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Modeling Household
Transmission of American
Trypanosomiasis
Joel E. Cohen
1
* and Ricardo E. Gu¨rtler
2
American trypanosomiasis, or Chagas disease, caused by the protozoan parasite
Trypanosoma cruzi and transmitted by blood-feeding triatomine bugs, is a
chronic, frequently fatal infection that is common in Latin America. Neither
adequate drugs nor a vaccine is available. A mathematical model calibrated to
detailed household data from three villages in northwest Argentina shows that
householders could greatly reduce the risk of human infection by excluding
domestic animals, especially infected dogs, from bedrooms; removing potential
refuges for bugs from walls and ceilings; and using domestically applied in-
secticides. Low-cost, locally practicable environmental management combined
with intermittent use of insecticides can sustainably control transmission of
T. cruzi to humans in rural Argentina and probably elsewhere.
Chagas disease, or American trypanosomiasis,
is endemic in Central and South America. An
estimated 16 to 18 million persons are infected
with Trypanosoma cruzi (Kinetoplastida:
Trypanosomatidae), the causative agent of Cha-
gas disease, and 100 million people (roughly
one-quarter of the population) are at risk of
infection (1). Despite decreasing rates of prev-
alence and incidence of T. cruzi infection (2, 3),
Chagas disease remains a serious obstacle to
health and economic development in Latin
America, especially for the rural poor.
The repertoire of control measures is limit-
ed. Two drugs are curative in the acute and
early chronic phase of infection but have ad-
verse effects and may not always eliminate T.
cruzi. No vaccines are available to prevent in-
fection. Transmission may be interrupted by
residual spraying to kill blood-feeding triato-
mine bugs (the vector of T. cruzi), screening
blood donors, and treating infected infants born
to infected mothers. A more controversial strat-
egy for interrupting transmission is to divert
bugs from humans by the use of animals that
are not susceptible to T. cruzi (4). This strategy,
called zooprophylaxis, is controversial for other
vector-borne diseases (5) as well and may re-
main so for T. cruzi because it cannot be tested
experimentally. Ethical considerations bar a
randomized prospective field study in which
the human prevalence of T. cruzi infection is
compared between households that do, and
those that do not, keep domestic animals in
bedroom areas in the presence of domestic tri-
atomine infestations.
Mathematical models of the transmission
of T. cruzi infection are the next best avail-
able tool to understand the effects of alterna-
tive control strategies (6–14 ) [Web supple-
ment gives further references (15)]. Here we
present a model of the transmission of T.
cruzi infection within an individual house-
hold. The model represents three vertebrate
populations (humans, dogs, and chickens),
the bug population, the parasite, and season-
ality. Although many existing models of the
transmission of infectious diseases use differ-
ential equations to represent changes in the
prevalence of infection and in the population
sizes of hosts and vectors, which are usually
assumed to be large, the discrete formalism of
the present model makes it easy to represent
the age structure of one household’s human
population, the small numbers of domestic
vertebrates, and the seasonal differences in
host composition and exposures to bugs.
This model was developed in close con-
nection with household data collected to sup-
port modeling in three rural villages, Amama´,
Trinidad, and Mercedes, in the province of
Santiago del Estero, northwest Argentina.
The villages are situated within 9 km of each
other in a semi-arid hardwood thorny forest
habitat. In 65 houses of Amama´ and Trinidad,
the median household in 1993 had five peo-
ple, about three infected dogs, no more than
one cat, and 8 to 27 chickens and ducks (16 ).
In Argentina, transmission of T. cruzi to
humans is minimal in fall and winter (April to
August). Below 16° to 18°C, bugs cease de-
velopment and feeding (17). In early spring,
people sleep indoors, chickens are maximally
present in bedroom areas (18), bugs are in-
creasingly active and feeding, and the domi-
ciliary bug population grows rapidly. In sum-
mer, chickens mainly roost outdoors, people
usually move their raised beds outdoors to
sleep on verandas or patios in front of their
bedrooms, and the size of the domiciliary bug
population is maximal.
Domestic triatomine bugs take blood meals
from household vertebrates to be able to develop
from each instar to the next and to lay eggs.
Keeping chickens in bedrooms in spring to pro-
tect them or their eggs from predation or theft
increases the domestic bug population size (18,
19), most notably in the following summer.
Because chickens cannot be infected with T.
cruzi, the more often a domestic bug feeds on a
chicken, the less likely the bug is to become
infected with T. cruzi. Therefore, keeping chick-
ens in bedrooms could decrease the prevalence
rate of T. cruzi in bugs. In contrast, keeping
infected dogs in the household increases both
the bug population size (20) and bug prevalence
of T. cruzi (21, 22). The summer population of
large and late-stage T. infestans bugs, increased
as a result of spring feeding on chickens, shifts
feeding from chickens to humans or infected
dogs in the hot summer season (16) when chick-
ens are largely absent. The presence of chickens
in bedroom areas decreased the prevalence of T.
cruzi in bugs but increased the density of T.
cruzi–infected bugs (23).
Mathematical modeling is required to un-
derstand the implications of these findings for
the human prevalence of T. cruzi infection
(4). We model only the transmission season,
spring (September to mid-December) and
summer (mid-December to March). The
model represents the population of the para-
site T. cruzi implicitly through prevalence
rates. The human (or dog) prevalence rate
gives the proportion of humans (or dogs) who
are infected with T. cruzi (as measured by
seropositivity) but does not distinguish
among different phases of infection. In these
calculations, all dogs are assumed to be in-
fected, as field data show. The bug preva-
lence rates (in spring and in summer) give the
fraction of bugs in each season that are in-
fected with T. cruzi. In the model and real
life, once a person, dog, or bug becomes
infected, that individual remains infected for
life (unless the person or dog is treated
promptly; there are no treatments for bugs).
The model represents explicitly four popu-
lations: humans, dogs, chickens, and bugs. Hu-
mans are represented by the numbers of indi-
viduals in 5-year age groups: under age 5, ages
5 to 9, ages 10 to 14, etc. Dogs represent all
mammalian domiciliary animals that are sus-
ceptible to infection with T. cruzi, provide
blood meals to T. infestans, and are more at-
tractive or accessible as a source of blood meals
than humans. Chickens represent all avian dom-
iciliary animals, which are not susceptible to
infection with T. cruzi but which may provide
blood meals to T. infestans and are more attrac-
tive or accessible blood sources than humans.
In the model and henceforth here, “bugs”
means exclusively fourth- and fifth-instar
nymphs and adults. These stages include almost
1
Laboratory of Populations, Rockefeller University and
Columbia University, 1230 York Avenue, Box 20, New
York, NY 10021, USA.
2
Laboratorio de Ecologı´a Gen-
eral, Facultad de Ciencias Exactas y Naturales, Uni-
versidad de Buenos Aires, Ciudad Universitaria,
C1428EHA Buenos Aires, Argentina.
*To whom correspondence should be addressed. E-
mail: cohen@rockefeller.edu
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27 JULY 2001 VOL 293 SCIENCE www.sciencemag.org694
all bugs that are infected (23, 24) and capable
of transmitting infection, and timed manual col-
lections of bugs yield samples biased toward
these stages.
Bugs are several times more likely to take
their blood meals from dogs and chickens than
from humans (20). For example, the ratio of
dog blood meals to human blood meals in the
engorged guts of bugs was 2.3 to 2.6 times the
ratio of the number of dogs to the number of
humans in a household in the spring-summer
period. Similarly, bugs selected chickens for
blood meals nearly five times as often as the
number of chickens relative to humans. These
ratios did not vary significantly among house-
holds with differing total numbers of humans,
dogs, and chickens (20). Dogs and chickens are
probably less attentive than humans in defend-
ing themselves against feedings by the bugs.
Also dogs and chickens sleep or nest in bed-
room places that are more accessible to the bugs
than the raised beds on which people typically
sleep, whether indoors or outdoors. We approx-
imate the ratio of feedings on dogs and chickens
relative to humans as 3, as a rough midpoint of
the large range of variation of empirical esti-
mates. The probability of infecting an initially
uninfected bug in one full blood meal from an
infected dog was 12 times that probability from
infected children and 200 times that from in-
fected adults (16).
The assumptions, variables, and formal
structure of the model are detailed in the Web
supplement (15). The model predicts how the
numbers of humans, chickens, and infected
dogs in a household and the physical-chemical
conditions affect the prevalence of T. cruzi
infection in humans and bugs, the number of
infected and uninfected bugs, and the distribu-
tion of feeding contacts, by season. Only one
parameter of the model was varied freely to
improve the quantitative fit between observa-
tion and prediction: the probability of transmis-
sion of infection from an infected bug to an
uninfected person (t
B3 H
⫽ 0.0008). This value
is close to the 0.0009 estimated from one field
study (9). For obvious ethical reasons, this
probability cannot be estimated experimentally
and is largely unknown.
