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Predicting mosquito infection from Plasmodium falciparum gametocyte density and estimating the reservoir of infection

  1. Thomas S Churcher  Is a corresponding author
  2. Teun Bousema
  3. Martin Walker
  4. Chris Drakeley
  5. Petra Schneider
  6. André Lin Ouédraogo
  7. María-Gloria Basáñez
  1. Imperial College London, United Kingdom
  2. Radboud University Nijmegen Medical Centre, The Netherlands
  3. London School of Hygiene and Tropical Medicine, United Kingdom
  4. University of Edinburgh, United Kingdom
  5. Centre National de Recherche et de Formation sur le Paludisme, Burkina Faso
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Cite as: eLife 2013;2:e00626 doi: 10.7554/eLife.00626

Abstract

Transmission reduction is a key component of global efforts to control and eliminate malaria; yet, it is unclear how the density of transmission stages (gametocytes) influences infection (proportion of mosquitoes infected). Human to mosquito transmission was assessed using 171 direct mosquito feeding assays conducted in Burkina Faso and Kenya. Plasmodium falciparum infects Anopheles gambiae efficiently at low densities (4% mosquitoes at 1/µl blood), although substantially more (>200/µl) are required to increase infection further. In a site in Burkina Faso, children harbour more gametocytes than adults though the non-linear relationship between gametocyte density and mosquito infection means that (per person) they only contribute slightly more to transmission. This method can be used to determine the reservoir of infection in different endemic settings. Interventions reducing gametocyte density need to be highly effective in order to halt human–mosquito transmission, although their use can be optimised by targeting those contributing the most to transmission.

https://doi.org/10.7554/eLife.00626.001

eLife digest

Malaria is one of the world’s most deadly infectious diseases. The most severe form is caused by the parasite Plasmodium falciparum, which can reside within red blood cells and thus evade the human immune system.

Plasmodium is transmitted between humans by mosquitoes. When a mosquito takes a blood meal from an individual infected with the parasite, the insect ingests Plasmodium gametocytes (i.e., eggs and sperm), and these go on to reproduce in the gut of the mosquito. These parasites then move to the mosquito’s salivary glands, to be injected into the next person whom the mosquito bites.

Although malaria is both preventable and curable, the mortality rates in many African countries remain high, especially among children. Reducing the transmission of malaria to mosquitoes is one of the primary goals in the global effort to control and eliminate the disease. While a range of drugs and vaccines that specifically try to reduce transmission are in development, non-medical interventions such as mosquito nets and insecticide spraying can quickly and effectively reduce infection rates.

Here, Churcher et al. examine the dynamics of human to mosquito transmission of P. falciparum, and report that the ease with which mosquitoes become infected is not directly proportional to the density of parasite gametocytes in human blood. They found that the transmission occurs readily at very low gametocyte densities. Moreover, the transmission rate remains relatively stable as the density increases, before increasing significantly when the density reaches around 200 cells per microlitre.

Churcher et al. also challenge the assumption that children are mostly responsible for transmitting the malaria parasite by suggesting that, in certain locations, there is a more significant role for adults than previously assumed. By identifying the groups that contribute most to transmission, and targeting resources to reduce gametocyte density in those individuals, it could be possible to greatly reduce the number of infected mosquitoes and, therefore, the number of infected humans.

https://doi.org/10.7554/eLife.00626.002

Introduction

The malaria parasite is transmitted among humans by anopheline mosquitoes. Male and female transmission stages (gametocytes) are ingested by the mosquito and reproduce sexually in its stomach before developing into oocysts. Once oocysts establish, it is assumed that mosquitoes will form infectious sporozoites, so the proportion of mosquitoes developing oocysts is used as a measure of mosquito infectivity. In the human malaria parasite Plasmodium falciparum, mosquito infection is thought to increase with the number of gametocytes ingested by the mosquito (Jeffery and Eyles, 1955; Graves et al., 1988; Bousema and Drakeley, 2011). However, several studies have failed to find an association (Boudin et al., 1993; Haji et al., 1996), and the precise shape of the relationship has never been rigorously quantified, particularly at very low gametocyte densities. Difficulties arise because estimates of gametocyte density have relied on microscopy, which may miss up to 80% of parasites (Dowling and Shute, 1966). More sensitive molecular methods such as Pfs25mRNA quantitative nucleic acid sequence–based amplification (QT-NASBA) have been developed (Schneider et al., 2004). Unlike conventional microscopy, this technique enables gametocyte densities to be quantified over the entire epidemiologically relevant range.

Transmission reduction is now a key component of global efforts to control and eliminate malaria (Alonso et al., 2011). A wide range of novel transmission-reducing drugs and vaccines are currently under development, which aim to reduce malaria incidence by restricting human to mosquito transmission. The transmission-reducing ability of front-line therapeutics (such as artemisinin-based combination therapies [ACTs], or potential combinations of ACTs with gametocytocidal drugs) is also gaining increased attention (WHO, 2012; White, 2013). It is likely that these interventions will be partially effective, but it remains unclear how much they need to reduce transmission from humans to mosquitoes in a particular location, and to which age groups they should be delivered in order to halt malaria transmission.

Mathematical models of malaria transmission tend not to include explicitly the relationship between gametocytaemia and mosquito infectivity, opting, for simplicity, to assume density-independent transmission probabilities (Smith et al., 2007; Griffin et al., 2010), or fit functions to the relationship between the number of asexual parasites and the probability of a blood-feeding mosquito becoming infected (Ross et al., 2006). Greater complexity, however, will be required to capture fully malaria population dynamics following the introduction of drugs and vaccines that specifically target transmission stages. For example, the World Health Organization (WHO) has recently recommended that in pre-elimination or elimination malaria programmes, single-dose primaquine (0·25 mg base per kg) with an ACT should be given to all patients with falciparum malaria except pregnant women and infants <1 year old (WHO, 2012). More detailed mathematical models will allow the full impact of this change in strategy to be assessed in a range of different endemic settings.

The optimal use of transmission-reducing interventions will also require a better understanding of the human reservoir of infection, which is likely to vary between different endemic settings. Various studies have attempted to estimate the relative contribution of different age groups to mosquito infection using skin or membrane feeding assays (Muirhead-Thomson, 1957; Graves et al., 1988; Boudin et al., 1991; Githeko et al., 1992; Drakeley et al., 2000; Bonnet et al., 2003). These studies are logistically complicated and expensive, so more efficient methods of determining the reservoir of infection are required in order to target interventions optimally. A thorough understanding of the factors determining mosquito infection would allow mathematical models to predict the relative contribution of different groups to transmission from cross-sectional surveys, both before and after the introduction of transmission-reducing interventions.

This article uses data from mosquito feeding assays conducted in Burkina Faso (Ouédraogo et al., 2009, Dryad: Ouédraogo et al., 2013) and Kenya (Schneider et al., 2007, Dryad: Schneider et al., 2013) to estimate the shape of the relationship between gametocyte density and mosquito infection (processed data available at Dryad, Churcher et al., 2013). Other covariates that may influence infection such as asexual parasite density (measured by microscopy) and host age were also included to improve predictions of human to mosquito transmission. These results are combined with data from a cross-sectional survey conducted in a high transmission setting in Burkina Faso to predict the relative contribution of different age groups to overall malaria transmission.

