Introduction

The World Health Organization (WHO)’s roadmap on neglected tropical diseases [1] has earmarked onchocerciasis for elimination by 2020 in selected African countries, and the Joint Action Forum (JAF) of the WHO African Programme for Onchocerciasis Control (APOC) proposed elimination in 80% of endemic countries by 2025 [2]. As onchocerciasis programmes based on mass drug administration (MDA) of ivermectin transition from morbidity control to parasite elimination [3], the usefulness of mathematical models will rest on our ability to identify and understand processes that may make parasite populations resilient to MDA and able to persist at low prevalence [4].

Density-dependent processes acting on various parts of parasite lifecycles are well recognised as an important aspect of helminth transmission dynamics, stabilising parasite populations and contributing to their resilience—their capacity to recover—during (and after) control interventions [5,6,7,8,9]. Positive or facilitative density dependencies (e.g. the mating probability in dioecious, separate sexes species) limit transmission at low parasite population densities and create so-called transmission breakpoints [10]. Negative or constraining density-dependent processes limit transmission at high parasite population densities and, as they are relaxed during intervention, enhance transmission at low population densities [6,7,11]. It follows that density-dependent parasite establishment may have important implications for the resilience of onchocerciasis to interventions [12, 13, 14]. Since parasites are typically overdispersed among hosts (i.e. the variance is substantially greater than the mean worm load), the severity of density dependence differs among individuals and, therefore, the net population-level effect is altered by the degree of parasite overdispersion [10,15]. A description of the biological motivation for accounting for a variety of density-dependent processes in onchocerciasis transmission models can be found in [6], [9], [16] and [17]. There are two representations of density dependence in the establishment of the parasite life stage in humans. The first was introduced by Dietz [18] and was later used in the deterministic precursor of the EPIONCHO model (see [22] for a review) and is also adapted for use (for individual human hosts) in this paper. The proportion of parasites that establish in humans is considered to be a function of the annual transmission potential (ATP = L3 × ABR, the number of L3 larvae potentially received per person per year, where L3 is the mean number of L3 larvae in the fly population and ABR is the annual biting rate (the average number of bites received per person per year)). This decreasing proportion of establishing parasites with increasing ATP has implications for parasite resilience during mass drug administration (Fig 1A). The red points indicate how the proportion of establishing L3 larvae may increase after treatment as the ATP declines. This increase in the proportion of establishing parasites constrains the ability of treatment to reduce transmission because—although the number of parasites in the fly population is reduced—a higher proportion of these parasites establish. Duerr et al. (2006) [9] presented evidence for density-dependent establishment using individual-level data. The equivalent output of Dietz’s function can be reproduced using data from the 14 villages with paired nodulectomy and ATP information in the OCP database, and the model in [9] (Fig 1B). Similarly, to Fig 1A, we see that the proportion of establishing adult parasites decreases as the ATP increases.

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Fig 1. Density dependence in the establishment of adult Onchocerca volvulus.

(a) Density-dependent establishment of worms in the human host (, as described by [18] and using the parameterisation in [6]). The red points represent hypothetical values of δ_H before and after treatment. δ_H0 is the proportion of L3 larvae developing to the adult stage within the human host, per bite, when ATP → 0; δ_H∞ is the proportion of L3 larvae developing to the adult stage within the human host, per bite, when ATP(t) → ∞ and c_H is the severity of transmission intensity-dependent parasite establishment. (b) The model and data in [9] are used to produce the equivalent of Dietz’s function [18], for the mean ATP values (black points), and their upper and lower confidence intervals (blue and red respectively) as reported in [9]. The O. volvulus nodule (onchocercoma) establishment rate per village is given by where NER_i = k_i/a_i; k_i is the number of nodules palpated in individual i at age a_i; Z is the number of people across all villages entering the analysis for which nodulectomy (and paired ATP) data were available for 14 villages in the Onchocerciasis Control Programme in West Africa (OCP) database; n is the population size of the village under consideration; n_a is the number of people aged a, and z_a is the number of people aged a across all villages. The incidence of adult worms (parasite establishment rate, PER) in a village is then, , where p is the proportion of nodules accessible to palpation, and m is the average number of adult female worms in a nodule. We set m (which was multiplied by 2 to find the total number of adult worms, assuming a balanced sex ratio) and p to values proposed by [34].

