Group Title: Malaria Journal 2009, 8:87
Title: Endemicity response timelines for Plasmodium falciparum elimination
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Title: Endemicity response timelines for Plasmodium falciparum elimination
Series Title: Malaria Journal 2009, 8:87
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Creator: Smith DL
Hay SI
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Malaria Jo nal ioed
Malaria Journal BiolVed Central


Research

Endemicity response timelines for Plasmodium falciparum
elimination
David L Smith*1 and Simon I Hay2


Address: 'Department of Zoology and Emerging Pathogens Institute, University of Florida, Gainesville, Florida, USA and 2Spatial Ecology and
Epidemiology Group, Tinbergen Building, Department of Zoology, University of Oxford, South Parks Road, Oxford, OX1 3PS, UK
Email: David L Smith* smitdave@gmail.com; Simon I Hay simon.i.hay@gmail.com
* Corresponding author



Published: 30 April 2009 Received: 25 February 2009
Malaria journal 2009, 8:87 doi: 10. 186/1475-2875-8-87 Accepted: 30 April 2009
This article is available from: http://www.malariajournal.com/content/8/1/87
2009 Smith and Hay; 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 scaling up of malaria control and renewed calls for malaria eradication have
raised interest in defining timelines for changes in malaria endemicity.
Methods: The epidemiological theory for the decline in the Plasmodium falciparum parasite rate
(PfPR, the prevalence of infection) following intervention was critically reviewed and where
necessary extended to consider superinfection, heterogeneous biting, and aging infections.
Timelines for malaria control and elimination under different levels of intervention were then
established using a wide range of candidate mathematical models. Analysis focused on the timelines
from baseline to 1% and from 1% through the final stages of elimination.
Results: The Ross-Macdonald model, which ignores superinfection, was used for planning during
the Global Malaria Eradication Programme (GMEP). In models that consider superinfection, PfPR
takes two to three years longer to reach I % starting from a hyperendemic baseline, consistent with
one of the few large-scale malaria control trials conducted in an African population with
hyperendemic malaria. The time to elimination depends fundamentally upon the extent to which
malaria transmission is interrupted and the size of the human population modelled. When the PfPR
drops below 1%, almost all models predict similar and proportional declines in PfPR in consecutive
years from 1% through to elimination and that the waiting time to reduce PfPR from 10% to 1%
and from I % to 0.1 % are approximately equal, but the decay rate can increase over time if infections
senesce.
Conclusion: The theory described herein provides simple "rules of thumb" and likely time
horizons for the impact of interventions for control and elimination. Starting from a hyperendemic
baseline, the GMEP planning timelines, which were based on the Ross-Macdonald model with
completely interrupted transmission, were inappropriate for setting endemicity timelines and they
represent the most optimistic scenario for places with lower endemicity. Basic timelines from PfPR
of 1% through elimination depend on population size and low-level transmission. These models
provide a theoretical basis that can be further tailored to specific control and elimination scenarios.







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Background
The Roll Back Malaria Partnership (RBM) recently called
for universal coverage with vector control and effective
drugs by 2010 [1], and major international agencies are
realigning their short-term goals and activities to achieve
the long-term goal of global malaria eradication [2-4].
Many countries are now contemplating malaria elimina-
tion, so there is a need to define plausible timelines for
planning. Much of the previous experience in setting
malaria elimination timelines comes from the Global
Malaria Eradication Programme (GMEP), which was
launched in 1955 and implemented in four phases called
planning, attack, consolidation, and maintenance [5,6].
The attack phase was based on three to four years of
indoor spraying with residual insecticides to completely
interrupt transmission; success was judged by several cri-
teria, one of which was that by the end of spraying, the
Plasmodium falciparum parasite rate (PfPR, the prevalence
of malaria infection) was reduced to less than 1% of the
pre-control baseline [7]. The GMEP prioritized this short
duration attack phase to reduce the opportunity for the
evolution of insecticide resistance in Anopheles vectors.
During the GMEP, expected PfPR declines within the
attack phase were based on the Ross-Macdonald model
[7].

The end of a successful GMEP attack phase marked the
beginning of a new control phase, called consolidation,
defined in part by new requirements for monitoring and


evaluation; when PfPR drops below 1%, parasite surveys
become an inefficient way of sampling malaria and meas-
uring progress [8]. The focus of malaria control pro-
grammes during consolidation shifted to developing
health and surveillance systems, identifying and eliminat-
ing residual transmission foci and preparing countries for
malaria free status certification [9]. While the GMEP
established well-defined timelines for the attack phase,
timelines for the consolidation phase were not outlined
[8].

