Required duration of mass ivermectin treatment for onchocerciasis elimination in Africa: a comparative modelling analysis

Background The World Health Organization (WHO) has set ambitious targets for the elimination of onchocerciasis by 2020–2025 through mass ivermectin treatment. Two different mathematical models have assessed the feasibility of reaching this goal for different settings and treatment scenarios, namely the individual-based microsimulation model ONCHOSIM and the population-based deterministic model EPIONCHO. In this study, we harmonize some crucial assumptions and compare model predictions on common outputs. Methods Using a range of initial endemicity levels and treatment scenarios, we compared the models with respect to the following outcomes: 1) model-predicted trends in microfilarial (mf) prevalence and mean mf intensity during 25 years of (annual or biannual) mass ivermectin treatment; 2) treatment duration needed to bring mf prevalence below a provisional operational threshold for treatment interruption (pOTTIS, i.e. 1.4 %), and 3) treatment duration needed to drive the parasite population to local elimination, even in the absence of further interventions. Local elimination was judged by stochastic fade-out in ONCHOSIM and by reaching transmission breakpoints in EPIONCHO. Results ONCHOSIM and EPIONCHO both predicted that in mesoendemic areas the pOTTIS can be reached with annual treatment, but that this strategy may be insufficient in very highly hyperendemic areas or would require prolonged continuation of treatment. For the lower endemicity levels explored, ONCHOSIM predicted that the time needed to reach the pOTTIS is longer than that needed to drive the parasite population to elimination, whereas for the higher endemicity levels the opposite was true. In EPIONCHO, the pOTTIS was reached consistently sooner than the breakpoint. Conclusions The operational thresholds proposed by APOC may have to be adjusted to adequately reflect differences in pre-control endemicities. Further comparative modelling work will be conducted to better understand the main causes of differences in model-predicted trends. This is a pre-requisite for guiding elimination programmes in Africa and refining operational criteria for stopping mass treatment. Electronic supplementary material The online version of this article (doi:10.1186/s13071-015-1159-9) contains supplementary material, which is available to authorized users.


ONCHOSIM: modelling the transmission and control of onchocerciasis
ONCHOSIM is a computer program for modelling the transmission and control of the tropical parasitic disease onchocerciasis, or river blindness. It was developed in collaboration with the Onchocerciasis Control Programme in West Africa (OCP, 1974(OCP, -2002, as a tool in the evaluation and planning of control operations. The African Programme for Onchocerciasis Control (1995-2015) also adopted the program as a tool. The model comprises a detailed description of the life history of the parasite Onchocerea volvulus and of its transmission from person to person by Simulium flies. The effects of different control strategies, based on larvicide application and chemotherapy (ivermectin), on the transmission and on the disease symptoms can be evaluated and predicted. Since its conception, the model has been used for multiple applications as described elsewhere [1-11].

ONCHOSIM as a variant of WORMSIM
The ONCHOSIM modelling framework for simulating transmission and control was first described by Plaisier et al in 1990 [1]. In subsequent years, Erasmus MC has developed similar models for lymphatic filariasis (LYMFASIM [12]), schistosomiasis (SCHISTOSIM [13]), and most recently also soil-transmitted helminthiasis [14]. The initial models were implemented in three separate, disease-specific computer programs (written in C++), although all had very similar features. Recently, WORMSIM was developed as a generalised framework for modelling transmission and control of helminth infections in humans, and it contains extra features to allow the simulation of these different infections. Through adjustment of input specifications on structural assumptions and the value of model parameters, WORMSIM can be made to represent onchocerciasis or other diseases. We continue to use the name ONCHOSIM to denote the WORMSIM model variant for onchocerciasis, because ONCHOSIM is well known in the field and we want to make clear that the model is still the same despite a different software implementation.

This document
In section 2, we briefly describe the general properties of the WORMSIM modelling framework, followed by a formal description in section 3. In footnotes we highlight details and alternative options that are not evident from the mathematical descriptions. In section 4, we provide an overview of the probability distributions, functional relationships, and parameter values that are used in ONCHOSIM. In section 5, we provide instructions for installing and running WORMSIM. In sections 6 and 7, we present annotated WORMSIM input and output files, respectively.

