integration of temporal and spatial models for examining the epidemiology of african trypansomiasis

ELSEVIER MEDICINE

Preventive Veterinary Medicine 24 ( 1995) 83-95

1:ntegration of temporal and spatial models for examining the epidemiology of African

trypansomiasis

Peter Yu *, Tsegaye Habtemariam, David Oryang, Mike Obasa, David Nganwa, Vinaida Robnett

Center for Computational Epidemiology, School of Veterinary Medicine, Tuskegee Universiry, Tuskegee,

AL 36088, USA

Accepted 12 December 1994

Abstract

A stochastic model was first developed to study the spatial dispersal of tsetse fly population, and subsequently integrated with a time oriented epidemiologic model. Such an integrated model was needed to understand better the epidemiology of cattle trypanosomiasis. Pre-existing data were used to determine the distributions of the random variables invoked in the model. We used the model to assess several alternatives of preventing the spatial progression of trypanosomiasis. We address the following question: what size of protective barrier is required to prevent the spatial progression of trypanosomiasis? The vector control alternatives considered in this study included insecticide applications, vegetation clearing, wild animal depopulation, use of tsetse traps, and combinations of the above methods within a protective barrier. Simulation results indicated that a protective barrier, about 1000 m wide, was effective in stopping the spatial spread of cattle trypanosomiasis.

Keyword.x Trypanosomiasis; Tsetse fly; Spatial model

1. Introduction

Trypanosomiasis is endemic in 36 countries of sub-Saharan Africa, and remains a threat to both humans and livestock in many rural communities. Because of trypanosomiasis,

nearly ‘7000000 km* of tsetse belt in sub-Saharan Africa is reported to be unavailable for raising livestock (International Laboratory for Research Animal Diseases (ILRAD) ), 1993). The disease is infectious, usually progressive and fatal if untreated. So far, vaccine production has not been successful.

-- *Corresponding author.

0167-5857/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDIO167-5877(95)00465-3

84 P. Yu et al. /Preventive Veterinary Medicine 24 (1995) 83-95

Tsetse flies are the natural vectors of African trypanosomiasis. The best method for controlling trypanosomiasis is to control the tsetse flies (World Health Organization (WHO) ) , 1986; ILRAD, 1993). A key to the fuller understanding of the epidemiology of trypanosomiasis, including the examination of feasible control alternatives, may rely on a quantitative study of the population dynamics of its vectors - tsetse flies. Habtemariam et al. ( 1983b, 1986) examined the probabilities of effective transmission of trypanosomes by

tsetse flies and developed a temporal model to evaluate the control alternatives for trypanosomiasis. However, a key element missing was a method to examine the spatial dimension of the spread of tsetse flies in a trypanosome risk region.

The population dynamics of tsetse flies have been studied over several decades in different regions of Africa using various methods including capture-recapture experiments (Laird, 1977; Dransfield et al., 1991). New technology such as remote sensing has been used to predict tsetse abundance over large areas of Africa (Rogers and Randolph, 1991). One of the trypanosome control strategies is to eliminate or minimize the fly-man or fly-host

contact by creating a protective barrier in which a control scheme is employed to prevent the spatial progression of trypanosomiasis (Buxton, 1955; WHO, 1986; Dransfield et al., 199 1) . For example, Zimbabwe reinforced the country’s tsetse barriers to protect the interior

of the country by restricting tsetse infestation mainly to non-farming border regions (ILRAD, 1993). A better understanding of the fly movements into tsetse-free regions would enable one to take early steps to deal with a threat, rather than to wait until an outbreak has

manifested. In this study, a stochastic model is developed to study the spatial dispersal of tsetse fly

populations. We then integrate the spatial model of tsetse flies with the time oriented

epidemiologic model of trypanosomiasis developed by Habtemariam et al. ( 1983a, 1986). Pre-existing data are used to determine the distributions of the random variables involved

in the model. We address the following questions: what size of a protective barrier is required to prevent the spatial progression of trypanosomiasis?; which vector control method is most efficient?; how would this affect the prevalence of cattle trypanosomiasis in a given area? To answer these questions, we use the integrated model to examine the following vector control alternatives for stopping the spatial progression of trypanosomiasis: insecticide applications (IA) ; vegetation clearing and wild animal depopulation (CD) ; use of tsetse traps (TT) ; and combinations of the above methods within a protective barrier.

