Transmission of Theileria: ILRAD's requirements
Transmission of trypanosomes: ILRAD's requirements
Models for Leishmania transmission
The transmission dynamics of Theileria parva
Spatial factors in the assessment of trypanosomiasis challenge
Session discussion
A.S. Young
International Laboratory for Research on Animal Diseases
P.O. Box 30709
Nairobi, Kenya
The International Laboratory for Research on Animal Diseases (ILRAD) carries out active research in this area being an essential part of the epidemiology of theileriosis, and thus may be able to generate information essential to the disease modeller. Attempts have been made to quantify the transmission processes for Theileria parva but results have shown that it is complex. Many of the processes involved in transmission are multifactorial; generally, they can be divided into those driven by the host, the parasite, tick and the environment. These factors do not necessarily operate in isolation. ILRAD's attempts to quantify some of these processes are illustrated as follows. A large database has been assembled on the laboratory infection of cattle by T. parva, related to the infections of feeding ticks and this has been extensively analysed. However, until recently it had not been possible to quantify the relationship between the infective stage of the parasite for the tick, the piroplasm, to the resulting levels of infection in the tick. This made the ILRAD requirement for the production of ticks with predictable infection levels difficult. It is evident that the vectorial capacity of ticks varies with the instars, the sex and the population of ticks. Furthermore, in any tick population examined, overdispersion of Theileria infection has been detected; a small proportion of the tick population becomes infected or highly infected with T. parva. Factors controlling the infectivity of T. parva populations and the susceptibility of tick populations are under investigation.
The mechanism by which cattle are infected with T. parva by feeding ticks is also being investigated with a view to the development of control strategies for the disease, particularly in designing novel vaccines. The survival of T. parva within the tick can be quite different under varying climatic conditions. A complication, both in cattle and ticks, is that one individual may be harbouring more than one species of Theileria. New methods for differentiating species within cattle and ticks are being developed. The nature of the host population, in their susceptibility to infection by T. parva, is important and needs further quantification. Progress on the in vitro feeding of ticks may allow study of transmission of Theileria without complication of the host factors. It is hoped that new data generated from these studies will be useful in the developing and improving of transmission models of T. parva infection which in turn may lead to better understanding of the epidemiology of theileriosis. However, there are very few longitudinal studies of cattle available in sitespecific situations. Data on such sitespecific situations has been shown to be useful in modelling the transmission of T. parva and therefore ILRAD's research efforts should be directed to support sitespecific studies in different epidemiological zones. ILRAD can be beneficial in the way of encouraging national programs to use the right techniques and the right approaches to provide the relevant information to disease modellers.
S.K. Moloo
International Laboratory for Research on Animal Diseases
P.O. Box 30709
Nairobi, Kenya
The epidemiology of animal trypanosomiasis in tsetseinfested areas of Africa involves the relationship amongst the four biological factors, namely trypanosomes, reservoir hosts, tsetse flies and livestock, operating within the physical environment, which determine the distribution and frequency of the disease in livestock populations in the endemic regions. Certain changes in this dynamic interaction may result in an epidemic of trypanosomiasis in livestock. The epidemiology of animal trypanosomiasis is therefore a complex ecology since it involves two types of mammalian hosts, i.e. species of wild animals and livestock, which differ with regard to their reservoir potential for trypanosomes and the level of susceptibility to Trypanosoma vivax, T. congolense and T. b. brucei stocks, and a diverse tsetse species having a somewhat different range of hosts and with differing vector competence for different stocks of the three parasite species.
The transmission of trypanosome infections to livestock by tsetse flies is but a component of this complex cycle of transmission, and hence it cannot be regarded as an independent entity. Yet, it might be possible to formulate an accurate predictive model of this essential component. To achieve this goal, ILRAD's efforts need to be directed towards research, in at least two trypanosomiasis endemic areas, one in East and the other in West Africa, to quantify the following factors: tsetse species present in the identified areas, their distribution and abundance; vector sex ratio, their feeding interval and survival rate; infection rates in tsetse by T. vivax, T. congolense, T. simiae and T. b. brucei; incidence of mixed trypanosome infections in the vectors; trypanosome transmission coeficients, from tsetse vectors to the mammalian hosts and vice versa; species of wild mammals present, their distribution, abundance and their possible trypanosome reservoir potential for tsetse; proportion of tsetse bloodmeals from different species of wild hosts; size of the livestock populations; proportion of tsetse bloodmeals from livestock; and incubation period, infectiousness and immunity in the mammalian hosts of tsetse to the three pathogenic trypanosome species. This study will probably lead to a better understanding of the relationship amongst the involved biological factors, and hence provide the basis for formulation of the model of transmission of trypanosome infections to livestock by tsetse flies.
C. Dye
Department of Medical Parasitology
London School of Hygiene and Tropical Medicine
Keppel Street, London WC1E 7HT, UK
Abstract
Introduction
Structural deficiencies of the vectorial capacity equation
Difficulties of parameter estimation
Comparative analysis
Absolute estimates of Leishmania transmission rate?
Conclusions
References
Discussion in this paper is restricted to the problem of using models as a basis for estimating the absolute transmission rate of leishmaniasis, in particular the basic reproduction number, R_{0}.
I begin by noting that it is beguilingly simple to adapt, in principle, the theory of mosquito vectorial capacity to phlebotomine sandflies, and to other insect vectors. However, vectorial capacity is rather intractable in practice, whatever system it is applied to. There are two reasons why this is so, with two consequences.
The first reason is that the basic formula, originally used in malariology, is structurally deficient. For example, it does not account for heterogeneous contact rates between vectors and hosts (nonrandom biting rates), or parasiteinduced mortality of vectors. Second, several of its parameters, such as vector mortality, are very difficult to estimate. There has been practically no effort to assess the confidence limits associated with published estimates of vectorial capacity, and many such estimates are not to be trusted.
The first consequence is that we need to be clear as to when vectorial capacity and related formulae are actually useful. I argue that effort will be expended more efficiently, and that conclusions will be more robust when such formulae are used to answer wellfocused, comparative questions.
Second, if absolute estimates of transmission rate are needed, they will be more credibly obtained by adapting methods which have been developed for directlytransmitted infections. I give one example where the formula Ro @ 1/s* (where s* is the equilibrium fraction of host animals susceptible) has been modified and applied to visceral leishmaniasis in dogs. I also describe the real difficulties involved in applying even such a simple formula to a parasitic infection, which are mainly to do with distinguishing between infection and infectiousness. The argument leads, in a tantalizing way, to the conclusion that the usual method of identifying animals to be treated or culled could result in effective disease control, but without having permitted an accurate assessment of the magnitude of the problem in the first place.
Macdonald's (1957) malaria model is one of the bestknown of all mathematical models of infectious diseases. One product of Macdonald's work was a definition of the basic case reproductive rate, R_{0}, for a disease transmitted by a mosquitolike vector. R_{0} is the average number of secondary cases which arise from each primary case when infection is introduced into a population consisting almost entirely of susceptibles. It is thus a mean maximum rate of spread of infection through a community. According to Macdonald:
R_{0} = ma^{2}bp^{n}/rlnp (1)
where m is the number of mosquitoes per person, a is the daily biting rate of a female mosquito on man (as opposed to other hosts on which bites are wasted from the viewpoint of Plasmodium), b is the probability an infectious mosquito actually transmits infection when biting, p is the daily mosquito survival rate, n is the time taken for plasmodia to mature to infectious sporozoites in mosquitoes (the extrinsic incubation period), and r is the daily human recovery rate from infectiousness. To focus the minds and hence the activities of entomologists, GarrettJones (1964) extracted all the transmission components of R_{0} as the vectorial capacity, C,
C = ma^{2}bp^{n}/lnp (2)
C is the number of secondary cases which arise from the bites taken by all mosquitoes on one infectious person in one day. Equations (1) and (2) have been enormously valuable in identifying the relative importance of the different components of transmission. A classic argument used by MacDonald was that, since p is raised to the power n in the numerator, and appears in the denominator too, C (and thus R_{0}) ought to be particularly sensitive to changes in vector survival rate. He made this proposition just as the appropriate tool became available to test it: DDT was hugely successful as a residual insecticide, even in many areas which were highly endemic for malaria.
But this is a qualitative argument. There are two main difficulties with using these formulae quantitatively. The first is that they are structurally deficient. The second is that most of their components are very hard to measure. These difficulties exist whether we are interested in mosquitoes and malaria, sandflies and leishmaniasis, or tsetse flies and trypanosomiasis (Dye, 1992). Thus, although this paper is primarily concerned with leishmaniasis, we can illustrate these general difficulties in the next two sections by drawing on examples from malariology too.
The following are four of the many assumptions made by equations (1) and (2).
