðsÞr12 þ R0 r3 g þ R0 r3 g½r1 ðt þ sÞ 1
;
r12
I20 ðtÞ ¼
1
ðeðr1 þ2r2 ÞðtþsÞ ðe2r2 ðtþsÞ r22 ½I10 ðsÞr12 þ R0 r3 g þ eðr1 þr2 ÞðtþsÞ r12 ½I10 ðsÞr22 þ R0 r3 g
r1 ðr1 r2 Þr22
I40 ðtÞ ¼
ð37Þ
þ eðr1 þ2r2 ÞðtþsÞ R0 ðr1 r2 Þr3 g½r2 þ r1 ðr2 ðt þ sÞ 1ÞÞÞ;
ð38Þ
r3 gðt þ sÞ2
:
2
ð39Þ
This approach is reasonably accurate when s ¼ 5 [8] and R0 6 7, which appears to be the relevant
range for R0 [14]. However, for longer intervention delays and/or higher values of R0 , it is important to capture the fact that I30 ð0Þ is roughly linear in R0 , rather than independent of R0 as in
(36). A refinement in these cases, which we do not pursue here, is to substitute (38) into (35) and
solve the latter to obtain a new estimate of g that yields a linear dependence of I30 ð0Þ on R0 .
3.2. Model transformation
We begin our TV analysis by defining a new set of variables. Let SðtÞ ¼ S 0 ðtÞ þ Q0 ðtÞ, I1 ðtÞ ¼
þ Q1 ðtÞ (note that S 1 ðtÞ ¼ I11 ðtÞ ¼ 0 because m0 ¼ m1 ¼ 1), and Ij ðtÞ ¼ Ij0 ðtÞ þ Qj ðtÞ þ Ij1 ðtÞ for
j ¼ 2; 3 be the total number of people in each disease stage,
susceptibles in queue are reP3 where
0
ðtÞ
denote the total number of
ferred to as being in disease stage 0. Let UðtÞ ¼ S 0 ðtÞ þ j¼1 IP
j
untraced people that are asymptomatic, and recall that QðtÞ ¼ 3j¼0 Qj ðtÞ. Then we have
dSðtÞ
n
¼ bI3 ðtÞSðtÞ lQ0 ðtÞ min 1;
;
ð40Þ
dt
QðtÞ
dI1 ðtÞ
n
ð41Þ
r1 I1 ðtÞ;
¼ bI3 ðtÞSðtÞ lQ1 ðtÞ min 1;
dt
QðtÞ
I10 ðtÞ
dI2 ðtÞ
¼ r1 I1 ðtÞ r2 I2 ðtÞ;
dt
ð42Þ
dI3 ðtÞ
¼ r2 I2 ðtÞ r3 I3 ðtÞ;
dt
ð43Þ
!
3
X
dU ðtÞ
U ðtÞ
¼ ½c pR0 ðtÞ
þp
kj ðtÞ r3 I3 ðtÞ r3 I30 ðtÞ;
dt
N
j¼1
ð44Þ
dQðtÞ
¼
dt
!
3
X
U ðtÞ
kj ðtÞ r3 I3 ðtÞ l minðn; QðtÞÞ r3 Q3 ðtÞ:
½c pR0 ðtÞ
þp
N
j¼1
ð45Þ
In contrast to the original model in Section 2, individuals can be in several compartments
simultaneously in the transformed model (40)–(45).
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3.3. Uncongested case
Our analysis of the uncongested case starts with a set of assumptions that simplify the model.
The simplified model is then analyzed to estimate the total number of deaths and the magnitude
and time of the maximum queue length.
3.3.1. Model simplification
Although we do not show the detailed simulations, the accuracy of these assumptions have
been confirmed for the base case values in Table 1.
Assumption 1. Nearly all random tracing is of susceptibles, and so assume jðtÞ ¼ 0 in (18).
Assumption 2. Nearly all uninfected people are untraced, and so replace S 0 ðtÞ by SðtÞ in (18).
Assumption 3. Nearly the entire queue is made up of susceptibles, and so replace Q0 ðtÞ by QðtÞ in (40).
Assumption 4. Nearly all people leave the untraced compartment via tracing, not disease symptoms, and so ignore the r3 I30 ðtÞ term in (44).
Assumption 5. Nearly all people leave the Q1 ðtÞ and QðtÞ compartments via vaccination, not
disease symptoms, and so ignore the r1 Q1 ðtÞ and r3 Q3 ðtÞ terms in (25) and (45), respectively.
Assumption 6. Because nearly all uninfected people are untraced ðS 0 ðtÞ=SðtÞ 1Þ and nearly all
untraced people are susceptible ðS 0 ðtÞ=U ðtÞ 1Þ, we assume that the ratio of the number of
susceptibles to the number of untraced people is one ðSðtÞ=U ðtÞ ¼ 1Þ.
Assumption 7. Nearly all the arrivals to Q1 ðtÞ are via tracing, not infection, and so ignore the
bI3 ðtÞQ0 ðtÞ term in (25).
P
3
Assumption 8. cU ðtÞ=N p
k
ðtÞ
R
ðtÞU
ðtÞ=N
in (44).
j
0
j¼1
Assumption 9. pR0 ðtÞI10 ðtÞ=N cI10 ðtÞ=N < pk1 ðtÞ in (25).
By Assumptions 1 and 2 and Eq. (18), we have
kj ðtÞ ¼
where
qj ¼ Pr
qj bSðtÞ
;
r3
(
j1
X
k¼1
Tk < T3 6
ð46Þ
j
X
k¼1
)
Tk
¼
j1
Y
k¼1
rk
r3
rk þ r3 rj þ r3
ð47Þ
and T3 is the time from when an index infects a (random) contact until the index is detected.
A key to our simplification is to assume that the queue composition Qj ðtÞ=QðtÞ is independent of
time, so that we can express (41) in terms of QðtÞ rather than Q1 ðtÞ; this simplification, combined
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15
with the other assumptions above, leads to a free-standing set of transformed equations.
To this
P3
end, let Aj ðtÞ be the time-dependent arrival rate to queue Qj ðtÞ and let AðtÞ ¼ j¼0 Aj ðtÞ be the
arrival rate to QðtÞ. Then
cI10 ðtÞ
I10 ðtÞ
pbSðtÞ
þ
q
1
r3
N
N
A1 ðtÞ
by Assumptions 1; 2 and 7;
ð48Þ
cU
ðtÞ
pbSðtÞ
U
ðtÞ
AðtÞ
þ
q
N
pq1 bSðtÞ
r3
cU ðtÞ
N
r3
N
by Assumptions 8 and 9;
ð49Þ
pq1 R0
by Assumption 6:
ð50Þ
c
The queues Q1 ðtÞ and QðtÞ have the same service rate by Assumption 5, the arrival rates are
multiples of each other by (50), and the queue departure rates are proportional to the relative
numbers in queue. Hence, it follows that Q1 ðtÞ=QðtÞ A1 ðtÞ=AðtÞ pq1 R0 =c.
Taken together, these assumptions allow us to re-express the core, or coupled portion, of our
model, as (by definition, minðn; QðtÞÞ ¼ QðtÞ in the uncongested case)
dSðtÞ
ð51Þ
¼ bI3 ðtÞSðtÞ lQðtÞ;
dt
dI1 ðtÞ
lpq1 R0
¼ bI3 ðtÞSðtÞ QðtÞ r1 I1 ðtÞ;
dt
c
ð52Þ
dI2 ðtÞ
¼ r1 I1 ðtÞ r2 I2 ðtÞ;
dt
ð53Þ
dI3 ðtÞ
¼ r2 I2 ðtÞ r3 I3 ðtÞ;
dt
dU ðtÞ
cU ðtÞ pbSðtÞU ðtÞ pqbSðtÞ
r3 I3 ðtÞ;
¼
þ
dt
N
r3 N
r3
dQðtÞ
cU ðtÞ pbSðtÞU ðtÞ pqbSðtÞ
r3 I3 ðtÞ lQðtÞ:
¼
þ
dt
N
r3 N
r3
ð54Þ
ð55Þ
ð56Þ
We make a final reduction by assuming that QðtÞ is in a quasi-steady state. While the service
rate is very fast, and this is not an unreasonable approximation, we are making this assumption
for purposes of analytical tractability. Setting Q_ ðtÞ ¼ 0 in (56), substituting in for lQðtÞ in (52),
and using Assumption 6 gives
13
2
0
SðtÞ
pR
q
0
N
dI1 ðtÞ
A5I3 ðtÞU ðtÞ r1 I1 ðtÞ:
¼ b41 pq1 @1 ð57Þ
c
dt
Assumption 8 applied to Eqs. (55) and (57) leads to the simplified SEIR-like model
dU ðtÞ
cr3
¼
I3 ðtÞU ðtÞ;
dt
N
ð58Þ
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E.H. Kaplan et al. / Mathematical Biosciences xxx (2003) xxx–xxx
dI1 ðtÞ
¼ bð1 pq1 ÞI3 ðtÞU ðtÞ r1 I1 ðtÞ;
dt
ð59Þ
dI2 ðtÞ
¼ r1 I1 ðtÞ r2 I2 ðtÞ;
dt
ð60Þ
dI3 ðtÞ
¼ r2 I2 ðtÞ r3 I3 ðtÞ:
dt
ð61Þ
3.3.2. Subcritical case
We first consider the subcritical case, where the post-intervention value of R0 , which is
R0 ð1 pq1 Þ, is less than one. In this case, we expect a very small fraction of people to get infected
or traced. Hence, we set U ðtÞ N for all t in (58) and (59), so that (59) becomes
dI1 ðtÞ
¼ r3 R0 ð1 pq1 ÞI3 ðtÞ r1 I1 ðtÞ:
ð62Þ
dt
P3
Eqs. (60)–(62) can be viewed as a system in which j¼1 Ij ð0Þ people initially reside and – upon
exiting compartment I3 – re-enter compartment I1 for another pass through the system with
probability R0 ð1 pq1 Þ. Because each pass through the system represents a symptomatic smallpox
case, we can sum the resulting geometric series to find that the total number of deaths in the
subcritical case is approximately
#
"
P3
j¼1 Ij ð0Þ
:
ð63Þ
Dð1Þ ¼ d I4 ð0Þ þ
1 R0 ð1 pq1 Þ
Hereafter, we assume that R0 ð1 pq1 Þ > 1.