2. Population dynamic model of cattle and tsetse flies

We consider two neighboring regions, tsetse-infested (0,) and tsetse-free (a). Try- panosomiasis is initially introduced to region 0,. A protective barrier is created between the two regions. Some control schemes are employed within the protective barrier in order to prevent the disease from spreading to region Q.. Let o be the width of a protective barrier. We investigate the effectiveness of various vector control methods for different values of w. It is obvious that the effectiveness of a vector control method can be measured by the prevalence of trypanosomiasis in region Q_.

In this study, cattle are considered to be in one of five states: susceptible; incubating; acute; chronic; immune; these are referred to by state 1, 2, 3, 4, and 5, respectively. The cattle population vector is denoted by:

P. Yu et al. /Preventive VeteriMty Medicine 24 (1995) 83-95 85

I Control Measures I

Fig. 1. Integration of temporal and spatial models for examing the epidemiology of trypanosomiasis.

Mi(t> = lmilCt) mi*(t) W3Ct) mi4(t> W5Ct> 1’ 1 (1)

where nlik( t) is the number of cattle of state k in region Q at time f, and the prime indicates vector transposition, i = 1,2, and k= 1,2, 3,4, 5. For simplification, we assume that cattle move within a given region, but not from one region to the other. Tsetse flies, the natural

vectors of African trypanosomiasis, are considered to be in one of three states: non-infected; incubating; infectious; these are referred to by state 1, 2, and 3, respectively. The tsetse population vector is denoted by:

NiCt) = [nir(t) ni*(t) +3Ct> 1’ 9 (2) where n!ik( t) is the number of tsetse flies of state k in region Q at time t, i = 1,2, and k= 1,

2, 3. The dynamics of cattle and tsetse populations in region LIi can be approximated by a

system of differential equations:

2 Mi(t) =Ai( t) %i(t) (3)

iNi =Bj(t) ‘N,(t) +Zi(t) -E,(t) (4)

where 14,(t) and B;(t) are the rate matrices, and Zi( t) and Ei( t) are the immigration and emigration vectors respectively, i= 1, 2. Note that the rate matrices A,(t) and Bi(t) are density dependent. The values of most elements of Ai( t) and B,(t) are taken from Habte- mariam et al. ( 1983a, 1986). The values of Zi( t) and Ei( t) are computed from the spatial dispersal model of tsetse fly populations discussed in Section 3. The model defined by Eqs. (3) and (4) is an extension of the model developed by Habtemariam et al. ( 1983a, 1986) to include spatial effects. Eqs. (3) and (4) are connected by the probabilities of transmission


of trypanosomiasis. Two systems of 0, and a are related through spatial dispersal of tsetse

fly populations (Fig. 1).

3. Spatial dispersal of tsetse fly populations

In sub-Saharan Africa, trypanosomiasis is spread mainly by the bite of trypanosome-

infected tsetse flies (Buxton, 1955; ILRAD, 1993). Dispersal of flies has an important bearing on control operations because of the potential of reinfestation of areas where control has been undertaken. In order to study the spatial spread of cattle trypanosomiasis, we need to keep track of both locations and health statuses of tsetse flies from time to time. To do this, let (I,,, yikn( t) ) denote the location of fly number n of state k in region 0,. at time t,n= 1,2 , . . . , nik( t) . In the text time interval, the fly moves a distance of p( t) m in a direction 0( t> given that it survives to time t + 1. The direction 6’(t) is expressed as an angle measured

counter-clockwise from east. WHO (1986) concluded that dispersal of flies was a random activity. Tsetse fly move-

ment, which appears to be random, is continually taking place within a limited area (Buxton,

1955). In this study, we assume that the direction O(t) is a random variable uniformly distributed within an interval (0, 2r). Suppose the location of fly number it of state k in region Q at time t is (Xikn( t), yikn( t) > , in the next time interval, the fly becomes number m

of state 1 in region 0, if

[xikn(t) +P(r) cos e(t), YiknCt) +P(t) sin e(t)lEf2j

and the location of the fly at time t + 1 is given by

xjln*(t+ l) =Xikn(f) +PCt> cos e(t)

(5)

(6)

Yjh*Ct+ 1) =Yi/m(t) +P(t> sin e(t) (7)

Flies may change their states in the process of movements. Given that a fly is in state k

at time t, the health status of the fly at time t + 1 is given by:

I= k, ifZ=O

k+l, ifZ=l (8)

where Z is a Bernoulli random variable with parameter pk(t), which is the transition probability from state k to state k + 1 at time t. The values of Pk( t) are calculated from the model proposed by Habtemariam et al. ( 1986).