https://doi.org/10.1371/journal.pntd.0007557.g001

In Onchocerca volvulus, density-dependent processes affect the proportion of microfilariae (mf, the progeny of adult worms) establishing in blackfly (Simulium spp.) vectors [16], the survival of infected blackflies [17], and the establishment of juvenile larvae as adult worms in humans [18,19,20]. A large amount of work has considered the impact of these density dependencies on the transmission dynamics of onchocerciasis using population-based deterministic models [6,21,22,23]. However, these approaches have not permitted investigation of the interactive effects of density dependence with heterogeneity in individuals’ exposure to blackfly bites.

Individuals may differ in their exposure to blackfly bites due to their attractiveness to flies [24], occupation [25,26] or age and sex [21,27]. Variation in exposure to vector bites has received theoretical and empirical attention in the context of lymphatic filariasis [28,29], schistosomiasis [30] and malaria [31,32] but is less well studied for onchocerciasis (although see [33]). In lymphatic filariasis, high levels of exposure heterogeneity are associated with increased resilience to MDA, allowing parasite persistence at low prevalence [28]. Moreover, it has been shown that identical prevalence values can be produced by different combinations of vector to host ratios (indicative of population average exposure) and levels of heterogeneity; high exposure heterogeneity reduces parasite prevalence [28] and therefore, a higher vector to host ratio is required to achieve a given prevalence. Thus, it has been suggested that prevalence alone should not determine expectations on intervention success [28]; a population with high prevalence and high exposure heterogeneity will be more resilient to treatment than a population with the same prevalence, but lower heterogeneity in exposure.

Here, we consider how the interaction between overdispersion in exposure to vector bites and density-dependent establishment of O. volvulus in human hosts affects the parasite’s population dynamics. We present a novel individual-based stochastic onchocerciasis transmission model, EPIONCHO-IBM (an analogue of the population-based deterministic EPIONCHO model (see [22] for a recent review)), which we parameterise using historical pre-intervention epidemiological data from savannah settings in Africa to capture the interaction between exposure heterogeneity and density-dependent parasite establishment within humans. We use the model to explore the impact of these interactive processes on O. volvulus population dynamics during MDA with ivermectin. We discuss how uncertainty in key parameters influences our projections and highlight data requirements for improving the accuracy of model-based predictions. In addition to individual-based models (IBMs) being suited to modelling heterogeneity in exposure (and potentially also in susceptibility) to infection, they are particularly useful in context of public health questions (for example, allowing the consideration of sensitivity and specificity to diagnostic tests; accounting for screen and treat protocols, among others). A broader aim of this work is, therefore, to develop a tool with which a wide range of questions (not within the scope of this paper) may be answered that is not possible with its deterministic precursor (EPIONCHO [22]).

Methods

There are three principal methodological components of this work: i) development of a novel individual-based onchocerciasis transmission model (EPIONCHO-IBM); ii) parameterisation of heterogeneity in exposure to blackfly bites and density-dependent establishment of worms within humans, and iii) simulations to investigate the role of these processes and their interactions on the dynamics of O. volvulus during MDA with ivermectin.

Results

Endemic microfilarial prevalence and intensity

Fig 2 shows the relationship between pre-control microfilarial prevalence, arithmetic mean microfilarial intensity and ABR, using the parameters obtained by LHS that achieved the minimum squared residuals compared with the observed data from [44, 45, 46] for each value of k_E. A value of k_E = 0.3 and the associated density dependence parameters (δ_H0 = 0.186, δ_H∞ = 0.003, c_H = 0.005) gave the lowest squared residuals.

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Fig 2. Pre-intervention Onchocerca volvulus microfilarial prevalence and intensity vs. the annual biting rate of simuliid vectors.

