# 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.

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**ABSTRACT:**Achieving a theoretical foundation for malaria elimination will require a detailed understanding of the quantitative relationships between patient treatment-seeking behavior, treatment coverage, and the effects of curative therapies that also block Plasmodium parasite transmission to mosquito vectors. Here, we report a mechanistic, within-host mathematical model that uses pharmacokinetic (PK) and pharmacodynamic (PD) data to simulate the effects of artemisinin-based combination therapies (ACTs) on Plasmodium falciparum transmission. To contextualize this model, we created a set of global maps of the fold reductions that would be necessary to reduce the malaria R C (i.e. its basic reproductive number under control) to below 1 and thus interrupt transmission. This modeling was applied to low-transmission settings, defined as having a R 0<10 based on 2010 data. Our modeling predicts that treating 93-98% of symptomatic infections with an ACT within five days of fever onset would interrupt malaria transmission for ∼91% of the at-risk population of Southeast Asia and ∼74% of the global at-risk population, and lead these populations towards malaria elimination. This level of treatment coverage corresponds to an estimated 81-85% of all infected individuals in these settings. At this coverage level with ACTs, the addition of the gametocytocidal agent primaquine affords no major gains in transmission reduction. Indeed, we estimate that it would require switching ∼180 people from ACTs to ACTs plus primaquine to achieve the same transmission reduction as switching a single individual from untreated to treated with ACTs. Our model thus predicts that the addition of gametocytocidal drugs to treatment regimens provides very small population-wide benefits and that the focus of control efforts in Southeast Asia should be on increasing prompt ACT coverage. Prospects for elimination in much of Sub-Saharan Africa appear far less favorable currently, due to high rates of infection and less frequent and less rapid treatment.PLoS Computational Biology 01/2014; 10(1):e1003434. · 4.83 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper, general SIS and SIRS models with immigration of human population for the spread of malaria are proposed and analyzed. Effects of natural as well as human population density related environmental and ecological factors, which are conductive to the survival and growth of mosquito population, are considered. It is shown in both the cases that as the parameters governing environmental and ecological factors increase, the spread of malaria increases. It is also found that due to immigration, this infectious disease becomes more endemic.Journal of Biological Systems 11/2011; 13(01). · 0.96 Impact Factor

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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

Page 2

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

Page 3

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)

Page 4

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-

Page 5

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

Page 6

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.

Page 7

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)

Page 8

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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