-
Ecological Modelling, 52 (1990) 249-266 249 Elsevier Science
Publishers B.V., Amsterdam
A population-dynamics simulation model of the main vectors of
Chagas' Disease transmission,
Rhodnius prolixus and Triatoma infestans
Jorge E. Rabinovich 1 and Patricia Himschoot
Instituto Nacional de Diagnbstico e Investigacibn de la
Enfermedad de Chagas "Dr. Mario Fatala Chab~n" Paseo Colbn 568
(piso 6), 1063 Buenos Aires (Argentina)
(Accepted 2 February 1990)
ABSTRACT
Rabinovich, J.E. and Himschoot, P., 1990. A population-dynamics
simulation model of the main vectors of Chagas' Disease
transmission, Rhodnius prolixus and Triatoma infestans. Ecol.
Modelling, 52: 249-266.
A simulation model was developed to represent the population
dynamics of two bugs (Triatoma infestans and Rhodnius prolixus),
which are the main vector species of Chagas' Disease in some Latin
American countries. The model includes the infection process of
bugs with the parasite causing the disease (Trypanosoma cruzi).
Human and animal hosts for bugs (with a preference factor for each)
were considered; bug migration between houses and between houses
and the wild environment was also modelled; the three types of
houses most common in rural areas were considered, The bugs'
population regulation mechanisms were modelled acting over natality
and mortality. The population parameters of the two bug species
that had no independent actual estimation were calibrated with a
field series available only for R. prolixus.
Populations of both species showed sustained oscillations under
certain conditions, with about two cycles per year for T. infestans
and about four cycles per year for R. prolixus, and proportional to
the length of each species' development time. Development-stage
class distribution alternations are associated with these
oscillations; adults are most abundant when population size is at
its minimum. The symmetry of the fluctuation seems to be affected
by female fecundity. Sensitivity analyses show that the main
parameters affecting the stability behavior of the vector
populations were: (a) the threshold nymphal density at which
irritation of the host starts, (b) female fecundity, (c) number of
hosts available, and (d) the emigration rate. The threshold nymphal
density at which irritation of the host starts, combined with
female fecundity, proved critical in producing a change in the
stability behavior of the population, from a stable point
equilibrium to a limit cycle. These results, if incorporated into
transmission control models, can be used to optimize control
strategies.
1 Present address: Departamento de Biologia, Facultad de
Ciencias Exactas y Naturales, Universidad de Buenos Aires, Arenales
3844(5B), 1425 Buenos Aires, Argentina.
0304-3800/90/$03.50 © 1990 - Elsevier Science Publishers
B.V.
-
250 J.E. R A B I N O V I C H A N D P. H 1 M S C H O O T
I N T R O D U C T I O N
Chagas' Disease is one of the most serious current health
problems in Latin America, affecting about 16 to 18 million people,
of whom approxi- mately 2.5 million are found in Argentina
(Moncayo-Medina, 1987). Diffi- culties in its control arise from
the ecological characteristics of the insect vectors and the
complex socioeconomic aspects of the human population at risk. The
parasite causing this disease (Trypanosoma cruzi) is transmitted by
two main vector species, Triatoma infestans and Rhodnius prolixus
(Reduvi- idae: Triatominae).
There are several mathematical models of the population dynamics
of these two vector species: life-table models were applied to T.
infestans (Rabinovich, 1971; Rabinovich and Dorta, 1973; Rodriguez,
1977), and a life-cycle model was developed for R. prolixus
(Soriano and Luis, 1977) as well as a logistic model (Rabinovich
and Rossell, 1976; Rabinovich, 1985). These models are strong in
dynamic relationships and parameter estimation, but poorly reflect
the actual environmental conditions of epidemiological importance,
such as the number of bug hosts and their type (human or animal),
and the type of dwelling the people inhabit.
We present here the results of a Chagas' Disease vector
population-dy- namics simulation model that deals with the above
weaknesses by incorpo- rating the following features: (a) hosts
differentiated as animal and human, with insects showing a certain
preference for each; (b) density-dependent vector population
regulation acting independently through bug mortality and natality;
(c) different kinds of houses; and (d) migration of bugs occurring
to and from houses, and to and from wild refuges. The model was
used to analyze the stability behavior of the vector population,
and to discuss its relevance to the epidemiology of the disease.
For the latter purpose the dynamics of the infection in the vectors
had to be modelled. This is the first case of a mathematical model
of Chagas' Disease transmis- sion where calibration was performed
with a set of data available for R. prolixus.