1. All mosquitoes feeding on a person carrying infectious parasites actually acquire infection. The probability that any vector picks up parasites from an infected host is in fact much less than one, and needs to be represented by including another parameter, say c, which has a value between zero and one (Nedelman, 1984).
2. Vectors bite hosts at random. This is almost never the case: for example, some people live nearer to breeding sites than others, and mosquitoes tend to bite adults rather than children because they are bigger (Port et al., 1980). The consequences of nonrandom host choice are that R_{0} will always be bigger than expected under the assumption of random biting (Dye and Hasibeder, 1986; Hasibeder and Dye, 1988).
3. Individual members of a vector population behave in a uniform way. There is growing evidence for genetically determined differences in host choice within mosquito populations, for which there must be epidemiological consequences. For example, V. Petrarca and J.C. Beier (cited by Coluzzi, 1992) have shown that the distribution of karyotypes among Anopheles arabiensis which have fed on man is different from those which have fed on animals.
4. Survival rate of vectors does not vary with age. Clements and Paterson (1981) have reviewed the evidence for both Anopheles and Culex mosquitoes and concluded that survival rate commonly declines with age. A graph of log numbers of individuals (captured by whatever method) against age is often not a straight line (indicating constant survival rate), but a curve with increasingly negative slope (Figure 1).
Among the components of equations (1) and (2), we focus on four problems of getting m, a and p from field data.
1. Manbiting rate, m is the number of vectors per person. Absolute vector density is very difficult to measure indeed. For example, different methods of interpreting markreleaserecapture data often give substantially different of vector population size (Sheppard et al., 1969). One solution is to measure m and a together in the product ma, which is the daily vector biting rate per person. In malariology, this has been done by trained baitcollectors spending entire nights, in shifts, inside and outside houses. In general, the results probably get close to the true mean biting rate. In recent years, the procedure has been euphemistically renamed the 'manlanding catch', but this does nothing to reduce malaria as an occupational hazard. Consequently, many research workers have now stopped using mainbaited catches, especially in areas where the parasites are drug resistant. The alternative is to use traps, such as CDC miniature light/suction traps, which can at least record proportional changes in vector biting rate.
2. Vector survival rate. The pattern suggesting agedependent survival seen by Clements and Paterson, and described above, has also been seen in a population of the sandfly Phlebotomus ariasi (Figure 2). But in this case we know that the shape of the curve actually reflects dissection error rather than survival rate falling with age (Dye et al., 1987). Do the mosquito data suffer from the same problem? A second difficulty is the assumption that parasites have no impact on vector survival rate. There is much laboratory evidence, now supported by field data (e.g. Lyimo and Koella, 1992), showing that parasites such as Plasmodium can significantly reduce vector survival. Of course, all field estimates of survival rate are obtained by working with the entire vector population, in which the prevalence of infection is generally very low.
3. Vector host choice. Parameter a in equations (1) and (2) includes the probability that a mosquito takes any of its bloodmeals on man. In malariology, this probability is measured by the Human Blood Index (HBI). Although HBI is simply the proportion of all bloodmeals taken on man, it is hard to measure because representative samples are needed from all mosquito resting sites (GarrettJones et al., 1980). That is, we need to be able to catch with the same efficiency in cattle pens, animal burrows, and vegetation etc. Since different sampling methods need to be used in these different circumstances, catches will rarely be comparable.
4. Vector biting rate (or the interval between bloodmeals, i). This is the other component of parameter a. It can be measured in markreleaserecapture experiments, but accuracy is often limited by low rates of recapture (Dye et al., 1991). A second technique, developed for mosquitoes but applicable in principle to other vectors too, uses time series analysis, and the fact that all parous mosquitoes found in a population at time t must have been produced by all female mosquitoes at time ti. The cross correlation coefficient between the parous time series and that for total females should reach a distinct maximum when the lag is i days (Birley and Rajagopalan, 1981). In practice, a distinct maximum is not always found (Charlwood et al., 1985). Underlying this technique is the further assumption that vectors are gonotrophically concordant, that is, they take one bloodmeal in each gonotrophic cycle. In fact, many bloodsucking insects take several bites during each cycle. The number of hosts an individual bites before acquiring enough blood to mature one batch of eggs may depend on the frequency with which feeds are interrupted.
Figure 1. Log numbers of fieldcaptured female mosquitoes (triangles), plotted as a function of calendar and physiological age (parous number), together with estimated agespecific mortality rates (per ovarian cycle, circles). (a) Culex quinquefasciatus, (b) Mansonia uniformis. These data, obtained by counting follicular relics in ovaries, suggest that mortality rate increases with age. From Clements and Paterson (1981).
Figure 2. Log numbers of Phlebotomus ariasi sandflies plotted against ovarian age, for three sites in the Cévennes, France. The curves are nonlinear because of dissection error; they are not evidence that mortality increases with age. From Dye et al. (1987).
Considering structural deficiencies and the problems of parameter estimation, the emergent conclusion is that equations (1) and (2) cannot easily be used to estimate the absolute transmission rate. The recommended alternative is to use entomological indices of transmission, like the vectorial capacity, in a comparative way. One example is given in the next section.
Transmissionblocking vaccines are a potentially important method of malaria control. Anticipating the experiment, Saul et al. (1990) developed a method of estimating the probability that a mosquito acquires infection at each bite, K. This is the quantity which would be reduced by vaccination. Their estimate is seven by:
K = D_{b} (1  P_{f})/[QP_{f} (1  D_{b})] (3)
in which D_{b} is the proportion of vectors infected in each biting catch, P_{f} is the probability of surviving each feeding cycle, and Q is the proportion of feeds taken on humans, the Human Blood Index discussed above. Parameters Q and P_{f} are the most difficult to measure accurately; D_{b} is made more accessible by the availability of immunoassays.
A useful index of vaccine efficacy would be the ratio v = K (after vaccination)/K (before vaccination). But if this is all we need then we note that Q and P_{f} are unchanged by vaccination, that 1  D_{b} @ 1, which gives v @ D_{b} (after vaccination)/D_{b} (before vaccination). Thus a robust estimate of v can be obtained by measuring just one relatively tractable parameter. Notice too that biases in D_{b} are unimportant, provided they are identical before and after vaccination. There may be occasions on which we need to know actual K's before and after vaccination, but we certainly do not always need to know them.
The principal message of the previous sections is that measurements of transmission should depend on few, tractable parameters. If we do wish to estimate R_{0}, there are more direct methods than that given by equation (1). One such is due to Dietz (1975) and to Anderson and May (1991) who have shown for directly transmitted viral infections that, if L is the life expectancy of the vertebrate host and A the average at which infection is acquired, then R_{0} = 1 + L/A. Where L >> A, R_{0} @ L/A. This formula has a simple, intuitive interpretation. On average, R_{0} > 1 is required for an epidemic to occur. The ratio L/A just says that an epidemic will occur provided hosts live long enough, on average, to get infected.
We can adapt this formula for canine leishmaniasis (due to Leishmania infantum or L. chagasi) but in doing so must account for the fact that this is a disease for which death, rather than immunity, follows a durable period of infectiousness. The result is (C. Dye and G. Hasibeder, unpublished data)
R_{0} = 1 +[L_{t}+P_{d}L_{i} (1 +E_{f})]/A (4)
Here P_{d} is the probability a dog survives the latent period, L_{t} is the expectation of the latent period, L_{i} is the expected duration of infectious life, and E_{f} is the expected number of bites taken by an infectious fly. Estimation of R_{0} for canine leishmaniasis by this route is therefore rather more awkward than for common childhood viral infections, and we move to an alternative.
There is a second simple but useful formula, also derived for stably endemic, directly transmitted infections in homogeneously mixing communities: R_{0} = 1/s*, where s* is the fraction of hosts susceptible to infection at equilibrium. Modifying this for a vectorborne disease like leishmaniasis gives R_{0} = 1/(s*u*) (Dye et al., 1992; Hasibeder et al., 1992). Here, u* is the fraction of vectors which are uninfected at equilibrium, but this is approximately equal to one, and in practice may be ignored.
Figure 3. The frequency distribution of antibody titres obtained from a population dogs on the island of Gozo, Malta. Sera were subjected to IFAT for L. infantum. The distribution is unimodal, giving no suggestion that one frequently used cutoff point (arrowed) decisively separates infected from uninfected animals. From Dye et al. (1992).
Less easily dismissed is the question of nonhomogeneous biting rates by sandflies. As already mentioned, mathematically convenient homogeneous biting rates are the exception rather than the rule. Assuming that a single population of sandflies bites at different rates in different patches of dogs, we get
_{}
Now we have n_{i} dogs in the ith patch. The l 's are, conveniently, the relative (rather than absolute) magnitudes of the forces of infection (instantaneous incidences) on dogs in different patches.
