Computers and electronics

ELSEVIER in agriculture

Computers and Electronics in Agriculmre 13 (1995) 155-175

Modelling movement and mortality: killing tsetse flies in the field

Brian Williams 1

Epidemiology Research Unit, Box 4584, Xphannesburg 2000, South Aftica

Accepted 19 April 1995

Abstract

Developments in the use of odour-ba.ited traps offer exciting prospects for the effective control of tsetse flies and the trypanosome disealses of which they are the vectors. Work carried out at Nguruman in south-western Kenya has led to the development of cheap and simple traps that can be made and serviced by local communities. Various models have been developed that enable us to understand the factors that determine the efficiency of traps which in turn will help us to determine the density of traps needed to achieve a set level of control and to design more effective traps. Four models are discussed here. The first is a model of the population dynamics of tsetse that relates the overall population loss rate to the mortality that we impose on adult flies. The second is a model of movement on a large scale that makes it possible to relate the adult mortality to the movement patterns and population dynamics of the flies, and to the properties of the trap. The third is a more speculative attempt to model the way in which individual flies locate traps once they are close enough to detect the odours. This should eventually make it possible to refine the parameters in the large scale movement models. The last is a model of invasions of flies into a cleared area. The development and testing of these models has relied extensively on the data collected in the field at Nguruman and in turn the models have helped us to interpret the data and to formulate new questions and experiments.

Keywords: Tsetse flies; Spatial dynamics; Modelling

. Introduction

Tsetse flies (Glossina spp.) are the vectors of trypanosomes that cause sleeping sickness in people and naguna in catt1.e and other domestic livestock, The Ugandan sleeping sickness epidemic in the early decades of this century is estimated to have killed 200,000 people (Ford, 1971). Nagana precludes the keeping of cattle and other domestic livestock over an area of Africa equal in size to the United States

016%1699/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved. SSDI 0168-1699(95)00022-4

156 B. Williams/Computers and Electronics RI Agriculture 13 (199s) 155-175

of America. In a continent afflicted by a host of vector borne diseases, the African trypanosomiases are widely regarded as the most devastating of all.

The thirty-six species and subspecies of tsetse flies may be divided into three groups: the morsitans group occurs mainly in Savannah areas; thepalpalis group has a riverine or lacustrine distribution; thefusca group comprises mainly forest species. Species of flies belonging to the morsztans andpalpalis groups are the most important vectors of trypanosomiasis. Sleeping sickness is restricted to defined area,s while nagana occurs wherever tsetse are present. Among cattle, the humpless Bos taigrus breeds such as the N’Dama and the West African Shorthorn, which arrived in E;gypt about 7000 years ago, tolerate trypanosome infections provided they are well fed, whereas the more common Bos indicus breeds such as the Zebu, which arrived in East Africa less than 2000 years ago, vary in their degree of tolerance (Epstein, 1971).

Attempts to understand the disease and to bring it under control have been many and varied. Chemotherapeutic drugs have been developed to treat the disease in cattle and in people. Attempts are being made to develop vaccines to protect aga.inst the trypanosomes. The genetic basis of trypanotolerance is being investigated. Insecticides sprayed both from the air and from the ground have been used to control flies. Insecticide impregnated targets have been effective in eradicating the flies and hence the disease in Zimbabwe. Dipping animals or pouring insecticides directly onto cattle is being tried in Tanzania. Traps have been developed that are effective against some species of tsetse, especially those of th.e j&a group, and have been used to good effect in some places. And yet, with an annual expenditure on research into the disease and on attempts to control or eradicate the flies that currently runs at several hundred million dollars per year, the map of the tsetse distribution in Africa is not significantly different now from what it was 100 years ago except at the margins of the distribution in northern Nigeria, South Africa and Zimbabwe (Williams and Williams, 1992).

If we are to succeed in the future where we have failed in the past, it is essential that we understand the reasons for our lack of success. An exhaustive analysis of the problem would involve political considerations, including the major social changes that occurred in the 1960s when African countries emerged from the colonial experience often with broken economies and with little experience of government. It would include social considerations, such a.s the reliance on foreign experts, and the tendency to regard local people as at best unimportant and at worst part of the problem. It would involve economic considerations in a continent whose economies are, even now, at the mercy of world markets largely beyond their control, and the resulting tendency to rely on foreign aid. But it would also involve specifically scientific questions for even allowing for the political, social and economic problems, our scientific knowledge has not had an impact commensurate with the scale on which it has been practised (Williams and Williams, 1992; Williams et al., 1993,1995).

2. Science and development

We cannot do science that is effective unless it is firmly grounded in an under- standing of the context in which it is to be applied. Until now the conventional

3. Williams I Computers and Electronics in ilgn’culture 13 (1995) 155-175 257

wisdom has been that we must eradicate the flies or the disease. Experience shows that neither of these is possible except in places, such as South Africa and Zimbabwe which are at the edge of the distribution and where a substantial infrastructure exists including effective extension services, access to foreign currency and well trained personnel. In most of Africa we need to look to control rather than eradication. This means that a sound understanding of the ecology of the flies and the epidemiology of the disease is an essential prerequisite for effective action. Control, rather than eradication is, by definition long term. This means that our technologies must be suited to local social, economic and biological conditions and that local communities must be seen as a key part of the implementation of the control.

If the aim of our research is to solve actual problems as they exist in the field rather than theoretical problems that we pose in the laboratory, the research and control must be closely integrated. Drugs, vaccines, behavioural studies and mathematical models are all important. But drugs are of no consequence if drug resistance develops rapidly, vaccines are of no consequence if they cannot be delivered, behavioural studies and mathematical models are of no consequence if they are only validated in the laboratory. While our research must be firmly grounded in field experience, it must also provide a conceptual framework within which we can integrate the results of our field studies and from which we can draw general lessons that can be applied in other places and at other times. The role of models is to provide this framework.