Implementation of WORMSIM in software
WORMSIM was programmed in Java using object-oriented principles. Individual people and mature worms are modelled as distinct objects. WORMSIM is event-driven, which means that time progresses as a result of events (although monthly events are used for most processes).
The main advantages of the implementation in Java are high code quality and therefore easier maintenance and extension. Model input parameters are specified in a structured XML-file, which is automatically validated using an XML Schema before the start of a set of simulations.
The WORMSIM framework is very flexible in that it allows the user to choose probability distributions for stochastic processes (Appendix I) and functional relationships for deterministic processes (Appendix II), and to change the associated parameter values. Table  A1-1 provides an overview of the probability distributions, functional relationships, and parameter values used in this study.

Human demography
The human population dynamics is governed by birth and death processes. We define F(a) as the probability to survive to age a. The cumulative survival for intermediate ages is obtained by linear interpolation.
The expected number of births (per year) at a given moment t is given by: with: , number of women in age group a at time t annual birth rate in age-group a.
number of age-groups considered.
Each month, is adapted according to the number of women and their age-distribution.
Once every year, the total number of human individuals is checked; if the total number is larger than a user-defined value, a fraction (also user-defined) is randomly removed from the simulation.
The population distribution resulting from the aforementioned parameters is illustrated in Figure A1

Transmission of infection
In WORMSIM, transmission between individuals is mediated by a conceptual cloud, which either represents a vector population or an environmental reservoir of infection. Individual human hosts are exposed to the infective material in the cloud at varying rates, given their age, sex, and personal factors. Vice versa, individual hosts contribute infective material (larvae or eggs) to the cloud, the amount depending on the number and reproductive statues of worms in the individual, as well as an individual host's contribution rate (depending on age, sex, and personal factors). The amount of infective material in the cloud is updated in discrete monthly time steps. Below, we describe how WORMSIM simulates a full transmission cycle: human exposure to infection and acquisition of worms, dynamics of infection within humans, contribution of infective material to the cloud, and within-cloud dynamics of infective material.

Exposure to infection and acquisition of new worms
First, we define the overall force of infection acting on the human population in month t as a function of the current absolute amount of infective material in the cloud : Here, (zeta) is a scalar representing the overall exposure rate, and v is the probability that an infective particle in the reservoir successfully develops into a parasite life stage that is capable of infecting a human host. a Next, we define the force of infection acting upon individual i of age a and sex s as: Here, is the relative exposure of an individual, taking into account age a and sex s, as well as personal factors: with: , " Relative exposure of person with age a and sex s, defined as a linearly interpolated function of user-defined exposure rates for a finite set of ages (for each sex).
Exposure index of person i, which captures personal factors related to e.g. behaviour and occupation. is assumed to follow a gamma distribution with mean 1.0 and shape and rate (or 1/scale) equal to # $% . The exposure index of a person remains constant throughout lifetime. b a is perfectly negatively correlated with transmission probability v, success ratio sr, relative biting rate rbr, and vector zoophily z. See also the section on contribution of infective material to reservoir. For filarial transmission, we set = 1, quantify v based on vector biology, set sr to a constant value, and calibrate transmission with rbr. For STH, we set = ' = " = 1 , and calibrate transmission with , which has a more natural explanation in the STH context (exposure to the reservoir) than rbr.
b If desired, other continuous probability function can be chosen.
Finally, a person i is assumed to become infected in month m, according to a Poisson process with rate equal to • " • ()) . Here, success ratio sr is a constant representing the probability that an inoculated infective particle will develop into a macroparasite. a Finally, ()) represents the impact of the host's immune response in montht t on incoming infections [2]: , -.+, = , + 0 1++ • , -.+, − 1 Here, # ++ is the effect of immunity and )) is an individual host's capacity to elicit an immune response, drawn from a positive bounded probability distribution with mean one (e.g. a gamma distribution with equal shape and rate (1/scale) parameters). , -.+, is the cumulatively experienced worm burden of host i in month t, , is the worm burden of host i in month t, and 0 1++ represents the immunological memory span (0 1++ = 2 3 45 6 /8 9:: , where ; ++ is the desired half-life (in months) of the immunological response; vice vera ; ++ = − ln 2 ln 0 1++ ⁄ ).

