Group Title: Malaria Journal 2007, 6:131
Title: Standardizing estimates of the Plasmodium falciparum parasite rate
CITATION PDF VIEWER THUMBNAILS PAGE IMAGE ZOOMABLE
Full Citation
STANDARD VIEW MARC VIEW
Permanent Link: http://ufdc.ufl.edu/UF00100281/00001
 Material Information
Title: Standardizing estimates of the Plasmodium falciparum parasite rate
Series Title: Malaria Journal 2007, 6:131
Physical Description: Archival
Creator: Smith DL
Guerra CA
Snow RW
Hay SI
Publication Date: 39350
 Record Information
Bibliographic ID: UF00100281
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: Open Access: http://www.biomedcentral.com/info/about/openaccess/

Downloads

This item has the following downloads:

01475-2875-6-131 ( PDF )


Full Text




Malaria Journal BioIed Cen



Research

Standardizing estimates of the Plasmodium falciparum parasite rate
David L Smith*l,5, Carlos A Guerra2,3, Robert W Snow3,4 and Simon I Hay2,3


Address: 'Fogarty International Center, National Institutes of Health, Building 16, 16 Center Drive, Bethesda, Maryland 20892, USA, 2Spatial
Ecology and Epidemiology Group, Tinbergen Building, Department of Zoology, University of Oxford, South Parks Road, Oxford OX1 3PS, UK,
3Malaria Public Health & Epidemiology Group, Centre for Geographic Medicine, KEMRI-Wellcome Trust-Collaborative Programme, Kenyatta
National Hospital Grounds, PO Box 43640-00100 Nairobi, Kenya, 4Centre for Tropical Medicine, John Radcliffe Hospital, University of Oxford,
Oxford, OX3 9DS, UK and 5Department of Zoology & Emerging Pathogens Institute, University of Florida, 223 Bartram Hall, PO Box 118525
Gainesville FL, USA
Email: David L Smith* davesmith@ufl.edu; Carlos A Guerra carlos.guerraloaiza@zoo.ox.ac.uk; Robert W Snow rsnow@nairobi.kemri-
wellcome.org; Simon I Hay- shay@nairobi.kemri-wellcome.org
* Corresponding author



Published: 25 September 2007 Received: 18 April 2007
Malaria journal 2007, 6:131 doi:10.1 186/1475-2875-6-131 Accepted: 25 September 2007
This article is available from: http://www.malariajournal.com/content/6/l/l3 1
2007 Smith et al; licensee BioMed Central Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.



Abstract
Background: The Plasmodium falciparum parasite rate (PfPR) is a commonly reported index of
malaria transmission intensity. PfPR rises after birth to a plateau before declining in older children
and adults. Studies of populations with different age ranges generally report average PfPR, so age is
an important source of heterogeneity in reported PfPR data. This confounds simple comparisons
of PfPR surveys conducted at different times or places.
Methods: Several algorithms for standardizing PfPR were developed using 21 studies that stratify
in detail PfPR by age. An additional 121 studies were found that recorded PfPR from the same
population over at least two different age ranges; these paired estimates were used to evaluate
these algorithms. The best algorithm was judged to be the one that described most of the variance
when converting the PfPR pairs from one age-range to another.
Results: The analysis suggests that the relationship between PfPR and age is predictable across the
observed range of malaria endemicity. PfPR reaches a peak after about two years and remains fairly
constant in older children until age ten before declining throughout adolescence and adulthood.
The PfPR pairs were poorly correlated; using one to predict the other would explain only 5% of
the total variance. By contrast, the PfPR predicted by the best algorithm explained 72% of the
variance.
Conclusion: The PfPR in older children is useful for standardization because it has good biological,
epidemiological and statistical properties. It is also historically consistent with the classical
categories of hypoendemic, mesoendemic and hyperendemic malaria. This algorithm provides a
reliable method for standardizing PfPR for the purposes of comparing studies and mapping malaria
endemicity. The scripts for doing so are freely available to all.




Background titative aspects of malaria epidemiology [2]. A commonly
The intensity of malaria transmission by its anopheline used index of malaria transmission intensity is the Plasmo-
vectors varies enormously [ 1], and this affects most quan- dium falciparum parasite rate (PfPR), the proportion of the


Page 1 of 10
(page number not for citation purposes)







http://www.malariajournal.com/content/6/1/131


population found to carry asexual blood-stage parasites.
In mathematical models, PfPR is related to the entomo-
logical inoculation rate (EIR), the number of bites on a
person by sporozoite positive vectors, at the steady state
[3,4]. The notion of a steady state has limited application
for PfPR, however, because PfPR follows a well-estab-
lished pattern as a function of age and transmission inten-
sity; PfPR rises during infancy and early childhood [5,6],
settles to a plateau in older children, and declines in ado-
lescents and adults as malaria immunity develops [7-10].
This pattern has been known for decades [11,12], but
there are no established standards for reporting PfPR, so
thousands of studies have reported crude PfPR, without
stratifying by age [13]. As a result, the different age-ranges
over which studies have reported PfPR make it difficult to
compare prevalence estimates at different times and
places. Assemblies of PfPR data aimed at describing the
current global distribution of malaria endemicity [13],
therefore, need a mechanism to standardize these data to
a consistent age-grouping in order to be meaningfully
summarized.