D E S C R I P T I O N OF THE M O D E L
The spatial unit was the human dwelling, the time unit was one
day and the time horizon 20 years. The species modelled were R.
prolixus and T. infestans, and their population dynamics were
simulated separately. Figure 1 shows the basic processes and
components considered in this model. The basic assumptions for
developing the model are given in Table 1. Figure 2 shows the
relationships between variables and factors of the three main
processes modelled: natality, mortality and migration; their
mathematical
-
VECTORS OF CHAGAS" DISEASE TRANSMISSION 251
N -° OF PEOPLE } - -
[ N~- OF ANIMALS I
ITYPE OF H O U S E ~
N -° OF VECTORS PER HOST
I BITIN 1 |FREQUENCY
NON-INFECTED ~ VECTORS
I ITVPEOFHOOSEI EMIGRATION I I MORTALITVl [EMIGRATIO~
BLOOD ] PREFERENCE
INDEX
l INFECTED l ) VECTORS
PROPORTION OF INFECTED HOSTS
I IM IORATIONt
Fig. 1. Block diagram of the basic components and processes of
the population-dynamics model of two Chagas' Disease vectors: R.
prolixus and T. infestans. It is applicable to each species
separately, with the only difference that palm-house immigration
does not occur in T. infestans.
relationships are shown in Table 2. The hosts were
differentiated as human and animal populations, with stable numbers
of 10 and 17, respectively (the latter constituted by 14 chickens
and three dogs from old houses with R. prolixus (Rabinovich et al.,
1980) and nine people and three dogs for houses with T. infestans
(Wisnivesky, personal communication, 1988). A biting- preference
index for blood from these hosts was used (Wisnivesky-Colli, 1987):
90% and 30% preference for humans for R. prolixus and T. infestans,
respectively, where the index expresses percent of meals on humans
with respect to total meals (human and animal).
Three kinds of houses were considered: (a) type 1, a typical
shack of mud walls and palm or branch roof; (b) type 2, a house
with mud walls and tin roof; and (c) type 3, a dwelling built of
bricks and cement. As the bugs hide in the roof and wall cracks,
the number of refuges available decreases from type 1 to 3. The
vector carrying-capacity of each type of house (Table 2) is related
to the amount and kind of refuges available.
The vector's population-density regulation mechanism is a
food-limited process but responds to accessibility of the hosts and
not to their numbers (Schofield, 1980a, b, 1982; Weir-L6pez, 1982;
Rossell, 1984). The higher the number of insect bites per person
per night, the higher the irritation
-
252 J.E. R A B I N O V I C H A N D P. H I M S C H O O T
produced to the human and animal host. Due to this irritation,
and the victims' response to it, the bug's blood ingestion is
interrupted, producing: (a) a reduction in fecundity (fewer eggs
per female per day); (b) an increase in mortality due to higher
predation by natural enemies (chickens, lizards,
J
@.
(~) variables (numbers that change during the simulation)
[ ] parameters (constant numbers or initial value of a
variable)
[ ~ constant coefficients (constant rates or proportions)
Eg variable coefficients (variable rates or proportions)
~ output (variables used as results for statistics)
flux (a variable changes by the influence of another variable,
with the latter being modified after the change)
- - ~ effect (a variable changes by the influence of another
variable, with the latter remaining unaffected)
-
V E C T O R S OF C H A G A S " D I S E A S E T R A N S M I S S I
O N 253
spiders) , a s soc ia ted with a h igher n u m b e r of feeding
a t t e m p t s ; and (c) an increase in m o r t a l i t y due to s
ta rva t ion . These three m e c h a n i s m s are repre- sen ted a
lgebra ica l ly in T a b l e 2. M e c h a n i s m s (a) and (c) d e
p e n d u p o n a th resho ld value: the n u m b e r of insects pe
r p e r s o n per night tha t t r iggers an i r r i ta t ive
response ; this va lue was ini t ial ly cons ide red to be 100 bi
tes pe r p e r s o n per night. Ho weve r , it was a s s u m e d
tha t the effect of r educed feeding due to i r r i ta t ion of the
host would be felt m o r e r ap id ly for (a) than for (c); thus,
the th resho ld was kep t at 50 bi tes pe r p e r s o n per night
for the effect u p o n fecundi ty . T h e f o r m of the func t ion
was the s ame for (a) and (c): a s t ra igh t l ine wi th the s a m
e in te rcep t wi th the x-axis .