3.3.3. Total deaths
Our primary performance measure is the total number of deaths. Let us consider (58)–(61)
along with
dI4 ðtÞ
¼ r3 I3 ðtÞ;
ð64Þ
dt
i
dX ðtÞ h cr3
ð65Þ
¼
bð1 pq1 Þ I3 ðtÞUðtÞ;
dt
N
where I4 ðtÞ is the cumulative
P4 number of people who become symptomatic by time t, and (65) is
chosen so that U ðtÞ þ j¼1 Ij ðtÞ þ X ðtÞ remains constant. Following the classical SIR analysis
(e.g., [20]), we divide (58) by (64) and solve to get
U ð1Þ ¼ U ð0Þ ecI4 ð1Þ=N :
ð66Þ
Dividing (65) by (58) implies that
X ð1Þ X ð0Þ R0 ð1 pq1 Þ
¼
1:
U ð1Þ U ð0Þ
c
ð67Þ
Note thatPthe total Ôpopulation sizeÕ of (58)–(61),
(64) and (65), initially (and forever after) is
P
S 0 ð0Þ þ 2 3j¼1 Ij0 ð0Þ þ I40 ð0Þ þ X ð0Þ ¼ N þ 3j¼1 Ij0 ð0Þ þ X ð0Þ, and hence
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U ð1Þ þ I4 ð1Þ þ X ð1Þ ¼ N þ
3
X
Ij0 ð0Þ þ X ð0Þ:
17
ð68Þ
j¼1
By (67) and (68), we can re-express (66) as (note that we never had to specify X ð0Þ––it cancels out)
!!
3
X
U ð1Þ
c
R
ð1
pq
Þ
0
1
¼ exp Nþ
Ij0 ð0Þ 1 ½U ð1Þ U ð0Þ U ð1Þ
:
ð69Þ
U ð0Þ
N
c
j¼1
Defining u~ ¼ U ð1Þ=U ð0Þ Uð1Þ=N, we have
!
3
c X
u~ ¼ exp R0 ð1 pq1 Þð1 u~Þ I 0 ð0Þ :
N j¼1 j
ð70Þ
Eqs. (32), (66) and (70) imply that the total number of deaths is approximately
dN
Dð1Þ ¼ f ð1 u~ÞN ln u~;
ð71Þ
c
where u~ solves (70). Eq. (71) implies that the number of deaths under TV scales with the population size N , with minimal dependence on the initial number of infections (which enters through
u~). This appears to be a fundamental property of TV, as we will explain later in Section 6.5.
3.3.4. Maximum queue length
We hypothesize that system performance degrades significantly when the system becomes
congested, i.e., when the maximum queue length exceeds the number of vaccinators. Hence, an
estimate of the maximum queue length in the uncongested regime can aid in staffing decisions. The
maximum queue length is estimated in two steps: we find an approximate relationship between the
maximum queue length Qmax and the maximum size of the I1 ðtÞ compartment, I1max , and then we
estimate I1max by forcing (58)–(61) into the classic SIR framework.
By (56) and Assumption 8, the maximum queue length occurs when
cr3
ð72Þ
I3 ðtÞU ðtÞ:
lQðtÞ ¼
N
Note also that I1 ðtÞ should hit its maximum at about the same time that QðtÞ does, because both
are receiving ÔarrivalsÕ at rate proportional to I3 ðtÞSðtÞ I3 ðtÞU ðtÞ by Assumption 6.
Setting the right side of (52) equal to zero and solving simultaneously with (72) yields, with
Assumption 6,
I1max ¼
lR0 ð1 pq1 Þ max
Q :
cr1
ð73Þ
max
1
1
, we transform
To estimate
1
P3 I1
P3 Eqs. (58)–(61) into a SIR framework via I3 ðtÞ ¼ r3 IðtÞ=r , where
1
r ¼ j¼1 rj and IðtÞ ¼ j¼1 Ij ðtÞ. Then (58)–(61) become
dU ðtÞ
c
¼ 1 IðtÞUðtÞ;
dt
Nr
ð74Þ
dIðtÞ
r1
1
¼ bð1 pq1 Þ 31 IðtÞU ðtÞ 1 IðtÞ:
dt
r
r
ð75Þ
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E.H. Kaplan et al. / Mathematical Biosciences xxx (2003) xxx–xxx
Using the classic SIR conservation law analysis (e.g., [20]), we divide (75) by (74) and integrate to get
R0 ð1 pq1 Þ
N
R0 ð1 pq1 Þ
N
UðtÞ ln UðtÞ ¼ Ið0Þ þ
U ð0Þ ln U ð0Þ
c
c
c
c
By (75), IðtÞ achieves its maximum when
N
U ðtÞ ¼
:
R0 ð1 pq1 Þ
IðtÞ þ
Substituting (77) into (76) gives
I
max
R0 ð1 pq1 Þ
N
N N
¼ Ið0Þ þ
U ð0Þ ln U ð0Þ þ ln
r3
c
c
c
N
:
R0 ð1 pq1 Þ
for all t:
ð76Þ
ð77Þ
ð78Þ
Ignoring the relatively tiny Ið0Þ term in (78), using I1max ¼ r11 I max =r1 and (73) gives the approximation
N
N
max
¼ 1
R0 ð1 pq1 Þ ln N 1 þ ln
:
ð79Þ
Q
lr R0 ð1 pq1 Þ
R0 ð1 pq1 Þ
In Section 6, we show that (79) accurately predicts that some of the model parameters influence
the number of deaths in a non-smooth manner. While it might seem surprising that Qmax is nearly
independent of c (incorporating Ið0Þ in (78) produces a slight dependence on c), this claim is
confirmed by simulation results (not shown here). Increasing c places more people in the queue
initially, but also limits the epidemic and causes less people to enter the queue in total.
3.3.5. Time of the maximum queue
With Eq. (79) in hand, we now approximate the time, tmax , when this maximum queue length is
achieved, and also construct an approximation for the entire queue length process. Eick et al.
[21,22] show that in an infinite-server queue with non-homogeneous Poisson arrivals, the time lag
between the maximum arrival rate and the maximum queue length is roughly the mean service
time, which is negligible in our case. Moreover, their analysis implies that the pointwise stationary
approximation (PSA), QðtÞ l1 AðtÞ, should be quite accurate in our setting. Hence, to estimate
the queue length process QðtÞ it suffices to construct an approximation to the arrival process AðtÞ.
To this end, we assume AðtÞ follows a Gaussian pdf (simulations reveal that AðtÞ is bell-shaped in t),
AðtÞ ¼ lQmax eaðtt
where
ln
a¼
lQmax
cr3 I30 ð0Þ
ðtmax Þ2
max Þ2
;
ð80Þ
ð81Þ
is chosen so that Að0Þ ¼ cr3 I30 ð0Þ, which follows from setting Uð0Þ N in (56) and imposing
Assumption 8, and Aðtmax Þ ¼ lQmax , which follows by the PSA. By the symmetry of AðtÞ and the
fact that roughly N ð1 u~Þ people are served in all, we find tmax by solving
Z tmax
N ð1 u~Þ
max 2
;
ð82Þ
lQmax eaðtt Þ dt ¼
2
0
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19
which yields
tmax
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
max N ð1 u~Þ ln crlQI 0 ð0Þ
3 3
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼
;
pffiffiffi
lQmax
max
pU
ln cr I 0 ð0Þ
lQ
ð83Þ
3 3
where UðÞ is the standard normal cdf. Substituting (83) and (79) into (80) and using QðtÞ ¼
l1 AðtÞ generates an approximate queue length process throughout the entire epidemic. While of
interest in its own right, this queue length estimate is also required to approximate the total
number of deaths in the congested analysis in Section 3.4.
3.4. Congested case
We now turn to the congested case, where the maximum queue size exceeds the number of
vaccinators. As in Section 3.3, we begin with a set of simplifying assumptions, followed by an
analysis of the number of deaths, and the size and time of the maximum queue length.
