# Mathematical Modeling of Malaria Infection with Innate and Adaptive Immunity in Individuals and Agent-Based Communities

Abstract

Agent-based modeling of Plasmodium falciparum infection offers an attractive alternative to the conventional Ross-Macdonald methodology, as it allows simulation of heterogeneous communities subjected to realistic transmission (inoculation patterns).
We developed a new, agent based model that accounts for the essential in-host processes: parasite replication and its regulation by innate and adaptive immunity. The model also incorporates a simplified version of antigenic variation by Plasmodium falciparum. We calibrated the model using data from malaria-therapy (MT) studies, and developed a novel calibration procedure that accounts for a deterministic and a pseudo-random component in the observed parasite density patterns. Using the parasite density patterns of 122 MT patients, we generated a large number of calibrated parameters. The resulting data set served as a basis for constructing and simulating heterogeneous agent-based (AB) communities of MT-like hosts. We conducted several numerical experiments subjecting AB communities to realistic inoculation patterns reported from previous field studies, and compared the model output to the observed malaria prevalence in the field. There was overall consistency, supporting the potential of this agent-based methodology to represent transmission in realistic communities.
Our approach represents a novel, convenient and versatile method to model Plasmodium falciparum infection.

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Mathematical Modeling of Malaria Infection with Innate

and Adaptive Immunity in Individuals and Agent-Based

Communities

David Gurarie

1.

, Stephan Karl

2,4*.

, Peter A. Zimmerman

3

, Charles H. King

3

, Timothy G. St. Pierre

2

,

Timothy M. E. Davis

4

1Department of Mathematics, Case Western Reserve University, Cleveland, Ohio, United States of America, 2School of Physics, The University of Western Australia,

Crawley, Western Australia, Australia, 3The Center for Global Health and Diseases, Case Western Reserve University, Cleveland, Ohio, United States of America, 4School of

Medicine and Pharmacology, The University of Western Australia, Fremantle Hospital, Fremantle, Western Australia, Australia

Abstract

Background:

Agent-based modeling of Plasmodium falciparum infection offers an attractive alternative to the conventional

Ross-Macdonald methodology, as it allows simulation of heterogeneous communities subjected to realistic transmission

(inoculation patterns).

Methodology/Principal Findings:

We developed a new, agent based model that accounts for the essential in-host

processes: parasite replication and its regulation by innate and adaptive immunity. The model also incorporates a simplified

version of antigenic variation by Plasmodium falciparum. We calibrated the model using data from malaria-therapy (MT)

studies, and developed a novel calibration procedure that accounts for a deterministic and a pseudo-random component in

the observed parasite density patterns. Using the parasite density patterns of 122 MT patients, we generated a large

number of calibrated parameters. The resulting data set served as a basis for constructing and simulating heterogeneous

agent-based (AB) communities of MT-like hosts. We conducted several numerical experiments subjecting AB communities

to realistic inoculation patterns reported from previous field studies, and compared the model output to the observed

malaria prevalence in the field. There was overall consistency, supporting the potential of this agent-based methodology to

represent transmission in realistic communities.

Conclusions/Significance:

Our approach represents a novel, convenient and versatile method to model Plasmodium

falciparum infection.

Citation: Gurarie D, Karl S, Zimmerman PA, King CH, St. Pierre TG, et al. (2012) Mathematical Modeling of Malaria Infection with Innate and Adaptive Immunity in

Individuals and Agent-Based Communities. PLoS ONE 7(3): e34040. doi:10.1371/journal.pone.0034040

Editor: Rick Edward Paul, Institut Pasteur, France

Received November 10, 2011; Accepted February 21, 2012; Published March 28, 2012

Copyright: ß2012 Gurarie et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits

unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Funding: DG was supported by National Institutes of Health Research Grants R01TW008067 and R01TW007872 funded by the Ecology of Infectious Diseases

Program of the Fogarty International Center. TMED is supported by an NHMRC practitioner fellowship. The funders had no role in study design, data collection

and analysis, decision to publish, or preparation of the manuscript.

Competing Interests: The authors have declared that no competing interests exist.

* E-mail: stephan.karl@physics.uwa.edu.au

.These authors contributed equally to this work.

Introduction

Many attempts have been made to describe the complex in-host

and population dynamics of malaria infection using mathematical

models. Classical population-based models developed by Ross and

MacDonald still provide the basis for many new approaches

[1,2,3]. These models are based on SIR (Susceptible/Infected/

Removed) methodology and sometimes aim at large-scale

epidemiological predictions such as in a recent paper describing

malaria dynamics in south-east Asia [4]. While any model may

omit or simplify some aspects of reality, SIR are less adequate for

infections like the one with Plasmodium falciparum [5]. Indeed, they

allow only a minimalistic account of the complex immune

processes within the human host. On a community level, SIR

type models typically assume homogenous populations and host-

vector interactions. Any kind of heterogeneity, such as multiple

parasite strains and vector species, variable human characteristics

(e.g. age, immunity and comorbidity), or type of intervention (e.g.

drug treatment and bed net usage), will automatically increase the

number of population strata and thus the number of variables and

parameters defining the SIR system [6]. Mathematically, this leads

to a substantial increase in the order and complexity of the system.

However, only the simplest, low dimensional SIR models are

amenable to algebraic manipulation and analysis.

Agent-based (AB) approaches can overcome some of these

drawbacks. Using AB methodology, individual agents are

represented by dynamic processes, describing in-host interactions

of the malaria parasite with target cells and host immunity. A

community of such agents can then be constructed and subjected

to realistic transmission in the form of inoculation patterns. Unlike

SIR systems, agent based (AB) communities are computationally

constrained by their size since computing time and resources

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increase with population size. This limitation is, however, more

than compensated by greater accuracy and versatility. For

instance, an AB community can be made completely heteroge-

neous, allowing multiple parasite strains and/or species, human

hosts with different age-dependent immunity and different

interventions, at little or no additional computational cost. Several

in-host models for malaria have been developed in previous studies

[7,8,9,10,11,12,13,14,15]. They vary in scope and detail depend-

ing on their objective. Many focus on theoretical aspects of

parasite interaction with the human immune system and the effect

of antimalarial interventions. In some studies models were

calibrated and validated using individual case clinical and

parasitological data, such as for the first wave of asexual

parasitemia [16], the full course of the infection based on informed

trial and error [17], and the transition of asexual parasites to

gametocytes [18,19]. Several other studies have also applied the

agent-based approach to a community level [20,21,22].