T h e s t a rva t ion func t ion ( m e c h a n i s m b) has an
'S ' shape wi th two cri t ical values: the base m o r t a l i t y
a n d a very s teep ver t ical b ranch . T h e base line had a d i
f fe ren t va lue for n y m p h s and adults , and also d i f fe
red for R. prolixus a n d T. infestans, a n d the s lope was ca l
ib ra t ed to regula te the
Fig. 2. Diagram of variable and parameter relationships, for one
unit day of simulation, of the Chagas' Disease vector
population-dynamics simulation model. Names in capital letters and
between parentheses are variables, parameters and coefficients from
Table 2. Interpreta- tion of figures: 1, Total number of bugs at
day 1 of simulation; 2, Total number of non-infected adult bugs at
day 1 of simulation; 3, Total number of infected adult bugs at day
1 of simulation; 4, Total number of adults at day 1 of simulation;
5, Total number of non-infected nymphs at day 1 of simulation; 6,
Total number of infected nymphs at day 1 of simulation; 7, Total
number of nymphs at day 1 of simulation; 8, Starvation
density-depen- dent mortality coefficient (XMN and TMA for nymphs
and adults, respectively); 9, Total number of dead adults from
starvation; 10, Total number of dead nymphs from starvation; 11,
Total number of emigrating adults (NE); 12, Emigration coefficient
(TE); 13, Total number of hosts; 14, Total number of immigrating
adults (NI); 15, Fraction of emigrating adults that will immigrate
to other houses (FR); 16, Number of passive immigrating adults
(associated to social visiting rate, see Table 2); 17, Number of
active immigrating adults (associated with palm emigration rate,
see Table 2); 18, Oviposition (number of eggs laid) (NH); 19,
Natality coefficient (TNATCH); 20, Female proportion (PH); 21,
Density-dependent factor affecting natality (FN); 22, Hatching
coefficient (EM); 23, Number of nymphs emerging from hatched eggs
(NNE); 24, Carrying capacity (Kj); 25, Basic adult mortality (MBa);
26, Basic nymph mortality (MBn); 27, Total number of dead adults by
predators and natural mortality; 28, Total number of dead nymphs by
predators and natural mortality; 29, Predator and natural mortality
coefficient (MNN and MNA for nymphs and adults, respectively); 30,
Number of nymphs moulting to adults; 31, Total number of adults at
day 2 of simulation; 32, Total number of nymphs at day 2 of
simulation; 33, Total number of bugs at day 2 of simulation; 34,
Total number of non-infected adults at day 2 of simulation; 35.
Total number of infected adults at day 2 of simulation; 36,
Fraction of infected adults; 37, Total number of non-infected
nymphs at day 2 of simulation; 38, Total number of infected nymphs
at day 2 of simulation; 39, Daily biting rate (FP); 40, Fraction of
infected nymphs; 41, Total number of infected hosts (PAR); 42,
Parasitaemic proportion of chronic human hosts; 43, Number of
parasitaemic animal hosts; 44, Number of parasitaemic human hosts;
45, Total number of infected bugs.
-
254 J.E. RABINOVICH AND P. HIMSCHOOT
TABLE 1
Implicit and explicit assumptions and simplifications used in
the Chagas' Disease vector population-dynamics simulation model, in
addition to the algebraic ones presented in Table 2
Explicit (1) The bug population is composed of eggs, an immature
stage (nymphs of any of the five
instars), and adults (2) The developmental times of eggs and of
the immature stage are considered constant (3) Adult bugs have a
pre-reproductive time of 10 days for both species (4) The sex-ratio
of the adult bug population is 50 : 50 (5) Only adult bugs migrate
(6) All non-human hosts have a constant population growth-rate of
10% per year (7) The host populations are considered to remain
constant after reaching ten, 14 and three
people, chickens, and mammals per house, respectively (8) The
bugs feed selectively upon animals and humans; the preference index
for humans is
90% and 30% for R. prolixus and T. infestans, respectively; for
definition of the index see text.
Implicit (1) Climatic effects and seasonality are not considered
(2) There are no individual differences between insects of the same
kind (3) The T. cruzi parasite has no effect upon mortality,
reproduction and migration of the
insect vector. (4) Probability of infection of bugs is the same
for nymphs and adults (5) The density-dependent functions that
involve the irritation of the vertebrate hosts are the
same for any kind of host
population at the average carrying capacity determined by the
type of house (2100, 700, and 120 for R. prolixus for house types
1, 2, and 3, respectively; 900, 350, and 100 for T. infestans for
house types 1, 2, and 3, respectively). The values of the
carrying-capacity-related constants that keep average populations
at those levels are shown in Table 2.
Field data (Rabinovich et al., 1980) showed that both emigration
and immigration were density-independent, and that emigration
occurred at a daily constant rate given as a fixed proportion of
the adult population size (Table 2). Immigration has two sources;
passive and active; the former takes place when the insects are
carried by people, and the latter occurs when adult insects
disperse by flight; this active immigration can originate from
other houses of the same village, or from wild environments (mainly
palms in tropical areas).