Given the complexity of the natural sys-
tem and the relative simplicity of any math-
ematical model, qualitative agreement is the
most that can be hoped for from the compar-
isons of model predictions and field observa-
tions. The following comparisons suggest
that the qualitative predictions of the model
give some insights into the real household
transmission of T. cruzi.
Empirically (Fig. 1A) and theoretically
(Fig. 1B), the bug prevalence rate increased
rapidly with the fraction of bugs that took
blood meals from infected dogs (or cats,
which were relatively rare) when the fraction
of bugs that took blood meals from chickens
stayed constant. The bug prevalence rate de-
creased slowly with an increasing fraction of
bugs that took blood meals from chickens
when the fraction of bugs that took blood
meals from infected dogs stayed constant.
Empirically (Fig. 1C) and theoretically (Fig.
1D), as the relative density of bugs collected per
unit of search effort at the end of summer
increased, the fraction of bugs that contained
blood meals from humans decreased while the
observed fraction of bugs that contained blood
meals from chickens increased. The model pre-
dictions are shown for spring and summer com-
bined because blood meals from both seasons
were probably detectable in bugs collected at
the end of summer (16 ). The source of a full
blood meal taken at an earlier instar is detect-
able for up to 3 months from blood taken from
the gut of a later instar of the same individual
bug. The intuitive explanation as to why both
field data and the model show that increasing
number of summer bugs are associated with
fewer human feeding contacts and more chick-
en feeding contacts is that, for any given num-
ber of dogs, the summer bug population in-
creases as a result of increasing availability of
chickens in spring, and chickens are more ac-
cessible or attractive than humans as blood
meal sources for the bugs.
The two principal predictions of the mod-
el are as follows:
Fig. 1. Comparison of observations (left) made
in March (end of summer in Southern Hemi-
sphere) from three rural villages in Santiago del
Estero, northwest Argentina, with model predic-
tions (right). (A) Bug prevalence rate as a func-
tion of the fraction of bugs that contained blood
meals from dogs (and cats, which were much
less frequent) and from chickens. A logistic re-
gression surface was fitted to data on individual
households. Reproduced by permission from Fig.
4a in (23), p. 754. (B) Predicted summer bug
prevalence rate as a function of the predicted
fractions of spring feeding contacts with dogs
and chickens. Although each feeding contact
had a single vertebrate blood source, each bug
may have had feeding contacts with more than
one species of vertebrate. A given bug may be counted more than once in (A), but each feeding
contact counts only once in (B). (C) Fraction of bugs that contained blood meals from humans (top)
and chickens (bottom) as a function of the bug density per unit of searching effort. Reproduced by
permission from Fig. 2, a and b, in (20), p. 706. Each point belongs to a different household. (D)
Predicted fractions of feeding contacts on humans (o) and chickens (x) in spring and summer
(combined because of the carryover of blood meals) as a function of the predicted number of
summer bugs for all combinations of numbers of dogs and chickens considered.
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www.sciencemag.org SCIENCE VOL 293 27 JULY 2001 695
1) The worst thing householders can do,
from the point of view of limiting T. cruzi
prevalence, is to keep roughly two infected
domestic dogs (Fig. 2I). This is precisely what
many households do. The intuitive explanation
why the model predicts that roughly two infect-
ed dogs should be pessimal for human preva-
lence is that each dog is as attractive as three
humans, and five humans are assumed in the
household, so two dogs is roughly equivalent to
the five humans from the bugs’ point of view.
As the number of infected dogs increases up to
two, the infected dogs are an increasing and
highly infective source of T. cruzi. As the num-
ber of infected dogs increases beyond two, the
additional infected dogs contribute infection to
many already infected bugs but also divert pro-
portionally more of the feeding bugs from hu-
mans. This intuitive account explains not only
the location of the peak human prevalence rate
around two infected dogs, but also explains
why the rise of human prevalence from zero to
two infected dogs is faster than the very gradual
fall of human prevalence as the number of
infected dogs increases beyond two.
2) Elimination of infected dogs from a
household with infected people is nearly suf-
ficient to extinguish transmission of T. cruzi,
barring reintroduction of infected dogs, chil-
dren, or bugs (Fig. 2, E, F, G, and I).
The model makes many additional predic-
tions (Fig. 2).
3.1) The number of spring bugs increases
with the number of dogs but is independent of
the number of chickens (Fig. 2A). Chickens,
kept in the household in spring, have no
immediate effect on spring bugs because the
model assumes a lag of one season between
the availability of blood sources and the re-
cruitment of late-instar and adult bugs.
3.2) The number of summer bugs increas-
es with both dogs and chickens in the previ-
ous spring (Fig. 2B), as 1988 –1989 data at
Amama´, Trinidad, and Mercedes showed
(18–20).
3.3) The fraction of spring bugs’ feeding
contacts with humans (Fig. 2C) decreases
symmetrically, rapidly at first and then more
slowly, with more dogs and chickens.
3.4) The fraction of spring bugs’ feeding
contacts with infected dogs (Fig. 2D) increas-
es at a diminishing rate with more infected
dogs and decreases with more chickens. The
fraction of spring bugs’ feeding contacts with
chickens (not shown) is obtained by exchang-
ing the axes of Fig. 2D for chickens and dogs.
3.5) The fraction of spring bugs infected
with T. cruzi (the spring bug prevalence rate)
increases at a diminishing rate with more
infected dogs and decreases slightly with
more chickens (Fig. 2E). Even when there are
no infected dogs and no chickens, the spring
bug prevalence rate remains positive because
infected people remain in the household (by
assumption in the model, and because of the
persistent presence of chronically infected
adults in the field). When there are no dogs,
an increase in the number of chickens diverts
bugs from feeding on humans to feeding on
unsusceptible chickens and gradually reduces
the bug prevalence rate.
3.6) The prevalence rate in summer bugs
(not shown) increases with the number of
infected dogs as in Fig. 2E. Observed bug
prevalence rates of T. cruzi infection in-
creased with the number of infected dogs in
1988 –1989 in Amama´(21). In Trinidad and
Mercedes, the mean bug prevalence rates
were 16, 41, 68, and 47% in households with
zero, one, two, and three infected dogs, re-
spectively (22). The positive mean bug prev-
alence rate in households with zero infected
dogs is due to the presence of infected cats.
Bug prevalence rates were 4% in households
with no infected dogs or cats and 35% in
households with an infected cat and no in-
fected dog.
3.7) The spring number of infected bugs
(Fig. 2F) parallels the spring bug prevalence
rate (Fig. 2E) because the spring bug popu-
lation size is independent of the number of
chickens (Fig. 2A).
3.8) For any fixed number of dogs, house-
holds with more chickens have slightly more
infected summer bugs (Fig. 2G) and substan-
tially more total summer bugs (Fig. 2B).
Thus, the presence of chickens in bedrooms
in spring decreases the number of infected
bugs in spring and increases the number of
Fig. 2. Predictions from a model
of the household transmission
of T. cruzi for selected combina-
tions of chickens and infected
dogs. (A) Number of bugs in
household in spring. (B) Number
of bugs in household in summer.
(C) Fraction of feeding contacts
from humans in bugs captured
in spring. (D) Fraction of feeding
contacts from dogs in bugs cap-
tured in spring. (E) Prevalence
rate in bugs captured in spring.
(F) Number of infected spring
bugs. (G) Number of infected
summer bugs. (H) Number of
feeding contacts per human per
year. (I) Human prevalence rate,
or fraction of humans infected,
at steady state.
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27 JULY 2001 VOL 293 SCIENCE www.sciencemag.org696
infected bugs in summer. The summer in-
crease in infected bugs is slightly smaller
than the absolute spring decrease in infected
bugs for any positive number of chickens in
spring.
3.9) The number of feeding contacts per
human per year (Fig. 2H) decreases rapidly
with increasing numbers of dogs and decreas-
es very slightly with increasing numbers of
chickens, because the chickens divert feeding
contacts only in spring.
3.10) The number of potentially infective
feeding contacts per human per year is the
product of the number of feeding contacts per
human per year (Fig. 2H) times a weighted
average of the spring (Fig. 2E) and summer
bug prevalence rates. The number of poten-
tially infective feeding contacts (not shown,
but similar in shape to Fig. 2I) rises to a peak
with an increase from 0 dogs to 1.5 infected
dogs, and then begins to decline very slightly
as further dogs are added to the household.
(When the continuous variables used to mea-
sure the sizes of host or bug populations
include fractions, the fractions may be inter-
preted as the average fraction of time indi-
viduals are present in the household.) With
additional dogs, more bugs are diverted from
feeding on humans because bugs prefer to
feed on dogs.