To compute the migration vectors Zi( t) and Ei( t) in Eiq. (4)) let uiknl( t) and uilonl( t) be 0 if the lly number m in state k in region pi stays in the same region and 1 if it moves, where 1 is the state of the fly at time t + 1. Then the kth element of emigration vector E, (t) is given

by

nlr(r) e&t) = c Ul!fnr(f) (9)

n,= I

which is a count of flies in state k that emigrate from region fi, to region a at time t, k= 1, 2, 3. The Zth element of immigration vector I1 (t) is given by

P. Yu et al. /Preventive Veterinary Medicine 24 (1995) 83-95 87

(10) k=l nz=l

which is a count of flies in state 1 that arrive in region 0, from a at time r. Note that flies may change their states in the process of emigration. The immigration vector Z2( t) of J& is not equal to the emigration vector E, (t) of 0, since the health status may change during time interval ( f, t + 1) . E2( t) and Z2( t) can be computed using formulas similar to Eqs. (9) and ( lo), respectively.

4. Control measures

There are two types of tsetse control strategies: one is the destruction of the tsetse flies in tsetse-infested regions, and the other is the breaking of the fly-man or fly-host contact by creating a protective barrier in areas ahead of an advancing wave. Studies show that up

to 95% reduction of tsetse flies can be achieved using the first strategy (WHO, 1986). However, to our knowledge, it has never proved possible to achieve a 100% reduction in a fairly large region. In this study, we examine the second strategy. The primary objectives

are to find the probabilities that a fly crosses a protective barrier and to estimate the width of the protective barrier (buffer zone) required to prevent the tsetse flies from advancing

to tsetse-free regions. The vector control alternatives considered in this study are insecticide application (IA), vegetation clearing and wild animal depopulation (CD), and use of tsetse

traps (IT). These three methods are widely used in tsetse control campaigns (Buxton,

1955; WHO, 1986; ILRAD, 1993). Sterile insect technique has been effective in pilot operaticsns (WHO, 1986)) but has not been applied on a large scale probably due to its high

cost. Note that massive vegetation clearing and wild animal depopulation were used to control ltsetse flies on a large scale before the early 196Os, which resulted in the destruction of ecosystems. It was realized that not all vegetation and animals were conductive to tsetse flies. Therefore selective vegetation clearing and wild animal depopulation were employed in tsetse control campaigns after the early 1960s. Furthermore, the selective clearing and depopulation were confined to protective barrier which separated tsetse infested areas from relative tsetse free areas.

4. I. Tsetse mortality due to the use of insecticides

Various insecticides have been used in tsetse control campaigns in several countries of Africa. WHO (1986) reported that repeated applications (e.g. once a week) of 25 gl-’ deltametrin could reduce fly density by 92-95% in 4-5 months. Insecticide resistance was widespread in arthropods of medical and veterinary importance, but had not been reported for tsetse flies although a potential for development of resistance existed (WHO, 1986). For simplification, we assume that the death of a fly from insecticide application is independent between time intervals, and that daily mortality rate is constant, say ‘y, when the same insecticide is used at the same level over days. Schaalji and van der Vaart ( 1989) studied time-dependent mortality due to pesticide application. Their model consists of

88 P. Yu et a/. /Preventive Veterinary Medicine 24 (1995) 83-95

differential equations for decomposition and degradation of pesticide in the environment, intake and clearance of pesticide by insects. This kind of model requires data on chemical decay dynamics and dose-response relations. Generally, such data are not available because

of the difficulty and cost of conducting related trials. Let y(x) denote the mortality rate in

x days, then 1 - y(r) is the survival rate. Thus, from the assumptions given above, we have

1 -y(x) = (1 -r)” (11)

If we know y( 7) = 5 for some T, then Eq. ( 11) gives y= 1 - [ 1 - [] “r. The mortality rate in x days is then

r(n) = 1 - [ 1 - 51”” (12)

In this study, we use y( 150) = 0.95 based on WHO’s report ( 1986). Let D,,(t) be the event that a fly dies before time b due to pesticide application. Since

the fly may move in and out the protective barrier where pesticide is applied, we let z denote the total number of days that afly stays in the protective barrier up to time r. The probability that a fly dies before time t due to insecticide. application can be approximated by

P]&(r) 1 = Y(Z)

where r(z) is given by Eq. ( 12).

