Introduction

Lymphatic filariasis (LF), a highly debilitating vector-borne macroparasitic disease of humans, has been targeted by the World Health Organization (WHO) for global eradication [1]–[5]. This has led to the rapid development and financing of large scale national-level intervention programs based primarily on annual mass drug administrations (MDA), which has led to impressive reductions in LF infection prevalences in endemic populations globally [6]. However, as infection levels have fallen in endemic communities, questions have been raised regarding the need for improved understanding of the dynamical processes that underlie infection persistence and extinction dynamics, and hence the controllability or eradicability of the disease [7]–[11]. A practical question in this regard revolves around the nature of LF extinction dynamics, including the numerical values of LF transmission/infection endpoints in endemic communities, the resolution of which is key to reliably determining when parasite transmission has been interrupted and interventions can therefore be stopped [12].

Our previous work has highlighted that both inherent complexity in transmission dynamics and parameter uncertainty can confound the prediction of extinction endpoints for complicated parasitic systems, such as LF [7], [9], [11]–[13]. In particular, this work has showed that complex ecological dynamics due to high sensitivity to initial conditions and other locally varying climatic and geographic factors mean that parasite transmission dynamics is likely to be highly variable in space and time [12], with the result that LF infection endpoints or breakpoints (e.g. threshold values of the microfilaria (mf) prevalence in humans below which infection cannot sustain itself) may vary considerably from site to site. Although it was shown that between-site heterogeneities in a few biological and socio-ecological parameters controlling the intensity and distribution of worm burdens in the human host population may underlie this variability, the results also demonstrated that a significant source of the variability reflects uncertainties surrounding parameter values of key transmission variables [12]. While a part of this uncertainty reflects epistemic uncertainty regarding transmission heterogeneity, it is clear that a major portion is also due to the limited quantity and quality of site-specific data used to develop/run existing models [12], [14], [15]. This implies that reducing parameter uncertainty will require the need for data-model integration methods that can assimilate information and update initial parameter values as new information regarding LF transmission dynamics becomes available either from different sites or over time from individual sites [7], [9], [11]–[13], [15]–[18].

In the context of the current initiative to eradicate LF, effective model updating using longitudinal infection data from sites undergoing interventions serves another crucial function, viz. providing an important means to investigate possible temporal patterns in the data. A particular focus in this regard of direct import to managing LF eradication is the determination of whether the updating of initial model fits (to baseline infection data) using follow-up data after the start of interventions will result in significantly altered estimates, particularly in the critical biological parameters underlying parasite transmission. If this occurs to the extent of bringing about dynamical behaviour changes, then it may raise the real possibility of the emergence of temporally varying endpoints as parasite interventions proceed. The outcome of such a qualitative change in overall system dynamics will, by affecting attractors, basins of attraction and stability, be that LF endpoints could become dynamic ‘moving’ targets. The implication of such changes for LF elimination may be the prolonging of interventions in some settings, while in others we may need to develop and implement other adaptive control strategies, including supplementation by vector control or with more frequent mass treatments. If on the other hand, model updates show that values of critical biological parameters underpinning transmission do not vary appreciably or are robust in response to the effect of interventions, then it might be possible to derive estimates of transmission endpoints from post-intervention monitoring data. This will be a significant step forward in the management of elimination efforts against LF since it would allow the estimation of target endpoints for interventions even in the absence of baseline infection data for those sites which have only post-intervention monitoring data. Such results would also facilitate uncovering of the environmental or other conditions that constrain or bound the transmission and extinction dynamics of LF to an endemic setting [19], [20]. This will in turn enable insights into countermeasures that can disrupt the local robustness in transmission, increase system fragility and more reliably push the locally adapted transmission system into extinction [19], [21]–[23].

Sequential calibration/updating of LF transmission models using post-intervention follow up data on infection levels could also produce additional information relevant to predicting the impacts of interventions. For example, such model updates will be important for a more direct estimation and evaluation of drug related parameters, such as worm and mf killing efficacies, reduction in worm fecundity, and drug lasting effects that impact the duration of suppression in mf production [10], [24]–[26]. Present estimates of these effects are still largely based on best guestimates [7]; better attempts to quantify such parameters using field data is thus a distinct need and will be critical to improving control predictions and hence intervention planning in endemic localities.

