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Disease Transmission Models for Public Health Decision-Making: Designing Intervention Strategies for Schistosoma japonicum

and .

Author Information and Affiliations

Modelling Parasite Transmission and Control, edited by Edwin Michael and Robert C. Spear.
© 2009 Landes Bioscience
Read this chapter in the Madame Curie Bioscience Database here.

The purpose of infectious disease transmission modeling is often to understand the factors that are responsible for the persistence of transmission, the dynamics of the infection process and how best to control transmission. As such, there should be great potential to use mathematical models to routinely plan and evaluate disease control programs. In reality, there are many challenges that have precluded the practical use of disease models in this regard. One challenge relates to the mathematical complexity of the models, which has made it difficult for field workers and health officials to understand and use them. Another challenge is that, despite their mathematical complexity, models typically do not have sufficient structural complexity to consider many of the site-specific epidemiologic and disease control details that the practicing health official routinely considers. Moreover, most modeling studies have not been sufficiently explicit or exemplary in explaining how field data may be incorporated into the models to impact public health decision-making. In this chapter, we start with a classic model of schistosomiasis transmission and relate its key properties to the more detailed model of Schistosoma japonicum model presented in Chapter by Remais and Chapter by Spear. We then discuss how various controls (e.g., chemotherapy, snail control and sanitation) may be evaluated via the detailed model. We then demonstrate in a practical manner, using S. japonicum data from China, how field data may be incorporated to inform the practice of disease control. Finally, we present a new model structure that considers how heterogeneous populations are interconnected, which has particular relevance to understanding disease control and emergence in today’s highly mobile world.

Introduction

Nearly two decades ago, Woolhouse reviewed the modeling literature for schistosomiasis and how various disease controls may be evaluated.1,2 We remark upon his rather depressing introduction, which stated that since Macdonald developed the first model in 1965 and subsequent reviews of the modeling literature in the 1977 and 1982 by Cohen and Barbour, respectively, the models had very limited impact on actual schistosomiasis control.3-5 One of Woolhouse’s reasons for this, which we tend to agree with, is that there exists a disconnect between disease modelers who focus their work on theoretical mathematics and field workers who make real-world public health decisions related to disease control. Another reason is that, while mathematically complex, the models still lacked the realism to capture local variations and logistical constraints that affect the implementation and efficacy of controls. Woolhouse argues that this does not need to be the case. Instead, models may be one of the tools that health planners use, in addition to field experience and epidemiology in designing control programs. With this as background, we begin where Woolhouse left off, using the same simple schistosomiasis ch41modeling framework to consider disease control. We discuss how the S. japonicum model presented in Chapters and 5 and 10 relates to the basic modeling framework and allows us to consider various disease controls, including chemotherapy, snail control, sanitation and health education. We then demonstrate in a practical manner, using S. japonicum data from China, how field data may be incorporated to inform the practice of disease control.

We also present a new development in schistosomiasis modeling that explicitly considers how heterogeneous populations are interconnected. These connections may occur via natural physical processes (e.g., hydrological connections via watersheds) and/or via social-economic processes (e.g., migrant labor or trading of buffalo). Such a model offers exciting new possibilities to consider how disease control may be best practiced in highly connected environments. As many areas are becoming increasingly connected due to economic development, it also provides opportunities to evaluate the impact of changes in connectivity due improvements to road and irrigation networks, for example, on disease transmission and (re)introduction.

Model Framework

Schistosomiasis infection persists in over 70 countries and remains a cause of great morbidity despite several opportunities in the lifecycle of the parasite to disrupt transmission.6 Figure 1 illustrates the lifecycle for S. japoncium in China and points of possible intervention, which include the treatment of human hosts via chemotherapy, snail control through molluscides or more permanent habitat removal, improvements in sanitation and reductions in exposure (e.g., via health education). We describe each of these controls in more detail below.

Figure 1. Schistosome lifecycle with main opportunities for control shown in parentheses.

Figure 1

Schistosome lifecycle with main opportunities for control shown in parentheses. 1) Humans become infected by the free-swimming cercarial form of the parasite from skin contact with contaminated surface water. 2) Once infected, adult schistosome worms (more...)

