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A viral load-based cellular automata approach to modeling HIV dynamics and drug treatment

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Journal of Theoretical Biology 253 (2008) 24–35
A viral load-based cellular automata approach to modeling HIVdynamics and drug treatment
Veronica Shi, Abdessamad Tridane, Yang Kuang
Ã
Department of Mathematics, Arizona State University, Tempe, AZ 85284, USA
Received 15 August 2007; accepted 2 November 2007Available online 17 November 2007
Abstract
We formulated a novel cellular automata (CA) model for HIV dynamics and drug treatment. The model is built upon realisticbiological processes, including the virus replication cycle and mechanisms of drug therapy. Viral load, its effect on infection rate, and therole of latently infected cells in sustaining HIV infection are among the aspects that are explored and incorporated in the model. Weassume that the calculation of the number of cells in the neighborhood which inﬂuences the center cell’s state is based on the viral load.This variable-cell neighborhood enables the simulation of an infection rate that is correlated to the viral load. This approach leads to anew and ﬂexible way of modeling HIV dynamics and allows for the simulation of different antiretroviral drug treatments based on theirindividual and combined effects. The results of the simulation show the three phases of HIV dynamics (acute, chronic, and AIDS) andthe additional drug response phase when drug treatment is added. The dynamics from the model qualitatively match clinical data. Drugtreatment combinations with reverse transcriptase inhibitors and protease inhibitors are simulated using various drug efﬁcacies. Theresults indicate that the model can be very useful in evaluating different drug therapy regimens.Published by Elsevier Ltd.
Keywords:
HIV; Cellular automata; Drug treatment; Viral load; Cell neighborhood
1. Introduction
In recent years, the combination of mathematicalmodeling with clinical research has greatly facilitated theunderstanding of the dynamics of HIV infection and theinteraction of the immune system with the virus. Muchwork has been done using ordinary differential equation(ODE) and partial differential equation (PDE) models(Arnaout et al., 2000; De Boer and Perelson, 1989; Dixitand Perelson, 2005; Kirschner, 1996; Wodarz and Nowak,1998). These models have explained different aspects of thedynamics of the virus–immune system interaction. How-ever, because HIV infection typically exhibits a three-phaseevolution (acute phase, chronic phase, and AIDS) (Panta-leo and Fauci, 1996), these continuous deterministic systemmodels may be insufﬁcient to describe the different timescales (days, weeks, and years) that are involved. Further-more, an additional drug treatment response phase is evenmore difﬁcult to simulate using these models, not tomention the different types of therapy and the correspond-ing responses.Recently, cellular automata (CA) models have beenused in HIV modeling (Sloot et al., 2002; Zorzenon dosSantos and Coutinho, 2001) and have shown greatpotential for simulating the temporal and spatial immuneresponses throughout the course of HIV infection. Thestrength of the CA model lies in its simplicity and, at thesame time, its ability to model complex systems (Gangulyet al., 2003). CA models also allow for more granularity of the simulation, which is hard to do with ODE and PDEmodels.CA were ﬁrst used byZorzenon dos Santos andCoutinho (2001)to model the evolution of HIV infection.A set of four simple rules was adopted to represent the lifecycles of the
CD
4
þ
T cells in terms of four states, namelyhealthy (T), infected stage 1 (A1), where the cell is infectedand able to spread the infection, infected stage 2 (A2), theﬁnal stage of an infected cell before it dies, and dead (D). Ithas been proposed that the lymphoid tissue is the primary
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www.elsevier.com/locate/yjtbi0022-5193/$-see front matter Published by Elsevier Ltd.doi:10.1016/j.jtbi.2007.11.005
Ã
Corresponding author. Tel.: +14809656915; fax: +14809658119.
E-mail address:
kuang@asu.edu (Y. Kuang).
