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The use of mathematical models to simulate control options for echinococcosis
Acta Tropica 85 (2003) 211 /221 www.parasitology-online.com
Review article
The use of mathematical models to simulate control options for echinococcosis P.R. Torgerson * Department of Veterinary Microbiology and Parasitology, Faculty of Veterinary Medicine, University College Dublin, Ballsbridge, Dublin 4, Ireland
Abstract In many parts of the world Echinococcus granulosus is a widespread infection in sheep and dogs with a consequential spill over into the human population. In the past, mathematical models have been derived to define the transmission dynamics of this parasite, principally in the sheep-dog life cycle. These models have characterized the cycles of infection as lacking in density dependent constraints in both the definitive or intermediate hosts. This suggested that there was little, if any, induced host immunity by the parasite in either host in natural infections. However, recent evidence from both Tunisia and Kazakhstan, where young dogs are the most heavily parasitised, suggests the possibility of significant definitive host immunity. This may have an effect on the control effort needed to destabilize the parasite. A preliminary computer simulation model (based on an Excel spreadsheet) to attempt to predict the results of a control programme has been written. This demonstrates that there could be significantly different results if there is indeed protective immunity in the dog than in the absence of immunity. In the former the parasite needs a greater control effort to push the parasite towards extinction than in the latter. The computer simulation is based on a mathematical model of the parasite’s life cycle and is flexible so that different values of parameters can be used in different situations where the transmission of the parasite may be at different levels. Because of the flexibility of the computer simulation it is anticipated that this programme can be applied in most situations, although initial parameters for a particular location or strain of the parasite will have to be first predetermined with base line field surveys and possibly experimental infections. The programme also has an additional flexibility to enable simulations if some parameters cannot be accurately estimated through Monte-Carlo techniques. In the latter situation, worst and best case scenarios can be estimated and likely frequency distributions of the unknown parameters can be included in the model. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Echinococcus granulosus ; Mathematical models; Monte-Carlo techniques
1. Introduction * Present address: Institute of Parasitiology, University of Zu¨rich, Winterthurerstrasse 266a, CH-8057, Zu¨rich, Switzerland. Tel.: /41-1-6358535; fax: /41-1-6358907 E-mail address: [email protected] (P.R. Torgerson).
The transmission dynamics of taeniid parasites depend on a number of factors. Gemmell (1990) summarized these as intrinsic, extrinsic and socioecological factors. Intrinsic factors depended on
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biological processes within the host or parasite. These include factors such as the parasite’s biotic potential and host immunity. Extrinsic factors included variables that were independent of the parasites and hosts such as climate and weather. Finally socioecological factors were largely maninduced such as stocking density or meat inspection practices. The interrelationships of these factors determine the extent to which these parasites can be transmitted between the animal hosts in the life cycle. Considerable work has been completed studying the transmission dynamics of Echinococcus granulosus . These transmission dynamics have been summarized in a mathematical model of the life cycle (Roberts et al., 1986). Ideally, mathematical models should be able to describe and predict the epidemiology of these parasites. Furthermore, for them to be useful and widely applicable, they should be simple to understand, but robust under a variety of different systems. To be of practical use models should be able to predict future events given the model and basic information. For example, models could be used to predict the outcomes of intervention such as a dog dosing programme to control cystic echinococcosis (CE). Assuming a model is based on theoretical ideas of parasite-host relationships, they can be tested using atual data and departures from the theoretical model can be used to define better models and thus gain a better insight into the parasite biology. To formulate a working model of the transmission dynamics of Echinococcus granulosus , a detailed knowledge of the parasite’s biology, together with observational studies of its epidemiology are required. Most of the data required for the construction of these models was derived from observational and experimental studies in Australasia, although some parameters have been modified in the light of published information from the former Soviet Union. The purpose of this paper is to demonstrate how a spread sheet model, constructed using equations describing the life cycle and transmission dynamics of the parasite, can be used to simulate various control options for echinococcosis.
2. The model The life cycle of E. granulosus is represented as dogs being the definitive hosts and sheep being the intermediate hosts (Fig. 1). Man is normally represented as an accidental intermediate host. The transition between different groups depends on the transmission dynamics of the system. For example, the transition between non-infected and infected dogs (E) depends on the frequency of feeding offal to dogs, the prevalence of infection in sheep and the age of the sheep being fed to the dogs. Likewise the transition from infected to noninfected dogs (D) depends on the life expectancy of the parasite, host immunity and the frequency of anthelmintic treatment. If the prevalence is constant (i.e. in a steady state) then the transition rate between non-infected dogs to infected dogs will equal the rate at which infected dogs lose their infection (D /E). The rate at which sheep become infected (A) will depend upon the prevalence in dogs, the intensity of infection in dogs, the contact rate between dogs and sheep and the level of host immunity in sheep. Once sheep are infected they generally remain infected for life. With E. granulosus , present information suggests that there is no effective naturally-induced immunity in sheep, the principal intermediate host for the most important zoonotic strain of the parasite (Gemmell, 1990; Roberts et al., 1986; Ming et al. 1992a; Cabrera et al., 1996; Torgerson et al., 1998; Lahmar et al., 1999). Therefore, sheep acquire increasing numbers of cysts as they become older. The relationship between age and mean numbers of parasites per sheep can be modelled as a straight-line relationship. The gradient of the line is dependent on the infection pressure, which in turn depends upon the level of infection in the definitive host, together with the contact rate between sheep and dogs and environmental conditions affecting the survival of eggs in the environment. 2.1. The definitive host The model described in this paper is based on a prevalence equation in dogs. The prevalence of infection in dogs can vary according to the infection pressure (b ), the probability of the dog
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Fig. 1. Representation of the life cycle of E. granulosus .
acquiring immunity on exposure (a ), the probability of an immune dog losing its immunity (g) and the life expectancy of the parasite in the dog. Roberts et al. (1986) reported that the prevalence (p) of infection in dogs varied according to age (t) as:
important in certain situations, and that in the presence of immunity, the asymptote is not merely lowered as suggested by Roberts et al. (1986), but that prevalence rates are lower in old dogs than in young animals. This suggested another form of the equation, which is used in this model:
p(t) p( )f1exp(mb)t)g
P(t)K1 [exp(mb)t]K2 [exp(mg)t]
(1)
K3 [exp(gab)t]K4 where b is the infection pressure (in terms of insults per unit time) and m is the rate at which hosts lose their parasite. The asymptotic prevalence p( )/b/(m/b ). In the presence of immunity the asymptotic prevalence is reduced by a factor, which is always less than 1. Roberts et al. (1986) suggested that immunity was not important in the definitive host and thus prevalence depended solely on the infection pressure and the life expectancy of the parasite. However, recent evidence from Tunisia (Lahmar et al., 2001) and from Kazakhstan (Torgerson, 2002) reports the highest prevalence and intensity of infection in younger dogs. This suggests that immunity may be
(2)
where K1, K2, K3 and K4 are dependent on a , b , g and m . When a /0, this equation simplifies to Eq. (1). 2.2. The intermediate host Roberts et al. (1986) has modelled the abundance of cysts in sheep. Both non-linear and linear forms of the equation have been described. There have been a number of reports of the variations in age intensity in sheep (Roberts et al., 1986; Ming et al., 1992a; Cabrera et al., 1996; Torgerson et al., 1998, Lahmar et al., 1999) from different geogra-
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phical locations. Each one suggested that there was insignificant immunity in sheep when naturally infected. Consequently the simple linear for of the equation was used in the model: m(t)ht
(3)
where m is the mean number of cysts at age t and h is the infection pressure in terms of numbers of cysts acquired each year. 2.3. Spread sheet model The spread sheet model was built round cell entries for prevalence of the dog population suggested by Eq. (2) and abundance of infection in the sheep population suggested by Eq. (3). Because it takes a period of time for cysts to become mature delay had to be built into the model. This was achieved by having a column representing the different age groups of sheep. For the purposes of this model this was taken as the age distribution of sheep described by Ming et al. (1992a) from China. However, this age distribution can be easily changed for any area based on base line surveillance data from that locality. The ability to calculate the proportion of each age group as a total of the population was included in the model. For any area this data would have to be obtained by field survey. From the average time to maturity of cysts together with the infection pressure, the average contribution of each age group to the infection pressure to dogs was calculated. From this the overall infection pressure to dogs was calculated. Delay was effectively modelled as any control programme would only prevent additional infection in sheep. Cysts that sheep had already acquired would only be removed when sheep infected with these cysts died.