When density-dependent regulation parameters (FN1, FN2, Fsl,
FS2, FS3, TEl, TM1, TM2 and TNATCH) had no field or experimental
estimates available, a numerical calibration experiment was
performed that produced parameter
-
TA
BL
E 2
V
ecto
r's
func
tion
s an
d a
lgor
ithm
s us
ed i
n th
e m
odel
(al
l ra
tes
are
in d
ay u
nits
, ex
cept
wh
en i
nd
icat
ed)
Pro
cess
F
un
ctio
n
Par
amet
er o
r va
riab
le
Un
it
Val
ue f
or
Val
ue f
or
Sou
rce
R.
prol
ixus
T.
inf
esta
ns
.r
>
Dev
elo
pm
ent
DT
= E
DT
+ N
DT
D
T
Tim
e fr
om e
gg l
aid
to
days
90
17
0 R
abin
ov
ich
(19
72),
ti
me
mat
ure
fem
ale
Ro
dri
gu
ez (
1977
) E
DT
T
ime
fro
m e
gg l
aid
days
15
20
R
abin
ov
ich
(19
72),
to
egg
hat
ched
R
od
rig
uez
(19
77)
ND
T
Tim
e fr
om
egg
da
ys
75
150
Rab
ino
vic
h (
1972
),
-q
hat
ched
R
od
rig
uez
(19
77)
to m
atu
re f
emal
e F
ecu
nd
ity
N
H =
TN
AT
CH
* F
N
FN
Nat
alit
y f
acto
r -
var
iab
le
var
iab
le
- •
VA
* P
H
FNI
Th
resh
old
den
sity
at
bu
gs/
ho
st
50
50
cali
bra
tio
n
FN =
[F
N2
- w
hich
irr
itab
ilit
y (D
/AL
)]
star
ts
/(F
N2
-- F
N1)
Hat
chin
g
NN
E
=
NH
* (1
-
EM
) E
°T
FN2
Den
sity
at
whi
ch
bu
gs/
ho
st
1000
10
00
cali
bra
tio
n
rep
rod
uct
ion
sto
ps
D
Den
sity
b
ug
s/h
ou
se
var
iab
le
vari
able
-
AL
F
oo
d s
ourc
es
No.
of
host
s v
aria
ble
v
aria
ble
-
NH
N
o. o
f eg
gs l
aid
eggs
v
aria
ble
va
riab
le
- T
NA
TC
H
Nat
alit
y r
ate
egg
s/fe
mal
e 1.
3 0.
77
Rab
ino
vic
h (
1972
) v
n
Ad
ult
po
pu
lati
on
b
ug
s/h
ou
se
var
iab
le
vari
able
-
Pn
Sex
rat
io (
~/a
du
lts)
-
0.5
0.5
Rab
ino
vic
h (
1981
) N
NE
Nu
mb
er e
ggs
egg
s/h
ou
se
vari
able
va
riab
le
- p
rod
uci
ng
n
ym
ph
s EM
D
aily
egg
mo
rtal
ity
-
0.00
3 0.
008
Rab
ino
vic
h (
1972
),
coef
fici
ent
Ro
dri
gu
ez (
1977
)
To
be c
onti
nued
...
-
to
o~
TA
BL
E 2
(co
ntin
ued)
Pro
cess
F
unct
ion
Par
amet
er o
r va
riab
le
Uni
t V
alue
for
V
alue
for
S
ourc
e R
. pro
lixu
s T.
inf
esta
ns
Nat
ural
M
NA
= ~
a M
NA
A
dult
mor
tali
ty r
ate
- va
riab
le
vari
able
-
mor
tali
ty
+ (1
-
r~m
a)
MB
a B
asic
adu
lt m
orta
lity
-
0.00
7 0.
001
Rab
inov
ich,
(i
nclu
des
/[1
+ (K
j/D
) 2°
] ra
te
(unp
ubli
shed
) pr
edat
ion)
M
NN
=
MB
n
+(1
+~
m.)
/[
1 +
( KJD
) 2°
]
Sta
rvat
ion
TMA
= [
FS2-
- m
orta
lity
(
D/A
L)]
/(rs
2-
Fsl
)
TMN
= [
FS
2-
(D/A
L)]
/(
FS
3 --
FS2
)
MN
N
MB
n
Kj
J TMA
rsl
FS2
TMN
FS3
Ny
mp
h m
orta
lity
rat
e
Bas
ic n
ymph
m
orta
lity
rat
e C
arry
ing
capa
city
- re
late
d co
nsta
nt
Hou
se t
ype
Adu
lt s
tarv
atio
n m
orta
lity
rat
e
vari
able
va
riab
le
0.00
9 0.