In the present study we developed a novel in-host agent model

that accounts for the most salient features and biology of parasite-

host interactions. This model and our calibration differ substan-

tially from earlier work. In particular, we paid special attention to

the parasite replication cycle (invasion and depletion of red blood

cells (RBC), immune stimulation and parasite clearance. Mathe-

matically, the model is implemented and run in discrete time steps

based on the 48 h parasite replication cycle. Such discrete models

behave in many aspects similar to models based on continuous

differential equations, but they can be often implemented and

simulated more efficiently, particularly for random processes. Our

model combines deterministic and stochastic components of in-

host dynamics, the latter resulting from random antigenic

variation (AV) of Plasmodium falciparum. The model was coded in

Wolfram Mathematica 7.

Our calibration procedure also differs from earlier related

models [13,17,18,19]. As in previous studies, we utilized individual

host histories from malaria therapy (MT) records. However, we

interpreted these MT histories in a different way. Rather than as

an accurate benchmark for parameter fitting, we view each history

as one of many possible random realizations of a stochastic AV

process. Therefore, our calibration procedure combines determin-

istic and stochastic steps. Through fitting of the model to a large

number of MT cases (n = 122), we created a pool of over 2000

parameter choices that serves as a basis for creating and simulating

AB communities. We conducted several numeric experiments by

subjecting these AB communities to realistic inoculation patterns

as reported from malaria endemic regions. In particular, we

studied the model predictions of malaria prevalence and compared

them to the reported field observations.

Methods

Biological assumptions

Several important biological factors are included in our model

of asexual parasitemia: (i) homeostatic production/loss of unin-

fected RBC; (ii) parasite replication (invasion of uninfected RBC

and release of merozoites); (iii) stimulation of innate and adaptive

immunity effectors by parasite density; (iv) parasite clearance by

immune effectors.

A schematic view of the within-host processes is shown in

Figure 1. We used the following notations: x– uninfected RBC

population (per mL of whole blood), y– infected RBC population

(per mL of whole blood), a– innate immune effector; b– adaptive

immune effector.

The normal RBC level x

0

(5610

6

mL

21

) is maintained through

stationary (homeostatic) production/loss terms. RBC are invaded

by the newly released merozoite population (M). The probability

of merozoite invasion depends on the available RBC pool per

merozoite, x/M. The parasite burden stimulates an immune

response consisting of innate and adaptive immune effectors aand

b. We view these effectors as simplified proxies of the effector

concentration (e.g., antibody titer) combined with the efficiency of

the effector to clear infection. The immune effectors reduce

parasite density by inhibiting parasite replication. Effectors are

stimulated by parasite density, or through parasite-immune

interactions above certain parasite density threshold levels.

The model was implemented using a discrete time-step based on

parasite replication cycle rather than a process based on

continuous differential equations. We utilize a novel approach to

parasite antigenic variation (AV) and a novel calibration

procedure.

P. falciparum has evolved several immune evasion strategies, most

notably AV, whereby it can vary (on each replication cycle) an

important class of surface proteins expressed on infected RBC.

These proteins play a double role. On the one hand they serve as

immunogenic targets with stimulation of antibody production and

consequent parasite clearance. On the other hand they mediate

adherence of infected RBC to endothelial cells in the microvas-

culature (sequestration) and thus promote parasite survival. AV of

P. falciparum has been the subject of considerable research

[23,24,25,26,27,28,29]. The process is controlled by the family

of var genes. Each parasite genome contains 50–60 of these var

genes [28]. During a replication cycle, each parasite expresses only

one var gene but can switch expression in the next generation [30].

So a typical parasite population may include several antigenic

variants present simultaneously. Therefore, every new infected

RBC generation exhibits an altered antigenic profile compared to

its predecessors, which has important implications for adaptive

immunity. If some of the new variants are sufficiently distinct from

their antecedents, the efficiency of previously developed adaptive

immunity would be weakened [31].

There are different ways to account for AV in mathematical

models. A direct approach to multi-variant parasite dynamics

assigns different population variables to each variant. In addition,

suitable ‘variability’ (exchange) patterns need to be set among

multiple variants and their possible immune interaction with

‘specific effectors’ assigned to each type. These two processes are

typically described by ‘mutability’ (switching) and ‘cross-reaction’

matrices. Such multi-dimensional approaches have been devel-

oped and utilized in previous studies (e.g. [19,32]).

In the present study, we propose a simpler way to account for

AV in a single infected RBC population. Assuming all variants are

nearly identical in terms of growth and invasion, the only essential

difference between them are their antigenic properties, i.e.

susceptibility to previously developed adaptive immune effectors.

As new infected RBC populations may differ antigenically from

the earlier ones, the effective adaptive immune response may be

reduced at each replication cycle. However, the magnitude of such

a change should diminish with time as the parasite gradually

depletes its repertoire of expressed var genes and the host develops

antibodies against all antigenically distinct variants. We account

for AV by random falls in adaptive effector bat each replication

cycle. Randomness is essential for describing a typical ‘switching’/

mutation’ process of any kind since, although they may not be

truly random by nature, we typically lack detailed knowledge of

their mechanisms and functions, and thus have to assume that they

are random variables or parameters.

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Model setup

A definition of all variables and parameters used in this study is

given in Table S1 in the supporting information. The basic system

at time t (measured in reproductive cycles) is described by variables

x,y,M,a,bðÞ:x(t) represents uninfected RBC (target cells) mea-

sured per mL of blood; y(t) represents infected RBC measured per

mL of blood, M~r:y(t)is the number of merozoites released; a(t)

and b(t) are dimensionless variables representing innate and

adaptive immune effectors. On each time step the transition from

the current state x,y,a,bðÞto the next state x0,y’,a’,b’ðÞis given by

the following equations.

x’~1{dðÞx0zdx{rM=xðÞx;

y’~2{azbðÞ

rM=xðÞx;

a’~fa:way=AðÞzsaa;

b’~fb:wbyazbðÞ=BðÞzsbb;