Mathematical models with slowly acquired immunity
qualitatively reproduce the empirical relationships
between age and PfPR [14-17]. The models suggest that
EIR (or the force of infection) determines both the rate
that PfPR rises in children and the level of the plateau, so
either measure would provide a reliable index of transmis-
sion intensity. The PfPR in children is correlated with EIR
[4], and the PfPR in children aged 2-10 has provided a
basis for the classical categorical measures of malaria
transmission: hypoendemic (<10%), mesoendemic (10-
50%), and hyperendemic (50-75%) [18]. Holoendemic-
ity refers to PfPR >75% in those less than 12 months old
[18].

The variable way of reporting PfPR also limits the number
of studies where PfPR can be used to index transmission
intensity, which poses a particular problem for recent
approaches aimed at using empirical PfPR data to model
the spatial distributions of malaria transmission intensity
at regional [19-21] or potentially at global scales [22]. For
example, maps of transmission intensity in Africa are
often based on studies of PfPR that include only children
[23,24,21,25]. The evidence base for mapping malaria,
the geographical extent and coverage of places where PfPR
has been measured directly, could all be greatly expanded
if it were possible to standardize crude PfPR in studies that
include adults. The consistent pattern apparent in age-
stratified PfPR data suggests that it is amenable to mathe-
matical or statistical modeling. A set of candidate algo-
rithms was derived and evaluated in an attempt to provide
a single evidence-based method for converting crude esti-
mates of PfPR to a standard age range for improving future


global comparisons of malaria risk using existing exten-
sive PfPR data [13].

Methods
Experimental approach and data
This analysis has the distinctive purpose of developing an
algorithm to age-standardize PfPR data or, in other words,
the purpose is to find a function, F, that transforms an
estimate of PfPR over any age range [x,y), into a PfPR over
a standard age range [L, U), i.e. F: PR[x,y)-PR[L, U). This
principle was achieved for a number of candidate statisti-
cal and biological deterministic models developed from
21 studies that report very detailed PfPR by age (the train-
ing set). The skill of the models was then evaluated by
inter-conversion of 121 pairs of crude PfPR estimates,
where both were taken from the same population but
aggregated over different age ranges (the testing set).

The data selection criteria and summary details are given
in Additional File 1 and Additional File 2 for the training
and testing sets, respectively. Briefly, the data spanned all
potential P. falciparum transmission intensity categories
[18]. The algorithms developed from the training set were
validated against the 121 pairs of the testing dataset by
comparing the conversions from one age group to the
other, usually from adults to children, and vice versa. A
selection was made of the best performing algorithm
using the goodness of fit, measured as the proportion of
the variance explained, and defined as the ratio of the var-
iance in residual error divided by the error in the observed
PfPR subtracted from one.

The algorithms all differ from each other in significant
ways. The linear regression algorithm was based only on
the testing set. The other algorithms were based on only
the training set, and did not use the testing set for their
development. The linear regression algorithm does not
predict a relationship between age and PfPR. The interpo-
lation algorithm predicts PfPR in the testing set based on
direct interpolation of the training set data; it does not
predict an age-PfPR relationship. The two parametric algo-
rithms were fitted to the training set, but they were used
only for prediction on the testing set, so parsimony is not
an important concern. The most important measure of
performance is their skill a more complicated model
that did better at standardizing PfPR would be preferred
regardless of the level of complexity, as long as the algo-
rithm had not been fitted to the testing set. Since the lin-
ear regression algorithm is a statistical description of the
testing set, it sets a standard for judging the skill of the
other algorithms. The predictions based on linear regres-
sion of the testing set, on the other hand, do raise impor-
tant issues of parsimony. Because the statistical analysis
was one of the candidate algorithms, the analysis was
repeated on a sub-sample of the testing set: two-thirds of


Page 2 of 10
(page number not for citation purposes)


Malaria Joumnal 2007, 6:131







http://www.malariajournal.com/content/6/1/131


the data pairs were used for fitting and one-third for vali-
dation. Based on these rules, one algorithm was selected
as the most appropriate evidence-based method to age-
standardize PfPR for future comparisons between studies
reporting parasite prevalence across different age ranges.

Statistical candidate algorithms
Linear regression
Linear regression was used to describe the relationship
between the pairs of PfPR estimates from the testing set. It
differs from the other algorithms in that it is based on a
statistical analysis of the testing set; it does not predict a
relationship between PfPR and age, and it does not rely on
the training set. Let PR, and PR2 denote the two estimates
made on the intervals [L1, U1) and [L2, U), respectively.
The full regression included the age limits: (i.e. PR, = co
PR2 + C1 L1 + c2 U1 + c3 L2 + C4 U2 + e). This formula some-
times returns values for predicted PR1 that are outside the
interval (0,1), so the analysis was repeated as non-linear
regression where the points outside this range were forced
to be either 0 or 1. The regression analysis included each
pair twice; each member was both PR1 once and PR2 once.
Linear regression of the full testing set provided a standard
for the other algorithms, and the predicted values from
the linear regression were also evaluated as a potential
algorithm.

Interpolation
A general class of interpolation algorithms was based on
the training set: (i) given the PfPR from a focal study,
PRf[x,y), compute the PfPR for the training sets over the
same interval, PRi[x,y), i = 1...21; (ii) let W, = I PRf[x,y) -
PRi[x,y) -z, and let oi = WJ/ZW, denote the weight of the
ith PfPR estimate and (iii) the standardized PfPR is then
XimiPRi [L, U). Linear interpolation was also considered
and was very similar to the general interpolation algo-
rithm with z = 6.