The predicted microfilarial prevalence (percent) and microfilarial intensity (mean no. of microfilariae, mf, per mg of skin) (from 2 skin snips) for the annual vector biting rates reported in the combined epidemiological dataset (i.e. fitting and validation data, solid colour circles), using the estimated parameters, are represented by solid lines. The best-fit parameter values of the density-dependent parasite establishment within humans were δ_H0 = 0.186, δ_H∞ = 0.003 and c_H = 0.005 for exposure heterogeneity parameter k_E = 0.3 (thick red line); δ_H0 = 0.385, δ_H∞ = 0.003, c_H = 0.008 for k_E = 0.2 (thin light blue line), and δ_H0 = 0.118, δ_H∞ = 0.002, c_H = 0.004 for k_E = 0.4 (thin dark blue line). The EPIONCHO-IBM predictions are based on a host population size of 500 and 100 runs for 80 years to reach endemic equilibrium (human demography was simulated to equilibrium before simulating the epidemiology to equilibrium). The error bars are binomial (Clopper-Pearson) 95% confidence intervals for prevalence and bootstrapped 95% CIs for intensity (for which the raw individual microfilarial intensity data were available [44]). The main vectors in each setting are Simulium damnosum s.s./S. sirbanum (African savannah in Benin, Burkina Faso, Cameroon, Côte d’Ivoire, Ghana, Guinea, Mali, Togo); S. guianense s.l. (Amazonian focus in southern Venezuela) and S. exiguum s.l. (Cayapas focus in Ecuador), all species are without armed cibaria and have vector competence features similar to those of S. damnosum s.s. [7]. Fitting data are from [44, 45, 46]; validation data are from [9, 48, 49, 50]. The prevalence and intensity data points for an annual biting rate of 81,000 were excluded from the fitting of the model as according to [45] a large number of flies were nulliparous.

https://doi.org/10.1371/journal.pntd.0007557.g002

When using k_E = 0.3, EPIONCHO-IBM captures well not only the pre-intervention observed microfilarial prevalence and intensity across the biting rates used for comparing the model fit to data, but also the other data sources used for validation (i.e. excluded from the least squares calculation) [9, 48, 49, 50]. A value of k_E = 0.2 predicted well the microfilarial intensity but tended to under predict microfilarial prevalence; conversely, a value of k_E = 0.4 underpredicted intensity but was better able to capture microfilarial prevalence across all ABRs than k_E = 0.2. Fig 2 shows that model predictions generated by values of k_E = 0.2, 0.3 and 0.4 encompass most of the prevalence and intensity data points in the range of ABR values explored (1,000 to 176,500). Figures showing microfilarial prevalence and intensity for all parameter sets from the Latin hypercube sample (for k_E = 0.3), as well as the influence of individual density dependence parameters on (endemic equilibrium) model outputs for a range of ABR values, are shown in S1 Text, Text B (Fig E and Fig F, respectively).

Threshold biting rate

Increasing the level of exposure heterogeneity (i.e. decreasing the value of k_E) decreases the threshold biting rate (TBR, the minimal ABR necessary for endemic onchocerciasis, i.e. for which the basic reproduction number of the parasite, R₀, is equal to 1 [16]), giving TBR values ranging from approximately 97 for k_E = 0.2 to 429 for k_E = 0.4, assuming that the proportion of bloodmeals taken on humans is 0.63 for S. damnosum s.s./S. sirbanum [50] (Fig 3).

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Fig 3. Threshold biting rates for each endemic microfilarial prevalence varying the exposure heterogeneity parameter k_E and associated best-fit parameters of the density-dependent parasite establishment within humans.

