Plasmodium falciparum parasitaemia described by a new mathematical model.

Abstract

A new mathematical model of Plasmodium falciparum asexual parasitaemia is formulated and fitted to 35 malaria therapy cases making a spontaneous recovery after primary inoculation. Observed and simulated case-histories are compared with respect to 9 descriptive statistics. The simulated courses of parasitaemia are more realistic than any previously published. The model uses a discrete time-step of 2 days. Its realistic behaviour was achieved by the following combination of features (i) intra-clonal antigenic variation, (ii) large variations of the variants' baseline growth rate, depending on both variant and case, (iii) innate autoregulation of the asexual parasite density, variable among cases, (iv) acquired variant-specific immunity and (v) acquired variant-transcending immunity, variable among cases. Aspects of the model's internal behaviour, concerning variant dynamics, as well as the respective contributions of the three control mechanisms (iii) - (v), are displayed. Some implications for pathogenesis and control are discussed.

Full text

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379
Plasmodium falciparum parasitaemia described by a new
mathematical model
L. MOLINEAUX?, H. H. DIEBNER?,?, M. EICHNER?, W. E. COLLINS?, G. M. JEFFERY?
and K. DIETZ?*
?World Health Organization (Retired)
?Department of Medical Biometry, University of Tu ? bingen, Westbahnhofstrasse 55, D-72070 Tu ? bingen, Germany
?Center for Art and Media, Lorenzstrasse 19, D-76135 Karlsruhe, Germany
?Centers for Disease Control and Prevention, US Public Health Service, Department of Health and Human Services,
Atlanta, GA, USA
?U.S. Public Health Service (Retired)
(Received 27 April 2000; revised 13 October 2000; accepted 15 October 2000)

A new mathematical model of Plasmodium falciparum asexual parasitaemia is formulated and fitted to 35 malaria therapy
cases making a spontaneous recovery after primary inoculation. Observed and simulated case-histories are compared with
respect to 9 descriptive statistics. The simulated courses of parasitaemia are more realistic than any previously published.
The model uses a discretetime-step of 2 days. Itsrealistic behaviour was achieved by the following combination of features
(i) intra-clonal antigenic variation, (ii) large variations of the variants’ baseline growth rate, depending on both variant and
case, (iii) innate autoregulation of the asexual parasite density, variable among cases, (iv) acquired variant-specific
immunity and (v) acquired variant-transcending immunity, variable among cases. Aspects of the model’s internal
behaviour, concerning variant dynamics, as well as the respective contributions of the three control mechanisms (iii) – (v),
are displayed. Some implications for pathogenesis and control are discussed.
Key words: Plasmodium falciparum, mathematical model, malaria therapy, PfEMP1.

Plasmodium falciparum malaria is a major cause of
morbidity and mortality, largely attributable to
asexual parasitaemia. A realistic simulation model of
the course of asexual parasitaemia in the human host
is, therefore, relevant for the planning and evaluation
of malaria control. We have recently reviewed
published intra-host models of malaria, questioning
the realism of their assumptions and behaviour, and
suggesting a more rigorous comparison with data,
the investigation of combinations of types of parasite
diversity and density regulation mechanisms, al-
lowance for individual variation, and the adoption of
discrete-time modelling (Molineaux & Dietz, 1999).
This paper presents a first implementation of those
suggestions. It is based on 35 cases of primary P.
falciparum infections inoculated for malaria therapy,
and classified as spontaneous cures from malaria. We
present (i) a statistical description of the data,
proposed as a simulation target, (ii) a new simulation
model of the course of P. falciparum asexual
parasitaemia in the human host, (iii) a comparison of
* Corresponding author: Department of Medical Bio-
metry, University of Tu ? bingen, Westbahnhofstrasse 55,
D-72070, Tu ? bingen, Germany. Tel: ?49 7071 29 72112.
Fax: ?49 7071 29 5075.
E-mail: klaus.dietz?uni-tuebingen.de
model outputs with the simulation target and (iv)
some aspects of the model’s internal behaviour.
  
The original data
Malaria therapy data have made a majorcontribution
to our knowledge of patterns of asexual parasitaemia,
for different species of human malaria parasites,
including P. falciparum. The data used here were
collected by the USPHS in the NIH Laboratories in
Columbia,SouthCarolina
Georgia, at a time when malaria therapy was a
recommended treatment for neurosyphilis. The
patients were Afro-American adult neurosyphilitics
with no history of prior exposure to malaria. The
parasiteofchoicefor
Plasmodium vivax, to which Afro-Americans were,
however, found to be refractory. So they were
treated with different strains of P. falciparum, under
close medical supervision. They were inoculated
either with sporozoites (generally through mosquito
bite) or with infected blood. Inoculations were
preceded by variable sequences of blood and mos-
quito passages of the strain. Microscopical exam-
ination of the blood was performed on an almost
daily basis. In principle 0?1 µl of blood was
examined,lessinthe
and Milledgeville,
malaria therapywas
caseofhighdensity,
ParasitologyParasitology (2001), 122, 379–391.
DOI: 10.1017?S0031182001007533
? 2001 Cambridge University Press
Printed in the United Kingdom