Pull & Grab-based Algorithm
Let P(A) denote the true prevalence and F(A) the sensitiv-
ity of the microscopy, a standard method for estimating
PfPR, as a function of age. The function F(A) was moti-
vated by the notion that sensitivity declines with age as
blood-stage immunity reduces parasite densities to a
point where they are often below the detection thresholds
of microscopy [26,27]. Therefore, the apparent PfPR, the
one that a study would find using microscopy, is p(A) =
P(A) F(A). Crude PfPR estimates also depend on the age-
distribution of the sampled population, S(A); when PfPR
is reported crudely, the age-distribution is generally not
reported, so it must also be inferred. Given p(A) and S(A),
an estimate of crude PfPR would be PRf[x,y) = Jyp(A) S(A)
dA/I YS(A) dA. The standardized estimate would be
PRf[L,U) = LUp(A) S(A) dA/lfUS(A) dA.


The estimate of S(A) was based on the age-distributions in
the training set (Figure 1). Each of the 21 studies reports
the number of people sampled by age or by age class. Typ-
ically, ages were binned by year through childhood, then
by 5-year age classes up to age 65, and finally the elderly,
which was arbitrarily truncated at age 85. When a study
binned several age classes, the observations for each year
were proportional to the total observed for that class. For
example, the sample for 27 year olds was counted as 1/5
of the total observations for 25-30 year olds. In sum, to
get an estimate of S(A): (i) for each study, the proportion
of all people sampled that belonged to each one-year age
class was computed and (ii) the proportion for each age
class was the average for all the studies, where each study
was weighted equally.

To generate p(A), the curves P(A) and F(A) were generated
separately. P(A) was based on Pull & Grab's equations
[28], which were motivated by the Ross model [29] and
previous work by Muench [30]. The change in PfPR with
age is described by the equation:

dP/dA = h (1-P) rP;

where h denotes the force of infection (i.e., the "happen-
ings" rate), and r is the rate at which infections clear.
When P(0) = 1, this equation has the solution:

P(A) = P' (1 e-bA).


Figure I
The average distribution of sample sizes, S(A), from 21 stud-
ies that report PfPR by age, A. The lines show, by age, the
mean (blue), the trimmed mean (red, with 10% of extreme
values eliminated), the median (thick black), the interquartile
range (thin black), and the 5th and 95th quantiles (dashed
lines).



Page 3 of 10
(page number not for citation purposes)


Malaria Journal 2007, 6:131







http://www.malariajournal.com/content/6/1/131


Here, P' = h/(h+r) is the PfPR at equilibrium, and b = h+r
describes the rate at which PfPR approaches equilibrium.

A three-parameter function was used for F(A): 1 s [1 -
min (1, e-c(A-a)) ]. Beginning at age a, this function declines
from 1, to 1-s; the parameter c describes the decline from
1 to 1-s as a function of age. Conceptually, this function
can be thought of as the decline in the probability of
detecting an active infection, although the real reason for
the decline in PfPR might be that immunity leads to real
declines in h or real increases in r. For the purposes of
standardization, the biological reasons for the decline are
not relevant.

The modified Pull & Grab model was fitted to all 21 data-
sets using maximum likelihood estimation (Figure 2). The
algorithm used the average estimates of b, a, s, and c. The
21 best-fit parameter values for a, b, c, and s were uncor-
related with P' (a scatter plot of the slope, b, is plotted
against the plateau, P', Figure 3). A few of the parameter
values were clearly extreme, but a careful examination of
the extreme values suggested that they were irrelevant (i.e.
changing their values by a factor of 10 did not change the
shape of the fitted curve because of the other fitted param-
eters), so the influence of these extreme but irrelevant val-
ues was removed by taking the trimmed means. This
generated a family of age-PR curves that depended only
on P'.

The modified Pull & Grab equations each generate a curve
that describes PfPR as a function of age for each value of
P'. To turn this function into a standardization algorithm,
the function need only be inverted. In other words, the
algorithm found a value P* such that PRf[x,y) = JYp(A|P*)
S(A) dAl/fxS(A) dA. The standardized value was then
PRf[L,U) = fUp(AIP*) S(A) dA/ILUS(A) dA.

The Garki model
Another parametric method was based on the Garki
model [14], using parameter values that were fitted during
the Garki project [31]. This model is based on a system of
seven coupled difference equations [14]. Here, P(A) was
computed using an analogue of the Garki equation with
seven coupled ordinary differential equations. To convert
the parameter values, -log(l-x), was used instead of the
parameter x. Instead of a fixed delay for the pre-patent
periods the pre-patent period was assumed to be distrib-
uted exponentially, but with the same mean time as in the
Garki model (i.e. 1/N). The differences between the ordi-
nary differential equation and difference equations are
negligible and are available upon request. Births and
deaths were ignored (i.e. 6 = 0) because the quantity of
interest was prevalence in the survivors; the initial value of
x, was set to 1, and all other variables were initially set to
0. The equations were:


dx1/dA = R y2 h x,

dx2/dA = h x x2/N

dy1/dA = x2/N l y1

dy2/dA= 1 1- (a2 + R) y2

dx3/dA = R2 y3 h x3

dx4/dA = h x3 x4/N

dy3/dA = 2 y2 + X4/N R2 y3

The function p(A) = qliy(A) + q2Y2(A) + q3y3(A), where
y,(A) were found by choosing a value for h and then
numerically solving the system of equations. The algo-
rithm used the age distribution from the training set, S(A).
All other parameters except h were taken from the original
paper (note that the values of R, and R2 are fixed by h and
other parameters). To turn this function into a standardi-
zation algorithm, the function again, need only be
inverted. In other words, the algorithm found a value h*
such that PRf[x,y) = Jyp(Alh*) S(A) dA/lJYS(A) dA. The
standardized value was then PRf[L,U) = fLUp(Alh*) S(A)
dA/ILUS(A) dA.

All the routines described were written in R [32], and are
available upon request.

