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Vector and helminth population dynamics


Tsetse vector population dynamics: ILRAD's requirements
Tick vector populations dynamics: ILRAD's requirements
Tsetse population dynamics
Simulation of tick population dynamics
Modelling of Rhipicephalus appendiculatus population dynamics
Host density and tick dynamics: The case of the vector of Lyme disease
Modelling helminth population dynamics
Session discussion


Tsetse vector population dynamics: ILRAD's requirements

S.G.A. Leak

International Laboratory for Research on Animal Diseases
P.O. Box 30709
Nairobi, Kenya

The International Laboratory for Research on Animal Diseases' (ILRAD) research on immunological methods for the control of African trypanosomiasis involves some work on parasite transmission by tsetse flies but none on tsetse population dynamics. However, ILRAD does have an interest in understanding the epidemiology of the disease and in assessing disease risk from tsetse flies in order to estimate the potential impact of a vaccine or other means of disease control which may be produced.

The disease incidence or prevalence and its seasonal variations in an area are determined primarily by the population dynamics of the tsetse fly vector. Trypanosome infection rates in tsetse species seem not to vary to the same extent as the apparent tsetse density and it is the latter which appears to be most responsible for determining the seasonal variations in disease risk or 'Tsetse challenge'. The relationships between tsetse population dynamics and the epidemiology of trypanosomiasis are discussed briefly, using data from the African Trypanotolerant Livestock Network (ATLN) which is coordinated by the International Livestock Centre for Africa (ILCA) in collaboration with ILRAD. These data show results achieved with minimal entomological input, employing standardized methods over a range of sites in Africa and collected over periods of seven years or more.

Seasonal changes in apparent density of tsetse are shown to be closely related to climatic parameters and also correlate closely with changes in disease prevalence in cattle. Attempts were made as early as the 1950s to predict changes in tsetse density from observed climatic parameters and to compare these predictions with field data. Climatic factors have their effects on tsetse population density through their effects on tsetse survival or mortality rates. These changes in survival rates may also affect the age structure of the population which in turn is related to the trypanosome infection rates in the tsetse population. Some models have already been produced to estimate mortality rates and to show the effects of seasonally varying mortality on tsetse population dynamics.

Although correlations between tsetse challenge and disease prevalence can be shown, the relationship is not precise due to complex interactions of the many factors determining tsetse challenge and the practical difficulties in obtaining accurate estimates of the variables and parameters involved. Such relationships may be demonstrated more easily in areas with marked seasonal changes in climate such as ATLN sites in Ethiopia or northern Cote d'Ivoire rather than, for example, in Gabon where the climate is less extreme and where there may be less marked seasonal fluctuations in tsetse populations due to climatic factors.

Climatic factors, in particular rainfall, temperature and relative humidity, have been shown to be important parameters determining tsetse distribution and population dynamics. The relationship between these factors and tsetse population dynamics is therefore a potential area for modelling and further development of existing models.

Tick vector populations dynamics: ILRAD's requirements

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) is primarily involved in developing improved methods for the control of tick-borne diseases; it therefore takes an active interest in research on tick vector population dynamics but does not lead research on this subject. It is realized that tick vector population dynamics is one of the main factors controlling the epidemiology of tick-borne diseases. ILRAD's focus, within the tick-borne diseases, has been research on the development of vaccines against Theileria parva, the cause of East Coast fever (ECF). Thus interest concentrates on the main field vector of T. parva, Rhipicephalus appendiculatus. The particular interests, in the population dynamics of this tick, are aspects of the biology which influences the epidemiology of ECF. This can be illustrated by some examples. The distribution of the tick vector controls the distribution of the disease and climatic models have been helpful in establishing the distribution of vector and parasite where hard data are not available. The distribution of the tick vector population is dynamic, changing continually in response to changes in climatic conditions as well as movements of hosts. The seasonality of the changes in tick vector population density plays a large role in the epidemiology of disease throughout its distribution. This can be well illustrated by the transmission of T. parva throughout the year in the Lake Victoria Basin compared with the seasonal occurrence of theileriosis in central and southern Africa (e.g. January Disease in Zimbabwe).

Strict seasonality of R. appendiculatus instars in central and southern Africa is controlled by behavioural diapause of the adult stage which results in univoltine population there and multivoltine populations further north towards the Equator. This is thought to be a survival strategy of the tick in the southern areas. Hence ILRAD is actively involved in research on diapause in tick populations to determine how this influences the epidemiology of ECF. Tick populations are difficult to quantify since the immature instars are small and the free-living stages on the ground are dispersed and difficult to sample. Therefore all data collected from site-specific areas, although useful, are all relative counts and hence these data are difficult to correlate with the epidemiology of theileriosis. An additional difficulty is the wide range of hosts of R. appendiculatus in all its instars; up to 35 host species, both domestic and wild mammals, have been recorded but only cattle, African buffalo (Syncerus caffer) and waterbuck (Kobus spp.) have been implicated as hosts of T. parva. These animals vary in their ability to act as hosts for ticks and in their resistance to tick infestations. Hence only ticks which have fed as larvae and nymphs on infected hosts play a part in the epidemiology of theileriosis. Hosts which are refractory to infection actually cleanse the tick vectors of their infections when they are infested. These examples illustrate the complexities of modelling tick vector population dynamics to a level where it can be useful in simulating epidemiology of theileriosis.

Tsetse population dynamics

D.J. Rogers

Department of Zoology
University of Oxford
South Parks Road
Oxford OX1 3PS
England

The paper presented describes the twin approaches of biological and statistical modelling of tsetse populations dynamics and distribution in space and time.

The biological approach is dealt with briefly. The requirements are full demographic (i.e. life-table) data for the tsetse combined with complete climatic data during the course of the sampling program. Extensive predictions that arise from the intensive studies at any one site are given with a cautionary note that vector dynamics elsewhere may differ because of the vectors' differing responses to local climate (i.e. abiotic effect) and because of the differing guilds of natural enemies throughout the range of the species concerned (i.e. biotic effect).

The statistical approach initially chooses to ignore local variations in the tsetse's response to abiotic conditions and seeks to model a distributional range through multivariate techniques such as discriminant analysis. This technique therefore requires extensive information on the presence and absence of a species throughout a region, and climatic data on an equally extensive scale. Whilst this approach is still rather 'broad-brush' in comparison with the biological approach, it does provide some indication of the likely regional variation in abiotic constraints on the presence or absence of a species, and thus can be used to target field investigations either to regions which show a variety of abiotic constraints (for comparative studies on fly ecology) or to regions where there is a false prediction of fly presence (flies may in fact be present, but inadequately surveyed), or absence (if confirmed, such areas indicate an inadequate understanding of climatic or other constraints). Examples are given of the distribution of Glossina morsitans and G. pallidipes in Kenya and Tanzania. Discriminant analysis identifies different climatic and vegetational constraints for these two species.

The above studies form part of an epidemiological interpretation of changes in disease risk in space. Another important element is changes in risk through time. Here the situation is complicated by a variety of suggested possible mechanisms for epidemic outbreaks of disease, and an acute shortage of data to distinguish between them.

It is concluded that past data sets are inadequate for the needs of present models, because the data were gathered in an epidemiological vacuum, i.e. without reference to any theory being tested. Today we are in a position to decide which data to gather, and how best to test alternative theories of trypanosomiasis transmission. As in other fields, however, modelling has made more and more rapid progress in the last few years, whilst field studies have declined almost to the point of extinction.

Simulation of tick population dynamics

D.G. Haile

United States Department of Agriculture
Agricultural Research Service
Medical and Veterinary Entomology Research
Laboratory
Gainesville, Florida, USA

Abstract
Introduction
Simulation approach
Software
Simulation of control
Research needs
References


Abstract

The Boophilus Cattle Tick Simulation Model is a computer program that simulates tick population dynamics and the processes involved in transmission of Babesia parasites. Tick population dynamics are controlled by climate, density and type of cattle and pasture type. Transmission of Babesia parasites is determined by the levels of infective ticks and cattle and various epidemiological parameters. Simulations can be run for either Boophilus microplus or B. annulatus ticks and either Babesia bovis or B. bigemina parasites. BCTSIM provides a tool to study tick density thresholds associated with disease transmission. Control technologies for the tick vectors are included in the model for comparison of various control strategies in a given geographic area. With proper regard for the limitations of the model, BCTSIM can serve as a knowledge source for planning an operational control program. The accuracy of model predictions can be determined only after extensive use in conjunction with actual control operations and field research.

Models developed using this approach provide a flexible framework for addition of biological details and control effects. Disadvantages of this approach include the difficult and time consuming process of model construction and programing, maintenance of program integrity and the extensive data requirements which are generally inadequate. Advantages include the biological realism of simulation results and the generally unlimited ability to program and simulate control technologies and strategies.

Introduction

Computer simulation models represent a powerful tool for study of vector population dynamics and integrated control strategies. Simulations can provide a comparison of the effectiveness of an unlimited number of control scenarios that would be impossible to conduct in the field because of the time and expense required. Effective use of simulation research will allow field research to concentrate on validation of model results and evaluation of the most promising control strategies.

Computer models for analysis of tick population dynamics and control have been developed for Amblyomma americanum, Dermacentor variabilis and Boophilus cattle ticks. These models all use a deterministic, dynamic-life-table approach with key biological parameters controlled by climate (temperature, humidity and rainfall) and other variables such as host type and density, habitat type and photoperiod. Components to simulate transmission of Rocky Mountain spotted fever by D. variabilis and babesiosis by Boophilus ticks were included in these models. Control technologies in the models include various acaricide application procedures, release of sterile hybrid ticks, and manipulation of hosts and habitat. Studies to assess the validity of the models and to compare various tick control technologies and strategies have been completed. The software for these models is written in Microsoft Professional BASIC for interactive operation on advanced microcomputer systems. The software design allows the models to be used for extensive simulation research and provides a useful demonstration and training tool.

For the purpose of this meeting I will cover details of only the Boophilus Cattle Tick Simulation Model (BCTSIM) because of the importance of this tick in Africa and because the model includes interactions between population dynamics and disease transmission within a cattle herd. This paper will review the major components and relationships in BCTSIM and discuss potential uses in development of control strategies.

