Aplicación a lo contenido en la Ley 1448 del 2011 en el municipio de Soacha 103

also be useful if the data are considered to be bi-modal. This allows the observed data to be derived from one of two groups; either the ‘infected’ continuous distribution or a group

with a true mean of zero which are truly uninfected (Martin et al.,2005). Alternatively, the

distribution could be used to approximate a bi-modal distribution with one high mode and one low mode if the majority of counts from the latter are expected to be zero.

Comparison of the inference made using each distributional assumption first requires that the optimal parametrisation for each of the models be defined. However, no such pre-defined models are readily available for use with software capable of implementing an analysis using

MCMC, such as WinBUGS (Lunn et al., 2000) and Just Another Gibbs Sampler (JAGS)

(Plummer,2008). The overall aim of the work described here was to compare the accuracy of several variants on each parameterisation using simulated data. The specific objective was to identify the formulations with the best defined properties, for use in later chapters of this thesis.

2.2

Defining parasite distributions

2.2.1 Introduction

There are three factors that are generally of interest when examining the distribution of parasites between and within hosts. The most obvious is the mean, which can either be the mean egg count or mean parasite count. This represents the expected number of parasites or parasite eggs in an animal, and is related to the total number of parasites or eggs by the number of animals in the group. It is therefore arguably a more useful measure of the mid-point of the distribution than the median or mode, neither of which are frequently used in the parasitology literature. The mean can either be thought of as the ‘population mean’, defined here as the mean of the distribution from which all animals in the infected group are derived, or as the ‘true sample mean’, defined here as the average of the true mean egg shedding rate, or parasite burden, of the sampled animals. The two measures are identical when all animals from the population are sampled, but when only a sample is taken from the population the two are distinct, as the former describes a population from which we only have a sample, and the latter describes the observed mean of the sample of animals. The uncertainty associated with estimating the population mean will always be greater than that for the true sample mean, as the population mean also includes animals which were not sampled. For the majority of the work here the mean referred to is the population mean; the

concept of the true sample mean will be revisited inChapter 6.

The second variable of interest is the variability, or dispersion in parasite burden or parasite egg shedding rate. This can be subdivided into the known Poisson variability due to the

2.2 DEFINING PARASITE DISTRIBUTIONS

counting process, variability between animals, the variability within individual animals over short periods of time, and the aggregation of eggs in a single faecal pile. Quantifying the amount of the total variability that is attributable to each of these factors is likely to be very difficult unless large numbers of repeat samples are taken, and is impossible if only one sample is taken per animal. Consequently, the variability is usually considered as a single value describing all the sources of variability between samples, excluding that due to the counting process.

When using zero-inflated distributions, the proportion of the population that produce zero counts in excess of that expected from the distribution used to describe the majority of the counts is also of interest. One interpretation of this value is as one minus the prevalence of infection in the population. It is possible for the prevalence to be 100%, so that there are no animals with a true mean of zero, in which case the zero-inflated distribution would collapse to the simpler distribution describing the infected animals given sufficient evidence for this in the data. In this situation, results obtained using a zero-inflated distribution and the non zero-inflated equivalent would be equivalent.

While the mean and zero-inflation of a distribution are more clearly defined, the variability can be measured in different ways. The most obvious is to examine the absolute standard deviation or variance between samples, but it makes more sense to use a measure that is standardised in terms of the mean of the distribution because the absolute variance of a sample is likely to be heavily affected by the mean. One possibility is to use the variance : mean ratio, also known as the coefficient of dispersion, as a measure of the variability. This has the advantage of describing the amount of the total variability between observed counts that can be attributed to the variability of the counting process. The variance of a Poisson process is equal to its mean; therefore if the variance between counts is greater than the mean then the distribution must be over-dispersed with respect to the Poisson. In practise, this measure is not particularly helpful as the vast majority of parasite distributions are already known

to be over-dispersed. It is also possible to alter this measure slightly so that it reflects

the ratio of the extra-Poisson variance, which is equal to the variance between samples, to the mean. In this case, any value close to zero represents a distribution with little over- dispersion. If using the gamma distribution to describe the variability between samples, the extra-Poisson variance : mean ratio can be derived as:

total variance − P oisson variance

mean =

λ + α × β2
_{− λ}

α × β = β

Where α and β are the shape and scale parameters of the gamma distribution respectively, and λ is the mean of the Poisson distribution. This measure is therefore convenient for the gamma distribution as it is exactly equivalent to the scale parameter, β.

2.2 DEFINING PARASITE DISTRIBUTIONS

The second possible measure of variability is to use the ratio of the extra-Poisson standard deviation to the mean, also known as the coefficient of variation (cv) . This measure is commonly used for data measured on a ratio scale, such as positive counts, although it has no meaning for data measured on an interval scale. Values of coefficient of variation (cv) of close to zero indicate a Poisson distribution, and the variability between samples increases as the cv increases towards Infinity. If using the gamma distribution, the cv would be equal to:

cv =standard deviation of gamma distribution

mean =

p

(total variance − P oisson variance) mean

= s

 total variance − P oisson variance

mean2  = s λ + α × β2− λ α2× β2 = r 1 α

The cv is therefore related to the shape parameter α, which is often used in the parasitology literature as the inverse measure of aggregation k.