By 1964, several GMEP control initiatives outside of Africa
had successfully reduced PfPR below 1% after three to
four years of spraying [7], which provided some valida-
tion for the GMEP model and timelines, but there were
fewer successes in Africa [10]. Before GMEP, there had
been a polarizing debate among malaria experts about the
feasibility of malaria control and elimination in hyperen-
demic Africa [11], which is defined by PfPR in children
aged 2-10 greater than 50% [12]. In Kenya and Tanzania,
the Pare-Taveta Malaria Scheme was launched in 1955 to
help address many points of contention [11]. The indoor
residual spraying campaign in Pare and Taveta lasted for
three and a half years during which the PfPR remained
high initially, but then fell from above 60% to around 5%
in older children [13,14] (Figure 1). The results of the
insecticide spraying when evaluated using traditional cri-
teria were not thought to have demonstrated complete
interruption of transmission [7]. Direct estimates sug-


incidence




clearance


I Ir_ I

Figure I
A diagram of human infection dynamics in the Ross-Macdonald model, which was used for planning during the
GMEP. Changes in the fraction of infected humans, denoted X, were described by two simple rules. The incidence rate of new
infections or "force of infection", denoted h, describes the per capital rate that uninfected humans would become infected.
Clearance of existing infections, the per capital rate that infected humans lose infections, was denoted r. Human infection
dynamics are described by an equation: X = h(I X) rX, where the superdot represents the derivative with respect to time.


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1-X

Uninfected


X

Infected


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gested the basic reproductive number for malaria [15] had
been reduced to below one [13], however, and that the
decay rates were consistent with a reproductive number
under control of 0.7 [7].

A result of intense exposure to malaria in populations
with hyperendemic malaria, such as those in Pare-Taveta,
is that new infections accumulate on top of uncleared
infections: a phenomenon called superinfectionn" [16].
These complex infections are comprised of multiple
founding parasite genotypes from multiple infectious
bites, and the number of distinct founding parasite geno-
types is called the multiplicity of infection (MOI). Theory
[16-19] and empirical studies [20] have suggested that
infections with high MOI take longer to clear. The Ross-
MacDonald model used by the GMEP assumed that infec-
tions were simple (the MOI was at most one) and, there-
fore, may have set inappropriate timelines for changing
endemicity from a hyperendemic baseline.

Superinfection had been discussed and modelled before
the GMEP [16,21], and several additional models have
been developed since [18,22,23]. These alternative mod-
els had not been used to re-evaluate response timelines to
intervention in areas where superinfection would have
been common, such as Pare-Taveta. It is clear that the
same theory can be also be used for predicting the results
of scaling up of malaria interventions today [1].

In this study, the theoretical basis for estimating endemic-
ity responses following intensive control across the
malaria transmission spectrum is critically reviewed and
extended. A range of simple mathematical models of
malaria transmission and parasite clearance were investi-
gated, to describe likely timelines for the changing PfPR in
hyperendemic areas from baseline to 1%. The theory is
further extended to establishing basic expectations about
the timelines for the consolidation phase, the likely time-
lines for the changes in PfPR from 1% to elimination.

Methods
The timelines for changing PfPR following malaria control
without mass drug administration are determined by the
waiting times to clear untreated human infections. The
Ross-Macdonald model, on which GMEP planning was
based, was used as a starting point for analysis. To evalu-
ate whether the Ross-Macdonald forms a good basis for
planning across the endemicity spectrum and during all
phases of malaria control and elimination, changes in
PfPR were simulated and compared using several other
models of malaria infections. These models are described
briefly in the following sections. A more detailed mathe-
matical description of the models is found in Additional
File 1.


Clearance rates
The waiting time to clear a simple infection is an impor-
tant parameter in these models. Estimates of the waiting
time to clear an infection come from several different
sources. One important source was data from the malaria
therapy of neurosyphilis patients, which estimated an
average duration of 220 days [24]. The 200-day waiting
time was also consistent with an older study in Puerto
Rico [25], and with recent studies that compared models
and estimated waiting time to clear infections of 150 days
in northern Ghana [26].

In these analyses, the expected changes in PfPR were
examined under the assumption that the average duration
of an infection is 200 days. This value was fixed by the
empirical evidence outlined above and numerous studies
conducted during the GMEP, where transmission was
considered by many criteria to be completely interrupted
and the documented declines in PfPR were broadly con-
sistent with an average duration of infection and maxi-
mum rate of decline of about 200 days [7].

Ross-Macdonald
The Ross-Macdonald model assumed that people were
either infected or not, and that once infected, infections
would clear at a constant per-capita rate, r. The assump-
tion applied regardless of how long a person had already
been infected or whether the person had acquired multi-
ple infections. The model thus implies that the waiting
time to lose an infection is exponentially distributed with
a mean waiting time of 1/r days. During the GMEP, r was
assumed to be 0.5% per day, which implied an average
waiting time to clear of 200 days. Figure 1 shows a dia-
gram of the human infection dynamics in the Ross-Mac-
donald model.

Queuing models
A challenge to the Ross-Macdonald model is that it does
not consider the time to clear malaria when superinfec-
tion occurs. A simple formula to describe the waiting time
to clear complex infections while transmission continues
was derived during the Garki Project [17], but the formula
converges rapidly to the Ross-Macdonald model if vector-
control measures reduce transmission (Additional File 1).