Simulation approach

Development and validation of the present version of BCTSIM are provided in Mount et al. (1991) and Haile et al. (1992). The model construction and simulation approach used for BCTSIM was the same as that used previously for other simulation studies on tick population dynamics, disease transmission and control strategies (Haile and Mount 1987; Mount and Haile, 1987, 1989; Cooksey et al., 1990; Haile et al., 1990).

The basic model structure allows simulation of the population levels of ticks and cattle in various age classes and stages of parasitic development. Interactions between susceptible and infective individuals in the tick and cattle populations can be quantified to simulate Babesia transmission.

A dynamic life table approach with weekly time steps is used to simulate tick population dynamics as influenced by climate and other variables. The major factors that influence tick population dynamics in BCTSIM are: 1) temperature-dependent development rates for eggs and for engorged females off the host, 2) fecundity rates for engorged females according to temperature and type of cattle, 3) density-dependent survival rates for ticks on the host varied by type of castle, 4) survival rates for free-living stages of ticks regulated by type of habitat, temperature, saturation deficit and precipitation and 5) host-finding rates for larvae dependent on host density, temperature and off-host larval density. Epidemiological parameters and relationships in the model include the reduction in fecundity of infected ticks, rate of transovarial transmission, effect of cattle type and inoculation rate on infectivity of cattle, variation of infected cattle recovery rate with age of infection, inoculation rate and species of parasite.

Several general assumptions were required in construction of the model. These assumptions are: 1) that every bovine animal of one type is equally susceptible to Boophilus tick infestation, 2) that all cattle of the same type are equally susceptible to Babesia infection and 3) that infected ticks survive equally long as non-infected ticks. Although these assumptions are not totally realistic, they are consistent with our objective of modelling mean tick populations and levels of disease transmission.

Software

The present version of BCTSIM is programmed in Microsoft BASIC version 7.1 (Professional Development System) for use on an IBM PC-AT or compatible microcomputer. A colour monitor (EGA or VGA) is desirable because the program makes use of colour in screen displays and graphics. Simulation time is improved with an advanced computer system based on an 80486 microprocessor or an 80386 microprocessor with a math co-processor.

BCTSIM was written for interactive operation and presents various menu options to specify input files and variable levels. The program also was designed to select from various methods for viewing and saving simulation results. The primary choices required for each simulation run include geographic location, basic biological data file for a tick species, climate file, cattle density, type of cattle, type of pasture and species of Babesia. A selection also allows the addition of Babesia infection in engorged female ticks or in cattle. The program is designed to present one year of simulation at a time and presents a post-simulation menu at the end of each year. The simulation can be continued for as many years as desired with the option to change selected parameters between years.

Each climate file contains values for mean weekly ambient temperature (°C), saturation deficit (mb), and precipitation (cm) for one year. Climate files for a given location may contain actual weekly average data for specific years or historical data which is the average for a number of years. For some climate files, interpolation was used to generate weekly data from monthly data.

The basic biological data files for BCTSIM contain 52 parameters and coefficients that define the tick life cycle and relationships between biological and environmental variables. A separate biological data file has been created for each tick species, Boophilus microplus (Canestrini) and B. annulatus (Say). These files are accessible from the main menu of the program and allow rapid adjustment of data during model development and refinement. An additional menu selection allows a choice of either Babesia bovis (Babes) or Babesia bigemina (Smith and Kilborne) as the parasite species for simulation. Although not in a data file, this choice allows on-screen review and editing of selected epidemiological parameters. The model allows simulations for one tick and one parasite species at a time.

Choices for the initial tick population include introducing eggs on a selected week during the first simulation year or introducing a population distribution of overwintering ticks saved as a data file at the end (week 52) of a previous simulation run. After initialization, simulations can continue for as many years as desired, with the initial population for each successive year being a continuation of the population from the previous year. The final population of any simulation year can be saved as an output data file for use as a future initialization file.

The primary output from BCTSIM is a graphics plot, presented on the monitor, of the weekly population of ticks on the host and the level of infection in cattle during each simulation year. An optional output is an animated life cycle display with an alternate screen presenting weekly data on major parameters in the model. A post-simulation menu provides options for graphic plots of climate data and levels of all life stages for each week of the yearly simulation. This menu also presents a choice to view a text summary screen which includes the annual mean numbers of each life stage on the host/hectare, off-host larvae/hectare and standard females/host/day. Epidemiological results on the summary screen include infected numbers of ticks off hosts and on hosts per hectares, percentage infection of ticks, numbers of cattle (susceptible, infected and recovered) per 1000 hectares, percentage of calves infected by nine months of age, and average daily inoculation rates of calves and cows.

BCTSIM provides an option to calculate growth rate per generation (R) and generation time (T) from the model output for a given set of input parameters. The growth-rate option accumulates all eggs produced by adult females rather than transferring them to the first egg stage; therefore, the total number of eggs produced over time from initialization by a single cohort of eggs can be determined. The growth-rate option accumulates egg production for each week and is programmed to compare the 500% accumulation with the initial number of eggs for a calculation of R. The week of 50% accumulation of eggs is also identified as a measurement of T.

Simulation of control

BCTSIM provides preliminary programing to simulate tick control by 1) acaricide applications to cattle and 2) release of sterile hybrid ticks. Interactive choices for acaricide applications include: 1) length of residual activity and effectiveness level for each week post-treatment, 2) number of treatments during the simulation year, 3) week of initial treatment and 4) treatment interval, weeks. For sterile hybrid releases, the present program assumes that hybrid larvae are released in pastures with menu choices for: 1) release level, number/hectare, 2) release interval, weeks, and 3) host finding effectiveness compared to wild larvae.

Extensive simulation studies on control strategies have not been conducted. This will be the subject of a future paper after refinement of the control sub-models and possibly further refinement of other aspects of BCTSIM.

The present program can be used to demonstrate control principles and the influence of the environmental variables on effectiveness at different geographic locations. The degree of control required to reach theoretical tick density thresholds for maintenance of Babesia in cattle or for inoculation of all new calves can also be analysed. Although eradication can be simulated in the model, confidence in a prediction of eradication in any given situation would be limited because of uncertainty concerning levels of input variables and the nature of a deterministic model with low numbers. Addition of stochastic elements to the model would improve realism with low numbers; however, the variables required for the additional complexity would increase the potential for errors and uncertainty. The overall confidence in eradication predictions would probably be about the same with either type of model. The most practical use of present models will involve relative comparisons of different control procedures to provide knowledge for planning and implementation of operational programs. Extensive experience with BCTSIM in conjunction with control operations and field research will be required to determine the predictive capability of the model.

Research needs

Improvement in the predictive capability of BCTSIM can only be accomplished with additional refinement and validation studies. These refinements will require quantitative field research on specific areas of the tick-host-disease system. This research includes detailed and long-term studies to measure the actual levels of tick density and parasite prevalence for comparison with simulation output. Other areas needing quantitative study include: 1) the effect of cattle type, age and previous exposure on death rates due to Babesia infection, 2) cross immunity between parasite species and tick species, and 3) effectiveness of new acaricides and control procedures. Additional effort and resources will be required for software design to improve interactive and 'user friendly' aspects of the program as refinements are incorporated into the model.

References

COOKSEY, L.M., HAILE, D.G. and MOUNT, G.A. 1990. Computer simulation of Rocky Mountain spotted fever transmission by the American dog tick (Acari: Ixodidae). Journal of Medical Entomology 27: 671-680.

HAILE, D.G. and MOUNT, G.A. 1987. Computer simulation of population dynamics of the lone star tick, Amblyomma americanum (Acari: Ixodidae). Journal of Medical Entomology 24: 356-369.

HAILE, D.G., MOUNT, G.A. and COOKSEY, L.M. 1990. Computer simulation of management strategies for the American dog tick, Dermacentor variabilis (Acari: Ixodidae). Journal of Medical Entomology 27: 686-696.

HAILE, D.G., MOUNT, G.A. and COOKSEY, L.M. 1992. Computer Simulation of Babesia bovis (Babes) and Babesia bigemina (Smith and Kilborne) transmission by Boophilus cattle ticks. Journal Medical Entomology 29: 246-258.

MOUNT, G.A. and HAILE, D.G. 1987. Computer simulation of area wide management strategies for the lone star tick, Amblyomma americanum (Acari: Ixodidae). Journal of Medical Entomology 24: 523-531.

MOUNT, G.A. and HAILE, D.G. 1989. Computer simulation of the population dynamics of the American dog tick, Dermacentor variabilis (Acari: Ixodidae). Journal of Medical Entomology 26: 60-76.

MOUNT, G.A., HAILE, D.G., DAVEY, R.B. and COOKSEY, L.M. 1991. Computer simulation of Boophilus cattle tick (Acari: Ixodidae) population dynamics. Journal of Medical Entomology 28: 223-240.

Modelling of Rhipicephalus appendiculatus population dynamics

D.L. Berkvens

Institute for Tropical Medicine
Nationalestraat 155
B-2000 Antwerpen, Belgium

Abstract
References


Abstract

Simulations models of Rhipicephalus appendiculatus population dynamics are driven by a direct cause-effect relationship between temperature and length of the development periods. The timing of cohorts of the various instars can be modelled accurately enough by this relationship, although it must not be forgotten that experiments to determine it were carried out at optimum relative humidities. Survival of the free-living stages is governed by one or more humidity parameters in the models. Accurate quantification of the relationship between survival and both intrinsic and extrinsic factors is as yet not feasible. Elucidating this crucial relationship deserves the necessary attention: there is little justification in using any of the existing models to test whatever hypothesis before survival can be predicted confidently in function of its regulating factors. The models attempt to simulate the need to synchronize the tick's lifecycle with seasonal variation in humidity: the ultimate factor determining the tick's phonology on the mammal host is to be found in the humidity requirements and tolerances of eggs and free-living unfed larvae. However, there is no indication that either eggs or larvae - or nymphae - possess the ability to delay development and/or questing.