Results

The relationship between the number of gametocytes in the blood and mosquito infection was found to be highly non-linear (Figure 1A). Plasmodium falciparum infects mosquitoes at the very low gametocyte densities that predominate in natural infections and may not be detected using standard microscopy (Bousema and Drakeley, 2011). Infection rises rapidly with increasing gametocyte density, and by 1 gametocyte per microlitre, ∼4% (95% Bayesian credible interval [CI], 3–5%) of all mosquitoes develop oocysts. The subsequent increase in infection with increasing gametocyte density is best described by the Gompertz model (deviance information criterion [DIC] = 1034), which gave a significantly better fit than the linear (DIC = 1073), power (DIC = 1059), or hyperbolic (DIC = 1062) functions (Figure 1A). The best-fit model predicts that increasing density from 1 to 200 gametocytes per microlitre does not appreciably increase infection. Beyond 200 gametocytes per microlitre, infection rises again to finally plateau at ∼18% infected mosquitoes. Gametocyte density on an arithmetic scale was a better predictor of mosquito infection than gametocyte density on a logarithmic scale. Including information on the host’s asexual parasite density significantly improved model fit. Children with asexual parasite densities between 1 and 1000 parasites per microlitre were on average 27% (CI, 19–58%) less infectious to mosquitoes than those with no detectable asexual parasites, whereas those with asexual densities >1000 µl−1 were 77% (CI, 14–140%) more infectious. Including age improves the fit of the model suggesting that age is an important confounder, although the Bayesian credible intervals include 0. Children >6 years old were 15% (CI, 24–64%) more likely to infect mosquitoes than those of younger ages, whereas no difference in the relationship between gametocyte density and mosquito infection was detected between Burkina Faso and Kenya.

The relationship between gametocyte density and mosquito infection and the impact this will have on the effectiveness of gametocytocidal interventions.

(A) The relationship between Plasmodium falciparum gametocyte density and the percentage of Anopheles gambiae mosquitoes that develop oocysts. Point colour, shading, and shape denote characteristics of the blood donor, such as location (blue = Burkina Faso; red = Kenya), asexual parasite density as measured by microscopy (no fill colour = none detectable, light shading = 1–1000 parasites per microlitre, dark shading ≥1000 µl−1), or host age (<6 years old = square, ≥6 years old = circle). The size of the point is proportional to the number of mosquitoes dissected. Coloured horizontal and vertical lines indicate 95% Bayesian credible intervals (CIs) around point estimates. The solid black line indicates the best-fit model, whereas the grey shaded area indicates the uncertainty around this line. The inset shows the relationship at very low gametocyte densities (on a logarithmic scale). The outputs show the shape of the relationship for a child with no detectable asexual parasites. A full description of data used to fit the model are given in Figure 1—source data 1. Regression coefficients and a measure of goodness-of-fit of the different models are given in Figure 1—source data 2. Panel (B) shows how efficacious transmission-reducing interventions would need to be (on the vertical axis) at decreasing gametocyte density in order to reduce human to mosquito transmission (contour lines). The best-fit line from (A) is used to illustrate the percentage reduction in mosquito infection that would be achieved according to the pre-intervention host’s gametocyte density (on the horizontal axis) for an intervention, which reduces gametocyte density by a given percentage (which is assumed to be constant over different gametocyte densities). The colours represent the percentage reduction in mosquito infection that would be achieved, ranging from red (low, 0–10% reduction) to darker hues of blue (high, 70–80% and 80–90% reduction, see legend). The 90–100% reduction in transmission is hardly visible and would correspond to nearly 100% efficacious interventions at the top of the graph.

https://doi.org/10.7554/eLife.00626.003

The complex shape of the relationship between gametocyte density and mosquito infection will influence the success of transmission-reducing interventions and may explain why their ability to reduce the proportion of infected mosquitoes has been shown to depend on the gametocyte density of the host (Churcher et al., 2012). For example, reducing gametocyte density by 99% in a host with 200 gametocytes per microlitre may not have much effect on their immediate contribution to transmission (Figure 1B), although it will probably reduce the duration of infectivity. The same intervention efficacy at reducing gametocyte density in a host with 300 gametocytes per microlitre would cause an appreciable reduction in human to mosquito transmission from that individual. Figure 1B can be used to estimate how a reduction in the number of gametocytes will equate to a reduction in the proportion of mosquitoes becoming infected, and hence mosquito to human transmission. How this relates to the incidence of malaria and subsequent disease will depend, in part, on the degree of immunity in the human population.

In the cross-sectional survey in Burkina Faso, children harbour more gametocytes than adults, with 10 year olds having five times as many gametocytes compared to the 50 year olds (Figure 2A). However, the relationship between gametocyte density and mosquito infection established here means that adults are only 37% less likely to infect mosquitoes (infecting on average 3.5% of them, Figure 2B). Results indicate that, at the time of the survey conducted in Burkina Faso, the peak in the reservoir of infection occurs at an earlier age than the peak in gametocyte density, which will improve the (cost) effectiveness of school-based programmes.

Age patterns of gametocyte density and estimates of the age profile of the human reservoir of infection.

(A) Results from a cross-sectional survey conducted in a high transmission setting in Burkina Faso showing how the mean number of gametocytes per microlitre of blood (including 0s) changes with host age. The distribution of gametocytes among hosts is highly overdispersed. A full description of data used to fit the model are given in Figure 2—source data 1 with the best-fit parameter estimates shown in Figure 2—source data 2. The relationship between gametocyte density and oocyst prevalence shown in Figure 1A is used to predict the percentage of mosquitoes that will become infected after biting a host of a certain age. This is shown in panel (B), which can be interpreted as the contribution of each age group towards the human to mosquito transmission. In both figures, the black solid line shows the best-fit model and the grey shaded area indicates the uncertainty (95% Bayesian credible Interval, CI) having fitted the model to the individual data (a total of 412 individuals). For illustrative purposes, the data are grouped into seven bins, namely 0–5, 5–10, 10–15, 15–20, 20–30, 30–40, 40–50, and ≥50 year olds, and the size of the point is proportional to the number of individuals in the group. In Figure 1A, the 10–15 and 15–20 year groups appear lower than the best-fit line due to sampling artefacts generated by the highly overdispersed data. Vertical lines indicate the 95% CI around grouped estimates.

https://doi.org/10.7554/eLife.00626.006

Discussion

The complex shape of the relationship between gametocyte density and mosquito infection elucidated here will influence our understanding of the population dynamics of falciparum malaria and will determine the success of transmission-reducing interventions. The efficiency with which P. falciparum gametocytes can infect mosquitoes means that transmission-reducing interventions, which reduce gametocyte density, will need to be highly effective in order to reduce human–mosquito transmission. Gametocytes can infect mosquitoes at very low densities, despite the mosquito needing to ingest both male and female parasites in the same blood meal. Substantial transmission was seen from hosts with <1 gametocyte per microlitre of blood. After this initial increase in infection, considerably more gametocytes are required in order to increase further the proportion of mosquitoes developing oocysts. The causes of this are unclear, but experimental systems may be informative. Data from the rodent malaria model Plasmodium berghei (Sinden et al., 2007) suggest that the phenomenon could be associated with parasite mortality during penetration of the mosquito gut wall or due to the mosquito’s immune response. Beyond 200 gametocytes per microlitre mosquito infection rises again (although QT-NASBA measurement error makes the exact gametocyte density for this transition relatively uncertain). If an intervention can keep gametocyte density beneath 200 gametocytes per microlitre in a low transmission area, it may be sufficient to push the basic reproduction number of malaria beneath one and eliminate the disease. However, the highly overdispersed distribution of gametocytes between hosts (adequately described by the negative binomial distribution) means that in the population from Burkina Faso, 30% of transmission comes from hosts with densities beneath the microscopy detection threshold of 16 gametocytes per microlitre. This means that evaluating interventions solely on their ability to reduce microscopically detectable gametocytes, as has been suggested (Graves et al., 2012), may give misleading results.