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L. Molineaux and others 380
Fig. 1. Five observed case-histories of daily asexual
parasitaemia. Selection: the 35 cases were ranked by
increasing number of local maxima (the density on day t
was declared a local maximum if it was higher than the
densities on days (t?6) to (t?1) and not lower than on
days (t ? 1) to (t ? 6)); the cases ranked 1, 9, 18, 27,
35 were selected. The number of local maxima is given
in parentheses, and their timing indicated by the
triangles on top of each graph. Missing observations are
substituted by log-linear interpolation, and the
interpolated values identified by lozenges.
occasionally more; the detection threshold was thus
about 10 parasitized red blood cells (PRBC) per µl
(Earle & Perez, 1932; Eyles & Young, 1951; Jeffery
& Eyles, 1954; Jeffery et al. 1959; Collins & Jeffery,
1999).
This investigation is restricted to 35 P. falciparum
infections classified as spontaneous cures, out of a
total of 334 primary inoculations. This classification
was made as follows (i) a small fraction (19?334 or
5?7%) of the cases were terminated by an early
curative treatment (most commonly a 3-day course
of chloroquine), started within days 1 to 21, in the
presence of fever and parasitaemia; they constitute a
non-random subset of relatively severe infections
and were excluded from this investigation, (ii) most
of the remainder were given a curative treatment at
the end of their initial period of continuous or nearly
continuous parasitaemia (i.e. after the period of
expected effect on the neurosyphilis) and (iii) an
apparently unselectedsubset
recruited in an investigation of the total duration of
infection in naive hosts (important at the time for the
malaria eradication strategy) and not given their final
curative treatment until they were continuously
negative for 1 month of quasi daily blood exam-
ination, plus 5 months of twice weekly examination;
they were then, as a rule, given a curative treatment
before discharge, but classified as spontaneously
cured; these patients are the subjects of the present
investigation. About half of the cases belonging to
categories (ii) and (iii) received, on the basis of their
clinical and parasitological findings, 1 or more low-
dose suppressive treatments (most commonly a
single dose of quinine). Among the 35 cases of the
present investigation, 16 received from 1 to 10 low-
dose suppressive treatments (median 2; quinine was
used 24 times, chlorguanide 21 times, chloroquine
twice). These treatments had only a limited and
short-lived effect on the course of parasitaemia and
are not taken into account in the analyses presented.
Eighteen patients were inoculated using infected
blood, 17 by mosquito-bite; the P. falciparum strains
inoculated were El Limon (Panama, 1948; 17 cases),
Santee Cooper (South Carolina, 1946; 17 cases), and
McLendon (South Carolina, 1940; 1 case).
Fig. 1 illustrates 5 contrasting observed case-
histories. The raw data show much variation, of
different kinds, both within and among individual
case-histories. We try to develop a model with
biologically plausible assumptions and parameter
ranges, which produces a population of simulated
case-histories that is statistically similar to the
population of observed case-histories. Therefore, we
need to reduce the raw data to a relatively short list
of variables whose distributions can be used as
quantified simulation targets. The following features
are apparent on inspection of the data (i) a variable
total duration of parasitaemia, including variable
periods of non-patency (in these data, patent para-
sitaemiasignifiesaparasite
parasites?µl), (ii) an initial period of increase, at a
rapid and variable rate, up to a first local maximum,
of variable height, (iii) a variable number of local
maxima, at irregular intervals, (iv) an almost mono-
tone downward trend of the local maxima, the first
local maximum being usually the absolute maximum
and (v) bouts of 2-day periodicity, in particular at
high parasite densities.
ofpatientswas
densityof
?10
The simulation targets
We opted not to take into account the 2-day

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P. falciparum intra-host model 381
Table 1. The variables used to summarize a case-history of asexual parasitaemia (observed on odd days or
simulated) and the results of paired t-tests (see Fig. 2)
VariableCode Explanation Mean (sim. ? obs.)P*
(1) Initial slopeinit. slope The slope of a linear regression line
through the log densities from first
positive slide to first local maximum
The density at time t is declared a local
maximum if it is higher than densities at
times (t?6), (t?4), (t?2) and not lower
than at times (t?2), (t?4), (t?6)
?0?0095 per day0?65
(2) Log density at first
local maximum
log 1st max.
0?0370?082
(3) Number of local maxima
(4) Slope of local maxima
No max.
slope max.
0?510?41
0?0021
The slope of a linear regression line
through the log densities of the local
maxima
?0?0035 per day
(5) Geometric mean (GM)
of the intervals between
consecutive local maxima
(6) SD of the logs of the
intervals between
consecutive local maxima
(7) Proportion of positive
observations in the first
half of the interval
between first and last
positive day
(8) Proportion of positive
observations in the
second half of the
interval between first
and last positive day
(9) Last positive day
GM interv.
?3?7 days0?030
SD log
interv.
0?00030.99
prop.?1stDay 1?first positive day; ‘positive’
means density?10?µl
0?115
?10−?
prop.?2nd0?086 0.040
last?day Difference between last and first
positive day
7 days0.64
* Before Bonferroni-Holm adjustment for the number of tests (see text)
periodicity. We wanted to explore to what extent we
could simulate the course of asexual parasitaemia by
a discrete-time model with 2-day steps. Accordingly
we read the data on alternate days (even or odd),
which had a smoothing effect. Table 1 lists and
defines the 9 descriptive variables used to summarize
a case-history, either observed or simulated. The 2
observed distributions (odd days, even days) were
compared by the paired t-test: the only significant
difference P?5%) was that the first local maximum
was higher on odd than on even days; inspection of
the data shows that this is probably related to the
frequent association of the initial slope with a 2-day
periodicity of the parasite density. Correlation
coefficients were calculated for the (9?8)?2?36
pairs of variables within each of the 2 observed
distributions (not shown); the odd-day and even-day
observations were strongly and positively correlated.
We opted to compare our simulations with the odd
day observations. The corresponding statistics are
given in Table 2. They constitute our first simulation
targets.
The model, its assumptions and their biological
justifications
The model equations and the definitions of variables
and parameters are given in the Appendix. The
model describes the time-course of asexual P.
falciparum parasitaemia following a single mono-
clonal infection of an individual human host without
previous exposure to malaria. It is a discrete-time
model with 2-day steps, corresponding to the
duration of P.f.’s erythrocytic cycle; time is
expressed in days, parasitaemia as PRBC per µl. The
population of PRBC is distributed among intraclonal
variantsofthemain
falciparumerythrocyte
expressed on their surface. It has been suggested that
the persistence of P. falciparum infections, at least in
naive hosts, depends largely on intraclonal variation
of antigen PfEMP1, displayed on the surface of
PRBC’s – a single variant being expressed per PRBC
– and crucial for their sequestration (Borst et al.
1995). In Plasmodium knowlesi and Plasmodium
chabaudi there is direct in vivo evidence of the
association between successive recrudescences of
parasitaemia and expression of different variants of
those parasites’ PfEMP1 analogues (Brown &
Brown, 1965; Phillips et al. 1997). The initially
estimated number of variants per P.f. genome was
50–150, but recent estimates are smaller, e.g. 41 in
clone 3D7 (Sutherland, 1998). Our simulations
antigen
membrane
(PfEMP1?P.
protein 1)