To describe the timelines for changing MOI when trans-
mission is interrupted, it is necessary to track changes in
the full distribution of MOI. Superinfection and dynamic
changes in PfPR were, therefore, simulated using a family
of queuing models [18]. Queuing models were extended
here to consider other biological scenarios, including
finite or infinite "strains," and infections that cleared
independently, or with competition that would increase
clearance rates of parasite types, or with facilitation that
would slow down clearance rates of parasite types (Figure


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2). The queuing model was further extended to include
heterogeneous biting; a queuing model described the
dynamics of MOI in each human population stratum,
defined by their relative rate of exposure.

Stochasticity
The Ross-Macdonald model and the queuing models are
deterministic: nothing happens by chance. Deterministic
models are good approximations when the law of large
numbers applies. In the approach to malaria elimination,
however, as the number of people who are infected
becomes small, it is no longer sensible to ignore chance.
During the consolidation phase, there must be a point
when the emphasis begins to shift from the proportion of
a population that is infected to the number infected. PfPR
defines the basis for switching malaria control phases, in
part because of the impractically large sample sizes
required to use changes in PfPR as a measure of progress.
When PfPR is 1%, the number of people who remain
infected is proportional to the population size. A PfPR of
1% means, for example, that one person is infected in a
population of 100 people, but that ten thousand people
would be infected in a population of a million. Starting
from a PfPR of 1%, the waiting time to elimination is,


XO
Uninfected
MOI=O


ho

incidence


clearance

Pi


X1
Infected
MOI=1


therefore, longer in larger populations because there are
more people who remain infected. To consider stochastic-
ity and the effect of population size on the time to elimi-
nation, stochastic analogues of the Ross-Macdonald
model and the queuing models were also developed and
the changes in PfPR were simulated in populations of var-
ious sizes.

Stage-structured models
The Ross-Macdonald model and the queuing models con-
sider the waiting time to clear an infection without regard
to the age of the infection. In the Ross-Macdonald model,
for example, a person who has been infected for just one
week and a person who has already been infected for 200
days are equally likely to remain infected for 200 more
days. This may not be realistic if clearance rates change
with the age of the infection. Stage-structured models are
a useful way to model infections that clear faster or slower
as they age [27], but it is computationally difficult to
model stage-structure and superinfection deterministi-
cally (Additional File 1). Aging infections were modelled
in stochastic models with superinfection, under the
assumption that infections clear independently. An
extreme version of this model was tested here in which


hi

incidence


clearance

P2


X2
Infected
MOI=2


h2
-------*
incidence


clearance

P3


Figure 2
A diagram of the queuing models, which extend the Ross-Macdonald model by tracking changes in the MOI.
The fraction of the whole population with MOI of m, denoted x,, changes when new infections occur or when existing infec-
tions clear, and in any short interval of time, one individual's MOI increments or decrements by one [18]. The rate that new
infections arise may depend on MOI, denoted h,. The rate of loss may depend on MOI, denoted p,, and pl = r denotes the
rate a simple infection is lost. Changes in the fraction of the population that is uninfected are described by an equation: to = -
hoxo + rxi. Changes in the fraction of people who are already infected with a given MOI are described by a set of equations: x,
= -hx, + h,_ ix,_ I px. + p,,+lx,,+. These equations describe a family of queuing models: each queuing model makes different
assumptions about infection and clearance. In "infinite strain" models, h, = h, and in "finite strain" models where M denotes the
maximum number of types, h, = h(I-m/M). Models considered parasite types that cleared independently, or with competition
or facilitation. For independent clearance, individual types were unaffected by concurrent infection with other parasite types,
so p, = rm. Competition and facilitation were modelled by letting p, = rm0, where o-> I described competition and o-< I
implies facilitation. Compared with independent clearance, per-strain clearance rates are faster with competition (i.e. p,> rm),
and slower with facilitation (i.e. p, occurred, the expected waiting time to lose all the existing infections would be the sum of times to progressively decrement
MOI: I/r+ /p2+ /p3+...+ /p,.



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infections progressed through n equal "stages," and were
cleared from the last stage. This is called senescence
because infection rates increase as infections age [27].

Ro and vector dynamics
Mosquito infection dynamics and the force of infection
were simulated in a minimal mosquito model (Addi-
tional File 1). In all these models, the intensity of trans-
mission by mosquitoes in the absence of control was
determined by the basic reproductive number, R0. The
intensity of transmission under control was determined
by the controlled basic reproductive number, Rc. The two
terms are directly analogous in every way but one;
whereas Ro is only defined in an environment that lacks
control, Rc describes potential transmission in a popula-
tion with some level of control. In the simulations, Ro sets
the baseline, which determines the PfPR at the start of a
simulation. In models with superinfection, Ro also sets the
equilibrium distribution of MOI. Rc determines the rate
of decline in PfPR and the waiting time to elimination
from that baseline. Transmission would be interrupted
eventually if malaria elimination were the endpoint in the
absence of imported malaria (Rc < 1), but it would be
completely and immediately interrupted if Rc = 0.