3. Control simulation Once the equations are written into the spreadsheet, precontrol parameters must be obtained. Some of these will be by local observations. By getting a representative sample of sheep of different ages the precontrol infection pressure to sheep can be calculated. Likewise the age at which sheep
are fed to dogs needs to be ascertained. For simplicity, it has been assumed this reflects the age structure of the sheep population and can be obtained from field data. The precontrol prevalence and the age structure of the dog population also needs to be obtained and this can give the likely precontrol values for the infection pressure to dogs and the likelihood of developing immunity. The average life expectancy of a patent infection has been assumed to be approximately 1 year (Aminjanov, 1975; Sweatman and Williams, 1963), but this may also be calculated from accurate surveillance data and fitting the data, by non-linear regression analysis, to Eqs. (1) and (2). Once baseline information is obtained for a particular locality, control can be simulated. For example, if dog dosing is undertaken every 6 weeks (the prepatent period) then the mean survival time of a patent infection (1/m) is going to be 0. Consequently the parasite will disappear, providing control is maintained until non-infected sheep have replaced all the sheep in the population. However, 100% compliance is rarely if ever achieved. In China, frequently less than 80% of dogs were treated in a control programme (Fen-Jie 1993), whilst in Turkana Kenya only 59% of dogs could be treated (MacPherson et al., 1986). Therefore assuming 80% of dogs in the target population are treated but 20% remain untreated it can be seen that the infection pressure to sheep will immediately drop by 80%. As the relationship between infection pressure and numbers of parasites acquired is linear, the numbers of cysts in sheep will begin to decline. This will not, however, change the infection pressure in dogs for some time. The mean time to maturity of cysts has been suggested as 6.63 years (Gemmell et al., 1986) or approximately 2 years by Aminjanov (1976) and Sweatman and Williams (1963). This time to cyst maturity acts as a notable delay in obtaining significant results after the control measures are implemented. One year after control, the number of mature cysts in sheep aged 1 year will be reduced, but since these animals contain only a small proportion of the mature cysts in the population there will only be a very small reduction in the infection pressure to dogs. However, in subsequent years, the reduc-
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tion in infection pressure becomes progressively greater. Once all old sheep have been removed, reinfection only occurs from dogs that have escaped treatment in the first place, thus prevalence rates in both life-cycle hosts start to drop quickly. The difference in the time for cyst maturity does have some effect on this. If the longer time period reported by Gemmell et al. (1986) is used then cysts from old animals result in a greater proportion of the infection pressure to dogs and the delay is greater than if a shorter time to maturity is used. Other control interventions can be similarly modelled. In the case of a vaccination campaign this can be modelled as a reduction in the numbers of new fertile cysts from the date of vaccination. Thus sheep that are vaccinated with a 100% effective vaccine will develop no further cysts from continued exposure to Echinococcus eggs. However, infection acquired before vaccination can still be further transmitted to dogs, thus there will be some delay until all the cysts disappear from the sheep population. Likewise an education campaign may reduce the rate at which offal is fed to dogs and this can be modelled as a reduction in the infection pressure in dogs. The parameters used in the model are illustrated in Table 1 and the parameters that are changed in control simulations illustrated in Table 2. As can be seen there is significant variability in some of the parameters suggested by different experimental work and in the possible capture rates of dogs in the populations. There will also be additional variability in the base line surveillance data obtained for the precontrol parameters. To
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Table 2 Modelling control through changes in the parameters Control intervention
Change in parameter
Anthelmintic treatment of dogs Vaccination of sheep
Reduction in life expectancy of patent infection Reduction in numbers of cysts in sheep from time of vaccination Reduction of infection pressure to dogs
Education programme
model this variability in the simulations, MonteCarlo routines have also been written into the spreadsheet model. Thus, variability in the capture rate of dogs between the best case (say 90% treated) and worse case (say 60% treated) can be modelled. This is particularly important in the absence of accurate dog population figures and would give best and worse case scenarios if a target capture rate or compliance rate of 75% was assumed. Consequently a continuous range of possibilities (one chosen at random for each simulation) can reduce the life expectancy of the parasite to reflect capture rates of between 60 and 90% (with a most likely rate of 75%). This randomization is undertaken with all the other parameters in the model in a similar manner. The results of the control in terms of best and worst case scenarios with the median likelihood can then be reported. A number of these simulations will now be illustrated. In a situation where there is relatively low infection pressure to dogs, there is assumed to be an initial prevalence in dogs of approximately 20%
Table 1 Baseline parameters Sheep Age related abundance of infection Infection pressure Time to maturity of cysts Acquisition and loss of immunity Age of feeding sheep to dogs
Dogs Surveillance data Abundance data 6.6 years (Gemmell et al., 1986); 2 years (Aminjanov, 1976; Sweatman and Williams, 1963) Negligible (Roberts et al., 1986) Surveillance data
Age related prevalence of infection Infection pressure Life expectancy of infection Acquisition and loss of immunity Age structure of dog population
Surveillance data Prevalence data 1 year (Aminjanov, 1975); can calculate from surveillance data Negligible (Roberts et al., 1986). Calculate from surveillance data Surveillance data
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and a precontrol prevalence in sheep of 50%. The sheep are under an infection pressure of approximately one new cyst per year and dogs are infected on average once every 2 years. In other simulations the dogs are under a much greater infection pressure and the sheep are more heavily infected. In these simulations dogs are infected between one and two times a year resulting in a prevalence approaching 40%. The prevalence in sheep is modelled as 80%. The life expectancy of the parasite, in the absence of control, is assumed to be 1 year in all cases. The model was constructed in an excel spreadsheet (Microsoft, Redmond, WA). The variability was modelled using Crystal Ball Software (Decisioneering, Denver, CO). One thousand simulations were undertaken for each control intervention with each variable being randomly changed along its range for each simulation.