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L. Molineaux and others382
Table 2. Observed statistics (odd days)
Variable*Code Minimum MedianMaximum
1
2
3
4
5
6
7
8
9
init. slope
log 1st max.
No max.
slope max.
GM interv.
SD log interv.
prop.?1st
prop.?2nd
last?day
0?188 per day
3?37
2
?0?074 per day
14?4 days
0?03
0?40
0?08
37 days
0?487 per day
4?79
10
?0?013 per day
20?0 days
0?20
0?88
0?46
215 days
0?872 per day
5?66
17
?0?0007 per day
77?8 days
0?47
1?00
0?94
405 days
* See Table 1.
assume 50 variants per clone, of which a variable
number are expressed in the course of a case-history.
The model dynamics (equation (1)) are driven by 2
forces: the switching among variants and the mul-
tiplication of a variant.
Switching dynamics among variants is expressed
by
where Pi(t) is the density of variant i, as PRBC?µl
blood and pi(t) is the probability that a switching
PRBC switches to variant i. A constant fraction (s) of
PRBC switch variants per 2-day-cycle. Initially they
switch into variant 1, 2, … , v according to a geo-
metric distribution with negligibly small prob-
abilities of switching into the higher-numbered
variants.Later,asvariant-specificimmuneresponses
develop, switching into variants against which there
is already a specific immune response is reduced
accordingly (equation (4)), and the probabilities of
switching into higher-numbered variants upgraded
through normalising the sum of residual proba-
bilities to 1. The model assumes that the host’s
variant-specific immune status affects variant ex-
pression in the course of intra-erythrocytic schizo-
gony, thus reducing subsequent immunoselection.
These assumptions concerning switching dynam-
ics are based on biological observations. In P.
falciparum the switch-rate has been estimated only in
vitro and the most commonly quoted estimate is 2%
per cycle (Roberts et al. 1992). In P. chabaudi the
switch-rate has been estimated in vivo, yielding
minimum estimates of 1?2–1?6% per cycle (Brannan,
Turner & Phillips, 1994). That variants can differ
greatly in their intrinsic probability of expression has
been shown directly in P. chabaudi in naive hosts
(Brannan et al. 1994). Modulation of variant ex-
pression by the host’s variant-specific immune status
in the course of intra-erythrocytic schizogony is
strongly suggested by findings inP.knowlesi (Brown,
1973; Barnwell et al. 1983).
Multiplication dynamics of a variant is expressed
by miSc(t)Si(t)Sm(t) in equation (1). A variant has a
baseline multiplication factor (mi), interpretable as
the number of successful merozoites produced per
PRBC in the absence of any controlling feedback, i.e.
A
B(1?s)Pi(t)?spi(t)?vj=?Pj(t)
C
Din equation (1),
in the very early stages of parasitaemia. For each
simulation of a case-history, variant-specific baseline
multiplication factors are randomly chosen from a
truncated normal distribution. Among 4 P. chabaudi
variants whose growth was compared in vivo, 1
showed a strong preference for reticulocytes and
yielded a much lower maximum parasitaemia
(Phillips et al. 1997). Our data show large variation
of the initial slope of the parasitaemia, which led us
to assume large variation of variant-specific growth
rates, plus random host-specific association between
a variant’s growth rate and its intrinsic probability of
expression, hence its probability of being expressed
early in a case-history. The fact that in the P.
chabaudi-mouse model, variation of growth rates
among hosts was negligible in comparison with
variationamongvariants,isnotanobjection,because
human populations are genetically much more
diverse than highly inbred laboratory mice.
Parasite multiplication is controlled by 3 inde-
pendent (and competing) immune responses, 1
innate and variant-transcending, 1 acquired and
variant-specific,1acquired
scending. The effects of the 3 immune responses
are represented by the functions Sc(t), Si(t) and
Sm(t). The existence and importance of an innate
autoregulation of density is well established; it is a
rapid and short-lived effect of the rupture of mature
PRBC’s via malaria toxin(s) and host cytokines
(TNF, etc) and their secondary effectors (fever, NO)
(Kwiatkowski, 1995). The wide variation in strength
of that mechanism among hosts is suggested by the
wide variation in density of the first local maximum
in the malaria therapy data (Jeffery et al. 1959). The
existenceandimportanceofacquiredvariant-specific
protective immunity are suggested by field obser-
vations (Bull et al. 1998). Most experts agree that
acquired protective immunity against the asexual
bloodstages has, in addition, some variant-trans-
cending component(s). The immune response’s
effects are formulated as success probabilities (i.e.
probabilities that the parasite escapes their effects),
in terms of their triggers (parasite densities), without
explicit modelling of their effectors (e.g. cytokines,
and variant-tran-