Serology is the usual method of estimating both s and l. However, the data are commonly hard to interpret. For many parasitic infections, frequency distributions of antibody titre are unimodal, showing no clear cutoff point between dogs which have been infected and those which have not (Figure 3). In a recent study of canine leishmaniasis on the Maltese island of Gozo, we explored the consequences of choosing three different but plausible cutoff points for ELISA, IFAT and DAT (Dye et al., 1992). As expected, estimates of l were rather insensitive to the choice of cutoff point. More surprisingly, they were insensitive to the assumption of homogeneous biting rates. But estimates of R_{0} varied from 1.6 to 11.1. The difference is extremely important: the lower estimate suggests that eradication could be achieved if the vector population were reduced by 38%, whereas the upper implies that a reduction of 91% would be needed.
To investigate further the performance of serological tests in leishmaniasis epidemiology, we have attempted to calculate sensitivity and specificity of IFAT during a longitudinal study on a cohort of 50 dogs in southern France (Dye et al., 1993). A combination of clinical signs and the success of efforts to isolate parasites were used as a 'gold standard', albeit an imperfect one. Figure 4 shows that, following infection during the transmission season in June and July, sensitivity took as long as eight to nine months to reach a satisfactory 80%, and this was maintained for just two months. Evidently, the IFAT will commonly underestimate infection rate in crosssectional studies.
The vectorial capacity has been a hugely important concept in medical entomology. As well as providing ammunition for advocates of control by adulticide, it also been used in 40 years of teaching to make plain the components of transmission by mosquitolike vectors. Quantitative application of formula (2) has, however, been less successful. The formula has important structural deficiencies, and its parameters are hard to measure accurately. Consequently, medical entomologists should not use it to measure absolute transmission rate. They should use it as a starting point for answering particular comparative questions. The index of relative vaccination success described above is one example. Others are (Dye, 1992): What is the most important vector in an area? What best explains geographic and temporal variation in the incidence of infection? Why did vector control have no impact on the prevalence of infection? Which of two alternative control methods is likely to have the greatest impact on prevalence?
But not all the important quantitative questions in epidemiology are comparative ones. Mindful of the difficulties of getting R_{0} via the vectorial capacity, we have tried to adapt two, more direct methods for canine leishmaniasis. One of these leads to a relatively simple formula which requires accurate estimates of incidence rate only. But even this simple formula is hard to apply in practice because serodiagnosis of Leishmania infection is an imprecise science. Our limited success in estimating absolute transmission rates underlines the general theme of this paper: in quantitative epidemiology, comparative questions are far more tractable.
Figure 4. Sensitivity (squares) and specificity (diamonds) of IFAT for L. infantum in a cohort of 50 dogs over 11 months following the 1989 transmission season. Sensitivity took eight to nine months to reach a maximum of around 80%. From Dye et al. (1993).
ANDERSON, R.M. and MAY, R.M. 1991. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press, 757 pp.
BIRLEY, M.H. and RAJAGOPALAN, P.K. 1981. Estimation of the survival and biting rates of Culex quinquefasciatus (Diptera: Culicidae). Journal of Medical Entomology 18: 181186.
CHARLWOOD, J.D., BIRLEY, M.H., DAGORO, H., PARU, R. and HOLMES, P.R. 1985. Assessing survival rates of Anopheles farauti (Diptera: Culicidae) from Papua New Guinea. Journal of Animal Ecology 54: 10031016.
CLEMENTS, A.N. and PATERSON, G.D. 1981. The analysis of mortality and survival rates in wild populations of mosquitoes. Journal of Applied Ecology 18: 37399.
COLUZZI, M. 1992. Malaria vector analysis and control. Parasitology Today 8: 113118.
DIETZ, K. 1975. Transmission and control of arbovirus disease. In: Ludwig, D. and Cooke, K.L., eds. Epidemiology. Philadelphia: Society for Industrial and Applied Mathematies, pp. 104121.
DYE, C. 1992. The analysis of parasite transmission by bloodsucking insects. Annual Review of Entomology 37: 119.
DYE, C. and HASIBEDER, G. 1986. Population dynamics of mosquitoborne disease: effects of flies which bite some people more frequently than others. Transactions of the Royal Society of Tropical Medicine and Hygiene 80: 6977.
DYE, C., DAVIES, C.R. and LAINSON, R. 1991. Communication among phlebotomine sandflies: a field study of domesticated Lutzomyia longipalpis populations in Amazonian Brazil. Animal Behaviour 42: 183192.
DYE, C., GUY, M.W., ELKINS, D.B., WILKES, T.J. and KILLICKKENDRICK, R. 1987. The life expectancy of phlebotomine sandflies: first field estimates from southern France. Medical and Veterinary Entomology 1: 417426.
DYE, C., KILLICKKENDRICK, R., VITUTIA, M.M., WALTON, R., KILLICKKENDRICK, R., HARITH, A.E., GUY, M.W., CAÑAVATE, M.C. and HASIBEDER, G. 1992. Epidemiology of canine leishmaniasis: prevalence, incidence and basic reproduction number calculated from a crosssectional survey on the island of Gozo, Malta. Parasitology 105: 3541.
DYE, C., VIDOR, E. and DEREURE, J. 1993. Serological diagnosis of leishmaniasis: on detecting infection as well as disease. Epidemiology and Infection 103: 647656.
GARRETTJONES, C. 1964. Prognosis for the interruption of malaria transmission through assessment of the mosquito's vectorial capacity. Nature 204: 11731175.
GARRETTJONES, C., BOREHAM, P.F.L. and PANT, C.P. 1980. Feeding habits of anophelines (Diptera: Culicidae) in 197178, with reference to the human blood index: a review. Bulletin of Entomological Research 70: 165185.
HASIBEDER, G. and DYE, C. 1988. Mosquitoborne disease dynamics: persistence in a completely heterogeneous environment. Theoretical Population Biology 33: 3153.
HASIBEDER, G., DYE, C. and CARPENTER, J. 1992. Mathematical modelling and theory for estimating the basic reproduction number of canine leishmaniasis. Parasitology 105: 4353.
LYIMO, E.O. and KOELLA, J.C. 1992. Relationship between body size of adult Anopheles gambiae s.l. and infection with the malaria parasite Plasmodium falciparum. Parasitology 104: 233237.
MacDONALD, G. 1957. The Epidemiology and Control of Malaria. London: Oxford University Press.
EDELMAN, J. 1984. Inoculation and recovery rates in the malaria model of Dietz, Molineaux and Thomas. Mathematical Biosciences 69: 209233.
PORT, G.R., BOREHAM, P.F.L. and BRYAN, J.H. 1980. The relationship of host size to feeding by mosquitoes of the Anopheles gambiae Giles complex (Diptera: Culicidae). Bulletin of Entomological Research 70: 133144.
SAUL, A.J., GRAVES, P.M. and KAY, B.H. 1990. A cyclical feeding model for pathogen transmission and its application to determine vectorial capacity from vector infection rates. Journal of Applied Ecology 27: 123133.
SHEPPARD, P.M., MacDONALD, W.W., TONN, R.J. and GRAB, B. 1969. The dynamics of an adult population of Aedes aegypti in relation to dengue haemorrhagic fever in Bangkok. Journal of Animal Ecology 38: 661702.
G.F Medley
Department of Biological Sciences
Warwick University
Coventry, CV4 7AL, UK
Abstract
Introduction
Description of model
Summary of results
Discussion
References
This paper describes a quantitative framework which allows epidemiological field observations to be analysed and used to examine the transmission dynamics of Theileria parva. Initially the situation in endemically stable areas (characterized by continuous activity of the tick vector, Rhipicephalus appendiculatus) is considered, as this allows consideration of transmission at a stable equilibrium. The data used are: 1) the rate at which cattle are infected; 2) the prevalence of infection of ticks; 3) the rate of infection of ticks fed on 'carrier' cattle (i.e. cattle that have been exposed to infected ticks and survived the initial infection).
A dynamic compartmental model of cattle infection is described. It incorporates five stages of cattle infection: susceptible (uninfected), incubating infection, clinically diseased cattle that will survive, clinically diseased animals that will die from infection and cattle that have survived the initial infection ('carriers'). Estimates of the rates at which cattle move through the model are taken from the literature. Initially, the rate of infection is taken from the field estimate and the model is shown to characterize the infection within a cohort of calves born into an endemically stable area.
The model is adapted to include a transmission term that describes the rate of infection of ticks by cattle. Using the model and data estimates above, the dynamic stable equilibrium is characterized. The model is used to demonstrate three conclusions. First, that 'carrier' animals must be responsible for the majority of tick infections. Second, that reduction in tick density only reduces the prevalence of infection in cattle appreciably when the effective tick feeding rate is reduced to about 20% of its equilibrium level. Third, that the creation of 'carrier' animals by infection and treatment immunization does not alter the equilibrium perceptively as the majority of animals are 'carriers' already.