4. Illustrative examples Fig. 2 represents the prediction of the decrease in prevalence of infection in sheep suggested by the model with a control intervention of 6-monthly anthelmintic treatments, in the absence of significant definitive host immunity. In the first example, the time for cyst maturity is assumed to be that reported by Gemmell et al., (1986). Six-monthly anthelmintic treatment is largely successful with control (as defined by reducing the prevalence in sheep to less than 1%) achieved within 20 years providing there is no protective immunity in dogs. This depends on the programme treating a target of 75% of the dog population. However, because of the inevitability of populations of strays and lack of compliance this represents the median outcome, with lower and upper 95% confidence limits based on a treatment rate of 60 /90% of the dogs and errors built into parameter estimates (upper and lower dotted lines). The main feature of this simulation is the delay before the prevalence starts to decrease in sheep. This is because most of the infection pressure to dogs is from old sheep, and young sheep, infected immediately before start of control, will still be infective to dogs some years later. If the time to cyst maturity
is reduced to 2 years, then the main difference is that the delay in reduction of prevalence to sheep is lessened, but it takes a little longer to achieve less than 1% of sheep infected. However, the model does illustrate that the time to maturity of sheep cysts is one of the least sensitive parameters in terms of final outcome. For all subsequent simulations illustrated a time to cyst maturity of 2 years is used. It may seem surprising that a modest intervention would be successful, but if there is no stabilizing effect of immunity in either the definitive or intermediate host then the parasite will have a relatively low reproductive ratio (Gemmell, 1990) and would require little force to destabilize it. Fig. 3 illustrates how protective immunity in the dog can affect the outcome. In this scenario, 6monthly anthelmintic treatment, whilst effectively reducing the prevalence in sheep, does not result in control. Eventually, there is a new equilibrium of approximately 20% prevalence in sheep reduced from 50%. This illustrates the important effect definitive host immunity can have on parasite stability. A lowering of infection pressure to dogs results in the dogs becoming more susceptible to infection and thus less overall control than might have been expected. This type of phenomenon is an important one in the host parasite relationships of helminth parasites and their population biology. Thus the presence of immunity acts as a stabilizing factor on the population (Anderson and May, 1985) In the presence of immunity even reaching 90% of the dog population still results in 5/10% of sheep remaining infected some 20 years after start of control. If an education programme is also put into place, there is a further reduction in prevalence in the sheep due to less feeding of offal. However, this still does not result in adequate control. It is also worth noting that, in the presence of definitive host immunity, education on its own (assuming 50% reduction in offal feeding) has very little effect because the lowered infection pressure to dogs is off set by the increased susceptibility to infection. However, education can be effective if there is no immunity in dogs. Of the three options (6-monthly anthelmintics, education or vaccination), vaccination appears to
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Fig. 2. Change in % prevalence of E. granulosus in sheep populations after commencement of control programme with no definitive host immunity. Top left: response to control of 6-monthly anthelmintic treatment in dogs with 6.6 years to cyst maturity; top right: with 2 years to cyst maturity; bottom left: an education programme resulting in a reduction of feeding offal to dogs and bottom right: a vaccination programme with a compliance rate of 75%. Median outcome ( */) and 95% confidence limits (---).
offer the best long-term solution as a sole intervention strategy, with a new equilibrium of approximately 8% prevalence in 20 years compared to 18% for anthlemintic treatment. However, immunization of the intermediate host results in a delay in the reduction in prevalence in the definitive host. This is because the vaccine only prevents new infections and will not prevent development of, or eliminate, cysts that are already present in sheep. Consequently there will be little immediate reduction in infection pressure to dogs, this only occurs as the generation of sheep, infected before vaccination, are replaced by animals vaccinated soon after birth. Thus, there will be a delay in reducing the parasite prevalence to dogs and hence transmission to humans. Whilst a number of intervention strategies will work in the absence of immunity, it requires a combined strategy in the presence of immunity. This can also be seen in Fig. 3. The scenario when there is both vaccination of sheep and 6-monthly anthelmintic treatment of dogs will result in good
control within about 15 years, whereas neither control strategy alone appears to have such an effect. This pattern is repeated in the situation where the parasite is intensely endemic (Fig. 4). In this example, it is assumed that dogs are infected between one and two times per year on average. In this example about 80% of sheep are infected and between 30 and 50% of dogs are infected (depending on the level of protective immunity assigned to the dogs). In this intensely endemic situation, immunity is much more likely to play a regulatory role in dogs because of the degree of exposure of the dog population. In this situation, vaccinating sheep (assuming a 75% vaccination rate) eventually results in about 23% of the sheep remaining infected. This of course represents the majority of the sheep that escaped vaccination. Thus, the new equilibrium state will result in a cycle between the non-immune sheep and the dog population. To obtain adequate control, at least 90% of the sheep need to be vaccinated. Likewise
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Fig. 3. Effect of definitive host immunity on control simulations. Top left: response to control measures using 6-monthly anthelmintic treatments. Top right: effect of vaccinating sheep with a target compliance of 75%. Bottom left: effect of education programme resulting in decrease in cysts fed to dogs by 50% and bottom right: integrated control using vaccination of sheep and anthelmintic treatment. Median outcome ( */) and 95% confidence limits (---).
anthelmintic treatment of dogs will reduce the initial prevalence in sheep from 80 to 25%, but at this level a new equilibrium state is achieved. Again at least 90% of dogs will need to be treated to control the parasite. Alternatively, a reduction in anthelmintic treatment intervals to 3 /4 months at most (with a capture rate of at least 75%) would also be effective. However, combining the two control measures will result in good control: in less than 10 years in the case of 90% of animals being treated to about 30 years if only 60% of animals are vaccinated or treated. If an education programme were also undertaken that reduced offal feeding to dogs by 50%, control could be achieved 3 /5 years earlier.
5. Conclusions A 6-monthly anthelmintic treatment interval was deliberately chosen in these examples as this is a relatively modest intervention compared to
many intervention strategies that have already been applied successfully. These include the 6weekly dog treatment programme, which is based on the prepatent period of the parasite in the definitive host. However, the time to re-infection is arguably a more important interval to take into account when designing anthelmintic treatment programmes. In addition, because it is very difficult to achieve treatment of every dog, and vaccination of every sheep, a target capture or compliance rate of 75% was assumed to illustrate the outcome. Using such simulations it should also be possible to undertake cost-benefit analysis of control of CE. All other things being equal, it will cost four times as much to undertake a 6-weekly anthelmintic dog treatment programme, than a 6-monthly programme. The excess costs may or may not justify the additional decrease in prevalence that is achieved with the programme using the more frequent dosing regimen. The decision may be based on the perceived compliance of the target
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Fig. 4. Control simulation in an intensely endemic region with dogs being infected one to two infections per year and an initial prevalence in sheep of 80%. Immunity in dogs is assumed. Top: anthelmintic treatment every 6 months; middle: vaccination of sheep with a 75% compliance rate. Bottom: vaccination of sheep and anthelmintic treatment. Median outcome ( */) and 95% confidence limits (---).
