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

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DOI: 10.1371/journal.pone.0034040 · Source: PubMed
  • 32.95 · Case Western Reserve University
  • 32.45 · The Walter and Eliza Hall Institute of Medical Research
  • 44.72 · Case Western Reserve University School of Medicine
  • 49.66 · University of Western Australia
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.
Modeling of Malaria Infection
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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,bðÞis 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.
doi:10.1371/journal.pone.0034040.g004
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PLoS ONE | www.plosone.org 6 March 2012 | Volume 7 | Issue 3 | e34040
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.
doi:10.1371/journal.pone.0034040.g009
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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
    • "[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.
    Full-text · Article · Oct 2016
    • "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. "
    File · Data · Nov 2015 · Journal of Theoretical Biology
    • "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.
    Full-text · Article · Sep 2015
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