This model requires extension in several areas. First, the biology of 'carriers' and their infection of ticks requires more study: carrier infection may be as a consequence of continual exposure and subclinical infection or as a consequence of survival of the original infection in immunologically privileged sites. The details of this interaction will have considerable impact on the transmission dynamics. Second, the model should be extended to seasonal areas, which will require explicit consideration of the tick population dynamics.
Theileria parva, an apicomplexan parasite, is the aetiological agent of East Coast fever (ECF) in cattle, and is transmitted by the ixodid tick Rhipicephalus appendiculatus (for a review of the epidemiology of T. parva see Norval et al., 1992). East Coast fever is an important disease of cattle in subSaharan Africa. It has been responsible for major epidemics in the past century (Norval et al., 1992) killing millions of cattle. Epidemics remain possible today when control methods are ceased, or the infection is introduced into previously uninfected areas. The economic losses to theileriosis are considerable (Mukhebi et al., 1992) and the effect of T. parva is to substantially reduce the productivity of cattle in Africa, both in terms of food and monetary value, and prevent the unconstrained introduction of improved breeds of cattle.
The transmission of T. parva is complex. The mammalian hosts of the infection are many, and there is considerable genetic diversity within populations of Theileria spp., complicating the classification of the parasite. Rhipicephalus appendiculatus is a threestage tick (larva, nymph and adult) which feeds three times during the life cycle on a variety of different mammalian hosts. Each species in this complex has its own biology, much of which is poorly understood. This paper continues a process of clarification of this situation with the eventual aim of developing models that are useful in the design of control programs against ticks and the infections they transmit. In order to be able to make some progress, it is necessary to make some gross, simplifying assumptions about the population dynamics.
This preliminary work focuses on the epidemiological state referred to as endemic stability. Within such areas there is a high level of infection in both cattle and ticks, but with little mortality from infection due, in the main it is thought, to the genetic constitution of the hosts, which have been subjected to T. parva infection for many generations (Perry et al., 1992). The details of the model framework and its analysis can be found in Medley et al., (1993). Here I confine myself to outlining the biological assumptions contained within the model, the results and the possibilities for developing the model. I also consider two aspects that were not discussed previously: the interaction of two control policies (tick control and immunization by infection and treatment) and the possible influence of host age.
The model is based on the division of the cattle population into five distinct categories (Figure 1). Calves are born susceptible to infection at a rate equal to total mortality in order to keep the host population size constant. On infection, cattle begin incubating the disease, the period of which lasts 15 days. During this period they are not infectious, but the infection is developing. After incubation, animals move into one of two acute disease stages, which corresponds to the onset of clinical symptoms. The division of the acute disease state into two groups is to make the model tractable, rather than being a biologically based assumption. One of these groups dies from the infection, with an average survival time of four days from onset of disease. The other group recovers with an average time of 15 days between onset of disease and recovery. On recovery, the animals enter an immune or carrier state, where they remain for life.
Estimates of the parameters described above were taken from the literature. The only parameter for which an estimate does not exist is the rate of infection. Data were taken from Moll et al. (1986) which describe the number of infected calves by age out of a cohort of 31. As endemic stability is assumed (i.e. there is no change in the transmission dynamics over time), age is synonymous with time. The estimate of the constant rate of infection was 0.016 per day, corresponding to a mean time from birth (= exposure to infected ticks) to infection of 63 days. The analysis showed that the rate of infection was agedependent, but the current model is not agestratified, so this result was not used in the model. The combination of the estimate of the rate of infection and the structure outlined above and in Figure 1 is a description of the infection and disease status of a cohort of animals in an endemic area.
Figure 1. Diagrammatic representation of the model used. Each box represents a disease state of cattle that can be assessed immunologically and/or parasitologically.
Figure 2 allows a comparison of the observed data and the model results. As the rate of infection is estimated from the same data shown in the figure, it is to be expected that the rise in infection with schizonts is well modelled. However, the correspondence with the development of piroplasms and the development of immune/carrier state is due to the model structure and assumptions. It would appear that the model provides a good description of the progress of infection through a cohort of cattle in endemic areas.
Figure 2. Results of the cohort model with agerelated rate of infection estimated from data given by Moll et al. (1986) (see Medley et al., 1992). The remaining parameters were taken from the literature (text and Figure 1). The figure shows the expected proportion of animals in each infection state compared with the observed state from Moll et al. (1986). The faster rising points (+) are the observed appearance of schizonts, and the lower points (x) are the first appearance of piroplasms. The infected curve includes all those animals not susceptible.
We can use the estimate of the rate of infection to gain some insight into tick activity. The rate of infection is the product of the rate at which ticks feed, the probability that a tick is infected and the probability that infection is passed from an infected tick to a susceptible host. Moll et al. (1986) give a direct estimate of the probability of a tick being infected as 0.0227 (95% confidence interval: 0.0150.033). If we assume that an infected tick has a high probability (90%) of passing the infection to the susceptible host it feeds on, then the number of tick feeds per day is 0.78. Equivalently, there is a successful tick bite every 1.28 days on cattle in the endemic area. This prediction could be tested empirically to validate the assumptions within the model thus far.
The model is developed to become a dynamic description of Theileria transmission, with the infection rate in cattle being dependent on the infection in ticks, which in turn is governed by the infection in cattle. Three of the cattle infection states (Figure 1) are allowed to be infectious: both acute disease stages and the immune/carrier state. This is done to appraise the influence that a carrier state has on the transmission. By carrier state, I denote cattle that have survived a primary infection as carriers. They are able to infect ticks (with some probability), but do not exhibit any clinical signs of disease. The assumption here is that their infectious status is maintained by continuous infection derived from the primary infection. However, it may be that these animals require continual infection to remain infectious. The effect of this will require further investigation. Throughout their life cattle are subject to mortality at a rate which corresponds to a life expectancy of four years at birth. This value was estimated from demographic data of cattle in an endemically stable area (Moll et al., 1984; Medley et al., 1993).
There is one dynamic equilibrium within the model, whereby the number of cattle entering each infection state is equal to the number leaving within a specified time period. I assume that this dynamic equilibrium is the state at which endemically stable areas are. The equilibrium results can be used with the rate of infection estimated above and the published estimates of progression through the infection classes (Figure 2) to assess the transmission potential of T. parva within the endemically stable area. The basic reproductive rate, R_{0}, is the most convenient parameter for encapsulating this potential (Anderson and May, 1991). In this case, the most appropriate definition of the basic reproductive rate is the number of infected ticks that would arise after one cycle of transmission if one infected tick were placed in completely susceptible populations of ticks and cattle. The value of the basic reproductive rate is 23, and the expected proportion of animals in each infection state is given in Table 1. When transmission occurs in parallel (i.e. there are three disease states in cattle transmitting infection simultaneously), then the basic reproductive rate can be separated into the components representing the contribution of each state to the overall transmission: the carrier state is the most important in terms of transmission, accounting for 95% of infection in ticks.
Table 1. Proportions of cattle in each infection class (Figure 1) calculated from the model. Note that the two acute compartments have been summed.
Infection Status 
Equilibrium Proportion 
Susceptible 
0.043 
Incubating 
0.01 
Acute 
0.01 
Carrier 
0.937 
There are field and laboratorybased estimates for all the parameters in the model except the infection rate of ticks fed on animals in the acute disease state. Given the estimates of the overall tick infection rate (0.0227, 95% confidence interval: 0.0150.033) and the proportions of cattle in each disease state derived from the model, the infection rate to ticks of each disease state of cattle is constrained. Young et al. (1986) estimated the infection rate of ticks fed on carrier animals to be 0.023 (95% confidence interval: 0.0160.030). The surprising result is that the infection rate of cattle undergoing acute infections is not strongly determined within the model structure (Medley et al., 1993). The best estimate of the infection rate to ticks of acute cattle is 10%, but values of 0.0% and 100% are compatible with the two observations and their confidence intervals. The reason for this is the comparison between the length of time that cattle reside in each infection state. The average time from commencement of overt clinical signs to recovery is of the order of 15 days, whereas the carrier state is probably lifelong. Consequently, the total number of ticks feeding on acutely infected cattle compared to carrier cattle at any one time is very small (15 days/4 years), and their infection rate is largely inconsequential in endemically stable environments.
Thus far, the model has concentrated on the results generated by considering the stable equilibrium condition. However, it is also instructive to consider the dynamics of infection, when the populations are not at the stable equilibrium, but change over time until that equilibrium is attained. The dynamic analysis is important because, first, it may give clues to the important processes in nonendemically stable areas (e.g. where tick activity is seasonal) and, second, the introduction of control is a perturbation that changes the stable equilibrium, and the populations must change to reattain the new equilibrium with control.