Results
The results focus on the PfPR timelines and decay rates
when the endpoint was elimination. The PfPR over time
was simulated numerically in all these models and com-
pared plotting the PfPR over time. Another useful way of
comparing models was to plot the decay rates versus PfPR
(Figure 3). The daily decay rate is the logarithm of the
daily change in PfPR, and over any interval, the decay rate
is the logarithm of the change in PfPR divided by the
length of the interval.

PfPR from baseline to I%
In all the queuing models, the waiting time to clear
malaria increases with MOI; it is the sum of waiting times
to clear a simple infection (1/r) plus the time to progres-
sively decrement MOI. With independent clearance, for
example, the expected waiting time to clear an infection
with m types was therefore 1/r + 1/2r + 1/3r + ... + 1/mr.
Queuing models with competition, independent clear-
ance, or facilitation each made quantitatively different
predictions about the decline of PfPR from baseline to 1%
(Figure 3a). Some model identifiability problems were
obvious. A model with four strains and facilitation was
very similar to a model with infinite strains and independ-
ent clearance, for example, and a model with infinite
strains and competition was similar to a model with two
strains and independent clearance. As a general rule, mod-
els with a large number of strains or facilitation had
longer timelines, while models with a small number of
strains or competition had shorter timelines.


The queuing models predicted different rates and patterns
of decline than the Ross-Macdonald model (Figure 3a, b,
c). In the Ross-Macdonald model, the decay rate in PfPR
was initially faster than in models with superinfection,
but the PfPR decay rate subsequently declined. In queuing
models in hyperendemic areas, the decay rate was initially
slow as MOI declined, but after average MOI had declined
to approximately one, PfPR decay rates then followed the
same pattern as the Ross-Macdonald model. The initial
rate of decline was related to the initial distribution of
MOI. Each one of the queuing models made a different
prediction about the distribution of MOI in relation to
baseline PfPR, and heterogeneous biting strongly influ-
enced this relationship (Figure 3d). When biting was
more aggregated on a few individuals, the average MOI at
a given level of endemicity was higher, and the initial rate
of decline was slower (Figure 3e). The average MOI
increased with PfPR, but it remained close to one until
malaria was hyperendemic, when it began to increase
sharply (Figure 3f). The transition from an area with pre-
dominantly simple infections to one with predominantly
complex infections was related to the degree of heteroge-
neous biting; for the same PfPR, higher Ro and higher MOI
were correlated. The queuing models did not differ sub-
stantially from the Ross-Macdonald model at low ende-
micity when MOI is typically close to one, which is
consistent with the GMEP experience outside Africa [7].
Because of the slow initial rate of decline, the waiting time
to reach a PfPR of 1%, starting from a hyperendemic base-
line, was two to three years longer than the Ross-Macdon-
ald model.

The initial rate of decline differed substantially among the
models, and it was also affected by the degree of biting
heterogeneity, but it was not affected by Rc. Holding all
else but Rc equal, the differences in PfPR over time were
not apparent until PfPR had fallen (Figure 3f). In other
words, the model determined the initial rate of decline,
but Rc determined the asymptotic rate of decline. In mod-
els that ignored the age of the infection, when Rc = 0, the
fastest decay rate possible was r. In other words, the GMEP
planning timelines, which used the Ross-Macdonald
model with completely interrupted transmission,
described the most optimistic timelines for changes in
PfPR from baseline to 1%.

Table 1 reports the waiting time to reach a PfPR of 1% in
the infinite queuing model with heterogeneous biting.
This model provides a good empirical fit to data describ-
ing the entomological inoculation rate and PfPR in Afri-
can children [28], and it also has other empirical support
[29]. The values are shown for PfPR baselines ranging
from 5% up to 80%, and for a range of Rc values. Also
shown are the waiting times to reach PfPR of 1% for Rc =
1.05, when the endpoint PfPR is 0.7%.


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0 1 2 3 4


Time (Years)


1000 600 400 300 200


Decay Rate


1000 600 400 300 20
1000 600 400 300 200


Decay Rate


0.0 0.2 0.4 0.6 0.8

PfPR

e)














0 1 2 3 4 5 6

Time (Years)

f)








,.