Results
Overall, the Pull & Grab-based algorithm ranked highest;
it explained 72% of the variance in the testing set, despite
being calibrated only on the training set. Linear regression
ranked second-best; it explained 68% of the variance.
Interpolation (with z = 1.5) ranked third; it explained
64% of the variance. The Garki model explained 58% of
the variance. The results are plotted in Figure 4, summa-
rized in Table 1, and discussed in detail in the following
sections. Because the algorithm based on Pull & Grab
ranked highest, and because it was never fitted to the test-
ing set, the issue of parsimony was not relevant. Had the
algorithm based on linear regression ranked highest, the
issue of parsimony would have been an important con-
cern.

Linear regression
By itself, PR2 explained only 5% of the variance in PR1 (i.e.
r2 = 0.05). Linear regression using the formula PR1 = a + b
PR2 improved the r2 to 26%. Linear regression with the
formula PR1 = a + b PR2 + c L, + d L2 + e HU + fH2 improved
the r2 to 68%, the same as with the slightly non-linear ver-
sion that always returned a value between 0 and 1. All of
the coefficients were statistically significant at the 95%
level except for c, the coefficient corresponding to L1.


Page 4 of 10
(page number not for citation purposes)


Malaria Journal 2007, 6:131









Malaria Journal 2007, 6:131


http://www.malariajournal.com/content/6/1/131


Eritrea
b = 32 99
1-s= 086

o





Vanuatu
b= 161
1-s= 049

o





Kericho
b= 101
/







Kissi
b= 02
1-s= 024







Sao Tome

,' >,' A-





o H I I I I1

Navrongo





. -


I- Mchenga

uro .


05 2 4 8 16 32


Tak
b= 017
1-s= 0







C Somalia
b= 113
1-s= 036


S Somalia
b= 053
1-s= 042







Ngerenya
b= 038
1-s= 03







Chonyi
b= 044
1-s= 022







I Linzolo





6b=3
1-s= 034

- I I I I I


,,
Nr


64 05 2 4 8 16 32


NE Somalia

1-s= 099







Kissi, Gucha
b= 026
1-s= 0







Orissa
b= 634
1-s= 035


Ndiop
b= 029
1-s= 037







Wosera
b= 049
1-s= 04





-I I I; i I


64 05 2 4 8 16 32


Figure 2
The 21 training sets, ordered by peak PfPR. The PfPR data (black) have been plotted against the root of age in years, along with
the confidence limits by the exact test for each proportion (grey). The maximum likelihood best-fit for the modified Pull &
Grab model (orange) was also plotted; the plateau (horizontal dashed red line) and the age when PfPR begins to decline (verti-
cal dashed red line) are indicated and the other fitted parameters are reported on each graph. The algorithm (blue) fitted to
the data (i.e. four parameters were fixed at their trimmed mean values and the fifth parameter was fitted).






Page 5 of 10
(page number not for citation purposes)


SDielmo






b= 201
I 1-s= 044








http://www.malariajournal.com/content/6/1/131


0








0
0 o o

0
0 o
0
S 000 o o 0 0

S 02 04 06 08
Plateau


Linear Regression

R2 -0 684 0o o
o-
oooo o o
o 0 00


o oo


00 02 04 06 08 10
PR

Garki


Pull and Grab, Modified

R2 = 721




o o
ooo i o o o





oo00 02 04 06 08 10
PR

Interpolation, z =1.5


Figure 3
Theory predicts that the slope of PfPR in young children (i.e.
b) and the PfPR in older children (i.e. the plateau, P') should
be correlated. The best-fit parameters describing these two
quantities are plotted here. Two extreme values were
excluded from this plot. There was no correlation with (p =
0.23) or without (p = 0.56) the extreme values.



Interpolation
Several different values of z were evaluated including z =
0.5, 1, 1.5, 2, 3, 4, 6, 10. Of these values, z = 1.5 mini-
mized the sum of squared errors and explained 65% of the
variance.



Table I: Comparing Algorithms


Data Subset All Random
(n=242) (n = 160182)

r2 st > 2 d > 3rd


L, 2-10 I U
(n = 87)

r2 (n=87) Cat
Err.


Pull & Grab
Regression
Interpolation
Garki
PR2


A summary of the skill of the four candidate standardization
algorithms. The columns summarize the proportion of variance
explained (r2), for the full data, the rankings of the four algorithms
fitted to 100 random sub-samples of the testing set (n = 160) and
tested on the remainder (n = 82), and the r2 and the proportion of
categorical errors for the subset of the testing set that was within the
standardization range. The table also includes an evaluation of the
paired PfPR estimate (PR2).


Figure 4
Predicted PfPR plotted against observed PfPR for the four
candidate algorithms for the testing set.



Pull & Grab
The trimmed mean value for a corresponded to a decline
in PfPR beginning around age 9.5. The trimmed mean
value for s corresponded to a decline in sensitivity to
approximately 36% of P', but the trimmed mean value of
c was low and implied that PfPR declines slowly, so that
the apparent PfPR is not close to 36% of the PfPR in chil-
dren between two and ten years of age until late in life.
There was substantial variability in b, but by the trimmed
mean, PfPR was within 90% of P' by age two. The param-
eter names, interpretations, and values are summarized in
Table 2. Some of the individual datasets differed from this
pattern, but the majority were within this value by age
two. The family of curves described by the algorithm is
illustrated in Figure 5. This algorithm explained 72% of
the variance.