ð1Þ

The first two terms of the x-equation, 1{dðÞx0zdx, account for

homeostatic production/loss of RBC that maintains its normal

level x

0

=5610

6

/mL, with survival rate d= 0.98/cycle (based on a

100 day life-span). The last term rM=xðÞxrepresents RBC loss

due to merozoite invasion. Specifically, a density-dependent

fraction rM=xðÞof xwould be invaded and turned into the next

generation of infected RBC (y9). Of those, only a fraction 2

2(a+b)

would survive through to the end of the cycle, depending on the

combined immune level (a+b). Factor rin merozoite equation

M~r:yrepresents the effective replication rate of the parasite and is

equal to the average merozoite progeny per infected RBC times

the maximal probability of RBC invasion by a merozoite. The

latter refers to optimal invasion conditions with large pool of

available target cells (x), or a small relative merozoite population

z~M=x. When fraction zgrows large, merozoites start competing

for available RBC, and the effective probability of invasion (or

invaded fraction of x) decreases according to

rzðÞ~1{e{zð2Þ

Derivation of function rzðÞis based on two assumptions regarding

the invasion process: (i) Poisson distribution of variable zabout

each (typical, average) RBC; (ii) exclusive competition where only

one merozoite (among competing pool z) can establish successful

invasion. This form of ‘invasion and resource depletion’ differs

from the standard continuous formulation (e.g. [5], [8]), given by

2

nd

order removal kinetics (dx=dt~0source0{k:x:M).

Immune regulation

Immune effectors (a, b) are dimensionless variables measured in

terms of clearance of y, so that a ‘unit effector’ would halve the

parasite population over the 2-day cycle (y0?y=2). Both are

stimulated by Hill functions (wa,wb)with suitable threshold

transition levels and maximal stimulation efficiencies 0vfavfb.

The innate stimulation fa:way=AðÞis triggered by the parasite level

relative to threshold A, it has relatively short life-span (approxi-

mately 4 days) or survival rate sa~:65, and lower efficiency f

a

.

Adaptive effector (b) has longer memory (100 days or survival

sb~:98) and higher clearing efficiency f

b

, but takes longer time to

develop in naı

¨ve hosts. Furthermore, production of (b) in our

model is triggered by the product of infected RBC density (y) and

the combined effector pool a+brelative to threshold B. Producty:a

serves as a primary trigger for the development of adaptive

responses, while y:baccounts for enhanced reactivation of b

through adaptive immune memory. The efficiency factors (f

a

., f

b

.)

represent the maximum stimulation rates of aand brespectively

under ‘high’ parasitemia. The corresponding maximal clearance

levels become

aM~fa

1{sa

;bM~fb

1{sb

The latter imposes some constraints on model parameters to allow

parasite clearance, namely

aMzbMwwlog rðÞ

The numeric simulations of dynamic system (1) can result in

arbitrarily low (unphysiological) parasite levels (y). Therefore, we

impose a lower cut-off on parasite density at y

c

=10

26

mL

21

in our

simulations, below which the parasite is considered cleared (y=0).

Antigenic variation (AV)

Equation system (1) represents the deterministic part of in-host

processes. To account for AV, we let the adaptive immune effector

(b) fall on each cycle by a random fraction q~qt(0,q,1). Thus,

the last equation of system (1) will take the form

b’~q:fb:wbyazbðÞ=BðÞzsb:bð3Þ

Figure 1. Schematic representation of the model. The population of uninfected red blood cells (x) provides the source for the infected

population (y). Level I immune effector (a) is stimulated by y. Level II immune effector (b) is stimulated by yinteracting with a+b.Mrepresents the the

number of merozoites, Srepresents an external source of inoculation.

doi:10.1371/journal.pone.0034040.g001

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The average severity of AV-induced reduction of bshould

diminish with each replication cycle, as the limited reservoir of

antigenically distinct variants is depleted and the repertoire of host

antibodies increases. In the present model we described this

random process (q= Random 0,q0t

½) by an exponential function

q0twith base

q0~2{1=mv1ð4Þ

dependent on the number of distinct variants and their cross-

reactive properties, and represented by half-life parameter m.It

should be noted that mdoes not represent the absolute number of

variants. It is rather related to the number of antigenically distinct

variant clusters. For example two var genes may encode for

surface proteins which exhibit so much similarity, that antibodies

developed specifically against one of them are effective to a certain

degree against the other as well.

Figure 2 illustrates an example of a dynamic pattern resulting

from the deterministic model (equation system (1)) and the

corresponding ensemble made up of 50 random realizations of

the stochastic AV process. We observe (Figure 2) that the

deterministic component is dominant at early stages of infection

(primary wave of parasitemia), while random AV variations

become more pronounced later in the course of infection.

For numeric simulations and calibration of equation system (1),

we fixed some model parameters, based on the available biological

data and estimates and allowed others to vary. Tables 1 and 2 give

a complete list of calibrated model parameters, and results of a

sensitivity analysis. They include the effective replication factor r,

innate and adaptive efficiencies (f

a

., f

b

.), immune stimulation

thresholds Aand Bfor aand brespectively and parameter m

(related to the number of antigenically distinct clusters of variants).

Equation system (1) is appropriate for asexual stage dynamics in

the absence of external infectious sources. External inoculations

can be added to (1) by augmenting the merozoite variable

M~r:yzSð5Þ

with the source term S– representing the number of merozoites

released from the liver per life cycle.

Calibration procedure

The model was calibrated with data from MT studies for

neurosyphilis, which provide a rare opportunity to examine host-

parasite interaction over extended periods of time. The complete

set of MT data used in the present study is given in Table S2 in the

supporting information. These data have been analyzed in detail

previously [33,34,35,36] and used for calibration purposes or as

‘direct input’ for agent-based communities [13,14,22,37].

The calibration procedure we propose involves two steps: a

deterministic fit to the first wave of parasitemia, and a second

stochastic step that attempts to accommodate irregular parasitemia

patterns following the initial wave. Based on these steps we select

‘best choices’ of in-host parameters.

The total number of datasets available was for 334 MT patients.

From these, we selected 122 for which patients were either

untreated or treated very late in the infection. The MT patients

exhibit highly irregular dynamic patterns of parasitemia for several

reasons. Firstly, due to cytoadherence of late stage parasites there

is an apparent oscillation with a 2-day recurring pattern. Since

these oscillations have little relevance for long-term infection

outcome, they were smoothed out by taking the maximum/

minimum parasitemia envelopes on consecutive odd and even

days, and computing their geometric mean curve, as illustrated by

a representative patient history in Figure 3. On longer time scales

there are recurrent irregular waves of parasitemia, often with

diminishing amplitude. These recurrent waves are not a result of

reinoculation (typical of a natural environment) as all MT hosts

received a single initial inoculum in strictly controlled clinical

experiments. Therefore these fluctuations in parasite density can

be attributed to AV of P. falciparum, whereby new variants help to

sustain infection over longer periods.

