ASIMULATION MODELOFTHEEPIDEMIOLOGY OF … · 492 FOCKS ANDOTHERS...

Am. I. Trop. Med. Hyg.. 53(5), 1995, pp. 489â€”506Copyright Â©1995 by The American Society of Tropical Medicine and Hygiene

A SIMULATION MODEL OF THE EPIDEMIOLOGY OF

URBAN DENGUE FEVER: LITERATURE ANALYSIS,

MODEL DEVELOPMENT, PRELIMINARY VALIDATION,

AND SAMPLES OF SIMULATION RESULTS

DANA A. FOCKS, ERIC DANIELS, DAN 0. HAILE, ANDJAMES E. KEESLINGModeling and Bioengineering Research Unit, Medical and Veterinary Entomology Research

Laboratory, Agricultural Research Service, U.S. Department of Agriculture, Gainesville, Florida;Department of Mathematics, University of Florida, Gainesville, Florida

Abstract. We have developed a pair of stochastic simulation models that describe the daily dynamics of denguevirus transmission in the urban environment. Our goal has been to construct comprehensive models that take intoaccount the majority of factors known to influence dengue epidemiology. The models have an orientation toward sitespecific data and are designed to be used by operational programs as well as researchers. The first model, the containerinhabiting mosquito simulation model (CIMSiM), a weather-driven dynamic life-table model of container-inhabitingmosquitoes such as Aedes aegypti, provides inputs to the transmission model, the dengue simulation model (DENSiM);a description and validation of the entomology model was published previously. The basis of the transmission modelis the simulation of a human population growing in response to country- and age-specific birth and death rates. Anaccounting of individual serologies is maintained by type of dengue virus, reflecting infection and birth to seropositivemothers. Daily estimates of adult mosquito survival, gonotrophic development, and the weight and number of emerging females from the CIMSiM are used to create the biting mosquito population in the DENSiM. The survival andemergence values determine the size of the population while the rate of gonotrophic development and female weightestimates influence biting frequency. Temperature and titer of virus in the human influences the extrinsic incubationperiod; titer may also influence the probability of transfer of virus from human to mosquito. The infection modelwithin the DENSiM accounts for the development of virus within individuals and its passage between both populations. As in the case of the CIMSiM, the specific values used for any particular phenomenon are on menus wherethey can be readily changed. It is possible to simulate concurrent epidemics involving different serotypes. To providea modicum of validation and to demonstrate the parametenization process for a specific location, we compare simulation results with reports on the nature of epidemics and seroprevalence of antibody in Honduras in low-lying coastalurbanizations and Tegucigalpa following the initial introduction of dengue-1 in 1978 into Central America. Weconclude with some additional examples of simulation results to give an indication of the types of questions that canbe investigated with the models.

This paper is the third in a series of articles documentingthe development, validation, and use of a pair of complementary simulation models describing vector population dynamics and the epidemiology of dengue viruses in the urbanenvironment. The subject of the first two articles was thecreation and evaluation of an entomologic model.' 2 Thismodel provides input to a transmission model, the subjectof the present article. As an introduction and to provide therationale for the need for comprehensive models that havean orientation toward site-specific epidemiologic data andoperational use, we begin with a survey of past and currentdengue control efforts. This is followed by a brief summaryof the entomology model as the usefulness of the transmission model is directly related to the accuracy of the entomologic inputs. We then present the development of thetransmission model. To provide an initial indication of modelvalidity and to demonstrate the parametenization process for

a specific location, we compare simulation results with reports on the nature of epidemics and seroprevalence of antibody in Honduras in low-lying coastal urbanizations andTegucigalpa following the initial introduction of dengue-1 in1978 into Central America. We conclude with some additional examples of simulation results to give an indicationof the types of questions that can be investigated with themodels by looking at the influence of virus titer and weatheron dengue transmission. We are currently working cooperatively with dengue control projects in the Caribbean, Cen

tral and South America, and Southeast Asia with the goalsof further validation and the transfer of the models to operational and research users. The results of this work will bepublished separately.

Globally, dengue viruses currently are considered to bethe most important arthropod-borne viruses transmitted tohumans, whether measured in terms of the number of humaninfections or the number of 4 Dengue viruses causea range of disease in humans from undifferentiated fever,dengue fever syndrome, to dengue hemorrhagic fever withand without@ During the last 40 years, all four serotypes of dengue virus have spread to virtually all receptiveareas of the tropical world including Africa, the Pacific, andthe Americas. In the 1950s, a more dangerous form of dengue fever involving hemorrhage and a shock syndrome(DHF/DSS) appeared in Southeast Asia. Primarily affecting

children of local populations in endemic areas, the untreated

fatality rate was sometimes as high as 30â€”40%.@Today, dengue viruses remain among the leading causes of childhoodhospitalization in many urban centers in Asia.6 In the pastdecade, DHFIDSS appeared in the New World,7 the first major epidemic occurring in Cuba in l981@ and the second inVenezuela in 1990.@ During the 1980s, intermittent DHF/DSS was confirmed or suspected in perhaps 14 additionalcountries within the hemisphere.'Â° II Recently, compelling

reasons have been voiced that DHFIDSS will increasinglybecome a serious public health issue in the Americas par

489

490 FOCKS AND OTHERS

alleling the experience of Southeast Asia in the l950s and19605.2 The deteriorating situation is attributed to popula

tion growth and urbanization without adequate provision for

reliable water supply and housing, the frequent introductionof virus in new areas through human movements, and thealmost universal absence of adequate control of the primaryurban vector, Aedes aegypti)@'5

The threat of urban yellow fever earlier this century wassufficient to prompt the eradication of Ae. aegypti from mostcountries in Central and South America.'6 This effort wasvertically orientated, a paramilitary operation largely devel

oped and funded by the Rockefeller Foundation with laterparticipation by the Pan American Health Organization.'7

The impetus for eradication was lost, however, with the discovery of the sylvan cycle of the virus and a growing realization that Ae. aegypti eradication efforts, unless global,were futile.'8 Gains made in the 1950s and l960s were notmaintained, and today, most formerly Ac. aegypti-free areashave become reinfested.â€• The developing situation of hy

perendemicity involving multiple viral serotypes and the resuiting specter of DHFIDSS in a growing number of areashas prompted a re-evaluation of dengue control strate

gies.'5 9 It is obvious that until a vaccine becomes available

for widespread use, control of dengue will continue to relyon control of the vector.

A number of problems have been identified with the wayAc. aegypti control programs have been conducted over thepast several decades.'5 On the technical side, too much reliance has been placed on insecticides. Spraying to controladult mosquitoes is often one of the stated mainstays of Ac.aegypti programs,6 yet this method is expensive for routine

use and is often ineffective in urban environments.20 Another

key element in many programs has been larvicides. However, growing reluctance on the part of residents to accepttheir use in potable water, their expense, the explosion in the

number of nonessential water-holding containers in the environment, and the increasing frequency of locked residences during the day has resulted in programs where applications are made too infrequently and with inadequate

coverage to effect long-term 21More substantive and intractable, however, are problems

associated with the perceptions and attitudes of endangered

populations)2 21.22 In most regions of this hemisphere, thereis no widespread appreciation or sense of urgency about thedeteriorating dengue situation. Moreover, dengue is seen as

one of many (not too serious) illnesses to be endured; therelationship between illness, mosquitoes, and sanitation islargely unknown. Even where DHF/DSS exists and people

commonly appreciate the mosquito connection, fatalistic attitudes and a belief that control is the responsibility of central government undermine individual action and a sense ofcommunity responsibility. It is understandable that these perceptions have not led to a clear mandate for government topursue a vigorous, necessarily intrusive, and sustainable control program. Nor is it surprising that control efforts havefaltered. National programs are typically underfunded andfrequently poorly managed; they often operate in isolationfrom other elements of health care delivery. Field workersare commonly poorly motivated and supervised; usuallyminimum wage earners, they lack the communication skills

and knowledge necessary to elicit or foster community co

operation and compliance. Perhaps most importantly, mostnational programs, whether directed nominally toward eradication or long-term suppression, did not transfer the re

sponsibility of Ac. aegypti control in the penidomestic environment to residents. When the vertically orientated programs could not compete with other pressing national needs,

vector control activities suffered.Some new developments are encouraging. The Rockefel

1cr Foundation is once again providing leadership and funding for Ac. acgypti control, this time directed toward dengue.'7 Recent efforts involving the Foundation, the JohnsHopkins University School of Hygiene and Public Health,and the Dengue Branch of the Centers for Disease Controland Prevention, and the Honduran and Mexican Ministriesof Health are attempting to develop realistic and sustainablemethods to control dengue via long-term vector suppression.Eradication or vertically orientated programs such as thoseof Cuba and Singapore are not being considered.2@25 Thefocus is primarily on public health education and the devel

opment of methods to promote source reduction by the localcommunity.26 27 The educational goals are not simply totransmit the details of dengue and mosquitoes, but to â€œ...

educate citizens of the community to be more responsiblefor their own health destiny. â€œ2 Education is seen as the essential element of sustainable community participation and

a requisite to an informed public that can provide the mandate required for effective governmental involvement)7

An example of a dengue control program that attempts todeal seriously with the present realities mentioned previouslyis the program in Puerto Rico and the U.S. Virgin Is

lands.'2 15What is envisioned is the development of an island-wide, early warning surveillance system for virus thatwill permit emergency vector control measures to limit nascent epidemics from spreading. The plan assumes the de

velopment of effective emergency control measures and thatepidemics are predictable with sufficient lead time and cer

tainty to mount emergency measures. The plan includes thedevelopment of an ongoing, community-based, integratedAc. acgypti control program designed to provide area-wide,

long-term vector suppression. The rationale is to reduce dengue transmission to a point where serious illness rarely occurs, in the words of the program's chief architect, Duane

Gubler, â€œ.. . we may have to live with a little dengue, butnot with F.'2 Other elements of the program includeeducation of the medical community and development of acontingency plan for hospitalizing large numbers of patients.