3.11) The human prevalence rate, or frac-
tion of humans infected (Fig. 2I), is positive
even in the absence of chickens and dogs
because infected humans are initially present
in the house. For any fixed number of infect-
ed dogs, the human prevalence rate declines
very slowly with more chickens. For any
fixed number of chickens, the human preva-
lence rate increases very rapidly as the num-
ber of infected dogs increases from 0 to
roughly 1.5 and then slowly declines with
additional infected dogs. In the study villag-
es, the adult seroprevalence of infection in-
creased from 24% in households with no
infected dogs or cats to 48% in households
with one to two infected dogs or cats and to
64% with three to five infected dogs or cats
(25). The predicted human and bug preva-
lence rates of T. cruzi are both approximately
consistent with earlier field data from 1982
and 1984, before the first professional insec-
ticide spraying (9).
To investigate the sensitivity of model
predictions to changes in key parameters,
Figs. 1 and 2 were recalculated after making
three separate changes, one at a time: (i) the
maximum number of fourth- and fifth-instar
nymphs and adult bugs that the physical in-
frastructure of the house will support, given
an unlimited food supply, was reduced to 150
from the baseline value of 500; (ii) the num-
ber of feeding contacts per bug per spring and
summer combined was increased to 10 in-
stead of the baseline value of 5; and (iii) the
relative preference of bugs for chickens was
increased to 6 while their preference for dogs
remained unchanged at the baseline value of
3. The predictions of the model were robust
to these changes. The first few infected dogs
in the household resulted in a large increase
in the number of infected summer bugs, the
number of infective feeding contacts per hu-
man, and the human prevalence rate. With 10
feeding contacts, keeping chickens in bed-
room areas in spring reduced the bug and
human prevalence rates even less than when
the number of feeding contacts was 5. Reduc-
ing the maximum number of bugs from 500
to 150 reduced the human prevalence rate
from 0.63 to 0.4.
This model indicates that an increase in
the domiciliary chicken population very
slightly decreases the human prevalence rate
but by an amount that would be undetectable
in practice. This marginal benefit for an in-
dividual household is accompanied by an in-
crease in the size of the infected summer bug
population. Because the bugs are most active
and most likely to disperse to other houses as
temperatures rise (26 ), the very slight reduc-
tion in the household prevalence rate with
more chickens may be outweighed by greater
risk of spreading both bug infestation and
infection with T. cruzi to the rest of the
village. These results too are robust with
respect to plausible variations of the under-
lying assumptions of the model.
Keeping domestic animals in bedroom ar-
eas entails health and economic hazards, in-
dependent of the effect on T. cruzi infection.
Domestic animals may attract or harbor other
potentially dangerous vectors (such as mos-
quitoes, sandflies, ticks, and lice) and are
associated with other pathogens infectious to
humans (such as influenza, Toxocara spp.,
and Echinococcus spp.). Chickens repeatedly
bled by bugs may be less valuable food for
householders. Zooprophylactic measures
may fail if vectors shift hosts (27).
The model produces a straightforward and
clear result from a complex system. Keeping
dogs and other highly infectious vertebrates
out of bedroom areas can effectively reduce
the bug and human prevalence rate, accord-
ing to the model and field data. Human be-
havior strongly influences the transmission of
T. cruzi infection, in addition to chemical and
environmental factors that are more common-
ly emphasized. Low-cost, locally practicable
environmental management strategies with
intermittent use of insecticides can control
human transmission of T. cruzi sustainably in
rural Argentina and probably elsewhere (28).
Historically, multiple simultaneous interven-
tions that included environmental manage-
ment measures proved successful in control-
ling malaria (29) and other vector-borne trop-
ical diseases (30). Community education and
continuous surveillance through a local
health post are key requirements for effective
use of the control strategies identified here.
The task force on applied research on
Chagas disease (31) sponsored by the United
Nations Development Program, World
Health Organization, and World Bank recom-
mends increased efforts to control triatomine
vectors that occupy domestic and other hab-
itats in the Andes and Central America. The
household model described here could be
extended to a spatially explicit form to take
account of interactions among households
and surrounding forests. A similar modeling
approach could prove useful in evaluating
zooprophylaxis of other infectious diseases
such as malaria, Japanese encephalitis virus,
visceral leishmaniasis in Brazil, and the red
grouse-hare-louping ill virus system.
References and Notes
1. A. Moncayo, World Health Stat. Q. 50, 195 (1997).
2. G. A. Schmun˜is, F. Zicker, A. Moncayo, Lancet 348,
1171 (1996).
3. E. L. Segura et al., Am. J. Trop. Med. Hyg. 62, 353
(2000).
4. D. M. Minter, in American Trypanosomiasis Research,
Pan American Health Organization Sci. Publ. 318
(PAHO, Washington, DC, 1976) , pp. 330–337.
5. M. W. Service, Rev. Sau´de Pu´blica 25, 165 (1991).
6. G. Da Silva, Rev. Sau´de Pu´blica 3, 23 (1969).
7. J. E. Rabinovich, in Chagas’ Disease Vectors,R.R.
Brenner, A. de la Merced Stoka, Eds. (CRC Press, Boca
Raton, 1987), vol. 1, pp. 119–146.
8. J. E. Rabinovich, P. H. Himschoot, Ecol. Model. 52,
249 (1990).
9. J. E. Rabinovich, C. Wisnivesky-Colli, N. D. Solarz, R. E.
Gu¨rtler, Bull. W.H.O. 68, 737 (1990).
10. S. N. Busenberg, C. Vargas, in Mathematical Popula-
tion Dynamics, Proceedings of the Second Interna-
tional Conference, Rutgers University, New Bruns-
wick, NJ, 17 to 20 May 1989, O. Arino, D. E. Axelrod,
M. Kimmel, Eds. (Dekker, New York, 1991), pp. 283–
295.
11. M. Canals, P. E. Cattan, Rev. Me´d. Chile 120, 1359
(1992).
12. P. H. Himschoot, thesis, University of Buenos Aires
(1993).
13. J. X. Velasco-Herna´ndez, Theor. Popul. Biol. 46,1
(1994).
14. M. B. Castan˜era, thesis, University of Buenos Aires
(1999).
15. Web table 1 and supplemental text are available at
Science Online at www.sciencemag.org/cgi/content/
full/293/5530/694/DC1.
16. R. E. Gu¨rtler, M. C. Ce´cere, D. P. Vazquez, R. Chuit, J. E.
Cohen, J. Med. Entomol. 33, 15 (1996).
17. S. Catala´, Med. Vet. Entomol. 5, 325 (1991).
18. M. C. Cecere, R. E. Gu¨rtler, R. Chuit, J. E. Cohen, Med.
Vet. Entomol. 11, 383 (1997).
19. M. C. Cecere, R. E. Gu¨rtler, D. Canale, R. Chuit, J. E.
Cohen, Bull. W.H.O. 76, 373 (1998).
20. R. E. Gu¨rtler, J. E. Cohen, M. C. Cecere, R. Chuit,
J. Appl. Ecol. 34, 699 (1997).
21. R. E. Gu¨rtler et al., Trans. R. Soc. Trop. Med. Hyg. 85,
741 (1991).
22. R. E. Gu¨rtler et al., Trans. R. Soc. Trop. Med. Hyg. 87,
12 (1993).
23. R. E. Gu¨rtler, J. E. Cohen, M. C. Cecere, R. Chuit, E. L.
Segura, Am. J. Trop. Med. Hyg. 58, 748 (1998).
24. L. C. Giojalas, S. S. Catala´, S. N. Asin, D. E. Gorla,
Trans. R. Soc. Trop. Med. Hyg. 84, 439 (1990).
25. R. E. Gu¨rtler et al., Am. J. Trop. Med. Hyg. 59, 741
(1998).
26. M. J. Lehane, P. K. McEwen, C. J. Whitaker, C. J.
Schofield, Acta Trop. 52, 27 (1992).
27. S. Hewitt, M. Kamal, N. Muhammad, M. Rowland,
Med. Vet. Entomol. 8, 160 (1994).
28. C. J. Schofield, R. Briceno-Leon, N. Kolstrup, D. J. T.
Webb, G. B. White, in Appropriate Technology in
R EPORTS
www.sciencemag.org SCIENCE VOL 293 27 JULY 2001 697
Vector Control, C. F. Curtis, Ed. (CRC Press, Boca
Raton, FL, 1990), p. 187.
29. U. Kitron, A. Spielman, Rev. Infect. Dis. 11, 391
(1989).
30. S. K. Ault, Am. J. Trop. Med. Hyg. Suppl. 50,35
(1994).
31. Special Programme for Research and Training in
Tropical Diseases, Work plan of the task force on
applied research on Chagas disease ( World Health
Organization, Geneva, 1999). Available at: www.
who.int/tdr/grants/workplans/chagas4.htm.
32. We thank A. E. Cohen, C. Dye, J. E. Rabinovich, C. J.
Schofield, and B. H. Singer for helpful comments on a
previous draft. We also thank the Rockefeller Foun-
dation for grant RF 91080, Allocation 133. J.E.C.
thanks the NSF for grants BSR 9207293 and DEB
9981552, and Mr. and Mrs. W. T. Golden for hospi-
tality during this work. R.E.G. thanks the University of
Buenos Aires, the Fulbright and Thalmann programs,
and CONICET of Argentina for grants. The authors
thank D. M. Canale, M. B. Castan˜era, I. Castro, M. C.