Current evidence indicates that adult ticks have the ability to diapause, i.e. are able to feed at the correct time of the year to ensure that the required minimum level of humidity is available for the immature stages. Given the uncertainty over the exact mechanism(s) available to adults to achieve this synchronization, simulation models can play an important role in this area to offer guidance with respect to the required research. It can be anticipated that other factors will assume more and more importance as models are developed further. These variables include host-dependent factors (host resistance to ticks and host density), tick population density-dependent factors and tick body size. Apart from the fact that we are not in a position to quantify any of the above relationships, they also introduce a degree of 'memory' into the system, which together with the absence of any form of selection in the models should urge for great caution when running the models over a period of several years. Simulation models of African tick population dynamics are likely to remain pure research tools in the foreseeable future and their further development should proceed in close collaboration with research, accompanied by the development of a flexible interface, ideally allowing the user to change not only the levels of the variables but also the actual relationships.

Rhipicephalus appendiculatus is a three-host tick. Briefly, a replete adult female drops off the host and produces a single egg batch after a pre-oviposition period. The eggs hatch into larvae. The larvae harden off and start to quest on pasture. After having been picked up by a host, the larvae feed and drop off the host upon repletion and moult into nymphs. The nymphs cycle through an identical series of events, resulting in adult ticks. It is accepted that adults, in particular females, have the ability to diapause in order to regulate host finding and feeding, thus ensuring that the most vulnerable stages, namely eggs and larvae, are exposed to favourable climatic conditions.

The lengths of the various development periods (pre-oviposition, pre-eclosion, larval and nymphal moults) depend to a large extent on temperature alone (Branagan, 1973a; Tukahirwa, 1976; Punyua, 1984; Short et al., 1989). Strictly speaking, unequivocal evidence does not exist for this direct cause-effect relationship. The laboratory experiments were carried out at optimum relative humidities, i.e. 85-87% (Branagan, 1973a) and 90% (Tukahirwa, 1976), and the field observations (Punyua, 1984; Short et al., 1989) do not allow the inference of cause-effect relationships. Nonetheless, we assume that the direct, single-variable cause-effect relationship between temperature and length of development period is correct, or at least sufficiently accurate, to allow simulation of the life cycle of the tick. Thus, there is little justification to direct a major research effort to fine-tune the details of this aspect of the tick's life cycle. The various approaches to quantify the relationship between temperature and development periods should be compared and tested, as was done by Byrom (1990) for the method developed by King et al. (1988). Testing criteria should include predictive power and complexity of the algorithm, both in terms of computing efficiency and collection of the required temperature data.

The survival of the free-living stages as well as hatching and moulting successes are regulated by humidity. The principal effect of humidity, and thus the limiting factor in relation to the geographic range of the tick, is thought to be the survival of the most vulnerable stages, particularly the eclosion of eggs and survival of larvae for a period sufficient to allow host contact (Branagan, 1973b; Hoogstraal, 1978). The ultimate factors determining the probability of instantaneous survival include the microclimate and the physiological age of the tick (Branagan, 1973b; Punyua, 1985). It is at present impossible to quantify the microclimate experienced by the ticks and, therefore, recourse must be taken to macroclimatological parameters. It is still a matter of debate whether it is soil moisture or air moisture or both that best predict the survival curves of the different instars. Furthermore, it has not been resolved whether humidity operates dependent or independent of temperature, i.e. whether survival and success should be expressed as function of relative humidities or vapour pressure deficits (Tukahirwa, 1976). Lastly, it must not be forgotten that humidity interacts with the physiological age of the tick.

Resolving the relationship between intrinsic and extrinsic factors on the one hand and survival on the other should be given priority. There is no point in looking for any other fundamental parameters and relationships that might influence the tick's population dynamics if we cannot predict with a certain degree of accuracy survival rates as a function of external factors. The same remark holds for index models that attempt to predict suitability of a certain area in relation to the tick's requirements and tolerances.

Although the survival of the most vulnerable stages is probably the ultimate factor that drives the tick's phonology, neither eggs nor larvae appear to have the ability to delay respectively development and questing to synchronize the life cycle with seasonal variation in humidity. It is now accepted that this synchronization, if required, must be achieved by the adult ticks, namely by either quiescence or behavioural diapause or a combination of both. A current research effort is designed to settle the question of whether or not the ability to diapause is confined to southern African R. appendiculatus populations or present in all populations. At the same time such studies will provide a better understanding of the mechanisms involved in induction, maintenance and termination of diapause to allow further real-life and computer experiments.

Quantifying the various interactions between the mammalian host and the tick remains fraught with difficulties. Broadly speaking, there are two components in the overall effect of the host, namely host density and host resistance to ticks. Host density has a direct effect on pick-up rates of the various instars and thus, together with survival rates, on the population level. However, our current knowledge about this complex relationship is largely qualitative and mainly confined to the cattle host. The effect of host resistance is probably far more important. The main problem here lies in the fact that although we have a whole body of experimental evidence relating to host resistance (Chiera et al., 1985a, 1985b; Fivaz and Norval, 1989; Jongejan et al., 1989), variation in host resistance in a field situation both seasonal and otherwise cannot at present be quantified.

Results from studies in the Eastern Province of Zambia, possibly a transition zone between areas with multivoltine and univoltine R. appendiculatus phenologies, reveal a distinct seasonal variation in body size of nymphae and adults, indicating that adult ticks collected at lower altitudes are smaller than those found at higher altitudes. Body size can be assumed to be correlated to the maximum possible generation interval, taking into account the need for diapause where required. Additional support for a correlation between voltinism and body size has been obtained by A.S. Young and R.G. Pegram (unpublished results). Clinal and seasonal variation in body size adds additional complexity to the model because of the different development and survival rates, and this aspect may require further research. Any heritable variation in body size and diapausing behaviour may complicate matters even more, especially when considering that host resistance has a direct effect on tick body size. This leads to an inherent problem when modelling, namely that certain parameters are built into the model at the start of a simulation run and that these parameters are kept constant and fixed throughout the entire simulation. This is hardly natural and will inevitably lead to erroneous conclusions if the horizon of the simulation exceeds anything but a few years.

To conclude, a computer simulation model of R. appendiculatus population dynamics very likely remains a research tool in the foreseeable future because of the lack of quantitative data required for the formulation of the relationships between certain aspects of the life cycle and their regulating factors.

References

BRANAGAN, D. 1973a. The development periods of the Ixodid tick Rhipicephalus appendiculatus Neumann under laboratory conditions. Bulletin of Entomological Research 63: 155-168.

BRANAGAN, D. 1973b. Observations on the development and survival of the ixodid tick Rhipicephalus appendiculatus Neumann, 1901 under quasi-natural conditions in Kenya. Tropical Animal Health and Production 5: 153-165.

BYROM, W. 1990. Simulation models for investigating East Coast fever and other parasitic diseases. Ph.D. Thesis, University of Strathclyde, 189 pp.

CHIERA, J.W., NEWSON, R.M. and CUNNINGHAM, M.P. 1985a. The effect of size on feeding and breeding performance of Rhipicephalus appendiculatus Neumann. Insect Science and its Application 6: 555-560.

CHIERA, J.W., NEWSON, R.M. and CUNNINGHAM, M.P. 1985b. Cumulative effects of host resistance on Rhipicephalus appendiculatus Neumann (Acarina: Ixodidae) in the laboratory. Parasitology 90: 401-408.

FIVAZ, B.H. and NORVAL, A. 1989. Observations on successive infestations of the rabbit host by the ticks Rhipicephalus appendiculatus and R. zambeziensis (Acari: Ixodidae). Experimental and Applied Acarology 7: 267-279.

HOOGSTRAAL, H. 1978. Biology of ticks. In: Wilde, J.K.H. ed. Tick-Borne Diseases and Their Vectors. Tonbridge: Lewis Reprints Ltd., PP. 3-14

JONGEJAN, F., PEGRAM, R.G., ZIVKOVIC, D., HENSEN, E.J., THIELEMANS, M.J.C., COSSÉ, A., NIEWOLD, T.A., ASHRAF EL SAID and UILENBERG, G. 1989. Monitoring of naturally acquired and artificially induced immunity to Amblyomma variegatum and Rhipicephalus appendiculatus ticks under field and laboratory conditions. Experimental and Applied Acarology 7: 181-199.

KING, D., GETTINBY, G. and NEWSON, R.M. 1988. A climate-based model for the development of the Ixodid tick, Rhipicephalus appendiculatus, in East Coast fever zones. Veterinary Parasitology 29: 41-51.

PUNYUA, D.K. 1984. Development periods of Rhipicephalus appendiculatus Neumann (Acarina: Ixodidae) under field conditions. Insect Science and its Application 5: 247-250.

PUNYUA, D.K. 1985. Longevity of hungry Rhipicephalus appendiculatus Neumann (Acarina: Ixodidae) under field conditions at Muguga, Kenya. Environmental Entomology 14: 392-395.

SHORT, N.J., FLOYD, R.B., NORVAL, R.A.I. and SUTHERST, R.W. 1989. Development rates, fecundity and survival of developmental stages of the ticks Rhipicephalus appendiculatus, Boophilus decoloratus and B. microplus under field conditions in Zimbabwe. Experimental and Applied Acarology 6: 123-141.

TUKAHIRWA, E.M. 1976. The effects of temperature and relative humidity on the development of Rhipicephalus appendiculatus Neumann (Acarina, Ixodidae). Bulletin of Entomological Research 66: 301-312.

Host density and tick dynamics: The case of the vector of Lyme disease

T.E. Awerbuch* and A. Spielman+

* Department of Biostatistics
Harvard School of Public Health
677 Huntington Avenue
Boston, Massachusetts 02115, USA

+ Department of Tropical Public Health
Harvard School of Public Health
665 Huntington Avenue
Boston, Massachusetts 02115, USA


Abstract
Introduction
The model
Results
Discussion
Acknowledgements
References


Abstract

To determine whether the abundance of deer and mice limits the abundance of deer ticks, we implemented a model representing the life cycle of these vector ticks using realistic parameter estimates taken from two field sites in coastal Massachusetts. The main inputs to the model were scanning capacity and host abundance. The equation for scanning capacity includes estimates of the density of questing ticks and applies an algorithm based on the density of ticks actually attached to hosts. In a site in which the abundance of mice varied from year to year, deer abundance remained constant, as did that of recently emerged larvae. When the density of mice was held at a level that corresponded to that in a year of exceptional mouse abundance, the ticks thrived. When such hosts remained scarce, tick abundance waned. A stable density of ticks accompanied an 'ordinary' density of mice. Deer abundance was reduced in the other site while mouse abundance fluctuated. Although deer density diminished by ¾ to about 100, tick density continued to increase. Our simulations suggested that the critical threshold of deer abundance is eight animals. We conclude that the abundance of deer ticks is sensitive both to the abundance of mice and of deer.