population. The greater the compliance, the fewer dogs are likely to escape treatment, thus the treatment interval could be lengthened. The simulations also illustrate the potential usefulness of the vaccine. For example, if immunity is present in dogs, the anthelmintic treatment interval must be shortened to achieve control if this is the only intervention strategy. Alternatively a very high
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capture rate is mandatory. Both options would prove costly. Ensuring 90% or greater capture rate is much more costly than 60 /70% capture rate. Decreasing treatment intervals to 3 months doubles the expense. However, vaccinating sheep on an annual basis and treating dogs at the illustrated compliance rates every 6 months appear to work which ever the scenario and would almost certainly be cheaper than trying to increase compliance rates or decreasing treatment intervals by using anthelmintics alone. Furthermore, should there be seasonal transmission then the anthelmintic treatment can be targeted at certain times of year. This will result in more rapid control than if there was continuous transmission. Interestingly, in Uruguay, studies suggest that dogs become reinfected between 2 and 4 months after treatment (Cabrera et al., 2001). Because of this a reduction in treatment intervals from every 6 weeks to every 12 weeks was subsequently shown to reduce the infection rate in sheep. This is in agreement with this model. Indeed, only a small percentage of the original dogs were represented for treatment at the end of the 2.6-year trial. This could indicate poor compliance and further indicates that less frequent treatments of dogs can be effective. Costs are important in regions where there are scarce financial resources. This model illustrates how treatment intervals can be lengthened if combined with other intervention strategies such as vaccination and thus reduce costs. Therefore, this model should be linked to economic models (Torgerson et al., 2001; Majorowski et al., 2001) to attempt to cost the benefits of a control programme in financial terms. This can then be used to lobby support for control. This paper illustrates that it should be possible to model the possible outcome of a control intervention against E. granulosus . This may provide a powerful tool for policy makers and it will give likely but, perhaps more importantly, worst case scenarios of a control intervention. It also illustrates that accurate estimation of precontrol parameters need not necessarily be known, but this will reflect the accuracy of the predicted outcomes. In particular it does suggest that parameters, such as time for cysts’ maturity, have little effect on the final outcome but will affect the
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initial rate of decline of echinococcosis in the population. Interestingly, in both the intensely endemic and less endemic scenarios, in the presence of definitive host immunity a combination of 6-monthly anthelmintics and sheep vaccination is likely to be successful, even with only a 60% compliance rate. In the presence of definitive host immunity neither vaccination nor dog treatment will work on its own (at the illustrated treatment intervals and compliance rates). Since the regulatory role of immunity by dogs in the population is not yet adequately understood, a minimum recommendation of 6-monthly anthelminitc treatments with sheep vaccination should be made. If only anthelmintics are available, the model indicates that treatment intervals should be no greater than 3 months. Obviously this is a generalization and should be modified in light of local transmission dynamics. The model also demonstrates that greater success is likely to be achieved if the parasite can be attacked in both hosts in the life cycle than if control is aimed at just one host. Logistically it is almost certainly easier to treat most sheep and most dogs than all dogs or all sheep and consequently is an important result. This is also in agreement with an earlier model (Harris et al., 1980) which also concluded the best results were obtained by treating both definitive and intermediate hosts. This model has mainly illustrated the likely outcome in the prevalence of echninococcosis in the sheep population and not the incidence in the human population. However, the behavior of the infection in the two species would be expected to be similar with perhaps a greater delay before human incidence starts to decline. In addition, education may have an additional positive effect as improvements in personal hygiene would be expected to lower transmission to man even in the absence of any change in prevalence in animals. This would be a positive effect of an education programme. One possible inaccuracy in the model is that it is based on a prevalence model in dogs. A model that is based on the abundance of infection would be more rigorous and is the subject of ongoing work. The reasons are because of the aggregation of
parasites within hosts. E. granulosus is highly aggregated in dogs (Roberts et al., 1986; Ming et al., 1992b; Lahmar et al., 2001) and can be modelled according to a negative binomial distribution. Thus the majority of the population of adult E. granulosus are in a small number of heavily infected hosts with a much larger number of lightly infected hosts. These heavily infected dogs, often with several thousand parasites per dog, may be responsible for the majority of the transmission with the lightly infected animals, often with fewer than 100 parasites, being less significant. A model needs to take this into account and particularly needs to account for the behavior of heavily infected individuals in response to control measures. In trichostrongylid infections of ruminant hosts, the negative binomial constant tracks changes in the mean burden of the parasites (Grenfell et al., 1995). Thus the higher the mean parasite burden, the lower the degree of aggregation. Modification of the simulation model may be required if the same is true of infection with Echinococcus in dogs or in sheep. Low mean burdens and low prevalences with a low k value, as can be predicted in such a scenario in the latter stages of the control programme, might lead to instability.
Acknowledgements The author would like to acknowledge the financial support of INTAS (INTAS-97-40311) and the NIH (1R01TW01565-01) in supporting his work.
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P.R. Torgerson / Acta Tropica 85 (2003) 211 /221 Cabrera, P.A., Haran, G., Benavidez, U., Valledor, S., Perera, G., Lloyd, S., Gemmell, M.A., Baraibar, M., Morana, A., Maissonave, J., Carballo, M., 1996. Transmission dynamics of Echinococcus granulosus , Taenia hydatidgena and Taenia ovis in sheep in Uruguay. Int. J. Parasitol. 25, 807 /813. Cabrera, P.A., Haran, G., Lloyd, S., Gemmell, M.A., Parietti, S., Correa, O., Valledor, S., Morana, A., 2001. Transmission dynamics of Echinococcus granulosus , Taenia hydatigena and Taenia ovis in sentinel lambs, Uruguay. Evaluation of three models of control. In: Altinas, N., Yolasigmaz, A., Coskun, E., Guunes, K. (Eds.), Proceedings of the XXth Internationl Congress of Hydatidology, Kusadasi, Turkey, p. 150. Fen-Jie, L. (1993) Prevalence of Echinococcus granulosus in dogs in the Xinjiang Uygur Autonomous Region, PRC. In: Anderson, F.L. (Ed.), Compendium on Cystic Echinococcosis with Special Reference to the Xinjiang Uygur Autonomous Region, The People’s Republic of China. Brigham Young University, Provo, UT, pp. 168 /176. Gemmell, M.A., 1990. Australasian contributions to an understanding of the epidemiology and control of hydatid disease caused by Echinococcus granulosus past, present and future. Int. J. Parasitol. 20, 431 /456. Gemmell, M.A., Lawson, J.R., Roberts, M.G., 1986. Population dynamics in echinococcosis and cysticercosis: biological parameters of Echinococcus granulosus in dogs and sheep. Parasitology 92, 599 /620. Grenfell, B.T., Wilson, K., Isham, V.S., Boyd, H.E.G., Dietz, K., 1995. Modelling patterns of parasite aggregation in natural populations: trichostrongylid nematode-ruminant interactions, a case study. Parasitology 111, S135 /S151. Harris, R.E., Revfeim, K.J.A., Heath, D.D., 1980. Stimulating control strategies for control of Echinococcus granulosus , Taenia hydatidgena and T. ovis . J. Hyg. 84, 389 /404. Lahmar, S., Kilani, M., Torgerson, P.R., Gemmell, M.A., 1999. Echinococcus granulosus larvae in the livers of sheep in Tunisia: the effects of host age. Ann. Trop. Med. Parasitol. 93, 75 /81. Lahmar, S., Kilani, M., Torgerson, P.R., 2001. Frequency distribution of Echinococcus granulosus and other helminths