Endemic prevalence values of 0 indicate annual biting rates (number of bites per person per year) that are below the so-called threshold biting rate (TBR) for transmission (indicating that the basic reproduction number R₀ <1). The EPIONCHO-IBM predictions are based on a host population size of 500, 1000 runs and for 80 years to endemic equilibrium, assuming 63% of blood meals are taken on humans [55]. Density dependence parameter values for each value of k_E are as in Fig 2 TBR values are 97, 239 and 429 for k_E = 0.2, 0.3 and 0.4, respectively.

https://doi.org/10.1371/journal.pntd.0007557.g003

Population and individual-level parasite establishment within humans

We see also that the population mean density dependence in the stochastic model developed here departs from that found in the simpler deterministic framework without exposure heterogeneity (when using the estimated density dependence parameters from the stochastic model) (Fig 4A). In the stochastic setting, density-dependent establishment of the parasite in an individual depends on their exposure to fly bites (Eq 2), which is determined by a gamma-distributed individual level exposure, as well as age- and sex-dependent exposure. This creates a distribution of Π_H curves in the host population (Fig 4B), which can vary depending on k_E (as well as on the host sex ratio and age distribution). Since (for the k_E values tested), most individuals in a population have low exposure (and therefore weak density-dependent establishment relative to the annual transmission potential), there is a higher mean proportion of establishing parasites in the human population for higher annual transmission potentials than in the deterministic model.

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Fig 4.

The EPIONCHO-IBM density-dependent parasite establishment within-humans using the best-fit estimates of the parameters δ_H0, δ_H∞ and c_h for each value of k_E. In panel (a), solid lines show the population mean density dependence with age-, sex- and individual-specific heterogeneity in exposure to vector bites. Dashed lines show the density-dependent establishment function without heterogeneity in exposure using the same parameters. Panel (b) shows the density dependence for 100 individuals in the stochastic setting (each line is for one individual) for each value of k_E. The mean of these lines (for each exposure heterogeneity value) gives the solids lines in panel (a). Density dependence parameter values for each value of k_E are as in Fig 2. Note that the y-axis label has been abbreviated for presentational purposes; specifically for panel (b) this is П_H(i), in panel (a), we represent the value given by taking the mean output of this function for all individuals in the population with П_H (solid lines), and represent the function without exposure heterogeneity using П_H also (dashed lines).

https://doi.org/10.1371/journal.pntd.0007557.g004

Dynamics of microfilarial prevalence under ivermectin treatment

Treatment dynamics indicate that resilience to MDA is markedly higher for k_E = 0.2 than for k_E = 0.3 and 0.4. This difference between the different levels of exposure heterogeneity tended to increase as the pre-intervention microfilarial prevalence increased but decreased with an increase in treatment frequency (Fig 5). Although this difference in resilience between the different values of k_E is in part due to the resulting exposure heterogeneity, it is also associated with different estimated strengths of density dependence (Fig 4). More heterogenous exposure leads to the estimation of stronger density dependence, principally due to a higher proportion of parasites establishing as ATP decreases. Since increasing exposure heterogeneity (reducing k_E) reduces the microfilarial prevalence, a higher proportion of parasites establishing (than when exposure is less overdispersed) is required to reach a given observed prevalence and intensity. Simulations assuming either a fixed level of exposure heterogeneity, and different strengths of density dependence, or a fixed level of density dependence and different levels of exposure heterogeneity are shown in S1 Text, Text B: Uncertainty and sensitivity analysis using the Latin hypercube parameter sets, Figs G and H.

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Fig 5.

Microfilarial prevalence dynamics during 25 years of annual (a–d) and biannual (e–h) mass drug administration (MDA) with ivermectin for various levels of endemicity, heterogeneity in exposure to vector bites (measured by k_E) and the associated density-dependent establishment parameters. The baseline microfilarial prevalence (30%, 50%, 60%, 70%, indicative of hypo-, meso-, hyper-, and high hyperendemicity, respectively) is modelled by increasing the annual biting rate. At each treatment round, 80% of the population receive treatment, excluding children aged <5 years who are ineligible to receive ivermectin. We assume 1% of the total population are systematic non-adherers (i.e. they never take treatment during the programme duration). The EPIONCHO-IBM predictions are based on a host population size of 500, 1000 runs and with 80 years to reach endemic equilibrium before initiating MDA. Parameters for the thick red lines (k_E = 0.3), thin light blue lines (k_E = 0.2) and thin dark blue lines (k_E = 0.4) are as in Fig 2.