Introduction

White-tailed deer serve as the main host for the adult stage of the deer tick (Ixodes dammini) (Wilson et al., 1985), the vector of Lyme disease in eastern North America (Piesman et al., 1979). Egg production follows successful feeding. The concept that deer are crucially important in the life cycle of these ticks is based on diverse kinds of evidence, including: (1) qualitative observations establishing that stable infestations of the tick are evident solely where deer are resident (Spielman, 1988), (2) geographical correlations between the density of larval deer ticks feeding on mice and the density of deer (Wilson et al., 1985) as well as (3) removal experiments indicating that the density of these ticks on mice in a site diminishes following local removal of deer (Wilson et al., 1988). Some subadult deer ticks also feed on deer, although they generally feed on mice or other small animals (Piesman et al., 1979). The abundance of deer ticks critically depends upon that of deer.

The immature stages of this tick appear to feed mainly on mice. Adult deer ticks, however, never parasitize these hosts. This complexity renders it difficult to define the density of hosts required for perpetuation of this tick. It may be that the density of both kinds of hosts determines the abundance of the tick.

To explore this hypothesis, we applied a mathematical model formulated to represent the life cycle of deer ticks (Sandberg et al., 1992) and implemented it with actual parameter estimates based on field observations in two sites in coastal Massachusetts. In one site, the deer population remained constant while the mouse population fluctuated during the eight-year period of the study. In the other site, the mouse population fluctuated mildly while deer were removed by limited hunting. We used computer simulations to explore the effect of realistic changes in mouse and deer abundance on the population biology of these ticks.

Description of Study Site and Field Methodology

To explore the effect of mouse abundance on the dynamics of the deer tick population, we used data derived from the Nantucket Field Station of the University of Massachusetts, on Nantucket Island, Massachusetts. This 32-hectare shoreline site was studied regularly during the period 1985 through 1991. The vegetation comprised equally of grassy meadows separating stands of dense coppice that mainly included bayberry, blueberry and scrub oak. Voles were most abundant in the meadows and white-footed mice in ecotonal situations. Although shrews, cottontail rabbits and domestic animals were abundant there, few animals of other kinds inhabit this restricted site.

Observations were made monthly between April and October. Deer density was estimated by direct observation and that of mice by a minimum-number-alive estimate (Wilson et al., 1988). The abundance of rodents was estimated by placing oat-baited live traps (Longworth Co, Abingdon, UK) in a permanent 7 x 7 grid in the field site. Each captured animal was marked by means of a numbered ear-tag. The minimum number alive method of estimating abundance was employed (Hilborn et al., 1976). The abundance of feeding ticks was determined by visually inspecting each trapped animal and animals were promptly released. Ticks were removed for subsequent identification.

The study on the effect of incremental removal of deer on the density of the deer tick was undertaken in Ipswich, Massachusetts, on a 567-hectare coastal site maintained by the Trustees of Reservations. Much of the property is comprised of a 9 km long barrier island characterized by beach, dunes, salt marsh and woodland. Lyme disease affects numerous residents of nearby sites (Lastavica et al., 1989). Mouse abundance and density of ticks feeding on mice was monitored as described. Deer abundance was reduced from an estimated 430 in 1983 to about 150 in 1991.

The model

Matrix Representation

Our present effort to apply a discrete mathematical model describing the life cycle of a particular population of vector ticks derives from a previously developed system of linked matrices constructed to represent the seasonal progression of the developmental stages of this tick (Sandberg et al., 1992). A separate matrix was used to represent each month of the year in a manner that captured the various events that punctuate the annual cycle of this tick. The main input variables were considered to be host density and efficiency of host-finding (scanning capacity) for each developmental stage of the tick. The original model was constructed around a set of arbitrarily chosen, seasonally variable biological parameters that remained constant from year to year. The present effort is designed to represent conditions actually observed during 1984-1991 in the Nantucket field site.

Table 1. Blood-feeding success of adult deer ticks feeding in the study site during 1983-1984.

Month

No. of deer present

No. of adults/deer

Days of activity/attached

No. of fed adults

Oct

10

86

15/10 = 1.5

1,290

Nov

10

62

30/10 = 3.0

1,860

Dec

10

9

25/10 = 2.5

220

Feb

10

11

15/10 = 1.5

165

Mar

10

11

30/10 = 3.0

330

Apr

10

11

15/10 = 1.5

165

Estimates representing feeding parameters of the tick population are entered as elements in a series of modified stage-structured matrices (Leslie, 1945; Caswell, 1989). The density of ticks present during each year is calculated as the product of the mathematical vector for the previous year and of the 12 transition matrices representing each month of the year in question:

where A represents an 11 x 11 matrix containing the monthly transition parameters and X (t) is the vector representing tick density during each of the 11 designated activity stages of the tick in year (t) (Sandberg et al., 1992). These include: eggs, nonfed larvae, fall-fed larvae, spring-fed larvae, nonfed nymphs (first year), fed nymphs (first year), nonfed nymphs (second year), fed nymphs (second year), nonfed adults, fall-fed adults and spring-fed adults.

Parameter Estimation

The two main input parameters that are used in the matrices are derived from the sequence of field observations actually conducted in the study site, as well as certain supplementary observations conducted elsewhere. One of these variables, host density, is determined by direct observation (of deer) and by mark-release-recapture (of rodents).

Table 2. Blood-feeding success of larval deer ticks feeding in the study site during 1984-1985.

Month


Kind of host


No. of hosts present


No. of larvae/host


Days of activity/attached


No. of larvae fed on

deer

mice

other

total

Aug



deer

10

200

15/3 = 5

10,000




mice

173

20



17,300



other

104

3




1,560

28,860

Sep



deer

10

178

30/3 = 10

17,800




mice

345

18



62,100



other

104

9




9,360

89,260

Apr



deer

10

8

15/3 = 5

400




mice

173

1



865



other

0

-




0

1,265

May



deer

10

84

30/3 = 10

8,400




mice

173

8



13,840



other

0

-




0

22,240

Jun



deer

10

121

30/3 = 10

12,100




mice

242

12



29,040



other

0

-




0

41,140

Jul



deer

10

31

15/3 = 5

1,550




mice

276

3



4,140



other

35

0




0

5,690

Total






127,285

10,920

188,455

The other main input parameter is estimated indirectly. Scanning capacity of a particular stage of the deer tick, questing in a particular month after a particular kind of host, is comprised of the following elements. The first two components, tick density on hosts and duration of activity that month, are derived from field observations. The third, duration of tick attachment, is determined in the laboratory (Piesman et al., 1979). Biological assumptions are required to estimate the fourth component in this formula, density of questing ticks. These assumptions are based on certain features of the seasonally discrete life cycle of this tick (Yuval and Spielman, 1990).

The developmental cycle of the stages of the deer tick proceeds as follows:

Egg ® Larva ® Nymph ® Adult ® Egg

Ticks feed once during each trophic stage of development before they moult to the next stage. The rigid seasonality of this feeding behaviour in the case of the deer tick punctuates its developmental cycle as follows: Eggs deposited in June and July hatch in August. Larvae abundantly quest for hosts in August through September and again in April through July. Nymphs abundantly quest in May through July, adults from mid-October through December and again in February through mid-April.

Density of feeding ticks is calculated, month by month, on the basis of the observed density of ticks per host and the observed abundance of that host. More than 4000 adult deer ticks appear to feed on deer in the site each year (Table 1), of which half are female. We assume that each of these becomes engorged and produces 2000 fertile eggs. We further assume that all eggs hatch to produce an equivalent number of questing larvae. In this case, more than four million questing larvae would be produced in the site each year. The cohort of ticks that hatches in the fall of 1984 will provide the basis of our analysis. Some of these larval ticks will attach to available hosts, and all of these will feed to repletion in three days. Those that fail to find a host during the fall of that year will resume questing during the subsequent spring (Table 2). We assume that all such engorged larvae ultimately moult to the nymphal stage and that all seek hosts beginning in the spring of 1985. Some of these nymphs will attach to available hosts as the opportunity presents itself during the spring and summer of 1985 and of 1986, and all of these will feed to repletion in five days (Table 3). We assume that all nymphs that engorge in 1985 moult to the adult stage and that all will seek hosts beginning in the fall of the same year. Some of these adults will attach to available hosts, and all of these will feed to repletion in ten days (Table 1). All engorged adults ultimately oviposit beginning in the summer of 1986. In this manner, the population of deer ticks completes its developmental cycle.

Table 3. Blood-feeding success of nymphal deer ticks feeding in the Nantucket study site during 1985.

Month


Kind of host


No. of hosts present


No. of nymphs/host


Days of activity/attached


No. of larvae fed on

deer

mice

other

total

Aug



deer

10

22

15/5 = 3

660




mice

173

2



1,038



other

0

-




0

1,698

Sep



deer

10

72

30/5 = 6

4,320




mice

173

7



7,266



other

0

-




0

11,586

Apr



deer

10

34

30/5 = 6

2,040




mice

242

3



4,356



other

0

-




0

6,396

May



deer

10

6

30/5 = 3

360




mice

276

1



1,656



other

35

0




0

2,016

Jun



deer

10

1

30/5 = 6

60




mice

242

0



0



other

0

-




0

60

Jul



deer

10

4

30/5 = 6

240




mice

345

0



0



other

35

0




0

240

Total





7,680

14,316

0

21,996

The density of questing deer ticks is estimated by combining the results of the monthly field observations with derivations based on these assumptions. The calculation begins with the assumption that about four million newly hatched larvae began to quest early in August of 1984 (Table 4). The activity of this cohort of ticks is followed until they mature to the adult stage and ultimately oviposit. We calculate the number of larvae questing at the beginning of each month by subtracting the calculated number of larvae that fed during the previous month from the number estimated to be questing during that month. In the event that no larvae were found on hosts during a particular month, we assume that diapause prevented questing activity. The number of questing nymphs and adults is calculated similarly. Larvae that engorge during the summer season are assumed to moult and to recommence questing as nymphs during the following season. These rules dictate the manner of calculating scanning capacity.