Including information on asexual parasite density (as measured by microscopy) significantly improved the fit of the model, with hosts harbouring intermediate densities having the highest infectivity. It is unclear whether asexual parasites directly influence parasite infectivity or are associated with other (here unmeasured) variables such as the multiplicity of infection (Nsango et al., 2012) or factors influencing the blood environment such as immunity (Bousema and Drakeley, 2011). Plasmodium falciparum has asynchronous waves of asexual and sexual parasites. Circulating gametocytes take 2–3 days to mature before they can infect mosquitoes (Lensen et al., 1999), which may explain the complicated relationship between infectivity and asexual density (i.e., intermediate asexual parasite densities might occur at the start of gametocyte production and have a relatively low proportion of infectious mature gametocytes). On average, older children were more infectious, supporting previous indications that transmission-reducing immunity may predominate in young children (Drakeley et al., 2006).

Figure 1A indicates that individual QT-NASBA gametocyte density estimates are relatively uncertain. Despite this, they are considerably more accurate than conventional microscopy. Molecular techniques may be highly precise in a state-of-the-art laboratory, but the practicalities of collecting and processing samples in non-ideal field settings increases the risks of incurring measurement error. Even though uncertainty is generally accepted by those conducting the experiments, measures of precision such as confidence intervals or standard errors around such density estimates are rarely reported in the scientific literature. Methods such as those presented in this article can allow robust quantitative insight to be gained from uncertain molecular methods. Care should be taken when interpreting gametocyte density estimates as it has been suggested that the marker used in this analysis (Pfs25 mRNA) might be parasite female specific (Schneider, 2006). Even if the current QT-NASBA method does only detect female gametocytes, it will neither change the qualitative conclusions of this study nor their application if other studies use the same technique. However, direct comparison of our results with those of studies using different methods for quantifying gametocyte density should be aware that the shape of the relationship may differ.

Gametocyte density measurement error is likely to cause the majority of the uncertainty seen in model outputs, although the complexity of the membrane feeding assay is also likely to contribute. The proportion of mosquitoes infected by Plasmodium-infected blood is known to vary substantially within and between studies. This is in part due to methodological issues with the membrane feeding assay (Bousema et al., 2013) but also related to biological differences in the parasite-vector combination and the blood environment (Bousema and Drakeley, 2011). Further standardising the membrane feeding assay, improving the accuracy of QT-NASBA technique and the inclusion of additional covariates (such as estimates of gametocyte maturity, which have been investigated in Plasmodium vivax; Chansamut et al., 2012), would improve the accuracy of the relationship between gametocyte density and mosquito infection. It would also permit the patterns described here to be checked for consistency across time and space, and enable a wider range of functional forms to be tested. Care should also be taken when interpreting the results of feeding assays as the mosquitoes used, and their biting behaviour is likely to be different from that seen in wild mosquitoes in field situations (Bousema et al., 2013).

Transmission reduction will become increasingly important as areas approach local elimination. Identifying host age groups that contribute most to the reservoir of infection will allow the optimal targeting of malaria control. In the high transmission site in Burkina Faso, young children harbour the majority of gametocytes but are only slightly more infective to mosquitoes than adults. The per person contribution of adults estimated here is considerably greater than that predicted by other mathematical models (Ross et al., 2006) and may increase further once age-dependent biting rates are taken into consideration (Carnevale et al., 1978; Ross et al., 2006). The contribution of different age groups to overall transmission will depend on local demography, age-dependent protection by malaria interventions (such as the use of bed nets or time spent in the house and therefore personally protected by indoor residual spraying), and human to mosquito contact patterns. The last two of these are difficult to measure and poorly understood, reducing our ability to accurately predict the sources of infection. The reservoir of infection is likely to vary between endemicity settings and over time so multiple cross-sectional surveys may be required. The relationship between gametocyte density and mosquito infection will allow estimates of the reservoir of infection from cross-sectional gametocytaemia surveys without the need for logistically complicated mosquito feeding assays. The results of this article should be included within mathematical models that capture changes in gametocyte density with time (Lawpoolsri et al., 2009) to evaluate the full impact that different transmission-reducing interventions will have on transmission and prioritise the most appropriate candidates according to transmission setting.

This article provides insights into the relationship between gametocyte density and mosquito infection that are needed to predict the outcome of transmission-reducing interventions. It shows that, given the ability of very low P. falciparum gametocyte densities to establish infection in A. gambiae, transmission-reducing interventions will need to be highly efficacious at reducing gametocyte density in order to halt human to mosquito transmission. The complex and non-linear shape of the relationship between gametocyte density and mosquito infection may explain why the ability of an interventions to reduce the proportion of infected mosquitoes has been shown to depend on the parasite load of the host (Churcher et al., 2012). Gametocyte density also changes with host age so the effectiveness of transmission-reducing interventions that target gametocytes will also vary with age. This should be considered in clinical trials of transmission-reducing candidate drugs and vaccines. The highly overdispersed distribution of gametocytes in the host population and the non-linear relationship between gametocytes/asexual parasite density and mosquito infection means that the impact of different transmission-reducing interventions on overall transmission at the population level will be far from intuitive.

Material and methods

Estimating gametocyte density and its associated uncertainty

Quantitative nucleic acid sequence-based amplification (QT-NASBA) is routinely used to estimate pathogen density. Like all diagnostic methods, it is prone to measurement error. To understand fully the associated uncertainty, it is important to appreciate the causes of the variability and how the quantification process might magnify uncertainty of point density estimates. Here, the uncertainty is quantified by repeatedly testing samples with known gametocyte density and fitting a hierarchical mathematical model.

Nucleic acids are extracted from 50 µl of blood and then amplified in the presence of a fluorescence probe. The assay measures time to positivity (TTP), which is the time it takes for the number of target amplicons detected to exceed a defined threshold (Schneider et al., 2004). The relationship between TTP and gametocyte density is estimated by fitting a linear regression to TTP estimates generated using a sample with known gametocyte density (a 10-fold dilution series of in vitro cultured gametocytes ranging from 106 to 101 gametocytes per millilitre). Let the observed TTP be denoted by Y then,

(1) Y=β0+β1lnx+ε,

where β0 and β1 are regression coefficients estimates, x is the (known) parasite density from the dilution series and ε represents a normally distributed random error with mean equal to 0 and constant variance, that is εN(0,σ2). A full list of the parameters is given in Table 1. The so-called statistical calibration or inverse regression problem (Osborne, 1991) concerns the issue of making statistical inference on the value of an unknown (log-transformed) gametocyte density), denoted as ln x′, from a new TTP observation, Y′ The classical approach (Eisenhart, 1939) involves rearranging the regression model (equation 1) so that

(2) lnx=Yβ0β1,

and substituting the regression parameters with their estimates β^0 and β^1 to yield an estimate lnx^. Here, we adapt this approach for use in a Bayesian hierarchical model. A number of different methods have been used to do this (see Hoadley, 1970 and Hunter and Lamboy, 1981 and for discussion see Osborne, 1991), each of which has a different method for dealing with the problem of high β^1 values (i.e., a gentle gradient, which when used in equation 2 can generate infinitely large confidence interval estimates). Here, we use the most parsimonious approach that does not require assignment of previous distributions to (the unknown) gametocyte densities. Rather, uncertainty in the estimated regression coefficients of equation 1 is propagated numerically via equation 2 to yield uncertainty in the estimated gametocyte densities. Since all the calibration line data have a relatively steep gradient, the difference between the different methods will be relatively minor, and all will generate sensibly tight confidence interval estimates.