https://doi.org/10.1371/journal.pntd.0007557.g005

Discussion

We have developed a novel stochastic individual-based onchocerciasis transmission model, EPIONCHO-IBM, based on the deterministic population-based analogue, EPIONCHO. EPIONCHO-IBM captures explicitly the interactive effects of heterogeneity among humans in exposure to blackfly vector bites and density-dependent processes that operate fundamentally at the individual host level and that are key determinants of the resilience of parasite populations to intervention. Modelling this interaction is, therefore, essential for predicting the likely impact of interventions on parasite population dynamics and ultimately the feasibility of elimination. This is the first study to explore this effect in the context of onchocerciasis using epidemiological data to parameterise and validate these interactive population processes. In doing so, we demonstrate that data collection on exposure heterogeneity may make an important contribution to reducing uncertainty when modelling parasite dynamics during MDA.

The range of exposure heterogeneity levels explored (k_E = 0.2 to 0.4) fits to the variation in pre-control microfilarial prevalence and intensity with ABR, encompassing most of the data points collated (including fitting and validation datasets), with k_E = 0.3 providing the best overall fit. A value of k_E = 0.2 tended to underestimate prevalence but fitted better low prevalence values, whilst k_E = 0.4 tended to overestimate prevalence but fitted better high prevalence. The results suggest that exposure heterogeneity may decrease as the annual biting rate increases (i.e. k_E increases with increasing ABR).

This decrease in heterogeneity with increasing ABR may represent that, in a small population of blackflies, the physical limit on how many times a fly can bite constrains the number of people that may be bitten. This would result in a high heterogeneity in exposure to bites. Conversely, a large population of flies is capable of biting many people, leading to lower levels of heterogeneity in exposure to bites. If such a relationship between exposure heterogeneity and ABR were supported by other data sources and incorporated into the model, it may have substantive implications for the predicted population dynamics during MDA. In particular, the degree of heterogeneity in exposure would be less in highly endemic settings with high prevalence and high ABRs. Consequently, the resilience to intervention in these settings would be reduced and the impact of MDA enhanced. The projected parasite population dynamics during treatment with ivermectin would likely be somewhat more closely aligned with the corresponding deterministic predictions of EPIONCHO, which incorporates a functional relationship between the degree of adult worm overdispersion and ABR [23]. The assumption of a fixed level of exposure heterogeneity (either across the wide range of annual biting rates collated, as in Fig 2, or for single modelled endemic communities, as in Fig 5) is unavoidable due to data availability. More likely, there will be a distribution of k_E values depending on the specific ecological and sociological context of a geographical area. Even if such a distribution allowed k_E to vary between 0.2 and 0.4, we might expect substantial spatial variation in population responses to treatment for a given baseline microfilarial prevalence.

The threshold biting rate decreased with increasing levels of exposure heterogeneity which is consistent with similar work on lymphatic filariasis [28]. Furthermore, the threshold biting rate for the best fit parameters was similar to previous deterministic versions of the model assuming the same proportion of blood meals taken on humans (380 in [6], 300 in [19]), but lower than a different model based on the same data (730 [20]).

A key distinction between EPIONCHO-IBM and its deterministic counterpart is that in the former, overdispersion is generated mechanistically via heterogeneity in exposure and density dependence operating at the level of the individual host. In EPIONCHO, the effects of parasite overdispersion on the severity of density dependence are modelled empirically [15, 21, 22, 23]. More generally, worm overdispersion in the host population is an input in deterministic population-based models, and an output of individual-based stochastic models (calculated by fitting a negative binomial distribution to predicted worm burdens), depending on exposure heterogeneity and potentially density dependent processes. This is a crucial difference and implies individual-based models are not simply approximations of their deterministic counterparts. How density-dependent processes influence the relationship between exposure heterogeneity and parasite overdispersion (of adult worms and microfilariae) in EPIONCHO-IBM is an important question which remains to be addressed and will be presented elsewhere.