Table 4. Seasonal abundance of questing deer ticks in the Nantucket study site, calculated by subtracting the observed number of feeding ticks in each stage (from Tables 1-3) from the number assumed to be present at the end of the previous month. This display represents the cohort that hatched as larvae in 1984.

Months


No. of larvae

No. of nymphs

No. of adults

questing

fed

questing

fed

questing

fed

1984

Aug

4,030,000

28,860





Sep

4,001,140

89,260





1985

Apr

3,911,880

1,265

188,455*

1,698



May

3,910,615

22,240

186,574

11,586



Jun

3,888,375

41,140

172,092

6,396



Jul

3,847,235

5,690

164,097

2,016



Aug



162,820

60



Sep



162,745

2,610



Oct





21,996

1,290

Nov





20,706

1,860

Dec





18,846

220

1986

Feb





18,626

165

Mar





18,461

330

Apr



70,335


18,131

165

May







*Includes 70,335 ticks from th Previous season.
Underscoring denotes groups of questing ticks containing individuals that engorged in a previous stage and emerged to quest at the designated month.

Scanning capacity of these ticks is calculated for a particular stage of the tick seeking a host in a particular month. That of an adult in October, for example (Tables 1 and 4), is:

(86 x 15)/(10 x 21,996) = 0.005865

which is the probability that an adult tick may attach to a deer. Thus, only 1 out of 171 adult deer ticks that are present in the site in October would find a host. A similar calculation is repeated for each of the other stages of the tick, for each month and for each kind of host. Scanning capacity is calculated and averaged over each of the 8 years (1984-1991) for which data are available.

Figure 1. Changes in the observed abundance of mice present in the field site and simultaneous changes in the abundance of larval deer ticks predicted for the first of December for each year of the study.

Implementation of the Model

To implement the model, two programs are constructed; one for calculating scanning capacity and the other for vector density. For convenience, both employ computerized spreadsheets (Lotus 1-2-3).

The program for scanning capacity is based on the set of data summarized in Tables 1-4. Because scanning capacity is calculated for each month of the year, a column in the spreadsheet is designated for each month. The rows receive the observed data on host and tick density as well as the equations for calculating scanning capacity. The output of each columnar calculation is placed in designated positions in the appropriate matrix, thereby enabling us to obtain numerical values when matrices are multiplied.

In constructing the program for tick density, difference equations represent matrix multiplication. The columns represent density of the various stages of the vector for each month of the year. Vector density during each month depends on density during the previous month and on a transition parameter that represents feeding, moulting or death of the tick. In this manner stage-specific vector density is calculated for each of the designated months of the year.

Results

Preliminary to our attempt to determine whether the abundance of mice may be critical in perpetuating the vector of Lyme disease, we analysed the pattern of annual variation in mouse abundance in the study site. Mouse abundance fluctuated annually, with a six-year rising trend between the years 1984 and 1989, followed by a two-year pattern of decrease (Figure 1). The abundance of mice varies from year to year in the study site.

We then determined whether the abundance of ticks may vary in parallel with the observed variations in the abundance of mice. Abundance of larvae during midwinter was chosen to represent the dynamics of the tick population because this phase of tick development directly reflects reproductive activity and is subject to few environmental variables. Abundance of questing larvae closely follows that of mice, generally lagging by a one- or two-year interval (Figure 1). The abundance of newly emergent larvae varies with the abundance of mice.

In order to examine the effect of a surfeit of hosts on the abundance of newly emerged larval deer ticks, we determined how many would develop if their rodent hosts remained as abundant as they were in an exceptional year. The year 1989, which was such a year, was chosen as the standard of such excess (Figure 1). The simulation was based on a model constructed so that one female deposited a clutch of 1000 fertile eggs in a site that had previously been devoid of ticks. Tick density increased exponentially, with a doubling time of 1.06 years (Figure 2). When hosts for the subadult stages of this tick are exceptionally abundant, the tick population increases rapidly and continuously.

We then determined how an exceptional paucity of hosts may affect the abundance of newly emergent larvae. Because mice were particularly scarce in the study site in 1990, that year was chosen as the standard of host scarcity (Figure 1). The simulation began, as in the previous simulation, with the deposition of a clutch of 1000 fertile eggs in a site that had previously been devoid of ticks. Tick density waned and ultimately disappeared (Figure 3). Under these conditions of exceptional mouse scarcity and a continuing presence of ten deer, deer ticks might become endangered.

Figure 2. Increase in the simulated abundance of larval deer ticks assuming that mice continuously remained exceptionally dense (as observed in 1989).

Figure 3. Decrease in the simulated abundance of larval deer ticks assuming that mice continuously remained exceptionally scarce (as observed in 1990)

To determine whether a particular level of mouse density may result in a stable density of ticks, we applied a similar process using the data of 1986, a year in which the density of mice noted in the study site somewhat exceeded that in the designated year of paucity, 1990 (Figure 1). After 1000 eggs were 'deposited in the empty field' of the study site, the tick population decreased and oscillated (Figure 4). Tick density ultimately stabilized at a density that was about half that of the first year of the simulation. Indeed, a stable density of ticks appears to accompany an 'ordinary' density of mice.

Finally, we compared the relative contributions of mice and deer to the abundance of deer ticks. Mouse density corresponding to that observed in 1985 was chosen for this simulation because this level seemed 'typical.' This year's abundance of mice in the study site (Figure 1) permitted simulated tick abundance to rise exponentially, with a doubling time of 2.7 years (Figure 5). We simulated removal of one deer from the site and, by graphical interpolation, found the density of mice required to maintain tick density at the original level (Figure 5). Under these conditions, an increment of 30 mice compensates for the absence of one deer.

Discussion

The dynamics of populations of deer ticks depends largely on host availability. Only a minute portion of the tick population proceeds from stage to stage, mainly due to a failure to find hosts (Sandberg et al., 1992). Longevity of these ticks seems to be closely programed in nature; a questing larva does not live longer than 11 months, a nymph 14 months and an adult 8 months (Yuval and Spielman, 1990).

The observed series of seasonal punctuations in the life cycle of these ticks insulate development from prolongation or abbreviation due to short-term weather conditions (Yuval and Spielman, 1990).

Successful development of a tick depends upon its scanning capacity. Although various ticks move toward their hosts (Semtner and Hair, 1975), deer ticks appear to 'quest' passively (Daniels et al., 1989). The ability of a tick to scan for a host, therefore, depends upon the nature of the terrain and the ability of the tick to resist drying. These variables are reflected in our parameter values.

Host abundance similarly affects the ability of a tick to find a host. Mouse density varies in a complex manner (Adler et al., 1984). They begin to reproduce in the spring and continue to increase until fall. This temporal coincidence facilitates feeding activity of deer ticks by matching maximum questing activity of larvae, which eclode in August (Yuval and Spielman, 1990), with maximum availability of their hosts. Few alternative hosts are endemic to this isolated island on which our analysis focuses. Years of relative mouse abundance tend to follow years of scarcity. This alternation buffers the abundance of deer ticks because the same cohort of ticks generally experiences both extremes in their different subadult stages of development which spans two years or more. Multi-year cycles also occur. The five-year cycle frequently observed in populations of these white-footed mice (Adler et al., 1984; Krohne et al., 1988) is consistent with the pattern observed in our study. The matrix structure of our model incorporates all of these crucial and temporally variable relationships.

Figure 4. Stabilization in the simulated abundance of larval deer ticks assuming that mice continuously remained moderately scarce (as observed in 1986).

Figure 5. Increase in the simulated abundance of larval deer ticks calculated on the assumption that mice continuously remained moderately dense (as observed in 1985); then calculated as though 1 deer were removed; then calculated as though 30 mice were substituted for that deer.

The derivation of scanning capacity combines estimates of potential turn-over of ticks on hosts and on efficiency of host-finding (Sandberg et al., 1992). Although separate calculations are made for each stage of the tick questing during each month of the year, our simulations are based on average estimates spanning the eight-year period of observation. Estimates of the abundance of questing ticks are required for directly calculating the scanning capacity of ticks in a particular site. Because field observations are not available for direct estimation of this parameter, our estimates are based on an algorithm that uses the density of feeding ticks on hosts as one of its major inputs.

The density of feeding ticks is more easily determined in the field than is that of questing ticks. Questing deer ticks generally are sampled by sweeping vegetation with a section of fabric intended to represent a surrogate host. Such an operation, however, is fraught with uncertainty; only a tiny fraction of the population can be sampled; rarely at night or in the rain and never from sequestered sites such as a mouse's nest. Indeed, the distribution of questing larval deer ticks clumps closely around the site in which a gravid female had deposited her eggs (Daniels et al., 1989), and questing larvae are far more abundant than are ticks in any other stage of development. Questing nymphal deer ticks are even more difficult to sample by means of a cloth flag because some variable portion of the population appears to emerge and quest within the nest of the host of the previous stage (Masher and Spielman, 1986). The number of questing ticks resident in a site vastly exceeds those that can be sampled on a flag or that actually succeed in finding a host and that are seen on hosts.

The presence of numerous deer generally is regarded as prerequisite to an abundant infestation of deer ticks (Spielman, 1988). Qualitative observations have established that stable infestations of the tick are evident solely where deer are resident; correlative observations demonstrate a relationship between the density of larval deer ticks feeding on mice and the density of deer (Wilson et al., 1985); and experimental evidence demonstrates that destruction of resident deer was followed by the virtual elimination of these ticks (Wilson et al., 1988). We now demonstrate that the abundance of mice also contributes to the abundance of these ticks and define a situation in which a paucity of mice eliminates the tick infestation. Under certain circumstances, one deer may be the equivalent of 30 mice.

This report constitutes the first representation of the population dynamics of the vector of Lyme disease in an actual endemic site. Our use of a sequence of assumptions combined with observational data makes it possible to construct an algorithm for calculating the density of questing ticks in the study site. We conclude that the abundance of deer ticks is sensitive to the abundance of mice and that these ticks may fail to perpetuate when mice are as scarce as they were in the field site in two of the eight years of observation.

Acknowledgements

Supported in part by grant 19693 from NIAID and the Office of Naval Research. This is a contribution of the Nantucket Field Station of the University of Massachusetts.