3. Models of fly movement and mortality

In recent years a number of models have been developed to help us to understand the spatial and temporal dynamics of tsetse fly populations and the epidemiology of trypanosomiasis. I shall concentrate on a series of models that have been developed as part of a community based tsetse control project on a Maasai ranch at Nguruman in south-western Kenya where Glossina pallidipes and G. longipemis are the vectors of nagana. The project, conceived and run by Robert Dransfield and Robert Brightwell, was designed from the start to provide a method of controlling tsetse flies that could be carried out by the Maasai community (Dransfield let al., 1991). With the chose involvement of the members of the ranch this led to the development of a cheap and effective trap that can be made entirely from locally available materials and now provides the basis of their control work (Brightwell et al., 1987, 1991; Kyorku et al., 1990). The traps are made of blue cloth that attracts flies and black cloth that elicits a landing response. At the top of the trap is a netti.ng cone. Flies that enter the trap from below are attracted up towards the light filtering through the netting. At the top of the cone is a small hole through which the flies enter a polythene bag where they are killed by heat stress. The traps are baited with cow urine and acetone. No insecticide is used.

The Nguruman project also led to detailed istudies of the ecology of the flies and to the development of a series of models that made it possible to determine the trap densities needed to obtain a given level of control, the optimal spacing of the traps in different habitats and the width of barriers that would be needed to prevent

1.58 B. Williams/Computers and Electronics in Agriculture 13 (1995) 155-I 75

reinvasions of flies from neighbouring areas. It has also been possible to identify factors that limit the efficacy of traps which in turn will determine the design of new traps.

I will consider different kinds of model. The population models were developed in order to determine the effective mortality that would be achieved provided. w-e can determine the mortality that is applied to flies of different ages. The move:ment models were developed in order to determine the effect of fly movement on the efficacy of control by traps or targets. The behavioural models are now being developed to help us to understand the way in which flies locate and find their hosts or the traps. The “consistency” model was developed in order to validate our interpretation of the consequences of the control as it was observed in practice.

4. Population models

Female tsetse deposit their first larva 17 days after they emerge from their pupae; thereafter they deposit one larva every 8 to 1.0 days. The larvae burrow into the soil and pupate. About one month later the pupae emerge as adults. As flies that lived forever would produce two larvae every 18 days, the maximum rate of increase of a tsetse population must be less than 3,9%/day. If we can impose an overall population mortality of more than 4%/day we will drive the population to extinction. This very low reproductive rate, at least by insect standards, immediately shows that traps might offer a viable strategy for the control of tsetse flies. Furthermore, the tight regulation of their reproductive cycle makes it possible to model their growth rate quite precisely.

The population model (Williams et al., 1990) proceeds from the continuous form of the Euler-Lotka equation which relates, Y, the growth rate of a population to m(t), the age-dependent mortality, and B(l), the age-dependent fecundity when the population has reached a stable age distribution, according to the following equation:

~mB(f)exp(-~ ) m(f) dt’ + rt dt = 1 (1)

Eq. 1 can be solved iteratively for the growth rate r given the fecundity p(t) and the mortality m(t) (Pielou, 1977). For tsetse flies Eq. 1 may be reduced to:

B exp[-(ma + r)& - (nzb + r)fb - (mc -k r)&] = 1 - exp(% + r>& (2)

At Nguruman the pupal period ta is about 27 days, the pre-adult period, tb, is about eight days and the interlarval period, t,, is about 9 days. An abortion rate of 5% gives a fecundity for female larvae, j3, of 0.475. Using Eq. 2 we can calculate the rate of decline of a tsetse population as a function of pupal mortality, m,, and adult mortality, mb and rn, assuming that the last two are equal (Williams et al., 1990). The results are shown in Fig. 1. Our best estimate of the pupal mortality is about l%/day (24% mortality over the whole pupal period) so that adult mortalities in the field must be about 2.8%/day for a stable population. From Fig. 1 we see

B. Williams/Computers and Electronics in .Agriculture I3 (199.5) 1.55-I 75 1.59

Rate of decline %/day 0 1 2 3 4 5 6 7

Adult mortality %/day

Fig. 1. Pupal mortality as a function of adult mortality for a range of growth rates assuming a stable

age distribution. The numbers along the top of the box give the population loss rate for each of the lines plotted. The two lines in the lower left-hand corner of the box are for growth rates of 1% and 2%/day. Other vital parameters are as given in the text.

that for adult mortalities less than 10% per day and pupal mortalities less than 5% per day the population growth or loss rate varies almost linearly with both of these parameters and we can approximate the population growth rate by:

r = 0.021 - O.O6m, - 0.41m P (3)

If we assume that the density dependent regulation operates on the adults and not the pupae, then at low densities the pupal mortality is still about l%/day but the adult mortality will be much reduced and the maximum rate of increase of a tsetse population is about 1.7%/day. The available field estimates give a value of l..5%/day (Turner and Brightwell, 1986). A 99.9% reduction in one year corresponds to a population loss rate of 1.9%/day so that to achieve this we need to impose an additional trapping mortality of about 1..7/0.6 or 3% per day.

At Nguruman, mark-release-recapture (MRR) experiments have been used to estimate the total population of flies in an area of about 100 km2 as between 10’ and lo6 flies (Dransfield et al., 1990) and we need to kill between 3000 and 30,000 flies per day to achieve a 3%/day adult mortality. Mean trap catches in the area averaged about 70 flies/trap/day (before the control operation started) so that we need between 40 and 400 traps in 100 km2 or between 0.5 and 5 traps per square kilometre. In the actual suppression operation, which began in 1987, 2 traps/km2 were used and these gave trapping mortality rates of 4-5%/day (Dransfield et al., 1991). Unfortunately, MRR population estimates are difficult to make and very time consuming and although they give required trap densities of the right order of magnitude, the range of population estimates is large. We would like to have a more direct way of estimating the den&y of traps needed to achieve a given adult mortality rate and hence a given rate of population reduction.

