
Author to whom correspondence should be addressed.Present address: Department of Biometrics, 432 WarrenHall, Cornell University, Ithaca, NY, 148537801, U.S.A.Email: jpa9
@
cornell.edu
J
.
theor
.
Biol
. (2000)
206,
327
}
341doi:10.1006/jtbi.2000.2129, available online at http://www.idealibrary.com on
Transmission and Dynamics of Tuberculosis on Generalized Households
J
UAN
P. A
PARICIO
*

, A
NGEL
F. C
APURRO
?A
AND
C
ARLOS
C
ASTILLO
C
HAVEZ
B
*Departamento de F
n
&
sica
,
Facultad de Ciencias Exactas y Naturales
,
;
niversidad de Buenos Aires
,
Pab
.
I
,
Ciudad
;
niversitaria
, 1428
Buenos Aires
,
Argentina
,
?
Departamento de Investigacio
H
n
,
;
niversidad de Belgrano
,
Zabala
1851,
Piso
12, 1426
Buenos Aires
,
Argentina
,
A
¸
aboratorio de Ecolog
n
H
a
,
;
niversidad Nacional de
¸
uja
&
n

CONICE
¹
,
Ruta
5
y
7, 6700
¸
uja
H
n
,
Argentina
,
Department of Biometrics & Mathematical and
¹
heoretical Biology Institute
,
Cornell
;
niversity
,431
=
arren Hall
,
Ithaca
,
N
>
148537801,
;
.
S
.
A
.,
and
B
Department of
¹
heoretical andApplied Mechanics
,
Cornell
;
niversity
, 317
Kimball Hall
,
Ithaca
,
N
>
148531503,
;
.
S
.
A
.(
Received on
1
June
1998,
Accepted in revised form on
20
June
2000)
Tuberculosis(TB) transmission is enhanced by systematic exposure to an infectious individual.This enhancement usually takes place at either the home, workplace, and/or school (generalized household). Typical epidemiological models do not incorporate the impact of generalized households on the study of disease dynamics. Models that incorporate cluster (generalizedhousehold) e
!
ects and focus on their impact on TB
'
s transmission dynamics are developed.Detailed models that consider the e
!
ect of casual infections, that is, those generated outsidea cluster, are also presented. We
"
nd expressions for the
Basic Reproductive Number
asa function of cluster size. The formula for
R
separates the contributions of cluster and casualinfections in the generation of secondary TB infections. Relationships between cluster andclassical epidemic models are discussed as well as the concept of critical cluster size.
2000 Academic Press
1. Introduction
Tuberculosis (TB) is an airborne transmitteddisease that, with some probability, infectsindividuals who inhale
Mycobacterium tubercu

losis
(Daniel, 1991). The reemergence of TB isa major source of concern all over the world. Thecauses behind the recently observed increases of
active
TB cases are the source of serious studiesand intense debate (CastilloChavez
et al
., 1997;CastilloChavez & Feng, 1996, 1997; Feng
et al
.,2000; Blower
et al
., 1995, 1996; Bloom, 1994;Reichman & Hersch
"
eld, 1993; Davis, 1998). TBtransmission is enhanced by the systematic andlong exposure of susceptible individuals to particular infectious individuals
*
a feature that isnot taken into account in the models that havebeen developed until now (Waaler
et al
., 1962;Brogger, 1967; ReVelle, 1967; ReVelle
et al
., 1967;Waaler & Piot, 1970; Azuma, 1975; Bermejo
et al
., 1992; Schulzer
et al
., 1992, 1994; Blower
et al
., 1995, 1996; CastilloChavez & Feng, 1996,1997; CastilloChavez
et al
., 1997; Feng
et al
.,2000). The focus of this article is on the impact of intense and long exposure to individuals withactiveTB on its transmission dynamics at population level.Rose
et al
. (1979) recommended priorities asa result of their study on the risk of transmission
0022
}
5193/00/190327
#
15 $35.00/0
2000 Academic Press
per contact. They classi
"
ed individuals at riskaccording to whether or not they lived in epidemiologically active households (a householdwith an actively infected individual). They considered household/nonhouseholdof TB contactsas a convenient measure of intimacy of exposure(household contacts being in general more conducive to infection). Rose
et al
. found that oftenthe estimated duration and/or proximity of exposure in some nonhousehold contacts was equalto that observed in households. Rose
et al
. proposed the use of
&&
close
''
and
&&
casual
''
rather thanhousehold/nonhousehold contacts to accountfor these e
!
ects. He encouraged the use of thisclassi
"
cation in the evaluation of the risk of transmission per contact (Rose
et al
., 1979).A
generalized household
or
cluster
is de
"
nedas a group of individuals who shared daily andprolonged contacts (e.g. people sharing a household, workplace, or a common locale intenselyand frequently). An
epidemiologically activecluster
is a generalized household with
infectious
individuals.When an individual becomes infectious thenthe status of his/her cluster changes. The risk of infection per individual in the cluster becomessigni
"
cant. The cluster moves from the inactiveto epidemiologically active category. Models aredeveloped to evaluate the relative importanceof TB transmission in populations with epidemiologically active clusters. Their study mayprove useful in evaluating their importance as an
epidemiological control unit
on public healthpolicy. A somewhat similar approach was developed for the study of gonorrhea transmissionvia the use of core groups by Hethcote & Yorke(1984).Our paper is organized as follows: Section 2gives a brief review of the epidemiology of TB;Section 3 introduces the basic cluster model[eqns (11)
}
(15)] and shows that the
Basic Repro