The phenomenological nature of density-dependent establishment of the parasite in the human host is an important limitation of this work. The use of population-level data to simultaneously estimate both density dependence and exposure heterogeneity allows the parameters involved to counteract, reducing identifiability and diminishing mechanistic interpretation. Since direct estimation of density-dependent parasite establishment within humans is not feasible experimentally (with the only available observational study on parasite establishment rates being that of [9]), data collection on heterogeneity in exposure to fly bites will be an important step in better resolving density dependence. This has been discussed previously [26] but available data remain limited. Since blackflies have very specific environmental requirements, with breeding sites varying in distance to human settlements, the estimation of individual-level variation in exposure to bites poses a substantial challenge.

Heterogeneity in exposure to mosquito bites has been estimated by comparing DNA extracted from (indoor resting) mosquito bloodmeals with DNA from individuals living in study households [32]. Since Anopheles mosquitoes breed in standing water, which may be littered throughout villages, this is a viable method for finding mosquitoes which have fed on study participants. In the context of onchocerciasis, this process may be impractical because blackflies do not bite and rest indoors, and outdoor resting sites are very difficult to find, so studies investigating bloodmeal origin rely on host-dependent and host-independent sampling methods [55]. The former (human landing catches) would require a large number of participants to find enough flies which that could be matched to the individuals on which they previously fed, and the latter (using oviposition traps in nearby breeding sites) may not sample sufficient flies which would have fed on the residents of the nearby communities [55]. Irvine et al. [29] have suggested there may be spatial variation in exposure heterogeneity to mosquito bites, adding an additional layer of complexity and uncertainty in obtaining generalizable estimates. Perhaps more suitable for estimating exposure to blackfly bites is the method which has recently been applied in an empirical study of Leishmania infantum transmission among dogs in Brazil [56]. Levels of (IgG) antibody responses to sandfly saliva were found to vary with the intensity of transmission, declining after periods of low transmission, as well as to vary greatly between dogs, correlating with the intensity of transmission experienced by individual dogs. Studies such as these require the development of anti-saliva antibody assays, which are currently lacking in simuliids (partly because their colonisation in the lab is very difficult). If these assays could be developed in the context of onchocerciasis, based initially on titres of antibody responses to crude blackfly salivary antigens, they could be used in population samples across all age and sex groups and may provide useful data for estimating heterogeneity in individual-level exposure. Such a study may also provide data to increase the accuracy of assumptions regarding age- and sex-dependent exposure. Currently this is estimated by fitting models to data on age- and sex-specific profiles of microfilarial prevalence and intensity [21]. However, these modelled age and sex exposure profiles are determined not by exposure to vectors and parasite infective stages alone, but also by processes involving parasite development and parasite fecundity within the host, as they rely on (downstream) microfilarial data. Reducing the uncertainty in age- and sex-dependent exposure patterns is particularly important for optimal selection of age (and potentially sex) groups for (serological) assessment of exposure to parasite antigens in foci thought to be nearing elimination [57]. Ideally, assays for anti-blackfly saliva could be combined with assays for exposure to O. volvulus to investigate both exposure to vector bites and parasite antigens (e.g. Ov16 [58]).

In addition to exposure, susceptibility to infection may also vary between individual hosts. Work on schistosomiasis has shown that there is marked variability in susceptibility to infection [59, 60], and that accounting for both exposure and susceptibility can better capture Schistosoma transmission to snails than accounting for exposure alone [61]. It is possible, that what we term exposure heterogeneity, does also account for individual variation in susceptibility, given the way it is estimated. However, as there is a lack of data on individual heterogeneity in exposure and susceptibility to infection, making this distinction explicit is unlikely to lead to better parameterisation or improved accuracy of the model. In addition, since density-dependent establishment of adult worms might represent immunological processes (which interact with exposure at the individual level in the model), it is not necessarily clear how variation in susceptibility (which is likely determined by genetic and immunological processes) should be accounted for. Therefore, although we do not account for variation in susceptibility explicitly, the proportion of incoming parasites which establish in an individual depends on their exposure, implying a relationship between exposure and susceptibility.