3.4.1. Model simplification
In addition to Assumption 1 in Section 3.3, we make the following assumptions that can be
confirmed by simulations of the R0 ¼ 6 case:
Assumption 10. Everyone is either successfully vaccinated or contracts symptomatic smallpox.
Assumption 11. r1 Q1 ðtÞ bI3 ðtÞQ0 ðtÞ for all t in (25).
Assumption 12.
Qj ðtÞ
QðtÞ
Aj ðtÞ
AðtÞ
pqj R0
c
I 0 ðtÞ
þ Uj ðtÞ for all t.
3.4.2. Integrating the ODEs
Our main concern is to estimate the total number of deaths. Assumption 10 implies that
Z 1
Z 1
n
n
Q0 min 1;
Q1 min 1;
dt l
dt :
ð84Þ
Dð1Þ ¼ fN þ d N l
QðtÞ
QðtÞ
0
0
Our analysis proceeds in two steps: first we
R 1integrate several of the ODEs to approximate
the integrals in (84) in terms of the integral 0 QðtÞ dt, and then we perform an analysis of the
congested period of the queue to estimate the latter integral, as well as the size and time of the
maximum queue.
R1
To find l 0 Q0 minð1; n=QðtÞÞ dt, we first integrate (19) and replace R0 ðtÞ by a time- averaged
value of R0 =2 (since SðtÞ drops from N to 0 during the epidemic), which yields
Z 1
N2
I3 ðtÞS 0 ðtÞ dt ¼
:
ð85Þ
R0 r3 1 p2 þ cr3
0
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Similarly, integrating (24) gives
Z 1
Z 1
Z 1
n
cr3 pb
b
I3 ðtÞQ0 ðtÞ dt þ l
Q0 min 1;
I3 ðtÞS 0 ðtÞ dt
dt ¼
QðtÞ
2
N
0
0
0
c p2 R0
N by ð85Þ:
¼
c þ 1 p2 R0
Assumption 12 implies that
Z 1
Z 1 0
Z
r1 pq1 R0 1
I1 ðtÞQðtÞ
dt:
r1
Q1 ðtÞ dt QðtÞ dt þ r1
U ðtÞ
c
0
0
0
ð86Þ
ð87Þ
ð88Þ
For the sake of analytical tractability, we ignore the last term in (88), which is the smaller of the
two terms, and use Assumption 11 and Eqs. (87) and (88) to get
Z
Z 1
c p2 R0
n
r1 pq1 R0 1
Q0 min 1;
N
QðtÞ dt:
ð89Þ
l
dt QðtÞ
c
c þ 1 p2 R0
0
0
R1
To estimate, l 0 Q1 minð1; n=QðtÞÞ dt, we integrate (25) and approximate R0 ðtÞ by R0 =2 to get
Z 1
n
Q1 min 1;
dt
ð90Þ
l
QðtÞ
0
Z
Z 1
cr3 pb r3 1 0
I ðtÞI3 ðtÞ dt þ pr3
k1 ðtÞI3 ðtÞ dt by Assumption 11;
ð91Þ
2 N 0 1
N
0
Z
Z 1
cr3 pb r3 1 0
I ðtÞI3 ðtÞ dt þ pq1 b
S 0 ðtÞI3 ðtÞ dt
ð92Þ
2 N 0 1
N
0
by Assumption 1; ð18Þ and ð47Þ;
!
Z
cr3 pb r3 1 0
pq1 R0
I ðtÞI3 ðtÞ dt þ
N
2 N 0 1
N
R0 1 p2 þ c
by ð85Þ:
ð93Þ
If we again ignore the smaller of the two terms in (93), which is the first term, then combining (84),
(89) and (93) gives
!
R1
c pR0 12 q1
r1 pq1 R0 0 QðtÞ dt
:
ð94Þ
þ
Dð1Þ ¼ fN þ dN 1 cN
R0 1 p2 þ c
3.4.3. Congested period analysis
Following
[23], we now analyze the congested period of the queue to
R 1 Chapter 2 of Newell
0
estimate 0 QðtÞ dt in (94). Let t be the time when QðtÞ ¼ n for the first time, let t1 be the time of
the maximum arrival rate, let t2 be the time of the maximum queue length, and let t3 be the time
when QðtÞ ¼ n again. Although the congested period starts at time t0 , we need to estimate the
number of people who are served before this time. For simplicity, we assume that AðtÞ ¼ Að0Þ eKt
for t 2 ½0; t0 , where Að0Þ cr3 I3 ð0Þ by (81) and Aðt0 Þ ¼ nl, and hence
nl
Kt0 ¼ ln
;
ð95Þ
cr3 I3 ð0Þ
but neither K nor t0 are known. The number of arrivals before time t0 is
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E.H. Kaplan et al. / Mathematical Biosciences xxx (2003) xxx–xxx
Z
t0
AðtÞ dt ¼
0
21
nl
:
K
ð96Þ
To estimate t0 such that Qðt0 Þ ¼ n (and hence K via (95)), we use the approximate queue length
process in the last sentence of Section 3.3, which yields
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
max 1
Q
t0 ¼ tmax ln
;
ð97Þ
a
n
where Qmax ; a and tmax are given by Eqs. (79), (81) and (83), respectively.
We now assume that all N people are vaccinated before time t3 , so that the length of the
congested period is
N nl
K
:
t3 t0 ¼
nl
We also assume that the arrival process AðtÞ is symmetric about t1 , so that
Z t1
N nl
AðtÞ dt ¼ :
2
K
t0
ð98Þ
R t1
0
AðtÞ dt ¼ N =2, or
ð99Þ
Following Newell, we assume that the arrival process during the congested period is quadratic,
with AðtÞ ¼ Aðt1 Þ kðt t1 Þ2 , where the curvature k is unknown. Newell shows that t0 ; . . . ; t3 are
equally spaced in time, so that the time lag, call it D, between ti and tiþ1 for i ¼ 0; 1; 2, is
N nl
K
D¼
:
3nl
ð100Þ
Integrating (99) gives
nlD þ
2kD3 N nl
¼ ;
2
K
3
ð101Þ
which gives
3
81ðnlÞ N6 2nl
3K
k¼
3 :
2 N nl
K
Newell shows that the queue length process in the interval ½t0 ; t3 is given by
t t0 2
:
QðtÞ ¼ kðt t0 Þ D 3
Moreover, because Aðt0 Þ ¼ nl, the maximum arrival rate is given by
Amax ¼ Aðt1 Þ ¼ Aðt0 Þ þ kD2 ;
3ðNK 4nlÞ
¼ nl 1 þ
by ð100Þ and ð102Þ;
4ðN K nlÞ
and, from Newell, the maximum queue length is
ð102Þ
ð103Þ
ð104Þ
ð105Þ
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4 3
kD ;
3
N 4nl
¼ 3
3K
which occurs at time
Qmax ¼ Qðt2 Þ ¼
tmax ¼ t0 þ 2D
1
¼
r
ln
R0 ð1 pq1 Þ 1
ð106Þ
by ð100Þ and ð102Þ;
nl
cr3 I3 ð0Þ
Finally, Newell also shows that
Z t3
9
QðtÞ dt ¼ kD4 ;
4
t0
9 N6 2nl
N nl
3K
K
¼
8nl
2 N nl
K
þ
3nl
ð107Þ
ð108Þ
by ð97Þ and ð100Þ:
ð109Þ
ð110Þ
by ð100Þ and ð102Þ:
ð111Þ
Because the queue
R 1length is very small before time t0 and after time t3 , we can substitute the right
side of (111) for 0 QðtÞ dt in (94) to estimate the total number of deaths in the congested case.
4. Approximate thresholds for initial epidemic control via TV interventions
For a variety of intervention strategies, we approximate in this section the time derivative of the
total number of infected asymptomatic people at the beginning of intervention. These calculations,
which appear in Sections 4.1–4.4, lead to threshold conditions (in terms of R0 ) to initially reduce the
number of infected but asymptomatic cases via intervention with various combinations of tracing,
imperfect vaccination and quarantine. While these results are of interest in their own right, they
also can be generalized to incorporate non-exponential disease-stage durations (Section 4.5) and
can be combined with results in Section 3 to approximate the total number of deaths and the size
and time of the maximum queue length under these more sophisticated strategies (Section 4.6).
4.1. Imperfect vaccination
Consider again the basic smallpox model in the absence of any interventions other than the
isolation of symptomatic cases (see Section 2.1). At the beginning of the epidemic, the total
number of infected but asymptomatic individuals grows as
d
ð112Þ
½I1 ðtÞ þ I2 ðtÞ þ I3 ðtÞ ¼ ðbS 0 ð0Þ r3 ÞI3 ð0Þ;
dt
t¼0
as can be seen from summing Eqs. (2) and (3). By (4), the net growth rate of new infections per
infectious individual is thus given by
c ¼ r3 ðR0 1Þ:
Hence, the epidemic initially increases if c > 0, which is the same condition as R0 > 1.