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P. falciparum intra-host model383
immune cells, antibodies). As the 3 immune
responses are independent and competing, the
probability that the parasite escapes the effects of all
3 is the product of the 3 success functions. All 3
success functions are of the general shape S?
(1?β)?1?[f(X)?X*]κ?−??β, where X is the trigger;
S is a sigmoid function, decreasing from 1 (for f(X)
?0) to β (as f(X)??), through S?(1?β)?2 for
f(X)?X*, a critical constant; the slope of the
sigmoid function at the inflection point increases
with the power κ; note that in this paper, β has been
set to zero in the equations for Sc(t) and Si(t), but not
in the equation for Sm(t), i.e. the innate and the
acquired variant-specific immune responses are
allowed to reach 100% efficacy in one infection,
whereas the maximum allowed efficacy of the
acquired variant-transcending immune response is
[100(1?β)]%. For the innate immune response, all
PRBC are triggers and targets; a given day’s parasite
density exerts innate autoregulation in the next 2-
day-interval only (no delay, no memory); the critical
constant P?
c
is case-specific and is assumed to be
proportional to the first local maximum parasite-
density of a case (equation (9)). For the acquired
variant-specific and variant-transcending immune
responses, the triggers and targets are, respectively,
the corresponding variant-specific PRBC and the
total PRBC; delays (δv, δm) are introduced to account
for the fact that it takes some time for antigens to
produceimmuneeffectors,anddiscountfactors(σ,ρ)
are introduced to allow immunity to decay; a
saturation level (C) for the acquisition of variant-
transcending immunity is introduced to prevent it
from growing too fast; the existence of a saturation
level is at least biologically plausible; the critical
constant P?
vis common to all variants and hosts; the
critical constant P?
mis case-specific, as suggested by
the large variation in total duration of patent
parasitaemia, and is assumed to be proportional to
the total duration of a case (equation (10)).
Initial conditions and extinction rules are as follows.
The initial parasitaemia is constituted by variant 1
and the initial density is set at 0?1?µl, corresponding
to 5?10? PRBC in an adult with a 5 l blood-volume,
or to a density 100 times smaller than the detection
level of 10?µl. A variant goes extinct when its density
drops below 10−??µl, i.e. when its number drops
below 50 in a adult blood volume of 5 l (it may
reappear through switching from other variants); the
infection goes extinct when all expressed variants
have gone extinct.
Method of fitting
We did not apply formal minimization but used
informed trial and error. Certain parameters were
fixed, e.g. the mean 2-day baseline multiplication
factor (µm?16) or the 2-day switching fraction (s?
0?02). Other parameters were varied, within ranges
based on the literature or on the raw data. Simu-
lations were conducted in 2 phases. In the first phase,
we performed sets of 35 simulations corresponding
to the 35 observed case-histories. Between sets, 1 or
more parameters were varied. Within each set, the
parameters P?
cand P?
mand the random allocation of
variant-specific baseline multiplication factors were
case-specific. Simulation outcomes were compared
to simulation targets (Table 2) by inspection, and a
‘best’ set was selected. In the second phase, starting
from that best set, we simulated each case 50 times,
reallocating randomly each time the variant-specific
baseline multiplication factors, and used χ? to
identify a ‘best’ case-specific matching between the
variants’probabilityofexpressionandmultiplication
factor.

Fitting the model to the data
Simulations were conducted in 2 phases. In the first
phase, we took, for each population of 35 case-
histories, the following steps (i) specify initial
conditions and parameter values for that population;
after preliminary exploration, the initial conditions
were set at Pi(0)?0?1 for i?1, and Pi(0)?0, for
i?2, 3, … , v; use the parameters kcand kmto
calculate host-specific values of P?
each simulation: randomly select variant-specific
values ofmiand conduct the simulationto extinction,
(iii) add the effect of measurement error, assuming a
Poisson distribution for the number of PRBC per
0?1 µl: for expected values less than 100 replace the
simulated result by a Poisson distributed random
number with the corresponding expectation; for
expected values above 100 sample a Poisson random
variable with mean 100 and multiply it with the
appropriate correction factor; this method assumes
that for high densities of PRBC only about 100 cells
are actually counted, (iv) after completion of the
simulated case-histories, summarize them by the
same statistics as used to summarize the observed
case-histories, and compare simulations to obser-
vations, in terms of minimum, median, maximum of
the 9 descriptive variables (see Table 2). Except for
the stochastic steps (ii) and (iii), the simulations are
deterministic. The parameter values that produced
the ‘best’ population of simulated case-histories are
given with the list of parameters in the Appendix.
In the second phase of simulations, for each host-
specific pair of P?
c
and P?
conducted, each with a new allocation of mivalues to
variants, while leaving the other parameters un-
changed. Within each set of 50 simulations, the fit
betweeneachsimulationandthecorrespondingcase-
history (i.e. the case used to calculate the case-
specific P?
c
and P?
minputs) was measured by a χ?
statistic which gave equal weight to the normalized
cand P?
m, (ii) for
m, 50 simulations were

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L. Molineaux and others 384
Fig. 2. Paired t-test comparisons of 35 simulated case-histories with the corresponding odd-day observed case-
histories, with respect to the 9 descriptive variables. Each simulated case-history is the best (by least χ?) out of 50
stochastic realizations. Variables, mean differences (between observed and simulated) and P values are given in
Table 1.
residuals of the 9 descriptive variables. The simu-
lation with the least χ? was designated best. The 35
best simulations were compared to the 35 observed
case-histories by the paired t-test, and the results are
shown in Fig. 2 and Table 1 (the table gives, for each
variable, the mean difference between simulated and
observed,andtheP-valueunadjustedforthenumber
of tests). For all of the 9 variables the ranges of
observed and simulated values are largely over-
lapping. After correction by the Bonferroni-Holm
procedure for the number of comparisons made, the
model outputs differ significantly (P?0?05) from the
observations with respect to 2 of the 9 variables (i)
the simulated slope of local maxima (slope max.) is
too steep, and (ii) the proportion of positive obser-
vations in the first half of the interval between first
and last positive day (prop.?1st) is too large. Note
that for those two variables, most differences be-
tween observed and simulated are relatively small,
but systematic, hence the low P values in the paired
t-test. We also compared the Spearman rank cor-
relation coefficients between pairs of descriptive
variables of the simulations with those of the
observations, as shown in Fig. 3; the agreement
between observed and simulated correlations is
remarkably good.