As MT hosts had no prior exposure to malaria, equation system

(1) was run starting with the naı

¨ve initial state (x

0

,y

0

,0,0), i.e.

normal RBC-level x

0

, initial inoculum y

0

= 0.001–0.01 mL

21

, and

no pre-existing immunity, a0~b0~0. For calibration purposes the

MT data set was divided into two groups: the first group consisted

of cases that exhibited only a single wave of parasitemia, after

which infection fell below the detection level (10 mL

21

) and was

presumed cleared. Such cases typically had a short duration of

patent parasitemia (10–20 days). These cases were calibrated using

the first (deterministic) step, as AV played no less important role in

the initial wave of infection, confirmed by our numeric

experiments.

The second group comprising long term irregular patterns with

multiple waves of parasitemia was subjected to two calibration

steps. In the first step, we collected 50 of the best-fit deterministic

parameters (r, A, B, f

a

,f

b

) (Table 1) from each suitable MT dataset

and a random ensemble of 25000 parameter choices. The ‘best-fit’

was confined to the first wave of parasitemia, and we used the

standard square-mean error between the observed and simulated

histories

E~Xlog yMT tðÞ{log ytðÞ½

2ð6Þ

As the MT data does not inform on inoculation dates, we

estimated the time lapse Tbetween inoculum and the first day

Figure 2. Deterministic pattern versus AV pattern. Panel A:

Typical deterministic parasite density pattern (solid blue line) as

predicted by the model. Also shown are innate immune effector a

(blue filled curve) and adaptive immune effector b(purple filled curve).

Panel B: Corresponding stochastically predicted mean parasite density

(solid green line) and minimum/maximum envelope (purple fill) for the

same deterministic solution (solid blue line) as shown in Panel A.

doi:10.1371/journal.pone.0034040.g002

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with observed parasite density and adjusted it in the fitting process

along with other parameters of Table 1. In the second calibration

step (for multiple-wave patterns) the best-fit parameter choices of

step 1 were further adjusted to account for an extended history

and random AV effects. The adjustment involved only immune

efficiencies (f

a

,f

b

), as those are primarily responsible for the long

term pattern of parasitemia and have only a minor effect on the

primary wave of parasitemia (Table 2).

We consider MT histories to be random realizations of a

stochastic AV process rather than unique ‘individual patterns’.

Therefore if the same host would be subjected to another

inoculation he/she may exhibit a different infection pattern. In

general, it is a challenging task to calibrate parameters of a

stochastic process from its single realization and obtain statistically

reliable results. The standard techniques would typically require a

sufficiently ‘long history’ and ‘simple stochasticity’ (e.g. linear

stationary process with ‘additive noise’). In the present model, we

are dealing with a highly nonlinear system (1) and non-stationary,

multiplicative noise, since the AV input is a decaying random

sequence qt&q0t. Our approach is to create, for each adjusted

choice r,A,B,fa,fb

ðÞwith fa~ufa;fb~vfb;1,u,2 and

1,v,4, a random AV-ensemble of 50 realizations, then try to

fit a given MT history within the ‘min/max ensemble envelop’ by

minimizing its distance from the ensemble mean curve (all y-values

are log-transformed). The best-choice values of factors (u*,v*) and

the adjusted 5-tuples r,A,B,fa,fb

ðÞare then selected to

represented a given host.

The computational codes were implemented and run in

Wolfram Mathematica 7. The code can be downloaded from:

http://www.cwru.edu/artsci/math/gurarie/Malaria/In host cali-

bration.nb (code) and http://www.cwru.edu/artsci/math/

gurarie/Malaria/Sorted%20mean%20hist%20210 (filtered MT

data in Mathematica format).

The Mathematica codes and procedures developed for the

model are very efficient and require only modest computational

resources. The efficiency is important for our calibration

procedures (computing large ensembles of hosts and histories)

and applications to agent based communities (AB communities).

Agent Based Communities

AB communities were assembled from the best parameter sets

resulting from the calibration process (50 best parameter sets for

each calibrated MT data set). The AB communities used in the

present study typically consisted of 1000 agents. The communities

(agents) were subjected to external inoculation (via the S-term in

Equation (6)), based on entomological inoculation rates (EIR) as

observed in field studies. External inoculation will produce

different recurrent patterns that combine in-host regulation with

external forces. In the present study, the inoculation patterns were

generated as series of binaries (0 and 1) with 1 being an inoculum.

The probability that an inoculation occurs on a specific time step is

dependent on the EIR, which represents mean number of

inoculations per agent in a specific time interval. Each agent was

subjected to an individual inoculation pattern based on the same

average EIR.

Model predictions were compared with the data from three

selected reports from Africa that correlated observed temporal

malaria prevalence patterns with observed EIRs. The reported

EIR patterns were used as input for the AB community model and

the model output compared to the data reported from these

studies. The first study was by Beier et al. from 1999 [38]. It

provided an overall correlation of EIR with malaria prevalence in

Table 1. Possible ranges of uncertain parameters of the model, and medians and interquartile ranges which resulted from the

fitting process.

Parameter Name Range Median (25%/75% Quartiles)

Invasion probability p0.5–1 0.66 (0.57/0.82)

Replication r15–50 23.91 (18.66/30.94)

Innate efficiency f

a

5–20 7.59 (6.12/9.22)

Adaptive efficiency f

b

20–120 38.48 (28.73/49.23)

Innate threshold A30–80 66.57 (53.32/73.96)

Adaptive threshold B10–50 27.46 (18.37/37.39)

Antigenically distinct variant clusters m3–20 12 (6/14)

doi:10.1371/journal.pone.0034040.t001

Table 2. Sensitivity analysis of in-host parameters and their contribution to dynamic patterns: strong ++; marginal +;no

contribution 2.