Control of theileriosis currently rests on two methods: reduction in the density of vectors by intensive use of acaricides, and immunization. Immunization, achieved currently by simultaneously infecting cattle and treating with chemotherapeutics to prevent acute disease, was shown to tee effective almost two decades ago (Radley et al., 1975). Following immunization, cattle are in the same infection state as cattle recovered from natural infection: they are immune to further disease, but able to support infections transmissible to ticks. These two control methods alter the population dynamics of the vector and infectious agent in a manner which is not easily predicted, and in addition their influence on transmission is opposite: tick control reduces the rate of infection and immunization creates infectious cattle.
The Carrier State
The major result of this work has been to demonstrate the importance of the carrier state in transmitting T. parva in areas which are endemically stable. The vast majority of infected ticks are derived from feeding on carrier animals. This can also been seen by comparing the infection rate of ticks in the field (2.27%) with the infection rate of ticks fed on carrier animals (2.3%). The infection rate of cattle showing acute clinical signs during their first experience of T. parva infection (the only parameter within the model for which there are currently no estimates) is undetermined.
Tick Infection Probabilities
The dynamic properties of the model can be evaluated by tracking the dissemination of infection through a population of susceptible cattle following the introduction of a single infected animal. The epidemic pattern is largely determined by the infection rate to ticks of cattle with acute clinical signs of infection. When this parameter is 1 (all ticks become infected), the epidemic is very peaked and susceptible cattle become infected quickly. When this parameter is 0 (no ticks become infected), then the dissemination of infection to susceptible cattle is dependent on carrier animals infecting ticks at a low rate, and is consequentially much slower. The net effect is that high infectivity of acute cattle produce a pronounced epidemic with morbidity and mortality confined to a short interval. As infectivity of acute cattle is reduced, then the epidemic becomes more extended and total mortality is reduced. The current understanding is that acutely infected cattle have a high rate of transmission to ticks, perhaps of the order of 70% (A.S. Young, personal communication). This understanding is in line with observed outbreaks of East Coast fever which have been characteristically associated with high morbidity and mortality in a short time (Norval et al., 1992).
Tick Control
This model is further used to evaluate the effect of different control measures on the transmission of T. parva. Theileria parva transmission can be halted by sufficient reduction in the feeding rate of the tick vector. The tick feeding rate required for eradication is the reciprocal of the basic reproductive rate, 0.034, which translates to an average time interval between successive tick feeds of 30 days. Figure 3 shows the effect of tick control on the equilibrium average at infection. With no tick control (relative tick feeding rate unity) the average age at infection is 63 days, and as the tick feeding rate is reduced, so the average age at infection rises. A reduction in tick feeding rate by half only increases the average age at infection to 130 days, and the feeding rate must be reduced to 16% of precontrol to increase the average age at infection to one year. Likewise the relationship between the proportion of animals infected with T. parva shows nonlinear relationship with a significant reduction not observed until the tick feeding rate is reduced to 20% of its equilibrium rate. The nonlinearity arises because as tick control is more intensively applied the infection rate in cattle is reduced and less cattle become infected, thus reducing the infection rate in the surviving ticks. This pattern explains why tick control has not been effective in reducing T. parva prevalence unless very intensively applied.
Figure 3. The equilibrium average age at infection and proportion of hosts infected with T. parva as functions of the relative tick feeding rate. These results are calculated taking the tick feeding rate estimated from the endemically stable area (0.78) as unity, and comparing the rate of infection when this is reduced by reduction in the tick population by acaricidal control.
The dynamic pattern of tick control is stable, with the overall level of infection reduced to the equilibrium shown in Figure 3 according to the level of control. The new equilibrium is attained over a period of about three years and is related to the life expectancy of carrier animals. The equilibrium in the presence of control is such that the proportion of susceptible cattle is increased. If tick control is ceased, the return to the precontrol equilibrium emulates the results observed for the introduction of T. parva into a susceptible population. The more acutely infected cattle are available to ticks, the faster subsequent dissemination of infection throughout the susceptible cattle population. The precontrol equilibrium is reattained within months of the cessation of control.
Immunization
Immunization has little direct effect on the dynamics of T. parva transmission in endemically stable areas. As the majority of cattle in these areas are carriers already, immunization reduces morbidity and mortality in direct proportion to the amount of immunization, but does not change the rate of infection to unimmunized animals. The dynamic changes following the commencement of an immunization program are very stable with the new controlderived equilibrium being attained within a year. On cessation of the control program, the precontrol equilibrium is reattained on the same timescale.
The value of the controlderived equilibrium is dependent on the infection probabilities of cattle to ticks. If tick infection relies more on cattle with acute clinical signs, and therefore less on carrier cattle, then immunization will have a more beneficial effect as it reduces the proportion of cattle with acute clinical signs, and consequently reduces the prevalence of infection in the ticks. This effect is, however, relatively marginal.
Tick Control and Immunization Combined
Tick control and immunization are commonly instituted in combination. The interaction between these two control methods is interesting as they have opposite effects. Tick control generally reduces the infection rate in cattle, consequently reducing infection to ticks, and further reducing infection in cattle. Immunization artificially creates infection, thus maintaining infection in ticks, and maintaining the tickderived infection rate in unimmunized cattle. The major effect of the combination is that tick control reduces natural infection rates, thus increasing the average age at natural infection (Figure 3), and so making immunization more effective. Increasing the average age at infection increases the 'window' between birth and tickderived infection during which immunization must occur to be effective.
Figure 4 shows the stable equilibrium situation for the combination of the two control policies. The graph charts the diseaseinduced mortality rate (per host per day) for different tick control intensities from no control (1) down to control which eradicates the tick (0) for five different immunization efforts (measured as the average age at immunization in days from 200 down to 25). Note that complete eradication of the tick is required to eradicate infection when immunization is operating, but that without control there is a tick density below which transmission cannot be sustained. The major effect is that even relatively modest immunization efforts do not allow the nonlinear pattern in tick control alone to develop. The effect of tick control is linearized, so that even less successful tick control policies are effective in reducing mortality. This is explained as tick control enhances the effect of immunization by increasing the average age at infection, and so makes immunization easier. The relative costs of each policy and the intensity with which they are applied would determine the best policy in terms of cost/benefit.
Figure 4. The effect on T. parvarelated mortality of simultaneous immunization and reduction in tick feeding. The lines are the instantaneous diseaseinduced mortality rates (per host per day) as a function of the relative tick feeding rate (as in Figure 3) for five different immunization strategies from no immunization down to an average age at immunization of 25 days.
Figure 5 shows the same relationship, but now charting diseaseinduced mortality as a function of immunization effort for different tick control intensities from no control (100% of precontrol feeding rate) down to 5% of precontrol tick feeding rate. The pattern can be explained by comparison with Figure 3. With no tick control, the average age at infection is approximately 60 days, but reducing the tick feeding rate to 50% increases the average age at infection to approximately 130 days (Figure 3), and it is at this point on Figure 5 that immunization becomes more effective, in that a slight increase in immunization effort reaps a higher reward in mortality reduction.
Figure 5. The same relationship as drawn in Figure 4, but the different lines show the diseaseinduced mortality as a function of the immunization effort for five different tick feeding rates relative to precontrol from 100% (= no tick control) down to 5%.
The dynamic introduction of a combined policy can be readily understood. Immunization prevents the creation of a large pool of susceptible animals. Consequently, at the cessation of control an epidemic is prevented in contrast to the situation with tick control alone.
This paper has extended the results derived from a model described in a previous paper (Medley et al., 1993) by considering the interaction of two control methods for T. parva, namely, reduction in the tick feeding rate and immunization by infection and treatment. The type of model used (a deterministic compartmental model based on ordinary differential equations) is useful for preliminary investigations. It allows robust analytical solutions, which can lead to general conclusions. In this case, we can use the model results to demonstrate the importance of carrier animals in endemically stable areas, but also show that in areas that are not stable, the importance of infection to ticks by animals suffering acute clinical signs. Immunization by infection and treatment does not alter the pattern of infection in endemically stable areas, but reduces death due to East Coast fever in a simple fashion related to the effort given to control. Reducing the tick feeding rate (by acaricide application or grazing management) does not have a significant effect until very low feeding rates are achieved, i.e., until the tick population is virtually nonexistent. However, reducing the tick attachment rate does increase the average age at which cattle experience tickborne infection, thus making immunization easier (in terms of increasing the age over which immunization may successfully be given). While these conclusions are robust within the assumptions of the epidemiology of T. parva in endemically stable areas, they may not be directly applicable to areas that do not fulfil these criteria, and several mechanisms may act to alter this pattern.