160 B. Williams I Computers and Electronics in Agriculture 13 (1995) 155-l 75

5. Movement models

We would like to be able to estimate the adult mortality imposed by traps or targets directly. This is important both for planning control operations and also if

we are to extend our results to other types of habitat and other species of fly, such as the forest and riverine species, that move much less than the Savannah species. The adult mortality imposed by traps or targets will depend on the rate at which flies encounter traps or targets and the proportion of flies that are then killed, as well as on the intrinsic growth rate of the population as illustrated in Fig. 2.

The available field evidence (Williams et al., 1992) suggests that, in uniform habitats, the Savannah species of tsetse flies move diffusively with a root-mean- square displacement of several hundred metres per day. (If flies are released from a central point their spatial distribution approximates to a Gaussian, the width of which increases as the square-root of the time. The root-mean-square displacement is the standard deviation of the Gaussian distribution.) The recovery of a tsetse population in an area from which they had been eradicated showed a logistic growth curve (Turner and Brightwell, 1986). The Fisher equation (Fisher, 1937) therefore provides a good starting point for the analysis (Williams et al., 1992) and the time rate of change of the population density is proportional to a diffusive term, a logistic population growth and a density independent mortality that allows for the effect of the traps. The change in density over space and time, p (r, t), is then:

ap tr, t> ~=U’o’,,(r.t)+rp(r,t)[l-~]-Sp(r.t) at

The key parameters in our model are the diffusion coefficient, CX, the population growth rate, r and the trapping mortality, 6.

Reliable estimates of cr for particular field situations can be obtained by mark- recapture experiments and many such studies have been published (see Rogers,

Fig. 2. Centre: Schematic diagram of a tsetse fly trap showing the radius of attraction, a, and the trapping mortality rate, 6. Left: Schematic diagram illustrating the step length, 1, and the net dispiacement after 16 steps, A. Strictly, k is the root-mean-square value of the net daily displacement.

Right: A graph showing the increase in the number of flies, N, with time for a population growing logistically at a rate I = 1.5%/day. Note that it takes about 5 months to recover from 50% to 90% oE the carrying capacity.

B. Williams/Computers and Eiectronics in Agriculture I3 (1995) 1.55-175 161

1977, and references therein, and Hargrove and Lange, 1989). The life-cycle of tsetse ffies constrains the possible values of r to a very narrow range and field measurements of the pupal, pre-adult and interlarval periods can be used to refine estimates of Y using models for the population dynamics as discussed above.

The parameter that is most difficult to estimate precisely is the trapping mortality. Williams et al. (1992) make a rough estimate of the trapping mortality as follows. We know that at any one time, some flies are inactive: those that have just fed will be resting while they digest their blood meal, heavily pregnant females willi be seeking larvae-position sites rather than hosts, and flies that have recently flown will be building up their reserves of proline. ‘We assume (1) that the proportion of flies that are potentially active at any one time is between 0.6 and 0.75; (2) that of the flies that enter the circle of attraction, the proportion that approach the trap is 0.8 (Vale, 1980); (3) that of the flies that approach the trap, the proportion that enter lies between 0.3 and 0.6 (Dransfield and Brightwell, unpublished data); iand (4) that the time spent in the vicinity of the trap is 5 min (Bursell, 1978). It then follows that the proportion of the flies within the circle of attraction that are killed in 5 min is between 0.14 and 0.36. If each fly spends 30 min per day flying (Bursell and Taylor, 1980) the equivalent instantaneous mortality rate is between 0.91 and 2.7Jday. This estimate of the instantaneous trapping mortality is clearly very crude but does give us an order of magnitude figure. from which to start. As we shall see, the most reliable estimates of the instantaneous trapping mortality are obtained by fitting the model that we develop here to field data and this gives a value of about lo/day.

We start by deriving analytical results for two limiting cases: very mobile flies and inefficient traps; relatively immobile fl.ies and very efficient traps. If the flies are very mobile and the traps are relatively inefficient, the rate at which the fly population is reduced is limited by the range of attraction of the traps and the trapping mortality; if the flies are relatively immobile and the traps very efficient, the rate of reduction is limited by the mobility of the flies.

5.1. Infinite difision: trap limited

If the flies are sufficiently mobile, the probability that a fly is within the radius of attraction of a trap at any time is the proportion of the area that is covered by the traps, na*/n, where n is the density of the traps. From the model of the population dynamics (Eq. 3) the effective mortality rate is then 0.6Slra2n and the density of traps needed to reduce the population at a rate s is:

r+s

n=0.6na26 (5)

We can now use field data from Nguruman to estimate the trapping mortality. During the suppression operation the average trap density was about 2 traps kme2. The population declined at a rate of about 2.6%/day during the dry season when few flies were moving into the suppression zone from the escarpment. With r equal to 1.7%/day and a radius of attraction equal to 100 m, this gives an estimate of the

162 B. Williams/Computers and Electronics m Agriculture 13 (1995) 155-175

trapping mortality 6 of l.l/day. This is in fact a lower limit on the estimate of the trapping mortality for if the flies are less than infinitely mobile the actual trap:ping mortality would have to be greater than the value calculated here. It is important to remember that the estimate of 100 m for the radius of the circle of attraction may well be in error so that, strictly speaking, Eq. 5 allows us to estimate a28 with slome confidence but not S on its own.

5.2. Efficient traps: difSusion limited

The other limiting case of interest is when the traps are very efficient but the flies move relatively little. In this case the rate at which flies are killed is limited by the rate at which they encounter traps. For some of the forest species of fly that move relatively little this might be close to the actual situation. In this case we imagine that the traps are equally spaced over a regular grid and approximate the first term on the right hand side of Eq. 4 over five points, covering four traps at the corners of a square and the point at the centre as shown in Fig. 3. (For details of the expansion see Williams et al., 1992.) This leads to an algebraic equation rela.ting the spacing between the traps, d, and the radius of attraction of the traps, a, to the root-mean-square displacement in one day, h (equal to 22/;;), the population growth rate, Y, and the rate of decline of the population, S, as:

Fig. 3. Schematic diagram of an array of traps to indicate the distances involved in calculating the loss rate of a fly population as a function of inter-trap spacing for perfect traps and relatively immobile flies. We assume that density of flies is zero within each circle of attraction and expand the diffusion equation over the five large solid dots.