I I I I I I
0 2 4 6 8 10


Time (Years)


Figure 3 (see legend on next page)


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Figure 3 (see previous page)
a) The decline in PfPR for six different models with completely interrupted transmission: the Ross-Macdonald
model (black), infinite strains, homogeneous biting, and independent clearance (red), two strains, homogene-
ous biting and independent clearance (blue), infinite strains, homogeneous biting and competition (purple),
four strains with homogeneous biting and facilitation (orange), and senescing infections (green). b)The daily
decay rates for several models have been plotted as a function of declining PfPR, which is plotted on the vertical axis. The decay
rate is the log of the PR ratio on consecutive days. Diamonds along these trajectories are plotted one year apart. The models
have the same colors as in the panel above: the Ross-Macdonald model (black), infinite strains, homogeneous biting and inde-
pendent clearance (red), and two strains, homogeneous biting and independent clearance (blue). The trajectories have also
been plotted for completely interrupted transmission as above and for two larger values of Rc (0.5, and 0.75), which reach dif-
ferent asymptotic decay rates. c)The same graph has been re-plotted on a logarithmic scale for PfPR to show that the decay
rates remain constant at r(R- I) after PfPR is below approximately 1%. d)The relationship between baseline PfPR and the pre-
dicted MOI for the model with heterogeneous biting at various levels: a = 2, dotted; a = 3, dashed; a = 4.2, solid; a = 6, dash-
dot; and homogeneous biting, red. The more aggregated the biting, the higher the average MOI for the same PfPR. e)The PfPR
over time for the same models as panel dand completely interrupted transmission starting from a baseline PfPR of 70%, and
compared with the Ross-Macdonald model (black). f)The decline in PfPR for a model with infinite strains, homogeneous biting,
and independent clearance, but with different values of Rc (solid, Rc = 0; dashed, Rc = 0.5; and dotted, Rc = 0.75).


From I% to elimination
After PfPR falls below 1%, the decay rate of PfPR in all the
models, except for those with senescing infections, was
approximately r(Rc-1) because superinfection is rare at
very low endemicity (Figure 3, bottom). In the determin-
istic Ross-Macdonald model, declines in PfPR from 1%
through to elimination are approximated by the formula
(SI):


ert(RC-1)


With a 200 day waiting time to clear infections, as the
GMEP model assumed, completely interrupted transmis-


sion would result in an 84% proportional reduction in
PfPR each year. Compared with completely interrupted
transmission, PfPR declines half as fast at Rc = 0.5, and a
fourth as fast at Rc = 0.75. The model predicts that the
ratio of PfPR in consecutive years remains constant
through elimination; the waiting time to reduce PfPR
from 10% to 1% is approximately the same as the waiting
time to reduce PfPR from 1% to 0.1%, or from 1,000
infected individuals to 100.

The expected waiting time to elimination, TE, can be
approximated using Eq. 1 as the time when only one per-
son in the population is infected, or in a population of
size H, when PfPR is less than 1/H:


Controlled Reproductive Number, Rc
Rc = 0.75 Rc = 0.9


Rc= I

6.2
7.6
8.2
8.5
8.8
9.1
9.3
9.6
9.8
10.1
10.3
10.6
10.9
11.3
11.5
11.7


Rc= 1.05

9.4
10.8
11.5
11.9
12.2
12.5
12.7
12.9
13.2
13.4
13.7
14.0
14.3
14.6
14.8
15.0


These values were computed using the queuing model with heterogeneous biting, infinite strains and independent clearance. The columns represent
waiting times for different values of the controlled reproductive number, Rc, that show representative waiting times from zero up to 1.05. The
projections assume a sudden reduction from R0 to Rc. Longer timelines are expected if Rc is reduced gradually.


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Table I: The expected waiting time (in years) to reach PfPR of I% from a range of different baselines.


Baseline PfPR

5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%


Rc= 0

0.9
1.4
1.8
2.0
2.2
2.5
2.7
2.9
3.1
3.3
3.6
3.9
4.2
4.5
4.9
5.2


Rc = 0.5

1.6
2.2
2.6
3.0
3.2
3.5
3.7
3.9
4.1
4.4
4.7
4.9
5.2
5.6
5.8
6.0


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SlogH-log00 (2)
r(l-Rc(0))

The larger H, therefore, the longer it takes to achieve elim-
ination. In a population of 1,000 humans with com-
pletely interrupted transmission (i.e. Rc = 0), It would take
2.5 years to eliminate malaria from a population of 1,000
people, but 5 years from a population of 100,000. The
waiting times also depend on Rc. Compared with com-
pletely interrupted transmission, the time to elimination
takes twice as long ifRc = 0.5 and four times as long if Rc
= 0.75. Sample values are shown in Table 2.

The waiting time to elimination was simulated in the sto-
chastic analogue of the models starting from a PfPR of 1%;
using Rc as an index of ongoing transmission, all of the
queuing models produced similar results, including those
with heterogeneous biting (see Additional File 1). As the
number of people who were infected becomes small
(<1,000), stochasticity introduces noticeable fluctuation
in the PfPR and this affects the time to elimination.

The average proportional reductions in PfPR from the
ensemble of simulations tracked the deterministic mod-
els, but there was noticeable variability when populations
became small. The waiting times to elimination were,
therefore, extremely variable. Similar results were pre-
dicted by all of the models, except those with senescent
infections, which produced much shorter and less varia-


ble waiting times to elimination when transmission was
completely interrupted (Figure 4). Again, there were
model identifiability issues; a model with senescence and
Rc = 0.5 was similar to a model with no senescence and
completely interrupted transmission (Figure 3b). Without
senescent infections, the sharpest annual decline in PfPR
that would be expected is 84%, but for models with senes-
cent infections, the PfPR decay rate would steadily
increase and much faster declines are possible.