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P. falciparum intra-host model 385
Fig. 3. Spearman’s rank correlation coefficients between
pairs of descriptive variables (numbered 1–9 as in Table
1) in the 35 best simulated case-histories and in the 35
observed case-histories.
To illustrate the variability within the sets of 50
simulations, and to show some ‘best fits’, we selected
the 1st, 3rd and 5th case of Fig. 1, representing a
wide range of numbers of local maxima. Fig. 4 shows
the distribution of the 3 sets of 50 simulations, with
respect to 3 of the 9 descriptive output statistics (init.
slope, No. max., slope max.); for each distribution,
the corresponding observation is added, and the
‘best’ simulation identified. The stochastic allo-
cation of growth factor (mi) to variants produces a
large variation in the outputs; the observed values lie
within the range of the 50 simulations in 7 of the 9
distributions shown, and just outside in the re-
maining 2. The best runs, within each of the 3 sets of
50, are shown in Fig. 5, in comparison with the
corresponding observations. The simulated case-
histories are rather similar to the corresponding
observed case-histories.
The model’s internal behaviour
It should be of interest to modellers and biologists to
find out how the model’s internal behaviour
generates the rather realistic patterns of asexual
parasitaemia obtained. This section and Figs 6–8
make explicit some aspects of that internal be-
haviour. The features displayed cannot be tested
with malaria therapy data, but might be tested with
data from other sources.
Fig. 6 illustrates the variants’ behaviour according
to the model; it shows, for the best run of case G408,
the total density plus the density of 10 selected
variants out of the 50 expressed in that run. Low-
numbered variants are expressed early because of a
high pi(0) (equation (4)), and compete through their
growth factors. At the first local maximum (Pc(14)?
Fig. 4. Distributions of 3 of the 9 descriptive variables:
(A) initial slope; (B) number of local maxima; (C) slope
of local maxima in 3 sets of 50 simulations,
corresponding to the 1st, 3rd and 5th case of Fig. 1,
representing a wide range of numbers of local maxima.
The P?
cand P?
mparameters are case-specific and fixed
for the 50 runs of the case; for each run within each set
of 50, the parameter miis randomly allocated to the
variants, from the distribution given in Table 2. For
each distribution, the corresponding observation is
added (?) and the best simulation (by least χ?)
identified (?).
207000?µl): (i) 13 variants are expressed, 5 of them
at?10?µl; (ii) variant 3, a fast grower (m??30) has
overgrown variants 1(m??1?3) and 2(m??12), and
makes up 91% of the total; (iii) the innate success
function reaches its lowest value in this run (Sc(14)

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L. Molineaux and others 386
Fig. 5. Three contrasting observed case-histories (odd days) and the best (by least χ?) out of 50 stochastic realisations
of the corresponding simulations.
Fig. 6. Total and selected variant-specific asexual parasite densities in the best simulation of case G408; 50 variants
were expressed, of which 10 (equally spaced) are shown, including those dominating the 1st and 2nd local maximum.
Simulation starts on day zero with variant 1 at density 10−??µl; local maxima are marked by triangles on top; the
horizontal line indicates the microscopic detection level (10?µl); the simulation results shown precede the application,
to the total, of a stochastic measurement error (see text), which, in this run, generated an 11th visible peak on day
178. The variants shown had the following stochastically allocated multiplication factors (rounded to the nearest
integer):
variant38 13 18 23 28 33 38 43 48
mi
30 23 14 33 23 15 27595