B. Williams/Computers and Electronics in Agriculture 13 (199.5) 1.55-l 75 163

For the parameter values used above, Eq. 6 shows that if the root-mean-square displacement in one day is 100 m then using perfect traps a mean spacing of a little more than one trap kmm2 would be sufficierrt to achieve a 99.9% reduction in one year.

It will be important to know whether the control of the flies is trap limited or diffusion limited. If the control is trap limited then it will pay us to design more efficient traps and to identify better odours to act as baits. If the control is diffusion limited then making more efficient traps will not help, and we should rather try to make cheaper traps and use them at higher densities. Combining Eqs. 5 and 6 shows that if the mean-square displacement per day is greater than half the area of the circle of attraction times the effective trapping mortality rate, the control is trap limited, otherwise it is diffusion limited. The model provides us with a set of measurable parameters that we can use to determine whether the control is trap- or diffusion-limited.

The complexity of the habitat and the variability of the flies response to it makes it difficult to simulate particular field situations in detail although this is a goal to which we might aspire. However, we can u,se simulation models to examine the precision of the analytical estimates given by IEqs. 5 and 6.

To keep the problem computationally tractable, simulations were done for lines of traps in which we imagine that the traps are very closely spaced within any one line and we are interested to know how the spacing between the lines of traps affects the rate of reduction. Eq. 4 can then be solved numerically in one dimension. Simulations were done for values of h ranging from 32 to 1000 m/day’i2 and trapping mortalities, S, ranging from 0.1 to 3.2/day. For all combinations of h and S we calculated the spacing required to achieve a 99.9% reduction in one year using the one dimensional equivalents of Eq. 5 or 6 depending on whether the control was trap limited or diffusion limited. These spacings were then also calculated by solving Eq. 4 numerically and adjusting the spacing between the lines of traps to give a 99.9% reduction in one year. The details of the simulations are given by Williams et al. (1992). Here it is sufficient to note that in all cases the analytical approximations give critical trap spacings that. are within 20% of the values obtained from the numerical solution of Eq. 4.

5.3. Other results

The model also leads to several other important results (Williams et al., 1992). First of all it allows us to relate apparent densities as measured by trap catches to actual densities. Secondly, although Savannah flies have been observed to move several kilometres in one day, the advance of a fly front into previously uninfested areas takes place at a rate of only about 5 km/year. From the Fisher equation it follows that the rate of advance of such a front is proportional to the square root of the mean-square displacement per day times the population growth rate. Although the former is large, the latter is very small and the rate of advance of a fly front is slow because it takes the population a long time to build up in a new area. Finally, our model allows us to estimate the width of barriers that are needed to exclude

164 B. Williams I Computers and Ekctronics in Agriculture 13 (1995) 155-l 75

tsetse from cleared areas (Williams et al., 1992). For the Savannah flies the barriers should be 10 to 20 km wide if they are to be effective. The practice of making narrow barriers, often only a few hundred metres wide, is clearly of little value.

6. Behavioural models

The most difficult parameter to estimate in our movement models relates to the trapping efficiency and the best estimates we can currently make are obtained by fitting the model predictions to the results of suppression campaigns. The movement model does not allow us to separate the range of attraction from the trapping mortality as these two parameters always enter the model in combination with each other. Very little direct evidence is available concerning the behaviour of flies as they locate and seek to find a trap although some progress has been made using video recording techniques (J. Brady, N. Griffiths and Q. Paynter, pers. commun., 1993). Given the lack of field data, models may provide useful insights into the factors that are likely to be important in addressing this question and help field ecologists to design their experiments.

It is known that tsetse flies turn upwind when they fly into an odour plume (Gibson and Brady, 1988) and there is evidence that they slow down and turn more rapidly when hying in rather than out of odours. If the wind is uniform and steady, an upwind anemotaxis is clearly a good way for a fly to find the source of an odour. However, in tsetse habitats the winds are often very light and variable (Brady et al., 1989) and the suggestion has been made that by slowing down and turning more often, the flies are executing a klinokinesis (Warnes, 1990).

In both models, anemotaxis and klinokinesis, we assume that the visual range of attraction of traps is 10 m and that the flies can detect odours from a distance of 100 m. The flies therefore start 100 m away from a trap and have found the trap once they are within 10 m of the trap. The fly can acljust its flight speed, the rate at which it samples the odour (under anemotaxis) and the turning rate (under klinokinesis). We then calculate the probability of locating the trap as we vary the directional bias (under anemotaxis) and the available Right time.

6.1. Anemotaxis

Our model of anemotaxis assumes that once the flies detect an appropriate hsost odour they turn upwind, hopefully in the direction of the source. I will let the flight speed be c and the step length be d so that the fly samples the wind direction every r = d/c s. I assume that the flies can only fly for 5 min in a single burst after which they need to rest for 20 min to replenish their supplies of proline which they use as a flight fuel (Bursell, 1978). The first thing that we need is a definition of bias which determines the precision with which they are able to determine the direction to the source of the odour. I assume that each time a fly samples the wind direction and corrects its course, it flies in a direction whose expected value is normally distributed about the true direction of the source with variance ,62. The probability of flying at an angle 0 to the true direction of the source is then:

B. Fi.i.tiwrs/ Computers and Electronics in Agriculture 13 (1995) 155-l 75 165

1 P(O) = - e -e2/2p2

p6

If we define the bias as:

S-J- +CU

B2/2n --oo cos(8) eMs212p2 dt? = e-p212 J’

we can show that after one step the expected displacement towards the source is:

and that the variance of this displacement is:

&(l - 62) v6-j = 2 WO

When /? = 0 the bias is 1 and the fly travels directly towards the source; when p == cc the bias is zero. With this redefinition we can reduce the problem to one-dimension in which we only consider the distance from the fly to the source of the odour after each step (Williams, 1994). The two-dimensional random walk is then equivalent to a one-dimensional random walk where the probability of moving towards the source is (1 + 6)/2 and the probability of moving away from the source is (1 - 6)/Z. The problem is then to find the first time ,at which the fly crosses the 10 m boundary.