In a similar vein, the sequential fitting of models to follow-up data could additionally offer the potential for allowing the backward estimation of pre-control baseline infection patterns. Reconstructions of historical baseline infection patterns (particularly in terms of age-profiles of mf prevalence levels) will be important because many LF intervention monitoring sites currently do not have such data even though this information is clearly essential for predicting trajectories in infection declines from baseline levels as a result of applied interventions [27], [28], which will be essential for evaluating if interventions are progressing as expected [27], [29].

Here, we report on our efforts to undertake a first systematic examination of each of the above issues in the context of a detailed field trial to evaluate the impact of MDA for reducing LF infection in endemic communities from the Dreikikir region of Papua New Guinea [30]. We begin by first describing the extension and use of a recently developed numerical modelling and Bayesian Melding data-model integration tool to sequentially fit the Anopheles-mediated LF transmission model [11] to baseline and follow-up human mf age-prevalence data recorded from each of our Dreikikir study communities. We then use the modelling results to address the following set of specific questions: 1) do biological parameters controlling LF transmission remain stable (with regards to baseline estimates) in the face of the specific interventions implemented in each community; 2) is it possible to use the fits to data from each intervention period to reliably perform backward extrapolation to reconstruct baseline age-infection patterns; 3) is it possible to use the post-intervention infection monitoring data to estimate LF transmission/infection breakpoints consistent with those estimated using baseline only data; and 4) can the modelling and fitting framework developed in this study allow better estimates of drug treatment-related parameters, including determination if such parameters may also vary in their values between treatment populations? We end by discussing the significance of the findings regarding the ecological factors that may underlie LF transmission and extinction dynamics in local settings, the importance of good quality intervention monitoring data, including drug coverage information, and the value of the applying the Bayesian model-data assimilation method used here, for guiding the WHO-directed program to achieve the eradication of LF globally.

Materials and Methods

Data

The data used in this analysis represent baseline and annual follow-up monitoring data on changes in lymphatic filariasis infection prevalences obtained from five communities (Peneng, Albulum, Yauatong, Nanaha and Ngahmbule) in the Dreikikir region of Papua New Guinea (PNG) [30]–[34]. These data were collected as part of an open-label field study to compare the population impacts of a single annual mass administration of diethylcarbamazine plus ivermectin (DEC+IVR) with that of a single annual dose of DEC alone for reducing LF infection transmitted by anopheline mosquitoes. The data on individual host mf status for the baseline period was collected in 1994 before the start of mass treatment intervention and thereafter at the end of one-year intervals for a period of 5 years [34]. Three of the five villages (Peneng, Albulum, and Yauatong) were classified as high transmission communities and the remaining as exposed to a moderate rate of transmission based on annual mosquito biting rates [30]–[32]. Annual population sizes, totals of individuals sampled, numbers of mf-positives out of these samples, yearly drug coverages and the drug regimen for each of these villages, are provided in Table 1. Infection status of individuals were assessed using 1 ml of blood, whereas prevailing vector abundances (in terms of annual biting rates per person) were determined using man landing catches in all communities [30], [31]. Note that as per WHO guidelines at the time of the field trials, individuals under five years of age were excluded from treatment with drugs during the MDA period in any of these villages [30].

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Table 1. Annual survey data for lymphatic filariasis (LF) mf prevalence for each of the five PNG study villages used in this work.

https://doi.org/10.1371/journal.pone.0067004.t001

Methods

Deterministic model of anopheline-mediated LF transmission.

We employed the recently developed Anopheles-vectored transmission model of LF to carry out the modelling work in this study [7], [9], [11], [12]. Briefly, the state variables of this hybrid coupled partial differential and differential equation model vary over age (a) and/or time (t), representing changes in the adult worm burden per human host the mf level in the human host modified to reflect infection detection in a 1 ml blood sample the average number of infective L3 larval stages per mosquito (L), and a measure of immunity developed by human hosts against L3 larvae. The state equations comprising this model are:

The above equations involve partial derivatives of three state variables (W - worm load; M - microfilaria intensity; I - immunity to acquiring new infection due to the pre-existing worm load), whereas given the faster time scale of infection dynamics in the vector compared to the human host, the infective L3-stage larval density in mosquito population is modeled by an ordinary differential equation. This basic coupled immigration-death structure of the model, as well as our recent extensions to reflect transmission by anopheline mosquitoes in endemic areas, such as in the Dreikikir region in PNG, have been extensively discussed previously [7], [11], [12], [35], [36]. The parameters and functions of the model and L^* are described in Table 2.