Author's personal copy
target and major reservoir of HIV infection (Fauci, 2003)in vivo, and a lymph node has a mesh structure that may beapproximated by a rough surface (Hood et al., 1984).Therefore, it is reasonable to model the interaction amongthe immune system cells in the lymphoid tissue usingsquare lattice. The basic dos Santos model produced resultsthat qualitatively matched the three-phase HIV dynamicsobserved from clinical data. Although their work presenteda novel approach to the modeling of HIV dynamics, criticsraised issues concerning the model’s sensitivity to para-meters (Strain and Levine, 2002). One particular issue wasthat the initial density of infected cells,
P
HIV
¼
0.05, wastoo large compared to the estimated value from clinicalexperiments. When
P
HIV
was set to less than 0.05, theinitial infection peak did not occur in the model, and therewere no distinct ﬁrst-phase dynamics.Based on Zorzenon dos Santos’ CA model,Sloot et al.(2002)developed a drug therapy CA model by modifyingone of the rules. Instead of infecting all eight neighbors of an infected cell, the number of neighbors to be infected wasset to
N
ð
0
p
N
p
7
Þ
with probability
P
resp
, and 8 withprobability
ð
1
À
P
resp
Þ
. The number
N
was used to mimicthe drug effectiveness, i.e., the smaller the
N
, the moreefﬁcient the drug. The probability
P
resp
represented thepatient’s response to the therapy: the higher the prob-ability, the better the treatment. Drug resistance wasmodeled by using a linearly decreasing
P
resp
.The weakness of this approach is multifold: (1) there isno quantitative and direct association of the number of neighbors to be infected to the efﬁcacy of the drug; (2) thenumber of neighbors to be infected is predeﬁned before thesimulation run and ﬁxed throughout the simulation run,which does not change with the progression of the disease;and (3) the model is not based on the principles of drugtreatment, and therefore, there is no differentiationbetween different types of drug treatments, such as aprotease inhibitor or a reverse transcriptase (RT) inhibitor.One important factor that was omitted in both of themodels bySloot et al. (2002)andZorzenon dos Santos and
Coutinho (2001)was the effect of viral load on thedynamics of the infection and drug treatment. We willshow later in this paper that our model has addressed allthese and many other important aspects.Here we describe a simple, ﬂexible, yet comprehensiveand practically useful CA model of HIV dynamics that isbuilt upon realistic biological processes. The model is basedon biological processes of virus replication, the principlesof drug treatment, and the different stages of the
CD
4
þ
Tcell as the immune system responds (Ho, 1997; Nowaket al., 1996; Nowak and May, 2000; Perelson et al., 1993,1996). To produce a model with biological ﬁdelity, manyaspects of the infection dynamics were incorporated,including the following: the fact that HIV is a retrovirusand its replication is through translation of viral RNAinto the host cell’s DNA by RT; that the latently andchronically infected
CD
4
þ
T cells play an important role insustaining viral replication and contributing to the eventualsharp rise in viral levels and collapse of the immune system;that the viral load is closely correlated to the progression of the disease; and that the basic principle of drug therapy isto disrupt virus replication, either through protease or RTinhibition. Once this model is tested and veriﬁed, it can beused to further understand this disease and help with theevaluation of different treatment strategies.
2. Model development
Zorzenon dos Santos’ CA model (Zorzenon dos Santosand Coutinho, 2001), which will be referred to as the basicCA model throughout the rest of the paper, serves as thestarting point of the model development. New cell statesand new rules are adopted to include many aspects of theviral dynamics and disease evolution.
2.1. Modeling latently infected cells
Recent studies have shown that a pool of latentlyinfected cells is established during the primary phase of HIV infection (Bonhoeffer et al., 1997; Grossman et al.,1998; Havlir et al., 2003; Lafeuillade et al., 2000; Mulleret al., 2002). These cells are infected cells in a resting stateand can be activated after a long time of dormancy toproduce infectious virus particles. The reservoirs of thesedormant infected cells play a critical role in sustainingactive infection (Bailey et al., 2004; Ramratnam et al.,2000). The process of the activation and transmission of these long-lived latently infected cells is the most plausibleexplanation of HIV propagation during the chronic phase.To incorporate this important aspect of HIV dynamics, anew cell state is added—the latently infected state (A0).When a cell is to become infected based on the srcinalrules, an additional rule is used to decide whether a cellshould go into the actively infected (A1) state with aprobability
P
inf
or the latently infected (A0) state with aprobability
ð
1
À
P
inf
Þ
. After a long time delay, the latentlyinfected cell may become activated with a probability
P
act
.