ð113Þ
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23
Now let us return to the model considered in Section 3. Recall that at the beginning of the
epidemic, each newly infected person produces R0 new infections, where R0 ¼ bS 0 ð0Þ=r3 bN =r3 .
At the time an index case enters stage 4 and is detected as symptomatic, a fraction qj of the
persons the index has infected are in disease stage j, where qj is defined in Eq. (47) (Assumptions 1
and 2 leading to (47) hold at time 0). We assume that any contacts infected by the index are
accurately named and located with probability p, and that any contact so traced is vaccinated. The
vaccine is effective with probability vj for people in disease stage j ¼ 0; 1, where – in contrast to
Section 3 – we allow mj < 1.
To mimic the analysis in (112) and (113) for this more general case, let us define
SðtÞ ¼ S 0 ðtÞ þ Q0 ðtÞ þ S 1 ðtÞ and Ij ðtÞ ¼ Ij0 þ Qj þ IJ1 ðtÞ be the total number of susceptibles and
people in disease stage j ¼ 2; 3; note that SðtÞ and I1 ðtÞ now include unsuccessfully vaccinated
people, which was not the case in Section 3.2. Summing the appropriate equations in Section 2
gives, at the time intervention begins,
d
½I1 ðtÞ þ I2 ðtÞ þ I3 ðtÞ ¼ bI3 ð0ÞSð0Þ r3 I3 ð0Þ m1 lQ1 ð0Þ:
ð114Þ
dt
t¼0
By Assumptions 5, 7 and 9 in Section 3.3, which certainly hold at time 0, and the fact that the
service rate is very fast, we can approximate the service completion rate lQ1 ð0Þ by the approximate arrival rate pk1 r3 I3 ð0Þ. These assumptions lead to a net growth rate of infections of
ð115Þ
cct ¼ r3 ðR0 1 pq1 m1 R0 Þ;
where the superscript ct stands for C ontact T racing. Eq. (115) reveals the approximate impact of
contact tracing at the beginning of the epidemic: with probability pq1 v1 , a contact of the index case
is named, found in stage 1 of infection, and successfully vaccinated. Hence, the last term on the
right side of (115) can be interpreted as an intervention-induced removal term of active infected
individuals. We initially control the epidemic if cct < 0, which is the same as the threshold
1
R0 <
:
ð116Þ
1 pq1 v1
Note that for our base case parameter values, q1 ¼ 1=2, and thus perfect recall ðp ¼ 1Þ and a
perfect vaccine ðm1 ¼ 1Þ cannot initially control an epidemic with R0 > 2 via TV.
Before proceeding to more complex interventions, it is important to note that these results are
neither necessary nor sufficient, but are approximate. Moreover, they ignore the positive effect of
random tracing (i.e., faster herd immunity) and the negative effect of congestion, both of which
may play a major role later in the epidemic.
4.2. Tracing contacts of contacts
A more intensive approach is to trace and vaccinate the contacts of an index caseÕs contacts.
Focusing on local tracing, we see that there is no immediate benefit from tracing the contacts of
susceptibles or infected persons in stage 1 or stage 2, because these people have yet to infect
anyone. However, tracing and vaccinating the contacts of infected persons in stage 3 is immediately useful, since some of these contacts of contacts will be infected (from the original indexÕs
infected stage 3 contact), but still in stage 1 and hence sensitive to the vaccine. Finally, the
contacts of the original index now in stage 4 themselves become indexes for tracing, and are already addressed by the model.
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E.H. Kaplan et al. / Mathematical Biosciences xxx (2003) xxx–xxx
We need to answer three questions. First, how many contacts of the original index are in stage 3
at the time the index was detected? This is just the fraction q3 of the R0 persons infected by the
index. Having located an index contact who is in stage 3, what is the probability distribution for the
length of time (s)he has already spent in stage 3? We need this, because we need to figure out how
many infections have already been transmitted (at the time we found the index case) by the stage 3
contact of the index. Finally, of these infections generated, how many infectees are still in stage 1
at the time we detected the index case? Those are the incremental infections we can prevent immediately via tracing and vaccination of contacts of contacts. Once we know the rate with which
we interdict infections traced from contacts of contacts, call this vcc (where cc indicates C ontacts
of C ontacts), we will subtract this from cct in Eq. (115) to obtain a new growth rate,
ccc ¼ cct vcc ;
ð117Þ
and see what must happen to force this growth rate negative.
We now turn to the second question. Let us begin with a newly detected index. We know that
with probability q3 , a person infected by the index is in stage 3 at the time the index is detected.
Given this, we seek the conditional distribution for the length of time the contact has already
spent in stage 3 by the time the index is detected. Denote this time by W .
Since the time from when an index infects a (random) contact until the index is detected is given
by T3 , we know that T3 represents the total elapsed duration of infection for the contact at the
time the index is detected. Recall that T3 is exponentially distributed with rate r3 due to the
memoryless property of the exponential distribution.
What must happen for W ¼ w? The contact is newly infected T3 time units before the index was
detected. From that point on, the contact must spend exactly T3 w time units traversing disease
stages 1 and 2, and remain in stage 3 for at least w time units. All this is conditional upon the
contact being in stage 3 at the time the index is detected (which is q3 ). We thus see that the density
function fW ðwÞ must equal
Z 1
1
r3 er3 x fT1 þT2 ðx wÞ er3 w dx:
ð118Þ
fW ðwÞ ¼
q3 w
We note that
Z x
Z x
er2 x er1 x
fT1 ðuÞfT2 ðx uÞ du ¼
r1 er1 u r2 er2 ðxuÞ du ¼ r1 r2
for x > 0
fT1 þT2 ðxÞ ¼
r1 r2
0
0
ð119Þ
and
r1
r2
r3
q3 ¼
;
ð120Þ
r1 þ r3 r2 þ r3 r3 þ r3
which yields
fW ðwÞ ¼ 2r3 e2r3 w
for w > 0:
ð121Þ
Eq. (121) implies that the infected contact will on average have spent 1=2r3 time units in stage 3 by
the time the index is detected. But note that this is not half of the time (s)he will spend there––after
the index is detected, on average a contact in stage 3 will stay for an additional r31 time units (the
memoryless property again), so that on average, index contacts found in stage 3 of their infection
are only one-third of the way through their infectious period.
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25
Now that we know how long the indexÕs direct contact has spent in stage 3, we can proceed to
the third question and ask how many persons have been infected by that contact who are in stage
1 of infection when the original index was detected. These are the contacts of contacts who are in
the vaccine-sensitive stage of infection. We can think of this as an infinite-server queueing problem
with deterministic arrivals and exponential service times. The customers are contacts of contacts
in stage 1 of disease. The arrival rate is bS 0 ð0Þ, which is the rate at which the indexÕs direct contact
is infecting susceptibles at the beginning of intervention. The service rate is r1 , which is the rate
with which infected persons leave stage 1. The number of persons infected by the indexÕs direct
contact who are in stage 1 of infection when the original index is detected is then exactly the
number of customers in an infinite-server queue that started as an empty system and has been
operating for W units of time. Let LðwÞ denote the mean number of customers (contacts of
contacts) who are in service (in the vaccine-sensitive stage of infection) after w time units. From
the theory of infinite-server queues we have
Z w
r 3 R0
bS 0 ð0Þ er1 ðwxÞ dx ¼
ð1 er1 w Þ
ð122Þ
E½LðwÞ ¼
r1
0
and consequently the expected number of contacts of contacts who would be found in stage 1 of
infection at the time the original index was detected is given by
Z 1
r3 R0
r 3 R0
2r3 e2r3 w
ð1 er1 w Þ dw ¼
:
ð123Þ
E½LðW Þ ¼
r
r
1
1 þ 2r3
0
We are now in a position to calculate the removal rate of infected persons via the tracing of
contacts of contacts. Recall that of all those infected by an index case, the fraction q3 will reside in
stage 3 when the index is detected. On average, each of the indexÕs direct contacts will have resulted in E½LðW Þ infected persons (contacts of contacts) who are in stage 1 when the original
index was detected. However, only a fraction p of the indexÕs direct contacts in stage 3 will be
found, and among those that are, only the fraction pv1 of the contacts of contacts still in stage 1
will be successfully located and vaccinated. Consequently, the incremental removal rate of infected persons due to contacts-of-contacts tracing is given by
r3 R0
r3 R0 :
ð124Þ
vcc ¼ q3 p E½LðW Þ pv1 r3 R0 ¼ p2 q3 v1
r1 þ 2r3
By Eqs. (115), (117) and (124), we have
p2 q3 v1 r3 R20
:
ccc ¼ r3 R0 1 pq1 v1 R0 r1 þ 2r3
ð125Þ
This epidemic is initially controlled by the tracing of contacts of contacts if ccc < 0, or equivalently
if
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2q m r
3 1 3
1 pq1 m1 ð1 pq1 m1 Þ2 4pr1 þ2r
3
R0 <
:
ð126Þ
2p2 q m r
3 1 3
r1 þ2r3
In the best case where we have a perfect vaccine and v1 ¼ 1, and perfect tracing and p ¼ 1, the
threshold in (126) equals 2.225 for our base case parameter values. From the standpoint of initial
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E.H. Kaplan et al. / Mathematical Biosciences xxx (2003) xxx–xxx
fraction of infectees named
1
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
2
2.5
Basic reproductive ratio
Fig. 2. The tracing accuracy p required to initially control an epidemic, as a function of the basic reproductive ratio R0 ,
for contact tracing (––) and contacts-of-contacts tracing ð Þ.
epidemic control, contacts-of-contacts tracing offers only a modest improvement beyond tracing
only the direct contacts of index cases, which could control epidemics with R0 < 2 if m1 ¼ p ¼ 1.