Table 1

Notation of statistical and mathematical models

https://doi.org/10.7554/eLife.00626.009
NotationDescriptionEquation
YTime to positivity (TTP) readout generated by QT-NASBAEquation 1
xKnown density of gametocytes per millilitre (generated using dilution series)Equation 1
lnx^Estimate of (unknown) gametocyte density on the logarithmic scaleEquation 2
x^Estimate of gametocyte density on the arithmetic scaleEquation 2
β0Intercept of the calibration line fitted to the dilution seriesEquation 1
β1Gradient of the calibration line fitted to the dilution seriesEquation 1
σ2Intra-assay variance measuring the accuracy with which the calibration line fits the TTP estimates from the dilution seriesEquation 1
gProportion of mosquitoes developing oocystsEquation 3
ϕ(x^,κ)Saturating function determining the shape of the initial relationship between gametocytes and the proportion of mosquitoes developing oocystsEquation 4
fi(x^)Function determining the shape of the relationship between gametocytes and proportion of mosquitoes developing oocysts. Subscript i indicates the functional form used, be it fα(x^)=α0+α1x^α2(1+α3x^α2), where constraining different parameters can generate a range of different shapes (constant α1 = 0, linear α2 = 1; α3 = 0, power α3 = 0, hyperbolic α1 > 0 α2 = 1 α3 > 0, or sigmoid α2 > 1) or fγ(x^)=γ0+γ1exp[γ2exp(γ3x^)],, which generates a Gompertz (sigmoid-like) functionEquation 3
μVector of regression coefficientsEquation 3
zVector of dummy variables denoting donor blood characteristics, z1 = asexual parasite density (0 = undetected, 1 = low, 2 = high), z2 = host age (0 = younger than 6 years old, 1 = 6 or older), z3 = study locale (0 = Burkina Faso, 1 = Kenya)Equation 3
h(A)Function describing how gametocyte density and the reservoir of infection change with host age (A). Shape determined by parameters τ, ψ, and ωEquation 5

Calibration lines can vary between runs and batches of reagents. Here, we define an experiment as being a single plate run with the same batch of reagents each of which will have its own dilution series and samples with unknown density. The accuracy with which the individual TTP estimates fit the log-linear calibration line can be used to estimate assay measurement error (the intra-assay variability, σ) by fitting a hierarchical mixed-effects model to multiple dilution series (allowing estimates of β0 and β1 to vary among experiments to determine whether this significantly improves the fit of the model). The accuracy of gametocyte density estimates can also be improved by running multiple assays on the same sample and then taking the mean of all the ln x′ estimates. The application of equation 2 yields an estimate of gametocyte density, x^, on the logarithmic scale. Estimates on the arithmetic scale are generated by taking the exponent, that is x^=exp(lnx^). These estimates are used in the functions below to determine the relationship between gametocyte density and the proportion of mosquitoes developing oocysts.

Mosquito infection

Plasmodium falciparum–infected blood was collected from children and fed through a membrane to A. gambiae sensu stricto mosquitoes that were dissected 7–9 days later to determine infection (oocyst carriage). Data from 171 mosquito feeds on patients’ blood conducted in Burkina Faso (Ouédraogo et al., 2009) and Kenya (Schneider et al., 2007) were combined and used to fit a quantitative relationship between gametocyte density, asexual parasite density, host age, and mosquito infection (proportion of mosquitoes developing oocysts). A full description of these data is given in Figure 1—source data 1. Gametocyte density was quantified using QT-NASBA, and the uncertainty in the density estimates (the intra-assay variability) was quantified by fitting a hierarchical model to the 16 independent dilution series. In the Burkina Faso dataset, multiple assays were carried out on the same blood sample (an average of 2.63 assays per unknown blood sample). The mean gametocyte density from these multiple assays was taken to increase the accuracy of the estimates. The precise shape of the relationship was determined by fitting a range of different functional forms (a modified constant, linear, power, hyperbolic, sigmoid, and Gompertz functions) to data presented in Figure 1—source data 1 and statistically determining which gave the best fit. The proportion of mosquitoes developing oocysts, denoted by g, can be described by equation 3

(3) g=ϕ(x^,κ)fi(x^)(1+μz1+μz2+μz3).

Function fi(x^) describes how infection changes with increasing estimated gametocyte density with subscript i indicating the different functions tested (whose equations are given in Table 1); μ is a vector of regression coefficients and z1, z2 and z3 are dummy variables denoting asexual parasite density, age of the blood donor for the membrane feeing assays, and location of provenance, respectively. Asexual parasite density (estimated by microscopy) was categorized as being either undetectable (z1 = 0), low (<1000 parasites per microlitre of blood, z1 = 1) or high (≥1000 parasites per microlitre of blood, z1 = 2). Hosts were classified into two different age categories (<6 years [children, z2 = 0] or ≥ 6 yr [older children, z2 = 1). Mosquito infection was allowed to vary between Burkina Faso and Kenya (z3 = 0 for Burkina Faso, z3 = 1 for Kenya). The function ϕ(x^,η) determines the shape of the relationship at very low gametocyte densities and is motivated by the observation that the malaria parasite can adjust its sex ratio to optimise transmission (Reece et al., 2008). Evidence indicates that transmission is possible even at very low gametocyte densities (Schneider et al., 2007; Ouédraogo et al., 2009), so ϕ(x^,η) allows infection to rise very quickly with increasing gametocyte density at a rate determined by parameter η,

(4) ϕ(x^,η)=1(1+x^2η)(η+1).

Equation 4 was originally derived to describe the probability that a host would contain both male and female parasites according to the mean number of parasites and the aggregation (overdispersion) parameter of the negative binomial distribution (May, 1977). Reproduction in malaria is more complex than in the (helminth) system for which equation 4 was devised; hence, parameter η is not biologically interpretable.

To investigate whether mosquito infection is best predicted by gametocyte density on the arithmetic or logarithmic scale, both hypotheses were tested in the model, substituting an estimate of gametocyte density on the logarithmic scale (lnx^) for x^ in equations 3 and 4 and comparing model fits. The model quantifying the uncertainty in gametocyte density estimates was fitted at the same time as the regression determining the relationship between gametocyte density (and other covariates) and the proportion of mosquitoes infected using Bayesian Markov Chain Monte Carlo methods. Fitting the models simultaneously enables the uncertainty in the gametocyte density estimates to be reflected in the uncertainty of the shape of the relationship.

Age profile of gametocyte density

Blood samples were randomly collected from 412 hosts from a single village in Burkina Faso (Ouédraogo et al., 2010). QT-NASBA assays were run on the samples, and the methods described above were used to convert assay results into estimates of gametocyte density. A description of these data is given in Figure 2—source data 1. To facilitate visual inspection of the age profile of gametocyte density, and to generate summary statistics, the function h(A) was fitted to these data to describe how the mean number of gametocytes per microlitre of blood changes with host age (A)

(5) h(A)=τ+(ψAτ)exp(ωA).

Parameters τ, ψ, and ω were estimated assuming a negative binomial error structure to account for the high degree of overdispersion (aggregation) observed in gametocyte density estimates (using an overdispersion parameter that did not change with host age or gametocyte density). As above, a hierarchical model was used to estimate the gametocyte density and its associated uncertainty at the same time as the age profile allowing the full uncertainty of the shape of the function to be expressed.