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in stray dogs in Tunisia. Ann. Trop. Med. Parasitol. 95, 69 / 76. MacPherson, C.N.L., Wachira, T.W., Zehle, E., Romig, T., Macpherson, C., 1986. Hydatid disease research and control in Turkana. IV. The pilot control programme. Trans. R. Soc. Trop. Med. Hyg. 80, 196 /200. Majorowski, M.M., Carabin, H., Kilani, M., Bensalah, A., 2001. Echinococcosis in Tunisia: an economic analysis. In: Menzies, F.D., Reid, S.W.J. (Eds.), Proceedings of the Society for Veterinary Epidemiology and Preventative Medicine, pp. 187 /201. Ming, R., Tolley, H.D., Andersen, F.L., Chai, J., Sultan, Y., 1992a. Frequency distribution of Echinococcus granulosus hydatid cysts in sheep populations in the Xinjiang Uygur Autonomous Region, China. Vet. Parasitol. 44, 67 /75. Ming, R., Tolley, H.D., Andersen, F.L., Chai, J., Chang, Q., 1992b. Frequency distribution of Echinococcus granulosus in dog populations in the Xinjiang Uygur Autonomous Region, China. Vet. Parasitol. 43, 233 /241. Roberts, M.G., Lawson, J.R., Gemmell, M.A., 1986. Population dynamics of echinococcosis and cysticercosis: mathematical model of the life cycles of Echinococcus granulosus . Parasitology 92, 621 /641. Sweatman, G.K., Williams, R.J., 1963. Comparative studies on the biology and morphology of Echinococcus granulosus from domestic livestock, moose and reindeer. Parasitology 53, 339 /390. Torgerson, P.R., 2002. Transmission dynamics of taeniid parasites in animal hosts. In: Craig, P.S. (Ed.), Cestode Zoonoses: an Emergent Global Problem. NATO Science Series, pp. 221 /235. Torgerson, P.R., Williams, D.H., Abo-Shehada, M.N., 1998. Modelling the prevalence of Echinococcus and Taenia species in small ruminants of different ages in northern Jordan. Vet. Parasitol. 79, 35 /51. Torgerson, P.R., Dowling, P.M., Abo-Shehada, M.N., 2001. Estimating the economic effects of cystic echinococcosis. Part 3: Jordan, a developing country with lower middle income. Ann. Trop. Med. Parasitol. 95, 595 /603.
Review article
The use of mathematical models to simulate control options for echinococcosis P.R. Torgerson * Department of Veterinary Microbiology and Parasitology, Faculty of Veterinary Medicine, University College Dublin, Ballsbridge, Dublin 4, Ireland
Abstract In many parts of the world Echinococcus granulosus is a widespread infection in sheep and dogs with a consequential spill over into the human population. In the past, mathematical models have been derived to define the transmission dynamics of this parasite, principally in the sheep-dog life cycle. These models have characterized the cycles of infection as lacking in density dependent constraints in both the definitive or intermediate hosts. This suggested that there was little, if any, induced host immunity by the parasite in either host in natural infections. However, recent evidence from both Tunisia and Kazakhstan, where young dogs are the most heavily parasitised, suggests the possibility of significant definitive host immunity. This may have an effect on the control effort needed to destabilize the parasite. A preliminary computer simulation model (based on an Excel spreadsheet) to attempt to predict the results of a control programme has been written. This demonstrates that there could be significantly different results if there is indeed protective immunity in the dog than in the absence of immunity. In the former the parasite needs a greater control effort to push the parasite towards extinction than in the latter. The computer simulation is based on a mathematical model of the parasite’s life cycle and is flexible so that different values of parameters can be used in different situations where the transmission of the parasite may be at different levels. Because of the flexibility of the computer simulation it is anticipated that this programme can be applied in most situations, although initial parameters for a particular location or strain of the parasite will have to be first predetermined with base line field surveys and possibly experimental infections. The programme also has an additional flexibility to enable simulations if some parameters cannot be accurately estimated through Monte-Carlo techniques. In the latter situation, worst and best case scenarios can be estimated and likely frequency distributions of the unknown parameters can be included in the model. # 2002 Elsevier Science B.V. All rights reserved. Keywords: Echinococcus granulosus ; Mathematical models; Monte-Carlo techniques
1. Introduction * Present address: Institute of Parasitiology, University of Zu¨rich, Winterthurerstrasse 266a, CH-8057, Zu¨rich, Switzerland. Tel.: /41-1-6358535; fax: /41-1-6358907 E-mail address: [email protected] (P.R. Torgerson).
The transmission dynamics of taeniid parasites depend on a number of factors. Gemmell (1990) summarized these as intrinsic, extrinsic and socioecological factors. Intrinsic factors depended on
0001-706X/02/$ - see front matter # 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 1 - 7 0 6 X ( 0 2 ) 0 0 2 2 7 - 9
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biological processes within the host or parasite. These include factors such as the parasite’s biotic potential and host immunity. Extrinsic factors included variables that were independent of the parasites and hosts such as climate and weather. Finally socioecological factors were largely maninduced such as stocking density or meat inspection practices. The interrelationships of these factors determine the extent to which these parasites can be transmitted between the animal hosts in the life cycle. Considerable work has been completed studying the transmission dynamics of Echinococcus granulosus . These transmission dynamics have been summarized in a mathematical model of the life cycle (Roberts et al., 1986). Ideally, mathematical models should be able to describe and predict the epidemiology of these parasites. Furthermore, for them to be useful and widely applicable, they should be simple to understand, but robust under a variety of different systems. To be of practical use models should be able to predict future events given the model and basic information. For example, models could be used to predict the outcomes of intervention such as a dog dosing programme to control cystic echinococcosis (CE). Assuming a model is based on theoretical ideas of parasite-host relationships, they can be tested using atual data and departures from the theoretical model can be used to define better models and thus gain a better insight into the parasite biology. To formulate a working model of the transmission dynamics of Echinococcus granulosus , a detailed knowledge of the parasite’s biology, together with observational studies of its epidemiology are required. Most of the data required for the construction of these models was derived from observational and experimental studies in Australasia, although some parameters have been modified in the light of published information from the former Soviet Union. The purpose of this paper is to demonstrate how a spread sheet model, constructed using equations describing the life cycle and transmission dynamics of the parasite, can be used to simulate various control options for echinococcosis.