Cecere, R. Chuit, A. Hurvitz, M. A. Lauricella, E. L.
Segura, and D. P. Va´zquez for sustained long-term
commitment which made possible the empirical
studies underlying the model.
12 March 2001; accepted 5 June 2001
R EPORTS
27 JULY 2001 VOL 293 SCIENCE www.sciencemag.org698
Cohen
Web Supplement
MS #1060638
In endemic areas, T. cruzi infection is transmitted on three distinct spatial scales: regionally
through human rural-to-urban migration and visitation between villages (1, 2); within a village
when domestic triatomine bugs disperse among households or from surrounding sites or forests to
houses, or when dogs, cats and humans visit among houses (3); and within each household,
through intimate contact among humans, dogs, cats, and bugs (4). Transmission is most intense
within the household. Hence, we developed a model of the seasonal transmission of T. cruzi in a
single household and focussed on the means of intervention available to householders.
This Web supplement specifies the assumptions, variables, and formal structure of the
model. It also lists the computing code used to generate and plot the numerical predictions of the
model.
Biological setting of the model
The model aims to represent transmission of T. cruzi in the domestic or domiciliary area of a
typical individual household in the Grand Chaco of northwestern Argentina. This area includes
human sleeping quarters and the veranda covered by the same roof, as well as the occupants at
night. It excludes the peridomestic area (corrals and buildings not connected to human sleeping
quarters) because peridomestic T. infestans exhibit marginal rates of contact with domiciliary
hosts and marginal T. cruzi infection rates (5, 6, 7). T. infestans does not have sylvatic foci in
most of its present distribution.
The model treats an individual household as a closed system. The model neglects the
introduction of T. cruzi into the household and its spread outward from the household. The model
assumes that the spatial distribution of people, dogs, chickens and bugs within the household is
unchanged from the situations in which the descriptive data were gathered. Changed spatial
arrangements could affect feeding contacts and transmission.
The model traces the household through two warm seasons within a single year, from the
onset of spring to the end of summer, assuming that transmission is negligible during the cold
seasons of fall and winter.
Counts of bugs refer exclusively to fourth- and fifth-instar nymphs and adult bugs. Late
instars at the end of summer pass through the winter with greatly reduced activity; recommence
feeding, molting, and egg-laying in spring; and die before summer. Most of the spring-born bugs
disappear before the next spring. Hence, the model assumes that all bugs live for two seasons
only, spring and the following summer, or summer and (after overwintering) the following spring.
Bugs acquire infection by feeding on infected humans and infected dogs (5, 6, 8). In the
spring but not the summer, chickens belong to the pool of blood meal sources for the bugs,
reducing the probability of a bug’s acquiring infection in the spring and increasing the bug
population size in the summer.
A feeding contact (or feeding) is defined as a full blood meal on a given individual host
obtained by a bug on one feeding trip or bout, regardless of the total number of bites delivered on
this host to secure the blood meal. “Full” is here used to match the experimental estimation of the
probability of a bug becoming infected in a single blood meal (8). Full blood meals imply an
increased chance of a bug defecating on the host’s skin and of potential parasite transmission.
When the continuous variables used to measure the sizes of host or bug populations
include fractions, the fractions may be interpreted as the average fraction of time individuals are
present in the household (e.g., 0.5 dog means a dog inside the household half time). Cats are
much less preferred by T. infestans than humans, dogs and chickens (5, 9) and are omitted from
the model.
Assumptions of the model
The input and output variables of the model are listed, defined and illustrated in table 1. The user
of the model specifies the number of humans by age groups as well as numbers of chickens,
infected dogs and uninfected dogs. These numbers are assumed fixed for any run of the model.
Here the modeled household contains one human aged 0 to 4, one aged 5 to 9, one aged 10 to 14,
one aged 25 to 29, and one aged 30 to 34 (three children, two parents). The base case assumes
two chickens, two infected dogs, and no uninfected dogs. Other household compositions can be
specified.
Chickens are assumed to nest or brood in bedroom areas in the spring but not the summer.
The number is given by the variable C, for "chickens." Humans, dogs and bugs are assumed to
live in bedroom areas in spring and summer. The user specifies how many of the household dogs
are infected with T. cruzi as the input variable DI (for "dogs infected"). All dogs are assumed
infected with T. cruzi in these calculations, as is nearly true in the highly infested houses (6), but
the model can take account of any number of uninfected dogs as well (DU). The total number of
dogs is D (for "dogs").
The number of bugs is treated as constant over time within each warm season (spring or
summer) but differs between seasons. The suffixes g and r denote spring and summer,
respectively. The size Bg of the spring bug population is a Monod function of the number of
vertebrate hosts Vr = H + R*D available the previous summer, where R is the user-specified
relative feeding index of dogs and chickens as sources of feeding contacts compared to humans.
With R = 3, one dog or chicken counts as three humans as far as the bugs are concerned, so a
household with one person and one dog supports as many bugs as a household with four people
only. The number Br of bugs in summer is a Monod function of the number of vertebrate hosts
Vg = H + R*(C + D) available in the previous spring.
The Monod function (see Bg, Br in table 1) is consistent with experimental evidence (10)
and field studies in experimental huts (11) and rural houses (12). The Monod function has two
user-specified coefficients, Blimit and Vhalf. The maximum number of bugs the physical
infrastructure of the house will support, Blimit, given an unlimited food supply, is higher if the
mud walls of the house are cracked, providing places for bugs to hide and lay eggs, and is lower if
the walls are smoothly plastered, if the roof of the house is built of materials resistant to bug
infestation or if the roofs and walls are treated with non-professional aerosol insecticides or
smokes. The value Blimit = 500 (late-instar and adult bugs) exceeds a census of 126 late instars of
T. infestans in bedrooms of a house demolished in Córdoba, Argentina (13), as well as two of the
three counts of 177, 682, and 137 late instars in three houses demolished in Goiás, Brazil (14),
and is approximately consistent with an estimate of the bug carrying capacity of mud and thatch
houses (15). The coefficient Vhalf (for "vertebrates halfway") is the number of vertebrate blood
sources (in human equivalents) sufficient to support Blimit/2 bugs.
The user also specifies five parameters of transmission of infection.
1) The number of feeding contacts per bug per spring and summer combined is M, for
“meals.” With M = 5 feeding contacts per late instar and adult bug per spring and summer
combined, most bugs are expected to feed on two or more vertebrate species (for example,
humans and dogs). The fractions of feeding contacts from humans, dogs and chickens calculated
in the model may correspond only approximately to the fractions of bugs whose guts have blood
meals from each of the sources (5, 16), because a bug may feed multiple times on a given
vertebrate host individual or species, and this repeat feeding is not detectable by routine
immunological tests. For the same reason, neither the fraction of bugs fed nor the fraction of
feeding contacts on a given species of vertebrate host can be interpreted as a surrogate for the
number of host individuals of that species in a household.
2) The fraction
α
of a bug’s M annual feeding contacts that a bug takes in the season in
which it is in instar 4 or 5 or adult reflects the prolonged life cycle of bugs, as estimated in
experimental huts under natural climatic conditions (11). The results of the model are insensitive
to the value of
α
in the range from 0.5 to 1.0, so we took
α
= 0.75. The model assumes that M
and
α
are independent of the number of vertebrate blood sources.
3) The bug-to-human transmission probability (t
B
→
H
) is the probability that, in one feeding
contact by one infected bug on an initially uninfected human, the human acquires infection. In all
simulations reported here, t
B
→
H
is assigned the value t
B
→
H
=0.0008. This value is based
on numerical experimentation with possible alternatives. This value is close to the 0.0009
estimated from one field study (17) but is smaller than 0.01 assumed in a theoretical model (18).
4) The human-to-bug transmission probability t
H
→
B
(here 0.03) is the probability that, in
one feeding by an initially uninfected bug on a seropositive human, the bug acquires infection.
5) The dog-to-bug transmission probability t
D
→
B
(here 0.49) is analogous to the human-to-
bug transmission probability. The probabilities 0.03 and 0.49 of infecting an initially uninfected
bug from a single full feeding on a T. cruzi-seropositive person or seropositive dog, respectively,
were estimated experimentally by xenodiagnosis: laboratory-reared uninfected bugs were fed
separately on seropositive people and seropositive dogs (8).
The user gives the model an arbitrary initial value of the household’s prevalence among
humans (pH) at the onset of spring (pH0). The model assumes that initially some humans or dogs
are infected, i.e., pH0 + DI > 0. Under this assumption, the value assumed for pH0 has no effect
on the computed equilibrial human prevalence.
Successive lines of code in "model 5" below compute as follows.
1. The total number D of dogs is DI + DU.
2. The total number H of humans is the sum of the number of humans of each age, that is, the sum
of the input vector Ha, which in these calculations is always [1, 1, 1, 0, 0, 1, 1].