ductive Number
is a linear function of averagecluster size and a bounded (nonlinear) function of the percapita risk of infection in epidemiologically active clusters; Section 4 looks at the addedimpact on TB dynamics of noncluster generatedsecondary infections (casual infections); Section 5uses marked di
!
erences on epidemiological timescales in the analysis of our general model; and,
"
nally the Conclusion introduces additionalevidence of the impact of clusters on diseasedynamics, restates our results, and suggestspossible avenues of future research.
2. Epidemiology of Tuberculosis
The number of bacilli excreted by most personswith pulmonary tuberculosis is small (Styblo,1991).Individualswho experienceintensecontactwith the TB bacilli in poorly ventilated areas arethe most likely to become infected. Long periodsof latency (
inactive
TB) imply that new cases of infectionare notclinically apparent and thereforego unobserved for some time. Progression fromlatent to active TB is uncommon. It is estimatedthat only about 5
}
10% of TBinfected individuals now develop clinical tuberculosis (albeit,this was not always the case, see Aparicio
et al
.,2000).Incidence of activeTB (new active cases peryear) in developed countries can be as low as 10per 100000 population (or less) while conservatives estimates of the value of this rate at thebeginning of the 20th century are in 300
}
600 per100000 population. Currently, most developingcountries have incidences of activeTB in the30
}
200 per 100000 population range. In someexceptional cases, incidence of activeTB is ashigh as 500 per 100000 population (for a recentrevision see Davis, 1998). These values are notconclusive as it is possible to have high prevalence of latent infections and low incidence of activeTB because TB is a disease with low progression rates.The likelihood of progression towards activeTB depends on age of infection (Comstock& Cauthen, 1993; Williamson County TB Study,1963; Comstock & Edwards, 1975) as well as onfactors that correlate well with socioeconomicstatus.Individualswho have a latentinfectionarenot clinically ill or capable of transmitting TB(Miller, 1993). The likelihood of adequate treatment is critical. Appropriately treated individualsbecome noninfectious quickly (Daniel, 1991)while latently infected, that is, infected but noninfectious individuals may be stopped from developing active TB by prophylactic therapy(Ferebee, 1970). Most exposed individuals mountan e
!
ective immune response to the initialinfection (Smith & Moss, 1994). This immune328
J. P. APARICIO
E
¹
A
¸
.
response limits proliferation of the bacilli leadingto what appears to be longlasting partial immunity against reinfectionor/a response capableof stopping reactivation of latent bacilli. Exposedindividuals may remain in the latent stage forlong and variable periods of time; in fact, mostdie without ever developing active TB. Consequently, age of infection as well as chronologicalage are important factors on disease progression(Castillo Chavez
et al
., 1997).Tuberculosismorbidity and mortality rates arestrongly a
!
ected by urban living conditions. Forexample, in the U.S.A. it was shown that the riskof tuberculosisincreases with population sizeandurban living conditions. In Central Harlem,a neighborhood in New York City, where incomeis low, TB incidence rates are several timesgreater than in NY City (Bloch
et al
., 1989). Theinfectiousness of the source case, the durationand frequency of exposure, and the characteristics of shared environments all contribute to theoverall risk of transmission per contact. TheCenters for Disease Control (CDC) estimatethat the followup of a
&&
typical
''
case of activetuberculosis results in the identi
"
cation of approximately nine potentially e
!
ective contacts(Etkind, 1993).A cluster becomes epidemiologically active assoon as one of its members progresses towardsactiveTB. It is assumed that all epidemiologically active clusters with a single infectious individual have the same risk of infection. Hence, thenumber of secondary infections produced in eachepidemiologically active cluster depends only onthe number of susceptible and infectious individuals in the cluster as well as on the averagelifespan of the epidemiologically active cluster.The same constant
per