2. The model The life cycle of E. granulosus is represented as dogs being the definitive hosts and sheep being the intermediate hosts (Fig. 1). Man is normally represented as an accidental intermediate host. The transition between different groups depends on the transmission dynamics of the system. For example, the transition between non-infected and infected dogs (E) depends on the frequency of feeding offal to dogs, the prevalence of infection in sheep and the age of the sheep being fed to the dogs. Likewise the transition from infected to noninfected dogs (D) depends on the life expectancy of the parasite, host immunity and the frequency of anthelmintic treatment. If the prevalence is constant (i.e. in a steady state) then the transition rate between non-infected dogs to infected dogs will equal the rate at which infected dogs lose their infection (D /E). The rate at which sheep become infected (A) will depend upon the prevalence in dogs, the intensity of infection in dogs, the contact rate between dogs and sheep and the level of host immunity in sheep. Once sheep are infected they generally remain infected for life. With E. granulosus , present information suggests that there is no effective naturally-induced immunity in sheep, the principal intermediate host for the most important zoonotic strain of the parasite (Gemmell, 1990; Roberts et al., 1986; Ming et al. 1992a; Cabrera et al., 1996; Torgerson et al., 1998; Lahmar et al., 1999). Therefore, sheep acquire increasing numbers of cysts as they become older. The relationship between age and mean numbers of parasites per sheep can be modelled as a straight-line relationship. The gradient of the line is dependent on the infection pressure, which in turn depends upon the level of infection in the definitive host, together with the contact rate between sheep and dogs and environmental conditions affecting the survival of eggs in the environment. 2.1. The definitive host The model described in this paper is based on a prevalence equation in dogs. The prevalence of infection in dogs can vary according to the infection pressure (b ), the probability of the dog
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Fig. 1. Representation of the life cycle of E. granulosus .
acquiring immunity on exposure (a ), the probability of an immune dog losing its immunity (g) and the life expectancy of the parasite in the dog. Roberts et al. (1986) reported that the prevalence (p) of infection in dogs varied according to age (t) as:
important in certain situations, and that in the presence of immunity, the asymptote is not merely lowered as suggested by Roberts et al. (1986), but that prevalence rates are lower in old dogs than in young animals. This suggested another form of the equation, which is used in this model:
p(t) p( )f1exp(mb)t)g
P(t)K1 [exp(mb)t]K2 [exp(mg)t]
(1)
K3 [exp(gab)t]K4 where b is the infection pressure (in terms of insults per unit time) and m is the rate at which hosts lose their parasite. The asymptotic prevalence p( )/b/(m/b ). In the presence of immunity the asymptotic prevalence is reduced by a factor, which is always less than 1. Roberts et al. (1986) suggested that immunity was not important in the definitive host and thus prevalence depended solely on the infection pressure and the life expectancy of the parasite. However, recent evidence from Tunisia (Lahmar et al., 2001) and from Kazakhstan (Torgerson, 2002) reports the highest prevalence and intensity of infection in younger dogs. This suggests that immunity may be
(2)
where K1, K2, K3 and K4 are dependent on a , b , g and m . When a /0, this equation simplifies to Eq. (1). 2.2. The intermediate host Roberts et al. (1986) has modelled the abundance of cysts in sheep. Both non-linear and linear forms of the equation have been described. There have been a number of reports of the variations in age intensity in sheep (Roberts et al., 1986; Ming et al., 1992a; Cabrera et al., 1996; Torgerson et al., 1998, Lahmar et al., 1999) from different geogra-
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phical locations. Each one suggested that there was insignificant immunity in sheep when naturally infected. Consequently the simple linear for of the equation was used in the model: m(t)ht
(3)
where m is the mean number of cysts at age t and h is the infection pressure in terms of numbers of cysts acquired each year. 2.3. Spread sheet model The spread sheet model was built round cell entries for prevalence of the dog population suggested by Eq. (2) and abundance of infection in the sheep population suggested by Eq. (3). Because it takes a period of time for cysts to become mature delay had to be built into the model. This was achieved by having a column representing the different age groups of sheep. For the purposes of this model this was taken as the age distribution of sheep described by Ming et al. (1992a) from China. However, this age distribution can be easily changed for any area based on base line surveillance data from that locality. The ability to calculate the proportion of each age group as a total of the population was included in the model. For any area this data would have to be obtained by field survey. From the average time to maturity of cysts together with the infection pressure, the average contribution of each age group to the infection pressure to dogs was calculated. From this the overall infection pressure to dogs was calculated. Delay was effectively modelled as any control programme would only prevent additional infection in sheep. Cysts that sheep had already acquired would only be removed when sheep infected with these cysts died.
3. Control simulation Once the equations are written into the spreadsheet, precontrol parameters must be obtained. Some of these will be by local observations. By getting a representative sample of sheep of different ages the precontrol infection pressure to sheep can be calculated. Likewise the age at which sheep
are fed to dogs needs to be ascertained. For simplicity, it has been assumed this reflects the age structure of the sheep population and can be obtained from field data. The precontrol prevalence and the age structure of the dog population also needs to be obtained and this can give the likely precontrol values for the infection pressure to dogs and the likelihood of developing immunity. The average life expectancy of a patent infection has been assumed to be approximately 1 year (Aminjanov, 1975; Sweatman and Williams, 1963), but this may also be calculated from accurate surveillance data and fitting the data, by non-linear regression analysis, to Eqs. (1) and (2). Once baseline information is obtained for a particular locality, control can be simulated. For example, if dog dosing is undertaken every 6 weeks (the prepatent period) then the mean survival time of a patent infection (1/m) is going to be 0. Consequently the parasite will disappear, providing control is maintained until non-infected sheep have replaced all the sheep in the population. However, 100% compliance is rarely if ever achieved. In China, frequently less than 80% of dogs were treated in a control programme (Fen-Jie 1993), whilst in Turkana Kenya only 59% of dogs could be treated (MacPherson et al., 1986). Therefore assuming 80% of dogs in the target population are treated but 20% remain untreated it can be seen that the infection pressure to sheep will immediately drop by 80%. As the relationship between infection pressure and numbers of parasites acquired is linear, the numbers of cysts in sheep will begin to decline. This will not, however, change the infection pressure in dogs for some time. The mean time to maturity of cysts has been suggested as 6.63 years (Gemmell et al., 1986) or approximately 2 years by Aminjanov (1976) and Sweatman and Williams (1963). This time to cyst maturity acts as a notable delay in obtaining significant results after the control measures are implemented. One year after control, the number of mature cysts in sheep aged 1 year will be reduced, but since these animals contain only a small proportion of the mature cysts in the population there will only be a very small reduction in the infection pressure to dogs. However, in subsequent years, the reduc-
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tion in infection pressure becomes progressively greater. Once all old sheep have been removed, reinfection only occurs from dogs that have escaped treatment in the first place, thus prevalence rates in both life-cycle hosts start to drop quickly. The difference in the time for cyst maturity does have some effect on this. If the longer time period reported by Gemmell et al. (1986) is used then cysts from old animals result in a greater proportion of the infection pressure to dogs and the delay is greater than if a shorter time to maturity is used. Other control interventions can be similarly modelled. In the case of a vaccination campaign this can be modelled as a reduction in the numbers of new fertile cysts from the date of vaccination. Thus sheep that are vaccinated with a 100% effective vaccine will develop no further cysts from continued exposure to Echinococcus eggs. However, infection acquired before vaccination can still be further transmitted to dogs, thus there will be some delay until all the cysts disappear from the sheep population. Likewise an education campaign may reduce the rate at which offal is fed to dogs and this can be modelled as a reduction in the infection pressure in dogs. The parameters used in the model are illustrated in Table 1 and the parameters that are changed in control simulations illustrated in Table 2. As can be seen there is significant variability in some of the parameters suggested by different experimental work and in the possible capture rates of dogs in the populations. There will also be additional variability in the base line surveillance data obtained for the precontrol parameters. To
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Table 2 Modelling control through changes in the parameters Control intervention
Change in parameter
Anthelmintic treatment of dogs Vaccination of sheep