Age profile of the infectious reservoir

The probability that a mosquito biting a host of a particular age will become infected can be estimated using the above model together with estimates of the host gametocyte and asexual parasite density. This was done using the cross-sectional data from Burkina Faso to generate a proxy for the human reservoir of infection at this specific time in this location (although it does not take into account how vector biting rate may vary with host age). It is important to use individual host estimates of gametocyte density instead of mean estimates as the high degree of gametocyte overdispersion among hosts may accentuate the non-linear relationship between gametocyte density and mosquito infection. For each of the 412 hosts, point estimates were obtained of the percentage of feeding mosquitoes (taking a single full blood meal) that would develop oocysts according to the age and the gametocyte/asexual parasite density of the host (equation 3). Equation 5 was used to fit a relationship between host age and their contribution towards human to mosquito transmission. This best-fit line was used to predict the percentage of overall transmission that originated from hosts who had gametocyte density estimates below the detection threshold of microscopy. Here, we define the detection threshold as the minimum (positive) density that can be estimated by sampling a certain volume of blood. It is assumed that gametocytes are counted against 500 white blood cells (WBC) and that there are on average 8000 WBC per µl of blood. This gives an average detection threshold of 16 gametocytes per microlitre of blood. This is a conservative estimate of the threshold density since densities above this value may still give false negative results both due reading errors or random sampling of gametocytes on the microscope slide.

Care should be taken when interpreting the results as mosquito membrane feeding experiments were not performed with blood from adult hosts. Evidence indicates that antibodies associated with transmission-blocking activity may be lower in older age groups (Drakeley et al., 2006), which could mean that adult hosts could have a greater potential to contribute to overall transmission.

Fitting procedure

Bayesian Markov Chain Monte Carlo techniques were used to fit the models in OpenBUGS (Lunn et al., 2009). This approach allows the uncertainties in the assessments of gametocyte density and mosquito infection rates (generated by different numbers of mosquitoes being dissected) to be taken into consideration, propagated into parameter posterior distributions. All parameters were assigned uninformative priors and were run until convergence was reached using standard methodology (Gelman and Rubin, 1996). The most parsimonious models were selected by comparing deviance information criteria (DIC) values (the lowest value giving the most parsimonious yet adequate fit) (Spiegelhalter et al., 2002). Uncertainly around the best-fit line is indicated by showing the 95% range of 10,000 runs randomly sampled from the posterior distribution. The distribution of gametocytes between hosts was highly overdispersed, with an aggregation parameter of the negative binomial distribution of 0.185 (95% Bayesian credible interval [CI], 0.16–0.21).

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Decision letter

  1. Prabhat Jha
    Reviewing Editor; University of Toronto, Canada

eLife posts the editorial decision letter and author response on a selection of the published articles (subject to the approval of the authors). An edited version of the letter sent to the authors after peer review is shown, indicating the substantive concerns or comments; minor concerns are not usually shown. Reviewers have the opportunity to discuss the decision before the letter is sent (see review process). Similarly, the author response typically shows only responses to the major concerns raised by the reviewers.

Thank you for sending your work entitled “Targeting Malaria Transmission Control: Predicting Mosquito Infectivity from Plasmodium falciparum Gametocyte Density” for consideration at eLife. Your article has been favorably evaluated by Prabhat Jha as Senior editor and 3 reviewers, one of whom was Vasee Moorthy.

The reviewers discussed their comments before we reached this decision and the Senior editor has assembled the following major substantive comments to help you prepare a revised submission.

1) Estimating Gametocyte Density From QT-NASBA Assays

I do not know the inverse regression problem well and can appreciate that there are alternative approaches. The supplementary file indicates some uncertainty in the choice of this method “for discussion see Osbourne, 1991” but this is not explained. It would be useful to give brief reasons why the particular method was chosen.

The authors say that QT-NASBA may detect only female gametocytes and elsewhere that the sex ratio of P falciparum can change to optimise transmission. This may alter the shape of the relationship between QT-NASBA results and the estimated gametocyte densities.

It would be useful to validate the model for estimating gametocyte densities from QT-NASBA assays for data other than that to what was fitted.

Figure 1A shows a double hump in the relationship between gametocytes in the blood sample and mosquito infectivity. This seems unintuitive for a biological process. I wondered to what extent this might be forced by the functions chosen. Does adding extra flexibility result in the same double hump?

Figure 1: An elegant presentation of this data. In the inset to part A there is a large clump of points on the x axis and it is not clear that the fit is good enough to be useful. For me this graph highlights the importance of understanding how different the line is in this model to other lines, and how much better this model fits, than the models that are the next best fit.

Please provide some discussion of this in the text (in addition to supplementary information). A listing of alternative models with their goodness of fit on a quantitative basis should be included in the manuscript itself.

The uncertainty range is huge, and ranges from very little increase in infectivity with increasing gametocyte density, to very steep increases in infectivity with increasing gametocyte density. What steps do the authors propose are taken to reduce the uncertainty range?

2) Mosquito Infectivity

Equation A4 includes a parameter called kappa. This is confusing since kappa is conventionally used to represent the infectious reservoir.

Results: The results for age are presented as if there was evidence of an effect, but the confidence intervals suggest otherwise.

The rationale for adjusting for asexual parasite density was not clear. It “improved model fit” but the aim here is not the best fitting model so much as to describe the relationship between estimated gametocyte density and infectivity. Including asexual parasite density as a main effect modifies the interpretation of the parameters.

3) The Paper Estimates the Relationship Between Gametocyte Densities and Age

The data comes from a cross-sectional survey of 412 people in a village in Burkina Faso. There is likely to be seasonal variation in parasite densities and the relative densities in adults and children may vary seasonally due to different levels of acquired immunity. It should be stated when the survey was carried out and also that a single survey is a potential limitation.

4) Efficacy Definition

A major issue is to be quite clear what is meant by efficacy throughout the paper. This lack of clarity permeates this entire technical area (i.e., not only this manuscript) and causes a lot of confusion. I believe most or all instances of use of the word efficacy in this manuscript refers to a % reduction in the gametocyte density. How does this relate to % reduction in proportion of infected mosquitoes and how will this translate to reduced incidence of human infection? This is a critical discussion that is omitted. Discussion of efficacy without some idea what this translates to in terms of reduction in human infection is rather misleading.

If the authors mean reduced transmission from humans to mosquito, I suggest they use a narrative phrase, such as this and do not use the word efficacy at all.

5) Data/Model Description

The data the model was fitted to was not well described (“repeatedly testing samples with known gametocyte density”). I also had trouble working out whether the fitting was simultaneous with the model for mosquito infectivity or not. The text mentions that the “estimates were used in the functions below” but the table presents all parameters together. My concern is that that the error in the estimated gametocyte densities, from the QT-NASBA results and also the chance of gametocytes being contained in the blood sample, is incorporated.

The model and parameters are not explained. The methods section merely mentions “a mathematical model” and the supplementary file gives an equation involving three parameters, but it is not clear how the equation was derived or what the parameters represent. A smooth curve is produced, but the relationship does not necessarily have to be smooth: for example, pregnant women may have higher parasitaemia and affect the age-curves.

Figure 2A suggests that the relationship does not fit well for two of the age groups. This should be acknowledged and possible reasons mentioned.

A negative binomial error structure was used. The Figure 2—source data 1 table suggests that there were some zeros in the data and that the densities are skewed. Did the residuals suggest a good fit? If there are many zeros, then a zero-inflated negative binomial model may be more appropriate.