Woolhouse's basic model framework is based on the simplification of the worm-snail-cercariae-miracidia 4-state system to a 2-state system of just mean number of schistosome worms per person m and the prevalence (as a proportion) of patent infections in snails y . This is reasonably assumed as the lifespans of the larval stages are considerably shorter than worms and snails (hours versus months and years). Hence, it is assumed that larvae are at equilibrium and we can simply model the remaining states via the following coupled differential equations (see Table 1 for a description of the model parameters):

Image ch4186eqn1.jpg
where each equation consists of a gain and loss term. For the worm burden equation, the gain term consists of the product of the number of snails N, prevalence of patent snail infection y and the rate of infection per person per patent infected snail α. The loss term consists of the product of per capita worm mortality γ and the mean worm burden per person m. For the snail infection prevalence equation, the gain term consists of the product of the number of humans H, mean worm burden per person m, the proportion of uninfected snails (1 -y ) and the per capita rate of infection of snails per schistosome β. The loss term consists of the product of per capita infected snail mortality μ and the patent snail infection prevalence y .

Table 1. Variables in the transmission models.

Table 1

Variables in the transmission models.

The simple model above has a number of assumptions that are inherent in the underlying Macdonald model, upon which (0.1) is based. First, it is assumed that the helminthes are bisexual and that the mating probability of these worms is incorporated in the β term. Moreover it is generally assumed that there is equal probability of male and female larval infectivity and worms are monogamously paired. More importantly, the basic model ignores possible density dependent effects that may arise through acquired immunity. Furthermore, it is assumed that the loss of miracidia from the environment due to snail infection is negligible. Snail numbers are assumed to be held constant, which does not account for seasonal variations or increased morbidity imposed by larval infection. Multiple infections of snails are ignored, as is the issue of infection latency due to a prepatency period. Additionally, human population numbers are assumed to be held constant, with homogeneous exposure to larvae occurring for the population. It also assumes humans are the single definite host. We comment on various extensions of the basic Macdonald model that relax some of these assumptions later.

Classically, the approach to studying the Macdonald system is first to evaluate the equilibrium solutions of the model. Equilibrium for each state denoted as m* and y* are reached when both the mean worm and patent snail infection prevalence reach a stable unchanging level (i.e., where dm/dt = 0 and dy/dt = 0). In discussing this equilibrium it is helpful to define the following terms:

Image ch4186eqn2.jpg

where TSH considers those parameters that relate to snail-to-human infection and TMS considers those that relate to human-to-snail infection.

The equilibia then can be described according to a quantity known as the basic reproduction number, R0:

Image ch4186eqn3.jpg

When R0 < 1 then the system does not sustain transmission and the equilibrium values are:

Image ch4186eqn4.jpg

However, when R0 > 1 there exists a positive equilibrium at:

Image ch4186eqn5.jpg

Hence, an important property of the basic schistosomiasis model is that endemic infection will exist for a population when the Basic Reproduction Number, R0 > 1.

We note that R0 is explicitly defined by the snail-to-human and human-to-snail infection parameters. However, these terms can be factored into more specific terms, as we saw in the Macdonald-like S. japonicum model of Chapter 10. Indeed, we note even in Macdonald’s early work, there was a distinction between biologic and site-specific variables (synonymous with the Psand Pb variables, respectively mentioned in Chapter 10). For instance, the probability of a cercaria finding and infecting a suitable host, (Macdonald’s so-called “exposure factor”) may depend upon a biologic constant representing the efficiency of cercarial penetration and a site-specific variable that relates to the number and characteristics of water contacts made by hosts.

In Macdonald’s work, considerable detail is spent in describing the effect of adding more detailed mating probability function Φ as one of the factors that makes up the β term. It is assumed that male and female schistosomes infect with equal probability, but mate at random, with at least two worms, one of each sex required for mating. The results of adding the mating probably function is the establishment of break point in the system, which concerns the resulting equilibrium when the parasite is introduced into a community. As described by Woolhouse,1 a constant mating probability results in any parasite introduction establishing endemic transmission (positive m* and y*). However, when a more realistic assumption that the mating function depends upon the aggregation of worms in hosts (typically regarded as negative binomial distributed), an unstable break point occurs in the system, such that parasite introductions below a threshold do not establish transmission (i.e., the system will fall to m* = 0 and y* = 0 equilibrium). Conversely, when there is a sufficient introduction of parasites, then endemic transmission will be established (i.e., the system will settle on the m* > 0 and y* > 0 equilibrium).