Rationale, goals, and overview of the dengue models.Increasingly, modeling is being used in the study and oftenthe management of complicated systems.28 Simulation research, the use of numerical techniques to conduct experi

mentation on the computer, has become an important adjunctto traditional methods of investigation. Comprehensive models serve as a repository of what is known about the dc

ments of a system and their interactions. Their constructionforces a clear and exhaustive consideration of the informa

tion from the various disciplines. As such, they represent aneffort that is just the opposite of the normal reductionistscientific process. An important facet of the systems ap

proach is that all factors within the system are consideredsimultaneously. For these reasons, we expect comprehensivemodels to be more rigorous and testable, more useful for the

491MODEL OF THE EPIDEMIOLOGY OF URBAN DENGUE FEVER

study of the real world than the corresponding mental mod

els many of us have about the systems we study from the

perspective of our own particular discipline. The range ofsystems that have been modeled spans from cosmology to

quantum mechanics.29 Epidemiologic models have a longand distinguished history providing insight, prediction, andoften, help in optimizing vaccination programs.3Â°32

A recent differential equation model of dengue transmission has developed an estimate of the basic reproductive rate

of dengue and confirmed the relative merits of control strategies based on insecticides and source reduction tech

niques.2Â° Our efforts seek to extend this type of work; aprincipal goal was to develop software sufficiently sophisticated to be of practical use to operational programs suchas the ones previously mentioned. With the Puerto Rican

program in mind, the following examples suggest the typesof questions than can be addressed with simulation. Theyillustrate some of the areas where an adequate model canprovide practical insight into and the regulation of a verycomplicated and nonlinear system. Given estimates of theage-specific seroprevalence of dengue antibody and theabundance of Ac. acgypti, how receptive is the populationof San Juan and other communities to the various dengueserotypes and how would this change with season, altitude,or with decreasing levels of herd immunity? How wouldreceptivity or the epidemic curve change if a community

based source reduction program reduced average vector production by 10, 20, or 50% within the city? Are there circumstances where insecticides could be used effectively ci

ther proactively or reactively? How much time would beexpected to lapse between introduction of the virus and peaktransmission levels and how would lag time change withseason? Following an epidemic of a particular serotype, what

would be the expected age-specific prevalence of individualswith histories of sequential infections and thereby presumably at elevated risk of serious illness? Given that Ac. acgypti, the urban vector in the Caribbean, is typically more

difficult to infect than other dengue vectors, should we cxpect selection for viruses that circulate in higher titers?5 Canwe understand the interepidemic maintenance of virus insmall communities apart from positing transovarial transmission in a second species such as Ac. mediovittatus (Coquillett)?33

The effort to model dengue transmission has been a twostage process. We began first in 1991 with the developmentof a comprehensive simulation model of the dynamics ofnondiapausing, container-inhabiting Acdes Meigen mosquitoes such as Ac. acgypti, Ac. albopictus (Skuse), Ac. mcdiovittatus, and Ac. polyncsiensis Marks) The container-in

habiting mosquito simulation model (CIMSiM) provides asecond model, the dengue simulation model (DENSIM), thesubject of the present paper, with entomologic inputs. Usingfield data from Israel, Thailand, and the United States,2 theCIMSIM has been developed and is currently being assessedoperationally or for research purposes in the United States,Thailand, Honduras, Trinidad, Colombia, Switzerland, andthe Netherlands. Together, CIMSiM and DENSiM attempt totake into account virtually all of the commonly recognizedfactors influencing the dynamics of these viruses in the urbansetting.

Overview of the entomologic model CIMSIM. Before

describing the development of DENSiM, we will briefly out

line the nature of CIMSiM because the adequacy of thetransmission model is directly related to the adequacy of theentomologic inputs. The entomologic model is a dynamiclife-table simulation model that produces mean-value estimates of various parameters for all cohorts of a single spe

cies of Aedcs mosquito within a representative one-hectare

(ha) area. Like the DENSiM, the entomology model is basically an accounting program; for each cohort, depending

on the life stage, the CIMSiM maintains information onabundance, age, and development with respect to temperature, weight of the individual, fecundity, and gonotrophicstatus. With few exceptions, the various processes are simulated mechanistically. The accounting is made dynamic bycalculating on a daily basis the number of each cohort thatwill pass to the next age or stage as a function of a host ofvariables and relationships. For example, development times

of eggs, larvae, pupae, and gonotrophic cycle are based ontemperature using an enzyme kinetics approach.34 The basisof larval weight gain, food depletion, and fasting are thedifferential equations of Gilpin and McClelland modified tocompensate for the influence of temperature.35 Fecundity ismodeled as a function of pupal size, which in turn is a function of the recent history of larval abundance, food, ternperature, and fasting in the container. All survivals are tiedto temperature, and for adults and eggs, saturation deficit ofthe atmosphere as well; larval survival is also a function offasting and fat body reserves.

The heterogeneity of the larval habitat is depicted by modcling the cohorts of eggs, larvae, and pupae within 1â€”9different containers, each of which serves to represent an irnportant type of mosquito-producing container in the field.For instance, in poorer neighborhoods of New Orleans, Louisiana, the most common containers include tires, one-gallon

(3.8-liter) buckets, and drink bottles.36 In Bangkok, Thailand,the common containers include the water storage (ong) jar,flower pot plates, and ant traps)7 The adult production fromthese representative containers is combined, with the outputof each type being scaled to reflect its relative abundance inthe environment. The model contains a database of containertypes from which the user may select and use with or with

out modification; it is also possible to describe entirely newcontainers. Containers are characterized by the specificationof parameters such as size, method of water loss and gain,location and abundance, and available larval food.

Because microclimate is a key determinant of survival anddevelopment for all stages, the CIMSiM also contains anextensive database of daily weather information for a nurnber of cities around the world for the past 25â€”40years. Adultmicroclimate is assumed to be the same as the daily localweather. For immature forms, however, the CIMSiM calculates daily water temperatures and water gains/losses foreach of the representative containers based on local weatherand container characteristics and location.

The CIMSiM assumes that the underlying mechanisms responsible for the salient biological characteristics of differentAcdcs are substantially similar. A third database in theCIMSiM contains biological profiles, one for each specieswhere data and analysis have permitted development. Selecting an existing biological profile sets the parameters inCIMSIM for that species; it is possible from within the pro


gram to create new profiles or use existing files as a basisfor new ones. The final database within the CIMSiM contains location-specific information. Here the user can associate with a location name, the species of mosquito to model,a beginning year and weather dataset for the simulation run,the containers to be used to describe the location, and localvertebrate host abundance and availability. Again, it is possible to edit the information on each location and create newlocations. In this manner, it is fairly straightforward to setupthe CIMSiM for use in any urban area.

The published validation of CIMSiM concluded that themodel met design goals.' Validation involved comparingsimulation results with several independent series of data onthe dynamics of Ac. aegypti that were not used in modeldevelopment.2 Validation data sets included laboratory workdesigned to elucidate the role of diet on fecundity and ratesof larval development and survival. Comparisons were alsomade with four field studies conducted in Bangkok, Thailandon seasonal changes in population dynamics and with a fieldstudy in New Orleans, Louisiana on larval habitat. Finally,predicted ovipositional activity of Ac. aegypti in seven citiesin the southeastern United States for the period 198 1â€”1985were compared with an unpublished data set developed bythe U.S. Public Health Service.

Overview of the transmission model DENSiM. Halsteadand Gubler have published recent reviews and discussionson the current understanding of the immunology and pathogenesis of dengue viruses.3@@@ Briefly, dengue viruses cxist in four antigenically-related but distinct serotypes. Infection with a particular serotype results in life-long immunityto that type (homologous immunity) and a brief period, ofsome months duration, of heterologous immunity to the other serotypes. It is apparently possible for an individual to besequentially infected with all four serotypes with the temporal sequence of infections being determined not only bythe timing of inoculations but also by the immune status(homologous and heterologous) of the individual.