Garki
The algorithm based on the Garki model [14] and cali-
brated during the Garki Project [31] was recently used to
generate endemicity maps [20]. The algorithm was there-
fore included as a candidate for comparison; the relation-
ship between age and PR is qualitatively very similar to the
modified Pull & Grab model. It was, therefore, used as-is,
without further fitting to the training set or the testing set.
It explained 58% of the variance, the lowest of the four
algorithms considered.


Page 6 of 10
(page number not for citation purposes)


Malaria Journal 2007, 6:131


00 02 04 06 08 10


00 02 04 06 08 10







http://www.malariajournal.com/content/6/1/131


0

-




0 20 40 60 80
Age

Figure 5
The age-PR relationship as generated by the algorithm.
Holoendemic areas are colored dark green, hyper-endemic
areas green, meso-endemic areas light green, and hypoen-
demic olive.



Analysis of the sub-sampled data
To evaluate the algorithms further, and to enable the lin-
ear regression analysis to be assessed as an algorithm, the
testing set was sub-sampled 100 times. The Pull-and-Grab
-based algorithm ranked 1st 80% of the time, 2nd or better
94% of the time, and 3rd or better 99% of the time. The
similar ranking vector for linear regression was 17%, 70%
and 95%, for interpolation 3%, 28%, and 50%. The
Garki-based algorithm ranked in the top three only 5% of
the time.

The algorithms were developed and evaluated for stand-
ardizing PfPR from any age range to any other; it is possi-
ble that they have different skill at standardizing to the
PfPR in 2-10 year olds. To further evaluate the algorithm
for standardization specifically to the 2-10 year old age
classes, 87 datasets were identified in which the age limits
were between two and 10 (Figure 6). The PfPR pairs were,
again, used as a standard for comparison. When PR1 was
used instead of its paired PfPR estimate, PR2, the categor-
ical description was wrong in 38% of the cases; virtually
all of the hyperendemic populations were misclassified as
mesoendemic, and many mesoendemic areas were mis-
classified as hypoendemic. By way of contrast, the stand-
ardized PfPR gave the wrong categorical description in
18% of the cases. Standardization reduced the number of


Table 2: Algorithm Parameters

b The slope, the rate that PfPR approaches the plateau (1.8 y-l)
P' PfPR in older children; the plateau (Variable)
a The age when PfPR begins to decline ( 9.5 years)
I-s The asymptotic sensitivity of microscopy (I-s 36%)
c The rate that PfPR declines with age, after age a (0.07 y-l)

The parameter names and interpretations in the algorithm based on
the modified Pull & Grab equations.


mesoendemic populations that would have been classi-
fled as hypoendemic, but it also misclassified two popula-
tions as mesoendemic that were actually hypoendemic.
Standardization correctly classified most, but not all, of
the misclassified hyperendemic populations, but two
mesoendemic populations were misclassified as hyperen-
demic. Given the natural scatter in the estimates, errors are
inevitable when continuous data are placed into catego-
ries. All other candidate algorithms performed more
poorly (Table 1).

Discussion
Four methods were developed and evaluated as potential
candidates for an algorithm that is suitable for standardiz-
ing crude PfPR data. Of the four methods, the selected
algorithm based on a modified version of Pull & Grab's
equations [28] was clearly superior to all others. The algo-
rithm explained 72% of the variance in PfPR estimates on
a set of independent data, and it was superior to the pre-
dictions obtained through linear regression of that data.
The mathematical model on which the algorithm was
based was biologically motivated and fitted to a set of
highly stratified PfPR data that spanned the range of P. fal-
ciparum transmission intensity. The analysis suggests that
the relationship between age and PfPR is fairly consistent
across the range of P. falciparum transmission intensity
and that it is strongly determined by the underlying biol-
ogy. The algorithm works better than regression because it
captures more information about the general patterns in
PfPR-age relationships than the 121 averaged PfPR pairs.

While the algorithm permits age-standardization of PfPR
for the heterogeneous way of reporting it, this would be
unnecessary if minimum standards were followed for
reporting PfPR data. Because PfPR varies by age, published
estimates of PfPR should always be stratified by age; the
21 training sets listed here have set a reasonable bar, with
some important caveats. This analysis suggests that report-
ing the average PfPR for the 2-10 year old age classes can
often be done without loss of information, but the data
should first be checked for the typical pattern. For exam-
ple, PfPR did not reach the plateau until age 5 in some
studies. The PfPR begins to decline in older children
sometime after age 10 because of blood stage immunity,


Page 7 of 10
(page number not for citation purposes)


Malaria Joumnal 2007, 6:131








http://www.malariajournal.com/content/6/1/131


00 02 04 06 08 10
Unstandardized PfPR (PR,)


V
V
V



e. *
4b%
%-


w
a
0

* a*.. . . . .
. .. .. . ..


00 02 04 06 08 10
Standardized PfPR (Algorithm)


Figure 6
The graphs plot a subset of 87 PfPR estimates (i.e. PRI) in the testing set that were already standard (i.e. LI > 2 and U I < 10)
and the predicted PfPR using either the PfPR pair (i.e. PR2, left) or the selected standardization algorithm (right). The dashed
lines show the cutoffs for the classical categories [18]. The colors highlight properly classified hypoendemic (blue), mesoen-
demic (grey), and hyperendemic (orange) populations, as well as misclassified populations. Some mesoendemic populations
were misclassified as hypoendemic (green) and some hyperendemic populations were misclassified as mesoendemic (orange).
After standardizing, a few populations were mistakenly misclassified moved from hypoendemic to mesoendemic cyann), or
from mesoendemic to hyperendemic (tan). In sum, the PfPR would have been misclassified by the paired estimate (i.e. PR2) in
33/87 cases (38%). Using the selected Pull & Grab modified algorithm, the standardized values were misclassified in 16/87 cases
(18%).


except possibly at very low PfPR, so PfPR data should be
stratified by age in older children and adults. A sensible
rule would be to bin by year at least through age 15, and
then to bin by at most 5-year age groups after age 20.