From a control perspective there are thus two theoretical goals in eliminating transmission. One goal may be to reduce the longer-term potential for transmission such that R0 < 1. Alternatively, another goal may be to perturb the system so much as to bring the system below the break point, whereby it will fall to the 0, 0 equilibrium. We note, however, that accomplishing either says nothing of the dynamics (how long it will take) for transmission to be eliminated. We deal with this issue of system dynamics later, after we first explore how the simple Macdonald model performs for real-world data.

A Crude Estimate of R0 for S. japonicum Based on Macdonalds Model

Based on the simple Macdonald model’s equilibria equations (0.5) for endemic transmission, we note a rather surprising finding: the only field data required to estimate R0 is snail infection data. Note that y*, the equilibrium patent snail infection prevalence (expressed as a proportion) is the sole estimator for R0:

Image ch4186eqn6.jpg

In China, as in many other places the proportion of snail surveyed that are patent infected typically does not exceed 0.05. In our study of 20 villages in Xichang county in Sichuan Province,7 we found the highest snail prevalence to be 0.03. Substituting 0.03 for y*, results in a solution for R0 of 1.03. Because R0 >1, the condition for endemic transmission is satisfied.

Control Needed to Terminate S. japonicum Transmission Based on Macdonalds Model

The above estimate of

Image ch4186eqn7.jpg
is surprising because it seems so small. One way to sustainably eliminate transmission is by reducing the potential for transmission by driving R0 < 1. We must effectively transition from a R0 of 1.03 to an effective reproduction number R (the common term for R0 after control has been implemented) of <1. We note that because of the following definition of R0, there are many possible control options to bring R < 1:

Image ch4186eqn8.jpg

In other words we need the following solution:

Image ch4186eqn9.jpg

where L represents a factor that reduces R0 sufficiently to bring R < 1. In this case, to terminate transmission, L must be <0.97. It is quite shocking that only a 3% reduction in R0 would terminate transmission. It is even more shocking when one considers that factor L could come from any of the possible controls and even combinations of controls.

Routine chemotherapy can be implemented as an increase in γ, the per capita mortality of schistosomes:8

Image ch4186eqn10.jpg

where g’ is the coverage rate and h the efficacy, both as proportions. The increased mortality essentially results in a decrease to TSH, thereby creating R < R0. And our goal is to obtain:

Image ch4186eqn11.jpg

Indeed, if we assume a fairly non-aggressive control strategy that consists of a coverage rate of single treatment every 2 years and 70% efficacy when the natural lifespan of worms is 3 years, working through the calculation suggests this strategy would be more than sufficient to eliminate transmission (at least eventually, at equilibrium)! Similar arguments could be made for relatively easy elimination through snail control by modifyingNN’, improved sanitation by modifying β ⇒ β’ and reducing exposure by modifying α ⇒ α’ (see2 for examples of how these additional controls may be implemented).

Problems Inherent in the Macdonald Model Assumptions

The above analyses suggest that with such low R0 (i.e., slightly above 1), it should be easy to terminate transmission. If only it were so easy in the real world! Indeed, because the Macdonald model is unrealistic in its assumptions, the equilibria conditions are thus unrealistic.

Barbour was quite vocal about the limitations of the basic Macdonald model, attempting to improve the model (or “rescue it” as he states) by accounting for various heterogeneities that may scale R0 upwards.9,10 Specifically, Barbour accounts for snail latency to patency by the addition of a full snail population submodel based on S. japonicum data,11 spatial heterogeneity in water contacts and age-dependent immunity.9 Despite various attempts to rescue the model, the values of R0 from variations on Macdonald’s model may still be unrealistically low.10

We note that in Chapter 10, for our calibrated Macdonald-like model for S. japonicum, we arrived at an approximate R0 (or R’ representing the prefiltering calibration criterion) of approximately 4. Moreover, as this is a prefilter criterion, it is an upper limit on R0! Again, using chemotherapy as an example, two treatments per year with 70% efficacy should eliminate transmission for this region of China—something that is already being conducted in many endemic areas of China. However, it is important to note that we still have not considered how long it would take to reach elimination, which we discuss later in terms of the dynamics of the system. We only note here that there may be benefits to combining chemotherapy with other more sustainable disease control strategies.