First is the effect of host age. As the median age at infection in endemically stable areas is of the order of 60 days, we have not assumed that there is any effect of host age on progression of disease. However, this may not be the case. The mammalian immune system continues to develop after birth, and it is likely that the age at which an animal first experiences T. parva will to some extent determine the outcome of the infection. In particular, the proportion of animals that develop a carrier state and the proportion of animals that die from the infection are both likely candidates for modulation by host age. In areas of seasonal tick activity, and therefore seasonal transmission, the cattle will tend to be older when they first experience infection than in areas with transmission all the year round. The same will be true when the tick population is reduced by some control measures. Obviously, more information on the biology of the effect of host age at infection is required to enable its impact on the transmission dynamics to be quantified. In particular, the use of immunization to create carrier cattle in areas of seasonal transmission may have counteractive effects.
Second is the influence of the tick that infects an animal. A tick carrying T. parva infection has one or more acini infected within the salivary glands. The distribution of infection amongst adult ticks is highly skewed, with most ticks being uninfected, most of those infected having a single acinus infected, and only a very small number having more than one acinus infected. It is already accepted that the greater the number of acini infected on the infecting tick, the more serious the clinical consequences of infection to the host. Further, the distribution of infection in ticks is likely to be influenced by the infection in the infecting host. Thus not only will the proportion of ticks infected by feeding on acute rather than carrier animals be higher, but also the distribution of intensity of infection may be altered.
Third is the effect of tick stages. Most of the epidemiological work to date has been concerned with the adult tick (that becomes infected as a nymph). However, the nymph (infected as a larva) can also transmit infection, but little is known about the prevalence or intensity of infection in nymphs, largely as a result of the difficulty in counting and collecting this stage in the field. It may be that infection from a nymph can induce different clinical consequences of infection than infection from an adult tick, and, again with respect to transmission dynamics, it is the diseaseinduced death and the development of the carrier state that are most important in this context. Larvae and nymphs feed for a much shorter time than adult ticks, but most acaricide application programs are designed to kill adult ticks and not prevent larval infection and nymphal transmission.
Finally are the effects due to the interaction between came and ticks. It is well accepted that cattle can mount a strong and effective immune response to R. appendiculatus that is acquired after past exposure to the ticks. What is not clear is the consequence of this response on the infection rate both to and from ticks. While the acquired immune response may reduce the feeding effectiveness and thereby reducing transmission to ticks, it may also result in increased concentrations of the lymphocytes at the biting site, facilitating their infection by T. parva sporozoites and increasing infection from ticks. Further the disease and clinical symptoms of animals suffering acute infections coincide with a reduced immune response, perhaps increasing the proportion of feeding ticks that become infected from acutely diseased animals.
A quantitative framework (mathematical model) is useful in two respects. First is allows observations on fairly disparate areas of research to be brought together in one structure. Thus, it is possible to evaluate the effect of transmission by carrier cattle within the context of all aspects of transmission. Second, the effects of interventions on the transmission dynamics may be counterintuitive and nonlinear, which suggests that only by examining the numerical details will the full ramifications of interventions be appreciated. The construction of the model also highlights the most important areas with respect to both future research and intervention possibilities.
The modelling approach adopted by Medley et al. (1993) and in this paper is powerful in that it can generate general principles. However, that generality is gained by adoption of the smallest possible number of assumptions required to generate the observed patterns. Before the results can be confidently applied, these assumptions require validation, and the importance of factors omitted, such as those outlined above, must be ascertained. As with all scientific methods, the current model has generated more questions than have been answered. However, the model does provide a good research tool in that it establishes a qualitative understanding of the transmission dynamics and produces a quantitative framework, both of which are essential for development of planned control policies. As more detail is incorporated into this model framework, so the model will become more useful in detailed design of the quantitative aspects of control policies, their effect on production, and their economic implications.
ANDERSON, R.M. and MAY, R.M. 1991. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press, 757 pp.
MEDLEY, G.F., PERRY, B.D. and YOUNG, A.S. 1993. Preliminary analysis of the transmission dynamics of Theileria parva in eastern Africa. Parasitology 106: 251264.
MOLL, G., LOHDING, A. and YOUNG, A.S. 1984. Epidemiology of theileriosis in the TransMara Division, Kenya. Husbandry and disease background and preliminary observations on Theileria in calves. Preventive Veterinary Medicine 2: 801831.
MOLL, G., LOHDING, A., YOUNG, A.S. and LEITCH, B.L. 1986. Epidemiology of theileriosis in calves in an endemic area of Kenya. Veterinary Parasitology 19: 255273.
MUKHEBI, A.W., PERRY, B.D. and KRUSKA, R. 1992. Estimated costs of theileriosis control in Africa. Preventive Veterinary Medicine 12: 7385.
NORVAL, R.A.I., PERRY, B.D. and YOUNG, A.S. 1992. Epidemiology of Theileriosis in Africa. London: Academic Press, 481 pp.
PERRY, B.D., DEEM, S.L., MEDLEY, G.F., MORZARIA, S.P. and YOUNG, A.S. 1992. The ecology of Theileria parva infections of cattle and the development of endemic stability. In: Munderloh, U. and Kurtti, T., eds. Proceedings of the First International Conference on TickBorne Pathogens at the HostVector Interface. St. Paul, Minnesota: College of Agriculture, University of Minnesota, pp. 290296.
RADLEY, D.E., BROWN, C.G.D., CUNNINGHAM, M.P., KIMBER, C.D., MUSISI, F.L., PAYNE, R.C., PURNELL, R.E., STAGG, S.M. and YOUNG, A.S. 1975. East Coast fever. 3. Chemoprophylactic immunization of cattle using oxytetracycline and a combination of theilerial strains. Veterinary Parasitology 1: 5160.
YOUNG, A.S., LEITCH, B.L., NEWSON, R.M. and CUNNINGHAM, M.P. 1986. Maintenance of Theileria parva parva in an endemic area of Kenya. Parasitology 93: 916.
T.J. Wacher*^{+}, P.M. Milligan^{++}, P. Rawlings*^{§} and W.F. Snow*
* International Trypanotolerance Centre
PMB 14, Banjul, The Gambia^{+} Zoological Society of London
Regent's Park, London NW1 4RY, UK^{++} Unit for Statistics and Epidemiology
Liverpool School of Tropical Medicine
Pembroke Place, Liverpool L3 5QA, UK^{ §}Institute for Animal Health
Pirbright Laboratory
Ash Road, Pirbright, Woking GU24 0NF, UK
Abstract
Introduction
Methods
Results
Discussion
Acknowledgements
References
The severity of the trypanosomiasis problem in a particular location is traditionally assessed in terms of the challenge index  the product of apparent density of tsetse, or trap catch per day, and infection rate  which is assumed to be proportional to the force of infection. However this index masks variation in the force of infection between herds and among individuals within herds. It is also not necessarily comparable between sites since the relative abundance of tsetse to hosts may vary. We have studied the spatial distribution of herds of livestock in relation to tsetse and calculated an index of challenge based on the ratio of vectors to hosts. This index is strongly correlated with estimates of the force of infection calculated from the incidence of infection in susceptible zebu cattle; and it provides information on heterogeneity in exposure of different herds to tsetse. We argue that spatial distribution of vectors to hosts is a prerequisite for the application of epidemiological models of vectorborne diseases to real field situations.
Trypanosomiasis challenge is traditionally estimated as the product of some measure of tsetse abundance and infection rate (i.e. the proportion of tsetse with mature infections). For example, the African Trypanotolerance Livestock Network of ILCA and ILRAD uses biconical trap catch per day x infection rate (Leak et al., 1988); this measure may be modified by including the proportion of tsetse blood meals taken from cattle (Leak et al., 1990). A challenge index such as this is a straightforward and inexpensive value to estimate in the field, which gives an indication of the risk of trypanosomiasis infection. Rogers (1985) found that the log of the challenge index was roughly linearly related to the Berenil index. The challenge index is assumed to be proportional to the force of infection, l, which is defined as the number of potentially infective bites per animal per day (Smith and Rennison, 1958). If l is constant during time t, 1exp(l t) is the risk of trypanosomiasis infection to susceptible animals in time t. 1/l is the average interval between potentially infective bites received by the host, and the average waiting time to first in susceptibles.
This challenge index suffers from two drawbacks: firstly, it is proportional to l only if the relative abundance of hosts, and the exposure of cattle to tsetse, are constant. The inclusion of proportion of cattle feeds in the index only partially addresses this problem, since differing preferences by tsetse amongst the varying range of alternative hosts, plus spatial heterogeneity in contact patterns at different sites, still affect the estimate. If these vary, estimates of the index will not be comparable between locations. Secondly, the index gives no indication of variability in l amongst individual animals, herds, or livestock species. Theoretical studies have indicated that for several systems heterogeneities in transmission rate can also have important consequences for the rate of spread of disease; recent field studies have addressed this issue for schistosomes (Woolhouse et al., 1991), but it has not previously been undertaken for trypanosomiasis.