Within-host dynamics of infection
For convenience, in this section we drop the subscript i for individual humans. The lifespan of male and female parasites within human hosts is a random variable: @ ~Weibull G HI , # HI , with mean G HI years and shape # HI . c Once parasites come of age (i.e. when they pass the prepatent age JJ), female worms can start producing larvae or eggs, and males can inseminate female worms. The reproductive capacity , of a patent female worm of age a at time t is calculated as follows (in absence of drug effects): with: L Potential reproductive capacity of a female worm, A years after reaching patency, defined as a linearly interpolated function of user-defined values for a finite set of ages.
) Mating factor at time t K The exponential fecundity coefficient at time t, defined as K = 2 3M ! 8 N , where W(t) is the number of adult worms (males and females) in a given host at time t, and ; O ∈ ℝ R quantifies the amount of negative density dependence. If ; = 0, there is no exponential saturation in egg production (K = 1).
To produce larvae or eggs, a female worm must be inseminated each reproductive cycle T, defined in terms of months. If insemination took place less than T months ago, then ) = 1. Otherwise, the probability of insemination or reinsemination U V in month t is given by: c For readers used to the other commonly used parameterization of the Weibull distribution in terms of shape k and scale λ, shape k is α Tl (as described in this document) and scale ; = G HI Γ 1 + 1 # HI ⁄ ⁄ .
with: , the number of male (, + ) or female (, ) parasite in the human at time t J the number of female worms that a male worm can inseminate per month d If no insemination takes place then ) = 0 and the female worm has a new opportunity to be inseminated in the next month t + 1. If insemination occurs in month t i then ) = 1 The density of larvae (e.g. per skin snip) or eggs (e.g. per gram faeces) " in a host at time t is calculated by accumulating the production of all female parasites over the past Tm months within that host: with: 2 the effective parasite load at time t. This intermediate variable describes the female parasite load obtained by weighting each worm according to the mfproductivity during the past Tm months.

b .
A function that returns the total amount of infective material produced by female parasites. For onchocerciasis, we assume that b . is a linear function through the origin with slope 7.6 mf per fully productive adult female worm. e e f dispersal factor of female parasite j. This is a random variable (mean 1.0) drawn for every "newborn" worm, and accounts for differences in the contribution of female worms to the density at the standard site of the body where samples are taken or vectors bite. @) (fixed) lifespan of larvae or eggs within the host in terms of months.
number of parasites alive during at least one of the months t-1,…,t-Tm.

Host contribution of infective material to the cloud
Given the density of larvae or eggs " in all host in month t, the total amount of infective material that is contributed to the cloud by the host is defined as d When the user specifies a negative value for male potential, female worms can produce larvae or eggs in the absence of male worms. e Alternatively, other functional relationships between el and sl can be defined. Saturating functionsshould not be used when @) > 1, as this will cause partial saturation of female worm productivity in month t, given the output in months − 1 through − @). This will be alleviated in a future version of WORMSIM by setting " Here, k' is the average contribution rate in month t (monthly biting rate for filarial infections), allowing the user to define a seasonal pattern (in absence of vector control). The relative biting rate rbr is used to scale this seasonal pattern to some desired level. a The function l . returns the amount of infective material taken up by the cloud given the density of eggs or larvae " in a host, possibly in a density dependent manner to represent e.g. limited vectorial capacity to transmit infection. f Last, m is the relative contribution of an individual, given age, sex, and personal factors: with: m , " Relative contribution of person with age a and sex s, defined as a linearly interpolated function of user-defined exposure rates for a finite set of ages (for each sex).
m Contribution index of person i, which captures personal factors related to e.g. behaviour and occupation. m is assumed to follow a gamma distribution with mean 1.0 and shape and rate (or 1/scale) equal to # no . The contribute index of a person remains constant throughout lifetime.
In WORMSIM default assumption is that m = , unless separate distributions are defined.

Dynamics of infective material in the cloud
For the dynamics of infective material in the cloud we define a deterministic, discrete model: Each month, new infective material is added to the cloud, and a fixed proportion p of the infective material from the previous month is carried over, assuming exponential survival of infective material. The average life span of infective material in the cloud is then defined as −1 ln p ⁄ months. To simulate filarial transmission, we set p = 0, such that the cloud represents a vector population in which larvae survive for much shorter than a month. To simulate hookworm or other STH infections, we set 0 < p < 1, such that the cloud represents an environmental reservoir of infection in which infective material survives for a nonnegligible time.

