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Transmission and dynamics of tuberculosis on generalized households

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Transmission and dynamics of tuberculosis on generalized households
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  - Author to whom correspondence should be addressed.Present address: Department of Biometrics, 432 WarrenHall, Cornell University, Ithaca, NY, 14853-7801, U.S.A.E-mail: 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 >  14853-7801, ; . S . A .,  and  B Department of  ¹ heoretical andApplied Mechanics ,  Cornell  ; niversity , 317  Kimball Hall ,  Ithaca ,  N >  14853-1503,  ; . 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 (gener-alized household). Typical epidemiological models do not incorporate the impact of generaliz-ed 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 air-borne transmitteddisease that, with some probability, infectsindividuals who inhale  Mycobacterium tubercu - losis  (Daniel, 1991). The re-emergence 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 (Castillo-Chavez  et al ., 1997;Castillo-Chavez & 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 par-ticular 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; Castillo-Chavez & Feng, 1996,1997; Castillo-Chavez  et al ., 1997; Feng  et al .,2000). The focus of this article is on the impact of intense and long exposure to individuals withactive-TB on its transmission dynamics at popu-lation 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 epi-demiologically active households (a householdwith an actively infected individual). They con-sidered household/non-householdof TB contactsas a convenient measure of intimacy of exposure(household contacts being in general more con-ducive to infection). Rose  et al . found that oftenthe estimated duration and/or proximity of expo-sure in some non-household contacts was equalto that observed in households. Rose  et al . pro-posed the use of   && close ''  and  && casual ''  rather thanhousehold/non-household 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 house-hold, 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 epi-demiologically active clusters. Their study mayprove useful in evaluating their importance as an epidemiological control unit  on public healthpolicy. A somewhat similar approach was de-veloped 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 per-capita risk of infection in epidemiologi-cally active clusters; Section 4 looks at the addedimpact on TB dynamics of non-cluster generatedsecondary infections (casual infections); Section 5uses marked di !  erences on epidemiological time-scales in the analysis of our general model; and, " nally the Conclusion introduces additionalevidence of the impact of clusters on diseasedynamics, re-states 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 TB-infected indi-viduals now develop clinical tuberculosis (albeit,this was not always the case, see Aparicio  et al .,2000).Incidence of active-TB (new active cases peryear) in developed countries can be as low as 10per 100000 population (or less) while conserva-tives 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 active-TB in the30 } 200 per 100000 population range. In someexceptional cases, incidence of active-TB is ashigh as 500 per 100000 population (for a recentrevision see Davis, 1998). These values are notconclusive as it is possible to have high preva-lence of latent infections and low incidence of active-TB because TB is a disease with low pro-gression 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 socio-economicstatus.Individualswho have a latentinfectionarenot clinically ill or capable of transmitting TB(Miller, 1993). The likelihood of adequate treat-ment is critical. Appropriately treated individualsbecome non-infectious quickly (Daniel, 1991)while latently infected, that is, infected but non-infectious individuals may be stopped from de-veloping 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 long-lasting partial im-munity against re-infectionor/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. Conse-quently, 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 character-istics of shared environments all contribute to theoverall risk of transmission per contact. TheCenters for Disease Control (CDC) estimatethat the follow-up 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 towardsactive-TB. It is assumed that all epidemiologi-cally active clusters with a single infectious indi-vidual have the same risk of infection. Hence, thenumber of secondary infections produced in eachepidemiologically active cluster depends only onthe number of susceptible and infectious indi-viduals 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  sus-ceptible 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    de-notes the average infectious period of the indexcase. The    value associated with an epi-demiologically active cluster depends on theinfectiousness of the active case and on the char-acteristics 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 non-dimensionalquantity q ,  , whichdeterminesthe percentageof secondary infections caused by the index caseduring all of its infections period, can be esti-mated from epidemiological surveys. This per-centage of the cluster contacts ranges between 40and 80% (Rose  et al ., 1979; Nardell  et al ., 1991;Catazaro, 1982; Riley  et al ., 1962). The low inci-dence 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 con-tacts in epidemiologically active clusters becausethe size of the subpopulation living in epi-demiologically 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 dra-matically on casual contacts rather than in gener-alized 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 popu-lations. It is also assumed that all infectious indi-viduals have the same degree of infectiousnessand, therefore, the same probability to transmitthe disease to susceptibles. The general form of TB models is as follows (Castillo-Chavez & Feng,1996; Castillo-Chavez  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 naturalper-capita mortality rate,  d  is the per-capita TB-induced mortality rate,     is the transmissionrate,  k  is the per-capita rate of progression toactive TB,  r  is the per-capita treatment rate, and  " r # d #   is the total per-capita 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 (epi-demiologically active clusters) on the globaltransmission dynamics of TB. Contacts are clas-si " ed into two categories: close, daily and pro-longed 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 pro-longed exposure of susceptibles to infectiousindividuals. Hence, members of an epidemiologi-cally active clusters are at risk of TB-infectionexclusivelybecause of close and frequent contactswith the infectious individual that de " nes it.Individuals either belong to epidemiologicallyactive clusters[size  N  ( t )]or theydo not [popula-tion 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, to-gether 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 non-infectiousindividuals in  N   and  N  " S  # E   those in the N  -population. It is assumed that epidemiologi-cally active clusters have only one infectious indi-vidual and no individuals belong to more thanone epidemiologically active cluster. These ap-proximations are justi " ed by the extremely lowvalues recorded of active-TB 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 def-inition 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 incorpo-ration 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 set-ting, 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  -popu-lation) individual becomes infectious the N   population increases by  n  (where  n  is theaverage cluster size associated with an infec-tious individual) while the  N   populationdecreases by  n # 1. When an infectious indi-vidual recovers after treatment or dies the  N  populationdecreases by  n  while the  N   popula-tion 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 com-ponent according to the susceptible and latentfractions  S  /  N   and  E  /  N  , respectively. There-fore, 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 pro-portion  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 propor-tionally into the populations  N  " N  # I  and N  " N  .   We assume that the  per capita  natural mortal-ity (  ), disease-induced mortality ( d ), and treat-ment ( r ) rates are also constant. Hence, thetotal  per capita  removal rate of infectious indi-viduals 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 active-TB implies that the N  -population is signi " cantly smaller than the N  " N   population. Prevalence of active-TB,that is  I /  N , is of the order of the incidence of active-TB. 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 approxi-mation) 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 full-model are com-pared 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
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