Page 9

P. falciparum intra-host model 387
?0?01) and all variants drop (equation (1)). After the
drop, variant 3 and 3 others (not shown) go on to
extinctionthroughvariant-specific
(equation (7), while most variants recover. High-
numbered variants can appear only after the host’s
variant-specific immunity has raised their selection
probability pi(t), while lowering the pi(t) of lower
numbered variants (equation (5)). Among the 11
peaks of this run (8 in the first 125 days), the number
of variants expressed varied from 3 (peak no. 11) to
15 (peak no. 4), and the contribution of the dominant
variant varied from 69% (peak no. 3) to 99% (peak
no. 10). Later peaks tend to be lower, with lower
innate immunity and higher variant-transcending
immunity, and to be composed of fewer variants,
with a stronger dominance of the dominant variant.
For example, at peak no. 8, on day 122, Pc?5769;
Sc?1?00; Sm?0?13; 9 variants are expressed, 4 of
them at?10?µl; the dominant variant makes up
95% of the total. Overall, the effects of differential
variant-specific baseline expression probabilities
(pi(0)) and multiplication factors (mi), of immuno-
modulation of expression probabilities, and of com-
petition through innate density regulation, are
strong.
Among the best runs of the 35 cases, the number
of variants expressed varied from 12 to 50 (median
45); the number expressed at?10?µl varied from 8
to 50 (median 42); the two numbers were strongly
correlated, and their ratio (i.e. the fraction of
expressed variants reaching?10?µl) varied from
0?53 to 1. The number of variants expressed
at?10?µl waspositively
(Spearman’s R??0?70; P?0?0001), with the
number of local maxima (R??0?58; P?0?0003),
and with time from first to last patent parasitaemia
(R??0?47; P?0?004). By linear regression we
found that the number of local maxima increased
with the number of variants expressed at?10?µl,
at the rate of 1 local maximum per 6 variants (in
Fig. 6 the first wave of parasitaemia ‘consumed’
4 variants).
Fig. 7 shows the relative contributions of the
model’s 3 control mechanisms in the 35 simulated
cases, cumulated either over the whole case-history
or only from onset to 6 days after the first local
maximum. The calculation assumes the following (i)
in the absence of control, the parasite population
would grow according to Pc(t?2)??vi=?Pi(t)mi; (ii)
the difference between that hypothetical Pc(t ? 2)
and the one calculated according to equations (1)–
(3) represents the number of parasites ‘controlled’ in
the interval (t, t ? 2); (iii) the 3 control mechanisms
compete to control variant i according to (1?Sk(t))?
[(1?Sc(t))?(1?Sm(t))?(1?Si(t))], where k?c, m,
i; (iv) summation over a period of time gives –
within that period – each controlled parasite an
equal weight. Over the whole case-history, the innate
variant-transcending immune response contributed
immunity
correlated withP?m
Fig. 7. Ternary plot of the relative contributions of the
model’s 3 control mechanisms in the 35 simulated cases,
cumulated either over the whole case-history (?) or
only from onset to 6 days after the first local maximum
(?). The contributions are calculated as outlined in the
text.
Fig. 8. The relative contributions of the model’s 3
control mechanisms (?, innate immune response;
O, acquired variant-specific immune responses;
P, acquired variant-transcending immune response) in
the control of the 11 successive peaks of parasitaemia of
the best simulation of case G408. Each peak is identified
by having a parasite density higher than the 3 preceding
values, and not lower than the 3 following values. The
contributions are calculated as outlined in the text,
including for each peak the 4 intervals from (t?4) to
(t?4), where t?time of peak.
31–56% (median 44) of the total control, the
acquiredvariant-specificimmuneresponses13–29%
(median 23), and the acquired variant-transcending
immune response 24–46% (median 33). For the

Page 10

L. Molineaux and others388
control of the early parasitaemia, up to 6 days after
the first local maximum, the relative contributions
of the 3 control mechanisms were quite different;
the innate immune response contributed 49–87%
(median 67) of the control, the acquired variant-
specific immune responses 2–35% (median 16), and
the acquired variant-transcending immune response
6–34% (median 15). Over the whole case-history
the relative contribution of the innate density
regulation is weakly negatively correlated with log
P?
c
(r??0?18; P?0?29); the relative contribu-
tion of variant-transcending immunity is negatively
correlated with P?
m(r??0?62; P?0?0001). The
relative contribution of the innate immune re-
sponse to the control of early parasitaemia is
weakly negatively correlated with log P?
P?0?14).
Fig. 8 shows the relative contributions of the 3
mechanisms to control the successive peaks of
parasitaemia of the best simulation of cases G408.
According to the model, the innate immune response
dominates during early control, and becomes pro-
gressively less important thereafter, while the
acquired variant-transcending immune response,
relatively unimportant for early control, becomes
dominant in the later stages of the infection.
c(r??0?26;

The work presented here is part of a larger project
aiming at the development of malaria (P. falciparum)
models more reliable than existing models for the
planning and evaluation of intervention trials and
control programmes, as well as for the discussion of
strategic options concerning the development of new
tools for malaria control. The course of asexual P.
falciparum parasitaemia in the individual human
host is a major node in the web of causation leading
to morbidity and mortality. Modelling that course
following a single inoculation in a non-immune indi-
vidual is an appropriate first step, for which the daily
follow-up of cases undergoing malaria therapy –
when the latter was a recommended treatment for
neurosyphilis – provide uniquely valuable data.
While the malaria therapy data are indeed
uniquely valuable, some constraints, related either to
their very nature or to the use made of them, should
be kept in mind. (1) The patients are (neuro-
syphilitic) adults, while in endemic areas the most
important group for the natural history of malaria
are (non-syphilitic) children. (2) As our purpose was
to simulate the natural history of P. falciparum intra-
host parasitaemia, we limited ourselves to cases that
could be classified as spontaneous cures, thus
excluding the small minority of more severe cases
that were, on clinical and?or parasitological grounds,
given early curative treatment. (3) Two modes of
infection (blood, sporozoites), and different parasite
strains were used; the strains used (2 from the US,
1 from Central America) represent a very small non-
random subset of existing P. falciparum populations;
among the 35 cases classified as spontaneous cures,
the subsets according to strain and mode of infection
are too small to allow an adequate stratified analysis.
However, study of the total number of primary P.
falciparuminfectionsconvinced
differences among the 3 strains concerned, or
between the 2 modes of infection were negligible in
comparison with differences among cases. (4) We
simulated the infections as if they were monoclonal,
with a single pool of 50 variants; the actual mono- or
polyclonality of the infections is unknown, although
the repeated artificial passages, in particular through
blood, may have driven the strains used towards
monoclonality. To our knowledge the strains used
have not been preserved, nor has any attempt been
made to assess possible clonal diversity from
preserved blood slides.
A review of published intra-host models of
malarial asexual parasitaemia left us rather dis-
satisfied with their lack of realism (Molineaux &
Dietz, 1999). We have the same reservation about
the new model of Hoshen et al. (2000). It is probably
fair to claim that, in comparison with those models
(leaving out those specifically designed to simulate
the 2-day periodicity of asexual parasitaemia) the
work presented here (i) sets more precise data-based
simulation targets, (ii) compares simulations and
observations more rigorously and (iii) produces more
realistic patterns of asexual parasitaemia. It is
important to consider whether the latter was
achieved with biologically plausible assumptions and
internal operations of the model.
With respect to the model’s assumptions, ex-
tensive simulation (of which only the more con-
clusive part is shown), leads us to conjecture, albeit
only through trial and error and incomplete sen-
sitivity analysis, that the 5 following factors or
equivalents, are necessary to achieve the level of
realism actually obtained. (i) Intra-clonal antigenic
variation, (ii) large variation of the variants’ baseline
growth rate, depending on both variant and case, (iii)
innate autoregulation of the asexual parasite density,
variable among cases, (iv) acquired variant-specific
immunity and (v) acquired variant-transcending
immunity, variable among cases.
We interpret the case-specific variation of factors
(ii), (iii) and (v) as host-specific (genetically de-
termined) variation in (a) growth rate of a given
variant (perhaps related to variation in quality,
quantity or distribution of cytoadherence receptors);
(b) strength of the innate autoregulation of density,
(related to variation in the cytokine response to a
given amount of malaria toxin) and (c) efficacy of
variant-transcending immunity, related to variation
inimmunosensitivity
antigens. We are currently analysing multiple inocu-
lations in the malaria therapy data to test the host-
usthatany
tovariant-transcending