Height 1

st

peak Day 1

st

peak peak height 2

nd

peak day 2

nd

peak clearing

p++ + 222

r++ + 222

A22 22+

B22 +++2

f

a

+ + ++ ++ ++

f

b

22 ++ ++ ++

m22 ++ ++ ++

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Africa in the form of a review combining data from different study

sites. To compare model predictions with these data, the AB

community was subjected to stationary biting sequences (deter-

mined by EIR) which corresponded to those reported in the study

by Beier. The two other studies were by Vercruysse et al. 1983

[39] and Gazin et al. 1988 [40]. Both correlated EIR and malaria

prevalence in seasonal transmission environments. The former

study of these two was conducted in an urban area of the Senegal,

the city of Pikine in 1979–1981. EIRs were reported in monthly

intervals from December 1979 to December 1980. Parasite

prevalence rates were reported at 7 time points, the first being

in November 1979 the last in January 1980, so that 2 of the 7 data

points fall outside the range in which EIR was measured. Gazin et

al. conducted their study in the village of Kongodjan in Burkina

Faso. Transmission there was also seasonal but the prevalence was

much higher than in the study by Vercruysse et al. EIRs were

reported in monthly intervals from January 1983 to January 1985.

Parasite prevalence was reported in 10 intervals from December

1982 to March 1985. For both comparisons, it was assumed that

the EIR observed at one time point changed linearly to that at the

next time point. In both studies, the reported EIR had the unit

infective bites/day but, in the model, this was converted to

infectious bites per asexual life cycle (48 h) by multiplication by 2.

The reported EIR patterns were then used as an input for the AB

community model. The malaria prevalence predicted by the

model was plotted in comparison to the reported prevalence.

Results

Deterministic and stochastic patterns produced by the

model

Depending on the parameters, the model can exhibit diverse

dynamic patterns of infection. The main patterns are shown in

Figure 4 in which the simulations depicted use one initial inoculum

into a naı

¨ve host. The deterministic oscillations result primarily

from the gradual decline of innate and adaptive immune effectors

aand bwhile the parasite density remains low but above the cut-

off threshold. As aand bweaken further, the restraints on parasite

growth diminish. Typically, the second and subsequent waves will

be much lower in parasite density and further expansion of b

during recrudescent phases may eventually clear the infection.

While the deterministic model can produce multiple oscillations

for certain parameter values, such deterministic waves look very

different from the observed MT cases. This observation confirmed

our hypothesis that simple deterministic models cannot account

for the complexities of MT cases, and that the proper calibration

procedure would require an additional stochastic component. We

used AV as stochastic component as proposed in the Methods

section.

Figure 2 demonstrates the principal effect of applying the

stochastic AV process to the deterministic model. Figure 2A shows

Figure 3. Typical MT-host with ‘odd-even’ envelope (purple shaded area) and its mean-curve (thick black line).

doi:10.1371/journal.pone.0034040.g003

Figure 4. Typical deterministic histories starting from an

immunologically naı

¨ve state with an initial inoculum. Blue solid

lines are parasitemia, the blue filled curve is immune effector a, the

purple filled curve immune effector b, and the blue filled curve at the

top are depleted resource cells. Deterministic histories can have single

(Panel A), double (Panel B) and multiple (Panel C) wave patterns.

However multiple waves patterns very rarely terminate and look very

different from the MT data.

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an example of a deterministic curve resulting from a specific set of

parameters. An example for an adjusted AV ensemble of 50

random realizations with its mean parasite density pattern and

minimum/maximum envelope is shown in Figure 2B. The AV

modified patterns resemble the MT data more closely. After the

initial growth phase, which is nearly identical for deterministic and

AV modified model predictions, AV retards the accumulation of

protective immunity. Hence there are delayed and higher

parasitemia peaks for the AV modified model predictions. For

instance, in Figure 2 where y

max

=2610

3

at day 13 for the

deterministic pattern (Panel A), y

max

= 2*10

3

to 3*10

3

at days or 13

to 15 for AV modified patterns (Panel B).

Calibration of the model using MT Data

Twenty-five datasets used for calibration in this study exhibited

single wave patterns of parasitemia. Figure 5 shows two typical

fitted simulated patterns along with the original MT data from

patients who had a single wave of parasitemia. A large number of

these fits can be found in the supporting information (Figures S1,

S2, S3, S4, S5, S6, S7, S8, S9, S10) In terms of duration,

maximum parasite density and day of maximum parasite density,

the deterministic calibration resulted in good curve fits for most

single wave MT cases.

For irregular patterns of longer duration, the qualities of the fits

varied. Although AV, as modeled here, can account for a large

number of irregular cases, some MT patterns are entirely different

and fall outside the current setup. A typical stochastic pattern

produced by our model exhibits an initial wave of parasitemia to a

peak, with a subsequent decline until clearance.

Figure 6 shows stochastic data fits resulting from the two-step

calibration method for hosts with irregular parasitemia. The fits in

Panels A and B are acceptable on two counts. First, the MT-data

are contained entirely within the envelope of the AV-ensemble.

The ensemble envelope represents the range of possible simulated

stochastic patterns and can therefore be viewed as a measure of the

ensemble variation about its mean. Second, the MT data pattern

falls within a reasonable range of the mean pattern of the

simulated ensemble. This is the case for most of the 97 irregular

patterned datasets calibrated with the two step method. Panels C

and D show two data fits of irregular patterns that exhibit greater

departure from the model behavior. The dataset shown in Panel C

(S-1249) starts with a plateau of parasitemia, while in most model

simulations we see a typical ‘initial wave’ of shorter duration. The

dataset shown in panel D (S-713) exhibits an irregular (fluctuating)

initial growth stage. While these cases depart from the ‘model

pattern’ at the early stage, their AV-envelops representing

stochastic patterns based on the 50 best parameter sets still cover

the remaining (long-term) trend reasonably well. The majority of

datasets (71/97) exhibited basic characteristics in concordance

with model output. Graphic representations of many of these fits

can be found in the supporting information (Figures S11, S12,

S13, S14, S15, S16).

For each dataset, parameter ensembles were collected that

resulted in the best fits of parasitemia patterns of MT patients. We

compared characteristic statistics (specifically the days and parasite

densities of the first and the last maximum) of the MT data set with

the same predicted characteristics from of a community generated

using the best parameters collected for each MT dataset in the

calibration process. The results of this comparison are presented in

Figure 7. A comparison to characteristic features of infection

patterns used in a previous study is included in the supporting

information (Table S3) [13].