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Table 2. Description the basic LF model parameters and functions used in the model.

https://doi.org/10.1371/journal.pone.0067004.t002

Note that the biological parameters of the model are classified into site-specific and site-invariant parameters, based on results from our previous work which suggest that it is useful to distinguish between biological parameters specific to the disease, and which therefore can be assumed to apply to parasite and vector populations that range over extended geographical areas, and those which vary between localities [12], [49].

Results

Model Fits to Annual Intervention Data

The age-profiles of mf-prevalence generated by the Anopheline LF model fits (grey curves) to the baseline and five annual post-intervention data from two of the high transmission study villages given either the DEC+IVR regimen (Peneng) or DEC alone (Yauatong) are shown in Figure 1. The corresponding model fits to the baseline and intervention data from the remaining three PNG villages are shown in Figure S1 in Supporting Information S1. Figure 2 shows the overall mf prevalences predicted by the model in comparison with the longitudinally observed baseline and post-intervention overall community mf prevalences recorded for each of the five study villages. Together, these results show clearly that the BM-based model-data assimilation method developed in this study is capable of reproducing the temporal changes in overall, and age-stratified, prevalences in mf, arising from the implemented mass drug treatments consistent with observed data in each of the study communities (Monte Carlo p-values >0.05 in each case (Table S1 in Supporting Information S1)), although as expected the fits to mf-age-prevalences are comparatively better for the study villages with the lowest variability in this infection measure, viz. Nanaha and Ngahmbule (Figure S1 in Supporting Information S1) owing primarily to their bigger sample sizes (Table 1).

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Figure 1. Predicted age-profiles of mf-prevalence from model fits to observed baseline and longitudinal post-intervention infection data.

Model outputs and observed data are shown for two of the PNG study villages exposed to high LF transmission intensity, and for both drug regimens used: Peneng (DEC+IVR) and Yauatong (DEC alone). Individual 500 best-fit model simulations are shown in grey while the thick blue line represents the median value of these curves. The observed data points (crosses) with 95% binomial credible intervals are shown at the mid-points of each 10-yearly age-group. The results for the remaining three study villages are exactly similar (Figure S1 in Supporting Information).

https://doi.org/10.1371/journal.pone.0067004.g001

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Figure 2. Comparison of the trends in overall community-level mf prevalence predicted by the model with the observed baseline and post-intervention infection data collected longitudinally during the mass treatment programme.

Model outputs and observed infection data are shown for all the five PNG study villages. The observed overall community prevalence collected at a yearly interval from each study site are depicted by empty red squares together with their corresponding 95% binomial credible intervals (bars). The baseline data (at the 0^th year in 1993–94) were collected soon before the start of the first mass drug administration round, and the five post-intervention mf prevalence data were collected annually a year after each mass drug treatment from the first through the 5^th MDA round (see Table 1). Note that there are 500 modelled data points (shown by empty cyan circles) from the 500 best fits.

https://doi.org/10.1371/journal.pone.0067004.g002

Stability of Model Parameters

As in our previous work [12], the results show that the Bayesian updating procedure used here when applied to baseline age-mf prevalence data can effectively refine initially assigned model parameter values. As indicated in Table S2 in Supporting Information S1, almost all the values initially assigned to the present model parameters were updated by the procedure, although as also noted previously [12], an important finding is that this was dependent on study sample size, with increasing changes in the posterior distributions occurring for those study communities (e.g. Nanaha, Ngahmbule) with the larger sample sizes. Across all study sites, however, only six parameters, namely, α, k_Lin, ψ₂s₂, c, I_C and S_C, which represent the fecundity rate per worm, the linear rate of increase in the aggregation parameter, the parasite establishment rate, the strength of acquired immunity to L3 larvae, the strength of immunosuppression and the slope of immunosuppression function, respectively, were consistently found to have their posterior distributions significantly altered from their assigned non-informative uniform prior distributions when the model was fitted to the baseline data (Figure 3). We next simulated the effects of each MDA regime using actually observed community drug coverage rates, and used the fits of these simulations to data from each study village to assess if the values of any of these biological/ecological parameters remained temporally invariant. The results demonstrated that these six parameters remained remarkably fixed throughout the intervention period in all study sites (Figure 4), with their posterior distributions retaining the same individual baseline distributional shapes in each case. A more formal test of how stable or robust to perturbations all the model parameters, including these six model parameters, were during the 5-year post-treatment period was carried out using the univariate KS test [58]. This indicated that almost all the biological parameters of the model remained at their initial baseline posterior distributions during the entire 5-year study period when infection in each of the five LF communities were going through their annual intervention-induced perturbations (Table S2 in Supporting Information S1).