The Pare-Taveta malaria scheme
Because of model identifiability issues and some uncer-
tainty about the underlying biology of P. falciparum trans-
mission, it is impossible to make strong conclusions
about whether malaria transmission had been interrupted
in Pare-Taveta based upon only the initial decay rate in
PfPR. Other evidence suggested that transmission had not
been interrupted, especially malaria infections in infants
who had never travelled outside the sprayed area [13,14].

The initial decay rates in Pare and Taveta, where malaria
was hyperendemic at the baseline, were slow initially and
then increased, consistent with models of superinfection
(Figure 5). Exponential decay models also provide a good
statistical fit to the data, but the rate is difficult to interpret
as a measure of ongoing transmission in the context of a
malaria transmission model [30]. One model of interest is
the queuing model with heterogeneous biting and inde-
pendent clearance, which provides a good fit to estimates
of the entomological inoculation rate and PfPR, and the


Table 2: The expected waiting time (in years) to elimination starting from PfPR = 1%.


Basic reproductive number under control (Rc)

0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90


H = 103

2.5
2.7
2.8
3.0
3.2
3.4
3.6
3.9
4.2
4.6
5.0
5.6
6.3
7.2
8.4
10.1
12.6
16.8
25.2


Population size
H = 104 H = 105


The columns represent waiting times for different human population sizes ranging in size from one thousand to one million. The rows show the
waiting times for Rc values from zero up to 0.9. These values were computed using Eqn. 6, which agrees with the median time to elimination for the
simulations when Rc = 0.5, but when Rc = 0.75, the median extinction times were slightly shorter (Figure 2).


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H = 106

6.3
6.6
7.0
7.4
7.9
8.4
9.0
9.7
10.5
1 1.5
12.6
14.0
15.8
18.0
21.0
25.2
31.5
42.1
63.1


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0 5 10 15 20 25


Time (Years)


0 5 10 15 20 25

0 5 10 15 20 25


LO
0



-


Senescent Infections







0 5 10 15 20 25

Time (Years)

d)












0 5 10 15 20 25


Time to Elimination(Years)


Time to Elimination (Years)


Figure 4
a) Using a stochastic model, PfPR was simulated over time starting from 1% and following through to elimina-
tion (defined as when no one remains infected) in a population of 100,000 people using the Ross-Macdonald
model for Rc= 0 (red), Rc= 0.5 (blue), and Rc = 0.75 (green). The solid line shows the median PfPR over time from the
ensemble, the dashed lines show the 5th and 95th quantiles, and the solid grey lines (sometimes covered by the median PfPR)
show the same results for the deterministic models. b)The same plots for ensembles of stochastic simulations using Gamma
distributed infectious periods (n = 4), for which infections effectively "senesce". The grey line (under the solid blue line) shows
the Ross-Macdonald model for completely interrupted transmission. c, d)For the same ensemble of 500 simulations corre-
sponding to the panels above, these violin plots show a kernel density plot for the distribution of extinction times, in years. The
white dots show the median time to extinction, the thick black lines show the inter-quartile range. In the left-hand panel, the
yellow dots show the predicted values from Equation 6, which are fairly close to the average waiting time to reach IIH from
the ensemble of simulations.


parameter estimates from the model fitting were consist-
ent with direct observations [28]. Because of the low
number of data points, however, there is not sufficient sta-
tistical power to test the models with any degree of confi-
dence. Nevertheless, superinfection does occur in
hyperendemic areas, and the models provide a better vis-
ual fit to the observed patterns [29].

Discussion
The Ross-Macdonald model was used during the GMEP
for planning endemicity response timelines during the


attack phase, when PfPR was reduced from the pre-control
baseline to 1%. A comparison of several different models
suggests that the Ross-Macdonald model remains useful
today in planning for the responses to massive scaling-up
of malaria control, but with two important caveats; the
GMEP timelines represent the most optimistic scenario in
areas where superinfection is uncommon, and in hyper-
endemic areas where superinfection is common, the
GMEP timelines are unrealistically fast. Queuing models
suggest that, because of superinfection, the initial decay
rates are very slow in hyperendemic areas. As PfPR



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0

0

0 -


Malaria Journal 2009, 8:87







Malaria Journal 2009, 8:87 http://www.malariajournal.com/content/8/1/87






Taveta Forest








C ..... ----









1955 1956 1957 1958 1959

Year




Taveta Village


cc
C0 _



0





















Od
CI
oc





























1955 1956 1957 1958 1959

Year


Figure 5
The parasite rate in two parts of Taveta [14]: the forest (top) and the village (bottom). The data track the PfPR in
C00























2-10 year olds starting from 1954 and extending through 1959 the grey shows the binomial confidence intervals by the exact
test. Spraying started late in 1955 with effects that extended through 1959 (shown in pink). The blue line shows the Ross-Mac-
donald model with Rc= 0 that was used for planning during the GMEP, the purple line shows the Ross-Macdonald model with
Rc= 0.7, the orange and red line shows the infinite queuing model with heterogeneous biting (a = 3) and Rc= 0 (orange) and
with Re = 0.7 (red).