To see this more vividly, Fig. 2 shows, for the case of a perfect vaccine, the tracing probability p
that would be required to control epidemics for different R0 values for both contact tracing and
contacts-of-contacts tracing. The improvement of contacts-of-contacts tracing over tracing only
the direct contacts of index cases is just the space between the two curves. However, recall that this
calculation ignores the impact of random tracing. The arrival rate to the tracing/vaccination
queue is about c2 people per index case under contacts-of-contacts tracing, which is significantly
higher than the c people per index case that arrive during contact tracing. Hence, contacts-ofcontacts tracing is likely to make better use of the tracing/vaccination resources than contact
tracing, but will also lead to more congestion.
4.3. Quarantine
Though we did not include quarantine beyond the isolation of symptomatic smallpox cases in
the model formulation and analysis reported above, it is easy to do so, and instructive to see how
the quarantine of traced contacts contributes to initial epidemic control. Focusing only on the
initial growth rate of infected persons early in the epidemic, we note that the impact of quarantine
is to lessen the amount of time during which an infectious person is mixing in the population and
thus able to transmit new infections. Quarantine will not always be effective: some infected persons will enter and leave quarantine before becoming infectious, and hence transmit the same
number of infections as would occur in the absence of quarantine, while others will be found in
quarantine for part or all of their infectious periods, resulting in reduced transmission.
As an illustration, suppose that once individuals are traced and vaccinated, they are quarantined
with probability h (alternatively, only a fraction h of contacts comply with residential quarantine
orders), and that the time spent in quarantine is exponentially distributed with mean a1 . If a
located contact is in disease stage 3 (which occurs with probability q3 ) and quarantined (which
occurs with probability h), then the remaining mean time during which such a contact is infectious
and mixing in the population is reduced from r31 (due to the memoryless property of the expo-
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27
nential distribution) to ar31 =ða þ r3 Þ (since with probability a=ða þ r3 Þ, quarantine ends before the
remaining time in stage 3, and by the memoryless property the remaining time in stage 3 once
released from quarantine again equals r31 ). Consequently, the reduction in the mean time spent
infectious and mixing due to quarantine for a contact found in stage 3 is given by ðr3 =ða þ r3 ÞÞr31 ,
and thus the rate with which such persons generate infections is reduced by the factor
w3 ¼
r3
r1
aþr3 3
r31
¼
r3
:
r3 þ a
ð127Þ
Quarantine therefore reduces the actual rate of new infections by the amount q3 hw3 bS 0 ð0Þ for
infected persons found in stage 3.
Similarly, if a located contact is in stage 2 and quarantined (an event that occurs with probability q2 h), the mean time during which such a contact will be both infectious and mixing in the
population is reduced from r31 in the absence of quarantine (for the contact would simply progress from stage 2 to stage 3 and then be infectious for the duration of stage 3) to
½a=ða þ r2 Þ þ ðr2 =ða þ r2 ÞÞða=ða þ r3 ÞÞr31 . To be infectious and mixing in the population, either
quarantine ends during stage 2 (with probability a=ða þ r2 Þ), and the contact is thus released and
infectious for the entire r31 mean duration of stage 3, or quarantine extends into stage 3 (with
probability r2 =ða þ r2 Þ) but ends during stage 3 (with probability a=ða þ r3 Þ), leaving the contact
infectious and mixing for an additional r31 time units on average (again due to the memoryless
property of the exponential distribution). For located contacts in stage 2 of infection, quarantine
thus reduces the mean time spent infectious and mixing by the factor
w2 ¼
r31 ½a=ða þ r2 Þ þ ðr2 =ða þ r2 ÞÞða=ða þ r3 ÞÞr31
r2
r3
¼
1
r3
r2 þ a r3 þ a
ð128Þ
and as a result, the actual rate of new infections is reduced by the amount q2 hw2 bS 0 ð0Þ for infected
persons found in stage 2.
A similar argument obtains for infected individuals found and quarantined in stage 1 of infection, though now one also must consider the probability v1 that such a person was successfully
vaccinated. Doing so reveals that the actual rate of new infections for infected persons found in
stage 1 is reduced by the amount q1 ðv1 þ ð1 v1 Þhw1 ÞbS 0 ð0Þ where
w1 ¼
3
Y
k¼1
rk
:
rk þ a
ð129Þ
Now recall that only a fraction p of all contacts are named and located. Combining our results,
we see that early in the epidemic, the growth rate of new infections per infectious individual incorporating (imperfect) contact vaccination and quarantine, cq , is given by
cq ¼ r3 ðR0 1 pR0 ½q1 ðm1 þ ð1 m1 Þhw1 Þ þ q2 hw2 þ q3 hw3 Þ;
ð130Þ
which yields the threshold for epidemic control of
R0 <
1
:
1 p½q1 ðm1 þ ð1 m1 Þhw1 Þ þ q2 hw2 þ q3 hw3 ð131Þ
As an extreme example, imagine that contact tracing and vaccination are perfect ðp ¼ v1 ¼ 1Þ
and that all vaccinated contacts comply with a quarantine averaging 14 days ða ¼ 1=14Þ. Under
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E.H. Kaplan et al. / Mathematical Biosciences xxx (2003) xxx–xxx
our assumptions governing the durations of disease stages, Eq. (131) suggests that such a draconian quarantine policy could initially control epidemics with R0 < 3:95. Given that the protection conferred by quarantine on susceptibles has not been taken into account, this threshold
understates the true effect. Of course, the actual number of persons in quarantine could grow quite
large, so that even allowing for residential (as opposed to institutional) quarantine, such aggressive quarantine would likely prove impractical [24].
4.4. CDC’s interim policy: contact tracing with febrile quarantine
Using approximations such as those described above, it is possible to construct ÔboutiqueÕ
thresholds for any particular policy of interest. As an important illustration, the CDCÕs interim
plan for the control of smallpox calls for case isolation, contact tracing and vaccination, and a 5day quarantine of all contacts found febrile [4]. To create an approximate threshold for this
policy, we (optimistically) presume that 90% of infected contacts found in stage 3 of infection are
febrile and remanded to the (average) 5-day quarantine (h ¼ 0:9 and a ¼ 1=5). Combined with
our base case assumptions that stages 1 and 3 of infection are exponentially distributed with mean
3 days ðr1 ¼ r3 ¼ 1=3Þ, that newly detected index cases accurately name half of their contacts
ðp ¼ 0:5Þ, and that the vaccine is effective 97.5% of the time when given to persons in stage 1 of
disease ðv1 ¼ 0:975Þ we obtain the growth rate
cCDC ¼ r3 ðR0 1 pR0 ½q1 v1 þ q3 hw3 Þ
ð132Þ
and the corresponding threshold
R0 <
1
¼ 1:36;
1 pðq1 v1 þ q3 hw3 Þ
ð133Þ
which suggests that the interim CDC policy can only initially control a smallpox outbreak if
R0 < 1:36 (as reported in [8]).
4.5. Non-exponential disease-stage durations
It is possible to account for non-exponential disease-stage distributions in determining the
probabilities that infected contacts are found in various disease stages, and use these probabilities
in the approximations discussed above. As an illustration, consider the threshold derived for basic
contact tracing in Section 4.1. A key ingredient is the probability q1 that early in the epidemic, an
infected contact is still in the vaccine-sensitive stage of infection at the time the infecting index case
is detected. When T1 and T3 , the durations of the vaccine-sensitive and infectious periods respectively, are exponentially distributed with rates r1 and r3 , from Eq. (47) we obtain
r3
:
ð134Þ
q1 ¼
r1 þ r3
If we relax the exponential assumptions and instead allow T1 and T3 to be arbitrarily (but independently) distributed, we can still derive the probability that an infected contact is vaccinesensitive when the infecting index is detected.
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To proceed, since an infected person is equally likely to transmit at any point in time during the
infectious stage 3 of infection, the remaining duration of infectiousness from the time of a random
infection until the end of stage 3, T3 , is distributed as a forward recurrence time with density given
by [25]
fT3 ðxÞ ¼
PrfT3 > xg
:
EðT3 Þ
ð135Þ
A randomly infected contact will thus still be in the vaccine-sensitive stage of infection if T1 > T3 ,
an event that occurs with probability
Z 1
PrfT3 > xg
EðminfT1 ; T3 gÞ
q1 ¼
PrfT1 > xg
dx ¼
:
ð136Þ
EðT3 Þ
EðT3 Þ
0
For example, if T1 and T3 are exponentially distributed with rates r1 and r3 respectively, Eq.