The etiology of serious illness is not completely understood. Infants born to mothers seropositive for dengue antibody possess protective levels of maternally acquired antibody that are catabolized to subneutralizing titers withinmonths of birth. Serious dengue illness, characterized by altered hemostasis and increased vascular permeability, is often associated with these infants when experiencing theirfirst dengue infection and with children more than one yearof age who are experiencing an infection with a second serotype. Recent research supports the notion that subneutralizing titers of heterologous dengue antibody facilitates theuptake of virus by mononuclear cells and that the incidenceof such infected cells plays a significant role in enhancingthe seriousness of dengue pathology, perhaps by the releaseof vasoactive mediators. Other factors suspected or knownto influence the severity of clinical illness include race, flutritional status, sex, age, and previous disorders such as asthma and diabetes. There is good evidence that different serotypes or strains of virus can differ in their pathogenesis.The present model accounts for serologic changes and thedynamics of infection; another model, currently under development, is designed to investigate the incidence or probability of serious illness based upon the DENSiM results and

known or hypothesized risk factors such as age, sex, race,and previous dengue infection.

Whereas the CIMSiM is basically a habitat- and weatherdriven accounting program of the population dynamics ofcertain Aedes mosquitoes, the DENSiM is essentially thecorresponding account of the dynamics of a human population driven by country- and age-specific birth and deathrates. In a typical simulation run of a small village or barrioin a developing country, over the course of 15â€”20years, thehuman population will grow from 10,000 to approximately17,000 people. An accounting of individual serologies ismaintained, reflecting infection and birth to seropositive

mothers. In the DENSiM, the entomologic factors passedfrom the CIMSIM, daily estimates of adult female survival,gonotrophic development, weight, and emergence on a perhectare basis, are used to define the biting mosquito population. The survival and emergence values dictate the dynamic size of the population within DENSiM while the gonotrophic development and weight estimates influence the rate

at which these females bite. Temperature and titer of virusin the human influences the extrinsic incubation period (EIP)in the mosquito; titer is also seen as influencing the probability of transfer of virus from human to mosquito. The infection model accounts for the development of virus withinindividuals and its passage between both populations. Wehave assumed that our current understanding of the factorsinfluencing dengue transmission is substantially correct, butas in the case of the CIMSiM, the specific values used forany particular phenomenon are on menus where they can bereadily changed. Also, like the entomology model, the userof the infection model is presented with an opening menuof several databases containing site- or species-specific information on weather, entomology, virus, seroprevalence of

antibody, and demographics; the program contains an editormaking it easy to add to or modify this information. TheDENSIM output includes demographic, entomologic, serologic, and infection information on a human age class and/or time basis in a graph and text format that can be printedor written to disk.' Both models are stochastic.

METHODS

The following sections describe briefly our analysis of theliterature and how a synthesis of this information was implemented within the DENSiM to allow simulation of thedynamics of dengue transmission.

Human and mosquito populations. Human demographics and initial seroprevalence. The DENSiM contains a dynamic life-table model of a variably sized human population.Using user-supplied information on a location's human agedistribution and age-specific birth and death rates, the modelinitially creates a population with the desired age distribution, and then on a daily basis, updates it to reflect births

â€˜TheDENSiM and the CIMSiM run on MS-DOSÂ®compatible computers with a minimum video graphic capability of VGA; a mathcoprocessor and 486 or later versions of CPU are strongly recommended. The software was developed using the MicrosoftÂ®BasicProfessional Development System (Microsoft, Redmond, WA).

MODEL OF THE EPIDEMIOLOGY OF URBAN DENGIJE FEVER

Cl) 1.0

@0.8

10.6

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493

and deaths.t Currently, the model can simulate upwards ofapproximately 20,000 people. Depending upon the size ofthe initial population and the growth rate, simulation runs of> 60 years can be made. The age classes used are as follows:

infants < 1 year of age belong to class 1, 1â€”5 years of ageclass 2, 6â€”10years of age class 3, . . . 76â€”80years of ageclass 17, and > 80 years of age class 18. Initial ages areassigned randomly within each class. The final item of inputfor demographic purposes is the initial number of humansper hectare. The size of the simulation area is the ratio ofthe number of humans and their density. Depending on theparticular situation, the simulation area could range fromtens of hectares to several hundred. Once the initial humanpopulation is set up, the model employs user-supplied infor

mation on the age classâ€”and serotype-specific prevalence ofdengue antibody in the population to randomly assign theinitial immune status to each individual. Often seroprevalence is poorly known and only what if scenarios can beevaluated, e.g., assuming an antibody prevalence profile ofsuch and such, how receptive is this area expected to be tofuture dengue virus introductions of different serotypes, etc.?

Adult mosquito population. In the DENSiM, two of thefour entomologic factors passed from the CIMSiM, dailyestimates of adult female survival and emergence, are usedto create a cohort-based mosquito population. The accounting in the transmission model is made dynamic over timeby replacing yesterday's cohort of newly-emerged femaleswith today's, yesterday's one-day-old females becoming today's two-day-old females, Each day, the number of newlyemerging females per hectare from the CIMSiM is scaledupwards to reflect the current size of the area being modeledin the DENSiM, and each day, the entire adult population isreduced to reflect adult daily survival.

Factors Influencing contact rates between mosquitoesand humans. Multiple blood meals during a single gonotrophic cycle or interrupted feeding attempts could significantly increase the potential for disease transmission provided the level of virus inoculum in the salivary fluid did notdecrease with subsequent host contacts'Â° â€˜@The followingdescribes some of the sources influencing the mosquito-hostcontact rate that have been incorporated into the DENSiMin addition to the simple ratio of mosquito and human abundance.

Adult size influences doublefeeding within the gonotroph

ic cycle. Two additional parameters are passed from theCIMSiM to the transmission model, adult female weight andthe proportion of gonotrophic development completed eachday (CDJ.

The CIMSiM was developed to estimate the weight oflarvae and pupae and thus the weight of adults as a functionof larval rearing conditions because size is known to or suspected of influencing fecundity, survival, blood-feeding suc

tSmall daily values for the probability of a death or birth within aparticular age class are accumulated until they exceed one, whereupon an individual is chosen randomly within the class to either dieor give birth. Good sources of demographic information include theCountry Demographic Profile Series of the Bureau of the U.S. Census, U.S. Department of Commerce (Washington, DC) and the annual Demographic Yearbook published by the Department of International Economic and Social Affairs, Statistical Office, UnitedNations (New York).

0.0

0.00 1.00 2.00 3.00 4.00Female wet weight (mg)

FIGURE 1. Proportion of females requiring two replete bloodmeals per gonotrophic cycle as a function of wet body weight.

cess, and frequency.42@5 In the entomology model, pupationis contingent upon the attainment of a minimum of physio

logic development and a minimal larval weight. Actually,weight may not be the controlling factor per se; rather, itmay be correlated with the real trigger, the amount or proportion of some reserve such as lipids or glycogen.@ As inthe case of physiologic development, this minimum weightis a function of temperature, but it is also a function of the

physiologic age of the larva.47 Under conditions of madequate food, development times are extended, weight gaintrajectories are lower, and subsequent pupation weights be

come progressively lower with increasing larval age. Undersevere conditions, dwarf adults are produced.2 35Our interestin the DENSiM with adult size stems from its influence onthe number of complete, replete feeds taken during a gonotrophic cycle. We assume that larval rearing conditions influence not only size but energy reserves as well so that thefrequency of taking more than one replete blood meal pergonotrophic cycle varies as a function of adult weight (Figure 1); females from well-fed larvae emerge with lipid reserves adequate to develop ovaries to stage II, with the resultthat the first blood meal is sufficient to complete oogenesis.42 48 Recent field work in rural Thailand has indicated that

multiple feeding within a gonotrophic cycle is very commonin Ac. aegypti and that its incidence may vary seasonallyand be positively correlated with the seasonality of denguetransmission.49 In the transmission model, replete meals are

sought on the second day of the first cycle and on the firstday of subsequent cycles; any second replete feed within acycle is considered to occur on the day following the firstfeed within that cyc1e.@Â°