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Figure 3. Posterior distributions of model parameters.

The results are shown for 6 model parameters, α, k_Lin, ψ₂s₂, c, I_C and S_C, whose posteriors changed significantly and consistently from their initially assigned uniform priors when the LF model was fitted to the baseline data in each study village. The results in the figure illustrate the estimated posterior parameter shifts for the village of Peneng. Dashed lines show the flat non-informative priors, while the bars denote the frequency distributions of the posterior values for each parameter. The parameter values along the x-axis are normalized for comparative purposes. The figures in square brackets at the top of each plot depict the actual range of the posterior values estimated for each parameter.

https://doi.org/10.1371/journal.pone.0067004.g003

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Figure 4. Stability in model parameters during intervention period.

Comparison of changes in the posteriors of the parameters α, k_Lin, ψ₂s₂, c, I_C and S_C obtained via sequential model fitting to annual infection data recorded over the 5-year intervention period in the high transmission villages of Peneng and Yauatong. Solid coloured lines denote of the relative frequencies of the posterior distributions of each of the six model parameters. The x-axis shows the scaled values of the posterior distributions for each parameter. The dashed line shows the initially assigned flat non-informative priors.

https://doi.org/10.1371/journal.pone.0067004.g004

Estimation of the Drug Efficacy Related Parameters

As noted above, the LF drug efficacy related parameters are (1) instantaneous worm killing efficacy: this is major parameter with values assigned previously based on either on expert advice [7], or limitedly estimated from the field data using a mechanistic transmission model (see [26]); (2) microfilariae killing efficacy: better information available from published community trials where values from 69.8% to 91.1% have been demonstrated previously [30], [60]; (3) suppressed production of mf by worms surviving MDA, and (4) the waning period: this is the short time-period that follows the commencement of mass drug administration during which the drug is assumed to remain effective, which is expected to range from between 3 to 9 months [7]. The summary statistics of their estimated posterior parameter distributions in this study indicate gains in knowledge for these parameters from the sequential updating of the LF model, with all parameters in general showing shifts from their assigned prior distributions as interventions progressed (Figure 5, Figures S2, S3, S4 and Table S3 in Supporting Information S1). The biggest and most informative posterior distribution shift, however, occurred for the worm killing efficacy parameter, with efficacy rates increasing to apparently settle around a peak mean rate of 74 to 77% (Figure 5). Mean efficacy rates also increased with sequential model updates for the microfilariae killing and waning period parameters to approximately 80% and 4 months respectively by year 5 post-intervention, although these increases were not found to be statistically significant. The least change occurred for the worm fecundity reduction parameter. Interestingly, the addition of IVR to DEC did not appear to significantly change the model estimated drug efficacy parameter distributions indicating that the major impact on reducing LF infection using the combination therapy is due to the effects of DEC. By contrast, an intriguing finding was that the mean worm killing rate became marginally higher with increasing cycles of intervention in the villages exposed to the low (Nanaha, Ngahmbule) compared to the high (Peneng, Albulum, Yauatong) transmission intensity (Figure 5).

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Figure 5. Peak shift in the estimated worm killing efficacy rate of the two drug regimens.

Horizontal lines depict the prior parameter distribution, which was assigned to vary from 10% to 85%. Bars represent the estimated relative frequencies of the parameter obtained by annual sequential model fits to data through the 5-year intervention period in each of the 5 PNG study villages. The vertical lines represent measures of the central tendency of the estimated posterior distributions: mean (broken line) and median (solid line).

https://doi.org/10.1371/journal.pone.0067004.g005

Backfitting to Reconstruct LF Baseline Data

We carried out this analysis to test the hypothesis that if the biological parameters of the model (i.e. not drug efficacy parameters) remained stable or are invariant during the intervention period for a site, then it will be possible to use the sequential fits to reconstruct that site’s baseline age-infection profile. The mf-prevalence curves generated by sequential model fitting/updates to the age-infection data following each of the five annual drug interventions from year 1 post-intervention (left panel), and their usefulness for reconstructing baseline infection (right panel) are shown in Figure 6 for the village of Peneng (similar findings were obtained for all the rest of the study villages and are shown in Figures S5, S6, S7, S8 in Supporting Information S1). All of the individual sets of l ( = 500) parameter vectors were used to simulate the baseline age-profiles of mf-prevalence, and the posterior individual curves, medians, and their 95% credible intervals from these simulations are plotted in the figure. The results depict that, although there are year-to-year variations in the backfitted age-mf profiles, the sequentially fitted parameter vectors from each individual year are able to recover the baseline situation remarkably well. Table 3 provides the age-stratified and overall Monte Carlo p-values for the fits of each the backward extrapolations to the example Peneng baseline infection data. The results indicate that model performance for recovering the baseline LF infection for this study site was excellent, when evaluated using age-aggregated data whereas when assessed against age-stratified baseline data discrepant results were obtained for only the youngest 0–10 age group. Similar results were obtained for the rest of the study villages (Table S4 in Supporting Information S1).