Still, there remain other heterogeneities that have not been widely considered. For one, more work is needed to consider age-acquired immunity, for which there is evidence for various Schistosome species, although the role of age-acquired immunity in S. japonicum, in particular, remains unclear.12 In addition, the presence of animal reservoirs may need to be considered for some environments particularly for S. japonicum, as over 40 different wild and domestic mammals have been shown to play a role in transmission.13 Stochastic effects and the transport of parasites between connected environments may also create heterogeneities that modify R0, the latter discussed in greater detail below. Space and time-varying effects, such as attenuations that may occur through the synchrony of exposure may also modify R0. In these respects, recent studies by various modeling groups may be considered (e.g., for S. japonicum, age-specific or population group-specific models in,14,15 models of parasite persistence in connected populations,16 time-varying factors (Chapter 5) and inclusion of animal hosts17).

Barbour has suggested an alternative to the Macdonald model (the so-called Ross model) based on human infection prevalence rather than worm burden.10 The rather obvious benefit of such a model is that it is based on human infection prevalence, which can be measured (or at least approximated from the currently best-available diagnostic tests), as opposed to worm burdens which cannot be directly measured in humans and are difficult to relate to egg count data. Under Barbour’s model:

Image ch4186eqn12.jpg

where, as before, y* is the equilibrium infected snail prevalence and now P* is the equilibrium prevalence of infection in humans. Real-world values of 3% snail infection and 50% human infection, immediately result in values of R0 twice as high as the Macdonald model without accounting for any heterogeneities. Indeed, there is some acceptance of this modeling approach by those working in China and the Philippines on S. japonicum.15,17

Modeling Control Dynamics

As we alluded to earlier, regardless of our choice of model (e.g., Macdonald or Ross), the equilibrium analyses of the models only tell us the eventual equilibrium values of the system, but not how fast the system will arrive at those equilibria. In the real-world, there is a practical need to understand the comparative performance of competing control programs, which often depends upon what can be done in short time frames (e.g., 5-10 year control programs) and how large of an impact might be expected. Such issues ultimately lead to questions about the dynamic properties of the system.

We note from our work, that when various heterogeneities and delays are incorporated into the system, analytical solutions are essentially not possible. Thus, numerical simulations are required to understand the equilibrium and dynamics of the system. Moreover, often quite complex models are numerically simulated and calibrated to field data, which can be computationally intensive. Thus, in Chapter 10, considerable attention was paid towards optimizing simulation times via tricks, such as prefiltering the parameter space based on R0. Indeed, in our work, we have spent considerable time to model the dynamic behavior of various combinations of disease controls (e.g., various combinations and levels of chemotherapy, snail control and sanitation).14 We note also that the dynamics of these controls may act differently. For instance, in contrast to the Anderson-May method of modeling a chemotherapy program as described above,8 we have modeled chemotherapy as a resetting of the worm burden state (i.e., an instantaneous drop in worm burden when chemotherapy is provided), which can lead to disease elimination when considered in terms of the break point phenomenon of the model system. In contrast, snail control may be implemented either as a instantaneous drop in snail populations, reduction in snail habitat, or additional mortality, which depends on how the snail control was implemented. Thus, some disease control strategies may be used to yield immediate effects on reducing disease morbidity (good public health benefit), while others may be aimed at long-term, sustainable disease control (moving towards disease elimination). This can be clearly seen in the time-series figures of that show sudden drops in worm burden due to chemotherapy, but gradual reductions in transmission over time due to combined control of snails and/or sanitation.14

Given the above dual objectives for disease control, it is interesting to comment upon the various stages of schistosomiasis control in the world. Global schistosomiasis control policy currently focuses strongly on the distribution of praziquantel. As the drug has become increasingly affordable, it has taken a leading role in disease control efforts.18 In contrast, 15 years ago, the World Health Organization placed health education, access to safe water supplies and sanitation at the center of schistosomiasis control efforts.19 Clearly, for some countries, drug treatment to control morbidity needs to remain the focus, particularly in environments where being able to reliably deliver the drug remains a challenge. However in China, where much of our work is focused, there is already good access to drugs, yet the disease persists. There is recent evidence of disease re-emergence and a rise in prevalence following a 10-year chemotherapy campaign in some endemic areas.20 The situation in China and in other middle-income countries dealing with parasitic diseases is transitioning away from asking the dynamic questions of how to achieve short term reductions in morbidity, but rather how to effectively reduce and/or eliminate transmission over the long term. Moreover, a related question concerns the threat of disease (re)emergence and how might models be best-used to evaluate this threat in terms of its probability, likely time-frame and place of occurrence. Current efforts in our group are focused on exploring these latter issues, particularly in the context of using models to control and provide early warning for emerging schistosomiasis.