Length of the gonotrophic cycle is influenced by temperature. The initial work in Bangkok, Thailand on the epidemiology of DHF was based on the faulty expectation thatseasonal changes in the density of the vector and the mcidence of DHF were correlated, primarily because diseasewas associated strongly with the wet season when rainfallwould presumably increase the number of breeding sitesand/or increase adult 375@Later, it was hypothesized

that temperature-related changes in the length of the gonotrophic cycle could be responsible for seasonal trends intransmission.52 The CIMSiM predicts the temperature-dependent rate of gonotrophic development in Ac. aegypti and thisparameter, the daily proportion of the cycle that is completed


each day (CD1), is passed to the DENSiM. A comparison ofthe lengths of observed and predicted first and second gonotrophic cycles in Bangkok during the wet, cool-dry, andhot seasons was part of the original validation of theCIMSiM.2 The first cycle, from emergence to oviposition, isconsidered to be completed on the day when the CD, > I .00.Subsequent cycles, oviposition to oviposition, are somewhatshorter (completed on the day when the CD, has increasedby an additional 0.58) because the ovaries are at stage II

following oviposition and not stage I as in the case of newlyemerged females.42

Interrupted feeding attempts increase mosquito-human

contact. Gubler attributes some of the efficiency of Ac. acgypti as a vector of dengue viruses to the habit of taking

partial blood meals or to interrupted feeding attempts, mdicating that it is not uncommon for a single female to probeand/or feed on several people within a room or house in the

course of obtaining a replete feed.@ Recent work on the epidemiologic significance of probing and interrupted attemptsindicated that neither multiple probing nor blood feeding reduced the ability of Ac. aegypti to transmit dengue viruseven after as many as 20 consecutive contacts.â€• In theDENSiM, the user specifies the average number of feedingattempts per replete feed and the probability that an interrupted attempt will be resumed on a different host; the current defaults are four interruptions and a probability of 30%,respectively.5

Alternate hosts. The probability of transmission is alsoinfluenced by the types of hosts bitten by the vector. Acdcsacgypti, because of its domestic habits and/or host prefer

ences, is usually considered a rather selective human feederalthough it is known to feed on a variety of other hosts.49@In contrast, Ac. albopictus, whether due to a wider host preference or simply due to the availability of more types ofhosts in the peridomestic/sylvan environment, is typicallyless selective than Ac. acgvpti.55 In the DENSiM, the user

specifies the proportion of feeds taken on humans; the default of 90% is probably reasonable for Ac. aegypti in manytropical locations.49

Virus infection, replication, and transmission. The usermay specify in the virologic database of the DENSIM anumber of serotype-specific parameters that can influencetransmission: virus titers and the duration of incubation and

viremic periods in humans, coefficients for the relationshipin mosquitoes between EIP and temperature and virus titer,and factors influencing the probability of virus passage fromhuman to mosquito and mosquito to human.

Virus introductions. Virus is introduced into the model via

viremic humans or infectious mosquitoes. The user specifiesthe number of humans and/or mosquitoes in each introduction and the number and frequency of the introductions byserotype. Human introductions are treated as random-agedimmigrants that are added to the existing population. Mosquitoes are assumed to be in their second gonotrophic cycleand bite on the day of their arrival.

Human infection and serologic responses. Following a

parenteral inoculation of virus (details presented below), susceptible humans are assumed to undergo invariant and serospecific periods of intrinsic incubation and viremia as specified in the database. The range of values reported for theduration of the incubation period is 3â€”14days;2Â°we use a

period of four days for the nominal default for all serotypes)' 3@We recognize that the period of detectable viremia(range 2â€”12days) may be longer than the period of infectiveviremia and use a default value of five days infective viremiafor all pe538 In the model, it is only during this viremic period that mosquitoes can imbibe potentially infectious doses of virus.

In the DENSIM, the newborns of dengue-immune mothersreflect, in the form of passively acquired antibody, the immune status of their mothers. Following birth, this maternally acquired antibody is considered to be catabolized frominitially neutralizing titers to subneutralizing and potentiallyenhancing (MAEA) titers. Currently, only neutralizing titersinfluence transmission in the model; the MAEA accountinghas been included to allow estimating the frequency of pri

mary infections occurring in infants with MAEA. As initialdefaults in the serology database, the duration of protectiveand enhancing titers currently is set to 0â€”90days and 90â€”270 days, respectively.3 56 Following the completion of the

viremic phase, a convalescing individual is assumed to have

developed lifelong homologous immunity3 and to have begun a brief period of heterologous immunity with a defaultduration of 60 days after the work of Sabin.57

Human-to-mosquito transmission. Each day, the transmis

sion model estimates the average number of contacts by different mosquitoes per person (a). By contact, we mean morethan the initial probing, rather, that some portion of a repletefeed is taken such that there is the possibility of virus transfer to the mosquito. We begin with the subset of femaleswithin the simulation area seeking a replete feed that daythe number of which, as indicated previously, is a functionof the number of females within the simulation area and theproportion attempting to bite, a function of the rate of gonotrophic development as a function of temperature, and adultsize as it influences the incidence of double feeds within a

single gonotrophic cycle. We multiply this number of seekers by the average number of interrupted attempts per repletefeed, and adjust for the probability that an interrupted attempt resumes on a different host and the presence of alternate hosts. Finally, this total number of different contacts isdivided by the number of people within the simulation areato give the estimate, a, the average number of different mosquito contacts per person per day.

We appreciate that imbibing some quantity of viremicblood does not invariably result in a subsequently disseminated infection with virus in the salivary glands. It has beensuggested and there is some evidence to support the notionthat the titer of virus in the blood meal could influence theprobability of subsequent infection.5@ We therefore assumethat the probability of a mosquito ingesting sufficient virusto subsequently become infectious after contact with viremichost (@3)is a function of the titer of virus in the blood meal.The current default relationship between the probability ofinfection and titer (Figure 2) gives a probability of 0.55 fora virus titer of l0@ mosquito infectious dose 50 (MID@)/ml;for a titer of l0@, the probability is 0.30.20.61 Note, as withall other default values in the various databases within theDENSiM, the user may edit these values to make infectionprobability some constant that is independent of titer or tochange the rate of change in probability with respect to titer.

The number of mosquitoes receiving a dose of a particular

0.00

3 4 5 6 7 8 9

Titer (logs of virus, MID50/mi)

FlGu@ 2. Probability of a disseminated infection resulting fromimbibing viremic blood of a specified titer after a suitable incubationperiod. MID@ = mosquito infectious dose 50.

serotype sufficient to result in a disseminated infection(should they remain alive during the EIP) each day, y, is theproduct of the contact rate a, the infection probability forthat serotype 13,and the number of viremic humans with thatserotype. We assume that oral susceptibility is independentof the age of the mosquito.62 Because infections are discreteand because we are often interested in the behavior of thesystem when the incidence of infection is low, we make themodel stochastic when -y falls below a threshold (default 5).We assume that the number of newly infected mosquitoeshas a Poisson distribution with mean of -y. When@ is large,the Poisson distribution tends toward symmetry and nothingis lost simply using the rounded integer portion of y deter

ministically. When â€˜yis small, however, we randomly selectan integer value to use from a Poisson distribution withmean @y.t

Adult survival and the extrinsic incubation period. The

daily survival of the vector and the EIP interact in a nonlinear fashion. Together, these two parameters exert an important role in the regulation of the dynamics of transmissionof arboviruses.63 Because the length of the EIP of dengueviruses is a significant proportion of the total adult life spanof Ac. aegypti, the type of underlying model assumed foradult survival may be significant in the context of vectorialcapacity. In both the CIMSiM and the DENSiM, we haveassumed an exponential model for survival, i.e., mortality isindependent of the age of the female (daily probability ofdeath is a constant).' Recently, Clements and Paterson haveshown for many species of tropical mosquitoes that an cxponential model is inaccurate and that the Gompertz model,where survival is proportional to (the logarithm of) age, ismore appropriate.TM They noted, however, that Ac. aegyptiwas an exception, with daily survival rates being independent of age of the female.

It has been known for more than 50 years that the EIP ofyellow fever virus in Ac. aegypti varies with temperature;40 years ago, a similar relationship was observed in Hae

@LetP (â€˜y,r) be the probability of r mosquitoes becoming infectedwhen the number of newly infected females is distributed Poissonwith a mean y. Numerically, this is P (-y, r) = @yre@/r!.To obtain arandom value for r from this distribution to be used that day for y,we begin summing this expression for r 0, 1, 2. . . until the sumjust exceeds the value of a uniformly distributed random variateranging between 0.00 and 1.00.