2.2. Viral load
The basic class of ODE HIV models (Culshaw andRuan, 2000; Nelson and Perelson, 2002; Nelson et al.,2000; Perelson and Nelson, 1999) generally includesvariables that represent the healthy cells T, infected cells
T
Ã
, and the viral load
V
. The viral load is an integral partof the system of equations that inﬂuences the dynamics of the healthy and infected cells.Conventional CA models are built with healthy andinfected cell states. Rules control the state transitions;therefore, the models autonomously account for thedynamics of the healthy and infected cells. However, noneof the CA models of previous studies included the viralload. The previous assumption was that the viral loadwas simply proportional to the number of infectedcells, effectively ignoring the role of free virions in the
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V. Shi et al. / Journal of Theoretical Biology 253 (2008) 24–35
25
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progression of the disease. To account for this importantfactor in the dynamics of the disease, a novel approach isdeveloped to include the effect of viral load on the infectionrate in the CA model.A difference equation can be written as shown below todescribe the dynamics of the viral load, assuming the timeinterval
D
t
¼
1
D
V
t
¼
d
T
Ã
t
À
cV
t
;
V
t
þ
1
¼
d
T
Ã
t
þ ð
1
À
c
Þ
V
t
, (1)where
d
is the average virion production rate per infectedcell and
c
is the virion decay rate.To incorporate the viral load into the CA model, Eq. (1)is calculated at each time step after all the cells in the latticehave been updated according to the rules. The viral load isthen used to adjust the number of cells in the domain of theneighborhood that will affect the state of the center cell inthe next time step. This effectively builds a feedback loop,which affects the infection rate at each time step. Thefollowing sections detail how the viral load is incorporatedinto the deﬁnition of neighborhood domain and its effecton infection rate.
2.3. Viral load and expanded neighborhood
The conventional CA models in previous studies all use aneighborhood of eight cells where the rules are applied todetermine changes in cell state. In CA models, the effect of the number of neighbors used is equivalent to the infectionrate since it determines how fast the infection spreads.Because previous studies used constant 8-cell neighbor-hoods, this meant that the infection rate did not change inthose models. A more realistic simulation would be toadjust the number of cells in the neighborhood based onthe viral load. The biological mechanism of the HIVinfection indicates that healthy T cells become infected inthe presence of the HIV virus. The higher the virusconcentration, the higher the probability of infection. Inorder to allow for the modeling of a higher infection rate,the domain of the neighborhood must be allowed toexpand beyond the eight adjacent neighbors. Based on theMoore neighborhood deﬁnition (Weisstein), the neighbor-hood of range
r
is deﬁned by
N
M
ð
x
0
;
y
0
Þ
¼ fð
x
;
y
Þ
:
j
x
À
x
0
j
p
r
;
j
y
À
y
0
j
p
r
g
. (2)The number of cells in the Moore neighborhood of range
r
is the odd square
ð
2
r
þ
1
Þ
2
. When the center cell isexcluded from the neighborhood, as in the CA model, thenumber of neighbors is
ð
2
r
þ
1
Þ
2
À
1. For the 8-cellneighborhood the corresponding range
r
is 1. A newconcept is to extend the deﬁnition of range
r
beyond theinteger domain to include real numbers, such that theneighborhood is still deﬁned in the same way. For example,a range
r
of 1.5 corresponds to a 15-cell neighborhood.The range
r
can be considered the equivalent of the rangeof infection, i.e., a healthy cell may be infected by aninfected cell within the neighborhood of radius
r
. Biologi-cally, long-range transmission of the infection is possibleunder high viral burden (Nowak and May, 2000). Allowingthe number of cells in the neighborhood to vary with theamount of viral load allows the modeling of changes ininfection rate, which leads to a more realistic model and aﬂexible way of incorporating different types of drugtreatment.In order to include the viral load in the CA model, weneed to ﬁnd the relationship between the viral load andthe number of cells in the neighborhood. Intuitively,the number of neighbors is proportional to viral load,because the higher the viral load, the more infectious it is,and thus the more possibilities for long-range infection.However, because of the discrete nature of the automataand the rules governing the infection of the healthy cells,the translation from viral load to the number of neighborsis more complicated than a simple linear or exponentialfunction.In all the previous studies, cell state change had beenbased on the cell states of the 8-cell neighborhood and a setof rules. Rule 1 of the basic CA model dictates that if oneof the eight neighbors of a cell is infected, then the cellbecomes infected. This rule governs how the numberof infected cells around an original infected cell grows,which is in the order of 1, 9, 25, 49, 81, etc. as time goes on.Table 1summarizes how the number of infected cells growsfor a Moore neighborhood range of 1 and 2.ExaminingTable 1leads to a generic formula for
r
¼
1:
D
T
Ã
n
¼
T
Ã
n
À
T
Ã
n
À
1
¼
n
Á
M
1
Á
T
Ã
0
and
M
1
¼
D
T
Ã
n
n
Á
T
Ã
0
. (3)Further examination of Table 1for
r
¼
2 reveals that thecoefﬁcients of
D
T
Ã
n
form an arithmetic series in the form of
a
n
¼
a
1
þ ð
n
À
1
Þ
d
, where
d
¼
1 and
a
1
¼
1 for
r
¼
1, and
d
¼
4 and
a
1
¼
3 for
r
¼
2, relative to
M
1
¼
8, i.e.,
M
2
¼
24
¼
3
Á
M
1
¼
3
Á
D
T
Ã
n
=
a
n
Á
T
Ã
0
.