Morbidity
The event of a person developing symptoms at age a depends on the accumulated parasite load (2 T) of a person: f For filarial infection, l . typically is a density-dependent function of " to represent limited vectorial capacity to transmit infection, whereas for STH, we take l . to be the identity function. Each person has a threshold level 2 T (denoted as T) at which a person goes blind. T follows a probability distribution: T~Weibull G $I-, # $I-, with mean G $I-and shape # $I-. Person i goes blind at age a when: At that moment the remaining lifespan at age a is reduced by a factor rl which follows a userdefined distribution on [0,1]. g

Mass treatment coverage and compliance
The primary characteristic of a certain ivermectin mass treatment w is the coverage m s (fraction of the population treated). However, a difficulty in calculating individual chances of participation is that there are several exclusion criteria for the drug. Moreover, compliance to treatment differs from person to person. Exclusion criteria can be either permanent (chronic illness) or transient (e.g. related to age or pregnancy). We define the eligible population as the total population minus a fractionthat is permanently (lifelong) excluded from. The coverage among the eligible population m′ s is now given by: Here, m′ s cannot be larger be than one (i.e. is capped off at one).
To capture transient contra-indications and other age-and sex-related factors for participation in mass treatment, we the age-and sex-specific relative compliance T u v, " . Note that in T u v, " only the ratio between the values for the different groups is relevant.
Now, the coverage T v, ", w in each of the age-and sex-groups at treatment round w is calculated as: with: v, ", w Number of individuals eligible to treatment in age-group k and sex s at treatment round w.
w Total number of eligible individuals at treatment round w.
Finally, the probability to participate in treatment round w for an eligible person i of agegroup k and sex s is given by: with: T Personal compliance index. This is considered as a lifelong property and is generated by a uniform distribution on [0,1] Note that for all k and s the average value of U ,s equals T v, ", w . Now, in WORMSIM we define 3 coverage models. In model 0, the probability to be treated is as given in equation g For STH modelling, we do not use this feature, and thus assume zero excess mortality from infection. (17). In model 1, the probability is equal to T v, ", w and the compliance index T is ignored. The simplest model is model 2 in which the treatment probability simply equals C' w . All models take account of a fractionof permanently excluded persons. Figure A1-2 illustrates the impact of different assumptions about compliance patterns on the proportion of the population that has never been treated after a certain number of treatment rounds.
Figure A1-2. Relation between compliance patterns and proportion of population that has never been treated. For simplicity, here we assume that compliance is independent of age and sex. Random compliance (solid line) means that eligible individuals participate completely at random (compliance model 1 or 2 in WORMSIM, depending on whether age and sex-patterns are required). Systematic compliance (dotted line) means that an individual either always participates (if eligible) or never (compliance model 1 or 2 in WORMSIM, combined with a fraction of excluded people equal to one minus the target coverage). The mixed compliance pattern (dashed line) means that some individuals are systematically more likely to participate than others (but everyone will participate at some point; compliance model 0 in WORMSIM).

Parasitological effects of treatment
In WORMSIM, drug treatment affects parasites in three main ways. First, a drug may instantly kill a proportion of larvae or eggs present in a host. This proportion is either fixed or a randomly drawn from a user-defined probability distribution for each host and treatment.
Second, a drug may instantly kill pre-patent and adult worms with probability ) in host i. A worm j dies when a random variate f on [0,1] (redrawn for every new treatment) is smaller than or equal to ) .
Third, a drug may temporarily and/or permanently (and cumulatively) reduce the reproductive capacity of female worm by a proportion e in host i. In case of a temporary effect, the reproductive capacity will restore within a period @ to its maximum value (in case of any concomitant permanent reductions, reproductive capacity will regenerate to the new, permanently reduced maximum value. The second and third effect are jointly defined as follows: f q c f , d Reproductive capacity of female worm j had person i not been treated at the last round, z months ago.

"
Shape parameter of the recovery function.
In addition to this, we explicitly consider that some persons (a user-defined random fraction of the treated population) do not at all react to the drug during a certain treatment due to malabsorption (e.g. due to vomiting or diarrhoea).

Vector control
Vector control is modelled as a reduction of the monthly biting rates during a given period of time. A period of vector control h is specified as the year + month of the beginning of the strategy and the year + month of the end of a strategy. If a certain month during a period of d days larvicides have been applied, then the reduction in k' in that month equals d/30 x 100%.