005
bu
gs/
ho
use
30
00j =
1
1300
j_ 1
60
0j=
2 50
0j=
2 15
%3
15%
3 1
,2,3
1
,2,3
-
vari
able
va
riab
le
Rab
inov
ich
(u),
R
abin
ovic
h (1
972)
R
abin
ovic
h (u
npub
lish
ed)
Thr
esho
ld d
ensi
ty a
t b
ug
s/h
ost
20
0 20
0 ca
libr
atio
n w
hich
irr
itat
ion
star
ts
Den
sity
at
whi
ch
bu
gs/
ho
st
1000
10
00
cali
brat
ion
ther
e is
no
surv
ival
N
ym
ph
sta
rvat
ion
- va
riab
le
vari
able
-
mor
tali
ty r
ate
Sam
e as
FS
1 bu
t fo
r b
ug
s/h
ost
10
0 10
0 ca
libr
atio
n ny
mph
s
-
Mig
rati
on
Vec
tor
infe
ctio
n
NI
=
(D
*T
E*
F
R)
+ (F
Vi *
PC
/360
) +
IP
NE
=
D
* T
E
PI
=
PP
* P
AR
* F
P
NI
TE
FR
Fv~
PC
IP
i NE
PI
PP
PA
R
FP
Imm
igra
ting
adu
lts
adu
lts/
ho
use
va
riab
le
Hou
se e
mig
rati
on r
ate
- 0.
15
Fra
ctio
n of
em
igra
nts
- th
at r
etu
rn
No.
of
bugs
per
vis
it
-
Soc
ial v
isit
ing
rate
P
alm
em
igra
tion
rat
e C
om
mu
nit
y ty
pe
Em
igra
ting
adu
lt
Pro
port
ion
of
bugs
tha
t be
com
e in
fect
ed
Bug
inf
ecti
on
prob
abil
ity
afte
r b
itin
g a
n
infe
cted
hos
t In
fect
ed h
osts
B
itin
g ra
te
vis
its/
yea
r b
ug
s/d
ay
adul
ts/h
ouse
host
s/ho
use
bit
es/n
igh
t
0.25
0.5
i =
2
0i-
3
6 1 1,2
,3
vari
able
va
riab
le
0.5
vari
able
0.
1
vari
able
0.
15
0.25
0.5i
= 2
0~
= 3
6 0 1,2
,3
vari
able
va
riab
le
0.5
vari
able
0.
1
Rab
inov
ich
et a
l.
(198
0)
Rab
inov
ich
(198
1)
Rab
inov
ich
(198
1)
Rab
inov
ich
(198
1)
Rab
inov
ich
(198
1)
scen
ario
Sch
enon
e et
al.
(1
977)
Rab
inov
ich
et a
l.
(197
9)
-
258 J.E. R A B I N O V I C H A N D P. H I M S C H O O T
values satisfying the best fit between simulated results and an
observed population-density time-series for R. prolixus (Rabinovich
et al., 1980).
The behavior of the vector population size with time as a
function of several variables and parameters was examined through a
program designed to perform a sensitivity analysis. Four variables
and parameters were selected as the ones that had an apparent
sensitive effect on the population dy- namics, because they appear
in a multiplicative or exponential form in the dynamic equations of
the model: the mortality density-dependent parameter (FS3), the two
basic mortality rates (TN1 and TN2), and the fecundity rate
(TNATCH). For the sensitivity analyses, three values were assigned
to each parameter: the standard one given in Table 2, plus values
50% above and below it; the parameter changes were applied in all
possible combinations simultaneously (34= 81 combinations). Two
indicators of sensitivity were used as output of 30-year
simulations for each combination: the vector average stable
population and its standard deviation. They were calculated by
averaging the daily total population of vectors per year, and then
computing the mean of means, and its standard deviation, for the
last ten years of simulation. Only eggs were not included as part
of the total population averages. The results of the 81 runs were
subjected to a stepwise regression analysis, with the model
parameters as independent variables and the two indicators of
sensitivity as dependent variables. The standardized partial
coefficients of regression were used to determine the relative
impor- tance of each independent variable in terms of the
explanation of the variance of the dependent variables.
The model was developed in a MICROVAX II system, and the
complete model took about 1 min to run for a 30-year simulation in
one house.
RESULTS
Effects of the initial conditions upon the final stable results
proved to be unimportant; as a consequence, simulations were
performed using the following arbitrary initial conditions: 5 eggs,
65 nymphs and 5 adults for R. prolixus, and 80 eggs, 60 nymphs and
25 adults for T. infestans.
The calibration simulation provided the values of the
density-dependent parameters FN1, FN2, Fsl, FS2, and FS3 for R.
prolixus (Fig. 3), and are given in Table 2. As there was no field
information for T. infestans, the same values were assigned to this
species.