(136) reduces to (134) as expected. Alternatively, if T1 and T3 are both uniformly distributed
between 0 and some upper limit u, then independently of u Eq. (136) yields 2/3. If T1 and T3 have
the same Weibull distributions, that is
PrfT1 > xg ¼ PrfT3 > xg ¼ eax
b
for x > 0 and a; b > 0;
ð137Þ
then Eq. (136) yields
q1 ¼ 21=b :
ð138Þ
This result is interesting, for it shows that with Weibull distributions, the probability that a contact
is vaccine-sensitive at the time the infecting index is detected can fall anywhere between 0 and 1,
depending upon the value of the shape parameter b; if b ¼ 1, then (138) coincides with (134).
4.6. Performance analysis
The extensions considered in this section make the direct analysis of TV very difficult. However,
all of the uncongested TV results in Section 3.3 depend explicitly on the Ôpost-intervention R0 ,Õ
which is R0 ð1 pq1 Þ, via Eqs. (63), (70) and (79). Hence, we propose the following approach to
analyzing TV under the generalizations in this section: Replace R0 ð1 pq1 Þ in Eqs. (63), (70), and
(79) by the corresponding post-intervention R0 derived in this section (e.g., with R0 ½1 pðq1 v1 þ
q3 hw3 Þ in the case of the CDC policy in (133)). There does not appear to be a similar approach to
generalizing the congested results in Section 3.4.
5. Mass vaccination
We now consider the case of mass vaccination (MV), where all asymptomatics enter the vaccination queue at time 0; this scenario can be viewed as a special case of TV, where the number of
names generated per index, c, is infinite. In view of the fact that nearly everyone is vaccinated, and
that QðtÞ < n only during the final minutes of vaccination, we assume that vaccination is completed at time T ¼ N =nl and that Q0 ðtÞ ¼ Q0 ð0Þ nlt for t 2 ½0; T . Hence, the infection term
bI3 ðtÞS 0 ðtÞ in Eqs. (19) and (20) is well approximated by r3 R0 ð1 t=T ÞI3 ðtÞ for t 2 ½0; T .
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5.1. Total deaths
To estimate the number of deaths, we keep track of how many people enter stage 2 throughout
the epidemic, and multiply this number times the
P smallpox death rate d. At time 0, which is when
mass vaccination begins, there will already be 4j¼2 Ij0 ð0Þ people who have entered stage 2. These
quantities are given in Eqs. (36), (38) and (39), recalling that at time 0 everyone in the Ij0 compartment moves to the Qj compartment. The people who are in disease stage 1 at time 0 become
symptomatic if they progress to stage 2 before they are vaccinated. From a probabilistic point of
view, a person in disease stage 1 at time 0 progresses to stage 2 at a time that is exponentially
distributed with parameter r1 , by the memoryless property of the exponential distribution. By our
assumptions in the last paragraph, each of these people will be vaccinated at a time that is uniformly distributed between 0 and T .
Hence, of these I10 ð0Þ people,
Z T
1
er1 T 1
0
r1 t 1
0
ð1 e Þ dt ¼ dI1 ð0Þ
ð139Þ
Tþ
dI1 ð0Þ
T
T
r1
0
of them will die, where I10 ð0Þ is given in (37).
Finally, we turn to the number of deaths of the S 0 (or Q0 ð0Þ) people that are susceptible at time
0. As mentioned above, new infections occur at rate r3 R0 ð1 t=T ÞI3 ðtÞ. As in Section 5.1, we
assume that I3 ðtÞ is linear with growth rate g derived in (36), which leads to reasonably simple and
accurate formulas, mainly because the susceptibles are dropping to zero linearly at rate nl and
most of the deaths under MV are of people who are already infected by time 0. Proceeding as
before, someone who is infected at time t 2 ½0; T progresses to stage 2 after an exponentially
distributed amount of time with parameter r1 , and is vaccinated at a time that is uniformly distributed between t and T . Hence, the number of people who are susceptible at time 0 and who die
during the epidemic is
Z T Z T
1
r3 R0 ðI30 ð0Þ þ gtÞ ð1 er1 ðutÞ Þ du dt:
ð140Þ
d
T
0
t
Integrating (140) and summing up over the people who are in stage 0, stage 1, and beyond stage 1
at time 0 gives the total number of deaths under MV:
4
X
er1 T 1
r3 R0 r1 T Ij ð0Þ þ I1 ð0Þ 1 þ
e
g 6 þ er1 T ðr13 T 3
þ
Dð1Þ ¼ fN þ d
3
r
T
6Tr
1
1
j¼2
!
3r12 T 2 þ 6r1 T 6Þ þ 3r1 er1 T ð2 2r1 T þ r12 T 2 Þ 2 I3 ð0Þ :
ð141Þ
The key parameters in the MV model are R0 and the vaccination capacity nl ¼ N =T . By (37),
(38) and (141), we predict that the number of deaths is roughly linear in R0 when this parameter is
in its practically relevant range, although the refinement discussed at the end of Section 3.1
suggests that I30 ð0Þ is linear in R0 , and hence the number of deaths is roughly quadratic in R0 ,
which becomes important when R0 > 7. Eq. (141) also reveals that while deaths under MV depend
only slightly on the population size N (for the vaccine death rate f is so small), they do scale with
the number infected at time 0 when the intervention is launched. Deaths under MV are therefore
heavily influenced by the number initially infected when the attack occurs. This stands in complete
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31
contrast to TV, where deaths scale with the size of the population with minimal dependence on the
initial attack size.
If vaccination is imperfect (i.e., m0 and m1 < 1), then simulation results suggest that about half of
the additional cases occur before time T and the other half occur after time T . Two changes are
required to capture the additional cases before time T : we replace 1 erðutÞ in the integral in
(140) by 1 m1 þ m1 ð1 er1 ðutÞ Þ because people now enter compartment I2 either by a failed
vaccination or by losing the race to trace (a similar substitution occurs in (139)), and we change
the infection term r3 R0 ð1 t=T ÞI3 ðtÞ that leads to (140) to r3 R0 ð1 m0 t=T ÞI3 ðtÞ because the susceptible population is decreasing at a reduced rate. However, the analysis of the additional cases
after time T requires an accurate estimate of the system state at time T , which we do not have.
6. Simulation results
This section assesses the accuracy of the analytical approximations in Sections 3–5 by comparing them to exact simulation results.
6.1. Pre-intervention
At time 0, the simulated number of infected individuals in the uncongested (i.e., R0 ¼ 3) scenario is 1336, where I10 ð0Þ ¼ 415, I20 ð0Þ ¼ 662, I30 ð0Þ ¼ 156, and I40 ð0Þ ¼ 103. Eqs. (36)–(39) predict
1346 infectees, with a breakdown of 410, 672, 144, and 120, respectively. In the congested (i.e.,
R0 ¼ 6) case, the actual number of infectees at time 0 is 1708 (Ij0 ð0Þ ¼ 660, 773, 168 and 107) and
the predicted number of infectees is 1684 (631, 789, 144, 120).
6.2. Uncongested TV
Fig. 3(a) shows the dynamics for the queue length process and the total number of infected
people for the uncongested base case. The total number of deaths is 105 000 and the maximum
queue length is 2864 on day 74, which are reasonably close to the predicted values of 116 000 deaths
and 2788 people on day 71 in (71), (79) and (83), respectively. The approximate queue length
process in Fig. 3(a) (see the last sentence of Section 3.3) is quite accurate throughout the epidemic.
A sensitivity analysis is shown in Fig. 4, which depicts the exact and approximated deaths
versus five different model parameters. Fig. 4(a) shows that our analysis accurately captures the
weak near-linear dependence of deaths versus initial attack size I10 ðsÞ. Fig. 4(b) and (c) depict the
uncongested and congested estimated number of deaths, versus the number of vaccinators n and
the tracing/vaccination rate l, respectively. Fig. 4(b) shows that the number of deaths is independent of the number of servers when n is greater than 2864. In this self-service regime, the queue
length is never greater than the number of servers and hence no named individuals wait to be
traced and vaccinated. We predict that the location of the kink in Fig. 4(b) that separates the selfservice regime from the congested regime is simply the right side of (79), which is 2788, quite close
to the actual value of 2864. However, our congested estimate significantly underestimates the
number of deaths when the system is highly congested. Substituting n for Qmax on the left side of
(79) and solving for l gives us the predicted location of the kink in the deaths vs. service rate curve
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3.5
Infected
Queue
Queue Approx
12
10
2.5
8
2
6
1.5
4
1
2
0.5
0
0
50
100
(a)
150
200
250
2000
Infected
Queue
Queue Approx
80
Number infected (104)
0
300
Time (days)
90
70
1800
1600
1400
60
1200
50
1000
40
800
30
600
20
400
10
200
0
0
(b)
3
Number in queue (103)
Number infected (104)
14
20
40
60
80
100
Number in queue (103)
32
0
120
Time (days)
Fig. 3. Population dynamics during the aftermath of a smallpox attack with the basic reproductive ratio (a) R0 ¼ 3 and
(b) R0 ¼ 6. The number of infected people, regardless of disease stage, and the number of people, whether susceptible or
asymptomatically infected, waiting in the vaccination queue, both exact and approximate.
in Fig. 4(c). The predicted value is l ¼ 28 and the actual value is 27. As in Fig. 4(b), the congested
estimate in Fig. 4(c) underestimates the impact of congestion on the number of deaths.