(13)

4.2. Tsetse mortality due to vegetation clearing and reduction of wild animal populations

Studies have been carrier out to assess the effect of vegetation clearing and wild animal depopulation in tsetse control campaigns (Buxton, 1955; Harley and Pilson, 1961). Veg- etation clearing and wild animal depopulation reduce the food available to tsetse flies. Brady

( 1972) mentioned that tsetse flies fed every m days, where 1 d m < 15. Habtemariam et al. ( 1986) assumed 4 <m G 10 in their models. Harley and Pibon ( 1961) observed that flies began to starve (the residual blood meal in abdomens was close to 0) 5 days after last feeding.

Let r be the last time at which a fly enters the protective barrier where no feed is available. Let D&t) be the event that a fly dies before time t due to vegetation clearing and wild

animal population reduction. The following function is used to compute the probability of

that event:

1, 0<6(t)<wt-r>tz

P[D,,(t)]= [(t-7)-t,]/(t*-t,),O O<S(t)<wt,dt-St* (14)

0, elsewhere.

where w is the width of a protective barrier, and 6(t) is the shortest distance from the fly to the far side of a protective barrier at time t. The implicit assumptions in Eq. ( 14) are: (a) the fly cannot survive if it continuously stays more than t2 days in a protective barrier where no food is available, (b) the fly will not die from starvation if it continuously stays less than t, days in a protective barrier. From the observations discussed above, we have t, > 5 and t2> 15. In this study, we use t, =7 and t,=21.

P. Yu et al. /Preventive Veterinary Medicine 24 (1995) 83-95

4.3. Tsetse mortality due to the use of tsetse traps

Interest in the use of tsetse traps was revived in the 1970’s. Tsetse traps are effective, simple, non-polluting and ideally suited for use by community health care workers (WHO,

1986). Animal host odors such as acetone, phenols and octenol are used with blue or black

cloth traps to attract tsetse flies. Traps can reduce an isolated fly population by 99% in 6 months (Dransfield et al., 1991; ILRAD, 1993). Let D&t) be the event that a fly dies before I:ime t due to the use of traps, then the probability that a fly dies due to the use of traps can be approximated using a formula similar to Eq. ( 12) with y( 180) = 0.99 (99% reduction in 6 months).

In order to examine the effect of combinations of these methods as well as individual method on the prevalence of cattle trypanosomiasis in region a, we let S(t) denote the event that a fly survives to time t. Then we have:

s(t) =&(t) n&(t) f&(t) f-60) (15) where 11,,(t) , D&t), Dn( t) and DN( t) are the events that the fly dies before time t due

to inset ticide applications, vegetation clearing and wild animal depopulation, use of tsetse

traps, and natural causes, respectively. Assuming that the four events D,,(t), DcD( t), D,(t) and DN( t) are independent of each other, the probability that a fly survives to time t is given by

PIS(t)l=I1-P[~,(t)l}.{1-P[~,,(t)l)

.(l-P[D,(t)l}.(l-P[D,(t)l} (16)

which is an element Of Bi( t) of Eq. (4).

5. Corn puter simulation and results

Monte-Carlo simulations of the model were carried out on a NeXT work station and Macintosh Quadra 800 with C as the programming language. Stochasticity was implemented

by allowing some of the key parameters to vary randomly. These parameters included the probability of effective transmission of trypanosomiasis, travel distance and direction, and the probabilities of birth and death of tsetse flies. Temperature and rainfall affected cattle and tsetse population dynamics. The monthly climate data from southwestern Ethiopia were used to generate daily temperature and rainfall based on Gaussian distributions. The simulations were time consuming due to the random movements, death and birth processes of tsetse flies, and random transmission of trypanosomiasis in cattle. Various control measures

were evaluated for the settings of protective barrier width w = 200 m, 300 m, 400 m, . . ., and 1000 m.

