Helminth parasite population models explore the impact of various population and regulatory processes regarding the rates of parasite establishment, development, mating, fecundity, survival, and transmission, as well as the impact of overdispersed parasite distributions among definitive and vector hosts [1]. These models can be fitted to baseline data and can be used to contrast hypotheses about the generation and consequences of infection heterogeneity, and the operation of density-dependent (including immunologically mediated) processes, among others, by comparing statistically the resulting fits [34]. These models can be updated as new data and knowledge become available. The interactions between positive (facilitating) and negative (constraining) regulatory processes (Text S1), and of these with parasite distribution and anthelmintic treatment, have been investigated [24], [35], [36].

Once the model has been developed and calibrated with appropriate parameter values (specific to the helminth–host system and location), it can be run until the endemic equilibrium steady state has been attained, after which interventions such as antiparasitic, antivectorial, snail control, and other measures [37] may be simulated. Among the antiparasitic measures that have been explored with models are the effects of chemotherapeutic treatment distributed either in the modality of mass drug administration (MDA), age-targeted (e.g., school-aged children), selective treatment (of given occupational groups) [24], [38], or vaccination [39], [40].

Assumptions regarding parasite life span and distribution of survival times, treatment efficacy, drug actions against various parasite life stages, coverage levels, and modalities of compliance can be incorporated and investigated [41]. Some models have also included acquired, protective immunity and looked at its interactive effect with control interventions [42], [43]. Model validation at this stage has been usually conducted by comparing how well model outputs reproduce observed data and resulting epidemiological trends during the intervention(s) under investigation [29]. Such model outputs may represent changes in worm burden, prevalence of infection and heavy infection, and changes in associated morbidity [44]. The latter itself requires careful investigation of disease outcomes related to infection, taking into account that co-infection with other parasites may be present, and that it is still unclear how present or past infection relates to measurable morbidity. However, a close fit between observed data and model outputs does not necessarily mean the model has captured the true underlying processes. The process of assessing structural validity, i.e., ascertaining that the model exhibits the right behaviour for the right reasons, is considered to be a more stringent measure of validation and has led to the devising of formal structural validity procedures. These include verification of structural assumptions (whether model structure is consistent with relevant and updated knowledge of the system being modelled) and parameter assumptions (model calibration), among others [45]. The literature on model validation is ample and controversial [46], as there are no fixed and universally agreed standards for selecting what test procedures or criteria to use for validation [47].

Stochastic models that incorporate parameter uncertainty can produce model outcomes ranging between upper and lower bounds, within which the data may be contained. Use and development of statistical methods for model fitting and comparison (both between alternative models, and between model outcomes and data) constitute an area of active research to be encouraged in the field of helminth mathematical models. This would strengthen our ability to understand and represent underlying processes and interpret modelling results. Although deterministic, population-based models may be suitable to investigate the average parasite population behaviour during the simulated control strategy, stochastic, individual-based models are more appropriate to investigate the probability of parasite elimination. However, Allee-type effects, introduced by facilitating density dependencies, allow elimination to occur in deterministic frameworks too [48].

With increasing use of advanced statistical methods (more recently including Bayesian approaches) that can now be implemented given the current availability of faster computing and more efficient algorithms, parasite epidemiology researchers are able to fit dynamic models to data for estimation of unknown parameters of interest. This has permitted estimation of parasite life span [49], [50]; treatment efficacy and drug effects on different parasite life stages [51], [52]; variation in host immune response to parasite life stages [53]; and transmission parameters such as parasite establishment rates [54], [55]. Such information is highly relevant but difficult to obtain by direct observation and/or experimentation. Recent examples in the field of schistosomiasis include measuring the transmission between hosts and snails in multi-host models of Schistosoma japonicum [56], and measuring the reductions in the force of infection (incidence of new infections) of S. mansoni resulting from large-scale implementation of praziquantel treatment [57]. However, there are also likely to be situations for which it is difficult to estimate separately highly correlated parameters, regardless of using the most advanced statistical methods. In these instances it may be necessary to either aggregate parameters in composite terms or seek data that may shed light on processes that are directly unmeasurable.