Fig. 4(d) shows that Eq. (70) accurately mimics the impact of the fraction of infectees named by
an index, p, on the number of deaths. Fig. 4(e) shows how the number of deaths varies as a
function of the number of names generated per index, c. The relationship predicted in (71) is fairly
accurate over the range of 10–70, where the number of deaths is roughly proportional to c1 .
Finally, to assess the accuracy of the approach suggested in Section 4, we consider quarantine
(with a ¼ (5 days)1 [4] and h ¼ 0:9) and imperfect vaccination (m0 ¼ m1 ¼ 0:975 [18,19]). Eqs. (70),
(71), (79) and (83), together with Eq. (131) for the post-intervention R0 , predict that the total
number of deaths is 103 000 and the maximum queue length is 2412 on day 78, compared to the
actual values of 96 500 deaths and 1629 people on day 98, respectively. This calculation suggests
that the approach proposed in Section 4.6 may be useful for assessing the impact of different
strategies on the number of deaths (we predict an 11.2% reduction in deaths relative to the base
case, i.e., from 116 000 to 103 000, while the actual reduction was 8.3%), but is too crude to
provide reliable estimates for the size and time of the maximum queue length.
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Number of deaths (10 )
250
Exact
Approx Uncong
Approx Cong
4
4
Number of deaths (10 )
14
33
12
Exact
Approximate
10
8
6
4
200
150
100
50
2
0
0
0
5000
(a)
10000
15000
20000
25000
120
1000
2000
3000
4000
5000
Number of vaccinators
Number of deaths (10 )
18
100
4
Exact
Approx Uncong
Approx Cong
4
Number of deaths (10 )
0
(b)
Number initially infected
80
60
40
Exact
Approximate
16
14
12
10
8
6
4
20
2
0
0
0
(c)
20
40
60
80
100
120
0
0.2
(d)
Tracing/vaccination rate (days-1)
0.4
0.6
0.8
1
1.2
Fraction of infectees named by infector
Exact
Approximate
4
Number of deaths (10 )
350
300
250
200
150
100
50
0
0
(e)
10
20
30
40
50
60
70
80
Number of contacts named
Fig. 4. Sensitivity analysis for uncertain model parameters under TV in the uncongested ðR0 ¼ 3Þ case. The exact and
approximate number of deaths versus (a) the initial attack size I10 ðsÞ, (b) the number of vaccinators n, (c) the tracing/
vaccination rate l, (d) the fraction of infectees named by an index p, and (e) the number of names generated per index c.
Figures (b) and (c) contain two analytical estimates, one for the uncongested case and one for the congested case.
6.3. Congested TV
Fig. 3(b) shows the queue length and infections in the congested case. The total number of
deaths is 336 000, and the maximum queue length is 1.74 million, which is achieved on day 50. The
predicted values are 311 000 (an underestimate of 7.4%), and a maximum queue length of 1.13
million on day 53. An investigation
of the inaccuracy of our estimated queue length process in
R1
Fig. 3(b) (and hence of 0 QðtÞ dt in (94)) reveals that the actual rush-hour arrival process is bellshaped rather than quadratic. Consequently, we attempted to use Eqs. (97)–(100) to fit a Gaussian
arrival
process; although our predictions for t0 (30.75 vs. 32 days), t1 (41.8 vs. 42 days) and
R t1
AðtÞ
dt
(3.35 · 106 vs. 3.4 · 106 people) appear to be reasonably accurate, this approach did not
t0
lead to a significantly better estimate for the total number of deaths in the congested TV case.
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5
0.8
0.7
0.6
0.5
0.4
0.3
0.2
4
3.5
3
2.5
2
1.5
1
0.1
0.5
0
0
0
20000
(a)
40000
60000
80000
100000
120000
2000
3000
4000
1.5
1
5000
6000
Number of vaccinators
Exact
Approx Cong
Approx Uncong
0.45
6
Number of deaths (10 )
6
1000
0.5
Exact
Approx Cong
Approx Uncong
2
0
(b)
Number initially infected
2.5
Number of deaths (10 )
Exact
Approximate
4.5
6
Number of deaths (10 )
Exact
Approximate
6
Number of deaths (10 )
1
0.9
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.5
0.05
0
0
0
(c)
20
40
60
80
100
-1
0
120
0.2
(d)
Tracing / vaccination rate (days )
0.4
0.6
0.8
1
1.2
Fraction of infectees named by infector
Exact
Approximate
6
Number of deaths (10 )
2.5
2
1.5
1
0.5
0
(e)
0
10
20
30
40
50
60
70
80
Number of contacts named
Fig. 5. Sensitivity analysis for uncertain model parameters under TV in the congested ðR0 ¼ 6Þ case. The exact and
approximate number of deaths versus (a) the initial attack size I10 ðsÞ, (b) the number of vaccinators n, (c) the tracing/
vaccination rate l, (d) the fraction of infectees named by an index p, and (e) the number of names generated per index c.
Figures (c) and (d) contain two analytical estimates, one for the uncongested case and one for the congested case.
Fig. 5 shows the exact and predicted (see (94) and (111)) number of deaths versus the five key
parameters. Our analysis is too crude to capture any dependence on the initial attack size in Fig.
5(a), although it is unlikely that attack sizes in the tens of thousands are feasible. As mentioned in
Section 6.2, the congested analysis underestimates the impact on the number of deaths if the
capacity nl is scarce, as seen in Fig. 5(b) and (c). The estimated number of deaths in Fig. 5(d) is
not accurate for low values of the fraction of infectees names by an infector, p. Eq. (79) predicts
that the onset of congestion occurs when l ¼ 64/day and p ¼ 0:86, which coincide reasonably well
with Fig. 5(c) and (d), respectively. As in Fig. 4(e), Fig. 5(e) shows that the number of deaths is
roughly proportional to the inverse of the number of contacts named per index.
Until now, we have only considered two values of R0 : 3 and 6. Fig. 6, which displays the
number of deaths as a function of R0 , provides three analytical estimates: Eq. (63) for the sub-
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20000
18000
Number of deaths (102)
16000
14000
12000
10000
8000
6000
4000
Exact
Approx Uncong
Approx Cong
2000
0
0
5
10
15
20
Basic reproductive ratio
100000
Number of deaths (102)
10000
1000
100
Exact
Approx Subcritical
Approx Uncong
Approx Cong
10
1
0.1
1
10
100
Basic reproductive ratio
Fig. 6. The exact and approximate number of deaths versus the basic reproductive ratio R0 , on both a (a) linear scale
and a (b) log–log scale. There are three approximations for the three different regimes (subcritical, uncongested,
congested).
critical case, Eq. (71) for the uncongested case, and Eqs. (94) and (111) for the congested case. The
log–log plot in Fig. 6(b) shows the appropriate range for each of the three approximations. The
approximations are quite accurate in their respective regimes when R0 < 6, but the congested
estimate significantly underestimates the total number of deaths when R0 > 6. The linear plot in
Fig. 6(a) shows the two kinks in the deaths versus R0 curve. The first kink corresponds to the value
of R0 such that the post-intervention R0 , which is R0 ð1 pq1 Þ in Section 3, equals 1, i.e., the kink is
at R0 ¼ ð1 pq1 Þ1 ¼ 1:33. The second kink corresponds to the value of R0 that causes the
maximum queue length to equal n. Substituting n for Qmax on the left side of (79), and solving for
R0 gives 4.6, compared to the actual value of 4.8.
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10
9
1.6
8
1.4
7
1.2
6
1
5
0.8
4
0.6
3
0.4
2
0.2
1
0
0
0
(a)
10
20
30
40
50
6
60
Time (days)
Number of deaths (103)
Infected
Queue
Number in queue (106)
Number Infected (103)
2
1.8
4
3
2
1
0
(b)
0
10
15
20
Basic reproductive ratio
50
Number of deaths (103)
Exact
Approximate
40
30
20
10
0
Exact
Approximate
1000
100
10
1
0.1
0
(c)
5
10000
60
Number of deaths (103)
Exact
Approx
Approx refined
5
20000
40000
60000
80000
Number initially infected
100000
(d)
0
1000
2000
3000
4000
5000
6000
Number of vaccinators
Fig. 7. The special case of mass vaccination. (a) The number of infected people, regardless of disease stage, and the
number of people, whether susceptible or asymptomatically infected, waiting in the vaccination queue during the aftermath of a smallpox attack. The exact and approximate number of deaths versus (b) the basic reproductive ratio R0 ,
(c) the initial attack size I10 ðsÞ and (d) the the number of vaccinators n. The refinement in Figure (b) is explained at the
end of Section 3.1.
6.4. MV
Fig. 7(a) shows the total number in queue and the total number of infected people under MV in
the R0 ¼ 3 case. The total number of deaths is 570, just one less than the estimated value of 571
from (141). Fig. 7(b)–(d) shows the exact and estimated number of deaths as a function of R0 , the
initial attack size, and the number of vaccinators n (a similar curve could be shown for the
vaccination rate l, since the number of MV deaths depends on the capacity nl), respectively. Eq.