Surveys
During the simulation, user-defined surveys will take place. During a survey, for all simulated individuals the actual number of male and female worms is recorded, and a diagnostic test is simulated to detect infective material (larvae, eggs). For the diagnostic test, the expected amount of infective material per sample (e.g. microfilariae per skin snip, or eggs per gram faeces) for an individual is given by " .
h Multiple periods of vector control can be specified, each with its own effectiveness.
The actual number of infective particles (microfilariae, eggs, etc.) in the sample is assumed to follow a discrete distribution like a Poisson or negative binomial distribution, with mean equal to " . i At each epidemiological survey a user-defined number of samples are taken from all simulated persons, for which the results are averaged (per simulated person). The results of such a survey are post-processed to arrive at age and sex-specific prevalences and intensities of infection.

Simulation warm-up
In general, before starting simulation of interventions in ONCHOSIM, a 200-year warm-up period is simulated, such as to allow the human and worm population to establish equilibrium levels, given the parameters for average fly biting rate and inter-individual variation in exposure to infection. At the start of the warm-up period, an artificial force of infection is simulated for a user-defined number of years, allowing worms to establish themselves in the human population (here: 4 worms per person per year for 7.5 years). After the 200 warm-up years, the simulated infection levels are no longer correlated with the initial conditions at the start of the warm-up period.

General transmission parameters
Relative biting rate (rbr) Varied between simulations to modify the annual biting rate.
Overall exposure rate of human hosts to central reservoir of infection ( ) Not applicable to onchocerciasis transmission ( = 1)

Seasonal variation in contribution to reservoir (mbr)
104%, 91%, 58%, 75%, 75%, 66%, 102%, 133%, 117%, 128%, 146%, and 105% times the average monthly biting rate (January-December) [16] Transmission probability (v), i.e. the probability that an infective particle in the reservoir successfully develops into a parasite life stage that is capable of infecting a human host = 0.07345; see reference for the derivation of this value, given parameters for fly biology and development of infective L3 larvae within the fly.

Individual relative exposure to flies
Variation in by age and sex (Exa) Zero at birth, linearly increasing between ages 0-20 from 0 to 1.0 for men and from 0 to 0.7 for women, and then constant from the age of 20 years onwards [17] Variation due to personal factors (fixed through life) given age and sex (# $% ) Gamma distribution with mean 1.0 and shape and rate equal to 3.5 [17], unpublished data from OCP

Individual relative contribution to infection in the fly population
Variation by age and sex (Coa) m = ; individual contribution and exposure to the cloud are perfectly correlated, given they are governed by the same fly bites.

Assumption
Variation due to personal factors (fixed through life) given age and sex (# no ) m = ; individual contribution and exposure to the cloud are perfectly correlated, given they are governed by the same fly bites.

Parameter Value Source
Exponential saturation of individual female worm productivity per worm present in host (; O ) ; O = 0, i.e. no exponential saturation. Assumption

Infection dynamics in the cloud
Cloud uptake of infectious material (l . ) Exponential saturating function with parameters a = 1.2, b = 0.0213, and c = 0.0861 (see appendix II for the definition of an exponential saturating function). [3], which refers to [28,29] Monthly cumulative survival of infective material in the central reservoir (p) 0%; i.e. the cloud represents a cloud of vectors that transmit infection within the same month.

Mass treatment coverage
Coverage (m s ) User-defined.

Vector control
Timing Not used.

Coverage
Not used.

Surveys
Dispersal factor for worm contribution to measured density of infective material (d) Exponential distribution with mean 1 [2] Variability in measured host load of infective material (eggs per gram faeces) Poisson distribution with mean "" [2] 5 Instructions for installing and running WORMSIM and run your shell script with:

Installing WORMSIM
.\my_test to do 100 runs and aggregate the output of these runs. and run your shell script with:

Mac OS X or Linux
./my_test.sh to do 100 runs and aggregate the output of these runs

Output options
The -d output option will make WORMSIM produce additional detailed output. This output is found in *X.txt and *Y.txt (for instance example_onchoX.txt and example_onchoY.txt).
The -n output option suppresses all output except the *.log output (e.g. example_oncho.log). Either output option can be added to the run command as follows: ./run.sh my_oncho.xml 0 99 -d or ./run.sh my_oncho.xml 0 99 -n

Annotated input file
The WORMSIM inputfile is an XML file that can be edited with any text editor or alternatively, with an XML editor (such as Oxygen XML Editor). The advantage of using the XML format is that any input file can be validated against an XML Schema (a formal specification of the grammar used in the specific XML dialect used for the WORMSIM input file).
The Wormsim input file is documented with an annotated example (annotated-example.xml). The XML Schema (wormsim.xsd) is documented in great detail in schema-documentationwormsim.pdf (provided with the software).
A copy of an annotated input file for ONCHOSIM is included below, split into fragments that cover the different elements of the input file (gray-shaded boxes). Together, these fragments constitute a complete input file.