6) Acknowledge Limitations

On a couple of occasions, the authors make statements that are not backed up by evidence from this paper.

Discussion. It is implied that this work gives direct evidence of the effectiveness of parasite strategies at low gametocyte densities such altering the sex ratio, but no direct information on strategies is obtained.

Although the paper does include an elegant sampling approach to quantify uncertainties, Although the paper does include an elegant sampling approach to quantify uncertainties, there should be a beefed up discussion of the uncertainties underlying this work. The authors should provide more discussion on what is known about which parameters in the input data are driving uncertainty, how much confidence they have that the model they chose is superior to alternative models, and on what basis they made this decision. There should also be a discussion on the limitations related to the differences between mosquitoes used in feeding assays, and wild-type mosquitoes.

Although uncertainty is presented in one of the figure panels, and this is very helpful, uncertainty ranges/confidence intervals should be quoted in the text where figures are provided.

We suggest that the authors consider how the paper can act as an aid to conceptualising the components underlying person-to-person transmission in field settings, and to gaining a better understanding of the uncertainties related to the component of transmission that their results speak to, and also to gain a better understanding of what the existing data gaps are that must be filled in order that the research community can determine how to test new transmission-reducing interventions, and quantify their effect on person-to-person transmission.

https://doi.org/10.7554/eLife.00626.010

Author response

1) Estimating Gametocyte Density From QT-NASBA Assays

I do not know the inverse regression problem well and can appreciate that there are alternative approaches. The supplementary file indicates some uncertainty in the choice of this method “for discussion see Osbourne, 1991” but this is not explained. It would be useful to give brief reasons why the particular method was chosen.

Inverse regression has been widely discussed in the statistical literature over the last 30 years. The majority of the controversy stems from when the gradient of the calibration line is very small (i.e., a small difference in TTP spans the whole range of gametocyte densities). This is not the case for our QT-NASBA results where the gradient is sufficiently large and no appreciable differences in the results are observed irrespective of the method used. To explain this fully to readers, we have included the following section in Supplementary file 1:

“Here we adapt this approach for use in a Bayesian hierarchical model. A number of different methods have been used to do this (see Hoadley (1970) and Hunter & Lamboy (1981) and for discussion see Osborne (1991)), each of which has a different approach for dealing with the problem of low beta1 values (i.e., a gentle gradient, which when used in equation A2 can generate infinitely large confidence interval estimates). Here we use the most parsimonious approach that does not require assignment of prior distributions to (the unknown) gametocyte densities. Rather, uncertainty in the estimated regression coefficients of equation A1 is propagated numerically via equation A2 to yield uncertainty in the estimated gametocyte densities. Since each of the calibration lines have a relatively steep gradient the difference between the different methods will be relatively minor and all will generate sensibly tight confidence interval estimates.”

The authors say that QT-NASBA may detect only female gametocytes and elsewhere that the sex ratio of P falciparum can change to optimise transmission. This may alter the shape of the relationship between QT-NASBA results and the estimated gametocyte densities.

The reviewers are correct in saying that if QT-NASBA does only detect female gametocytes then the relationship between gametocyte density (as estimated using another method which does not use Pfs25 mRNA such as microscopy) and mosquito infectivity might be different. However, we would suggest that in such a situation the number of female gametocytes would be a better prediction of infectivity than total gametocytes as each male gametocyte can fertilise up to 7 female gametocytes and the sex ratio will likely change to maximise female fertilization. Therefore, female gametocyte density represents the maximum potential infectivity.

We have changed the text accordingly:

“Even if the current QT-NASBA method does only detect female gametocytes, it will not change the qualitative conclusions of this study, nor their application if other studies use the same technique. However, direct comparison of our results with those of studies using different methods for quantifying gametocyte density should be aware that the shape of the relationship may differ.

It would be useful to validate the model for estimating gametocyte densities from QT-NASBA assays for data other than that to what was fitted.

We are unsure of the exact purpose of this validation procedure. If the reviewers meant to validate the QT-NASBA against another highly accurate method of determining gametocyte density, then we agree that this is a necessary process for us to have full confidence in the technique and our method of correcting for its uncertainty. However, we feel that our dilution procedure (whereby a gametocyte sample is diluted 5 times and each generates a TTP) provides a strong validation procedure within the model. In this example the fit of the calibration line to these dilutions is relatively poor, which is the reason behind the uncertainty round the individual gametocyte density estimates being so high. We are in the process of refining the nucleic acid extraction and amplification technique and we are generating increasingly precise QT-NASBA estimates, as validated by the dilution procedure. Since this has been done on samples not related to the current study we feel that including these here would confuse the message. If the reviewers are suggesting that we repeat the experiment with a different dataset then we do agree, and we have highlighted this in the text. However, to our knowledge no dataset currently exists.

Figure 1A shows a double hump in the relationship between gametocytes in the blood sample and mosquito infectivity. This seems unintuitive for a biological process. I wondered to what extent this might be forced by the functions chosen. Does adding extra flexibility result in the same double hump?

We would argue that the double hump is consistent with our current knowledge of the biological process and what has been observed in laboratory models. The first hump is a necessity for the line to go through point 0,0 (which it has to do as zero gametocytes cannot generate any oocysts) since ∼4% of mosquitoes are infected at very low gametocyte densities. The second hump is less certain since there are relatively few data points at very high gametocyte densities. There is evidence of a second plateau in these data using differently shaped functions, which is why the power and hyperbolic functions give better fits than the linear function. This is also consistent with what has been observed in animal models (see Sinden et al. 2009). That said, we agree that the functions chosen might influence the shape and that it could be more complicated than shown here.

More complex functions were tested (such as asymmetrical logistic functions), though they failed to converge adequately due to the additional parameters and the high uncertainty in gametocyte density. We have added a comment to stress that a wider range of functions should be tested when the accuracy of the data allows:

Discussion: “Further standardising the membrane feeding assay, improving the accuracy of QT-NASBA technique and the inclusion of additional covariates (such as estimates of gametocyte maturity which have been investigated in P. vivax (Chansamut et al., 2012)) would improve the accuracy of the relationship between gametocyte density and mosquito infectivity. It would permit to see if the patterns described here are consistent across time and space, and enable a wider range of functional forms to be tested.”

Figure 1: An elegant presentation of this data. In the inset to part A there is a large clump of points on the x axis and it is not clear that the fit is good enough to be useful. For me this graph highlights the importance of understanding how different the line is in this model to other lines, and how much better this model fits, than the models that are the next best fit.

Please provide some discussion of this in the text (in addition to supplementary information). A listing of alternative models with their goodness of fit on a quantitative basis should be included in the manuscript itself.

We agree that this information should be presented in the main text and we have added the following sentences to the Results section:

“Infectivity rises rapidly with increasing gametocyte density and by 1 gametocyte µl-1 ∼4% of all mosquitoes develop oocysts. The subsequent increase in infectivity with increasing gametocyte density is best described by the Gompertz model (deviance information criterion, DIC = 1034), which gave a significantly better fit than the linear (DIC=1073), power (DIC=1059) or hyperbolic (DIC=1062) functions (Figure 1A). The best fit model predicts that increasing density from 1 to 200 gametocytes µl-1 does not appreciably increase infectivity.”

The uncertainty range is huge, and ranges from very little increase in infectivity with increasing gametocyte density, to very steep increases in infectivity with increasing gametocyte density. What steps do the authors propose are taken to reduce the uncertainty range?