In this paper we examine the importance of space utilization by N'Dama cattle (T.J. Wacher and W.F. Snow, unpublished data; Rawlings et al., 1994) in relation to tsetse distribution in determining the challenge rate to village herds in two study sites in The Gambia. The results are discussed in terms of the ratio of tsetse numbers to host numbers, which is referred to as tsetse exposure. The central objective is to measure the extent to which exposure to tsetse varies with spatial factors at the level of herd and season, and to compare this with the naive estimate from the challenge index in routine use which assumes uniform distribution of cattle and tsetse, as well as uniform infection rate amongst tsetse. Data on tsetse infection rates are available in the current study (Rawlings et al., 1990), but not at a scale of resolution adequate to determine spatialand temporal variation, so this factor has not been used in our estimate of exposure.
Field work reported here was carried out at two study sites in The Gambia, Keneba and Niamina East (Figure. 1). At each place cattle ranging and ecological studies were carried out in conjunction with livestock censuses, warthog transect counts and tsetse population studies. At each location a small herd of sentinel zebu cattle was maintained in order to provide an independent estimate of force of infection.
The unit of measurement for cattle range use was the daily grazing trail of one animal from selected herds, mapped by grid references taken every five minutes throughout the day. This was done on 820 days for each study herd in each fourmonth season. The sum of grid references falling within each 500 x 500 m grid unit of the study area provides an estimate of the space utilization distribution for each herd, which is weighted according to the number of animals in that herd.
Stock censuses and warthog transect counts were carried out monthly. Other wild hosts of G. m. submorsitans, while known to be present, were seen so infrequently that no population density estimates were possible, and they are not taken into account in analysis.
A small herd of 10 sentinel zebus was established for one year during the study period at each site, bled weekly and treated with Berenil when infected with trypanosomiasis (Claxton et al., 1992). These data were used to provide an estimate of the frequency with which cattle received fresh infections.
Tsetse populations were monitored at each site by semisystematic arrays of modified F3 blue box traps (Green and Flint, 1986), set for three days each month. A markreleaserecapture experiment was conducted to provide an estimate of trap efficiency. Trap catch data were converted to absolute densities of flies using this estimate, and the results interpolated using an inverse distance squared function in the program SURFER (Golden Software Ltd.) to convert point estimates to a regular 500 x 500 m grid.
These data were analysed to examine the effect of spatial heterogeneity on the vector to host ratio over each season. Analysis proceeded by calculating for each cell of the 500 x 500 m grid the ratio of tsetse to all hosts (composed of all cattle herds, small stock and warthogs) and for each herd summing the contribution of each cell weighted by the proportion of grazing time by that herd at that location. This spatially weighted estimate of tsetse/host ratio was compared with the 'crude' estimate which was derived from the simple ratio of total number of tsetse and total number of hosts in the study area.
The analysis assumes that spatial and temporal scale factors in recording intervals were appropriate, that all members of the herds used the seasonal ranges in an equivalent way and that range use was constant throughout each season. It was also assumed that warthogs were evenly distributed in woodland and scrub habitats and effectively absent elsewhere.
Cattle and Tsetse Distributions
The patterns of spatial and temporal distributions measured in this system at Keneba are summarized in Figure 2. These maps illustrate the data sets that were used to calculate exposure and show the seasonal changes between tsetse distribution and abundance and cattle grazing range. They indicate that at Keneba cattle range utilization was heavily focused on the village; through the early dry season of 1988, all village herds exploited the fields around the village and woodland areas to the south, while tsetse were relatively abundant in the woodlands to the north. Through the late dry season, village cattle made use of the field areas surrounding the village and began to exploit the woodlands to the north, where the tsetse population collapsed. In the wet season, village cattle were herded into the woodlands to the north of the village, keeping them out of the planted crops, when the tsetse population began to show signs of recovery.
Stock Censuses
The combined results of censusing the major host animals showed that domestic stock were more numerous than warthogs in both study sites, with the exception of the wet season period at Niamina East. The village herds at Keneba showed a marginal increase in numbers through the study period, but the site was affected by the substantial increase in cattle overall through the introduction of large numbers of ITCowned cattle.
The Niamina East site was characterized by a small, stable number of resident cattle, annually augmented by an influx of dry season migrants taking advantage of the swamp grazing (T.J. Wacher and W.F. Snow, unpublished data), resulting in an approximately 20fold annual variation in cattle stocking density. Transect counts for warthog at Keneba indicated an overall annual average of 11/km².
Tsetse Exposure
The estimated number of tsetse per head for each study herd of cattle at each study site over one year are shown in comparison to the 'crude' estimate (which assumes spatially homogeneous cattle and tsetse distributions) in Tables 1 and 2. Results at Keneba, suggest that tsetse exposure may have varied for individual herds by a factor of up to five or more from the crude estimate and by a factor routinely between 5 and 10 fold, occasionally much more, between herds living in the same village, once variation in range utilization has been taken into account. In early dry season of 1988, for example, Herd 3 received at least seven times the tsetse exposure of any other herd (Table 1). Herd 4 at Keneba achieved an exposure level consistently below the crude estimate; this was the ITCowned group which in the study period reported spent a high proportion of time in a cleared field area near the village.
At Niamina East a similar order of variation (up to sevenfold) in tsetse exposure was indicated between individual herd values and the crude estimate. Betweenherd variations were less extreme than at Keneba, though still frequently in the vicinity of a factor of five to seven.
Note that in analysis the grazing distribution is further broken down to the range of each herd separately for comparison with the tsetse map.
Herd 3 at Niamina East, based in the village of Misira (Figure 1) was estimated to suffer particularly heavy tsetse exposure. Although this herd suffered the greatest exposure, it did not experience the greatest number of tsetse within its range. The extreme values estimated are due to a combination of the small herd size (only 18 animals), and the fact that they spent time grazing on a rice field closely bounded by a locally increased density of tsetse in an area little used by other herds.
Table 1. Estimated numbers of tsetse/head experienced by villagemanaged N'Dama cattle from four different herds at Keneba, The Gambia, over three seasons. Note that the crude estimate refers to the number of tsetse/head over the whole study site, while the values below indicate the number of tsetse/head for each herd after taking into account herd movement and tsetse distribution.

Early dry 88 
Late dry 88 
Wet season 88 
Crude 
4.7 
3.3 
1.0 
Herd 1 
3.6 
6.5 
1.8 
Herd 2 
3.8 
4.9 
3.6 
Herd 3 
28.8 
3.6 
4.7 
Herd 4 
1.0 
0.7 
0.04 
Sentinel Herds
Comparison of the mean tsetse catch per trap per day for each season with the estimated force of infection to the sentinel zebu herds at each site shows a nonsignificant positive relationship (Table 3). The catch per trap per day (CTD) acts as a more convincing index when corrected for total number of hosts available in each season. The relationship between our estimate for tsetse exposure and the force of infection shows good correlation and the highest proportion of the variance is accounted for when comparing the force of infection in zebus with the estimated tsetse/head experienced by those zebus.
Table 2. Estimated numbers of tsetse/head experienced by villagemanaged N'Dama cattle from three different herds at Niamina East, The Gambia, over three seasons. Note that the crude estimate refers to the number of tsetse/head over the whole study site, while the values below indicate the number of tsetse/head for each herd after taking into account herd movement and tsetse distribution.

Early dry 88 
Late dry 88 
Wet season 88 
Crude 
158.9 
59.2 
204.7 
Herd 1 
83.2 
32.8 
269.6 
Herd 2 
139.6 
42.0 
322.5 
Herd 3 
432.6 
88.1 
1159.0 
In this paper we report data on the spatial distribution of tsetse and their hosts and have calculated an index of exposure of livestock to tsetse attack based on the ratio of the two. This index is validated by demonstrating close correlation to the force of infection independently assessed in studies of susceptible sentinel zebu herds. The index suggests that individual herds based at the same village or in the same general areas may experience a 5 to 10 fold, sometimes greater, variation in the degree of exposure to tsetse challenge.
Table 3. Correlations between estimated force of infection in sentinel zebu herds and mean catch per trap per day (CTD), corrected for number of hosts, and the estimate of exposure, which incorporates grazing space utilization in relation to tsetse population distribution.

r^{2} 
F ratio 

Mean CTD 
48.0 
4.61 
>0.051 
Mean CTD/head 
88.3 
37.0 
<0.011 
Exposure of zebus 
97.6 
203.26 
<0.0011 
The order of magnitude in this variability was similar at two sites despite considerable differences in stock management conditions at each location. The relative density of hosts was lower at Keneba; exposure was also much lower. Domestic stock numbers were essentially stable from season to season at Keneba, but through the course of this study the expansion of ITCowned herds affected this pattern. The seasonal migration of cattle at the Niamina East site provides strong confirmation that host numbers should be a component of challenge index, if such an index is to provide a comparable measure under all conditions and related to the force of infection.