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6a










1955 1956 1957 1958 1959

Year


Figure 5
The parasite rate in two parts of Taveta [14]: the forest (top) and the village (bottom). The data track the PfPR in
2-10 year olds starting from 1954 and extending through 1959 the grey shows the binomial confidence intervals by the exact
test. Spraying started late in 1955 with effects that extended through 1959 (shown in pink). The blue line shows the Ross-Mac-
donald model with Rc = 0 that was used for planning during the GMEP, the purple line shows the Ross-Macdonald model with
Rc = 0.7, the orange and red line shows the infinite queuing model with heterogeneous biting (a = 3) and Rc = 0 (orange) and
with Rc = 0.7 (red).




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declines and MOI approaches one, after two to three years
of intensive control, the decay rates will resemble places
that started from a lower baseline. This pattern was con-
sistent with the response to malaria control observed in
Pare-Taveta. Reasonable "attack" phases to bring PfPR to
below 1% could last two to three years longer than the
GMEP timelines. During the GMEP, the three to four year
response timelines were inadequate for hyperendemic
Africa, including Pare-Taveta. Also of note is that interven-
tions during the Garki Project lasted only 18 months [31].

These models also generate useful timelines for the con-
solidation phase. Many studies have suggested that Rc < 1
represents a threshold for elimination, but this analysis
suggests that the timelines for control and elimination are
extremely long unless Rc is reduced substantially below
one. High levels of control must also be sustained for sev-
eral years after PfPR falls below 1%. The waiting time to
fall from 10% to 1% is the same as the waiting time to fall
from 1% to 0.1% or from 1,000 infected individuals to
100 infected individuals. After reducing PfPR below 1%,
therefore, the waiting time to complete elimination can
be longer than the waiting time to reach 1%, depending
on the population size and the degree to which malaria
transmission is interrupted.

An important recommendation is that malaria control
planning timelines for consolidation after PfPR drops
below 1% should be based on the number of people who
are infected, not the PfPR. Large populations must wait
longer than small populations to achieve elimination
because when PfPR is 1%, more people remain infected in
the larger population. After reaching a low endemicity of
1%, the waiting time to elimination ranges from 2.5 addi-
tional years in a population of a thousand humans to 6.3
additional years in a population of a million, assuming
transmission is completely and immediately interrupted.
When there is some low level transmission, populations
of a million people can reasonably expect to wait a decade
or more until the endemic reservoir of parasites has
cleared.

These timelines are largely based on models that ignore
the age of malaria infections. If infections do "senesce"
such that older infections clear faster than new ones, the
time to elimination would be much shorter. There has
been some mixed evidence for senescence [29,32], but
PfPR decay rates that are broadly consistent with non-
senescing infections were found often during GMEP expe-
riences [7]. In the documentation of successful attacks
during the GMEP, decay rates were found that were faster
than the predicted 84% maximum decline, but only when
programmes used drugs, or when PfPR was very low and
sampling error was an important concern [7]. The GMEP
studies were also generally of short duration (three to four


years), so that senescence might not have become appar-
ent in the data. When there is some ongoing low-level
transmission on the way to elimination, infections in the
population will be a mix of young and old infections, and
senescence may not be important or apparent. Because of
uncertainty about senescence, however, projections based
on the Ross-Macdonald model should be considered a
worst-case scenario for the consolidation phase, because
shorter elimination timelines can be expected if infections
senesce.

There is some uncertainty about the waiting time to clear
an infection, and the timelines described here would
change if the duration of an infection were actually closer
to 150 or 250 days. This analysis has focused instead on
comparing models, and it demonstrated that the same
patterns could arise from differences in either r, in Rc, or
in the model. A basis exists for predicting Ro based on
PfPR and PfEIR [28,33], and there is also a basis for pre-
dicting Rc in relation to usage of insecticide treated nets
[34,35], so it is possible to develop a predictive theory for
malaria control endpoints and timelines. Regardless of
the uncertainty, most models give broadly similar results
when transmission intensity is reduced below the thresh-
old for elimination, but not completely interrupted (i.e.
when R, < 1).

One way to resolve the identifiability issues is to design
studies that concurrently estimate r, PfPR, MOI, and Rc.
The analysis thus suggests that a field-estimate of MOI
could be an important complement to PfPR. After fixing
PfPR, biting heterogeneity, R0, and the time to reach PfPR
of 1% all increase with MOI. A gap currently exists, how-
ever, between the notion of MOI in these models and
MOI in studies.