For children younger than 2, the PfPR should be reported
at a grain that is fine enough to describe changes in PfPR,
subject to the social, logistic and ethical concerns of
recruiting sufficient infants and young children to the sur-
vey. PfPR increases with age in very young children as they
acquire their first infections, develop clinical malaria, and
then either clear the infections with antimalarial drugs
and await reinfection, or recover from symptoms and
maintain an asymptomatic infection. Reinfection and
clinical malaria continue throughout life, but the fre-
quency and severity of clinical malaria decline as clinical
immunity develops [33]. As a consequence, so does the
rate at which parasites are cleared by antimalarial drugs.
Thus, the expected time to clinical malaria gets longer as
children grow older, and this can affect the use of antima-
larial drugs. The rate at which children are bitten also


increases as their body size increases, and they thus absorb
a greater proportion of the bites in the household [34].
Thus, the expectation is that PfPR increases rapidly during
these years.

This analysis suggests that after the first few years of life,
PfPR settles to a plateau where it remains fairly constant
until the onset of adolescence. The plateau reflects the
original steady state that was the motivation for using
PfPR to index transmission intensity, a population
dynamic equilibrium at which parasite clearance is bal-
anced by new infections, and simultaneously not biased
by detection issues associated with immunity and micros-
copy [26,27]. The rise in PfPR with age has been suggested
as an alternative index of transmission intensity [35], but
this analysis found substantial variability in the slope that
was uncorrelated with the PfPR in 2-10 year olds (Figure
3). The discrepancy between the two raises the question of
which measure provides a better index of transmission
intensity.




Page 8 of 10
(page number not for citation purposes)


N=17
V
V
V VV
Vy f
V V
V V V
..
ato

%


Malaria Joumnal 2007, 6:131








http://www.malariajournal.com/content/6/1/131


For practical and epidemiological reasons, the PfPR in
older children provides a good index of transmission
intensity, and the age limits from the classical categories
are appropriate for standardizing PfPR. There are three
justifications for standardizing reported PfPR to 2-10 year
olds for use as an index of transmission intensity. First,
because PfPR remains relatively constant between the ages
of about two and 10, the average is therefore meaningful.
Second, because the frequency of clinical malaria declines
in older children, the PfPR over these ages may be least
influenced by drug treatment [36]. PfPR in older children
is also relatively unaffected by immunity, so it can be
argued that the PfPR in older children comes closest to
reflecting the steady state relationship predicted by math-
ematical models. It is, therefore, most likely to provide a
good index of transmission intensity. Third, it is consist-
ent with the historical approaches to defining endemicity
[18].

Using the rise in PfPR in very young children to index
transmission has some advantages, but it also has several
disadvantages. Although young children are of great inter-
est for malaria control, the slope of PfPR, not the average
PfPR in young children, is an index of transmission inten-
sity. From a practical point of view, it is more difficult to
measure a slope than an average. The ideal measure of
transmission intensity would be the infant conversion
rate, characterized by the waiting time to a patent infec-
tion in uninfected children. A slightly less direct measure
of the force of infection would be any exposure to malaria
vs. age, a measure that would be found in an appropri-
ately designed study of seroprevalence [37,38]. PfPR is a
distant third because it reflects a balance between expo-
sure and clearance, including natural clearance and radi-
cal curative therapy with antimalarial drugs. The
frequency of clinical malaria is highest in young children,
before they develop functional immunity, so the need for
antimalarial drugs is also likely to be correlated with age.
The rise in PfPR reflects both exposure and clearance, so it
can be quite a volatile measure. For example, the rise in
PfPR in young children would dramatically underestimate
transmission intensity in populations where effective anti-
malarial drugs had been used frequently and properly to
clear infections. Because gametocytes clear slowly, it has
also been argued that measuring gametocyte prevalence
directly is a more appropriate measure of exposure, but
considerable technical difficulties still limit its wider
application [39].

The relationship between age and PR follows a fairly con-
sistent pattern across the natural range of malaria trans-
mission intensity that can be described by a relatively
simple, biologically motivated mathematical model. The
model, fitted to age-stratified data, accurately describes
the rise and fall in PfPR with age, but it does not resolve


any questions about the biological causes of this pattern,
since it was not designed to do so. The algorithm based on
this model can be used to reliably standardize crude esti-
mates, but some important residual variance remains
unexplained. This could be due to a variety of factors that
were not considered here, including sampling, seasonal-
ity, heterogeneous biting, the prevalence of P. vivax, and
variable use of effective anti-malarial drugs. These will be
the focus of future research effort. Despite these caveats,
the algorithm explained 72% of the variance, suggesting
that age is a dominant source of heterogeneity in PfPR
estimates from places with similar transmission intensity.
The algorithm thus provides a useful way of standardizing
PfPR to 2-10 year olds for comparing studies and ulti-
mately for the mapping of malaria risk.