Agent based communities

As a first test of validity of our calibration procedure, we created

two AB communities. The first AB community was built from our

collection of fitted parameters the second from random parameter

choices within the calibration ranges (Table 1). The predicted

parasite prevalence using calibrated parameter sets differed

considerably from the predictions using random parameter sets

over a wider range of EIR. Figure S17 in the supporting

information illustrates the observed differences. Thus we conclude

that MT calibration does provide a meaningful selection of model

parameters, in a statistical sense.

The comparison with the data from Beier et al 1999 [38] is

shown in Figure 8. There is a good agreement between reported

and predicted values, however our range of EIR was limited to

182.5 based on 2-day cycle) compared with .700 per year as

recorded in the publication. The comparison with the studies of

Vercruysse et al. 1983 [39] and Gazin et al. 1988 [40] are shown

in Figure 9. Figure 9 A and B show the observed and simulated

EIR pattern, and the observed and predicted malaria prevalence

for the comparison with the data from Vercruysse et al. 1983 [39].

Figure 9 C and D show the same data for the comparison with the

study of Gazin et al. 1988 [40]. There is good agreement between

the observed and predicted parasite prevalence dynamics over a

wide range of time points.

Discussion

AB modeling of malaria requires the construction of a

reasonable in-host model as a foundation. Depending on its

intended use, the model is a compromise between reasonable

simplicity and the complex mechanisms it seeks to describe on the

in-host epidemiological scales. In this report, we present an agent

based model that can be used to reproduce MT data with

reasonable accuracy and is computationally efficient so that

Figure 5. Two typical single wave datasets. The solid gray lines are parasitemia. The curve with light gray fill is the model prediction.

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Figure 6. Long term, multiple wave datasets calibrated with the present model. Panels A and B depict cases where the calibration resulted

in a reasonable fit. The datasets are suitable because they exhibit an initial wave of parasitemia which contains the global maximum of the entire

history. Panels C and D depict cases which are less suitable because they are missing a pronounced first wave of parasitemia. The blue solid lines are

the original MT data, the dashed blue lines are the best fits to the first wave of parasitemia (1st calibration step), the purple shaded areas are the AV

envelopes (2nd calibration step) with its mean curve (solid green line).

doi:10.1371/journal.pone.0034040.g006

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communities of 1000 or more agents can be run on a desktop

computer without a long processing time.

Several types of within-host malaria models have previously

been developed [3,12]. Some use continuous differential equa-

tions, with very limited account of immunity and make no attempt

to accommodate AV or fit unknown parameters [7,8,9,10,11].

Other studies have developed models which use a discrete-time

step approach for single or multi-variant parasite densities

[13,14,16,17,18]. Multi variant modeling approaches typically

require more computational power and involve larger sets of

uncertain parameters for model calibration. It might be difficult to

relate their setup and output to epidemiological data and

considerations. Besides, community level effects may become

insensitive to the detailed structure of multi-variant systems.

Some previous models were fitted to MT data by either formally

fitting specific characteristic features (e.g. the first-wave of

parasitemia, duration of infection, maximum parasitemia, various

slopes, number of local maxima, etc.) or using informed trial and

error methodology [13,16,17].

To our knowledge, no previous study has developed a formalism

to fit the entire length of MT histories and utilized the resulting

fitted parameters to generate AB communities for simulations and

data analysis. We proposed a novel calibration method in this

study that differs considerably from earlier approaches. We

assumed that all parasitemia patterns beyond the first wave are

the result of a stochastic AV process with multiple uncertain

contributing factors. Therefore, each of these patterns is a single

realization of the process and subjecting the same host to another

inoculation with the same parasite and the same number of

sporozoites may result in a different pattern of parasite density.

However we also assumed that these stochastic patterns fall within

certain boundaries. Ideally, a calibration procedure for stochastic

patterns should use stable output characteristics and statistics, and

reconstruct the unknown model parameters based on them. In

case of the MT data, these output characteristics could be the

duration of the infection, the day of maximum parasite density, the

maximum parasite density and so on. These outputs are, however,

themselves random quantities. Thus, to give a statistically reliable

prediction, such a calibration procedure would require more than

a single individual realization of the stochastic process. The MT

data do not provide this information since each patient was

Figure 7. Comparison of characteristic statistics between MT data and model prediction. Panel A: Day of the first maximum; Panel B: Day

of the last maximum; Panel C: Parasite density at first maximum; Panel D: Parasite density at last maximum; None of these characteristic features

where significantly different between model and MT data.

doi:10.1371/journal.pone.0034040.g007

Figure 8. AB communities subjected to stationary EIR and

comparison to field data. The data points (black) are taken from a

review conducted by Beier et al. (1999) [33]. The black line is the curve

fit also given in that reference. The colored lines are model predictions

based on different numbers of antigenically distinct variant clusters (m).

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inoculated only once. The standard methods of parameter

estimation for stochastic processes are inappropriate in our

context, as the equations (1)–(4) are nonlinear, and randomness

(AV) enters in a complicated nonlinear fashion. Therefore, we

proposed that MT patterns are random realizations of the

stochastic process and tried to ‘optimally’ fit them within a

suitable ensemble envelope.

The majority of the MT datasets exhibit a dominant first wave

of parasite density followed by recurrent, diminishing waves. Our

calibration procedure resulted in reasonable fits for all MT

datasets exhibiting these basic features. We validated our scheme

by comparing AB communities based on our calibrated parameter

sets with purely random parameter choices, and found substantial

differences in their predicted outputs (Figure S17 in the supporting

information). Furthermore characteristic features in the MT data

were reproduced reasonably well by the model. We therefore

concluded, that the calibration to a certain extent allows a

meaningful selection of in-host parameters. We then subjected the

calibrated AB community to random inoculations at prescribed

rates (EIR) based on several field studies and found reasonably

good agreement.

There are several limitations in our approach and results. On

the in-host side, it may be desirable to account for variant diversity

and more detailed structure of immunity. In the current setup all

variants were considered to have identical growth and clearance

characteristics. Furthermore, the immune regulation takes an

abstract form represented by only two effector variables (aand b).

A further extension of the in-host model would include more

detailed structure of the immune system with different B- and T-

cell populations and relevant processes (activation, proliferation,

effector function, parasite clearing and memory maintenance),

along with antibodies, cytokines and fever.