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Figure 6. Sequential model backfitting to baseline infection data.

Comparisons of sequential model fits to observed annual declines in mf age-prevalence (left panel), and of back predictions of each yearly fit to observed baseline infection data for the village of Peneng (right panel). Sequential parameter updating was accomplished by using posteriors from the model fit to a previous year as priors for each successive subsequent year, starting with the posteriors obtained by model fitting to year 1 intervention data (see text). The thick (blue) line represents the median value of the SIR selected 500 prevalence curves. The observed annual declines in mf age-prevalence (left panel) and baseline mf age-prevalence (right panel) respectively are shown by crosses with 95% CIs. The dashed lines in the right-panel plots represent the 95% bounds (the 2.5^th and 97.5^th percentile values) of the simulated mf-prevalence curves shown in grey. Similar results were obtained for the remaining 4 study villages (Figures S5, S6, S7, S8 in Supporting Information).

https://doi.org/10.1371/journal.pone.0067004.g006

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Table 3. Carlo p-values for the backfitted models to the Peneng baseline data.

https://doi.org/10.1371/journal.pone.0067004.t003

Worm Breakpoint Comparisons

Figure 7 depicts and compares the mf breakpoint values obtained from the stability analyses of the directly (i.e. fitted only to baseline data) versus 1) sequentially fitted LF models to the year 1, 3 and 5 post-MDA data, and 2) a situation reflecting either a 25% or a 50% overall reduction in initial annual biting rate over the entire 5-year MDA period, both for the study village of Peneng. The results from the plotted quadratic discriminant analyses show clear separation between breakpoint values estimated retrospectively using backfitted models compared to those calculated from the direct fits to baseline data only in the case when the initial biting rate is reduced by 50% over the entire MDA period. Reduction of annual biting rate by 50% also raised worm breakpoint values and generated lower threshold biting rates (Figure 7F). The formal KS test underscored these results indicating a significant difference (p = 0.0042) only for the direct versus backfitted estimates in the case of the 50% reduction in the biting rate scenario. Similar findings were also obtained for the rest of the study villages (data not shown).

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Figure 7. Comparison of results from carrying out a quadratic discriminant analysis on worm breakpoints and threshold biting rates estimated from direct- and back-fitted LF models to baseline mf age-prevalence data for Peneng village.

In (A) the endpoint data (worm breakpoints and biting thresholds) estimated by direct model fits to the baseline mf age-prevalence data (shown by 0) are compared against endpoint estimates derived from the reconstructed baseline mf age-prevalence obtained via model fitting to the first year intervention data (shown by 1). Similarly, in (B) and (C) the directly fitted model estimates are compared against those obtained from backfitted mf age-prevalences using, respectively, the third (denoted by 3) and fifth (denoted by 5) years’ intervention data. In the right panel, the endpoint estimates from the model fitted directly to baseline infection data are compared against those estimated from the backfitted models using the fifth year intervention data, but with the original annual biting rates increased by 25% (shown by 6) in (D), reduced by 25% (shown by 7) in (E), and by 50% (shown by 8) in (F). The edges between the two coloured regions in each graph depict the separation boundaries for the direct and back-fitted endpoint estimates in each case. The two solid dots in each plot depict the means of the distributions of the endpoint estimates obtained from the direct- and the back-fitted models to the baseline data. Note that the two dots begin separating from each other but still remaining in the same coloured region in (E), with full separation between the two estimated distributions (i.e. falling in a distinct coloured region) occurring only in (F).

https://doi.org/10.1371/journal.pone.0067004.g007

Discussion

In our previous work, we had highlighted the importance of assimilating locality-specific LF infection data into process-based mathematical models as a strategic framework for determining LF elimination endpoints that best reflect the realities of transmission in local environments [12]. Here, we extend that work by investigating if the biological parameters underlying LF transmission estimated by calibrating mathematical models using baseline pre-intervention data in a field site are robust to infection perturbations induced by drug regimens applied sequentially at that site. This is an important question of practical significance to the current drug-based LF elimination program, because if these model parameters are found to be dynamically variable then parasite infection dynamics, including endpoint values, are likely to also vary as interventions proceed. This would make predictions of when elimination may be achieved commensurately more difficult, besides indicating a need for adaptively revising intervention schedules (e.g. duration, frequency of MDAs) periodically as programs progress. On the other hand, if the parameters in a site undergoing interventions are found to be stable in the face of perturbations, then this robustness would likely enhance achievement of LF elimination since it would imply that endpoints estimated from either baseline or monitoring data during interventions are exchangeable, and therefore either could serve as reliable targets for determining parasite extinction.