Reduction in life expectancy of patent infection Reduction in numbers of cysts in sheep from time of vaccination Reduction of infection pressure to dogs
Education programme
model this variability in the simulations, MonteCarlo routines have also been written into the spreadsheet model. Thus, variability in the capture rate of dogs between the best case (say 90% treated) and worse case (say 60% treated) can be modelled. This is particularly important in the absence of accurate dog population figures and would give best and worse case scenarios if a target capture rate or compliance rate of 75% was assumed. Consequently a continuous range of possibilities (one chosen at random for each simulation) can reduce the life expectancy of the parasite to reflect capture rates of between 60 and 90% (with a most likely rate of 75%). This randomization is undertaken with all the other parameters in the model in a similar manner. The results of the control in terms of best and worst case scenarios with the median likelihood can then be reported. A number of these simulations will now be illustrated. In a situation where there is relatively low infection pressure to dogs, there is assumed to be an initial prevalence in dogs of approximately 20%
Table 1 Baseline parameters Sheep Age related abundance of infection Infection pressure Time to maturity of cysts Acquisition and loss of immunity Age of feeding sheep to dogs
Dogs Surveillance data Abundance data 6.6 years (Gemmell et al., 1986); 2 years (Aminjanov, 1976; Sweatman and Williams, 1963) Negligible (Roberts et al., 1986) Surveillance data
Age related prevalence of infection Infection pressure Life expectancy of infection Acquisition and loss of immunity Age structure of dog population
Surveillance data Prevalence data 1 year (Aminjanov, 1975); can calculate from surveillance data Negligible (Roberts et al., 1986). Calculate from surveillance data Surveillance data
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and a precontrol prevalence in sheep of 50%. The sheep are under an infection pressure of approximately one new cyst per year and dogs are infected on average once every 2 years. In other simulations the dogs are under a much greater infection pressure and the sheep are more heavily infected. In these simulations dogs are infected between one and two times a year resulting in a prevalence approaching 40%. The prevalence in sheep is modelled as 80%. The life expectancy of the parasite, in the absence of control, is assumed to be 1 year in all cases. The model was constructed in an excel spreadsheet (Microsoft, Redmond, WA). The variability was modelled using Crystal Ball Software (Decisioneering, Denver, CO). One thousand simulations were undertaken for each control intervention with each variable being randomly changed along its range for each simulation.
4. Illustrative examples Fig. 2 represents the prediction of the decrease in prevalence of infection in sheep suggested by the model with a control intervention of 6-monthly anthelmintic treatments, in the absence of significant definitive host immunity. In the first example, the time for cyst maturity is assumed to be that reported by Gemmell et al., (1986). Six-monthly anthelmintic treatment is largely successful with control (as defined by reducing the prevalence in sheep to less than 1%) achieved within 20 years providing there is no protective immunity in dogs. This depends on the programme treating a target of 75% of the dog population. However, because of the inevitability of populations of strays and lack of compliance this represents the median outcome, with lower and upper 95% confidence limits based on a treatment rate of 60 /90% of the dogs and errors built into parameter estimates (upper and lower dotted lines). The main feature of this simulation is the delay before the prevalence starts to decrease in sheep. This is because most of the infection pressure to dogs is from old sheep, and young sheep, infected immediately before start of control, will still be infective to dogs some years later. If the time to cyst maturity
is reduced to 2 years, then the main difference is that the delay in reduction of prevalence to sheep is lessened, but it takes a little longer to achieve less than 1% of sheep infected. However, the model does illustrate that the time to maturity of sheep cysts is one of the least sensitive parameters in terms of final outcome. For all subsequent simulations illustrated a time to cyst maturity of 2 years is used. It may seem surprising that a modest intervention would be successful, but if there is no stabilizing effect of immunity in either the definitive or intermediate host then the parasite will have a relatively low reproductive ratio (Gemmell, 1990) and would require little force to destabilize it. Fig. 3 illustrates how protective immunity in the dog can affect the outcome. In this scenario, 6monthly anthelmintic treatment, whilst effectively reducing the prevalence in sheep, does not result in control. Eventually, there is a new equilibrium of approximately 20% prevalence in sheep reduced from 50%. This illustrates the important effect definitive host immunity can have on parasite stability. A lowering of infection pressure to dogs results in the dogs becoming more susceptible to infection and thus less overall control than might have been expected. This type of phenomenon is an important one in the host parasite relationships of helminth parasites and their population biology. Thus the presence of immunity acts as a stabilizing factor on the population (Anderson and May, 1985) In the presence of immunity even reaching 90% of the dog population still results in 5/10% of sheep remaining infected some 20 years after start of control. If an education programme is also put into place, there is a further reduction in prevalence in the sheep due to less feeding of offal. However, this still does not result in adequate control. It is also worth noting that, in the presence of definitive host immunity, education on its own (assuming 50% reduction in offal feeding) has very little effect because the lowered infection pressure to dogs is off set by the increased susceptibility to infection. However, education can be effective if there is no immunity in dogs. Of the three options (6-monthly anthelmintics, education or vaccination), vaccination appears to
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Fig. 2. Change in % prevalence of E. granulosus in sheep populations after commencement of control programme with no definitive host immunity. Top left: response to control of 6-monthly anthelmintic treatment in dogs with 6.6 years to cyst maturity; top right: with 2 years to cyst maturity; bottom left: an education programme resulting in a reduction of feeding offal to dogs and bottom right: a vaccination programme with a compliance rate of 75%. Median outcome ( */) and 95% confidence limits (---).
offer the best long-term solution as a sole intervention strategy, with a new equilibrium of approximately 8% prevalence in 20 years compared to 18% for anthlemintic treatment. However, immunization of the intermediate host results in a delay in the reduction in prevalence in the definitive host. This is because the vaccine only prevents new infections and will not prevent development of, or eliminate, cysts that are already present in sheep. Consequently there will be little immediate reduction in infection pressure to dogs, this only occurs as the generation of sheep, infected before vaccination, are replaced by animals vaccinated soon after birth. Thus, there will be a delay in reducing the parasite prevalence to dogs and hence transmission to humans. Whilst a number of intervention strategies will work in the absence of immunity, it requires a combined strategy in the presence of immunity. This can also be seen in Fig. 3. The scenario when there is both vaccination of sheep and 6-monthly anthelmintic treatment of dogs will result in good
control within about 15 years, whereas neither control strategy alone appears to have such an effect. This pattern is repeated in the situation where the parasite is intensely endemic (Fig. 4). In this example, it is assumed that dogs are infected between one and two times per year on average. In this example about 80% of sheep are infected and between 30 and 50% of dogs are infected (depending on the level of protective immunity assigned to the dogs). In this intensely endemic situation, immunity is much more likely to play a regulatory role in dogs because of the degree of exposure of the dog population. In this situation, vaccinating sheep (assuming a 75% vaccination rate) eventually results in about 23% of the sheep remaining infected. This of course represents the majority of the sheep that escaped vaccination. Thus, the new equilibrium state will result in a cycle between the non-immune sheep and the dog population. To obtain adequate control, at least 90% of the sheep need to be vaccinated. Likewise
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Fig. 3. Effect of definitive host immunity on control simulations. Top left: response to control measures using 6-monthly anthelmintic treatments. Top right: effect of vaccinating sheep with a target compliance of 75%. Bottom left: effect of education programme resulting in decrease in cysts fed to dogs by 50% and bottom right: integrated control using vaccination of sheep and anthelmintic treatment. Median outcome ( */) and 95% confidence limits (---).
anthelmintic treatment of dogs will reduce the initial prevalence in sheep from 80 to 25%, but at this level a new equilibrium state is achieved. Again at least 90% of dogs will need to be treated to control the parasite. Alternatively, a reduction in anthelmintic treatment intervals to 3 /4 months at most (with a capture rate of at least 75%) would also be effective. However, combining the two control measures will result in good control: in less than 10 years in the case of 90% of animals being treated to about 30 years if only 60% of animals are vaccinated or treated. If an education programme were also undertaken that reduced offal feeding to dogs by 50%, control could be achieved 3 /5 years earlier.