3. The number Vg of spring vertebrates, in human equivalents, is H + R*(D + C), where C is the
total number of chickens.
4. The number Vr of summer vertebrates, in human equivalents, is H + R*D.
5. The number Bg of spring bugs is a Monod function of the number of previous summer
vertebrates, Blimit*Vr/(Vhalf + Vr).
6. The number Br of summer bugs is a Monod function of the number of previous spring
vertebrates, Blimit*Vg/(Vhalf + Vg).
7. The number of infected humans is NH = H*pH0.
8. The probability that, in one full feeding contact, an initially uninfected bug acquires T. cruzi
infection is Tg = (0.03*NH + 0.49*R*DI)/Vg in the spring and Tr = (0.03*NH + 0.49*R*DI)/Vr
in the summer. The latter denominator omits chickens because they are not available to bugs in
the summer. The model assumes that each bug selects the host for its next feeding contact at
random among the available vertebrate blood sources (after allowing for their different host
preferences). Data which have been offered as relevant to this assumption have varying
interpretations (19).
9. The prevalence rate pBg of T. cruzi infection in spring bugs is 1 – ((1 – Tg)^(M*
α
))*((1 –
Tr)^(M*(1 –
α
))), assuming that the bug has M*
α
feeding contacts in spring and M*(1 –
α
)
feeding contacts in the previous summer and that the transmission of infection is independent
among feeding contacts. The prevalence rate pBr of T. cruzi infection in summer bugs is
computed in the identical way, with spring and summer exchanged.
10. The spring number of infected bugs NBg is the product Bg*pBg of the number of spring bugs
times the prevalence rate in spring bugs. The summer number of infected bugs NBr is likewise
Br*pBr.
11. The average number of feeding contacts per person per year (spring and summer combined)
Bitespy is the sum of the average number of feeding contacts per person in spring plus the average
number of feeding contacts per person in summer. Each season-specific average number of
feeding contacts per person is the quotient of the number of feeding contacts from all bugs in that
season divided by the number of all vertebrate hosts in that season. For example, the spring
average number of feeding contacts per person is (Bg*
α
+ Br*(1
−
α
))/Vg because spring bugs
deliver Bg*
α
feeding contacts to the household in spring, whereas summer bugs deliver Br*(1
−
α
) feeding contacts to the household in spring. These feeding contacts are uniformly distributed
among the human-equivalent vertebrate hosts, and each human counts as one human-equivalent.
12. The average number of feeding contacts by T. cruzi-infected bugs per person per year
Infbitespy is the sum of the average number of feeding contacts from infected bugs per person in
spring plus the average number of feeding contacts from infected bugs per person in summer. The
formulas are the same as for Bitespy except that the numbers of bugs, Bg and Br, are replaced by
the numbers of infected bugs, NBg and NBr respectively.
13. The average number Ca of feeding contacts with an infected bug a human of age a has
experienced over his or her lifetime thus far is the product of the person’s age times the average
number of feeding contacts with infected bugs per person per year, a*infbitespy. In these
calculations, the input vector a = [2.5, 7.5, 12.5, 17.5, 22.5, 27.5, 32.5].
14. The prevalence rate pHa in humans of age a is the complement of the probability that none of
the feeding contacts with an infected bug has transmitted infection to the human, that is, pHa = 1 -
(1 - t
B
→
H
)
Ca
.
15. The number NHa of infected humans of age a is Ha*pHa, the product of the number of
humans of age a times the prevalence rate in humans of age a.
16. The average (over all ages) of the prevalence rate of infection in humans is the sum of the
number NHa of infected humans of all ages, divided by the total number of humans of all ages.
This quantity is pH1, the net result of the operation of the model through the spring and summer
seasons starting from the initial prevalence rate pH0 at the onset of spring.
17. The fraction of feeding contacts with humans, dogs and chickens in spring and summer are
calculated, using similar reasoning, in the concluding lines of model5.
Additional assumptions of the model are:
1) Transmission processes and probabilities are the same in spring and summer. The only
difference is that chickens are present in bedroom areas in spring and absent in summer.
2) T. cruzi infection does not affect significantly the vital parameters (feeding, growth,
survival, and reproduction) of hosts and bugs.
3) Susceptibility to T. cruzi infection is independent of bug stage (for fourth- and fifth-
instar nymphs and adult bugs), age and sex of mammal hosts, and ambient temperature.
4) The probability of bug infection after a single full blood meal on an infected host is
independent of host age and ambient temperature.
5) Transmission mediated by T. infestans is the sole route of infection. Vertical
transmission of T. cruzi from an infected human mother to her infant and transmission by
transfusion are ignored.
6) The probability of transmission to an uninfected human per feeding contact is
independent of (a) the intensity of bug infection [number of trypomastigote parasites (infectious
forms of T. cruzi) per bug], (b) bug stage, (c) previous bloodmeal sources, (d) absolute or relative
(per host) density of bugs, and (e) ambient temperature.
7) Host bloodmeal sources do not affect significantly the bugs’ vital parameters,
bloodmeal size, and acquisition or development of T. cruzi infection.
8) Individuals within each species of hosts, bug and parasite do not differ significantly in
their course of infection.
9) The population dynamics of hosts, bugs and parasites are not significant for the
equilibrium human prevalence of infection with T. cruzi, that is, it is sufficient to consider the
effect of the steady-state population sizes on human prevalence.
Of these assumptions, 1, 2, and 7 have supporting evidence or are a reasonable
approximation given current knowledge. Only partial evidence is available regarding assumption
6(d). A multiple linear regression (20) showed that, when T. infestans was interrupted while it
was biting, the time to the first fecal drop was inversely and significantly related to blood intake
and was directly and significantly related to how long the bug was starved before feeding and how
much the bug weighed at the start of feeding. Although previous experiments using restrained
mice or unrestrained chickens held in small boxes showed that T. infestans blood intake was
negatively density dependent, it does not necessarily follow that a feeding contact with a single
bug at high bug population density has a lower risk of transmitting infection to an uninfected
mammalian host than a feeding contact with a single bug at a low bug population density. The
relation between bug population density and the probability of transmission by a single feeding
contact with an infected bug remains to be assessed empirically. Mice have been infected with as
few as 25 T. cruzi trypomastigotes inoculated intraperitoneally (21), whereas field-collected T.
infestans had millions of infective trypomastigotes per milliliter of bug feces (22). Assumptions 3,
4, 5, and 8 are simplifications of convenience, against which there is some contradictory evidence.
For example, contrary to assumption 4, children have many more trypomastigotes in their blood
than older people with chronic infections and are therefore more infectious than adults (8). Dogs
also may be slightly more infectious in the early than in the late chronic phase (8). Assumption 9 is
appropriate for an equilibrium model applied to a chronic endemic disease transmitted by a K-
strategist insect vector living close to equilibrium abundance (3, 10).
The prevalence rates neglect the latency between entry of T. cruzi into a susceptible host
and that host’s becoming infectious, because latent periods of bugs and hosts (1 to 3 weeks) are
negligible compared to the remaining average lifetime of a dog, human or bug at the time when
each acquires the infection.
Analysis of the model
The steady-state condition pH0 = pH1 means that the human prevalence rate at the end of
summer equals the human prevalence rate at the onset of spring. This condition has exactly one
mathematical solution, so the output of the model is uniquely defined, apart from limitations of
numerical analysis in finding the mathematical solution. This steady-state solution pH is computed
approximately by numerical iteration (repeatedly replacing pH0 with pH1) by the Matlab function
iterate5 until the initial human prevalence rate pH0 at the onset of spring differs from the final
human prevalence rate pH1 at the end of summer by no more than the user-specified criterion
δ
for convergence, here taken as 0.001; i.e., the iteration continues until |pH0
−
pH1| <
δ
= 0.001.
The Matlab function chidogtable5 (for "chicken-dog table") investigates how the assumed
numbers of infected dogs and chickens affects the steady-state human prevalence of T. cruzi
infection and other characteristics of the household system, such as bug population sizes by
season. This function is the computational alternative to ethically impossible field experimentation.
Fifteen numerical values [0, 0.1, 0.2, 0.3, 0.4, 0.6, 0.8, 1.0, 1.5, 2, 2.5, 3, 4, 5, and 6] were
assigned in succession to the number of chickens C, and, for each value of C, the same 15
numerical values were assigned in succession to the number of infected dogs DI. Thus, model
outputs were produced for 225 = 15
×
15 combinations of the numbers of chickens and infected
dogs. These values for the input variables "chickens" and "dogs" were selected to display clearly
how the model behaves in the region where output variables change rapidly, rather than to reflect
the actual distribution of the numbers of chickens and infected dogs per household in the villages
studied. All other parameters remained constant at the values shown in table 1.