We agree that there is considerable uncertainty in the shape of the fitted curves reflecting the uncertainty in gametocyte density and the variability seen in mosquito feeding assays. We have included a few sentences outlining how the uncertainty can be minimised:

“The infectivity of mosquitoes to Plasmodium-infected blood is known to vary substantially within and between studies. This is in part due to methodological issues with the membrane feeding assay (Bousema, Churcher, Morlais, & Dinglasan, 2013) but also related to biological differences in the parasite-vector combination and the blood environment (Bousema & Drakeley, 2011). Further standardising the membrane feeding assay, improving the accuracy of QT-NASBA technique and the inclusion of additional covariates (such as estimates of gametocyte maturity which have been investigated in P. vivax (Chansamut et al., 2012)) would improve the accuracy of relationship between gametocyte density and mosquito infectivity.”

2) Mosquito Infectivity

Equation A4 includes a parameter called kappa. This is confusing since kappa is conventionally used to represent the infectious reservoir.

We have changed this parameter to eta to avoid confusion.

Results: The results for age are presented as if there was evidence of an effect, but the confidence intervals suggest otherwise.

Including age significantly improves the fit of the model though we appreciate that the 95% Bayesian credible intervals span 0. We have augmented the text in order to reflect this: “Including age improves the fit of the model suggesting age is an important confounder though the Bayesian credible intervals include zero.”

The rationale for adjusting for asexual parasite density was not clear. It “improved model fit” but the aim here is not the best fitting model so much as to describe the relationship between estimated gametocyte density and infectivity. Including asexual parasite density as a main effect modifies the interpretation of the parameters.

We appreciate that adjusting for asexual parasite density (as with age) may alter the shape of the relationship between gametocytes and mosquito infectivity, as asexual parasite density changes with increasing gametocyte density. However, the relationship between the density of gametocytes and asexual parasites is unlikely to remain constant (certainly not after certain drug treatments), so we feel that it is important to understand the impact of increasing gametocyte density without the influence of confounding variables. It also allows more accurate predictions of mosquito infectivity from cross-sectional surveys where these covariates are collected (such as in Burkina Faso). Since asexual parasite density and age are routinely collected we felt that the extra complexity of the model was warranted. We thank the reviewers for bringing this to our attention and we have changed the end of the Introduction accordingly to stress the rationale for doing so: “Other covariates that may influence infectivity such as asexual parasite density (measured by microscopy) and host age were also included to increase the ability of the model to predict human to mosquito transmission.”

3) The Paper Estimates the Relationship Between Gametocyte Densities and Age

The data comes from a cross-sectional survey of 412 people in a village in Burkina Faso. There is likely to be seasonal variation in parasite densities and the relative densities in adults and children may vary seasonally due to different levels of acquired immunity. It should be stated when the survey was carried out and also that a single survey is a potential limitation.

Thank you for highlighting this oversight. We have included a sentence to highlight the dangers of cross-sectional surveys in the main text. In the Results: “In the crosssectional survey in Burkina Faso children harbour more gametocytes than adults, with 10-year olds having 5 times as many gametocytes compared to 50-year olds (Figure 2A).” In the Discussion: “The reservoir of infection is likely to vary between endemicity settings and according to local malaria control practices. It is also likely to vary over time so multiple cross-sectional surveys may be required.”

4) Efficacy Definition

A major issue is to be quite clear what is meant by efficacy throughout the paper. This lack of clarity permeates this entire technical area (i.e., not only this manuscript) and causes a lot of confusion. I believe most or all instances of use of the word efficacy in this manuscript refers to a % reduction in the gametocyte density. How does this relate to % reduction in proportion of infected mosquitoes and how will this translate to reduced incidence of human infection? This is a critical discussion that is omitted. Discussion of efficacy without some idea what this translates to in terms of reduction in human infection is rather misleading.

If the authors mean reduced transmission from humans to mosquito, I suggest they use a narrative phrase, such as this and do not use the word efficacy at all.

We appreciate the ambiguity of the term “efficacy” in the literature and we thank the reviewers for highlighting it here. As suggested we have changed the terminology so that each time the word efficacy is used we clarify which life-stage is targeted using a narrative phrase. We have also included the following sentences in the Discussion, stating:

Figure 1B can be used to estimate how a reduction in the number of gametocytes will equate to a reduction in the proportion of mosquitoes becoming infected, and hence mosquito to human transmission. How this relates to the incidence of malaria and subsequent disease will depend, in part, on the degree of immunity in the human population.”

5) Data/Model Description

The data the model was fitted to was not well described (“repeatedly testing samples with known gametocyte density”). I also had trouble working out whether the fitting was simultaneous with the model for mosquito infectivity or not. The text mentions that the “estimates were used in the functions below” but the table presents all parameters together. My concern is that that the error in the estimated gametocyte densities, from the QT-NASBA results and also the chance of gametocytes being contained in the blood sample, is incorporated.

The uncertainty in the QT-NASBA technique includes the variability in the number of gametocytes in the blood sample. We have added a number of sentences stressing this and that the model for quantifying gametocyte density was fit at the same time as the model determining mosquito infectivity. This allows the uncertainty in gametocyte density to be reflected in the uncertainty of the best-fit line (and other covariates).

We have changed the “Mosquito infectivity” section of the methods:

“Gametocyte density was quantified using QT-NASBA and the uncertainty in the density estimates (the intra-assay variability) was quantified by fitting a hierarchical model to the 16 independent dilution series. In the Burkina dataset multiple assays were carried out on the same blood sample (an average of 2.63 assays per unknown blood sample). The mean gametocyte density from these multiple assays was taken to increase the accuracy of the estimates. The precise shape of the relationship was determined by fitting a range of different functional forms (a modified constant, linear, power, hyperbolic, sigmoid and Gompertz functions) and statistically determining which gave the best fit (a full description of the model is given in the Supplementary files). The model quantifying the uncertainty in gametocyte density estimates was fitted at the same time as the regression determining the relationship between gametocyte density (and other covariates) and the proportion of mosquitoes infected using Bayesian Markov Chain Monte Carlo methods. Fitting the models simultaneously enables the uncertainty in the gametocyte density estimates to be reflected in the uncertainty of the best fit model.”

The model and parameters are not explained. The methods section merely mentions “a mathematical model” and the supplementary file gives an equation involving three parameters, but it is not clear how the equation was derived or what the parameters represent. A smooth curve is produced, but the relationship does not necessarily have to be smooth: for example, pregnant women may have higher parasitaemia and affect the age-curves.

We have explained the age profile model in greater detail in the main text. We appreciate that certain groups, such as pregnant women, may have higher parasite densities, though we do not have more detailed information from the cross-sectional survey to include additional covariates within the model. Nevertheless, we feel that looking at things at a population level using smoothed curves is important for policy makers. The age profile takes into account how these heterogeneities, such as pregnancy status, changes with age and will result in elevated parasite levels in particular age groups.

The following section was included in the “Age profile of gametocyte density” section of the methods:

“A three parameter functional form was used which can describe an age profile which increases with age or peaks at an intermediate age. As above, a hierarchical model was used to estimate the gametocyte density and its associated uncertainty at the same time as the age profile allowing the full uncertainty of the shape of the function to be expressed (a full description of the data and model is given in the Supplementary files).”

Figure 2A suggests that the relationship does not fit well for two of the age groups. This should be acknowledged and possible reasons mentioned.