1000

800

600

400

200

0

LI

.@

0

01@

0â€˜a

a

0

a

0

11 16 21 26 31 36Temperature (Â°C)

FIGURE 3. Length of the extrinsic incubation period (EIP) of dengue virus in Aedes aegypti as a function of temperature. This nominal value for EIP may subsequently be modified by the titer of virusin the infecting blood meal.

magogus Williston, leading to speculation that environmen

tal temperature was influential in the endemicity of the virus.65 66The most recent hypothesis regarding the seasonalityof DHF in Bangkok posits a mechanism of seasonal changein EIP (and gonotrophic cycle length) as a function of temperature (Burke DS and others, unpublished data).52 In theDENSIM, we relate EIP to temperature using the enzymekinetics model of Sharpe and DeMichele;@â€• this model as

sumes that other factors not limiting, the rate of developmentis determined by a single rate-controlling enzyme that is denatured reversibly at high and low temperatures. The formof the kinetics equation and the statistical methods used toestimate parameter values for it from observed data werepresented in the description of the CIMSiM.' Without takinginto account the titer of virus in the infective meal (see below), we estimate parameter values for the enzyme kineticsmodel (Figure 3) to be @)@5@)3.359 X l0@, T,,@H â€”2.176x l0@Â°,L@HAâ€•1.500 x l0@,and i@HH 6.203 x 1021basedon observations by McLean and others67 and Watts and others.6' This relationship (Figure 3) gives estimates of EIPsthat are very similar to those given by Gubler5 and Halstead.38

Bates and Roca-Garcia conjectured that the EIP of yellowfever virus also varied as a function of the titer of the mosquito-infecting dose.@' Watts and others reported a similarrelationship with dengue in Ac. aegypti (Figure 3); the EIPat 30Â°Cwas I 2 and 25 days for mosquitoes infected withhigh and low doses, respectively.6' Provision has been madein the DENSiM to allow evaluating the consequences of anEIP-titer relationship; the temperature-based estimate of EIP(Figure 3) may be simply scaled by a factor that is a functionof infecting virus titer (Figure 4).

Mosquito to human transmission. For each serotype present, DENSiM begins by looking for how many mosquitoeswith virus have completed the E@ Of these females, themodel determines how many will be seeking a replete feedon that day and then calculates how many different peoplewill be bitten by each taking into account alternate hosts,interrupted feeding attempts, and the probability that inter

rupted feeds will be resumed on a different host. The productof the number of potentially infectious mosquitoes and thenumber of different people per mosquito is then multiplied

Honduras, El Salvador, Guatemala, and Mexico. Most epidemics in Honduras occurred during 1978 and 1979 andwere largely confined to the communities of the coastal lowlands and interior valleys. The best characterized epidemicoccurred in San Pedro Sula where cases of dengue illness,first seen in June, peaked in August 1978; the proportion ofrandom sera taken by mid-October that were positive fordengue-1 hemagglutination-inhibition (HI) antibodies was61 % . In other nearby towns, the seroprevalence of dengue1 HI antibodies ranged from 37% (El Progreso) to 8 1% (Villanueva). It was later realized that in some communities, theserosurvey was conducted before the end of epidemic transmission and, therefore, prevalence data may have underestimated the true extent of some epidemics.

Parameterization process. The use of the CIMSiM andDENSiM in the field involves collecting certain site-specificinformation to allow parameterization. The number and typeof mosquito-producing containers in the environment andestimates of their associated productivity (standing crop ofpupae or adult emergence) are the important entomologicinputs; these data can be collected by mosquito controlworkers. This information is used to create the representativecontainers within the CIMSiM; food inputs are fit by simulation (trial and error) by comparing model projections ofpupal abundance in the various containers and observed values from the field. Both models need daily weather information (usually available from the National Weather Services) and density estimates of humans and other hosts. Themajor requirements for the DENSiM include anti-dengueseroprevalence and demographic information discussed earher. With the exception of dengue antibody prevalence andcontainer productivity, most dengue control projects haveestimates (of various quality) of the required information.The cost of obtaining such information would be relativelytrivial in comparison with the total cost of an ongoing control program.

Following parameterization, initial simulations are conducted and comparisons are made between various observedand predicted values. Possible entomologic comparisons indude adult size and biting rates, the seasonal presence andabsence of water in containers, and for each of the containertypes, daily mosquito production and the average numbersof larvae and/or pupae per container. Validation of the tansmission model is fundamentally different than the validationof the entomologic model. In most locations where denguetransmission is an ongoing problem, vector populations,while varying seasonally, are continuously present. Theirpresence is not contingent on reintoductions such as in thecase of Philadelphia where Ac. aegypti could not overwinterbut was often reintroduced via sailing ships each spring or

mm7273 J@ contrast, while for many locations dengueviruses are endemic, e.g., Bangkok, for many other locationstransmission is contingent on the unpredictable arrival ofvirus from elsewhere. An example of this would be the1978â€”1980 epidemic in Honduras; presumably this site hadbeen receptive to virus introduction (low herd immunity,presence of competent Ac. aegypti, etc.) for a number ofyears prior to the introduction. In such cases, we can predictthe receptivity of an area to virus introduction or make estimates of the rate of transmission or the expected extent ofan epidemic, but we cannot predict the arrival of virus.


2 3 4 5 6 7 8 9

Titer (logs of virus, MIDso)

FIGURE 4. Modification of the nominal extrinsic incubation period of dengue virus scaling factor to reflect the influence of virustiter in the infecting blood meal. The current defaults are shown.MID@ = mosquito infectious dose 50.

by the probability that the mosquito contact, ranging between initial probing and taking a replete feed, will result inthe transfer of sufficient virus to initiate an infection shouldthe host be susceptible. The model is stochastic here afterthe fashion described for â€˜Ii.

What is a reasonable value for the probability of sufficienttransfer to effect infection in susceptible hosts? Gubler andRosen observed that probing prior to feeding by dengueinfected Ac. albopictus could result in the transfer of between 300 and 1,000 MID50 of virus to a hanging drop offluid and that a feed to repletion was accompanied by thetransfer of up to 2 X l0@ MID@ Kraiselsurd and othersestimated that it took 9.5 and 22.0 MID@ of dengue-2 anddengue-4, respectively, to infect 50% of susceptible rhesusmonkeys; they suggested that a dose of 100 MID@ of eithervirus would give infection rates of 100%.@ Based on thesestudies, we believe a reasonable default to be 90%.

â€œ4,,,

.@

bO

1.50

1.25

1.00

0.75

0.50

RESULTS AND DISCUSSION

The following section will briefly describe how the models are set up for use in a specific location. This location isSan Pedro Sula, a town representative of several urbanizations situated in the coastal lowlands of northwestern Honduras that experienced their first ever epidemics of dengue1 in 1978â€”1980. We will use the Honduran situation to provide examples of entomologic, demographic, and epidemiologic output from the models and to provide somepreliminary validation of the CIMSiM/DENSiM. We condude with some examples of simulation studies, again usingthe Honduran situation, designed to illustrate the use of themodels in investigating research and operational questions.

Model parameterization, examples of output, and validation. Overview of the Honduran epidemic of 1978â€”198O.@Â°@â€˜Epidemic dengue-l first appeared in the Americasin Jamaica in February 1977. It soon spread to other baations including Cuba, Puerto Rico, the Bahamas, Dominica,Grenada, Surname, and Venezuela. This virus was introduced into Central America via the Honduran Bay Islands,probably by means of holiday travelers visiting Jamaica; thefirst observable epidemic in the Bay Islands occurred Febmary 1978. The virus soon spread from the Bay Islands into

MODEL OF THE EPIDEMIOLOGY OF URBAN DENGUE FEVER 497

100

80

C) 60

a;;2. 40

20

0

1 61 121 181Day of year

FIGURE 5. Weather data from the La Mesa Station at San Pedro

Sula, Honduras for 1977. The upper and lower continuous seriesdepict relative humidity (%) and saturation deficit (mBars), respectively. The inner two continuous series, ranging between approximately 20 and 30, indicate maximum and minimum temperatures(Â°C).The height of the vertical bars indicates daily rainfall (mm).

Transmission validation is also hampered by a lack of gooddata on the incidence of infection. The DENSiM estimatesthe incidence and prevalence of infection but not the commonly reported evidences of dengue transmission such ashospitalizations and reports of dengue-like clinical illness.

For demonstration and validation proposes, we will modelthe 1978â€”1979 dengue-l epidemic in San Pedro Sula andthe surrounding towns using entomologic data from the nearby town of El Progreso and the serologic data of Figueroaand others.7Â°We assume an initial population of 10,000 anda human density of approximately 240/ha; values used forthe human age distribution and birth and death rates werethose reported for Honduras in 1976. The year is 1978 andthere is no immunity to dengue viruses.70 A single serotypeof low-titering virus (l0@ MID50) is introduced each monthinto our representative urbanization via an individual who isassumed to become viremic on the day of arrival. For boththe CIMSiM and DENSiM, we use weather data from SanPedro Sula, Honduras (Figure 5).

Larval habitat. Mosquito-producing artificial containers,

similar in type and abundance to other Honduran towns inthis region, include outdoor laundry sinks (50/ha), 55-gal(207-liter) drums (15/ha), discarded automobile tires (10/ha),and animal watering dishes (10/ha); food inputs into thesefour representative container types were determined by simulation2 so that the adult mosquito productivity as estimatedby the CIMSiM would be similar to that observed in thesecontainers in various communities in El Progreso, Honduras(Fernandez E, Director, Proyecto Control Integrado de Dengue, Ministerio de Salud de Honduras, Division de Enfermedades Transmitidas por Vectores, Tegucigalpa, Honduras,unpublished data). The tires are filled only by rainfall andtherefore subject to drying out (Figure 6); a piped supplyserves as the water source for the other containers and theirdepths, with the exception of animal watering dishes, whichare also subject to rainfall, are essentially constant. Watertemperatures vary in response to ambient air temperatures,the size of the container, and exposure to sun. The CIMSiMwas developed to estimate water depths and temperatures(Figure 7) because of their influence on hatch, larval development and survival.'