Although the goal of some programmes is that of morbidity control and elimination of the public health burden of the diseases (STHs, schistosomiasis and, up until recently, onchocerciasis in Africa), others aim at interruption of transmission and eventual elimination of the parasite reservoir (LF, onchocerciasis through vector control/elimination in Africa, onchocerciasis in Latin America, dracunculiasis). The models to inform parasite elimination should generally be stochastic because as parasite density decreases, stochastic variations (and stochastic fade-out) will be more important than the mean behaviour. (This is partly due to demographic stochasticity; worms are individuals, not fractions.) As the simulated intervention programmes reach their end points, the parasite population will die out in some model runs but not in others. By undertaking many model runs, each model run with a different set of parameter values randomly chosen from a plausible range, statements can be made regarding the probability of parasite elimination that will result from a given intervention or combinations thereof [41], [58]. This would permit investigation of the influence of factors such as initial endemicity level, transmission intensity, host heterogeneity (exposure, susceptibility, predisposition), parasite overdispersion, vector or intermediate host competence and vectorial capacity (the latter including vector density and biting rate on humans), treatment frequency, duration and coverage required, synergistic effects of vector/snail control, etc.

Whereas programmes aimed at morbidity control may benefit from age-targeted chemotherapy of those hosts at higher risk of acquiring heavy infection and subject to the greatest morbidity, parasite elimination will require prolonged mass treatment of all infected individuals at all endemicity levels, and in all endemic communities if parasite eradication is the goal. (For definitions of parasite control, elimination, eradication, and extinction see [59].) Such elimination strategies will substantially shrink the size of susceptible parasite refugia (populations of untreated parasites, not subjected to drug pressure [60]), an important factor influencing the spread of anthelmintic resistance. Parasite elimination programmes that rely on chemotherapy alone must therefore put in place careful surveillance systems for prompt detection of transmission resurgence, suboptimal parasite clearance rates, and monitoring of drug efficacy (parasite susceptibility). The same applies for those programmes relying on vector/snail control because of the possibility of insecticide/molluscicide resistance.

Transmission breakpoints should not be confused with the threshold for transmission, known as the basic reproduction number or R₀, or related parameters such as the threshold biting rate (TBR) in vector-borne infections [20]. Transmission breakpoints refer to finite parasite densities below which the parasite population would not be able to maintain itself; the basic reproduction number is, by definition, density independent. R₀ represents the threshold condition for parasite invasion and persistence, as it has to be greater than 1 for the parasite population to reach its endemic state. It is possible to rearrange the equations of R₀ for each helminth infection to derive threshold population sizes of definitive, intermediate, and vector hosts and demonstrate, for instance, that STHs can persist in human populations of much smaller sizes than those required for viral infections such as measles [61]. For the (vector-borne) filarial nematodes it is also possible to calculate TBR values below which the infection would not persist. These depend, in part, on the proportion of bites that vectors take on humans [20], [54], [62], emphasising the important role of measures that reduce vector density, measured by the annual biting rate (ABR), and vector-human contact. However, R₀ is a somewhat idealised, parasite density–independent entity. In reality, many transmission processes depend on parasite density, so the quantity of interest becomes the effective reproduction ratio (R_E), the composite parameter that reflects the changes in the transmission potential of a parasite with changes in parasite density [35], [63]. The value of R_E will be equal to 1 at endemic equilibrium (each female worm in the population replaces itself) and also at the so-called unstable equilibrium, the “elusive” transmission breakpoint or “holy grail” of parasite elimination. For dioecious parasites (those with separate sexes) and in those host–parasite systems with facilitating types of density dependence of the type described in Text S1, there will be unstable equilibrium parasite densities below which the parasite population would, in principle, become locally extinct (because females will not be mated and/or parasites will not establish within humans or vectors), and above which the parasite population will return to endemic equilibrium (Text S2 and Figure S1).