Sensitivity analyses
The printed paper reports a sensitivity analysis with respect to the parameter values of
Blimit, M, and R (see table 1 for definitions). In addition, four related models that differed in
detailed structure were investigated numerically. These other models differed in the absence of
seasonality, and (among the seasonal models) in the seasonality of keeping chickens indoors, in
the dependence of the bug population size on the number of vertebrate hosts in the present versus
prior season, and in other minor respects. Although the quantitative predictions of the models
varied, the major conclusions for the control of T. cruzi transmission within households were
remarkably robust.
If the model were simplified by replacing the age-structured human population with a
single age group, the human prevalence rate could be significantly overestimated. Because the
curve of the human prevalence rate as a function of age is concave in this model, using a single
age group in which the age is the average age Abar = sum(a*Ha)/H will increase the predicted
human prevalence rate as a mathematical consequence of Jensen’s inequality. Numerical
calculations not shown here indicate that ignoring age structure could introduce substantial bias.
Other related work
Two independent simulation models assessed the effects of chickens on the household
transmission of T. cruzi. In the presence of humans and dogs infected with T. cruzi, maintaining
10 or 20 chickens in domiciliary areas depressed bug and human prevalence rates of infection
substantially (23). The presence of a single brooding chicken in domiciliary areas in both spring
and summer decreased the daily rate of potentially infective feeding contacts experienced by
humans but did not significantly reduce the human incidence rate (24).
An existing model of zooprophylaxis in malaria (25) omits seasonality. A model for the
insect-transmitted African horse sickness includes seasonality (26) but deals with only two
susceptible host species that differ in infectivity and pathogen-induced mortality. In the red
grouse-hare-louping ill virus system (27), the nonviremic hosts can amplify the tick (vector)
population and cause the virus to persist or can cause the infection to die out. None of these
models made any comparison with field data.
Computing software
All numerical calculations reported here were carried out on a Dell personal computer using
MATLAB 5.2 (28). The three Matlab functions used to implement the model, namely,
chidogtable5, iterate5, model5, are listed below. The user first defines numerical values for the
variables
a, Ha, C, DI, DU, R, Blimit, Vhalf, M, t, pH0, alpha,
delta, chickens, dogs
, which are defined in the internal comments of chidogtable5, then
commands:
out=chidogtable5(a,Ha,C,DI,DU,R,Blimit,Vhalf,M,t,pH0,alpha,
delta,chickens,dogs);
The function chidogtable5 invokes the function iterate5, which in turn invokes model5. The
function iterate5 determines a steady-state human prevalence rate by iteration of model5. The
function chidogtable5 computes the steady-state human prevalence rates associated with different
combinations of numbers of chickens and dogs, and plots the results.
Function model5
function
[pH1,NHa,Bg,Br,pBg,pBr,NBg,NBr,bitespy,infbitespy,humbloodg,do
gbloodg,chibloodg,humbloodr,dogbloodr,chibloodr]=model5(a,Ha,C
,DI,DU,R,Blimit,Vhalf,M,t,pH0,alpha)
%[pH1,NHa,Bg,Br,pBg,pBr,NBg,NBr,bitespy,infbitespy,humbloodg,dogb
loodg,chibloodg,humbloodr,dogbloodr,chibloodr]=model5(a,Ha,C,D
I,DU,R,Blimit,Vhalf,M,t,pH0,alpha)
%Model of Chagas transmission in a household: 2 seasons (spring,
summer)
%Bug population size is set in previous season
%Bug prob. of acquiring infection and bug prevalence rate are set
in season of adulthood
%Suffix g = spring, r = summer
%a =vector of midpoints of age groups
%Ha=vector of number of people in each age group in household
%C =chickens
%DI=dogs infected
%DU=dogs uninfected
%R =relative attractiveness of dogs and chickens as blood meals
wrt humans
%Blimit=maximum number of bugs given unrestricted food
%Vhalf=no.of vertebrate blood sources (in human equivalents)
sufficient to support Blimit/2 bugs
%M =meals per bug per summer
%t =prob. that, in one bite by one infected bug on initially
uninfected human, human acquires infection
%pH0=initial prevalence rate of Chagas infection in humans
%alpha = fraction of blood meals that adult bugs take during
season of adulthood
%27 December 1999; 9, 14, 25, 29 March 2000
D=DI+DU; %dogs
H=sum(Ha); %humans
Vg=H+R*(D+C); %spring vertebrate blood sources, in human
equivalents, chickens indoors
Vr=H+R*D; %summer vertebrate blood sources, in human
equivalents, chickens outdoors
Bg=Blimit*Vr/(Vhalf+Vr); %spring IV-V-adult bugs in household set
by summer vertebrates
Br=Blimit*Vg/(Vhalf+Vg); %summer IV-V-adult bugs in household set
by spring vertebrates
NH=H*pH0; %initial prevalence in humans
Tg=(0.03*NH+0.49*R*DI)/Vg; %spring prob. that an initially
uninfected bug acquires infection in one blood meal
Tr=(0.03*NH+0.49*R*DI)/Vr; %summer prob. that an initially
uninfected bug acquires infection in one blood meal
pBg=1-((1-Tg)^(M*alpha))*((1-Tr)^(M*(1-alpha))); %spring
prevalence rate in spring-late-instar and adult bugs
pBr=1-((1-Tr)^(M*alpha))*((1-Tg)^(M*(1-alpha))); %summer
prevalence rate in summer-late-instar and adult bugs
NBg=Bg*pBg; %spring number of infected spring-late-
instar and adult bugs
NBr=Br*pBr; %summer number of infected summer-late-
instar and adult bugs
bitespy=(Bg*alpha+Br*(1-alpha))*M/Vg+(Bg*(1-
alpha)+Br*alpha)*M/Vr; %average number of bites per person
per year
infbitespy=(NBg/Vg+NBr/Vr)*alpha*M + (NBg/Vr+NBr/Vg)*(1-alpha)*M;
%average number of bites by infected bugs per person per year
Ca=a*infbitespy; %ave.no.times human aged a has been bitten
by infected bug
pHa=1-(1-t).^Ca; %prevalence rate in humans aged a
NHa=Ha.*pHa; %prevalence in humans aged a
pH1=sum(NHa)/H; %prevalence rate in humans
humbloodg=alpha*H/Vg+(1-alpha)*H/Vr; %human blood meals of
spring-late-instar and adult bugs
dogbloodg=alpha*R*D/Vg+(1-alpha)*R*D/Vr; %dog blood meals
of spring-late-instar and adult bugs
chibloodg=alpha*R*C/Vg; %chicken blood meals of spring-
late-instar and adult bugs
humbloodr=alpha*H/Vr+(1-alpha)*H/Vg; %human blood meals of
summer-late-instar and adult bugs
dogbloodr=alpha*R*D/Vr+(1-alpha)*R*D/Vg; %dog blood meals
of summer-late-instar and adult bugs
chibloodr=(1-alpha)*R*C/Vg; %chicken blood meals of summer-
late-instar and adult bugs
return
Function iterate5
function
[pH1,PHa,Bg,Br,pBg,pBr,NBg,NBr,bitespy,infbitespy,humbloodg,do
gbloodg,chibloodg,humbloodr,dogbloodr,chibloodr,aveage,aveinfa
ge]=iterate5(a,Ha,C,DI,DU,R,Blimit,Vhalf,M,t,pH0,alpha,delta)
%[pH1,PHa,Bg,Br,pBg,pBr,NBg,NBr,bitespy,infbitespy,humbloodg,dogb
loodg,chibloodg,humbloodr,dogbloodr,chibloodr]=iterate5(a,Ha,C
,DI,DU,R,Blimit,Vhalf,M,t,pH0,alpha,delta)
%27 December 1999, rev. 2 Jan 2000, 9, 14, 25 March 2000
pH1=pH0;
pH2=model5(a,Ha,C,DI,DU,R,Blimit,Vhalf,M,t,pH0,alpha);
while abs(pH2-pH1)>delta
pH1=pH2;
pH2=model5(a,Ha,C,DI,DU,R,Blimit,Vhalf,M,t,pH1,alpha);
end
[pH1,PHa,Bg,Br,pBg,pBr,NBg,NBr,bitespy,infbitespy,humbloodg,dogbl
oodg,chibloodg,humbloodr,dogbloodr,chibloodr]=model5(a,Ha,C,DI
,DU,R,Blimit,Vhalf,M,t,pH0,alpha);
aveage=sum(a.*Ha)/sum(Ha); %average age of humans
aveinfage=sum(a.*PHa)/sum(PHa); %average age of infected humans
return
Function chidogtable5