New Model Developments: Incorporating Population Heterogeneity and Connectivity

Quite early on it was recognized that one of the limitations of the simple schistosomiasis model was the assumption of population homogeneity, which led subsequently to adjustments such as those for worm aggregation and differences in exposure (e.g., different risk groups in our work). Still it was assumed that transmission occurred such that host populations were isolated from other hosts and parasite populations were isolated from other parasite populations. The real world, however, is made up of a complex mixture of populations that are defined by natural and social and political boundaries. Moreover, while parasites may be transported via hydrological or social networks, it is political boundaries that often govern how control is conducted in the real world. In China, for instance, each county organizes their own local control strategy against S. japonicum. In any given year, a county may decide to perform surveillance or mass control in some villages, but not in others.

Increasing evidence suggests that transmission actually occurs in groupings of heterogeneous populations that are in fact connected to one another in varying degrees. Connections may occur via numerous processes, most notably via hydrological channels that transport the larval stages of the parasite between environments and via socio-economic connections, such as host movement between environments. Generally, this understanding has led to the following model as described by Guarie and Seto16 (modifying their notation slightly to match our’s):

Image ch4186eqn13.jpg

in which different heterogeneous populations are represented in matrix notation as M, a vector of village worm burdens and Y , a vector of infected snail prevalences. The model accounts for differential social contacts through the matrices Θ, Ω, per capita cercarial and miracidial outputs πC, πM and spatially explicit transport for cercariae and miracidia that incorporate hydrological connectivity, survival and mortality in the matrices TC,TM and different vectors of human H and snail N populations.

Various elaborations of this system have been explored by our group for S. japonicum.16,21 The more recent theoretical analysis of the system by Gurarie and Seto describes both the general characteristics of the system and a few surprising aspects that have practical disease control implications. First, we define the terms:

Image ch4186eqn14.jpg

which represent the contribution of social and hydrological connectivities for snail-to-human and human-to-snail transmission, respectively. The product of these factors and the “internal potential” ρ results in the “Basic Reproduction Matrix”:16,22

Image ch4186eqn15.jpg

Similar to the R0 > 1 inequality for the simple Macdonald system (0.1), this matrix analog, R conditions sustained transmission for the distributed-connected system (0.10). Specifically, transmission is sustained when the largest eigenvalue of R satisfies the condition:22

Image ch4186eqn16.jpg

We note that (0.12) and (0.13) precisely define the role that “internal potential” and “connectivity” play in transmission. Both are important in sustaining transmission. The underlying parameters that make up internal potential ρ relate to the local potential for transmission within each village in the absence of connectivity (i.e., R0). In fact, in the absence of connectivity, the matrices simplify in such a manner that each individual village’s R0 conditions transmission.

The practical implications of the system on control are large. We can show that due to connectivity, infection may be high in certain villages. Yet focusing treatment on these high-infection villages may be inefficient. Instead, careful exploration of the network of connectivity and choosing key nodes that contribute to downstream infection may be much more useful in achieving region-wide control. Indeed, it can be shown that the structure and characteristic of various social and hydrological connectivities can determine whether region-wide schistosomiasis transmission will occur optimally, or not at all (Fig. 2).

Figure 2. Network of connected villages with hydrological connections illustrated via arrows and each village’s local R0 shown in increasing shades of grey (left).

Figure 2

Network of connected villages with hydrological connections illustrated via arrows and each village’s local R0 shown in increasing shades of grey (left). Region-wide schistosomiasis transmission across this network can be shown to be a function (more...)

We have observed this in our work in 20 villages in Xichang County located in Southwest China: the village with the highest infection prevalence (73% of residents were infected in 2000) was located downstream from an endemic village that pertained to a different county. High levels of cercariae were detected in the irrigation channels through which water entered the village suggesting local infection may not only be dependent on village conditions but the influx of parasites from upstream.7 Despite treatment of all infected residents in 2000, infection prevalence was 52% only two years later. We surmise that disease control efforts in this village that do not address the sources of parasite import will yield transient reductions in infection. The connected model confirms this hypothesis and suggests a more regional perspective to control activities may be useful, particularly in environments like those in China where rapid re-infection occurs and re-emergence is common. This understanding has led most recently to spatial analyses aimed at identifying connected watersheds that maybe used by local disease control agencies to better coordinate regional control strategies (Fig. 3).