<Simulation>
The <simulation> element specifies the start year of the simulation, the timing of surveys (i.e. output moments), the number of skin snips taken at each survey and the age classes for output.

<Demography>
The <demography> element defines life tables for the male and female population, the maximum population size (above which random persons will be removed), a fertility table, and the initial population size and age distribution. See comments below.

<Blindness>
The <blindness> element defines the parameters for development of morbidity ("blindness" as originally developed for ONCHOSIM, where we specify a threshold of cumulative exposure to microfilaria) and the effect of morbidity on the remaining life expectancy.

<Exposure.and.contribution>
The <exposure.and.contribution> element defines the parameters for the exposure of humans to a vector (or infectious reservoir) and the contribution of humans to the vector cloud (or infectious reservoir). See comments below.

<Worm>
The <worm> element defines parameters for worm lifespan, prepatent period, mating between M and F worms, age-dependent production of microfilaria, mf density per worm and skin dispersal.

<Fly>
The <fly> element defines parameters that determine the successful uptake and development of L1 larvae into infective L3 larvae and also determins the fly biting rate.

<Mass.treatment>
The <mass.treatment> element defines parameters for the timing of mass treatment rounds, individual compliance (permanent, temporary and age dependent), and effects of ivermectin on mature worms, mf production by F worms and on mf.

Annotated output file
The WORMSIM standard output (e.g example_oncho.txt) is a tab delimited text file with the following columns: year : time (years) N : nr examined N20 : nr examined > 20 yrs old mf+ : percentage with positive skin snip mf+20 : percentage with positive skin snip (> 20 yrs) mfPr : age/sex standardized mf prevalence aNmf : arithmetic mean nr mf per skin snip aNmf20 : arithmetic mean nr mf per skin snip (> 20 yrs) gNmf : geometric meain nr mf per skin snip CMFL : geometric meain nr mf per skin snip (> 20 yrs) bl : percentage blind bl20 : percentage blind (> 20 yrs) blPr : age/sex standardized prevalence of blindness w+ : percentage with at least one adult female worm w+20 : percentage with at least one adult female worm (> 20 yrs) aNw : arithmetic mean nr of adult female worms per person aNw20 : arithmetic mean nr of adult female worms per person (>20 yrs) mbr : monthly fly biting rate in previous month mtp : monthly transmission potential in previous month L1 : mean nr of L1 larvae per 1000 biting flies in previous month L3 : mean nr of L3 larvae per 1000 biting flies in previous month foi : mean force of infection (nr of new adult worms per person) in prev. month The WORMSIM log output (e.g example_oncho.log) is a tab delimited text file with output for each simulation run in the following columns: seed : the seed of the random number generator (i.e. run nr) of that specific run year : time (years) mf+ : fraction with positive skin snip mf5+ : fraction with positive skin snip > 5 yrs w+ : fraction with at least one adult female worm N : nr examined aNmf : arithmetic mean nr mf per skin snip aNmf20 : arithmetic mean nr mf per skin snip (> 20 yrs) N20 : nr examined > 20 yrs old CMFL : geometric meain nr mf per skin snip (> 20 yrs) The detailed *X.txt output (e.g example_onchoX.txt) contains sex and age specific output with the following columns: year : time (years which results in the following output columns: -1 : percentage of M in that age group with average skin snip count < 0.5 -1 : percentage of M in that age group with 0.5 <= average skin snip count < 1 -2 : percentage of M in that age group with 1 <= average skin snip count < 2 -4 : percentage of M in that age group with 2 <= average skin snip count < 4 -8 : percentage of M in that age group with 4 <= average skin snip count < 8 - 16 : percentage of M in that age group with 8 <= average skin snip count < 16 -32 : percentage of M in that age group with 16 <= average skin snip count < 32 -64 : percentage of M in that age group with 32 <= average skin snip count < 64 -128 : percentage of M in that age group with 64 <= average skin snip count < 128 -256 : percentage of M in that age group with 128 <= average skin snip count < 256 -512 : percentage of M in that age group with 256 <= average skin snip count < 512 -1e9 : percentage of M in that age group with 512 <= average skin snip count < 1e9 followed by the same categories for F.