function
out=chidogtable5(a,Ha,C,DI,DU,R,Blimit,Vhalf,M,t,pH0,alpha,del
ta,chickens,dogs)
%out=chidogtable5(a,Ha,C,DI,DU,R,Blimit,Vhalf,M,t,pH0,alpha,delta
,chickens,dogs)
%Model of Chagas transmission in a household: 2 seasons (spring,
summer)
%Bug population size is set in previous season
%Suffix g = spring, r = summer
%a =vector of midpoints of age groups
%Ha=vector of number of people in each age group in household
%C =chickens
%DI=dogs infected
%DU=dogs uninfected
%R =relative attractiveness of dogs and chickens as feeding
contacts wrt humans
%Blimit=maximum number of bugs given unrestricted food
%Vhalf=no.of vertebrate blood sources (in human equivalents)
sufficient to support Blimit/2 bugs
%M =feeding contacts per bug per spring and summer combined
%t =prob. that, in one bite by one infected bug on initially
uninfected human, human acquires infection
%pH0=initial prevalence rate of Chagas infection in humans
%alpha = fraction of feeding contacts that late-instar and adult
bugs take during season when they become late-instar or adult
%delta=threshold for quasi-convergence of iteration: steady-state
iff |pH2-pH1|<delta
%chickens=vector of numbers of chickens to assume in household
%dogs=vector of numbers of dogs to assume in household
%out level 1 number of chickens in household
% level 2 number of dogs in household
% level 3 pH prevalence rate in humans
% level 4 Bg late-instar and adult bugs in household in spring
% level 5 Br late-instar and adult bugs in household in summer
% level 6 pBg prevalence rate in bugs in spring
% level 7 pBr prevalence rate in bugs in summer
% level 8 NBg number of infected bugs in spring
% level 9 NBr number of infected bugs in summer
% level 10 bitespy average number of bites per person per year
% level 11 infbitespy average number of bites by infected bugs
per person per year
% level 12 humbloodg fraction of feeding contacts on human blood
in spring
% level 13 dogbloodg fraction of feeding contacts on dog blood
in spring
% level 14 chibloodg fraction of feeding contacts on chicken
blood in spring
% level 15 aveage average age of humans
% level 16 aveinfage average age of infected humans
% level 17 humbloodr fraction of feeding contacts on human blood
in summer
% level 18 dogbloodr fraction of feeding contacts on dog blood
in summer
% level 19 chibloodr fraction of feeding contacts on chicken
blood in summer
%29 December 1999, rev. 2 January 2000, 25 March 2000, 7 May
2000; 23 May 2000; 6 May 2001
fnt=22; %font size for lettering of figures
fntic=18; %font size for labeling of tic marks
lc=length(chickens);
ld=length(dogs);
[chigrid,doggrid]=meshgrid(chickens,dogs); %grid for each
combination of chickens & dogs
out=cat(3,chigrid,doggrid); %first 2 layers of output
array ’out’
out=cat(3,out,zeros(ld,lc,17)); %holder for remaining
17 layers of ’out’
chibloodrmat=zeros(ld,lc);
for i=1:ld %for each combination of dogs(i) and chickens(j),
get steady-state behavior
for j=1:lc
[pH1,PHa,Bg,Br,pBg,pBr,NBg,NBr,bitespy,infbitespy,humbloodg,do
gbloodg,chibloodg,humbloodr,dogbloodr,chibloodr,aveage,aveinfa
ge]=iterate5(a,Ha,chickens(j),dogs(i),DU,R,Blimit,Vhalf,M,t,pH
0,alpha,delta);
chibloodrmat(i,j)=chibloodr;
temp=[pH1,Bg,Br,pBg,pBr,NBg,NBr,bitespy,infbitespy,humbloodg,d
ogbloodg,chibloodg,aveage,aveinfage,humbloodr,dogbloodr,chiblo
odr];
for k=1:17
out(i,j,k+2)=temp(k); %load output values into
’out’
end
end
end
%Figure 1B, bloodmealssummerbugprevrate
figure(14)
h=plot3(out(:,:,14),out(:,:,13),out(:,:,7),’k.-’);
grid
set(gca,’fontsize’,fntic)
title(’summer bug prevalence rate’,’FontSize’,fnt)
xlabel(’chicken feedings’,’FontSize’,fnt),ylabel(’dog
feedings’,’FontSize’,fnt),zlabel(’bug prevalence
rate’,’FontSize’,fnt)
print -deps -tiff bloodmealssummerbugprevrate.eps
%Figure 1D, humchibloodmealsvsummerbugs
figure(16)
x=[out(:,:,5),out(:,:,5)]; % late-instar and adult bugs in
household in SUMMER ONLY!!!!
y1=[out(:,:,12),out(:,:,17)]; %humbloodg and humbloodr
y2=[out(:,:,14),out(:,:,19)]; %chibloodg and chibloodr
h=plot(x(:),y1(:),’o’,x(:),y2(:),’x’);
set(gca,’fontsize’,fntic)
title(’spring & summer feeding contacts’,’FontSize’,fnt)
xlabel(’summer bugs in household’,’FontSize’,fnt),ylabel(’human
(o), chicken (x) feeding contacts’,’FontSize’,fnt)
%print
print -deps -tiff humchibloodmealsvsummerbugs.eps
%Figure 2A springbugsinhousehold
figure(1)
mesh(chigrid,doggrid,out(:,:,4))
colormap([0 0 0])
title(’spring bugs in household’,’FontSize’,fnt)
xlabel(’chickens’,’FontSize’,fnt),ylabel(’dogs’,’FontSize’,fnt),z
label(’spring bugs’,’FontSize’,fnt)
set(gca,’fontsize’,fntic)
%print
print -deps -tiff springbugsinhousehold.eps
%Figure 2B summerbugsinhousehold
figure(2)
mesh(chigrid,doggrid,out(:,:,5))
colormap([0 0 0])
title(’summer bugs in household’,’FontSize’,fnt)
xlabel(’chickens’,’FontSize’,fnt),ylabel(’dogs’,’FontSize’,fnt),z
label(’summer bugs’,’FontSize’,fnt)
set(gca,’fontsize’,fntic)
%print
print -deps -tiff summerbugsinhousehold.eps
%Figure 2C humanbloodmeals
figure(3)
mesh(chigrid,doggrid,out(:,:,12))
colormap([0 0 0])
title(’spring feeding contacts with humans’,’FontSize’,fnt)
xlabel(’chickens’,’FontSize’,fnt),ylabel(’dogs’,’FontSize’,fnt),z
label(’fraction of feeding contacts with
humans’,’FontSize’,fnt)
set(gca,’fontsize’,fntic)
%print
print -deps -tiff humanbloodmeals.eps
%Figure 2D springdogbloodmeals
figure(4)
mesh(chigrid,doggrid,out(:,:,13))
colormap([0 0 0])
title(’spring feeding contacts with dogs’,’FontSize’,fnt)
xlabel(’chickens’,’FontSize’,fnt),ylabel(’dogs’,’FontSize’,fnt),z
label(’fraction of feeding contacts with dogs’,’FontSize’,fnt)
set(gca,’fontsize’,fntic)
%print
print -deps -tiff springdogbloodmeals.eps
%Figure 2E prevrateinspringbugs
figure(6)
mesh(chigrid,doggrid,out(:,:,6))
colormap([0 0 0])
title(’prevalence rate in spring bugs’,’FontSize’,fnt)
xlabel(’chickens’,’FontSize’,fnt),ylabel(’dogs’,’FontSize’,fnt),z
label(’prevalence rate in bugs’,’FontSize’,fnt)
set(gca,’fontsize’,fntic)
%print
print -deps -tiff prevrateinspringbugs.eps
%Figure 2F infectedspringbugs
figure(8)
mesh(chigrid,doggrid,out(:,:,8))
colormap([0 0 0])
title(’infected spring bugs’,’FontSize’,fnt)
xlabel(’chickens’,’FontSize’,fnt),ylabel(’dogs’,’FontSize’,fnt),z
label(’number of infected bugs’,’FontSize’,fnt)
set(gca,’fontsize’,fntic)
%print
print -deps -tiff infectedspringbugs.eps
%Figure 2G infectedsummerbugs
figure(9)
mesh(chigrid,doggrid,out(:,:,9))
colormap([0 0 0])
title(’infected summer bugs’,’FontSize’,fnt)
xlabel(’chickens’,’FontSize’,fnt),ylabel(’dogs’,’FontSize’,fnt),z
label(’number of infected bugs’,’FontSize’,fnt)
set(gca,’fontsize’,fntic)
%print
print -deps -tiff infectedsummerbugs.eps
%Figure 2H feedingsperhumanyear
figure(10)
mesh(chigrid,doggrid,out(:,:,10))
colormap([0 0 0])
title(’feeding contacts per human’,’FontSize’,fnt)
xlabel(’chickens’,’FontSize’,fnt),ylabel(’dogs’,’FontSize’,fnt),z
label(’feeding contacts per human’,’FontSize’,fnt)
set(gca,’fontsize’,fntic)
%print
print -deps -tiff feedingsperhumanyear.eps
%Figure 2I humanprevalencerate
figure(12)
mesh(chigrid,doggrid,out(:,:,3))
colormap([0 0 0])
title(’human prevalence rate’,’FontSize’,fnt)
xlabel(’chickens’,’FontSize’,fnt),ylabel(’dogs’,’FontSize’,fnt),z
label(’human prevalence rate’,’FontSize’,fnt)
set(gca,’fontsize’,fntic)
%print
print -deps -tiff humanprevalencerate.eps
return
Web table 1. Parameters, calculated variables and descriptive statistics of the model.