References

ADLER, G.H. and TAMARIN, R.H. 1984. Demography and reproduction in island and mainland white-footed mice (Peromyscus leucopus) in south-eastern Massachusetts. Canadian Journal of Zoology 62: 58-64.

CASWELL, H. 1989. Matrix Population Models. Sunderland: Sinauer.

DANIELS, T.J., FISH, D. and FALCO, R.C. 1989. Seasonal activity and survival of adult Ixodes dammini (Acari: Ixodidae) in southern New York State. Journal of Medical Entomology 26: 610-614.

HILBORN, R., REDFIELD, J.A., and KREBS, C.J. 1976. On the reliability of enumeration for mark and recapture census of voles. Canadian Journal of Zoology 54: 1019-1024.

KROHNE, D.T., MERRITT, J.F., VESSEY, S.H. and WOLFE, J.O. 1988. Comparative demography of forest Peromyscus leucopus. Canadian Journal of Zoology 66: 2170-2176.

LASTAVICA, C.C., WILSON, M.L., BERARDI, V.P., SPIELMAN, A. and DEBLINGER, R.D. 1989. Rapid emergence of a focal epidemic of Lyme disease in coastal Massachusetts. New England Journal of Medicine 320: 133-137.

LESLIE, P.H. 1945. On the use of matrices in certain population mathematics. Biometrika 35: 183-212.

MATHER, T.N. and SPIELMAN, A. 1986. Diurnal detachment of immature deer ticks (Ixodes dammini) from nocturnal hosts. American Journal of Tropical Medicine and Hygiene 35: 182-186.

PIESMAN J., SPIELMAN, A., ETKIND, P., REUBUSH, T.K., and JURANEK., D. 1979. Role of deer in the epizootiology of Babesia microti in Massachusetts, USA. Journal of Medical Entomology 15: 537-540.

SANDBERG, S., AWERBUCH T.E. and SPIELMAN, A. 1992. A comprehensive multiple matrix model representing the life cycle of the tick that transmits the agent of Lyme disease. Journal of Theoretical Biology 157: 203-225.

SEMTNER, P.J. and HAIR, J.A. 1975. Evaluation of CO2-baited traps for survey of Amblyomma americanum Koch and Dermacentor variabilis Say (Acarina: Ixodidae). Journal of Medical Entomology 12: 137-138.

SPIELMAN, A. 1988. Lyme disease and human babesiosis: evidence incriminating vector and reservoir hosts. In: Englund, P.T., and Sher, A. eds. The Biology of Parasitism. New York: Alan R. Liss, pp. 147-165.

WILSON, M.L., ADLER; G.H. and SPIELMAN, A. 1985. Correlation between abundance of deer and that of the deer tick, Ixodes dammini (Acari: Ixodidae). Annals of the Entomological Society of America 78: 172-176.

WILSON, M.L., TELFORD III, S.R., PIESMAN, J. and SPIELMAN, A. 1988. Reduced abundance of immature Ixodes dammini (Acari: Ixodidae) following elimination of deer. Journal of Medical Entomology 25: 224-228.

YUVAL, B. and SPIELMAN, A. 1990. Duration and regulation of the developmental cycle of Ixodes dammini (Acari: Ixodidae). Journal of Medical Entomology 27: 196-201.

Modelling helminth population dynamics

G. Smith

University of Pennsylvania
School of Veterinary Medicine
Department of Clinical Studies
New Bolton Center, 382 West Street Road
Kennett Square, Pennsylvania 19348, USA

Abstract
Introduction
A brief history of helminth models
The parasitic phase of the life cycle of Fasciola hepatica
The parasitic phase of the life cycle of the common trichostrongylid nematode parasites of cattle and sheep
Other regulatory processes
The significance of parasite frequency distributions
Summary
Acknowledgements
References


Abstract

The population dynamics of the common helminth parasites of cattle and sheep have been the focus of much study over the last three decades. The dynamics of the accessible free-living stages is well understood, but there is still considerable uncertainty concerning the processes which regulate and control parasite numbers in the ruminant host. For example, it has long been supposed that resistance to Fasciola infections in cattle develops between the sixteenth and twentieth week after infection and is manifested in the rejection of established flukes. On the other hand, recent reanalyses of experiment infections show that the mortality of established flukes is essentially constant over this period and the supposed sudden rejection of the parasite burden is, in fact, illusory. In the case of the trichostrongylid gastrointestinal nematodes, there is a plethora of hypotheses concerning the causation of the observed changes in parasite burden during single- or trickle-infection experiments. Since these results are very similar irrespective of the parasite-host combination under study, it seems likely that eventually a single hypothesis should be able to account for all the observed patterns. It is suggested here that the proportion of ingested larvae that becomes established declines in sigmoidal fashion and that the mortality of established worms increases to some asymptotic value as the host's experience of infection increases.

Investigations of the population dynamics of helminth infections are useful in that they lead to mathematical models that can be used to design and communicate rational strategies for parasite control. Examples of such models include the UNIVERSE model for Trichostrongylus infections in sheep and PARABAN, a model for the common gastrointestinal parasites of cattle.

Introduction

This paper will deal with some recent empirical models for the population processes that occur in the parasitic phase of the life cycles of the common liver fluke, Fasciola hepatica, and several of the gastrointestinal nematode parasites of cattle and sheep. It will be argued in both cases that changes in the rate of parasite establishment provide the best explanation of observed patterns of infection. It will also be argued that density dependent regulation of parasite fecundity, though important in certain species, is by no means ubiquitous amongst the parasites considered here. Finally, the paper will address the problem of aggregated parasite distributions and the different extent to which these have to be taken into account in models of fascioliasis and parasitic gastro-enteritis.

The models described below deal exclusively with parasites of veterinary importance. Nevertheless, they owe a substantial debt to models that were first elaborated in a medical context and so we begin with a brief survey of the history of helminth models in general.

A brief history of helminth models

There are numerous mathematical models for the population biology of parasitic helminths (cestodes, trematodes, nematodes) of medical and veterinary importance. Most of these models are built according to a formal structure that evolved in a series of papers by Kostitzin (1934), Macdonald (1965), Tallis and Leyton (1966), Gordon et al., (1970) and Nasell and Hirsch (1972).

Between 1965 and the early 1980s, the principal focus was on parasitic helminths causing disease in humans (e.g. nematode infections - Anderson, 1979b, 1980a, 1982, 1985; Anderson and May, 1982, 1985a; Anderson and Medley, 1985; Dietz, 1982 - trematode infections - Hairston, 1965; Sturrock and Webbe, 1971; Lewis, 1975; Cohen, 1977; Barbour, 1978; Anderson and May, 1979a; Coutino et al., 1981 - and cestode infections - Ghazal and Avery, 1974; Keymer, 1982). A copious theoretical literature accumulated over the same period. This literature dealt in particular with the representation and understanding of the processes that regulate and control parasite abundance (e.g. Leyton, 1968; Crofton, 1971; Bradley, 1972; Bradley and May, 1978; Anderson 1976, 1978, 1979a, 1979b, 1979c, 1980a, 1980b, 1982; Anderson and Gordon, 1982; Anderson and May, 1978a, 1978b, 1978c, 1979a, 1979b, 1982, 1985a, 1985b; Anderson and Medley, 1985, Smith, 1984a).

From the mid 1970s onward, more and more of the models dealt with helminth infections of veterinary importance. Some early examples include Gettinby (1974), Gettinby et al. (1974), Hope-Cawdery et al. (1978), Williamson and Wilson (1978), Thomas (1978), Gettinby et al. (1979), Gettinby and Paton (1981), Smith 1982, Wilson et al. (1982), Paton and Gettinby (1983), Paton and Thomas (1983), Paton et al., 1984. These studies and those that followed depended in large part on methodologies that evolved during the mid 1960s and early 1970s in the course of modelling helminth infections in humans and on models of physiological development first elaborated in the zoological literature in the mid 1950s (Grainger, 1959; see also Gettinby and Gardiner, l 980 and Gardiner et al., 1981).

The most recent phase of modelling parasitic diseases of veterinary importance has seen a burgeoning literature on models for nematode infections of ruminants in cattle and sheep. For example, there are at least five models for Haemonchus contortus (Talks and Donald, 1964; Tallis and Leyton, 1969; Leyton, 1968; Gordon et al., 1970; Tallis and Donald, 1964; Smith, 1988, 1990; Coyne, et al., 1991a, 1991b; Coyne and Smith, 1992; Leathwick, 1992), four for Ostertagia circumcincta (Paton and Gettinby, 1983, 1984; Paton and Thomas, 1983; Paton et al., 1984; Paton, 1987; Gettinby et al., 1989; Callinan end Arundel, 1982; Callinan et al., 1981; Smith and Galligan, 1988, Smith, 1989), two for Ostertagia ostertagi (Gettinby et al., 1979; Gettinby and Paton, 1981; Grenfell and Smith, 1985; Smith et al., 1986, 1987a, 1987b; Grenfell et al., 1986, 1987a, 1987b) and one for Trichostrongylus colubriformis (Barnes et al., 1988, Dobson et al., 1990a, 1990b, 1990c; 1990d; Barnes and Dobson, 1990a, 1990b). Over the same period, there was a parallel surge in the development of mathematical models for cestode and trematode infections of sheep (Mizraji et al., 1980; Hernandez et al., 1983; Correa et al., 1983; Gemmell et al., 1986a, 1986b, 1987; Roberts et al., 1986, 1987; Meek and Morris, 1981; Smith, 1982, 1984a, 1984b, 1984c).

Broadly speaking, the early veterinary helminth models were conceived as tools to help predict damaging disease outbreaks. There was considerable emphasis on the effect of weather on the development of the free-living stages of each parasite species because it was believed that climate held the key to the epidemiological patterns observed in the field (Gettinby et al., 1974, 1979). This, together with the fact that the free-living stages were often more accessible than the parasitic stages, engendered a gigantic literature (not dealt with here) on the development and mortality of the free-living stages. As a result, by the early 1980s, modelling the demography of the free-living stages of any parasite was limited only by the availability of appropriate data. The methodology was well established and has since been subject to relatively few refinements.