Provided the step length is sufficiently small the probability distribution of first arrival times for a biased random walk follows an inverse Gaussian (IG) distribution (Chhikara and Folks, 1989) for which, the probability density function is:

where

the expected duration of flight, increases with increasing distance to the source, a, or decreasing flight speed, c, and bias S. If the: energetic cost of flight is independent of flight speed, as is the case for honey-bees, for example, then tsetse should fly as fast as possible to reduce the expected duration of flight. We can use Eq. 12 to determine an approximate lower limit for the bias. If we restrict the available flight time to 5 min at a flight speed of 5 m/s (Gibson and Brady 1988), then the bias S must be greater than about 0.06 if the fly is to find the source of the odour with a probability of at least 50%.

The shape of the distribution is determined by A, which in terms of the parameters used here is:

a2 h=

c?t(l - 62)

166 B. Williams f Computers and Electronics in Agrkulture 13 (1995) 155-I 75

For small values of h the distribution peaks near the origin and is highly skew to the right. For large values of h the distribution becomes symmetrical and peaks at the expected duration of flight.

The variance of the inverse Gaussian distribution is T3/h (Chhikara and Folks, 1989) so that:

a(1 - 62) V(f) = y3- ‘(14)

The variance of the arrival time also increases with a and decreases with c and 6. The variance also increases with t, however, so that if the sampling time is small (the sampling rate is high) the flies will arrive at the source in a well-defined time and we will refer to this as a “careful-navigation” strategy. If the sampling time is long some of the flies will arrive well before the expected duration of flight while some will only arrive well after the expected duration of flight and we will refer to this as a “point-and-shoot” strategy. We are concerned to determine the conditions under which each of the two strategies is best.

Provided the sampling rate is sufficiently high, the distribution of arrival times agrees well with the predictions of the inverse Gaussian distribution. However, when the sampling rate is low the predictions of the inverse Gaussian distribution break down and we resort to simulations. Simulations were run in which 1000 flies were released from a point 100 m from the source of the odour with biases of 72%, 18%, 9% and 4.5%. Fig. 4 gives the cumulative probability of reaching the source, as a function of time, from a distance of 100 m with the flies sampling the wind direction every 0.01, 0.1, 1 and 10 s. When the bias is relatively high (Fig. 4a, b) the best strategy is to sample the wind as frequently as possible. Provided the wind direction is sampled at least every second, the flies reach the source within 300 s (the maximum flight time in a single burst) with probability close to 1. As the bias decreases a “point-and-shoot” strategy becomes more favourable (Fig. 4c, d) depending on the precise amount of time that is available for flight. For example, if the bias is 5% and the flight time is limited to 300 s, the best strategy is to sample the wind every 1 s and flies will reach the source with about 30% probability. If they sample the wind more frequently they follow a very convoluted path and run out of flight fuel before they have reached the source while if they sample the wind less frequently they may go past the source and never reach it. With a sampling interval of 0.1 s, for example, the probability of reaching the source in 300 s is about 18%. A fly can then, of course, rest for a time and replenish its reserves of proline but if it waits for too long the host animal might well have moved away. With a bias of 5% and a sampling interval of 10 s a fly has a 12% probability of reaching the source in about 100 s but is quite likely to miss the source altogether and the probabilitij of reaching the source in 300 s is only 20%.

6.2. Hinokinesis

Anemotaxis provides an efficient strategy for flies to locate the source of a host odour provided the directional information carried by the wind, and therefore the

I?. Williams / Computers and Electronics in Agriculture 13 (1995) 155-l 75 161

a 100

90

00

70

. 60

5 50

40

30

20

10

0 0 40 80 120 160 200 240 280

Time/l3

c 00

90

80

70

60

50

40

30

20

10

0 0 40 00 120 160 200 240 280

Time/9

i a d

”

0 40 80 120 160 200 240 280 TilUdEi

100

90

80

70

60

50

40

d

0 40 80 120 160 200 240 280 rime/a

Fig. 4. Cumulative distribution functions showing the probability that a fly reaches an odour source

before the given time. The fly starts 100 m from the sourmce of the odour at time zero and flies at 5 m/s. Once the fly is within 10 m of the source it is assumed to have found the source. The bias is 72%, 18%, 9% and 4.5% in (a), (b), (c) and (d), respectively. In each figure calculations are done for a range of

values for the sampling interval, the time interval at which the fly changes direction, as indicated on the figures in units of seconds.

bias, is sufficiently high. However, winds in typical tsetse habitats are often light and variable, having to pass through dense thickets, and it may be that there is insufficient directional information in the wind for anemotaxis to be useful. An alternative strategy is to use a klinokinesis by reducing the flight speed and increasing the turning rate. It is therefore of interest to consider the conditions under which klinokinesis might facilitate host finding by tsetse flies. The potential for using a kinesis to locate the source of an lodour is more difficult to assess than the effect of bias. IIere we are concerned with a situation in which the fly is able to detect the presence or absence of an odour but with no directional information. We need a model of fly movement in which the flies decide at successive moments in time to turn to the left or to the right, relative to their current direction. If we let the time intervals go to zero in a suitable fashion we will then have a model in which the flies can vary their flight speed or their turning rate and we can relate this to the overall displacement of the flies and determine the effect of changes in flight speed and turning rate on the ability of the fl.ies to find a source of odour.