If using the lognormal distribution with mean and variance on the log scale to describe the variability between samples, the cv is related to the standard deviation of the log scale

distribution alone (Limpert et al.,2001). This can also be shown from the following derivation.

The variance of the lognormal distribution is given by (exp (lµ))2×exp lσ2
_{× exp lσ}2
_{− 1}

and the mean by exp (lµ) × exp 

lσ2 2



, where lµ is the mean and lσ the standard deviation

of the distribution on the log scale (Evans et al.,2000). The ratio of the standard deviation

to mean (cv on the exponent scale) is therefore given by:

cv = q

(exp (lµ))2× exp (lσ2_{) × (exp (lσ}2_{) − 1)}

exp (lµ) × exp  lσ2 2  =exp (lµ) ×pexp (lσ 2_{) × (exp (lσ}2_{) − 1)} exp (lµ) × explσ₂2 (2.1) =pexp (lσ 2_{) × (exp (lσ}2_{) − 1)} pexp (lσ2₎ =pexp (lσ2_{) − 1} 25

2.2 DEFINING PARASITE DISTRIBUTIONS

In deciding which of these measures to use, it is important to consider what is being quantified. In this case, it is most relevant to measure the ‘difference’ in worm burden or egg shedding rate between samples, due to the differences in host susceptibility and worm survival and fecundity. Therefore, the measure of variability that would be most useful would remain constant when the larval challenge changes, but differences in animal susceptibility and worm survivability are constant. Which of the different measures of variability best fulfils these requirements is assessed using a rudimentary simulation model to describe, in very basic terms, the variability in worm burden between animals. This is based on each host animal being assigned a probability of removing a larva before development to an adult parasite, and each ingested larva being assigned a probability of developing to an adult. These probabilities are varied between hosts and larvae to simulate variability between animals and worms, and stochastic variability in the number of adult worms between animals given a fixed larval challenge is examined.

2.2.2 Materials and methods

A mathematical model representing the processes discussed was written in the R statistical

programming language (R Development Core Team,2009). The number of ingested larvae

for each of 10,000 animals was fixed at the same value for each simulation. The variability in host immune defence was simulated using a beta distribution, so that each animal was given a distinct probability of eliminating each larva before development to an adult parasite. The variability in worm resistance to the host immune response was modelled similarly. The survival of each larva was then modelled using a Bernoulli trial using the combined probabilities derived from host immunity and worm resistance, and the resultant number of adult parasites in each host was calculated. The observed coefficient of variation and

coefficient of dispersion were calculated as σ_µ and σ_µ2 respectively, where σ is the standard

deviation of the number of adult worms per animal, and µ is the mean number of adult worms per animal. The model was run a total of three times, with the second and third simulations using altered values for the larval challenge and altered values for the variation in host immunity and worm resistance respectively.

2.2.3 Results

The effect of doubling the larval challenge while retaining the same values for host and worm susceptibility distributions, simulating the effect of moving the same group of animals onto a

field with a different larval challenge, is shown inFigure 2.1. The mean number of worms per

adult animal is increased, but the distribution retains a similar shape when the x-axis scale is increased proportionately with the increase in larval challenge. In effect this can be thought

2.2 DEFINING PARASITE DISTRIBUTIONS

of as doubling the expected number of parasites in each animal within the population, with- out changing the proportion of worms in each individual animal. Conversely, altering the distributions describing the host and worm susceptibilities while retaining the same value for

larval challenge alters the shape of the distribution, but not the mean (Figure 2.2). In this

case the variation in the proportion of worms in each animal is changed, altering the shape of

the distribution. InFigure 2.3, this effect is shown using an empirical cumulative distribution

function, plotting the proportion of the total worm population within each animal. The cor-

Number of adult worms per animal

100 200 300 400

0

100

200

300

Number of adult worms per animal

200 400 600 800

0

100

200

300

(a) Distribution of adult worms with (b) Distribution of adult worms with

low larval challenge high larval challenge

Figure 2.1: The effect of altering larval challenge on the distribution of adult worms in a mathematical model of the distribution of parasites between hosts - the

x-axis scale is increased proportionately to the increase in larval challenge

Number of adult worms per animal

100 200 300 400

0

100

200

300

Number of adult worms per animal

100 200 300 400

0

200

400

600

(a) Distribution of adult worms with (b) Distribution of adult worms with

high host / worm variability low host / worm variability

Figure 2.2: The effect of altering the distribution of host and worm variability on the distribution of adult worms in a mathematical model of the distribution of

parasites between hosts - the x-axis scale is fixed as the larval challenge is unchanged

Proportion of adult worms per animal

Empir

ical CDF

Figure 2.3: The effect on the proportion of adult worms in each individual host of altering different parameters in a mathematical model of the distribution of parasites between hosts - the effect of changing larval challenge (orange) and the distribution

of host and worm susceptibility (blue) shown relative to the baseline parameters (black).

Table 2.1: The effect of altering either larval challenge (Dataset 2) or the distribution of host and worm susceptibility (Dataset 3) compared to the baseline

(Dataset 1) on the distribution of worms in a mathematical model of the distribution of parasites between hosts

Coefficient of variation Coefficient of dispersion

Dataset 1 0.220 12.158

Dataset 2 0.220 24.285

2.3 SELECTING A MODEL DEFINITION FOR THE ZERO-INFLATED GAMMA