The major result of this study viz. that the biological parameters underpinning LF transmission may indeed remain dynamically stable at least over a moderately long MDA program (5 years) in individual endemic localities, may therefore represent an outcome of import to the current LF elimination initiative. This result suggests: 1) that LF extinction dynamics and hence elimination endpoints in different sites are unlikely to vary over the durations of MDA expected to bring about parasite elimination in most endemic communities [7], [12], 2) that endpoint values may be estimated equally reliably from either baseline pre-MDA or from intervention monitoring data depending on data availability in a site, and finally, 3) that monitoring data during a MDA may be used to reliably back calculate or hindcast initial baseline age-infection patterns, thereby allowing generation of the expected system trajectories required for evaluating the progress made towards LF extinction for all those sites without such data [27], [29]. Taken together, these new findings enhance the prospects of LF elimination as they signify that once endpoints and original infection are known/estimated for a site, initially specified optimal intervention schedules and parameters (e.g. coverage/duration of MDA) for crossing such endpoint targets may be followed without alteration until the end of the program, thereby significantly simplifying the management of parasite control [27], [29].

The finding of temporal invariance or dynamic stability in key biological parameters in this study has also shed intriguing new insight into the nature of LF transmission in endemic communities. In particular, it supports recent work exploring the design of biological systems, which suggests that such invariance in biological parameters could be a direct function of the structural adaptation of a complex biological system, such as LF, to initial ecological conditions of a locality so that maintenance of system function (parasite transmission in the present case) against perturbations are preserved locally [19]–[23], [61]. This perspective suggests that the remarkable observed persistence of LF transmission in the face of the drug perturbations carried out in the PNG study sites could be a direct outcome of the complex biological architecture which underpins the transmission of vector-borne diseases - viz. diverse regulatory feedback loops that are able to compensate for changes in ecological conditions, diversity and modularity in overall transmission function by involving two hosts and heterogeneous mosquito larval habitats, which can help contain perturbations locally to minimize effects on the whole system, and different parasite stages exhibiting markedly different life spans to buffer interactions against perturbations [21], [23] - adapted to be robust to normal variations in the values of key transmission-related ecological factors prevailing in a given locality. This implies that if LF transmission parameters are stable, then initial conditions will set the boundaries of local transmission dynamics [21], [23], [61]. Recent studies in schistosomiasis modelling have also highlighted the impact and need for estimating site-specific internal determinants of parasite transmission for gaining a better understanding of the population dynamics and control of Shistosoma japonicum in endemic communities [52], [56], [62]. This finding, together with the present results, indicate firstly that robustness to initial conditions in a site may be a fundamental structural feature of vector-borne, and possibly other, parasitic transmission systems, and secondly that gaining a better understanding of the mechanisms underlying such dynamics will be fundamental to identifying the faults and hence the effective countermeasures required to disrupt parasite transmission reliably.

However, it is clear that a trade-off arising from such dynamics is that locally robustly adapted systems may also face devastating fragilities in the face of novel environmental conditions or perturbations leading to catastrophic system failure [21], [23], [61]. Here, we have demonstrated that one major initial external factor that may confine LF dynamics over time in an endemic locality is vector abundance (Figure 7). The available data on changes in entomological variables over the 5 year MDA period in each village showed yearly fluctuations that were most prominent for the high transmission villages studied here [30]. Our analysis of the impact of these fluctuations depicted in Figure 7 shows that LF transmission robustness could be sensitive to such vector abundance changes, but only when >50% change occurred in this key variable over an intervention period. By contrast, when fluctuations in vector biting rates came close to but not >50% (as actually recorded for the high transmission study villages examined here [30]), LF transmission robustness was maintained. This finding indicates that model calibration to key local variables, such as vector biting rates, may allow detection of the boundaries or thresholds of the specific initial constraining conditions that confine dynamics to a site. A major question in this regard is then whether once the values of this confining variable traverse such thresholds, this change might shift the transmission dynamics (and associated attractors, basins of attraction and stability) of the historically adapted LF system into a new and possibly more fragile transmission state and push the system into extinction [20]. Our investigation of this question with regard to changes in vector biting rate has indicated that this outcome might be a distinct possibility in the case of LF. This is illustrated in Figure 7F, which shows that if the initial bounding vector abundance variable in a site is reduced over time by a large degree (>50%), fragile LF transmission regimes may indeed emerge as demonstrated by the shifts to higher worm endpoints and lower TBR values in the figure. This finding presents a new mechanism in support of our previous conclusion from modelling intervention dynamics regarding the role of vector control in LF elimination, viz. that including vector control into MDA programs can significantly accelerate the achievement of parasite elimination by altering both vector transmission and worm breakpoint thresholds negatively [7], [9], [13].