Understanding the behaviour of the host–parasite system in the vicinity of these transmission breakpoints is a priority area of research that requires the concourse of cross-disciplinary approaches such as mathematical analysis, knowledge of vector–host–parasite interactions, parasite population biology, and epidemiology. The values of helminth transmission breakpoints are themselves complex dynamic entities influenced by the nature and magnitude of vector- and host-specific density-dependent processes, the local characteristics of vector competence and vectorial capacity in different vector species, the degree of parasite overdispersion among hosts in the population, and the interactions of these with the intervention(s) deployed [48], [63]–[65]. This highlights the problems faced in trying to obtain a single, “one size fits all”, infection breakpoint value that can be applied across a variety of epidemiological settings, suggesting that the end game in parasite elimination programmes will have to respond flexibly and adaptively to the locale-specific microepidemiology of infection in endemic communities [66].

Recently, mathematical models of parasite population dynamics such as those summarised in Table S1 have been modified to incorporate parasite genetic structure with regard to drug susceptibility [67]–[69]. These models have permitted, for example in the filarial nematodes, theoretical exploration of the spread of putative resistance alleles under various assumptions of the genetics of drug resistance (how many loci would be involved, whether or not these are linked [inherited together], whether anthelmintic resistance is conferred by recessive alleles) and parasite inbreeding.

As part of the M&E strategies, models can also assist, in critical ways, in the design of treatment efficacy and effectiveness studies; phenotypic characterisation of responses to treatment [70]; and design of sampling protocols for the study of parasite genetic structure under treatment, thereby facilitating prompt detection of anthelmintic resistance [71].

With regard to schistosomes, a model including time delays, mating structure, multiple resistant strains, and additional biological complexity associated with the parasite's life cycle has been used to explore the impact of drug treatment on resistant strain survival. This model suggests that time delays make it more likely for drug-resistant strains to spread in a parasite population [72], [73]. Other models, in which resistance has a fitness cost, in terms of reduced reproduction and transmission, have been used to infer the impact of drug treatment on the maintenance of schistosome genetic diversity. The likelihood that resistant strains will increase in frequency depends on the interplay between their relative fitness, the cost of resistance, and the degree of selection pressure exerted by the drug treatments [72], [74].

Although many of the populations afflicted by helminthiases are polyparasitised or co-infected with other pathogen species, most models consider the dynamics of single-species parasite populations. The majority of models also ignore spatial structure and are confined to closed populations of hosts, parasites, and vectors. More recently, however, models for investigation of the population dynamic consequences of co-infections [79], [80], of multiple, spatially heterogeneous populations [81], and of connected (meta-)populations [82] are starting to receive attention. The further development of these frameworks will constitute important scientific advances for our understanding of the effects of interventions affecting some parasite/vector species or zoonotic reservoirs more strongly than others; the effectiveness of integrated neglected tropical disease (NTD) control; and the ability of some parasites, pathogens, intermediate hosts, or vectors to invade/occupy niches previously used by those species that are most vulnerable to particular interventions.

Epidemiological and risk mapping integrates observed, georeferenced data and predictive, remote-sensing-derived environmental variables into model-based geostatistical approaches to indicate areas with different probabilities of infection presence and severity across chosen geographical scales, aiding national control programmes to evaluate the extent of the public health problem posed by helminth infection and deploy appropriate anthelmintic strategies [83]–[88]. Readers are referred to the Global Atlas of Helminth Infections (which at present provides an open-access information resource on the distribution of STHs and schistosomiasis in Africa) at http://www.thiswormyworld.org/.

The effectiveness of integrated NTD control programmes depends on the degree of geographical overlap between such diseases. However, in spite of being co-endemic at the country level, different helminth species may in certain settings exhibit limited geographical overlap at sub-national scales, necessitating a more geographically targeted approach [89], [90]. Thus, it will be important to devise optimal strategies for rapidly and simultaneously assessing the epidemiology of multiple helminth infections so as to effectively implement integrated control approaches. In addition to mapping single infections, risk and prediction of co-infection mapping should be developed to aid integrated and cost-effective control [91], [92]. Efforts should also be devoted to linking statistical epidemiological mapping with dynamic epidemiological modelling such that the outcomes of interventions over various geographical scales can be simulated and their impact evaluated.