The density-dependent within-human parasite establishment function used in EPIONCHO and EPIONCHO-IBM, derived from the work in [18], is a phenomenological representation of a possible immune-mediated response to infective (L3) larvae driven by the intensity of exposure to these parasite stages. In areas of low ATP (or in individuals with little exposure), low levels of exposure to L3 larvae may be responsible for a poor development of protective immune responses, leading to high parasite establishment rates. Conversely, in areas of high ATP (or in individuals highly exposed), stronger immune responses against incoming worms may decrease parasite establishment rates. The operation of anti-L3 responses in putative immunes living in areas of hyperendemic onchocerciasis has been shown by [62, 63]. In the Teladorsagia circumcincta–Scottish Blackface sheep parasite–host system, priming of the immune system early in the season by exposure to (ingestion of) infective L3 larvae in pasture, reduces parasite establishment and growth and, therefore, faecal egg counts in a density-dependent fashion, not unlike our own parasite establishment function (see Figs 2 and 3 of [64]).

Density-dependent processes may also (or instead) act on parasite fecundity [65]. Here we assume for simplicity, that one worm of each sex is required for reproduction, i.e. that one male worm is sufficient for all adult female worms to be fertilised and produce mf [10, 34], and that increasing female worm density does not reduce the per capita fecundity rate [19]. However, if in addition to density-dependent parasite establishment, the fecundity rate was to decrease as the density of adult parasites increases, we might expect additional resilience during treatment, since the fecundity rate per worm would increase as MDA reduces the number of parasites in the population. Of interest, previous work indicates that, when investigated separately, density-dependent fecundity contributes less to the rate of bounce back following treatment than density-dependent adult worm establishment [8].

The EPIONCHO-IBM projections indicate that there is a disproportionate increase in resilience with increasing pre-intervention prevalence for k_E = 0.2. That is, as pre-intervention prevalence increases, particularly from low hyperendemic (microfilarial prevalence ≈ 60%) to high hyperendemic (microfilarial prevalence ≈ 70%) settings, the O. volvulus population becomes disproportionally more resilient to MDA. This can be partly understood in the context of the strongly nonlinear prevalence vs. ABR relationship, in which prevalence begins to saturate at ABR values exceeding 7,000 (daily biting rates > 20/person, ~70% mf prevalence). This phenomenon would also be found for lower levels of exposure heterogeneity, as the prevalence vs. ABR relationship begins to saturate, albeit at higher ABRs. Another contributory factor to resilience to MDA-based interventions in our model is that our parasite establishment function is not affected by the potential interaction between microfilaricidal treatment and the immune response. Therefore, as treatment progresses and the ATP declines, the parasite establishment rate inevitably increases. However, ivermectin-facilitated immunity in onchocerciasis has been reported [66, 67]. Human immunological studies have demonstrated that filarial parasites induce a state of hypo-responsiveness in the host that is associated with the presence of circulating mf (patent infection), and O. volvulus is no exception [68]. The reversal of this mf-associated immune-suppression, following clearance of skin mf due to ivermectin treatment, may contribute to controlling O. volvulus infection [66]. In the O. ochengi–cattle system, animals treated with ivermectin and exposed to blackfly bites under natural transmission conditions did not develop patent infections whilst treated but recovered their susceptibility and acquired infection (at rates higher than untreated counterparts) once treatment stopped [69]. Similar observations have been made in a large pharmaco-epidemiological study of Dirofilaria immitis in dogs [70]. A better understanding of how ivermectin-facilitated immunity could impact the establishment of incoming worms in human onchocerciasis would greatly improve the modelling of parasite establishment rates during the implementation and after cessation of MDA programmes.