20

@15U

0.

â€˜.4

a

10

0

Day of year

241 301 361 1 61 121 181 241 301 361

FIGURE 6. Daily water depths in representative tires under shaded (exposed to direct sunlight 10% of daylight hours; thinner line)and exposed (30%; thicker line) conditions. Hydrologic calculationsin the container-inhabiting mosquito simulation model make evaporative losses a function of (among other things) sun exposure.2 Notethe slower water losses and greater duration of the presence of waterin the shaded tire. Rainfall into full tires is lost to overflow. Projections were based on 1977 weather.

Entomologic results. Oviposition is a function of the presence of water, the size of a container, and the numbers andtypes of other containers in the environment; larval and pupal abundance (Figure 8) within wet containers, in turn, arefunctions of oviposition, food, competition, temperature, andfluctuations in water 2 Briefly, larval competition inanimal watering dishes, a consequence of a relative paucityof food in the face of high levels of oviposition from adultsarising from other, more productive containers, results in lowlarval survival; larvae in the dishes are often present withoutthe production of pupae. This is in contrast with the situationin tires, which have higher survival rates. Both shaded andunshaded tires show the results of drying out during thespring. Larval survival in the laundry sinks and drums isintermediate. As a result, the annual total production of females from the animal dishes is appreciably less than the

32

@â€˜30

@ 282

@ 2604E 240)

@22

@:20

18

1 61 121 181Day of year

241 301 361

FIGURE 7. Daily water temperatures (minimum and maximum)

in a representative laundry sink (thicker lines) and in a tire with30% sun exposure (thinner lines). Because of the sink's larger sizeand shaded location, daily temperatures fluctuate and range lesswidely than in the tire; the average temperature in the sink is alsoslightly lower. None of the depicted temperatures would be expectedto reduce the survival of Aedes aegypti.2 Projections were based on1977 weather.

Animal dish

DUJs,M â€˜IS!.

:K'Kn /J@f@\f

,@ @i F'mak' crncrgcncc

@@ â€˜.i ,, .


C) a) C)

c@@

8 80

6 60

4 40

2

0

8

6

4

2

0

Day of year

C)

8

6

4

2

0

8

6

4

2

0

FIGURE 8. Estimated numbers of larvae (thinner lines) and pupae (thicker lines) in the four types of representative containers over a one

year period. Projections were based on 1977 weather.

20

0

1 61 121 181 241 301 361 I 61 121 181 241 301 361

other containers (Figure 9). When the abundance of the containers within the environment is taken into account, totaladult female production by type ranges from approximately100/ha/year from animal dishes to approximately 2,900/ha/year from the laundry sinks (Figure 9). While drums, sinks,and tires are approximately equal in productivity, the abundance of sinks makes them especially significant contributorsto the biting population of Ac. acgypti. Finally, the total dailyemergence of all females, the first of four data series passedfrom the CIMSiM to the DENSiM, averages approximately14/ha. The resulting female population averages approximately 125/ha and varies seasonally, somewhat in responseto those containers influenced by rainfall (Figure 10). Notethat for every adult female Ac. aegypti, there are about 25

larvae (Figure 10). The CIMSiM estimates adult daily survival (not shown) to be essentially 0.89 during the entireyear, with conditions being neither too cold or hot nor toodry to adversely influence adult mortality.

The two last series to be passed from the entomologicmodel are the daily gonotrophic development rate as a function of temperature and an average of adult female wetweight, a function of larval competition and temperature.While the development rate is relatively constant at this latitude, approximately 22% per day (Figure 11), femaleweight is more variable. The spikes in weight are primarilydue to bursts of emergence of large females from tires, emergence that was associated with rainfall ending periods ofdryness that had allowed an accumulation of larval food.

Demographic results. During the course of 1978 and

1 61 121 181 241 301 361

Day of year

250225

@ 200.@ 175

I!t@ 5Â°

250

6000

5000 0)â€˜.4a

4000@â€˜a

3000@04

2000

1000@

0

FIGURE 10. Area-wide estimates (per hectare) of Aedes aegypti

larvae, daily emergence of females, and the adult female populationover a one-year period. Projections were based on I977 weather.

Dish Drum Sink Tire

70 3000

@60 2500@

@ .@

@ 2000wLI â€˜aI4'+u0) 0)04 1d@J@J 04

@30a iooo'@

@20 E

114 io 500

0 0

Container type

FIGURE 9. Annual production of adult female Aedes aegypti from

individual representative containers (bars) and from all containersof that type within a one-hectare area (line). Projections were basedon 1977 weather.

0 Births b@'@igecIa'@'ot mother

0 Deaths by age'@Hn@@

i

MODELOF THE EPIDEMIOLOGYOF URBANDENGUEFEVER 499

0.7

I'@0.6a@ 0.50)

40

E0 0.20)

0.0

â€” t'cnial@ weight

@ â€˜(initriiphi

1 61 121 181 241 301 361 1 2 3 4 5 6 7 8 9 101112131415161718

Day ofyear Age class

FIGURE 1 3. Age class-specific number of births and deaths with

in a single year. Births are given by the age class of the mother.

approximately I 8%, respectively, of their population. Thesedifferences have the potential to influence the dynamics oftransmission.

Initial epidemic. The monthly introductions of a viremic

individual did not result in any local transmission duringJanuary and February, 1978 (Figure 14). Two cases occurredin March but the virus was lost. The April introduction resuIted in I I cases and the persistence of the virus betweensubsequent introductions. The monthly number of cases increased rapidly between May and September (60, 195, 510,988, and 1,598) with the epidemic peaking in October with2,122 additional cases. Defining, as suggested by Newtonand Reiter,2Â°an epidemic to be detectable when the prevalence exceeds I % of the population, the epidemic began July21 . It peaked 92 days later on October 21 and then decreasedover the next 61 days to less than 1% on December 19; theduration of the detectable epidemic was 153 days. At itspeak, 430 people were viremic, a little more than 4% of thetotal population.

Because stochastic events are sometimes important in theearly phases of a mass-action phenomenon such as epidemics, repeated simulations with the same starting conditionsdo not always result in similar outcomes; this is most likelyto occur as conditions become intermediate between favor

FIGURE 1 1 . Proportion of total development of the first gono

trophic cycle (CD) occurring each day and average adult Aedesaegypti female wet weight (mg) upon emergence over a one-yearperiod. Projections were based on 1977 weather.

1979, the transmission model estimates the population of oururbanization to grow at an annual rate of 3.8% from 10,000to 10,772; total deaths, births, and immigrants (the viremicintroductions) for the two years were 258, 1,006, and 24respectively. The age distribution at the end of the year isvery similar to the beginning (Figure 12); for longer runs of15 or 25 years, this is not always the case because the currentage distribution reflects past and sometimes substantially different age-specific birth and death rates. The model also provides estimates of deaths by age of individual and births bythe age of the mother (Figure 13); these results are in veryclose agreement with U.S. Commerce Department estimates.74 Because the birth and death processes are stochastic,repeated runs with the same inputs give slightly differentresults.

The relatively high birth and death rates observed in Honduras result in a population skewed toward the younger ages;here infants and children less than one and 10 years of age,respectively, account for approximately 5% and 35% of thepopulation, respectively. In contrast, the annual growth ratein Puerto Rico is approximately I .2% and infants and children less than 10 years of age account for less than 2% and

2000

I:15000 1000

I5000

1 2 3 4 5 6 7 8 9 101112131415161718Age class

FIGURE 12. Number of people within the representative town by

age class at the beginning and end of a two-year simulation. Class1 is infants < I year of age, class 2 includes children spanning fouryears in age from 1 to < 5 years of age; the other classes spanfiveyearsofage.

Djanuarv 1, 1978

@ December 31, 1979

4.5

0 4.

a 3@5

04 25

@2:0

a 15a

@ 1.0â€˜-40)

04.0.0

-,..@

Jan-78 May-78 Aug-78 Dec-78 Apr-79 Aug-79 Dec-79

Date

FIGURE 14. Prevalence of viremic individuals (line) and mci

dence of new cases (boxes) resulting from the monthly introductionof a single viremic individual into a naive population of approximately 10,000. The first three introductions can be clearly seen duringJanuaryâ€”March1978.

3001â€•25004 200

0@ 150

â€˜@ 100

z

0


100

__800)

.4@4

C,)004 400

20

0


Date

FtGuRi@ 17. Projected seroprevalence of dengue antibody during1978 and 1979. The lines labeled MANA and MAEA indicate, respectively, the prevalence of maternally acquired neutralizing antibody and maternally acquired enhancing antibody among infants.The other two lines indicate seroprevalence in the 1â€”4-and 5â€”80-year-old age groups. The dengue simulation model provides similarestimates (not shown) for each age class.