capita
risk of infection perunit of time is assumed for all epidemiologicallyactive clusters. Hence, the expected number of secondary infections produced by a source casein an epidemiologically active cluster with
S
susceptible individuals equals
S
(1
!
e
) (see alsoKelling & Grenfell, 2000). Here,
denotes themean per susceptible risk of infection per unit of time in an epidemiologically active cluster whenthere is only one infectious individual while
denotes the average infectious period of the indexcase. The
value associated with an epidemiologically active cluster depends on theinfectiousness of the active case and on the characteristics of the epidemiologically active cluster.Estimates for the length of the infectious periods(
) are di
$
cult to obtain because the time of TBactivation (age of active intection) is di
$
cult todetermine. Fortunately, the nondimensionalquantity
q
,
, whichdeterminesthe percentageof secondary infections caused by the index caseduring all of its infections period, can be estimated from epidemiological surveys. This percentage of the cluster contacts ranges between 40and 80% (Rose
et al
., 1979; Nardell
et al
., 1991;Catazaro, 1982; Riley
et al
., 1962). The low incidence of TB disease suggest that the probabilityper unit of time that a susceptible individual, whodoes not belong to any epidemiologically activecluster, has a close contact with an infectiousindividual is very low. Hence, the
per

capita
riskof infection of individuals who are only exposedto casual contacts is signi
"
cantly smaller to thatof those who risk infection in epidemiologicallyactive clusters. Nevertheless, the total number of secondary infections produced by casual contactsmay still be greater than those produced by contacts in epidemiologically active clusters becausethe size of the subpopulation living in epidemiologically active clusters is signi
"
cantlysmaller than the total population size. Hence, itwould not be surprising to
"
nd out that thedynamics of tuberculosis at the population levelin cities (high casual contact rates) depends dramatically on casual contacts rather than in generalized households contacts. We will elaboratethis point further in the Conclusion section.
3. The Basic Cluster Model
Classical deterministic epidemiological modelsfor the transmission dynamics of an infectiousdisease are built on homogeneously mixing populations. It is also assumed that all infectious individuals have the same degree of infectiousnessand, therefore, the same probability to transmitthe disease to susceptibles. The general form of TB models is as follows (CastilloChavez & Feng,1996; CastilloChavez
et al
., 1997; Feng
et al
.,2000; Blower
et al
., 1995, 1996).d
S
d
t
"
!
bS
!
S IN
, (1)
TB DYNAMICS ON GENERALIZED HOUSEHOLDS
329
d
E
d
t
"
S IN
!
(
#
k
)
E
, (2)d
I
d
t
"
kE
!
I
, (3)d
¹
d
t
"
rI
!
¹
, (4)where
is the recruitment rate,
is the naturalpercapita mortality rate,
d
is the percapita TBinduced mortality rate,
is the transmissionrate,
k
is the percapita rate of progression toactive TB,
r
is the percapita treatment rate, and
"
r
#
d
#
is the total percapita removal ratefrom the infectious class. All of these rates areassumed to be constant.
S
,
E
,
I
and
¹
, representpopulation numbers of susceptible, latent,infectious, and recovered (treated) individuals,respectively.
N
"
S
#
E
#
I
#¹
is the totalpopulation size.The
Basic Reproductive Number
R
, de
"
ned asthe mean number of secondary cases producedby one infectious individual in a fully susceptiblepopulation is given by
R
"
k
(
#
k
). (5)The above expression for
R
depends linearlyon both, the e
!
ective transmission rate
, andthe mean infectious period 1/
. When
R
(
1 anepidemic is not possible and the disease diesout. Therefore, classical public health controlmeasures are usually based on methods that, if e
!
ective, would lower
R
. Earlier models for TBdynamicshave not incorporatedlocale
!
ects (epidemiologically active clusters) on the globaltransmission dynamics of TB. Contacts are classi
"
ed into two categories: close, daily and prolonged contacts, that is, contacts in a
cluster
(generalized household) and close but infrequentcontacts, that is,
casual contacts
. In our
"
rstmodel, casual contacts are deliberately ignored.To illustrate our ideas in the simplest possiblesetting, we consider a homogeneously mixingpopulation where, at
"
rst, TB transmission isdriven exclusively by the systematic and prolonged exposure of susceptibles to infectiousindividuals. Hence, members of an epidemiologically active clusters are at risk of TBinfectionexclusivelybecause of close and frequent contactswith the infectious individual that de
"
nes it.Individuals either belong to epidemiologicallyactive clusters[size
N
(
t
)]or theydo not [population size
N
(
t
)]. To simplify matters, clusters are
not
followed through time. We only follow thedynamics of the aggregatedpopulations
N
(
t
) and
N
(
t
). When latently infected individuals of the
N
population develop active TB and becomeinfectious (disease progression) they move, together with the members of their clusters into the
N
population. Conversely, when an infectiousindividual recovers, then he/she returns, with allthe members to his/her associated cluster, intothe
N
population.We let
N
,
S
#
E
denote the noninfectiousindividuals in
N
and
N
"
S
#
E
those in the
N
population. It is assumed that epidemiologically active clusters have only one infectious individual and no individuals belong to more thanone epidemiologically active cluster. These approximations are justi
"
ed by the extremely lowvalues recorded of activeTB prevalence (
I
/
N
)and by the extremes low progression rates fromlatent to active TB. In fact,
I
/
N
is typically under100 per 100000 population in most developingcountries and signi
"
cantly lower in developedones (Davis, 1998). Hence, it is reasonable toassume further that the
N
population is signi
"
cantly smaller than the
N
population. The definition of epidemiologically active cluster and theabove assumptions imply that
N
"
nI
and that
N
"
(
n
#
1)
I
, where
n
is the mean generalizedhousehold size.The above approach neglects the contributionto TB dynamics of treated classes. Their incorporation is straightforward and does not producesigni
"
cant di
!
erences in the resulting qualitativedynamics (see Appendix B). Albeit their role iscritical in the evaluation of control policies wherethe epidemiological unit is the epidemiologicallyactive cluster.In order to look at the simplest possible setting, it is further assumed that the process of epidemiologically active cluster formation has
&&
no memory
''
. That is, it is assumed that whena latent individual develops active TB, he/she hasno prior information about the cluster from330
J. P. APARICIO
E
¹
A
¸
.
where he/she caught the disease. This assumptionis justi
"
ed assuming long periods of latency.These assumptions will be weakened later on.The development of the equations that describethe simplest cluster or generalized householdmodel follows from these additional observationsand assumptions:
When an exposed (belonging to the
E
population) individual becomes infectious the
N
population increases by
n
(where
n
is theaverage cluster size associated with an infectious individual) while the
N
populationdecreases by
n
#
1. When an infectious individual recovers after treatment or dies the
N
populationdecreases by
n
while the
N
population increases by
n
.
If
k
is the progression rate to active TB, thenthere are
kE
new infectious individuals perunit of time and, the rate of change of the
N
population is increased by
nkE
. This rateis budgeted into a susceptible and latent component according to the susceptible and latentfractions
S
/
N
and
E
/
N
, respectively. Therefore, the gain terms are (
S
/
N
)
nkE
and(
E
/
N
)
nkE
, for the
S
and
E
classes.
If
is the total
per