Comparisons of cattle trypanosomiasis prevalence in the initially trypanosome-free region a under various control methods are shown in Figs. 2-4. The simulations began with initial values of ten incubating cattle and 1000 susceptible cattle in region R1, and 1010 susceptible cattle in region a. Both 0, and a are of 100 km’. These initial values were chosen such that the: cattle and tsetse fly densities were comparable with those in the study by Habte-


40

35 I

30 I

2 25

B I =

2 20

s ; 15

1

10 $

0 t /

0 700 1400 2100

Time (days)

2800 3500

Fig. 2. Prevalence projections of cattle trypanosomiasis for various vector control measures with a protective barrier width of 200 m: (a) no control; (b) insecticide application; (c) vegetation clearing and wild animal depopulation; (d) tsetse traps; (e) insecticide + clearing; (f) clearing + tsetse traps; and (g) insecticide + tsetse traps.

mariam et al. ( 1983a). For a protective barrier of width w= 200 m, the cattle trypanosomiasis prevalence in region a reached a maximum of 36.2% after 3 years and 2 months if insecticides were applied (IA), then decreased to a steady level of about 24%. When clearing techniques (CD) or tsetse traps (‘IT) were used, the prevalence was not importantly

different from that when insecticides were applied. However, the cattle trypanosomiasis prevalence remained below 20% before the end of the fifth year if both insecticide application and clearing techniques (IA + CD) were used, and reached a maximum of 30.5% after 6 years and 5 months, then decreased to a level of 25%. The combinations CD + ‘IT and IA+lT were less efficient than IA +CD. In the long run, the cattle trypanosomiasis prevalence under any control measures described above was not importantly different from that when no control measures were used (Fig. 2).

When the width of a protective barrier was increased to 600 m, the cattle prevalence in region L$ reached a level of about 5% after 1 year and 4 months if insecticides were applied (IA), and an epidemic of trypanosomiasis occurred shortly before the end of year 10. A similar pattern was observed if clearing techniques were used (CD). However, if tsetse traps were used (IT), no epidemic of cattle trypanosomiasis was observed in region Q during a lo-year period. For all the combinations, IA + CD, CD + ‘IT, and IA + IT, the cattle prevalence in region L& remained below a level of 5.5% throughout a lo-year period (Fig. 3).

For a protective barrier of width w= 1000 m, the cattle trypanosomiasis prevalence in region J& reached a maximum of 5.2% after 1 year and 5 months if insecticides were applied (IA) in the protective barrier, then decreases to a level of about 2.5%. The prevalence decreased to almost zero by the end of year 10 if both insecticides and clearing techniques

P. Yu et al. /Preventive Veterinary Medicine 24 (I 995) 83-95 91

30

8 = 20 2 P g 15

IO

0 700 1400 2100

Time (days)

2800 3500

Fig. 3. Prevalence projections of cattle trypanosomiasis for various vector control measures with a protective

barrier w:th of 600 m: (a) no control; (b) insecticide application; (c) vegetation clearing and wild animal

depopulation: (d) tsetse traps; (e) insecticide + clearing; (f) clearing + tsetse traps: and (g) insecticide +

tsetse trarls.

t c P

20

9 g 15

10

5

0

0 700 1400 2100

Time (days)

2800 3500

Fig. 4. Prevalence projections of cattle tqpanosomiasis for various vector control measures with a protective

barrier wldh of 1000 m: (a) no control; (b) insecticide application; (c) vegetation clearing and wild animal

depopulalion; (d) tsetse traps; (e) insecticide + clearing; (f) clearing + tsetse traps; and (g) insecticide +

tsetse traps.

(IA + CD) were used. There was no significant difference between the other control methods. In these cases, the prevalence in region Q remained below a level of 2.5% most of the time during a IO-year period (Fig. 4).

92 P. Yu etal. /Preventive Veterinary Medicine 24 (1995) 83-95

6. Discussion

A spatial model of tsetse fly populations is integrated with a time oriented epidemiologic model of cattle trypanosomiasis. This type of integrated model provides a useful tool for examing the epidemiology of cattle trypanosomiasis in both temporal and spatial dimen- sions. The integrated model is used to evaluate the vector control alternatives of preventing

the spatial progression of trypanosomiasis. The simulation results appear to indicate that a protective barrier of about 1000 m wide is effective to stop the spatial progression of cattle trypanosomiasis if a proper vector control method is used. Generally, using combinations of different control measures is more effective than employing one method alone. The results suggest that control measures other than vegetation clearing and wild animal depopulation may be better in terms of technical efficiency and environmental preservation. This

raises an important question whether destruction of plants and animals should be encouraged at all in the fight against trypanosomiasis if less environmentally destructive options such

as tsetse traps can be used. However, a word of warning is necessary about the interpretation of the prevalence

simulated as above. Notice that stochasticity was introduced in the simulations via some

key parameters. The prevalence does not remain the same, even for the same initial conditions due to the random properties of the model. Furthermore, there exists some degree of uncertainty in other parameters such as efficacy of insecticide applications and other control measures. Therefore the quantitative behavior of the trypanosomiasis prevalence should be interpreted with caution.