Symbol Meaning Formula or illustrative value in
base case at equilibrium
a Age groups of humans in household (vector of midpoints of ages,
measured in years)
[2.5 7.5 12.5 17.5 22.5 27.5
32.5]
Ha Number of humans in each age group (vector of values) [1 1 1 0 0 1 1] (meaning 1
person aged 2.5 years, 1 aged
7.5 years, 1 aged 12.5 years,
etc.)
H Number of humans in household sum of Ha
Ha = 5 people
DI Number of dogs infected with T. cruzi DI = 2 infected dogs
DU Number of dogs uninfected with T. cruzi DU = 0 uninfected dogs
D Number of household dogs DI + DU
D = 2 dogs
C Number of household chickens C = 2 chickens
R Relative feeding index of dogs and chickens as sources of feeding
contacts compared to humans
R = 3 dog or chicken
feedings per human feeding
Vg, Vr Number of vertebrate blood sources, in human equivalents, in
spring (Vg) and summer (Vr)
Vg = H+R*(D+C), Vr =
H+R*D
Vg = 17 vertebrates in
spring,
Vr = 11 vertebrates in
summer
Blimit Maximum number of fourth- and fifth-instar nymphs and adult
bugs the physical infrastructure of the house will support, given
an unlimited food supply
Blimit = 500 large and late-
stage bugs
Vhalf Number of vertebrate blood sources (in human equivalents)
sufficient to support Blimit/2 bugs
Vhalf = 7 vertebrates
Bg, Br Number of fourth- and fifth-instar nymphs and adult bugs in
spring (Bg) and summer (Br) that can be supported at steady-
state by the vertebrate blood sources in the household
Br=Blimit*Vg/(Vg + Vhalf)
Bg=Blimit*Vr/(Vr + Vhalf)
Br = 354.2 bugs in summer,
Bg = 305.6 bugs in spring
M Number of feeding contacts per fourth- or fifth-instar nymph or
adult bug per spring and summer combined
M = 5 feeding contacts per
bug per summer and spring
pH0 Human prevalence rate in late winter (fraction of all humans
infected with T. cruzi)
initial pH0=0.8
pH0=pH1 at equilibrium
t
B
→
H
Transmission probability to human = probability that, in one
feeding by one infected bug on an initially uninfected human, the
human acquires infection
t
B
→
H
= 0.0008
t
H
→
B
Probability that, in one feeding by an initially uninfected bug on
an infected (seropositive) human, the bug acquires infection
t
H
→
B
= 0.03
t
D
→
B
Probability that, in one feeding by an initially uninfected bug on
an infected (seropositive) dog, the bug acquires infection
t
D
→
B
= 0.49
Tg, Tr Transmission probability to bug in spring (Tg) and summer (Tr)
= probability that, in one blood meal, an initially uninfected bug
acquires infection
Tg=(0.03*NH +
0.49*R*DI)/Vg,
Tr=(0.03*NH +
0.49*R*DI)/Vr
Fraction of feedings a bug takes in the season the bug is in instar
4 or 5 or adult
pBg, pBr Bug prevalence rates in spring (pBg) and summer (pBr)
(fractions of bugs infected with T. cruzi in spring and summer)
pBr=1-((1-Tr)^(
pBg=0.6839 in spring
pBr=0.7702 in summer
NBg, NBr Number of infected bugs in spring (NBg) and summer (NBr) NBg=Bg*pBg, NBr=Br*pBr
NBg=209.0 infected bugs in
spring, NBr=272.8 infected
bugs in summer
Bitespy Number of feeding contacts per human per year (
( Vr
Bitespy=248.9 feeding
contacts per human per year
Infbitespy Number of potentially infective feeding contacts per human per
year
(NBg/Vg+NBr/
(NBg/Vr+NBr/
Infbitespy=182.9 infective
feeding contacts per human
per year
Ca Contacts by age a = average number of times a human in each
age-group a has had a feeding contact with an infected bug
(vector of values)
a*Infbitespy
pHa Prevalence rate in humans in each age-group a (vector of
values)
pHa = 1-(1-t
B
→
H
)^Ca
Nha Expected number of infected humans in each age-group a
(vector of values)
Ha*pHa
NH Total number of infected humans Sum of Nha
pH1 Human prevalence rate calculated through one cycle of the model pH1 = NH/H
pH1=0.7572 at end of
summer if pH0=0.8 at onset
of spring
Humblood
g
Fraction of feeding contacts from humans in bugs collected in
spring
Humbloodg
Vr
Dogbloodg Fraction of feeding contacts from dogs in bugs collected in
spring
Dogbloodg
Vr
Chibloodg Fraction of feeding contacts from chickens in bugs collected in
spring
Chibloodg
Humbloodr Fraction of feeding contacts from humans in bugs collected in
summer
Humbloodr = Vr + (1-
Dogbloodr Fraction of feeding contacts from dogs in bugs collected in
summer
Dogbloodr = Vr + (1-
Chibloodr Fraction of feeding contacts from chickens in bugs collected in
summer
Chibloodr
Abar Average age of humans (years) Abar = Sum(a*Ha)/H
Abar = 16.5 years old
AI Average age of infected humans (years) AI = Sum(a*Nha)/NH
AI = 19.9391 years old
References
1. E. H. Bucher, C. J. Schofield, New Scientist 92, 321 (1981).
2. M. A. Miles, Trans. R. Soc. Trop. Med. Hyg. 70, 521 (1977).
3. R. Zeledón, J. E. Rabinovich, Annu. Rev. Entomol. 26, 101 (1981).
4. K. E. Mott et al., Am. J. Trop. Med. Hyg. 27, 1116 (1978).
5. R. E. Gürtler, M. C. Cécere, D. P. Vazquez, R. Chuit, J. E. Cohen, J. Med. Entomol. 33, 15
(1996).
6. R. E. Gürtler, J. E. Cohen, M. C. Cecere, R. Chuit, E. L. Segura, Am. J. Trop. Med. Hyg. 58,
748 (1998).
7. M. C. Cecere, M. B. Castañera, D. M. Canale, R. Chuit, R. E. Gürtler, Pan Am. J. Publ. Health
5, 392 (1999).
8. R. E. Gürtler et al., Am. J. Trop. Med. Hyg. 55, 24 (1996).
9. D. M. Minter, in American Trypanosomiasis Research, PAHO Sci. Publ. 318 (Washington
D.C., 1976), pp. 330-337.
10. C. J. Schofield, Ann. Soc. Belg. Med. Trop. 65 (Suppl. 1), 149 (1985).
11. M. C. Cecere, thesis, University of Buenos Aires (1999).
12. M. C. Cecere, R. E. Gürtler, D. Canale, R. Chuit, J. E. Cohen, Bull. WHO 76, 373 (1998).
13. R. A. Ronderos, J. A. Schnack, J. M. Ghilini, G. R. Spinelli, Ecosur 8, 1 (1981).
14. P. D. Marsden et al., Rev. Inst. Med. Trop. Sao Paulo 21, 13 (1979).
15. J. E. Rabinovich, P. H. Himschoot, Ecol. Model. 52, 249 (1990).
16. R. E. Gürtler, J. E. Cohen, M. C. Cecere, R. Chuit, J. Appl. Ecol. 34, 699 (1997).
17. J. E. Rabinovich, C. Wisnivesky-Colli, N. D. Solarz, R. E. Gürtler, Bull. WHO 68, 737
(1990).
18. J. E. Rabinovich, O. Rossell, in American Trypanosomiasis Research, PAHO Sci. Publ. 318
(Washington, D.C., 1976), pp. 359-369.
19. D. W. Kelly, C. E. Thompson, Parasitology 120, 319 (2000).
20. E. V. Trumper, D. E. Gorla, Trans. R. Soc. Trop. Med. Hyg. 85, 800 (1991).
21. P. Cabeza-Meckert, J. G. Chambo, R. P. Laguens, Agents Chemother. 32, 241 (1988).
22. L. C. Giojalas, S. S. Catalá, S. N. Asin, D. E. Gorla, Trans. R. Soc. Trop. Med. Hyg. 84, 439
(1990).
23. P. H. Himschoot, thesis, University of Buenos Aires (1993).
24. M. B. Castañera, thesis, University of Buenos Aires (1999).
25. T. Sota, M. Mogi, Med. Vet. Entomol. 3, 337 (1989).
26. C. C. Lord, M. E. J. Woolhouse, J. A. P. Heesterbeek, P. S. Mellor, Med. Vet. Entomol. 10,
19 (1996).
27. R. Norman, R. G. Bowers, M. Begon, P. J. Hudson, J. Theor. Biol. 200, 111 (1999).
28. MathWorks Inc., Matlab 5.2 Product Family New Features (Natick, MA, 1998).