Additional material


Additional file 1
The training set. A description of the . ,i;. .1 PfPR data from 21
studies.
Click here for file
[http://www.biomedcentral.com/content/supplementary/1475-
2875-6-131-S1.doc]

Additional file 2
The testing set. A description of the pairs of PfPR estimates from 121 stud-
ies. The estimates were taken from the same population over ir. i .-
ranges.
Click here for file
[http://www.biomedcentral.com/content/supplementary/1475-
2875-6-131-S2.doc]


Acknowledgements
We are extremely grateful to all those individuals who contributed PfPR
data and who are acknowledged on the Malaria Atlas Project (MAP) web-
site [40]. We additionally thank those who have supplied (directly to the
authors or indirectly through exemplary documentation in publication)
very high resolution age-stratified PfPR data: Waqar Butt, Verena Carrara,
Helen Guyatt, Marie Claire Henry, Akira Kaneko, Kojo Koram, Akiko Mat-
sumto, Bruno Moonen, D. Muller, Tabitha Mwangi, Francois Nosten, Chris-
tophe Rogier, Dennis Shanks, David Sintasath, Tom Smith, Jean-Francois
Trape and Dejan Zurovac. We also thank F. Ellis McKenzie, Andy Tatem,
Anand Patil, and three anonymous reviewers for their helpful comments
and suggestions.

DLS conducted this research at the Fogarty International Center, National
Institutes of Health, and at the Emerging Pathogens Institute, University of
Florida. CAG is supported by a Wellcome Trust Project grant (#076951)
and also acknowledges support from the Fundaci6n para la Ciencia y Tec-
nologia (FUNDACYT), Quito, Ecuador. RWS is a Wellcome Trust Principal
Research Fellow (#079080). SIH is funded by a Senior Research Fellowship
from the Wellcome Trust (#079091). RWS and SIH acknowledge the sup-
port of the Kenyan Medical Research Institute (KEMRI). This paper is pub-
lished with the permission of the director of KEMRI. This work forms part
of the output of the MAP, principally funded by the Wellcome Trust, UK.



Page 9 of 10
(page number not for citation purposes)


Malaria Joumnal 2007, 6:131








http://www.malariajournal.com/content/6/1/131


References
I. Smith DL, McKenzie FE, Snow RW, Hay SI: Revisiting the basic
reproductive number for malaria and its implications for
malaria control. PLoS Biol 2007, 5:e42.
2. Snow RW, Marsh K: The consequences of reducing transmis-
sion of Plasmodium falciparum in Africa. Adv Parasitol 2002,
52:235-264.
3. Dietz K: Mathematical models for transmission and control of
malaria. In Malaria, Principles and Practice of Malariology Edited by:
Wernsdorfer WH, McGregor I. Edinburgh: Churchill Livingstone;
1988:1091-1133.
4. Smith DL, Dushoff j, Snow RW, Hay SI: The entomological inoc-
ulation rate and Plasmodium falciparum infection in African
children. Nature 2005, 438:492-495.
5. Gupta S, Snow RW, Donnelly C, Newbold C: Acquired immunity
and postnatal clinical protection in childhood cerebral
malaria. Proc R Soc Lond Ser B-Biol Sci 1999, 266:33-38.
6. Gupta S, Snow RW, Donnelly CA, Marsh K, Newbold C: Immunity
to non-cerebral severe malaria is acquired after one or two
infections. Not Med 1999, 5:340-343.
7. Baird JK, Jones TR, Danudirgo EW, Annis BA, Bangs MJ, Basri H,
Purnomo Masbar S: Age-dependent acquired protection
against Plasmodium falciparum in people having two years
exposure to hyperendemic malaria. The American Journal of
Tropical Medicine and Hygiene 1991, 45:65-76.
8. Baird JK, Purnomo Basri H, Bangs MJ, Andersen EM, Jones TR, Mas-
bar S, Harjosuwarno S, Subianto B, Arbani PR: Age-specific preva-
lence of Plasmodium falciparum among six populations with
limited histories of exposure to endemic malaria. The Ameri-
can journal of Tropical Medicine and Hygiene 1993, 49:707-719.
9. Trape JF, Rogier C, Konate L, Diagne N, Bouganali H, Canque B,
Legros F, Badji A, Ndiaye G, Ndiaye P, et al.: The Dielmo project:
a longitudinal study of natural malaria infection and the
mechanisms of protective immunity in a community living in
a holoendemic area of Senegal. The American Journal of Tropical
Medicine and Hygiene 1994, 51:123-137.
10. Baird JK: Host age as a determinant of naturally acquired-
immunity to Plasmodium falciparum. Parasitol Today 1995,
11:105-111.
II. Bagster-Wilson D: Malaria incidence in central and south
Africa. In Malariology A comprehensive survey of all aspects of this group
of diseases from a global standpoint Volume II. Edited by: Boyd MF. Phil-
adelphia and London: W. B. Saunders Company; 1949:800-809.
12. Garnham PCC: Malaria immunity in Africans: effects in infancy
and early childhood. Ann Trop Med Parasitol 1949, 43:47-61.
13. Guerra CA, Hay SI, Lucioparedes LS, Gikandi P, Tatem AJ, Noor AM,
Snow RW: Assembling a global database of malaria parasite
prevalence for the Malaria Atlas Project. Malarj 2007, 6:17.
14. Dietz K, Molineaux L, Thomas A: A malaria model tested in the
African savannah. Bull World Health Organ 1974, 50:347-357.
15. Aron jL, May RM: The population dynamics of malaria. In Pop-
ulation dynamics and infectious disease Edited by: Anderson RM. Lon-
don: Chapman and Hall; 1982:139-179.
16. Aron JL: Mathematical modeling of immunity to malaria.
Mathematical Biosciences 1988, 90:385-396.
17. Smith T, Hii JL, Genton B, Muller I, Booth M, Gibson N, Narara A,
Alpers MP: Associations of peak shifts in age-prevalence for
human malarias with bednet coverage. Trans R Soc Trop Med
Hyg 2001, 95:1-6.
18. Metselaar D, van Thiel PH: Classification of malaria. Trop Geog
Med 1959, I 1:157-161.
19. Gemperli A, Sogoba N, Fondjo E, Mabaso M, Bagayoko M, Briet OJ,
Anderegg D, Liebe J, Smith T, Vounatsou P: Mapping malaria
transmission in West and Central Africa. Trop Med Int Health
2006, 11:1032-1046.
20. Gemperli A, Vounatsou P, Sogoba N, Smith T: Malaria mapping
using transmission models: application to survey data from
Mali. Am ] Epidemiol 2006, 163:289-297.
21. Omumbo JA, Hay SI, Snow RW, Tatem AJ, Rogers DJ: Modelling
malaria risk in East Africa at high-spatial resolution. Trop Med
Int Health 2005, 10:557-566.
22. Hay SI, Snow RW: The Malaria Atlas Project: developing global
maps of malaria risk. PLoS Med 2006, 3:e473.
23. Kleinschmidt I, Bagayoko M, Clarke GPY, Craig M, Le Sueur D: A
spatial statistical approach to malaria mapping. IntJ Epidemiol
2000, 29:355-361.