capita
removal rate in theinfectious class then there are
n
I
people goingout of the population
N
per unit of time. It isassumed that
S
/
N
proportion of this ratereturns to the susceptible class while the proportion
E
/
N
of the same rate returns to thelatent class. The relation
nI
"
N
imply that(
S
/
N
)
n
I
"
S
and (
E
/
N
)
n
I
"
E
.
The constant
#
ux of susceptible individuals (
)is also assumed to be distributed proportionally into the populations
N
"
N
#
I
and
N
"
N
.
We assume that the
per capita
natural mortality (
), diseaseinduced mortality (
d
), and treatment (
r
) rates are also constant. Hence, thetotal
per capita
removal rate of infectious individuals is given by the constant
"
b
#
d
#
r
.These assumptions and observations lead tofollowing model for transmission in generalizedhouseholds:d
S
d
t
"
N
N
!
(
#
#
)
S
#
S
N
nkE
, (6)d
E
d
t
"
S
!
(
#
)
E
#
E
N
nkE
, (7)d
I
d
t
"
kE
!
I
, (8)d
S
d
t
"
N
N
!
S
#
S
!
S
N
nkE
, (9)d
E
d
t
"
E
!
(
#
k
)
E
!
E
N
nkE
. (10)Low prevalence of activeTB implies that the
N
population is signi
"
cantly smaller than the
N
"
N
population. Prevalence of activeTB,that is
I
/
N
, is of the order of the incidence of activeTB. In developing countries, we must havethat this prevalence is of the order of 10
whilein developed ones it is of the order 10
. Becausecluster size is of the order 10 we have that
N
/
N
"
(
n
#
1)
I
/
N
is of the order 10
or less.Onthe otherhand, themean infectiousperiod 1/
is of the order of a year while the life expectancy,1/
, is of the order of 60 years, that is,
<
. It istherefore reasonable to neglect (as a
"
rst approximation) recruitment and natural mortality intothe
N
population. This last assumption leads tothe simpli
"
ed basic cluster model:d
S
d
t
"!
(
#
)
S
#
S
N
nkE
, (11)d
E
d
t
"
S
!
E
#
E
N
nkE
, (12)d
I
d
t
"
kE
!
I
, (13)d
S
d
t
"
!
S
#
S
!
S
N
nkE
, (14)d
E
d
t
"
E
!
(
#
k
)
E
!
E
N
nkE
. (15)The above approximation appears reasonablewhen the dynamics of the fullmodel are compared to the dynamics of the above model insimulations based on realistic parameter valuesthat cover a wide range of scenarios. Figure 1
TB DYNAMICS ON GENERALIZED HOUSEHOLDS
331