(Figure 17); the distribution of these titers reflects the portionof mothers converting, at approximately 80%, and the relative lengths of time required for the acquired antibody todecrease from neutralizing to enhancing to trivial concentrations. The DENSiM also provides less detailed estimates ofthe class-specific number and percentages of individuals infected (Figure 18).

The model indicates that without new introductions of vin's from the outside following the primary epidemic, thevirus would typically persist in the village for only a fewmonths following the peak of the epidemic. A few simulations resulted in the virus disappearing by December 1978,more commonly, the virus was lost during Januaryâ€”March1979, although in one instance, virus persisted until August1979, nine months following the peak of the epidemic. Withthe passage of time, herd immunity decreases, and the population, led by the younger age classes, progressively becomes more receptive to new introductions.

Correspondence between some observed and predicted

facets of the epidemic. The entomobogic results presented

here serve to illustrate model output only; we have no fielddata other than that used to parameterize the CIMSiM, con

100

Date

30

25

20

15

10

5


Date

Cl)

a

04

80

60@0)

CO

4Â°.@0)

0)

20@

0

1 2 3 4 5 6 7 8 9 101112131415161718Age class

FIGURE 1 8. Distribution of infections by age class. The bars and

line indicate the number of individuals and proportion of the ageclass infected, respectively, at the end of 1979.

FIGURE 16. Estimated extrinsic incubation period of virus inmosquitoes based on temperature and virus titer (see text for details).

80.@

60E

40

20

FIGURE 15. Plot of the number of female Aedes aegypti per per

son and the number of infectious mosquitoes (females with virus intheir salivary glands) within the approximately 40-hectare area ofthe representative town.

ing the loss of or rapid transmission of virus. Repeated runs,however, typically gave similar results regarding dates ofonset and duration of the epidemic and extent of infectionwithin the population. This was principally due to the factthat seasonal temperature changes resulted in a rapid iransition from conditions favoring the loss of introduced virusto epidemic conditions. Cool temperatures early in 1978lowered substantially the chance of local transmission resuiting from the early introductions. The date of onset of theepidemic had little to do with fluctuations in the abundanceof female Ac. aegypti (Figure 15); rather, the onset was triggered by the precipitous decrease in the EIP associated witha brief time of high temperatures (Figure 16).

Following the onset of the epidemic, conditions remainedpermissive, allowing transmission to increase until the depletion of susceptible individuals began to limit the epidemic(Figure 17); in the epidemic shown, this class decreased toabout 20% by the end of 1978. The seroprevalence of antibody to dengue-l decreased in the 1â€”4-year-old age classlate in 1979 because of their movement without replacementby dengue-immune infants into the next age class and thelack of substantial transmission following the epidemic morethan a year earlier. The prevalence of maternally acquiredneutralizing and perhaps enhancing titers of antibody increased following the increase in seropositivity of mothers

35

1@@Â°@


1200

4@)1000

.@800

@600

@ 400

200

0'


tamer types and abundances along with their associated mosquito productivities, with which to compare it. We can sayonly that the entomobogic inputs to the transmission modelwere adequate to allow simulating the epidemic of San PedroSula and surrounding towns with some degree of fidelity.

Temporally, the points of correspondence between modelpredictions and the epidemic in San Pedro Sula include thedate of the onset of the epidemic, July 1978 and June 1978,respectively, and the date of the epidemic peak, October1978 and August 1978, respectively. While the length of theepidemic was not reported, the time between first reports ofcases in San Pedro Sula and the epidemic peak was approximately 2.5 months; the corresponding model estimate wasapproximately 90 days.

In terms of seroprevalence of antibody, the model projected some 80% of the population to have become infectedin the epidemic. The observed average seroprevalence ofdengue-l antibody for San Pedro Sula and neighboringtowns experiencing epidemics in 1978 was 59% (range 37â€”81%).

Figueroa and others7Â°reported that not all communities indepartments that were affected by dengue had epidemics andsuggested that because there was a positive correlation between Ac. aegypti indices and seroprevalence of dengue antibody, that low mosquito densities were prophylactic. Simulation studies with lower mosquito abundances indicate thatthis explanation is plausible. Densities, when reduced to75% of the levels described above, still regularly supportedepidemic transmission, albeit, at slower rates, sometimesspreading over two years with a slowing of transmissionduring the winter months. However, densities of 50% of norma! precluded epidemic transmission; introduced virus underthese conditions would often lead, especially in the warmermonths, to some locally acquired cases before the virus waslost. Under these conditions, seroprevalence of dengue antibody would increase at a rate of only a few percent peryear.

As a final point of correspondence between model andobserved, we note that the model permits a low level oftransmission and the maintenance of virus through the wintermonths in the coastal lowland as was observed in Honduras.7Â°

Some examples of simulation studies addressing research or operational questions. How would the cooler climate of inland, more elevated locations be expected to influcncc transmission in Honduras? Tegucigalpa, at an dcvation of approximately 1,000 m, has average maximum andminimum temperatures that are approximately 1.2Â°C and4.0Â°C lower than the coastal lowlands, respectively. UsingTegucigalpa weather, the CIMSiM estimates the averageweight of females to increase by approximately 7%, anamount insufficient to significantly change the proportion offemales taking double replete feeds per gonotrophic cycle.However, the summertime length of the gonotrophic cyclewas increased from 4.6 to 6.3 days; this would lower theexpected daily biting rate in Tegucigalpa to approximately72% of the coastal lowland rate. Another important climaterelated change occurred in the length of the EIP; annual

estimates for the high and low elevations were 19.9 and 15.8days, respectively. Assuming a daily adult female survivalrate of 0.89 (as indicated by the CIMSiM), this 4.1-day in

crease in EIP would result in approximately 38% fewer females surviving the duration of the EIP at the cooler climate.

What impact would these changes be expected to have ontransmission? We have no information on the density of Ac.aegypti in Tegucigalpa. If we assume container types andabundances are similar to coastal communities, then the temperature-related changes would be expected to result in verylow and sporadic rates of local transmission; in most simulation runs, the prevalence of cases rarely exceeded the detectable threshold rate of 1%. At the end of 1979, modelprojections of seroprevalence of antibody ranged between2% and 7%. How well do these projections line up with theepidemic experience of Tegucigalpa? For the 1978-1979 epidemic, Figueroa and others7Â°reported that the capital wasnot significantly affected and had only sporadic and mostlyimported cases; seroprevalence in September 1979 was given as 11%. Later, after a hiatus of almost 10 years, denguetypes 1, 2, and 4 were isolated in 1989â€”1991 from the metropolitan area, but the reported case rates were very low,257 per 100,000 (0.3%), suggesting only low levels of tansmission (Honduran Ministry of Health, unpublished data).

Could the dampening influences of cooler temperatures becompensated by increased mosquito densities? If we increasemosquito abundances by 50%, the DENSiM projects the occurrence of epidemics in Tegucigalpa as significantly morelikely with seroprevalences at the end of 1979 expected torange between 35% and 50%. A doubling mosquito abundance ensures that epidemics will occur with resulting seroprevalences typically in the 80% range. These results illustate the expected attenuation of transmission with altitudethat can be modulated by mosquito densities; they are consistent with field reports of such a 776

What is the abundance of infectious mosquitoes during an

epidemic and should we expect to detect them ? The averagenumber of Ac. aegypti females in our simulation village wasapproximately 120/ha (Figure 10) or 0.5/person (Figure 15).This is substantially lower than the levels reported for Bangkok, Thailand of 700â€”1,000/ha and 3â€”5/person.2 37 At thepeak of the Honduran epidemic, there were only about 100infectious mosquitoes (roughly one per every 100 people)within the approximately 40-ha village (Figure 15), roughly2% of the approximately 5,000 female mosquitoes withinthe village. It is not surprising then that dengue virus is rarely detected in field-collected Ac. aegypti from endemic areas.49

What pattern oftransmission would be expected following

the primary epidemic? Figure 19 presents the projected number of cases and the seroprevalence of dengue antibody for16 years following the primary epidemic of 1978; we haveassumed a continuation of the monthly introductions of aviremic individual. For five or six years the introductionsresulted in few locally contracted infections due to herd immunity and the relatively low abundance of the vector. Asthe immune population aged, the younger age classes progressively became more susceptible. As a consequence, mostof the subsequent infections occurred primarily in theseclasses as indicated by the increase in seropositivity amonginfants during the summers of 1983 and 1985. Seropositivityin infants, as indicated by the DENSiM, generally runs higher than the overall rate (Figure 19) because the accountingincludes maternally acquired as well as infection-induced an


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FIGURE 19. Projected seroprevalence of dengue antibody and the number of cases of dengue infection during the years following the virginsoil epidemic of 1978. The results are based on a continuation of the monthly introductions of a single viremic individual circulating a virustiter of l0@mosquito infectious dose 50. The human population grows during the 1978â€”1994period from 10,000 to almost 19,000. Weatherdata for 1993 were used for 1994.

tibody. If this scenario is run for decades, the age-specificdistribution of seroprevalence settles down to that seen between 1989 and 1994, with sporadic, small epidemics involving at most a few hundred (primarily young) individualswith the overall prevalence of antibody averaging approximately 70%.