A major gap in helminthiases research is the development of statistical and dynamic models linking infection and morbidity. Rigorous evaluation of programmes aiming at elimination of helminthiases as a public health problem hinges on assessing the point at which infection levels have been reduced below those that no longer represent a disease burden to the individual or the population. This is an area of ongoing and much needed research. Recent progress has been made on the use of statistical modelling to ascertain the relationship between microfilarial load and blindness incidence as well as excess mortality in onchocerciasis [93], [94], and the relationship between infection and morbidity indicators in schistosomiasis [95]. In dynamic models, morbidity has been modelled as a variable depending on the density and distribution of adult worms [29], [31] or of transmission stages (eggs or larvae) [2], depending on which stages are responsible for most pathology. More research is needed to ascertain how morbidity relates to present, lagged, and/or cumulative experience of infection and co-infection, and to link dynamical models of infection and disease into the estimation of disease burden and cost-effectiveness analysis of interventions.

Cost-effectiveness analysis using parameterised dynamic infection models enables the long-term effectiveness of an intervention to be estimated [96]–[98], avoiding the limitation of so-called static economic models [97], which consider only the effectiveness of an intervention at a particular point in time (e.g., [99], [100]) or over a limited period of follow-up (e.g., [101], [102]). This permits a more comprehensive assessment of effectiveness and allows interventions that elicit different dynamics to be compared fairly. For example, ivermectin (a microfilaricide) elicits a pronounced yet transient reduction in the numbers of Onchocerca volvulus microfilariae in human skin [52], while doxycycline (a macrofilaricide) causes a gradual but sustained reduction [103]. A fair comparison of the effectiveness of these drugs must account for the markedly different durations over which they act. Linking infection models to the prevalence of disease and associated morbidities [29], [30] further improves the capacity to capture the full benefits of an intervention [97], [98], and the predictive capability of models permits a priori comparison of a range of intervention strategies under various scenarios [96], [98]. Cost-effectiveness analysis using dynamic infection models needs to be further developed and made more accessible as a decision-making tool for the planners and implementers of control initiatives.

Although it is well recognised that the transmission of helminthiases is strongly conditional on biotic and abiotic environmental factors, the latter including temperature, relative humidity, rainfall patterns, and hydrology, there are scarce data documenting the effects of these factors on life history traits of parasites, vectors, and intermediate hosts. Therefore, the impact of environment-driven changes on population dynamics and direct and indirect effects on transmission is poorly understood. This incomplete mechanistic understanding of environment–helminth disease interactions is reflected in the fact that mathematical models for such diseases have seldom included the effects of environmental processes on transmission dynamics. Recent modelling work on schistosomiasis japonica is addressing these deficiencies [104]–[106], which constitute an important research gap in the modelling of human helminthiases.

Linked to an increased awareness of the environmental determinants of infectious disease transmission, the effect of global warming on human health is an important topic that has received much interest in recent years. However, the precise effects of climate change on vector competence, duration of extrinsic incubation periods, survival of vectors, intermediate hosts, and reservoirs, and parasite transmission cycles in general remain poorly understood for the helminthiases [107].

There are two principal strategies for managing or reducing the risks of environmental change: mitigation and adaptation. The former seeks to reduce the presence and strength of anticipated risk factors (when these are known). The latter accepts that some degree of environmental change is inevitable and seeks to limit its negative impacts by encouraging and investing in preparedness. Both mitigation and adaptation will require detailed assessment of the existing distribution of the infections, their vectors, and intermediate/definitive hosts, and of the environmental determinants to which these are sensitive, including temperature, humidity, rainfall, vegetation cover, changes in the distribution and nature of water bodies, and modifications to agricultural and husbandry practices, among others [108]. The combined impact of these determinants on the transmission cycles of and rates of exposure to helminth infections of humans is poorly understood, and relevant information remains scattered in the literature, calling for systematic phenology reviews and experimental investigation. The results of these will help parameterise models with which to predict the consequences of climate change on the incidence and severity of human helminthiases. Among the few available modelling studies are those assessing the potential impact of rising temperature on the transmission of schistosomiasis [109]–[112]. Table S1 reveals a striking paucity of models for the transmission dynamics, control, and morbidity due to cestode infections, which needs to be addressed in light of climate change and its impact on agricultural and farming practices.