168 B. Williams I Computers and Electronics in Agriculture 13 (1995) 155-I 7.5

Let us therefore consider a discrete model of movement in which the flies take a series of steps, each of length d. Suppose that the first step makes an angle $1 to the x-axis in a Cartesian co-ordinate system; the next step makes an angle &. to the direction of the first step and so on. We then assume that successive angles & are chosen from a normal distribution >with mean zero and variance a2 and we let the time step tend to zero. The expected value of the square of the displacement at time t is then:

E(g) = T - !g [l _ .n%2] W

where cr2 is the turning rate measured in radians squared per unit time. [Skellam (1951), Dunn, (1983) and Williams (1992) discuss the measurement of sinuosity in correlated random walks.] In the limit t --+ 0, Eq. 1.5 becomes:

E(r2) = c2t2 - c2a2t3/6 (16)

so that for very small times, while the path is straight, the root-mean-square displacement is simply the velocity times the time. As the time increases and the path begins to curve, the second term in Eq. l6 becomes significant and the expected displacement falls below the value given by the first term. When cr’t is greater than about 4 the exponential term in Eq. 15 is no longer significant and E(r2) is equal to the first term minus a constant sh.ift. Once 02t, the mean-square angle turned, is greater than about 20 rad2/s the shift becomes insignificant and the fly will appear to move according to a classical random walk with mean-square displacement per unit time given by:

A2 = !$ (17) (Note that the Central Limit Theorem allows us to relax the assumption that the distribution of turning angles is Gaussian and we only demand that the probability of turning left is the same as the probability of turning right.)

Fig. 5 shows how the path of the fly cha.nges as the turning rate changes. For a mean-square turning rate of lop4 rad2/s, the flies path is almost straight and it covers a distance of about 1.5 km in 5 min (Fig. 3a). Increasing the mean-square turning rate to 1tY2 rad2/s the flight path begins to curve so that it covers the ground more thoroughly. Increasing the mean-square turning rate further to 1 rad2/s the flight path becomes very convoluteld but it now only covers the ground out to a distance of about 150 m. Finally, with a mean-square turning rate of 100 rad2/s the movement rapidly becomes diffusive and the fly only covers the ground out to 35 m in 5 min.

Provided a2t is sufficiently large we can treat the correlated random walk as a standard two-dimensional random walk. We need to proceed cautiously, however. If a fly moves a distance d in a direction uniformly distributed around a circle ,then the expected displacement from a point a distance a away increases by an amount d2/4a (Williams, 1994). A two-dimensional random walk with no bias is therefore equivalent to a one-dimensional random walk with a negative bias where the bias is:

B. Williams/Computers and Electronics in Agriculture 13 (1995) 155-l 75 169

a 0.3

0.0 -

0.7

0.6 -

E 0.5 Y 1 II.4

0.3 -

0.2 -

0.1

0 0 0.2 0.4 0.6 0.a 1 1.2

x/km

b 10.9 -

13.8

0.7 -

0.6

1 0.5 -

2. 0.4

0.3 -

11.2

0.1 -

0 0 0.2 0.4 0.6 0.B 1

x/km

c d 170 35 160 ::: 30

130 ::x 25

100 20 e

‘x 2 e 70

'"R 15

2 10

% 5

:x 0

-18 -180

-5 -140 -100 -60 -20 20 60 -4 0 4 a 12 16 20 24

r/m x/m

Fig. 5. Correlated random walks with mean-square turning rates equal to (a) 0.0001 rad’is; (b) 0.01

rad2/s; (c) 1 rad*/s; and (d) 10 rad2/s. As the mean square turning rate increases the sinuosity of the path decreases.

&2cz aaZ

and the variance is:

(18)

(19)

For a random walk with negative bias, Chhikara and Folks (1989) show that the probability of reaching the source in any time less than infinity is:

P(t c: 00) = e2”lV = e-l = 0.3’7 (20)

In other words, if the flies execute a two-dimensional correlated random walk under the conditions defined above, the probability of reaching the source can never exceed 37% even if there is no limit on the available flight time.

Provided the turning rate is sufficiently high the inverse Gaussian approximation can be used to determine the cumulative distribution function but for the values of interest it is again necessary to carry out simulations. As before we “release” a fly at a distance of either 100 m from an odour source and assume that once it is within 10 m of the source it has effectively found the source. Once the odour has

170 B. Williams I Computers and Electronics in Agticulture 13 (1995) 155-I 75

a d

Fig. 6. Cumulative distribution functions showing the probability that the fly reaches an odour source before the given time. The fly starts 100 m from the source at time zero and flies at 5 m/s. Once the fly is within 10 m of the source it is assumed to have found the source. There is no directional information

in the signal. Cumulative probabilities are shown for mean-square turning rates of 0.01, 0.1, 1 and 10 rad*/s as indicated on the figure.

been detected we assume that the fly maintains a constant turning rate even if it subsequently moves more than 100 m away from the source. Fig. 6 then shows Ihe probability of finding the source of the odour within 5 min.

When the turning rate is very small (0.01 r2/s) the flight path is almost straight and the probability of finding the sou.rce, 3.2 %, is given by the solid angle subtended by the ten metre circle at 100 m divided by 2~. The flies either reach the source in d/c = 1X s or they never reach it. As the turning rate is increased to 1 r2/s the probability of reaching the source very quickly drops but since they search the area more thoroughly the cumulative probability of reaching the source in 3~00 s increases. However, if the turning rate is increased to 10 r2/s the path that .they follow becomes very convoluted and the probability of reaching the source even in 300 s drops.

We see from Fig. 6 that if the flight time is limited to 300 s, the highest probability of reaching the source that can be achieved is about 14% when they start 100 m away. The same probabilities of success can be achieved using a biased random walk when the bias is only 5% even if the flies only sample the wind direction every 110 s. As for the biased random walks the optimal strategy depends critically on the time available for flight and if the flight time is limited a “point-and-shoot” strategy is better than an “intensive-search” strategy.

As expected, anemotaxis provides a very efficient way of locating the sourcse of an odour but only if the bias is greater than about 5%. Klinokinesis is considerably less efficient and at best enables the flies to locate the source of the odour with a probability of about 1.5%. Both models show that this result depends critically on the frequency with which the flies are able to sample the odour and on the time that they have available for flight. If th.e bias is very low and the Aight time

B. Williams / Computers and Electronics in Agriculture 13 (1995) 15.5-175 171

is limited a “point-and-shoot” strategy turns out to be optimal while if the bias is high or the flight time is not limiting a “careful-navigation” or an “intensive-search” is best.