(141) accurately captures the impact on deaths of the initial attack size and the capacity, and is
accurate for moderate values of R0 . The extension discussed at the end of Section 3.1 provides a
reasonable approximation for higher values of R0 .
6.5. Mixing versus scaling in comparing TV and MV
Halloran et al. [13] reported a detailed microsimulation model of smallpox transmission in a
small community of 2000 persons. They reported that deaths under TV exceeded deaths under
MV by a factor of 2-to-1, which is smaller than the factor of 200 reported by Kaplan, Craft and
Wein in their base case analysis of 1000 initial infections in a city of 10 million [8], and attributed
this difference to the highly structured, non-random mixing assumed in [13] versus the mass-action
free-mixing employed in [8] (and also this analysis). However, recall that our mathematical
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37
Table 2
Deaths per 1000 estimated from Halloran et al. as reported in the first column of Table 2 in [13] and the model of
Kaplan et al. [8] for the inputs used by Halloran et al. [13]: a population of 2000, a single initial infection, R0 ¼ 3:2, 80%
vaccination coverage, and response delays of 7, 27, and 37 days to match the detection of smallpox after the first, 15th,
and 25th case, as discussed by Halloran et al.
Deaths per 1000
Halloran et al. [13]
Kaplan et al. [8]
80% MV after
1 case
15th case
25th case
0.9
9.4
13.7
0.4
6.4
17.8
80% TV after
1 case
15th case
25th case
10.9
19.6
28.2
8.8
12.0
33.9
analysis has shown that deaths under (uncongested) TV scale with the population size (Eq. (71)),
while deaths under MV scale with the number infected at the time intervention begins (Eq. (141)).
Given that both the population and initial attack sizes are so different in these two analyses,
scaling alone could account for the differences reported rather than population mixing patterns.
Table 2 compares the number of deaths that result from these competing models when they are
compared using the inputs reported by Halloran et al. The results clearly indicate that the two
models are in strong agreement for both policies. In particular, note that having greatly reduced
the size of the population, deaths under TV are only a factor of two larger than deaths under MV
for both models. Thus, it is not the case that the difference in previously reported results came
about due to the different population mixing assumptions employed. Consistent with the analysis
of this paper, the different results obtained can be explained by scaling alone.
7. Discussion
For the interim CDC Ôtraced vaccinationÕ (TV) plan [4] to successfully control a bioterror
smallpox attack, it needs to be both accurate and fast. Our detailed modeling of both the race to
trace and the queueing of contacts waiting to be traced and vaccinated allows us to assess the
importance of TVÕs efficacy and efficiency, and compare it to the more efficient (but less efficacious) mass vaccination (MV). Because the complexity of the model appears to preclude an exact
analysis, we resort to analytical approximations in an attempt to understand how the number of
deaths is affected by the parameters describing the tracing and queueing processes. Although we
make liberal use of approximations, the overall accuracy of our results – typically within 10–15%
in moderately congested cases, sometimes less accurate in highly congested cases that generate
hundreds of thousands of deaths – suffices for purposes of gaining qualitative insight, comparing
various policies, and capacity planning.
Our analysis includes three sets of results, which are given in Section 3–5. First, an approximate
analysis of the TV model in Section 3 – which uses a detailed understanding (gleaned from
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simulations) of the relative magnitudes of the flow terms in the TV model, a transformation of the
state variables, and some classic results from epidemic theory and non-stationary queueing theory –
generated a closed-form estimate of the total number of deaths and the queue length process.
These results were derived in both an uncongested regime, where the maximum queue length is
less than the number of available vaccinators, and a congested regime.
While the fraction of deaths in a traditional SIR model depends only on R0 (see Eq. (5)), the
number of deaths in the TV model also depends strongly on three tracing parameters ðc; p; q1 Þ and
two logistical parameters (n and l, via the capacity nl). More specifically, the number of names
generated per index, c, measures the efficiency of TV, and is due to what we call random tracing:
The higher c is, the faster herd immunity is reached. The efficacy of TV, which is due to what we call
local tracing, is the product of the accuracy p, which is the probability that an infected contact is
correctly named by his infector, and the probability that the race to trace is won (i.e., the contact is
located and vaccinated while still vaccine-sensitive), which is q1 in the absence of congestion.
We have three main TV results: the number of deaths in the uncongested case, the maximum
queue length in the uncongested case, and the number of deaths in the congested case. In the
uncongested regime, Eqs. (70) and (71) show that tracing essentially reduces the post-intervention
basic reproductive ratio to R0 ð1 pq1 Þ, and the number of deaths is roughly proportional to c1 .
Eq. (70) also shows that the number of deaths increases with the number of asymptomatic infecteds at the time intervention begins, but only in a weak manner. Eq. (71) also shows that under
(uncongested) TV, the number of deaths scales with the population size N.
The significance of our approximate expression for the maximum queue length in the uncongested regime in (79) is that it quantifies the transition from the uncongested to the congested
regime. Fig. 4 confirms that Eq. (79) accurately predicts the kinks in the curves of deaths versus
the basic reproductive ratio, the number of vaccinators, and the vaccination rate. The degradation
in performance characterized by the kinks in these curves can be explained in terms of the
probability of winning the race to trace, conditioned on a contact being successfully named by his
infector. In the uncongested case in Section 3.3, we find that the race to trace is won if T1 , which is
the duration of the infecteeÕs vaccine-sensitive period, is less than T3 þ V , which is the remaining
time in the infectorÕs infectious period plus the tracing/vaccination time V . Because T1 and T3 are
exponentially distributed with means r11 and r31 respectively, and because the tracing/vaccination
rate satisfies l r3 (see Table 1), the probability of winning the race in this case is r3 =ðr1 þ r3 Þ, as
shown in (47). However, in the congested regime, the race to trace pits T1 against T3 þ V þ Wq ,
where Wq is the waiting time in the tracing/vaccination queue. If the queue length is QðtÞ > n, then
Wq ðQðtÞ nÞ=nl. Hence, a negative feedback loop is generated when QðtÞ exceeds n: the
probability of winning the race to trace is reduced, the resulting infectees who evade the TV intervention become infectious and then symptomatic, generating R0 ðtÞ new infections and c names,
thereby further increasing both the spread of the epidemic and the queue length, which in turn
makes it even more difficult to win the race to trace. In Fig. 3(b), this feedback loop causes the
queue length to skyrocket to 1.74 million people, and this vicious cycle is not broken until herd
immunity (due to disease recovery and vaccination) kicks in, and results in nearly the entire
population being traced.
Eqs. (94) and (111) provide an approximate expression for the number of deaths in the congested regime. While this result is less transparent than in the uncongested case, it again shows
that the number of deaths is roughly proportional to c1 . However, our approximations are less
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39
accurate here because of our inability to accurately predict the area under the queue-length curve;
e.g., we are unable to capture the effect of the initial attack size. Also, although the capacity nl
dictates whether the system is uncongested or congested, our analysis predicts that the number of
deaths in the congested regime is only weakly affected by capacity (i.e., deaths is c1 þ c2 =ðnlÞ,
where the constants c1 and c2 satisfy c1 c2 ), whereas Figs. 4(b),(c) and 5(b),(c) show that the
actual impact is greater.
The second set of results are approximate thresholds (in terms of the basic reproductive ratio
R0 ) for initially containing the epidemic. These results allow us to assess the impact of imperfect
vaccination, quarantine and contacts-of-contacts tracing. A main conclusion from Section 4 is
that contacts-of-contacts tracing offers only a slight enhancement in tracing efficacy (i.e., successfully vaccinating people who were infected by contacts of the index case) over contact tracing
(see Fig. 2); its primary benefit is its significant increase in efficiency – by generating c2 rather than
c names per index case – which hastens the effect of herd immunity.
Our third set of results is for mass vaccination (MV). A probabilistic analysis provides a closedform expression for the number of deaths, and reveals that the number of deaths is roughly linear in
R0 over the practically relevant range [14] of R0 , and shows how the number of deaths decreases as
more vaccination capacity is added. Hence, in addition to enabling the comparison of TV and MV,
Eqs. (79) and (141) provide a capacity planning tool for tracing/vaccination resources in the event
of a smallpox attack. We also showed that under MV, the number of deaths is heavily dependent on
the number infected at the time intervention begins, and is not influenced heavily by the population
size. A companion paper [8] carries out a detailed comparison of TV and MV, and concludes that
serious consideration should be given to changing the interim CDC response plan from TV to MV.
Acknowledgements
Supported by the Societal Institute for the Mathematical Sciences via grant DA-09351 from the
National Institute on Drug Abuse (E.H.K.), a grant from the Singapore–MIT Alliance (L.M.W.),
and contract 263-MD-210207 from the Fogarty International Center of the National Institutes of
Health (L.M.W.). We thank Ellis McKenzie of the Fogarty International Center for his comments
and interest in our work. L.M.W. also thanks the Department of Management, University of
Canterbury, New Zealand for their hospitality while this work was performed.
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