During the 1980s, veterinary helminth models became much more like medical helminth models in that they were more concerned with the efficacy of proposed parasite control strategies. The utility of the early predictive models lay in the extent to which they could exactly predict pasture larval contamination at specific sites. They assumed that if one knew what the pattern of pasture larval contamination was likely to be, one would also know how best to apply anthelmintics. The more recent models and, indeed, modified versions of the predictive models, embody the view that the most efficient use of anthelmintics remains to be determined and that models can be instrumental in guiding that decision. At first modellers were interested only in what had to be done to keep parasite burdens low; later, several of the models were modified to address the question how can one keep parasite burdens low and simultaneously impede the spread of anthelmintic resistance (Gettinby et al., 1989; Barnes and Dobson, 1990b; Smith 1990). This increasingly ambitious purpose meant that the model elements dealing with the parasitic phase became more sophisticated. Unfortunately, there was much less information on the demography of the parasitic stages of helminths of veterinary importance than there was on the free-living stages and veterinary models differed from one another not so much in the mathematics of their construction (which was fairly standard) but in the nature of the host-parasite interactions that were actually modelled. For example, there are (still) at least two competing hypotheses concerning the regulation of trichostrongylid infections in ruminants: the 'threshold' hypothesis first suggested by Dineen et al. (1965), Dineen and Wagland (1966) and Wagland and Dineen (1967) and the competing 'turnover' hypothesis developed in a series of papers by Michel (1963, 1969a, 1969b, 1970). Variants of both hypotheses can be found in all the existing trichostrongylid models. Nor is there any agreement on the processes which regulate F. hepatica infections in cattle. For example, Doyle (1972) argued that calves given a single infection of F. hepatica metacercariae will begin to reject the flukes between 20 and 24 weeks after infection, whereas Hope-Cawdery et al. (1977) stated that the fluke survivorship curve following a single infection was a simple exponential decline indicating a constant attrition of flukes throughout the course of the infection.

The parasitic phase of the life cycle of Fasciola hepatica

The Problem

Fascioliasis is the cause of serious production losses worldwide. Often considered in the context of ovine infections, where production losses are directly, but non-linearly, related to intensity of infection (Hawkins and Morris, 1978), modelling fascioliasis has proved to be a relatively simple task. In sheep there appears to be no significant acquired immune response to infection: parasite numbers are regulated by a severe density dependent constraint on parasite fecundity and density dependent, parasite-induced host mortality (Smith, 1982). Modelling bovine fascioliasis, on the other hand, has proved to be much more problematic. In the Americas, where bovine fascioliasis is more important than ovine fascioliasis, the infection accounts for millions of dollars of lost profit annually. In North America, bovine fascioliasis is endemic in the gulf coast region and north western regions of the USA and in eastern Canada (Malone et al., 1982; Bouvry and Rau, 1986). In the Caribbean, bovine fascioliasis has been a serious long-term problem in Cuba, Jamaica and Puerto Rico (Frame et al., 1979; Bundy et al., 1983). It occurs also in many South American countries (Ueno et al., 1982; Griffiths et al., 1986). Precise estimates of the scale of the production losses are difficult to come by. Many studies are methodologically flawed because they fail to consider the cost and production consequences of attempting to control the infection (Morris, 1969). One of the most thoughtful though is the study by Bundy et al. (1983) on the losses incurred in Jamaica between 1979 and 1980. They estimated total annual losses of over two million (Jamaican) dollars for a national herd of about 300,000 animals. Anthelmintics directed against bovine fascioliasis can bring about profitable increases in production (Genicot et al., 1991) but over an area as geographically diverse as North and South America the optimum treatment strategy is often by no means obvious. One solution is to construct a model that is sensitive to details of climate and management and, in particular, accurate with respect to its representation of the population of flukes within the host. This last is particularly important because most common flukicides are not equally efficacious over all age classes of flukes. Unfortunately, as mentioned above, there is no consensus on what happens in the parasitic phase of the life cycle in cattle.

A Model for the Parasitic Phase

The outcome of the well known single infection experiments by Doyle (1971, 1972) dominated much of the subsequent literature on the dynamics of the parasitic phase of the F. hepatica life cycle in cattle. Doyle found that most of the fluke burden was rejected between weeks 20 and 24 after infection. Hope-Cawdery et al. (1978) disagreed and suggested that the 'decline in the numbers of fluke in the liver follows an exponential decay curve' but offered no supporting data. An opportunity to discriminate between these conflicting claims arose during the construction of a mathematical model for bovine fascioliasis intended to assist in the design of effective anthelmintic strategies in the Americas. A survey of the literature revealed a number of studies in which calves had been infected with 1000 metacercariae and the flukes counted at varying intervals after infection. Combining these data produced a survivorship curve spanning 400 days (Figure 1).

The compilation showed that there is no reason to reject the simpler hypothesis that the survivorship curve was indeed exponential and that fluke mortality was constant throughout the course of the infection. In other helminth infections, parasite mortality following single infections is non-linearly related to the number of infective stages administered at the outset. Although such a model has been fitted to the data in Figure 1, there is no convincing evidence that fluke mortality varies in this way. However, challenge experiments clearly show that calves can develop an acquired resistance to infection with F. hepatica and the question arises how is this resistance expressed? Doy and Hughes (1984a) make a convincing argument that resistance to reinfection in cattle occurs at around the time when the juvenile flukes attempt to penetrate the liver. The evidence is twofold. The lack of liver damage in immune challenged animals suggests that very few flukes enter the liver capsule (or are killed soon after). On the other hand, there is no significant difference between the numbers of flukes recovered from the body cavity of newly challenged immune calves and newly challenged susceptible animals. Because juvenile flukes are already penetrating the liver capsule in large numbers four to seven days after excystment (Doy and Hughes, 1984b), it seems that resistance to further infection is expressed very early in the developmental phase of each cohort of flukes.

Figure 1. Upper axis: the survivorship curve of Fasciola hepatica in calves infected once only with 1000 metacercariae. Lower axis: the average mortality of established F. hepatica in calves infected once only with varying numbers of metacercariae (data from Pomorski, 1980; Wensvoort and Over, 1982; Kendall et al., 1978; Furmaga et al., 1983; Malone et al., 1984; Herlich, 1977; Flagstad et al., 1972; Dargie et al., 1972; Dickson, 1964; Oldham, 1985; Hope-Cawdery et al., 1978).

Doyle (1973) demonstrated that this resistance to challenge infection takes some time to develop. He found no significant resistance to infection in calves infected seven weeks previously, but substantial resistance in calves infected with the same number of parasites 12 weeks previously. The simplest function describing this pattern of change in the proportion of flukes gaining access to the liver is a declining sigmoidal curve of the form e (a+bi)/(1+e (a+bi)) (where a and b are constants and i is the infection rate). The parameters of such a curve were estimated by a non-linear least squares method (Berman and Weiss, 1978) from trickle infection data in Van Tiggle (1978). Van Tiggle infected calves with between 50 and 400 cysts per day. The rate of change in the proportion of flukes gaining access to the liver capsule varied with the daily rate of infection (Figure 2a). The steepness of the curves was determined by parameter b, which was inversely proportional to the infection rate (Figure 1b). When the complete model for the parasitic phase of the F. hepatica life cycle in calves was then tested against results of the trickle infection experiment (20 cysts per day) reported by Burden et al. (1978) it was found to slightly overestimate the observed burdens (Figure 1c). Reinspection of the relationship between parameter b and infection rate indicates that it may be curvilinear rather than linear as shown (Figure 1b). If this were the case, backwards extrapolation of the curve in Figure 1b to give the appropriate value of parameter b would give a much better fit to the data in Figure 1c.

The parasitic phase of the life cycle of the common trichostrongylid nematode parasites of cattle and sheep

The Problem

There are over a dozen different models for the parasitic phase of trichostrongylid gastrointestinal nematode parasites of cattle and sheep. What each of these models has in common with the others is that it represents a hypothesis about what happens to the parasites after the ingestion of the third larval stage and there appear to be almost as many hypotheses as there are models. Some workers suggest that this multiplicity of frameworks represents a real level of diversity between the different species of trichostrongylid nematodes. This is a view which Smith (in press) has contested suggesting instead that a single model adequately accounts for the observed dynamics in all of the common species. There are practical as well as biological implications in this suggestion. Natural infections are usually mixed species infections and, since no trichostrongylid species is ubiquitous in the cattle-and sheep-producing regions of the world, the mix of species changes from place to place. It would be far simpler if it were indeed true that a single model framework would work for all the species of interest (one species being differentiated from the rest merely in terms of the numerical values assigned to the constants that determine development and mortality rates). Such a framework would have universal application anywhere in the world.

Figure 2. a) The proportion of flukes becoming established in the liver parenchyma at intervals during a trickle infection (estimated from data in Van Tiggle, 1978). b) Variations in the parameter b in trickle infection experiments involving 50 to 400 cysts per day (estimated from data in Van Tiggle, 1978). c) Predicted (solid line) and observed (·) fluke burdens in which calves were infected with 20 metacercariae each day (data from Burden et al., 1978).

A Model for the Parasitic Phase

There are two pieces of evidence that a single framework is appropriate for all the common trichostrongylid species. First, the qualitative outcome of trickle and single infection experiments is always the same irrespective of which species is used. Second, a single model framework has been shown to account adequately for the observed patterns in all the instances it has been tested (Smith, in press). The important features are these. Established nematodes (fifth stage worms) die at a rate which varies with the hosts' experience of infection. Parasite mortality increases non-linearly with exposure, but only up to a point. After that point it remains constant irrespective of any further exposure (Figure 3). In the most sophisticated versions of the model the mortality of the fifth stage worms declines in the absence of further exposure (Coyne, 1991), but the basic model works very well in the absence of this refinement provided its use is restricted to prebreeding hosts (Smith and Guerrero, 1993). Meanwhile, the proportion of newly ingested third stage larvae that become established in their predilection sites declines with time (Figure 3).

The pattern of immune exclusion occurring in the first 24 hours after the infective larvae have been ingested follows a declining sigmoidal curve in the same manner as the fluke model discussed above but, unlike the fluke model, it has so far proved impossible to demonstrate that the rapidity of the decline varies with the host's experience of infection (Smith, in press).