Another limitation of the present model is the 2-day replication

cycle, which limits the dynamics process and specific features of

different (young/mature) parasite stages. Furthermore we modeled

only a single parasite strain and a useful extension of the present

model would include multiple strains with different fitness and

drug susceptibility.

The calibration procedure was limited mostly by the availability

of data. Ideally, unperturbed (baseline) parasite density patterns

from populations exposed to different endemicity levels and with

different ages should be used to calibrate the model. However such

data are not available. It remains to be determined whether MT-

based calibration results are applicable to field data and how it

should be modified to account for the important (e.g. individual or

age-dependent) differences in malaria infection.

A major limitation of the AB community part of the present

model is that there is no transmission. Therefore, the AB

community is assumed to be a very small part of a larger

transmission environment and changes in malaria prevalence

within the AB community have no effect on this transmission

environment. Future model developments and applications will

include gametocyte production as a function of the asexual

Figure 9. Comparison of model prediction to field observations

from areas of seasonal malaria transmission. Panel A: EIR as

reported by Vercruysse [34] (solid purple line), and reproduced as input

for the model (solid blue line). Panel B: Malaria prevalence as reported

by Vercruysse (solid blue line) and model prediction as monthly average

(solid purple line) and envelope of monthly minima and maxima (olive

fill) using as input the EIR pattern from Panel A. Panel C: EIR as reported

by Gazin [35] (solid purple line), and reproduced as input for the model

(solid blue line). Panel D: Malaria prevalence as reported by Gazin (solid

blue line) and model prediction as monthly average (solid purple line)

and envelope of monthly minima and maxima (olive fill) using as input

the EIR pattern from Panel C.

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parasite density and calibrate gametocyte density using MT data

or field data [41].

When comparing model predictions to observed field data it

should be noted that the calibration proposed in the present study

was conducted using data from adults who had never been exposed

to malaria before. In reality, this is almost never the case. In the

studies by Vercruysse and Gazin, only children were enrolled and

we have to assume that nearly all of thesechildren had been exposed

to malaria before these studies commenced. Furthermore there was

considerable usage of bed nets, chemoprophylaxis and insecticide

spraying over the course of these studies. These are factors that are

not taken into account by the present model. Extended versions of

the model should address these issues. Nevertheless, it is evident that

the model predictions and the observed prevalences are very similar

over a wide range of compared data. We therefore concluded that

this model can make some useful predictions and serve as a basis for

future development.

Supporting Information

Figure S1 Graphic representations of the six best fits to

the first wave of parasitemia for datasets 35, 37, 38, 39,

40, 41, 42 and 43 (A–H). X axes are days, y axes are decadic

logarithms of parasite density. The numbers above the graphs are

the errors calculated using equation (6).

(TIF)

Figure S2 Graphic representations of the six best fits to

the first wave of parasitemia for datasets 44, 45, 46, 48,

50, 51, 52 and 54 (A–H). X axes are days, y axes are decadic

logarithms of parasite density. The numbers above the graphs are

the errors calculated using equation (6).

(TIF)

Figure S3 Graphic representations of the six best fits to

the first wave of parasitemia for datasets 55, 56, 57, 58,

59, 60, 61 and 62 (A–H). X axes are days, y axes are decadic

logarithms of parasite density. The numbers above the graphs are

the errors calculated using equation (6).

(TIF)

Figure S4 Graphic representations of the six best fits to

the first wave of parasitemia for datasets 63, 64, 67, 69,

70, 71, 73 and 74 (A–H). X axes are days, y axes are decadic

logarithms of parasite density. The numbers above the graphs are

the errors calculated using equation (6).

(TIF)

Figure S5 Graphic representations of the six best fits to

the first wave of parasitemia for datasets 76, 77, 78, 79,

80, 81, 82 and 83 (A–H). X axes are days, y axes are decadic

logarithms of parasite density. The numbers above the graphs are

the errors calculated using equation (6).

(TIF)

Figure S6 Graphic representations of the six best fits to

the first wave of parasitemia for datasets 84, 85, 86, 87,

88, 89, 90 and 91 (A–H). X axes are days, y axes are decadic

logarithms of parasite density. The numbers above the graphs are

the errors calculated using equation (6).

(TIF)

Figure S7 Graphic representations of the six best fits to

the first wave of parasitemia for datasets 92, 93, 94, 95,

96, 97, 98 and 99 (A–H). X axes are days, y axes are decadic

logarithms of parasite density. The numbers above the graphs are

the errors calculated using equation (6).

(TIF)

Figure S8 Graphic representations of the six best fits to

the first wave of parasitemia for datasets 100, 101, 102,

103, 105, 106, 107 and 109 (A–H). X axes are days, y axes are

decadic logarithms of parasite density.

(TIF)

Figure S9 Graphic representations of the six best fits to

the first wave of parasitemia for datasets 110, 111, 114,

115, 116, 117, 118 and 119 (A–H). X axes are days, y axes are

decadic logarithms of parasite density. The numbers above the

graphs are the errors calculated using equation (6).

(TIF)

Figure S10 Graphic representations of the six best fits

to the first wave of parasitemia for datasets 120 and 121

(A–B). X axes are days, y axes are decadic logarithms of parasite

density. The numbers above the graphs are the errors calculated

using equation (6).

(TIF)

Figure S11 Best ensemble fits to the entire course of

infection for data sets 37, 39, 40, 41, 44, 45, 46, 48, 50,

51, 52 and 54 (A–L). Blue lines are the MT data, green lines are

the ensemble means and shaded purple areas are ensemble

envelopes. X axes are days, y axes are decadic logarithms of

parasite density.

(JPG)

Figure S12 Best ensemble fits to the entire course of

infection for data sets 55, 56, 57, 58, 59, 60, 62, 63, 64,

67, 69 and 70 (A–L). Blue lines are the MT data, green lines are

the ensemble means and shaded purple areas are ensemble

envelopes. X axes are days, y axes are decadic logarithms of

parasite density.

(TIF)

Figure S13 Best ensemble fits to the entire course of

infection for data sets 71, 73, 74, 76, 77, 78, 79, 80, 81,

82, 83 and 84 (A–L). Blue lines are the MT data, green lines are

the ensemble means and shaded purple areas are ensemble

envelopes. X axes are days, y axes are decadic logarithms of

parasite density.