5. Conclusions A 6-monthly anthelmintic treatment interval was deliberately chosen in these examples as this is a relatively modest intervention compared to
many intervention strategies that have already been applied successfully. These include the 6weekly dog treatment programme, which is based on the prepatent period of the parasite in the definitive host. However, the time to re-infection is arguably a more important interval to take into account when designing anthelmintic treatment programmes. In addition, because it is very difficult to achieve treatment of every dog, and vaccination of every sheep, a target capture or compliance rate of 75% was assumed to illustrate the outcome. Using such simulations it should also be possible to undertake cost-benefit analysis of control of CE. All other things being equal, it will cost four times as much to undertake a 6-weekly anthelmintic dog treatment programme, than a 6-monthly programme. The excess costs may or may not justify the additional decrease in prevalence that is achieved with the programme using the more frequent dosing regimen. The decision may be based on the perceived compliance of the target
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Fig. 4. Control simulation in an intensely endemic region with dogs being infected one to two infections per year and an initial prevalence in sheep of 80%. Immunity in dogs is assumed. Top: anthelmintic treatment every 6 months; middle: vaccination of sheep with a 75% compliance rate. Bottom: vaccination of sheep and anthelmintic treatment. Median outcome ( */) and 95% confidence limits (---).
population. The greater the compliance, the fewer dogs are likely to escape treatment, thus the treatment interval could be lengthened. The simulations also illustrate the potential usefulness of the vaccine. For example, if immunity is present in dogs, the anthelmintic treatment interval must be shortened to achieve control if this is the only intervention strategy. Alternatively a very high
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capture rate is mandatory. Both options would prove costly. Ensuring 90% or greater capture rate is much more costly than 60 /70% capture rate. Decreasing treatment intervals to 3 months doubles the expense. However, vaccinating sheep on an annual basis and treating dogs at the illustrated compliance rates every 6 months appear to work which ever the scenario and would almost certainly be cheaper than trying to increase compliance rates or decreasing treatment intervals by using anthelmintics alone. Furthermore, should there be seasonal transmission then the anthelmintic treatment can be targeted at certain times of year. This will result in more rapid control than if there was continuous transmission. Interestingly, in Uruguay, studies suggest that dogs become reinfected between 2 and 4 months after treatment (Cabrera et al., 2001). Because of this a reduction in treatment intervals from every 6 weeks to every 12 weeks was subsequently shown to reduce the infection rate in sheep. This is in agreement with this model. Indeed, only a small percentage of the original dogs were represented for treatment at the end of the 2.6-year trial. This could indicate poor compliance and further indicates that less frequent treatments of dogs can be effective. Costs are important in regions where there are scarce financial resources. This model illustrates how treatment intervals can be lengthened if combined with other intervention strategies such as vaccination and thus reduce costs. Therefore, this model should be linked to economic models (Torgerson et al., 2001; Majorowski et al., 2001) to attempt to cost the benefits of a control programme in financial terms. This can then be used to lobby support for control. This paper illustrates that it should be possible to model the possible outcome of a control intervention against E. granulosus . This may provide a powerful tool for policy makers and it will give likely but, perhaps more importantly, worst case scenarios of a control intervention. It also illustrates that accurate estimation of precontrol parameters need not necessarily be known, but this will reflect the accuracy of the predicted outcomes. In particular it does suggest that parameters, such as time for cysts’ maturity, have little effect on the final outcome but will affect the
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initial rate of decline of echinococcosis in the population. Interestingly, in both the intensely endemic and less endemic scenarios, in the presence of definitive host immunity a combination of 6-monthly anthelmintics and sheep vaccination is likely to be successful, even with only a 60% compliance rate. In the presence of definitive host immunity neither vaccination nor dog treatment will work on its own (at the illustrated treatment intervals and compliance rates). Since the regulatory role of immunity by dogs in the population is not yet adequately understood, a minimum recommendation of 6-monthly anthelminitc treatments with sheep vaccination should be made. If only anthelmintics are available, the model indicates that treatment intervals should be no greater than 3 months. Obviously this is a generalization and should be modified in light of local transmission dynamics. The model also demonstrates that greater success is likely to be achieved if the parasite can be attacked in both hosts in the life cycle than if control is aimed at just one host. Logistically it is almost certainly easier to treat most sheep and most dogs than all dogs or all sheep and consequently is an important result. This is also in agreement with an earlier model (Harris et al., 1980) which also concluded the best results were obtained by treating both definitive and intermediate hosts. This model has mainly illustrated the likely outcome in the prevalence of echninococcosis in the sheep population and not the incidence in the human population. However, the behavior of the infection in the two species would be expected to be similar with perhaps a greater delay before human incidence starts to decline. In addition, education may have an additional positive effect as improvements in personal hygiene would be expected to lower transmission to man even in the absence of any change in prevalence in animals. This would be a positive effect of an education programme. One possible inaccuracy in the model is that it is based on a prevalence model in dogs. A model that is based on the abundance of infection would be more rigorous and is the subject of ongoing work. The reasons are because of the aggregation of
parasites within hosts. E. granulosus is highly aggregated in dogs (Roberts et al., 1986; Ming et al., 1992b; Lahmar et al., 2001) and can be modelled according to a negative binomial distribution. Thus the majority of the population of adult E. granulosus are in a small number of heavily infected hosts with a much larger number of lightly infected hosts. These heavily infected dogs, often with several thousand parasites per dog, may be responsible for the majority of the transmission with the lightly infected animals, often with fewer than 100 parasites, being less significant. A model needs to take this into account and particularly needs to account for the behavior of heavily infected individuals in response to control measures. In trichostrongylid infections of ruminant hosts, the negative binomial constant tracks changes in the mean burden of the parasites (Grenfell et al., 1995). Thus the higher the mean parasite burden, the lower the degree of aggregation. Modification of the simulation model may be required if the same is true of infection with Echinococcus in dogs or in sheep. Low mean burdens and low prevalences with a low k value, as can be predicted in such a scenario in the latter stages of the control programme, might lead to instability.
Acknowledgements The author would like to acknowledge the financial support of INTAS (INTAS-97-40311) and the NIH (1R01TW01565-01) in supporting his work.
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