The effects of different house types for 10 years of simulation
are shown in Fig. 4 for T. infestans and R. prolixus; the bug
populations fluctuate at the average levels of the carrying
capacities of each house type. Both species display sustained
oscillations; T. infestans shows about two cycles per year,
-
VECTORS O F C H A G A S ' D I S E A S E T R A N S M I S S I O
N
(X lO0)
..J
6 0 I 2 3 4 ,5 6
T I M E ( M O N T H S )
Fig. 3. S imula ted ( b r o k e n line) and field (sol id line)
p o p u l a t i o n s o f Rhodnius prolixus.
259
( X IO0)
, o ........ ! - - " .~ ....
6 . . . . . . ! . . . . . . . . i . . . . . . . . . i . . . . .
. . . .
4 . . . . . . . . . . . . . . . . ! . . . . . . . . . 4 . . . .
. . . . .
a . . . . . ! . . . . . . . . ; . . . . . . . . i . . . . . . .
. .
~ o i . . . . (..9 6 ......... i . . . . . . . . i . . . . . . .
. . . ', . . . . . . . . ~ ) , , ,
4 . . . . . . ! . . . . i . . . . . . " . . . . . .
,, 2 ...... i ........ ~ ......... ~ ....... o
0 , . , . . . . . . . . . . . . . .
m oF . . . . . . . i ~ " , 1 0 I 2 4
Z ( X I 0 0 0 )
. j (X I 00 )
I - - 8
6 . . . . . . . i . . . . . . . ~ . . . . . . . . . . . .
4 . . . . . . . . ~ . . . . . . . . . . . . . . . . . . . . . .
. .
2 . . . . . . . . - . . . . . . . . . . . . . . . . . . . . . .
. . . o51
0 ~ , t O 0 2 0 0 3 0 0 4 0 0
(X I 0 0 0 )
3, ' i . . . . i . . . . i i:i i
1 . 5 . . . . . . -, . . . . . . . . . ~ . . . . . . . . . , . .
. . . . .
0 . 5
~ _ _ ~ . ! : ~ b z I . . . . . . . . . . ; . . . . . . . . . ~
. . . . . . . . . ~- . . . . . . . . .
oL , , ' . . . . i , , , ! ,.
O r ;
0 I 2 3 4
(X IO00)
( X l O 0 0 )
2.5 ,, ; ; .
1.5
' 212221121111 211122211111-ii-- 0.5
0 IOO 2 0 0 3 0 0 4 0 0
T I M E ( D A Y S )
Fig. 4. Popu la t i on osc i l la t ions of s imu la t ed
Triatoma infestans (a) and Rhodnius prolixus (b) popu la t ions .
Sub ind ices 1, 2, and 3 c o r r e s p o n d to house types 1, 2,
and 3, respect ively; sub ind ices 4 a n d 5 c o r r e s p o n d to
n y m p h a l a n d adul t popu la t i ons , respect ively, a f ter
day 3000 ( tha t is, af ter r each ing s table osci l la t ions)
for house type 1.
-
260 J .E. K A B I N O V I C H A N D P. H I M S C H O O T
[ - 1 5 I e4 I
[ 4 z I
F I ~ I 41 I
I 54
[13 I 39 I
I1 2
49.
[ e5
~ 2 2 l I 96 I
I
I 96 I
I s 5 I I ~ 1 I 9o I
n 3 I 6 5 I
I ' " 3 2 J 1 7 0 I
['-'] 8 I 5 2 I
1 4 , J
Fig. 5. Developmental-stage pyramids for Triatoma infestans
(left side) and Rhodnius prolixus (right side). The base represents
a pool of nymphs I, II, III and IV, the intermediate section,
nymphs V, and the upper section the adult population. The numbers
associated with each section are the percentage of the respective
developmental-stage class in the total population. Each pyramid
shows (from the top to the bottom) the population structure every
30 days after entering the tenth year of simulation.
. s t . . . . I . . . . I . . . . I ' ' i d~1 4 . . . . . . J .
. . . . . . . . . L . . . . .
1:3
" ' , - "I"r . . . . . . . . . . o o r . . . . . . . . LIJ h z
.3o~ . . . . ~ . . . . i . . . . i " ( b i ! 8 . 2 ~ . . . . . . ~
. . . . . . . : . . . . . . . ~ . . . . . . . -
g . 2 o . . . . . . . i . . . . . . . . i . . . . . . . --i ~
.15 . -
~ ,io . . . . i
~- .o5 ~ - - ! o [ _ , , J , , _ i . . . . ! , , ,
0 I 2 3 4 (X I 0 0 0 )
T I M E (DAYS) Fig. 6. Proportion of infected bugs [(a) Triatoma
infestans and (b) Rhodnius prolixus)l in house type 1 with one
initial infected dog.
-
V E C T O R S O F C H A G A S ' D I S E A S E T R A N S M I S S
I O N 261
R. prolixus about four per year, with adults lagging after
nymphs by 150 and 75 days, respectively.