Several simulations were conducted to study the sensitivity of prevalence projections of

cattle trypanosomiasis in relation to changes in some key parameters and initial conditions. The simulations were confined to the use of tsetse traps within a protective barrier of 1000 m wide. Fig. 5 shows the prevalence projections of cattle trypanosomiasis in region flZ when the efficacy of tsetse traps was set at (a) the estimated value, (b) 10% less than the

estimated value, and (c) 10% greater than the estimated value, respectively. The cattle trypanosomiasis prevalence reached a maximum of 4.9% after 1 year when the efficacy of tsetse traps was set at 10% less than the estimated value, then decreased to a level below

2% after 4 years. If the efficacy of tsetse traps was set at 10% greater than the estimated value, the cattle trypanosomiasis prevalence reached a maximum of 2.7% after 1 year, then decreased to a level below 1% after 4 years. Fig. 6 shows the prevalence projections of cattle trypanosomiasis in region L$ for different initial infected cattle populations in region 0,. The prevalence in Q decreases to almost zero by the end of year 10 if the initial infected cattle was set at 0.5% in region fl, although the prevalence remained below 3% during a

lo-year period if the initial infected tattled was set at 1% and 2% respectively. Fig. 7 shows the prevalence projections of cattle trypanosomiasis in region a when the initial infected tsetse flies in region a, were set at l%, 2% and 4%, respectively. All of the projected prevalence in @, remained below 2.5% during a IO-year period for the three different initial values. The simulation results indicated that tsetse traps were effective to stop the spatial progression of cattle trypanosomiasis if they were used at a greater efficiency in a protective barrier of about 1000 m wide.

Genetic control by release of sterile males has been effective in pilot operations (WHO, 1986). However, such a biologically viable technology has not been applied on a large


5

4.5

4

3.5 ‘3 !; 3

:: ‘: 1Y 2.5

!I !D 2 OL

1.5

1

0.5

0

0 1400 2100

Time (days)

Fig. 5. Prevalence projections of cattle trypanosomiasis when the efficacy of tsetse traps is set at: (a) the estimated

value; (b) 10% less than the estimated value; and (c) 10% greater than the estimated value. The protective barrier

width is 1000 m.

scale probably due to its high cost. Anaman et al. ( 1994) conducted a benefit-cost analysis of sterile insect technique to eradicate screwworm fly using a dynamic spatial bioeconomic model. This type of analysis may be incorporated in this model to evaluate different tsetse- control strategies including sterile insect technique. In the future, this model may be

3-

2.5 --

2 -- .w G

8 = -- .L 1.5

B e IL 1 --

0.5 --

0-l :” -._ 0 700 1400 2100 2800 3500

Time (days)

Fig. 6. Prevalence projections of cattle trypanosomiasis with the initial trypanosome-infected cattle population at:

(a) 0.5%; (b) 1%; and (c) 2%. Tsetse traps are used within the protective barrier of 1000 m wide.

94 P. Yu et al. /Preventive Veterinary Medicine 24 (1995) 83-9s

0 +_-+_-_--,~-

0 700 1400 2100 2800 3500

Time (days)

Fig. 7. Prevalence projections of cattle trypanosomiasis with initial trypanosomekfected tsetse fly population at:

(a) 1%; (b) 2%; and (c) 4%. Tsetse traps are used within the protective barrier of 1000 m wide.

expanded to include a geographical information system module with detailed spatial distri-

bution of cattle population. We suggest that such an integrated dynamic model can be useful as an aid in planning and decision making for the control of trypanosomiasis.

Acknowledgements

This work was partially supported by grant no. RII-9005621 from the National Science

Foundation (NSFIRIMI), and grant no. 2G12 RR 03059-06 from the National Institute of Health (NIH/RCMI) .

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integration of temporal and spatial models for examining the epidemiology of african trypansomiasis

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