The use of herd movement data to assess the range utilization distribution of domestic stock has also involved a number of explicit assumptions, most notably that all members of a herd use the described seasonal ranges in an exactly equivalent way. Variation in range use by individual cattle within the same herd can only be considered once repeated tracking of the same sample of animals have been carried out. This was not done in the current study, since it was initially decided that sampling from the full pool of adult females in the herd would give a more accurate picture of overall herd ranging than a small subset that might include unknown individual biases in movement patterns.
The data have been used in a way that assumes the cattle ranges are used in a constant manner by each herd throughout each seasonal period. In practice this is unlikely to be so; a herd will on occasion spend several days visiting one area, before graduating to a new zone. There is thus potential for considerably different outcomes in real tsetse exposure according to the interaction of spatial patterns in tsetse and hosts and the relative timing with which this occurs.
For the warthog data, it has been necessary to assume a uniform distribution within the woodland and fallow habitats. This is unlikely to be a good estimate, and other mammalian hosts for G. m. submorsitans have been ignored altogether. This is an area of significant neglect, since the data clearly reveal that the periods of seasonal peak in tsetse numbers do not necessarily coincide with high overlap between cattle and tsetse. Hence in periods of maximum tsetse abundance it is clear that tsetsewild host dynamics are of great importance and merit more detailed attention; warthogs (Phacochoerus aethiopicus) are a major host of G. m. submorsitans in The Gambia (Snow and Boreham, 1979; ITC Entomology Program, unpublished data). Similarly, the methodologies developed here can be adapted to other livestock categories, notably small stock, but also groups subject to distinctive management regimes, such as draft animals.
The review of assumptions and simplifications involved in making these estimates of exposure has shown that the analysis used is likely to have given a conservative estimate of the real variability in individual exposure, since it was necessary to treat individuals as if details of their behaviour, beyond the spatial variability measured, was uniform. In practice this is unlikely to be true.
It is likely that the type of heterogeneity reported here will prove to be characteristic of most, if not all, situations involving tsetsetransmitted trypanosomiasis in Africa. The general consequences of such heterogeneities in host exposure for vectorborne disease transmission systems are known to result in an increase in the estimated transmission rate of the disease (Dye and Hasibeder, 1986). A full understanding of the dynamics between parasites and hosts, particularly mechanisms by which trypanosome species sustain their populations, will only be achieved when the importance of the spatially heterogeneous patterns of tsetsehost contact are considered in models which are being developed to describe the epidemiology of African animal trypanosomiasis.
We wish to thank the directors of the International Trypanotolerance Centre in The Gambia and the Liverpool School of Tropical Medicine for providing facilities to carry out research and for administrative support. These studies were funded by the Overseas Development Administration of the British Government, partly through the Natural Resources Institute, Chatham. Dr. Milligan was also supported by the Wellcome Trust. We are grateful for the full support of many field assistants who often worked in uncommonly arduous conditions to obtain the data reported here.
CLAXTON, J.R., LEPERRE, P., RAWLINGS, P., SNOW, W.F. and DWINGER, R.H. 1992. Trypanosomiasis in cattle in Gambia: Incidence, prevalence and tsetse challenge. Acta Tropica 50: 219225.
DYE, C. and HASIBEDER, G. 1986. Population dynamics of mosquitoborn disease: effects of flies which bite some people more than others. Transactions of the Royal Society of Tropical Medicine and Hygiene 80: 6977.
GREEN, C.H. and FLINT, S. 1986. An analysis of colour effects in the performance of the F2 trap against Glossina pallidipes Austen and G. morsitans morsitans Westwood (Diptera: Glossinidae). Bulletin of Entomological Research 76: 409418.
LEAK, S.G.A., AWUOME, K., COLARDELLE, C., DUFFERA, W., FREON, A., MAHAMET, B., MAWUENA, K., MULONGO, M., NANKODABA, G., ORDNER, G., PELO, M., SHERIA, M., TIKUBET, G., TOURE, M. and YANGARI, G. 1988. Determination of tsetse challenge and its relationship with trypanosome prevalence in trypanotolerant livestock at sites of the African Trypanotolerant Livestock Network. In: The Africa Trypanotolerant Livestock Network: Livestock Production in TsetseAffected Areas of Africa. Proceedings of a Meeting Held 2327 November 1987, Nairobi, Kenya. Nairobi: ILCA/ILRAD, pp. 4354.
LEAK, S.G.A., COLLARDALE, C., COULIBALY, L., DUMONT, P., FERON, A., HECKER, P., d'IETEREN, G.D., JEANIN, P., MINENGU, M., MINJA, S., MULATU, W., NANKODABA, G., ORDNER, G., ROWLANDS, G.J., SAUVEROCHES, B., TIKUBET, G. and TRAIL, J.C.M. 1990. Relationships between tsetse challenge and trypanosome prevalence in trypanotolerant and susceptible cattle. Insect Science Application 11 (3): 293299.
RAWLINGS, P., DWINGER, R.H. and SNOW, W.F. 1990. An analysis of survey measurements of tsetse challenge to trypanotolerant cattle in relation to aspects of analytical models of trypanosomiasis. Parasitology 102: 371377.
RAWLINGS, P., WACHER, T.J. and SNOW, W.F. 1994. Cattletsetse contact in relation to the daily activity patterns of Glossina morsitans submorsitans in The Gambia. Medical and Veterinary Entomology 8: 5762.
ROGERS, D.J. 1985. Trypanosomiasis 'risk' or 'challenge': a review. Acta Tropica 42: 532.
SMITH, I.M. and RENNISON, D.B. 1958. Some factors concerned in trypanosome challenge. In: Proceedings of 7th Meeting of the International Scientific Committee on Trypanosomiasis Research, pp. 6366.
SNOW, W.F. and BOREHAM, P.F.L. 1979. The feeding habits and ecology of the tsetse fly Glossina morsitans submorsitans Newstead in relating to nagana transmission in The Gambia. Acta Tropica 36: 4751.
WOOLHOUSE, M.E.J., WATTS, C.H. and CHANDIWANA, S.K. 1991. Heterogeneities in transmission rates and the epidemiology of schistosome infection. Proceedings of the Royal Society of London B 245: 109114.
Transmission of Theileria
The transmission model developed by Drs Medley, Perry and Young was considered to be a useful first step in the development of more comprehensive models which represent the transmission of Theileria spp. throughout Africa. It was agreed that such models could be of great value in enhancing the knowledge on the epidemiology of Theileria infections and for testing the effects of different control interventions.
Some of the areas which require further studies in order to generate data sets necessary for the development of such models were:
· The relative roles of acute infections of clinical cases, and low level infections of carrier animals, in maintaining a source of infection to cattle and as a cause of different tick infection rates under different epidemiological situations.· The maintenance of T. parva infections within different populations.
· The relative roles of larval to nymphal transmission and nymphal to adult transmission.
· The seasonality of tick populations and T. parva transmission, as influenced by the occurrence (or lack) of diapause in R. appendiculatus populations at different latitudes.
· The relative role of wildlife in maintaining tick populations and T. parva infections in ticks.
It was proposed that studies using in vivo feeding of ticks could result in a greater insight into factors which control the transmission of T. parva by ticks. There was general support for the idea that other methods of modelling Theileria transmission should be considered, including matrix analysis, and that models developed for other parasites such as malaria should be evaluated as to their relevance to Theileria. The lack of specialized data sets, particularly those derived from longitudinal studies of calf populations under different epidemiological conditions, were considered a constraint for the development and validation of models.
TRANSMISSION OF TRYPANOSOMES
The possibility of strengthening decisionsupport systems for the development of control programs for tsetsetransmitted trypanosomiasis in livestock through modelling weighing the four options, namely tsetse control, chemotherapy (including chemotherapeutic and chemoprophylactic agents), trypanotolerance, and vaccines (when they become available) was discussed.
A potential area where modelling could be applied is in trypanosomiasisendemic areas where trypanosusceptible cattle breeds exposed to tsetse challenge and under constant trypanosomiasis risk may develop a degree of trypanotolerance. For example, in eastern Africa the Orma Boran breed has been shown to be relatively trypanotolerant compared with other zebu cattle.
It was stressed that when studies of the spatial distribution of tsetse vectors and zebu cattle were designed, all the important variables involved in the transmission of trypanosome infections to cattle, such as the disease status, reservoir hosts, the tsetse flies and livestock, should be included in order to generate reliable and appropriate data sets for modelling. It is important that such studies are conducted in several trypanosomiasis endemic regions of Africa. The resulting data may serve to validate or modify the existing models of the transmission of trypanosome infections to livestock by tsetse flies. Such models will provide a useful basis for the identification of optimal control interventions for tsetsetransmitted trypanosomiasis in livestock in different epidemiological situations.