During the oscillations the population development-stage pyramid
changes (Fig. 5). In general, adults represent a small fraction of
the total population (between 2% and 5%).
Although the population was divided only into eggs, nymphs and
adults, knowledge of the average development time of the five
nymphal stages for each vector species allowed the grouping of
densities by nymphal stages for the purpose of showing changes in
the developmental structure of the population with time. The bug
population's infection fluctuates, oscillating between 30% and 60%
for T. infestans (Fig. 6a) and between 5% and 20% for R. prolixus
(Fig. 6b).
Z
~ ( X I0C 24
.J
.D 0. 2O 0 r, 16
Z
( X I 0 0 ) ~ ~ ~ ~ ~ 1
21 19
12 17
8 120 120 4 ~ _ _ . . _ . [ ..... / I(30 I S k ~ , ~ " - ' Y - -
/ I ( 3 0 Oo o; °,
8 0 • . L . .19 , r
400 500
~z~=
" - " ~°°[I 19 °" ~°°t- I / t - ,~o ~~~_-,:N--~ ,oo ,oo ~ ~ ! '
_ ' I - - /80 ~ a o
AL TEl
Fig. 7. Effect of simultaneous changes in AL (number of hosts)
(a) or TEl (insect adult house emigration rate) (b) and ES3
(threshold nymphal density at which irritation of the host starts),
on the stability behavior of simulated Rhodnius prolixus
populations. Subindices 1 and 2 represent the average stable
population and its coefficient of variation, respectively.
-
262 J.E. RABINOVICH AND P. HIMSCHOOT
SENSITIVITY ANALYSIS
The stepwise regression of the 81 runs for sensitivity analysis
showed that the two basic mortality rates (TM1 and TM2) had a
negligible effect on the vector's mean stable population size (MSP)
and its coefficient of variation (cv), leaving only the mortality
density-dependent parameter (FS3) and the natality rate (TNATCH) as
the most sensitive parameters. The standardized partial regression
coefficients for TNATCH and FS3 were 0.807 and 0.348 for MSP, and
0.850 and 0.054 for cv, respectively. However, having observed from
previous simulations that the total number of hosts (AL) and the
emigration rate (TEl) had an important effect on the MSP, they were
also incorporated into simultaneous combinations with TNATCH and
FS3 in a second, more detailed, sensitivity analysis.
The parameter values used for both species were (each triplet
represents the minimum, the maximum, and the step of change in that
order): for FS3,
'°k , ~ 1 1 I
o r r ~ 7 - ~ - -._Lj.'q----/80 4f ~ , , _,,1---~/~8o
• 1.7 2
g 6 o o f ~ ~ ~ ~ z 4 o ~ . .-~ooE- i - ~ r ~ - - - _ / , i
~oo~- I ~ I I
~{ 3oo~ J-tiLL II I ,2% ~ I I ~> zoo[- /-~ ~ ' i ~ ~ ~2o 8oL
/ i ~ . ' : ...... 4 1 ~ ,20 °~ , o 4 / ~ ? -,~oo , , o ~ - / ~ s ~
~ l : ~ ..... / ,oo
o~V--,~: . . . . . ~ r--× 8o o ~ ~ ~ . i t : - / ~ o ~ O o ~
~__~o~%~ .~ .~ ~ ~ o ~o
TNATCH 4 1.4 1.7 2 TNATCH
Fig. 8. Effect of simultaneous changes in TNATCH (natility rate)
and FS3 (threshold nymphal density at which irritation of the host
starts) on the stability behavior of simulated Rhodnius prolixus
(a) and Triatoma infestans (b) populations. Subindices 1 and 2
represent the average stable population and its coefficient of
variation, respectively.
-
VECTORS O F CHAGAS" DISEASE T R A N S M I S S I O N 263
40, 105, 5; for TEl, 0.12, 0.18, 0.03; for AL, 10, 70, and 10.
As fecundity is different for each species (Table 2), the TNATCI-I
values used for R. prolixus were: 0.5, 3.1, 0.2; and for T.
infestans, 0.57, 1.97, 0.2. This gives 2058 combinations for T.
infestans and 4116 combinations for R. prolixus, giving a total of
6174 runs of the simulation model.
The changes in MSP and c v for simultaneous variations of FS3
and AL, and also in FS3 and TEl for R. prolixus are shown in Fig.
7a and b, respectively. They display a steep interaction when FS3
and AL vary simultaneously, but a smooth one for variations in FS3
and TEl. The simultaneous variations in FS3 and TNATCH also show a
steep interaction in the MSP and c v results for R. prolixus (Fig.
8a), but a smooth one for T. infestans (Fig. 8b).