The microsimulation model for onchocerciasis ONCHOSIM [22, 33, 41, 71], which also uses a gamma distribution to model individual-level variation in exposure, uses parameter k_E as 1 or 3.5, giving substantially less exposure heterogeneity than in EPIONCHO-IBM (k_E = 0.3). It has been shown that EPIONCHO (the deterministic analogue of EPIONCHO-IBM) predicts more resilience to intervention than ONCHOSIM under a range of treatment scenarios (using k_E = 3.5 for ONCHOSIM) [13, 14]. Although our treatment dynamics were simulated under high (and likely unrealistic) levels of therapeutic coverage and adherence, the programme duration (25 years) was motivated by that in the Vina valley of northern Cameroon (from which the data used to estimate the parameters investigated in this paper originated [44, 45]). In Cameroon, there was no interruption of transmission after 15 [72, 73], 17 [74], 18 [75] or 25 years [76] of annual ivermectin MDA in some communities. Although this may be due to a number of factors including lower than reported coverage, decreased ivermectin efficacy [77], movement of infected individuals between foci and spatial variation in exposure heterogeneity, it may indicate that the more pessimistic predictions of EPIONCHO-IBM for hyperendemic populations, are not necessarily out of touch with observed trends. It is noteworthy that EPIONCHO-IBM can mimic the behaviour of ONCHOSIM (i.e. less pessimistic treatment dynamics) by removing density-dependent parasite establishment within humans (absent in ONCHOSIM) and increasing k_E. This process—and its interaction with heterogeneity in exposure—is a critical determinant of the different population dynamics predicted by the two models (to be formally discussed elsewhere).

EPIONCHO-IBM has a wider scope of application than its population-based predecessor, albeit at the cost of reduced tractability (a general drawback of many individual-based stochastic models). A particular advantage is the capacity to model individual-based interventions. For example, individuals co-infected with O. volvulus and Loa Loa are at risk of severe adverse events (e.g. encephalitis) following treatment with ivermectin if harbouring heavy loiasis microfilaraemia [78]. Consequently “test and treat” (testing for O. volvulus) or “test and not treat” (testing for L. loa) strategies have been proposed and trialled for control in areas co-endemic with onchocerciasis and loiasis [79]. EPIONCHO-IBM, as an individual-based model, can simulate such strategies, incorporating performance features (i.e. sensitivity, specificity) of a variety of diagnostics and detailed information on observed screening, coverage and adherence patterns among demographic groups. It follows that co-infection dynamics may also be modelled, although the within-host interaction between O. volvulus and L. loa and helminth species more generally is poorly understood (but see [80]).

Conclusions

In conclusion, we have developed a novel individual-based stochastic onchocerciasis transmission model, EPIONCHO-IBM, based on the well-established deterministic analogue, EPIONCHO. We have used EPIONCHO-IBM to better understand how density-dependent processes—in particular the density-dependent establishment of newly acquired O. volvulus parasites—and heterogeneity in individual human exposure shape both the relationships between microfilarial prevalence, microfilarial intensity and ABR, and the resilience of onchocerciasis to MDA with ivermectin. In future, EPIONCHO-IBM will be used to model the control and elimination of onchocerciasis using current and alternative interventions, including the use of anti-Wolbachia therapies [81], moxidectin [82], ground-based vector control [83] and new macrofilaricidal therapies [84]. Our work also highlights the importance and uncertainty in the key and interactive population processes of density dependence and heterogeneity in exposure to blackfly vectors. Novel approaches for generating data on exposure heterogeneity and anti-L3 immunity during and after the cessation of ivermectin MDA programmes would be particularly valuable in helping to resolve outstanding uncertainty on their relative importance to the population dynamics of O. volvulus. Combined with more statistically advanced approaches for estimating the parameters of individual-based models [85], and fitting such models to longitudinal epidemiological trends, such data would greatly enhance the reliability and accuracy of onchocerciasis modelling projections.

Supporting information

Acknowledgments

We are grateful to Alfonso Renz for providing data, and Gaby Harrow for data entry. We thank Simon O’Hanlon, Wilma Stolk and Sake de Vlas for helpful discussions during model development.