Other regulatory processes

Parasite-Induced Host Mortality

When an infected host dies its parasite burden dies with it. If the probability of host death increases with the intensity of infection, parasite mortalities due to that cause are density dependent and potentially regulatory. Parasite-induced host deaths in sheep infected with F. hepatica are still an important mechanism by which populations of flukes are regulated (Smith, 1982). For example, in the Rio Grande Do Sul in Brazil, acute fascioliasis causes mortality rates of 15-20% (500-800 sheep per flock, Ueno et al., 1982). In the case of ovine fascioliasis, the relationship between host mortality rates and current parasite burdens is relatively easy to discern. That does not seem to be true of F. hepatica in cattle or trichostrongylid infections in cattle or sheep. Anecdotal reports suggest that liver dysfunction associated with chronic bovine fascioliasis may be implicated as an accessory factor in cattle deaths (Bundy et al., 1983) and acute fascioliasis in calves may be fatal, but I can find no systematic investigation of the relationship between fluke burden and host mortality rates in cattle.

Figure 3. Upper axes: population processes regulating the abundance of Ostertagia ostertagi in cattle (Smith, in press). Lower axes: observed (solid line) and predicted (·) worm burden in calves infected daily with either 500 or 1000 larvae per day (data from Michel, 1969a).

Similarly, there seems to have been no systematic attempt to examine parasite-induced host mortality in trichostrongylid infections. Such information that exists is generally inconsistent. Deaths attributable to natural infections with trichostrongylid parasites have been reliably reported (Anderson et al., 1969; Al Saqur, 1982) but in order to get information on worm burdens at time of death we have to turn to experimental infections. In the case of bovine parasites, administration of over 300,000 L3 larvae either in a single dose or as the culmination of smaller doses has frequently caused the deaths of a proportion of the calves (Michel, 1963; Herlich, 1959, 1962) but neither the proportions affected nor the worm burdens at time of death show any between-study consistency. Parasite-induced host mortality is a feature of only one of the models of trichotrongylid biology published to date. Barnes and Dobson (1990a) utilized the 'lethal level' concept of Crofton (1971) to mimic sheep deaths due to T. colubriformis infections. They assumed that parasite burdens could be best described by the negative binomial distribution and 'killed' any hosts having an estimated number of 50,000 or more adult worms based upon the presumed value of k for the flock (where k is the exponent of the negative binomial distribution).

Regulation of Fecundity

There is good direct evidence that the fecundity of F. hepatica is regulated in sheep (Smith, 1982) and reasonable inferential evidence that the same thing happens in cattle (Van Tiggle, 1978). The regulation of trichostrongylid fecundity is much less clear cut. Michel (1969b) believed that the constraints on the fecundity on O. ostertagi in calves were so formidable that one could expect the same fecal egg output irrespective of the actual worm burden. This was to overstate the case, but Smith et al. (1987a) were able to show that O. ostertagi fecundity was significantly smaller in hosts infected for extended periods with large numbers of parasites. Paton (1987), quoting work by Jackson and Christie (1979) and Gibson and Everett (1978), argued that the fecundity of T. circumcincta was similarly regulated. However, other analyses indicated that constraints on the fecundity of this parasite were probably feeble at best (Callinan and Arundel, 1982; Symons et al., 1981; Gibson and Parfitt, 1977; Smith and Galligan, 1988). The cautionary paper by Keymer and Slater (1987) made workers more critical about what constituted evidence for density dependent regulation of fecundity. For example Coyne et al. (1991b) reinvestigated claims for constraints on the fecundity of H. contortus infections in sheep (e.g. Roberts and Swan, 1981; Coyne et al., 1991b). They measured parasite fecundity in natural and experimental infections and in neither case was there any need to invoke regulatory processes to explain observed patterns of fecal output (Figure 4). Furthermore, G. Smith (unpublished work) has been unable to find evidence that fecundity is regulated in bovine infections of C. oncophora or T. axei.

This kind of investigation is complicated by a number of factors. The actual measurement of fecundity (eggs/female worm/day) requires that both eggs and female worms can be counted accurately. These parameters are often estimated based upon assumptions about recovery rates, sex ratios and amount of faeces produced per day in growing animals. Furthermore, it is necessary to be able to identify mature (egg laying) worms. Fecundity is normally calculated on the assumption that all fifth stage female worms are mature, but this is not the case and in the initial stages of an infection measured levels of fecundity are frequently less that actual levels because immature fifth stages predominate. Some models take explicit account of this maturation phase (e.g. Barnes and Dobson, 1990a; Coyne et al., 1991a, 1991b) in order to lend verisimilitude to frameworks that are meant to be used by non-modellers. While the realistic patterns so obtained inspire confidence in lay users, it is debatable whether the increased complexity in model format is worth the small improvement in overall model performance.

Figure 4. Fecundity of Haemonchus contortus. Upper axis: The relationship between estimated fecundity and total worm burden. Lower axis: The predicted (solid line) and observed (·) mean total output of eggs per day; the dashed lines show the 95% confidence limits for the observations. (Redrawn from Coyne et al., 1991a).

The significance of parasite frequency distributions

In a single host the severity of the constraint depends simply on the intensity of infection. However, in a community of hosts, the constraints on survival and fecundity summed over the whole parasite population depend upon the frequency distribution of parasites per host. Parasite frequency distributions are frequently aggregated. Such distributions are conveniently described using the negative binomial frequency distribution. This distribution is completely characterized by two parameters: the mean of the distribution and an exponent, k (the degree of aggregation being inversely proportional to the value of k). Aggregated parasite distributions enhance the ability of regulatory processes to maintain parasite populations at or near their equilibrium level but incorporating parasite frequency distributions in realistic models of parasite population biology is fraught with difficulty. Indeed, it is not always possible to incorporate parasite frequency distributions and the question arises as to whether this compromises model performance. Smith and Guerrero (1992) investigated this problem with respect to models of trichostrongylid population dynamics. They found that trichostrongylid populations are aggregated but that the estimated value of k for such populations is usually greater than one. Unlike helminth parasites of wildlife, where many hosts may escape infection altogether, helminth parasites of domestic species maintained at the highest possible stocking rates are usually found in high number in every host (Figure 5). Although the variance to mean ratio in such cases is invariable much greater than unity, indicating an aggregated distribution, the estimated values of k tend to be one or greater. Smith and Guerrero (1993) went on to show that when the degree of aggregation is such that k > 1, the results of a trichostrongylid model which recognizes parasite frequency distribution are insignificantly different from the results of a model which ignores parasite frequency distributions (Figure 5). It is not known whether this happy state of affairs is likely to apply to bovine fascioliasis.

Smith (1982) estimated the value of k for flukes in sheep flocks and found it varied over a range of values from very much below one (in which case the distribution must be recognized by the model) to values very much above one (in which case the distribution can be ignored). There are almost no studies of bovine fascioliasis which provide sufficient information to estimate k but a group of naturally infected calves autopsied by Nansen (1975) had a mean fluke burden of 82 and an estimated value for k of 2.5 (G. Smith, unpublished data).

Summary

This paper presents a rationalization of the population processes occurring in the parasitic phase of two of the most important classes of helminth parasites infecting domestic cattle. Both F. hepatica and the common trichostrongylid gastrointestinal nematode parasites seem to be principally regulated by events early in the infection process. This is of clear benefit to the host because it is subsequent invasive events that give rise to the pathology usually associated with the infections (e.g. parenchymal migration in the case of F. hepatica and emergence from the mucosa in the case of the trichostrongylids). It is not yet clear what is the role of other potentially regulatory processes. Parasite-induced host mortality occurs in both cases but its importance is unknown. Fecundity appears to be regulated in some of these parasites but not in others. Whether there is some systematic relationship between the regulation of fecundity and the survival of the free living stages has yet to be considered. It appears that modellers can gratefully relinquish the burden of trying to write tractable formulations incorporating empirical descriptions of parasite distributions, at least in the case of the trichostrongylids. Whether this is true of bovine fascioliasis remains to be seen.

Figure 5. Upper axes: distribution of helminth parasites in wildlife and domestic host species respectively. Lower axis: the effect of parasite aggregation on the rate of change in the number of parasites (dP/dt) in a model assuming that parasite mortality was a function of the rate of infection in any given host. There is no significant difference between the trajectories of a model in which k is 1 and a model in which it is assumed that all hosts contain exactly the same number of parasites ('regular'). (Redrawn after Smith and Guerrero, 1993).

Acknowledgements

The author gratefully acknowledges the expert technical assistance of Ms. C. Tallamy and Ms. R. Owen. Portions of this work were funded by a University Research Foundation Grant (Penn) and grants from MSD AGVET, a division of Merck and Co., Inc., USA.

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Session discussion

A general principle and requirement in vector research is to predict the distribution and abundance of the vectors in order to determine the disease risk in a population. It is important to identify both the set of factors which predict disease risk and the set which determine the population dynamics if the goal is to control diseases per se. The number of factors varies for different diseases but the overall opinion was that few variables are often responsible for the majority of the variance; such is the case for factors affecting the distribution of tsetse fly populations, where two or three variables account for most of the variance.

Concern was raised with regard to the availability of reliable data sets. Older data sets often were not developed with the needs of modelling in mind, and the negative effect of this is enhanced by the fact that fewer field projects are initiated to collect new data these days, reducing the possibilities of establishing new, reliable and useful data sets. It was proposed that literature reviews may supply certain initial data sets which, in combination with well designed field research for collection of information on outstanding questions, could contribute to better model development. It was generally agreed that models need to be validated and the lack of data sets is a severe constraint for testing the validity of the rather advanced vector models available.

The use of models to predict distributions of tick vectors was discussed and the importance of identifying the appropriate variables for inclusion in these models was debated. Climatic data are clearly crucial, and their availability and resolution, both as long-term average and real time data, are improving. Other factors likely to be important but yet to be fully evaluated in models include physiological age of ticks, host resistance to ticks, diapause and the role of wildlife hosts.

A factor which has not been considered important in the current tick and helminth vector models is the phenomenon of 'overdispersion' and it was pointed out that little is known about this. The distribution of different levels of parasitism within a population will determine levels of disease and economic loss, and overdispersion, in which low numbers of individuals carry high parasite burdens while the majority of the population carries low burdens, is likely to affect these significantly. The possible contributory roles of genetic and behavioural factors to overdispersion were discussed.


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