Page 11

P. falciparum intra-host model389
specificity of variation in the innate and acquired
variant-transcending control mechanisms. None of
the published intra-host malaria models has the
above combination of 5 factors; a trypanosomiasis
model (Agur & Mehr, 1997) has the 5 factors,
without mention of case-specific variation of any of
them; on the other hand, in contrast to most intra-
host malaria models, the model presented here does
not model explicitly uninfected RBC, merozoites, or
immune effectors (Molineaux & Dietz, 1999).
Biological justifications for the model’s assump-
tions have been presented above. They are, ad-
mittedly, not compelling, and our simplifying
assumptions should be critically reviewed. With
respect to antigenic variants, the main difficulty
encountered was to balance consumption and
spacing by means of biologically plausible mech-
anisms. Consumption is achieved through variant-
specific immunity with a steep success function
(equation (7), κv?3). The following assumptions
contribute to spacing. (i) A baseline probability of
expression following a geometric probability law
(equation (4)), sufficiently steep (q?0?3) to ensure
that, initially, many variants have a negligibly low
probability of expression. (ii) Strong modulation of
the probability of expression by the host’s variant-
specific immune status (equation (5)). (iii) Wide
variation of the variants’ baseline multiplication
factors, independently of their baseline probability
of expression (the main stochastic element of the
simulations; it can have a large effect). (iv) Com-
petition through innate autoregulation of density
(early in the infection).
What kind of – biologically plausible – mechanism
of variant expression is likely to be selected by
evolution? Intraspecific evolution of P. falciparum is
likely to take place mainly in areas of intense
transmission, because of large mutation denomin-
ators, high rates of genetic recombination in the
vector, and intense selection pressure from human
immune responses. In such areas, P. falciparum is
inoculated into human hosts with very diverse
repertoires of variant-specific immunities. If (i)
PfEMP1 is crucial for sequestration, (ii) successful
sequestration is crucial for the parasite’s growth and
(iii) variant-specific immunity prevents successful
sequestration, then the parasite’s fitness would be
increased by a mechanism allowing the parasite, on
entering a new host’s blood-stream, to explore its
own variant repertoire fast enough to find rapidly
some variant(s) likely to be successful in that host at
that time (i.e. with a sufficient baseline growth rate,
and not too strongly affected by the host’s repertoire
of variant-specific immunities), while saving some
variants for long enough to adapt the period of
potential gametocytogenesis (out of asexual para-
sites) to the time-dependent vectorial capacity (e.g.
to bridge more or less extensive periods in which
vectorial capacity is very low or zero). Such a
mechanism is assumed in the model proposed here,
but nature’s solution may be quite different, and we
arecurrentlycomparingalternativemodelsofvariant
expression, in terms of assumptions, inputs, internal
behaviour, outputs and measures of fit to data.
One of the more problematic assumptions of the
model presented is the strong case dependence of the
basic multiplication factor of a given variant,
identified (numbered) by its baseline probability of
expression. This case-dependence is such that the
same variant may be the fastest growing – out of 50
– in one simulation, and the slowest growing in
another. This allowed realistic simulation of the
wide variation, among cases, of the initial slope of the
parasite density, but may biologically be question-
able. Also crucial in our simulations are the
case-specificity of the strengths of innate density
regulation and of variant-transcending acquired
immunity and the estimation of the corresponding
parameters (P?
c, P?
m) from related case-specific ob-
servations (the parasite density at its first local
maximum and the total duration of parasitaemia, re-
spectively). Our modelling of variant-transcending
acquired immunity is admittedly a crude caricature,
lumping its presumably multiple components into a
single dimension. The model’s assumptions and
internal behaviour (e.g. concerning variant-dynam-
ics and relative contributions of different mech-
anisms of density regulation), not testable with the
data used, are likely to be challenged by the rather
rapidly expanding knowledge of the biology of P.
falciparum.
Implications about pathogenesis and control can
only be speculative and conditional. If the model
captures correctly the main features of P. falciparum
infection, a major determinant of severe disease and
death may be the inability to control the first local
maximum of the asexual parasite density, and that
inability may largely be due to the weakness of the
innate cytokine response (TNF etc) to malaria toxin.
The role of the cytokine response (TNF etc) involves
an apparent paradox: on the one hand, a high TNF
production per parasite should keep the parasite
density down. On the other hand according to
clinical observations, severity of disease, parasite
density, and TNF concentration are positively
correlated with each other (Grau et al. 1989;
Kwiatkowski et al. 1990; Kremsner et al. 1995). This
would imply a negative correlation between pro-
duction of TNF per parasite and production (hence
concentration) of TNF per host. A negative cor-
relation was indeed observed between patients’ TNF
concentration and their white blood cells’ TNF
production in vitro in response to PHA (phyto-
haemaglutinin, used as a proxy for malaria toxin)
(Kremsner et al. 1995). Weakness of the innate
cytokine response to toxin could explain why only a
minority of the exposed develop severe malaria, but
could not, however, explain why some of them do so