7. Consistency models

The study site is an area of about 100 km2 of dense forest bounded on the west by the Rift Valley escarpment which rises steeply by about 500 m and beyond which is the Maasai Mara. To the east the land is dry and barren. The suppression of tsetse flies at Nguruman started in January 1987 (Dransfield et al., 1990) with a trap density in the suppression zone of about l-2 traps per square kilometre.

Fig. 7 shows the changes in the numbers of female tsetse flies for various ovarian age categories. The effect of the traps during the first year of the suppression was not altogether as expected. During the first month of the suppression, when it was still dry, there was a substantial reduction of about 90% for both male and female Glossina pallidipes, the dominant species in the area, but the numbers increased slightly during the rains from April. to June. During the following dry season, the numbers again fell steadily, the reduction reaching 99% in October. Then in November, when the short rains started, the numbers increased dramatically over a few days with 90% of the increase being due to female flies. The numbers soon declined under the trapping pressure but a similar increase was again observed in January.

The picture that emerged from this study was as follows. The traps were placed mainly in thicker vegetation where the flies tend to concentrate during the dry season so that the effective trapping mortality is higher during the dry season than it is during the rains. Furthermore, when the rains come flies spread out rapidly from the escarpment leading to the invasions that were observed in November and January of the following year. The females spread more rapidly than the males and the older females more rapidly than younger females.

In order to decide if this description was reasonable we developed what, for want of a better term, I will call a “consistency model” (Dransfield et al., 1990). During the routine population monitoring, the flies had been sexed and the age of the females determined by identifying their ovarian age classes. In the model the flies were then divided into pupae (with a pupal period of 27 days), ovarian class 81 (0 to 18 days after emergence), classes 1 to 3 (each covering 9 days) and all flies in age classes four and above. Once the flies reach age class 1 we assume that they deposit one larva every 9 days with an abortion rate of 5%. We then varied the following parameters in our model: the initial number of flies, the mortality, the trapping efficiency and the proportion of invading flies in each age class, as well as the number of flies that invade in each week of the year. We have 162 independent data points but 69 degrees of freedom.

With this model we were able to get good fits to the data, as shown by the dotted lines in Fig. 7, with the following parameter values. Of the invading flies 25% are in age class 0, 48% in age classes 1 to 3 and 27% in age classes 4 and above. The trapping efficiency was about the same for flies in all age classes but the overall

L72 B. Williams I Computers and Electronics in Agriculture 13 (1995) 155-l 75

a

;, 01, , ( , , , , , , , , , , , JFMAMJJASONDJF

Month

I I I I I I1 , I I I

J F M A M J .I A S 0 N D J F

Month

Month

Fig. 7. Apparent densities from trap catches of female $7. pallid&es (solid lines) of (a) nulliparous flies (up to 18 days after emergence); (b) young parous flies (18 to 45 days after emergence); and (c) old

parous flies (more than 4.5 days after emergence. Dotted lines are model predictions.

mortality increased from ll%/day for flies in age class 0, to 13%/day for flies in age classes 1 to 3, to 36%/day for flies in age classes 4 and above. The model gives high rates of invasion during the rains and very little or no invasion pressure during the dry season.

B. Williams/Computers and Electronics in Agrkulture 13 (1995) 155-I 75 173

With so many parameters we should certainly hope to get a good fit to the data but of course with so many parameters the minimum is very flat and each! of the best fit values, could be varied by up to 50% without increasing the residual sums-of-squares by more than a factor of two. The point of the model is then not to make precise predictions or even to obtain accurate parameter estimates but simply to confirm that the descriptive model that we developed from our knowledge and experience could be quantified in a way that provided reasonable estimates of the various parameters.

8. Summary

The models that have been developed in relation to the Nguruman tsetse control project have served several distinct roles. The population models allowed us to determine the level of adult mortality that we needed to impose in order to achieve a given level of control. The movement models allowed us to estimate the trap density that we need in order to achieve the desired level of adult mortality. The actual suppression campaign provided field tests of the models that enabled us to refine some of the parameter values that were used. The behavioural models provide the first steps towards developing a more detaiied understanding of the factors that determine trap efficiency. The consistency model allows us to confirm the validity of our understanding of a rather complex series of events involving intermittent reinvasions from neighbouring areas.

These models also show the need to measure various parameters thzt have not previously been measured. For example, the movement model shows that we should carefully determine both the radius of attraction of our traps and the diffusion rate of the flies in order to decide if the control is going to be trap limited or diffusion limited. Incidentally, it provides an explanation for the very slow rate of advance of a tsetse fly front (a few kilometres per year) when we know that individual flies can cover tens of kilometres in a day. It also allows us to determine the widths of barriers that would be needed to prevent reinvasions of cleared areas and to explain why previous attempts at establishing barriers have often failed. The behavioural model shows that if we are to understand the way in which flies locate traps or hosts in the field we need to know the time for which a fly can sustain flight in the field. This time is believed to be about 5 min but the evidence is weak slight. This model also shows that we need to determine the frequency with which a fly can sample an odour and that we need to think about ways to estimate the effective bias presented by the wind-born odour to tsetse flies in the field.

The modelling work at Nguruman has helped us to plan our control effectively and the results of the control have helped us to refine our models. The models have also enabled us to think about the effect of traps on populations of tsetse flies in a more structured way and have led LIS to consider several new aspects of the flies behaviour. Hopefully, the results of our modelling efforts will make it possible to generalize the experience gained at Nguruman to other places and to other species of flies provided we can measure the relevant biological parameters.