In all cases there is evidence that c v (the coefficient of
variation, here used as an indicator of stability behavior) jumps
from null values to relatively high ones (except in Fig. 8b),
suggesting a change in the stability
4 0 0
300
2 0 0
I O O
o o
la. o t r t~ 4 0 0 El
D z 5 2 0 I
f ' ' ' L ' ' ' 1 . . . . . ' ' ' 1 . . . . I . . . . _
24060 ! i 0
0 05 I 1.5 2 2.5 3 ( X IO00)
NUMBER OF NYMPHS
Fig. 9. Phase diagrams of nymph and adult numbers of Rhodnius
prolixus simulated populations for Fs3 (threshold nymphal density
at which irritation of the host starts) values of 100 (upper graph)
and 50 (lower graph).
-
264 J.E. R A B I N O V I C H A N D P. H I M S C H O O T
behavior from a stable point to a limit cycle. This change in
behavior is better exhibited if population values are displayed in
a phase diagram of nymphs against adults, as shown in Fig. 9 for
changes in FS3 from 50 to 100 in R. prolixus. With all other
parameters in their standard values, FS3 ---- 50 produces a phase
diagram with a 'closing-up' spiral trajectory indicating an
equilibrium point around 1500 nymphs and 65 adults; the trajectory
of the phase diagram for vs3 = 100 shows a progressively
'opening-up' spiral until it reaches a limit cycle confined to
about 1300-2600 nymphs and about 10-400 adults.
DISCUSSION
Our model shows that populations of both insect species
fluctuate with periodicities that reflect their development times:
R. prolixus, with a total development time (eggs, nymphs, and
pre-reproductive adults) of 90 days has about four peaks per year,
while T. infestans, with a total development time of 170 days, has
about two cycles per year.
The development time is also associated with the temporal
alternation of developmental stages' abundances: the peak of the
adult population lags behind the nymphal peaks in proportion to the
development time of each species (cross-correlation analysis shows
an average lag of 73 and 132 days for R. prolixus and T. infestans,
respectively). This periodic change in the developmental-stage
structure also affects the percentage of infected bugs; these
percentages fluctuate with peaks of infection that coincide with
peaks of adult bug populations. This is so because the acquisition
of the infection by the bugs is a time-cumulative process: the
percentage of infected bugs increases with developmental stage.
The asymmetry of the population oscillation seems to be related
to fecundity: the growing arm is much steeper than the falling one
in the oscillation of the nymph population in the case of R.
prolixus with a fecundity of 1.3 eggs per female daily, while it is
symmetric in the case of T. infestans with a fecundity of 0.77 eggs
per female.
The densities of the simulated populations show two types of
behavior: a stable equilibrium point and a limit cycle. The
limit-cycle behavior is determined by the non-linear natural
mortality function. However, as sug- gested by sensitivity
analyses, the change in behavior results from the interaction
between the two sources of mortality: the natural mortality (a
sigmoid density-dependent function related to the carrying
capacity) and the starvation mortality (a linear density-dependent
function related to host irritation). When the parameter FS3 of the
starvation-mortality linear func- tion has a low value (e.g., 55-65
for R. prolixus), population regulation takes
-
V E C T O R S O F C H A G A S ' D I S E A S E T R A N S M I S S
I O N 265
place at a p o p u l a t i o n dens i ty tha t does not al low
the act ion of the non- l inear na tu ra l -mor t a l i t y funct
ion .
It is known that the switch be tween a stable po in t and an
osci l la tory behav io r is m o d u l a t e d by bo th the f ecund
i ty and the d e v e l o p m e n t t ime (Has t ings and Cons tan t
ino , 1988). Rhodnius prolixus has egg and n y m p h a l d e v e l
o p m e n t t imes shor te r and fecundi ty larger than T.
infestans; the dif- ferences in the p o p u l a t i o n stabil i ty
behav io r of the two species ana lyzed here p r o b a b l y
reflect these two biological parameters ,
This knowledge abou t the stabil i ty behav io r of these two
species vectors will be of value in the appl ica t ion of models
for the design of the o p t i m u m con t ro l s t ra tegy of
Chagas ' Disease t ransmiss ion; cont ro ls should aim at shif t
ing popu la t ions to a s table po in t equi l ibr ium (thus avoid
ing p o p u l a t i o n explosions) at their lower possible
values.
ACKNOWLEDGEMENTS
We gra tefu l ly acknowledge the Ins t i tu to Nac iona l de
Diagn6s t i co e In- vest igaci6n de la E n f e r m e d a d de
Chagas 'Dr . Mar io Fa ta la Chab6n ' for k indly p rov id ing free
c o m p u t i n g t ime for model d e v e l o p m e n t and simula-
tions, and to Marce lo Berre ta who he lped with the drawings.
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