We agree that these two data points could be closer to the line, though the confidence intervals of the point estimate and the model do overlap. The functional form is flexible enough to pass much closer to these points though doing so generates a worse fit to these data. We feel that the main reason for the apparent discrepancy is the sampling from the highly overdispersed data, where missing the rare very high parasite densities by chance would result in lower mean estimates. Neither of these data points have particularly large sample sizes, which increases the probability of this happening. This overdispersion is taken into account in the fitting process and though other methods of central tendency could be presented in the figure (such as the geometric mean or median), since the model is showing the arithmetic mean we felt this was the best comparison. To highlight this we have added the following sentence to the legend of Figure 2: “In Figure 1A the [10-15) and [15-20) groups appear lower than the best fit line due to sampling artefacts generated by the highly overdispersed data.”

A negative binomial error structure was used. The Figure 2—source data 1 table suggests that there were some zeros in the data and that the densities are skewed. Did the residuals suggest a good fit? If there are many zeros, then a zero-inflated negative binomial model may be more appropriate.

Though there are a lot of zero counts, this is adequately described by the simple negative binomial distribution. A zero-inflated negative binomial distribution was fitted to the estimates of gametocyte density (used to generate the gametocyte age profile) using maximum likelihood, though the improved fit didn’t warrant the extra complexity (likelihood ratio test p value = 0.7). Diagnostics on the residuals indicate no major issues other than the high level of uncertainty. We have changed the following section of the Discussion to reflect this: “However, the highly overdispersed distribution of gametocytes between hosts (adequately described by the negative binomial distribution) means that in the population from Burkina Faso…”

6) Acknowledge Limitations

On a couple of occasions, the authors make statements that are not backed up by evidence from this paper.

We have removed the misleading statements such as those highlighted below.

Discussion. It is implied that this work gives direct evidence of the effectiveness of parasite strategies at low gametocyte densities such altering the sex ratio, but no direct information on strategies is obtained.

We were trying to suggest that transmission at very low gametocyte densities is evidence that the strategies of the parasite to maximise transmission were working. However we agree that this sentence could be misleading and so we have deleted it from the Discussion.

Although the paper does include an elegant sampling approach to quantify uncertainties, there should be a beefed up discussion of the uncertainties underlying this work. The authors should provide more discussion on what is known about which parameters in the input data are driving uncertainty, how much confidence they have that the model they chose is superior to alternative models, and on what basis they made this decision. There should also be a discussion on the limitations related to the differences between mosquitoes used in feeding assays, and wild-type mosquitoes.

We agree that the uncertainties and the difference between laboratory and wild mosquitoes should be stressed further. We have added the following section to the Discussion:

“Gametocyte density measurement error is likely to cause the majority of the uncertainty seen in model outputs though the complexity of the membrane feeding assay is also likely to contribute. The infectivity of mosquitoes to Plasmodium infected blood is known to vary substantially within and between studies. This is in part due to methodological issues with the membrane feeding assay (Bousema, Churcher, Morlais, & Dinglasan, 2013) but also related to biological differences in the parasite-vector combination and the blood environment (Bousema & Drakeley, 2011). Further standardising the membrane feeding assay, improving the accuracy of QT-NASBA technique and the inclusion of additional covariates (such as estimates of gametocyte maturity which have been investigated in P. vivax (Chansamut et al., 2012)) would improve the accuracy of the relationship between gametocyte density and mosquito infectivity. It would also permit to see if the patterns described here are consistent across time and space, and enable a wider range of functional forms to be tested. Care should also be taken when interpreting the results of feeding assays as the mosquitoes used and their biting behaviour is likely to be different from that seen in wild mosquitoes in field situations (Bousema et al., 2013).”

Although uncertainty is presented in one of the figure panels, and this is very helpful, uncertainty ranges/confidence intervals should be quoted in the text where figures are provided.

We agree and have added 95% Bayesian Credible Intervals where appropriate.

We suggest that the authors consider how the paper can act as an aid to conceptualising the components underlying person-to-person transmission in field settings, and to gaining a better understanding of the uncertainties related to the component of transmission that their results speak to, and also to gain a better understanding of what the existing data gaps are that must be filled in order that the research community can determine how to test new transmission-reducing interventions, and quantify their effect on person-to-person transmission.

We very much agree with the reviewers that this is an important topic. Person-to-person transmission is highly complex due to the complexities of the human immune response. Since this is not the topic of this manuscript, we concentrate our answers on the reservoir of infection (i.e., transmission, not necessarily incidence of malaria). We have augmented the following section of the Discussion to reflect our opinions on the subject:

“The contribution of different age groups to overall transmission will depend on local demography, age-dependent protection by malaria interventions (such as the use of bednets or time spent in the house and therefore personally protected by indoor residual spraying), and human to mosquito contact patterns. The last two of these are difficult to measure and poorly understood, reducing our ability to predict accurately the sources of infection. The reservoir of infection is likely to vary between endemicity settings and over time so multiple cross-sectional surveys may be required.”

https://doi.org/10.7554/eLife.00626.011

Article and author information

Author details

  1. Thomas S Churcher

    Department of Infectious Disease Epidemiology, Imperial College London, London, United Kingdom
    Contribution
    TSC, Conception and design, Analysis and interpretation of data, Drafting or revising the article
    For correspondence
    thomas.churcher@imperial.ac.uk
    Competing interests
    The authors declare that no competing interests exist.
  2. Teun Bousema

    1. Department of Medical Microbiology, Radboud University Nijmegen Medical Centre, Nijmegen, The Netherlands
    2. Department of Immunology and Infection, London School of Hygiene and Tropical Medicine, London, United Kingdom
    Contribution
    TB, Conception and design, Drafting or revising the article
    Competing interests
    The authors declare that no competing interests exist.
  3. Martin Walker

    Department of Infectious Disease Epidemiology, Imperial College London, London, United Kingdom
    Contribution
    MW, Acquisition of data, Drafting or revising the article, Contributed unpublished essential data or reagents
    Competing interests
    The authors declare that no competing interests exist.
  4. Chris Drakeley

    Department of Immunology and Infection, London School of Hygiene and Tropical Medicine, London, United Kingdom
    Contribution
    CD, Conception and design, Drafting or revising the article
    Competing interests
    The authors declare that no competing interests exist.
  5. Petra Schneider

    Institutes of Evolution, Immunology and Infection Research, University of Edinburgh, Edinburgh, United Kingdom
    Contribution
    PS, Acquisition of data, Drafting or revising the article
    Competing interests
    The authors declare that no competing interests exist.
  6. André Lin Ouédraogo

    Department of Biomedical Sciences, Centre National de Recherche et de Formation sur le Paludisme, Ouagadougou, Burkina Faso
    Contribution
    ALO, Acquisition of data, Drafting or revising the article, Contributed unpublished essential data or reagents
    Competing interests
    The authors declare that no competing interests exist.
  7. María-Gloria Basáñez

    Department of Infectious Disease Epidemiology, Imperial College London, London, United Kingdom
    Contribution
    M-GB, Conception and design, Drafting or revising the article
    Competing interests
    The authors declare that no competing interests exist.

Funding

European Commission FP7 Collaborative Project (HEALTH-F3-2008-223736)

  • Thomas S Churcher
  • María-Gloria Basáñez

AFIRM grant from the Bill & Melinda Gates Foundation (OPP1034789)

  • Teun Bousema
  • Chris Drakeley
  • André Lin Ouédraogo

The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.

Reviewing Editor

  1. Prabhat Jha, University of Toronto, Canada

Publication history

  1. Received: February 11, 2013
  2. Accepted: April 9, 2013
  3. Version of Record published: May 21, 2013 (version 1)

Copyright

© 2013, Churcher et al.

This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.

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