The sequential BM fitting of the anopheline-mediated LF model to longitudinal mf data from each of the study villages has also shown that the temporal invariance observed for the biological parameters of this model was compensated for by the occurrence of significant changes in all of the corresponding drug-related parameters relevant to modelling the impact of the drug regimens used in each site, viz. DEC alone or combined DEC/IVR. The estimated posterior distributions for each of four drug parameters –1) worm killing efficacy, 2) mf killing efficacy, 3) reduced fecundity of surviving worms and 4) the waning period – showed variable shifts from their initially assigned prior distributions in each site, with the most informative change observed for the worm killing efficacy parameter, the mean rates of which appearing to settle around values of between 74 to 77% for the two drug regimens by the end of 5 MDA cycles in these study villages (Figure 5). Similarly, mean efficacy rates for mf killing and the waning period parameter increased to peak values of around 80% and 4 months respectively over the same intervention period. These sequential model estimates indicate that while previous estimates of the mf killing and waning rates for the present drug regimens appear to have been generally well defined [7], [26], the worm killing rate, by contrast, has been grossly underestimated, suggesting that previous modelling results of the impact of MDA may have been overly pessimistic. Furthermore, the results also show that the addition of IVR to DEC does not appear to markedly change the estimate for this parameter in comparison to the effects of the DEC only regimen. Thus, our results indicate that the major impact of the combination therapy in reducing LF infection in field settings is attributable mostly to the effects of DEC. This finding implies that regimens with IVR but without DEC, such as the ivermectin/albendazole (IVR/ALB) combination regimen advocated for use in many parts of Africa with co-occurring onchocerciasis [1], will have a comparatively lower impact - indeed there is some evidence that ALB would also further lower the efficacy of this combination by an inhibitory effect on worm killing [63] - in reducing LF transmission, thus making parasite elimination more difficult to achieve in such areas. Note that such an inhibitory effect may also reduce the worm killing efficacy of DEC in the DEC/ALB combination, which would clearly have implications for LF elimination in other areas; more biological details of this effect or data from sites using the DEC/ALB combination regimen will need to be modeled if we are to better estimate such inhibitory effects. A further complication highlighted by our results regarding the effectiveness of MDA is that there may also be a density-dependent effects on worm killing by both the applied DEC and DEC/IVR regimens, with the mean worm killing rate estimated for each regimen being marginally lower at each cycles of intervention in the villages exposed to the high compared to the lower transmission intensity (Figure 5). The import of this result is clear: worm killing efficacy of anti-filarial drug regimens will vary with the intensity of infection in a community. These findings further emphasize the crucial requirement for estimating transmission and intervention parameters that reflect local realities of transmission if LF is to be reliably eliminated from endemic communities. They also underscore the importance of including supplementary measures, such as vector control, to MDA, this time both to overcome the effects of using a less efficacious macrofilaricidal regimen in DEC contra-indicated regions and to overcome the possibility of density-dependent worm killing effects, if we are to achieve LF elimination globally.