Perhaps the most important lesson from the Nguruman experience is that ‘we

174 5. Williams JComputers and Electronics in Agriculture 13 (19951 155-I 75

should no longer regard research and control as mutually exclusive activities. The development of the control programme at Nguruman has benefited greatly from, our models but only because they were models of the actual situation pertaining in the field. Furthermore, the models have helped LIE to identify some of the key biological factors that determine efficacy of the control programme and these should in turn help us to generalize the results to other habitats and other species of flies. However, the control has also benefited the research. In particular, it has led to the discovery of very rapid, synchronized invasions from the escarpment down into the study area by flies 90% of which are females. Although we have not yet begun to understand the nature of these invasions they represent a new and unexpected form of tsetse fly behaviour. Finally, the control of tsetse at Nguruman is now being carried out by the Maasai who live there as part of the Olkirimatian and Shompole Community Development Project, the very first example of a successful tsetse control project that is run entirely by a local community.

Acknowledgements

I thank David Rogers and Sarah Randolph for their advice, help and support and the Royal Society for a Guest Fellowship at Oxford University during which this paper was written. I thank Robert Dransfield and Robert Brightwell who were responsible for the development of the Nguruman project and are now involved with the Olkirimatian and Shompole Development Project, for teaching me about tsetse and the members of the Olkirimatian Group Ranch, especially Joel Larinkoi, for their support and friendship without which none of this work could have been done.

eferemces

Brady, J., Gibson, G. and Packer, M.J. (1989) Odour movement, wind direction and the problem of host-finding by tsetse flies. Phys. Entomol., 14: 369-380.

Brightwell, R., Dransfield, R.D., Kyorku, C., Golder, TK., Tarimo, S.A. and Mungai, D. (1987) A new

trap for Glossina pallidipes. Trop. Pest Man., 33: 151-159. Brightwell, R., Dransfield, R.D. and Kyorku, C. (1991) Development of a low-cost tsetse trap and

odour baits for Glossina pallidipes and Glossina lwgipennis in Kenya. Med. Vet. Entomoll., 5: 153-164.

Bursell, E. (1978) Quantitative aspects of proline utilizarion during flight in tsetse flies. Phys. Entomol., 3: 265-272.

Bursell, E. and Taylor, P. (1980) An energy budget for Glossina (Diptera: Glossinidae). Bull. Entomol.

Res., 70: 187-196. Chhikara, R.S. and Folks, J.L. (1989) The Inverse Gaussian Distribution: Theory, Methodology and

Applications. Marcel Dekker, New York, N.Y. Dransfield, R.D., Brightwell, R., Kyorku, C. and Williams, B.G. (19SO) Control of tsetse P,y (Diptera:

Glossinidae) populations using traps at Nguruman, south-west Kenya. Bull. Entomol. Res.; 80:

265-276. Dransfield, R.D., Williams, B.G. and Brightwell, R. (1991) Control of tsetse flies and trypanosomiasis:

myth or reality? Parasit. Today, 7: 287-291. Dunn, G.A. (1983) Characterizing a kinesis response: the average measures of cell speed and

directional persistence. Agents Actions Suppl., 12: 14-33.

B. Williams I Computers and Electronics in Agriculture 13 (1995) 155-175 175

Epstein, H. (1971) The Origin of Domestic Animals, Vol. 1. African Publishing Corporation, New

York, N.Y.

Fisher, R.A. (1937) The wave advance of advantageous genes. Ann. Eugen., 7: 335-369. Ford, J. (1971) The Role of the Trypanosomiases in African Ecology. Clarendon Press, Oxford. Gibson, G. and Brady, J. (1988) Flight behaviour of tsetse flies in host odour plumes: the initiai

response to leaving or entering odour. Phys. Entomol., 13: 29-42. Hargrove, 9. and Lange, K. (1989) Tsetse dispersal viewled as a diffusion process. Trans. Zimbabwe Sci.

Assoc., 64: l-8.

Kyorku, C., Brightwell, R. and Dransfield, RD. (1990) Traps and odour baits for the tsetse fly, Glo.rsina iongipennis (Diptera: Glossinidae). Bull. Entomol. Res., 80: 405-415.

Pielou, E.C. (1977) Mathematical Ecology. Jcshn Wiley, New York, N.Y., p. 60. Rogers, D.J. (1977) Study of a natural population of GJossina fisclpes fuscipes Newstead and a modei

of fly movement. J. Anim. Ecol., 46: 309-330. Skellam, J.G. (1551) Random dispersal in theoretical populations. Biometrics, 38: 196-218.

Turner, D. and Brightwell, R.D. (1986) An evaluation of a sequential aerial spraying operation ag,ainst Glossina pallidipes in the Lambwe Valley of Kenya: aspects of post-spray recovery and evidence of

natural population regulation. Bull. Entomol. Res., 64: 545-588. Vale, G.A. (1980) Flight as a factor in the host finding behaviour of tsetse flies (Diptera: Glossinidaej.

Bull. Entomol. Res., 70: 299-307. Warnes, M. (1990) The effect of host odour and carbon dioxide on the flight of tsetse flies (Glossina

spp.) in the laboratory. J. Inst. Phys., 36: 607-611. Williams, B.G. (1992) The measurement of slnuosity in correlated random walks. J. Theor. Biol., 155:

437-442.

Williams, B.G. (1994) Models of trap seeking by tsetse flies: anemotaxis, kinesis and edge detection. J. Theor. Biol., 168: 105-115.

Williams, B.G. and Williams, G.P (1992) Science for Development. Perspect. Biol. Med., 36: 64-78. Williams, B.G., Dransfield, R.D. and Brightwell, R. (199’0) Tsetse fly (Diptera: Glossinidae) population

dynamics and the estimation of mortality rates front life-table data. Bull. Entomol. Res., 80: 479-

485. Williams, B.G., Dransfield, R.D. and Brightwell, R. (1992) The control of tsetse flies in relation to By

movement and trapping efficiency. J. Appl. Ecol., 29: 163-179. Williams, B.G., Dransfield, R.D., Brightwell, R. and Rogers, D.J. (1993) Trypanosomiasis: where. are

we now? Health Pollut. Plan., 8: 85-93.

Williams, B.G., Campbell C.M. and Williams, R. (1995) Broken houses: science and development in

the African savannahs. Agric. Human Val., 12: 29-38.