MOI in the models is based on the notion of a parasite
"brood," the number of founding parasite genotypes.
Superinfection can also arise if a person is simultaneously
bitten by many infectious mosquitoes [36], or if a bite
transmits several parasite genotypes. The problems of
defining broods and relating these to empirical estimates
of MOI are complicated by parasite populations that are
highly genetically polymorphic and infections that are
highly variable parasite population densities and fre-
quencies can fluctuate by several orders of magnitude over
short periods of time. Similar counting problems arise in
ecology, where there has been a long-standing interest in
estimating species diversity [37]. Ecologist measure diver-
sity in two ways: richness is the total number of different
entities present, and evenness is the probability that any
two random individuals would have the same type.
Sequencing methods that sample random parasites tend
to measure evenness, not richness, because they resample
common types. Sequencing methods that identify genetic


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polymorphisms can't infer the genotypes of individual
parasites, which sets an upper limit on the ability to esti-
mate MOI. MOI in the models is based on the notion of
richness, so the relevance of existing MOI estimates not
clear. If MOI could be developed as a standardized and
interpretable field-measure of transmission, it would rep-
resent a highly useful complement to PfPR as a basis for
setting timelines.

Timelines for elimination can be advanced with antima-
larial drugs. Mass drug administration can rapidly reduce
the number of infected people and advance the timelines
for elimination, but the expense and risks may not be jus-
tified since the declines would happen regardless. The use
of drugs to cure malaria infections can prevent ongoing
transmission, reduce Rc and speed up elimination; other
models suggest that using drugs for treating clinical
malaria controls transmission better in low endemicity
settings [27]. Active case detection to find, treat and cure
nearby asymptomatic infections can speed up progress
towards elimination. Prompt treatment of a high fraction
of clinical malaria, case-based investigations, and active
case detection become increasingly important for reduc-
ing ongoing transmission from imported malaria in the
end phases of consolidation and after malaria has been
eliminated [38], so implementing such policies during the
consolidation phase provides an opportunity to train and
build capacity.

This modelling framework was designed to establish basic
expectations and a template that can be tailored to local
situations. Planning for malaria elimination must also
consider the risk of imported malaria and ongoing trans-
mission from imported cases. This is of great concern dur-
ing elimination campaigns, and it would continue to be a
concern after achieving elimination, but it is a separate
issue from the initial goal of eliminating the endemic res-
ervoir. Other important considerations are natural fluctu-
ations in mosquito population densities, changes in the
coverage levels or effectiveness of various interventions,
and the spatial structure of populations. These factors
would lead to quantitative changes and increased variabil-
ity in the expected waiting times to elimination, but so
long as there is not a temporal trend in these factors, the
basic expectations established here would still be useful.
Given the large number of factors and possible permuta-
tions of those factors, it would be impractical to consider
every scenario prospectively. Expansion of the basic mod-
els may be best done in case studies, which can consider
these other factors in country-specific malaria elimination
plans.

Several models were used to evaluate decay rates in PfPR
in response to malaria control; the analysis demonstrates
that many different plausible models produce very similar


patterns. The timelines described here can help to estab-
lish reasonable planning horizons for countries that are
contemplating elimination. A substantial body of further
work is required to investigate the most locally appropri-
ate and effective suite of interventions needed to affect
that goal. This modelling framework, when combined
with contemporary global maps of P. falciparum preva-
lence [39] and summaries of existing intervention cover-
age [40], provides a basis for setting plausible timelines
for changing malaria endemicity in Africa and this is the
subject of ongoing work.

Abbreviations
(GMEP): Global Malaria Eradication Programme; (PfPR):
Plasmodiumfalciparum parasite rate; (MOI): multiplicity of
infection; (Ro): basic reproductive number; (RJ): basic
reproductive number under control.

Competing interests
The authors declare that they have no competing interests.

Authors' contributions
DLS wrote the first draft of the manuscript and outlined
the models. SIH contributed to the refining of the models
and the writing of the manuscript.

Additional material


Additional file 1
Supplementary Online Information: Endemicity response timelines for
Plasmodium falciparum elimination. A fuller description and analysis
of the mathematical models.
Click here for file
[http://www.biomedcentral.com/content/supplementary/1475-
2875-8-87-S.1pdf]


Acknowledgements
Thanks to Eili Klein, F. Ellis McKenzie, Abdisalan M. Noor, Robert W. Snow
and Andrew J. Tatem for comments on the manuscript. The R code used
to generate all of the figures is freely available upon request. DLS is sup-
ported by a grant from the Bill & Melinda Gates Foundation (#49446) and
funding from the RAPIDD program of the Science & Technology Directo-
rate, Department of Homeland Security, and the Fogarty International
Center, National Institutes of Health. SIH is funded by a Senior Research
Fellowship from the Wellcome Trust (#079091). The authors acknowledge
the support of the Emerging Pathogens Institute, University of Florida. This
work also forms part of the output of the Malaria Atlas Project (MAP, http:/
/www.map.ox.ac.uk), principally funded by the Wellcome Trust, UK.

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