Page 12

L. Molineaux and others 390
after successfully controlling several or even many
prior inoculations. That observation could be
explained by a parasite factor (intrinsic virulence)
and?or by a host–parasite mismatch (Molineaux,
1996), and either of these explanations might be
applied to differences among clones or among
variants within a clone. If innate density regulation
is as important for the early control of acute malaria
in non-immunes, as suggested by the simulations
presented, then a purely anti-toxic vaccine may
indeed be a two-edged sword, as conjectured by its
opponents and as suggested – indirectly – by the
results of 2 anti-TNF antibody trials (Kwiatkowski
et al. 1993; Boele van Hensbroek et al. 1996).
Whether a given gain in model realism balances
the corresponding cost in model complexity is not
easy to decide. We conjecture that, for the fore-
seeable future, control measures, including vac-
cination, will not eradicate P. falciparum but modify
the host–parasite balance, at the individual and
population levels. If so, a simulation model, in order
to be useful, should probably be rather realistic
biologically. The work presented here may be a
useful step in that direction, with respect to a crucial
part of the parasite’s life-cycle. We are currently
working on a systematic sensitivity analysis of this
model, including alternative or simpler assumptions,
detection and understanding of the model’s weaker
points, and exploration of the implications for
pathogenesis and control.
Support by the European Commission is acknowledged
(Project No. IC 18-CT97-0242(DG 12-SNRD). We thank
the participants of the Concerted Action ‘Mathematical
models of the immunological and clinical epidemiology of
P. falciparum malaria’ for helpful discussions. We also
thank Adrian Luty for his help with improving our
English.

The model’s equations, variables, and parameters
The model is a discrete-time model, with step-size of
2 days; times are in days, concentrations per µl.
Equations
Pi(t?2)??
A
B
(1?s)Pi(t)?spi(t)?
v
j=?
Pj(t)
C
D
?miSc(t)Si(t)Sm(t),
if Pi(t?2)??10−?,
otherwise
(1)
Pi(t?2)?
1
2
3
4
Pi(t?2)?
0
(2)
Pc(t)? ?
v
i=?
Pi(t), (3)
pi(t)?
1
2
3
4
0
qiSi(t)
?v
j=?qjSj(t)
,
if
otherwise,
Si(t)?0.1
(4)
Sc(t)?
E
F
1?
E
F
1
P?c
Pc(t)
G
H
kc
G
H
−?, (5)
Si(t)?
E
F
1?
E
F
1
P?v
?
t−δv
τ=?
Pi(τ) e−σ(t−τ−δv)
G
H
κv
G
H
−?,(6)
Sm(t)?(1?β)
E
F
1?
E
F
1
P?m
?
t−δm
τ=?
P ?c(τ)e−ρ(t−τ−δm)
G
H
κm
G
H
−??β,
(7)
P ?c(t)?
1
2
3
4
Pc(t),
C
if
otherwise
Pc(t)?C,
(8)
P?c?kc?(first local max. density)
P?m?km?[(last pos. day)?(first pos. day)]
mi?N(µm, σ?m) truncated to ensure mi?1,
(9)
(10)
(11)
Variables
Pi(t)
? density of variant i, as PRBC?µl
blood
? total parasite density, as
PRBC?µl blood
? probability that a switching
PRBC switches to variant i
? probability that a parasite of
variant i escapes control by the
innate IR (IR?immune
response), acquired variant-
specific IR, and acquired variant-
transcending IR, respectively, in
the interval (t, t?2)
Pc(t)
pi(t)
Sc(t), Si(t), Sm(t)
Parameters
[parameter
values used in
the simulations]
[v?50]v
? number of variants
per clone
? fraction of parasites
switching among
variants per two-day
cycle
? parameter of the
geometric distribution
of switching
probabilities
? basic multiplication
factor, per two-day
cycle, of variant i; mi
?N(µm, σ?m?mi?1)
? two host-specific
critical densities
? two constants
allowing calculation of
P?c, P?mfrom host-
specific data
? critical density of a
variant, common to
all variants
? stiffness parameters
for saturation of
innate IR, acquired
variant-specific IR,
and acquired variant-
transcenting IR,
respectively
? maximum daily
antigenic stimulus,
per µl, of the acquired
variant-transcending
IR
s
[s?0?02]
q
[q?0?3]
mi
[µm?16;
σm?10?4]
P?c, P?m
kc, km
[kc?0?2;
km?0?04]
P?v
[P?v?30]
κc, κv, κm
[κc?3,
κv?3,
κm?1]
C[C?1]

Page 13

P. falciparum intra-host model 391
σ, ρ
? decay parameters, per
day, of the acquired
variant-specific and
variant-transcending
IR’s, respectively
? delay parameters, in
days, of the acquired
variant-specific and
variant-transcending
IR’s, respectively
? minimum value of Sm
[σ?0?02;
ρ?0]
δv, δm
[δv?δm?8]
β
[β?0?01]
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