Figure 3. 3-dimensional rendering of watersheds (bounded by white lines) in Xichang County, Sichuan Province, China, computed in a geographic information system and displayed in Google Earth.

Figure 3

3-dimensional rendering of watersheds (bounded by white lines) in Xichang County, Sichuan Province, China, computed in a geographic information system and displayed in Google Earth. These visualizations were created to assist the county in better planning (more...)

While it has not yet been explored extensively for S. japonicum, we note that the dynamics of such a system maybe particularly interesting from the perspective of emerging disease and disease spread. Because population connectivities are explicitly modeled in the system, it would be possible to consider the impacts of increasing population connectivity that might occur via improved infrastructure, such as road and water resource development. In some cases, as we have shown in our work, increased social dispersion may actually reduce the potential for transmission by moving people away from hotspots in the network that sustain infection.16 The practical analog to this is the increasing migration of labor away from rural environments to urban ones. Moreover, stochasticity may be incorporated into the model to consider the effects of chance perturbations to the network, including, for example, the occasional introduction of infected hosts into areas with low or no transmission. Such a model would allow for the identification of environments particularly favorable to the (re)initiation of transmission.

Concluding Remarks

Given almost half a century of model development for schistosomiasis, surprisingly little use of models occurs in the planning and implementation of disease control programs. As described above, simple models and crude R0-based analyses allow for easy calculations of the effect of chemotherapy and other control strategies, however, such calculations may lead to misleading results because of the over-simplified assumptions of these models. Accounting for various heterogeneities in the transmission of the parasite has led to more complex models, which have suggested that the disease may be more difficult to control than suggested by simpler models. Indeed, there is a trend towards increasingly complex models because simple models are not able to mimic real-world field data. Another reason for increasing model complexity may also be less justified, simply because there exists computing power to consider such complex models via numerical simulation. Unfortunately, the trend towards more complex models and numerical simulation is pushing the field of schistosomiasis modeling outside of the realm of practical field use at the local or regional level. Indeed, after over a decade in our group of developing our S. japonicum model for China, there has not been the knowledge transfer necessary for our complex model to be run at a provincial-level Chinese research institute.

However, our modeling work has demonstrated some encouraging applications. The most important application has been the ability to use the models as an independent tool based on field data to assess whether current control policies seem reasonable. In this regard, we have been able to confirm regional and national-level policies promoting combinations of disease control measures as having the best chance at eliminating transmission. Much of this knowledge has come about from studying the more complex dynamics of the model, rather than the equilibria of the system, which is consistent with the practical question of how long it takes to reach appreciable levels of morbidity reduction from various combinations of controls. Even more encouraging are the recent developments in modeling heterogeneous, yet connected populations that are beginning to tackle some of the most challenging aspects of schistosomiasis control, such as how to reduce the potential for re-emergence. In our increasingly connected society, these connected models also have great potential to identify the impacts of changes in connectivity on transmission of parasitic diseases such as schistosomiasis.

Although many practical field decisions regarding schistosomiasis control continue to be based on experiential knowledge, there exists a new class of problems, for which there is little to no field experience. This is the case with disease emergence and re-emergence. In our group, the current work on developing connected models may be used to design surveillance systems and examine predictors of disease emergence. Such models may prove to be very useful in exploring the possible impacts of (re)emerging disease, making full use of the ability to run computer simulations for a variety of hypothetical scenarios when very little experiential knowledge is available from the field. Our group is also continuing our work on complex site-specific models to explore the effect of various mixtures of control strategies, which is very much needed to combat endemic transmission as well as to prevent disease re-emergence.

Acknowledgements

The authors are supported by the National Institutes of Health, National Institute for Allergy and Infectious Disease (NIH R01 AI68854). We thank the Sichuan Institute of Parasitic Disease for their collaboration on fieldwork that has led to the parameterization of our models.

Modelling Parasite Transmission and Control, edited by Edwin Michael and Robert C. Spear. ©2009 Landes Bioscience.

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