Our use of the BM methodology to integrate sequential intervention-related infection data with the LF model to predict the effects of MDA on infection dynamics show that this Monte-Carlo based Bayesian calibration method can yield results that efficiently approximate reality as close as possible. As highlighted previously [51], [54], the innovative benefit using this approach rather than simple Monte Carlo simulations lies in the ability to attach prior distributions and calculate likelihoods for both model input and the output, which improves the use of data to inform initial conditions, update model parameters, and constrain a model during simulation so to match observed data closely. However, one drawback of using sampling filters such as the SIR algorithm used in this study to generate the posterior probability distributions of model parameters and predicted state variables (e.g. endpoints) is that it may lead to sample impoverishment, wherein the posterior sample may have a large number of repeated copies due to the weighted resampling carried out based on likelihoods of parameter vectors [64]–[66]. Although this did not represent a major problem in the present study, future applications of this approach to extended datasets and perhaps fitting of models to data from longer MDA cycles than studied here will need to take this technical issue into consideration. Parasite system dynamics, particularly in the case of vector-borne diseases, are also likely to be strongly influenced by weather and climate changes via effects on vector population dynamics. As shown in this study, such vector abundance changes can critically impact parasite transmission dynamics in a locality, meaning that future studies may require considering statistical frameworks, such as Markov Chain Monte Carlo (MCMC) methods or hybrid sequential Monte Carlo sampling/MCMC approaches, for joint estimation of time varying parasite system states and parameters [57], [64], [66], [67]. We are currently extending our LF model to explicitly include mosquito population dynamics (rather than using a fixed ABR value to represent such dynamics as performed here) as a first step to developing such statistical methods. Note also that if it is confirmed that local LF transmission dynamics can be largely constrained by the prevailing vector abundance in a site, another outcome of such a finding is that it may also allow the application of our model, and hence transferability of results, across different settings based on calibration to available joint spatial data on vector biting rates and mf prevalences.

A further improvement to our modelling approach is to directly include infection diagnostic uncertainties into the model calibration approach so that true prevalences can be quantified from joint estimates of test specificities and sensitivities [68], [69]. This will clearly be important in correcting for endpoint values estimated using different types of infection indicators as well as for correctly predicting the corresponding system trajectories due to the application of MDA in different localities [27]. Such work will improve understanding of how indicator dynamics are linked to the underlying parasite transmission dynamics, and in turn guide the choice of the best indicators that closely match worm dynamics for monitoring the impact of interventions on LF infection/transmission in communities [27], [29].

Despite these limitations related to our current model and its calibration, the present study has nonetheless advanced new understanding regarding LF transmission and the prospects for its control using currently available MDA methods. Firstly, our findings underscore the need to understand the biological robustness of parasite transmission, and the factors that underlie such robustness, in local settings if we are to develop reliable countermeasures to achieve parasite extinction. Our preliminary analysis of LF transmission robustness to initial conditions has highlighted the impact that fluctuations in vector abundance can play in pushing a locally robustly adapted vector-borne parasite system to a different state with its own dynamics, and hence endpoints. This result corroborates previous theoretical and experimental work in demonstrating that adaptability or robustness of biological systems, including complex parasitic systems, to one set of conditions can enhance their fragility to abrupt changes in such conditions [19]–[23], [61]. We suggest that learning about these trade-offs between transmission robustness and fragility has now become central if we are to successfully accomplish the elimination of LF, and indeed other parasitic infections, both locally and globally. Secondly, results presented here also suggest that progress in this next generation of work will not only require the development of modelling methods to analyse and evaluate biocomplexity and robustness of parasitic systems, but also the development of data-driven inferential techniques that can reliably allow the joint determination of model states, structure and calibration using local data. A key need here is the development of modelling frameworks that can assimilate spatially distributed data on key ecological conditions relevant to parasite transmission from a variety of sources, couple with climate and weather dynamics, and integrate with the available spatio-temporal datasets on human infection prior, during and following interventions [57], [67]. Constructing such models of everywhere is no doubt a daunting challenge, but we suggest that with the rapid improvement in computing power, advent of new intelligent eco measurement devices and observation networks, and development of new flexible state space data modelling and assimilation approaches, these challenges can be met [57], [67], and the notion of using models as an effective management tool for achieving parasite control or eradication becomes an attractive, acceptable option. Lastly, a theme that can be identified running throughout this work is a need for control programs to focus on obtaining good quality infection monitoring data. Data will be required to characterize local dynamics, to evaluate and update model predictions, extrapolate predictions to sites without or poor quality data, and monitor changes in coverage and system response as interventions progress. We end by indicating that a closer alignment of modelling work with well-collected field data constitutes an urgent operational requirement if the current global program to eradicate LF is to successfully achieve its laudable goal.

Supporting Information

Acknowledgments

We would like to thank Professor Lance Waller, Department of Biostatistics, Emory University, for help with the calculation of Monte Carlo p-values. A part of model runs was carried out using the Matlab Parallel Computation Toolbox available on the Compute Clusters of the University of Notre Dame’s Center for Research Computation. We would also like to thank two anonymous reviewers for their comments and suggestions that improved the first draft of this manuscript.