(TIF)

Figure S14 Best ensemble fits to the entire course of

infection for data sets 85, 86, 87, 88, 89, 90, 91, 92, 93,

94, 95 and 96 (A–L). Blue lines are the MT data, green lines are

the ensemble means and shaded purple areas are ensemble

envelopes. X axes are days, y axes are decadic logarithms of

parasite density.

(TIF)

Figure S15 Best ensemble fits to the entire course of

infection for data sets 97, 98, 99, 100, 101, 102, 103, 105,

106, 107, 109 and 110 (A–L). Blue lines are the MT data, green

lines are the ensemble means and shaded purple areas are

ensemble envelopes. X axes are days, y axes are decadic

logarithms of parasite density.

(TIF)

Figure S16 Best ensemble fits to the entire course of

infection for data sets 111, 114, 115, 116, 117, 118, 119,

120 and 121 (A–I). Blue lines are the MT data, green lines are

the ensemble means and shaded purple areas are ensemble

envelopes. X axes are days, y axes are decadic logarithms of

parasite density.

(TIF)

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PLoS ONE | www.plosone.org 11 March 2012 | Volume 7 | Issue 3 | e34040

Figure S17 6 panels comparing random versus best

parameter based community predictions of the model.

The panels on the left hand side are community runs using

random parameters. The panels on the right hand side are

community runs using parameters from the model calibration.

Community size is n = 2000. Panels A and B compare the

community prevalences at an EIR of 1 per parasite reproductive

cycle (182.5 per annum), Panels C and D compare the community

prevalences at an EIR of 0.1 per cycle (18.3 per annum), and

panels E and F compare community prevalences at an EIR of 0.01

per cycle (1.83 per annum). The dotted black lines denote fraction

of uninfected RBC, the dashed black denotes iRBC, and the solid

gray denotes infected but below limit of detection by light

microscopy (10 parasites per microliter).

(TIF)

Table S1 Description of all variables, parameters and

indices used in the model.

(DOC)

Table S2 Overview over the MT data used in the

present study and allocation of numbers to each set of

MT data. The columns labeled ‘#’ are the numbers assigned to

each dataset over the course of this study to facilitate data

processing.

(DOC)

Table S3 Comparative statistics between the output of

our model, the MT data we used for calibration and data

presented by Gatton et al 2006 [13].

(DOC)

Acknowledgments

The authors thank V. Ganusov, A. Perelson and E. McKenzie for

stimulating discussions and T. Smith for making available the MT data

sets.

Author Contributions

Conceived and designed the experiments: DG SK PAZ CHK TGSP

TMED. Performed the experiments: DG SK. Analyzed the data: DG SK

PAZ CHK TGSP TMED. Contributed reagents/materials/analysis tools:

DG SK. Wrote the paper: DG SK PAZ CHK TGSP TMED.

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Modeling of Malaria Infection

PLoS ONE | www.plosone.org 13 March 2012 | Volume 7 | Issue 3 | e34040

- CitationsCitations11
- ReferencesReferences42

- "[62] Similarly, the model does not account for clone specific acquisition of immunity in the human population. [43, 63] However, the aim of this study was to show the general effects of spatial heterogeneity on MOI and therefore these features were not considered essential. As with previous vector borne disease models, the present model assumes a fixed spatial distribution of humans and mosquitoes in which humans and mosquitoes are predominantly associated with specific locations. "

[Show abstract] [Hide abstract]**ABSTRACT:**As malaria is being pushed back on many frontiers and global case numbers are declining, accurate measurement and prediction of transmission becomes increasingly difficult. Low transmission settings are characterised by high levels of spatial heterogeneity, which stands in stark contrast to the widely used assumption of spatially homogeneous transmission used in mathematical transmission models for malaria. In the present study an individual-based mathematical malaria transmission model that incorporates multiple parasite clones, variable human exposure and duration of infection, limited mosquito flight distance and most importantly geographically heterogeneous human and mosquito population densities was used to illustrate the differences between homogeneous and heterogeneous transmission assumptions when aiming to predict surrogate indicators of transmission intensity such as population parasite prevalence or multiplicity of infection (MOI). In traditionally highly malaria endemic regions where most of the population harbours malaria parasites, humans are often infected with multiple parasite clones. However, studies have shown also in areas with low overall parasite prevalence, infection with multiple parasite clones is a common occurrence. Mathematical models assuming homogeneous transmission between humans and mosquitoes cannot explain these observations. Heterogeneity of transmission can arise from many factors including acquired immunity, body size and occupational exposure. In this study, we show that spatial heterogeneity has a profound effect on predictions of MOI and parasite prevalence. We illustrate, that models assuming homogeneous transmission underestimate average MOI in low transmission settings when compared to field data and that spatially heterogeneous models predict stable transmission at much lower overall parasite prevalence. Therefore it is very important that models used to guide malaria surveillance and control strategies in low transmission and elimination settings take into account the spatial features of the specific target area, including human and mosquito vector distribution.- "Stochasticity is incorporated ad hoc into the models by the emergence of new variants (which are not recognized by immune system) at random times, usually driven by a Poisson process (see Nowak and May, 2000 and references therein). Very recently Gurarie et al. (2012) implemented a discrete time computer model for the case of malaria. This modeling approach, termed agent-based, consists of a set of coupled difference equations that describe the transition between successive iterations of the parasite population (i.e. "

- "Very recently Gurarie et al. (see ref. [18]) implemented a discrete time computer model for the case of malaria. This modeling approach, termed agent-based, consists in a set of coupled difference equations that describe the transition between successive iterations of the parasite population (i.e. "

[Show abstract] [Hide abstract]**ABSTRACT:**We present a novel model that describes the within-host evolutionary dynamics of parasites undergoing antigenic variation. The approach uses a multi-type branching process with two types of entities defined according to their relationship with the immune system: clans of resistant parasitic cells (i.e. groups of cells sharing the same antigen not yet recognized by the immune system) that may become sensitive, and individual sensitive cells that can acquire a new resistance thus giving rise to the emergence of a new clan. The simplicity of the model allows analytical treatment to determine the subcritical and supercritical regimes in the space of parameters. By incorporating a density-dependent mechanism the model is able to capture additional relevant features observed in experimental data, such as the characteristic parasitemia waves. In summary our approach provides a new general framework to address the dynamics of antigenic variation which can be easily adapted to cope with broader and more complex situations. Copyright © 2015. Published by Elsevier Ltd.

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