24. Kleinschmidt I, Omumbo j, Briet O, van de Giesen N, Sogoba N, Men-
sah NK, Windmeijer P, Moussa M, Teuscher T: An empirical
malaria distribution map for West Africa. Trop Med Int Health
2001, 6:779-786.
25. Kazembe LN, Kleinschmidt I, Holtz TH, Sharp BL: Spatial analysis
and mapping of malaria risk in Malawi using point-refer-
enced prevalence of infection data. Intj Health Geogr 2006, 5:41.
26. Mayor A, Aponte JJ, Fogg C, Saute F, Greenwood B, Dgedge M,
Menendez C, Alonso PL: The epidemiology of malaria in adults
in a rural area of southern Mozambique. Malarj 2007, 6:3.
27. McKenzie FE, Sirichaisinthop J, Miller RS, Gasser RA Jr, Wong-
srichanalai C: Dependence of malaria detection and species
diagnosis by microscopy on parasite density. The American jour-
nal of Tropical Medicine and Hygiene 2003, 69:372-376.
28. Pull JH, Grab B: A simple epidemiological model for evaluating
the malaria inoculation rate and the risk of infection in
infants. Bull World Health Organ 1974, 51:507-516.
29. Ross R: The prevention ofmalaria London: John Murray; 1910.
30. Muench H: Catalytic models in epidemiology Cambridge: Harvard Uni-
versity Press; 1959.
31. Molineaux L, Storey J, Cohen JE, Thomas A: A longitudinal study
of human malaria in the West African Savanna in the
absence of control measures: relationships between differ-
ent Plasmodium species, in particular P. falciparum and P.
malariae. The American journal of Tropical Medicine and Hygiene 1980,
29:725-737.
32. R Development Core Team: R: a language and environment for
statistical computing. In Book R: a language and environment for sto-
tistical computing City: R Foundation for Statistical Computing; 2006.
33. Snow RW, Omumbo JA, Lowe B, Molyneux CS, Obiero JO, Palmer
A, Weber MW, Pinder M, Nahlen B, Obonyo C, et al.: Relation
between severe malaria morbidity in children and level of
Plasmodium falciparum transmission in Africa. Lancet 1997,
349:1650-1654.
34. Port GR, Boreham PFL, Bryan jH: The relationship of host size to
feeding by mosquitoes of the Anopheles gambiae Giles com-
plex (Diptera: Culicidae). Bulletin of Entomological Research 1980,
70:133-144.
35. Smith T, Beck HP, Kitua A, Mwankusye S, Felger I, Fraser-Hurt N,
Irion A, Alonso P, Teuscher T, Tanner M: Age dependence of the
multiplicity of Plasmodium falciparum infections and of other
malariological indices in an area of high endemicity. Trans R
Soc Trop Med Hyg 1999, 93:15-20.
36. al-Yaman F, Genton B, Reeder JC, Anders RF, Smith T, Alpers MP:
Reduced risk of clinical malaria in children infected with mul-
tiple clones of Plasmodium falciparum in a highly endemic
area: a prospective community study. Trans R Soc Trop Med Hyg
1997, 91:602-605.
37. Snow RW, Molyneux CS, Warn PA, Omumbo J, Nevill CG, Gupta S,
Marsh K: Infant parasite rates and immunoglobulin M sero-
prevalence as a measure of exposure to Plasmodium falci-
parum during a randomized controlled trial of insecticide-
treated bed nets on the Kenyan coast. The American journal of
Tropical Medicine and Hygiene 1996, 55:144-149.
38. Snow RW, Shenton FC, Lindsay SW, Greenwood BM, Bennett S,
Wheeler J, Delgiudice G, Verdini AS, Pessi A: Sporozoite antibod-
ies and malaria in children in a rural area of The Gambia. Ann
Trop Med Parasitol 1989, 83:559-568.
39. Drakeley C, Sutherland C, Bouserna T, Sauerwein RW, Targett GAT:
The epidemiology of Plasmodium falciparum gametocytes:
weapons of mass dispersion. Trends Parasitol 2006, 22:424-430.
40. Malaria Access Project (MAP) [http://www.map.ox.ac.uk]


Page 10 of 10
(page number not for citation purposes)


Malaria Joumnal 2007, 6:131




University of Florida Home Page
© 2004 - 2010 University of Florida George A. Smathers Libraries.
All rights reserved.

Acceptable Use, Copyright, and Disclaimer Statement
Last updated October 10, 2010 - - mvs