If this scenario is run again with the titer of the introducedvirus increased from l0@to l0@MID@IJ(Figure 20), the initialepidemic is more acute than that shown in Figure 14, beingshorter in duration and involving approximately 95% of thepopulation. The nature of transmission following the primaryis different as well, with transmission being more intense,and the sporadic and small ensuing epidemics are less fre

quent but involve more people and produce higher levels ofimmunity than those associated with the lower titering virus.

What is the probability of an epidemic following a single

introduction ? A number of factors determine the fate of in

troduced virus. We mentioned several earlier the key role ofstochastic events in the early phases of transmission, temperature, and herd immunity. Obviously, many of the parameters included in the models are potentially influential. Asanother example of the type of questions that can be investigated with the models, we ask: how receptive is the villageto a single introduction occurring at various times of theyear? Would this be modified by titer of the virus, given theinfluence of titer on the probability of infection and EIP in

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the mosquito? Perhaps, less ambitiously, we could couch thequestion in terms of parameter sensitivity: if conditions arenear the threshold permitting transmission, would factorssuch as seasonality in mosquito abundance, size, and temperature be sufficiently influential against the backdrop ofother factors so as to significantly alter the probability of anepidemic and would we expect this to be substantially modified by the titer of introduced virus?

Simulation results (Figure 21) indicate that at the low titer(10-i MID50) used in our example, seasonal changes result inan almost three-fold difference in the probability of an epidemic resulting from a single introduction (30â€”35% in Dccember and January versus 80% in Aprilâ€”May). That is tosay, a introduction in the winter is about one-third as likely

to cause an epidemic as one occurring in the spring or sum

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FIGURE 20. Projected seroprevalence of dengue antibody and the number of cases of dengue infection under the conditions described forFigure 19 except that the virus titer was l0@mosquito infectious dose 50.

mer. The results, while suggesting that many introductionsinto a naive population would be lost and not produce anepidemic, also indicate that a single introduction is capableof producing an epidemic any time of the year; recall theinitial dengue-l epidemic in Central America that occurredon the Bay Island of Roatan in February.7Â° This is interestingin light of the rapid and global movement of dengue viruses.5Simulations also indicate that introductions of high-titeringvirus more frequently lead to epidemics than introductionsof the lower type. Associated with the higher titering virusis a reduction in the magnitude of role seasonal influences

play; summer introductions are only approximately 1.5 timesmore likely to cause an epidemic than winter introductionsof the same virus. The difference between the ability of thetwo viruses to cause an epidemic is most pronounced during


@- ::@â€˜@â€˜@ers are equally productive of adult mosquitoes. Without running a single simulation with the DENSiM, the parameterization process of the CIMSiM identifies the important Ac.aegypti-producing containers in an area and allows significant epidemiologic comparisons, on a females-per-personbasis, to be made between different barrios or adjacenttowns.

How effective in reducing Ac. acgypti must a communitybased source reduction efforts be in this part of the world?We are fairly comfortable with our estimates of approximately 0.5 Ac. aegypti females per person in El Progreso,Honduras (Figure 15), the CIMSiM is fairly well validated2and the field observations on containers and productivitiesare extensive. We know that this ratio has recently supporteddengue transmission (Honduran Ministry of Health, unpublished data) in El Progreso and probably was sufficient forthe dengue-! epidemic of 1978.@Â°Thus, while we do nothave an absolute threshold value, it is arguably somethingless than 0.5 females per person; this would correspond toapproximately 0.25 pupae per person.t An ongoing programin another nearby town with vector densities averaging twofemales per person cannot be expected to be effective if itcan only halve adult emergence.

It must be emphasized, however, that this is just an estimate and only an estimate of an upper bound for the transmission threshold. Moreover, it is only for a certain location,being predicated on local temperatures and a naive population, among other factors. As we use the models in morelocations, a series of estimates like this one will be developed. We would also like to draw attention to the utility ofmeasuring mosquito densities, not in terms the traditionalindices, but rather in terms of adults or pupae per person.

We have attempted with the CIMSiM and DENSiM toprovide an essentially complete accounting of the currentstate of our understanding of the factors influencing denguetransmission in urban areas. The results presented here, theongoing validation work in various locations, and the recentmodeling work of Newton and Reiter@Â°suggest that the present knowledge of this system is adequate to permit modeldevelopment for the goals given earlier. We have only begunthe validation process for the DENSiM, but the good correspondence between the model projections and the reportedepidemics following the initial introduction of dengue intoHonduras in the late 1970s@ is encouraging. We hope thatthe use of these models will further dengue control effortsworldwide.

Acknowledgments: Information from El Progreso, Honduras, on thenumbers of different kinds of containers per hectare, as well as theaverage numbers of pupae in different kinds of containers, was graciously provided by Dr. Eduardo Fernandez (Director) and CatalinaSherman (Medical Entomologist of the Proyecto Control Integradode Dengue, Ministerio de Salud de Honduras, Division de Enfermedades Transmitidas por Vectores, Tegucigalpa, Honduras), Ger

tWhat would the corresponding estimate be in terms of pupae perperson? Assume a steady-state population, the sex ratio in pupae tobe approximately 1:1, and the length of the pupal period and theadult life span to be approximately 2.0 and 8.6 days, respectively.â€•2the conversion factor would be (2.0 days/8.6 days) X (2 pupae ofboth sexes per female pupa) = 0.465 pupae per adult female. Rounding to 0.5 pupae per female, 0.5 females per person would correspond to about 0.5 X 0.5 or 0.25 pupae per person.

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FIGURE 21. Probability that the introduction of a single viremicindividual will result in an epidemic in the representative town as afunction of the date of the introduction and the titer of the virus (inunits of mosquito infectious dose 50).

the cooler months when the high-titering virus is approximately two times more likely to result in an epidemic thanlower-titering introduction. These results are typical of others where different factors or combinations of factors become key regulatory factors under different conditions ofweather, presence of antibody, demographics, and mosquitocharacteristics.

From an operational perspective, what significance wouldreports of low-level transmission have ? Regarding the pos

sibility of sufficiently early detection of the circulation of anew virus to allow emergency vector control prior to anepidemic, the models indicate for the San Pedro Sula epidemic (Figure 14) that continuous autochthonous transmission was occurring for almost three months in 1978 prior tothe prevalence exceeding 1%. During this time, about 570cases (approximately 6% of the population) occurred. Theseresults would seem to provide support for the recommendations of Gubler and Casta-Valez regarding the utility ofproactive surveillance in a control program.'5 However, theinterpretation of reports of a few sporadic cases of denguewould be problematic; their significance would depend uponconvictions held regarding how receptive the area is to anepidemic of a particular serotype. Objectively, this would bea function of the seroprevalence of antibody, weather, timeof year, the ratio of mosquitoes to humans, and some ideaof the serotype causing the current cases. For example, whileSan Pedro Sula was certainly less receptive in June 1979than during June 1978 to dengue-1, it was fully susceptibleto an introduction of another serotype. Our point is that surveillance and decision making in an ongoing control program could be well served by integrating demographic, entomologic, virobogic, and serologic data using models suchas the CIMSiM/DENSiM.

What can be said about transmission thresholds ? The parameterization process for the CIMSiM forces one to lookat Ac. aegypti abundance in absolute terms; how many containers per hectare and their associated standing crops ofpupae. Once these data are integrated with the CIMSiM, theyallow describing the entomologic situation in terms of thenumber of females per person; this development is in markedcontrast to the usual Ac. aegypti indices, all of which arehampered by the predication that all larval-positive contain


ardo Borjas (National University of Honduras, Tegucigalpa), and Dr.Gerald Marten (Thlane University, New Orleans, LA). As far as weare aware, this is currently the only dengue control project that canevaluate their interventions or compare different barrios in terms ofepidemiologically significant measures such as reductions in adultor pupal production or female Ac. aegypti per person or per hectare.

Authors' addresses: Dana A. Focks, Eric Daniels, and Dan 0. Haile,Modeling and Bioengineering Research Unit, Medical and Veterinary Entomology Research Laboratory, U.S. Department of Agriculture, P0 Box 14565, Gainesville, FL 32604. James E. Keesling,Department of Mathematics, University of Florida, Gainesville, FL32611.

Reprint requests: Dana A. Focks, Medical and Veterinary Entomology Research Laboratory, U.S. Department of Agriculture, P0 Box14565, Gainesville, FL 32604.

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