ARTICLE IN PRESS
Table 1Growth of number of infected cells around an srcinal infected cellTime step,
t r
¼
1
r
¼
2Number of
T
Ã
D
T
Ã
Number of
T
Ã
D
T
Ã
1
ð
8
þ
1
Þ Á
T
Ã
0
8
Á
T
Ã
0
¼
M
1
Á
T
Ã
0
ð
24
þ
1
Þ Á
T
Ã
0
24
Á
T
Ã
0
¼
3
Á
M
1
Á
T
Ã
0
2
ð
24
þ
1
Þ Á
T
Ã
0
16
Á
T
Ã
0
¼
2
Á
M
1
Á
T
Ã
0
ð
80
þ
1
Þ Á
T
Ã
0
56
Á
T
Ã
0
¼
7
Á
M
1
Á
T
Ã
0
3
ð
48
þ
1
Þ Á
T
Ã
0
24
Á
T
Ã
0
¼
3
Á
M
1
Á
T
Ã
0
ð
168
þ
1
Þ Á
T
Ã
0
88
Á
T
Ã
0
¼
11
Á
M
1
Á
T
Ã
0
4
ð
80
þ
1
Þ Á
T
Ã
0
32
Á
T
Ã
0
¼
4
Á
M
1
Á
T
Ã
0
ð
288
þ
1
Þ Á
T
Ã
0
120
Á
T
Ã
0
¼
15
Á
M
1
Á
T
Ã
0
V. Shi et al. / Journal of Theoretical Biology 253 (2008) 24–35
26
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Therefore Eq. (3) can be generalized as
M
r
¼
s
Á
D
T
Ã
n
a
n
Á
T
Ã
0
. (4)Rule 2 of the basic CA model also speciﬁes that aninfected-A1 cell becomes infected-A2 after
t
time steps,where
t
¼
4. The infected-A2 cells die in the next time stepaccording to rule 3. Therefore, for
r
¼
1 we have
T
Ã
5
¼ ð
120
þ
1
Þ Á
T
Ã
0
À
T
Ã
0
¼
15
Á
M
1
Á
T
Ã
0
,
T
Ã
6
¼ ð
168
þ
1
Þ Á
T
Ã
0
À ð
8
þ
1
Þ Á
T
Ã
0
¼
20
Á
M
1
Á
T
Ã
0
and
D
T
Ã
6
¼
T
Ã
6
À
T
Ã
5
¼
5
Á
M
1
Á
T
Ã
0
and Eq. (4) becomes
M
r
¼
s
Á
D
T
Ã
n
a
n
Á
D
T
Ã
0
;
a
n
¼
a
1
þ ð
n
À
1
Þ Á
d
;
n
o
t
;
a
n
¼
5
Á
r
2
;
n
X
t
:
(
(5)Now assuming that the viral load
V
is strictly propor-tional to the number of infected cells
T
Ã
(i.e., the decay of the virion can be ignored for the moment), then the numberof neighbors
M
r
can be expressed as
M
r
¼
s
Á
D
T
Ã
n
a
n
Á
T
Ã
0
¼
s
Á
d
Á
D
T
Ã
n
a
n
Á
d
T
Ã
0
¼
s
Á
D
V
n
a
n
Á
d
T
Ã
0
. (6)Although Eq. (6) is derived based on the expansion of the infected cells, it is assumed that it still holds true duringthe declining phase of the infection. Biologically, theinfection rate is not only related to the amount of virionsin the system, but also to the spatial distribution of the virion concentration. In other words, the growthrate of the infected cells is proportional to the peaks of the local concentration of virions, not to the averageconcentration throughout the system. Eq. (6) indicates thatthe number of neighbors is related to the change of theviral load, which is consistent with the biological char-acteristics. Without an elaborate algorithm to ﬁgure outthe geometric distribution of the infected cell clusters, wewill make a simple approximation by assuming that thechange of number of neighbors,
D
M
, is a fraction of
M
andis proportional to the change of the total viral load:
D
M
¼
M
=
h
¼ ð
s
=
h
Þ Á
D
V
n
=
a
n
d
T
Ã
0
. Then by choosing aproper coefﬁcient
h
and combining
h
with the scale factors
s
and
a
n
, 1
=
k
¼
s
=
ha
n
, and we have
D
M
t
¼
V
t
À
V
t
À
1
k
d
T
Ã
0
;
M
t
þ
1
¼
M
t
þ
D
M
t
. (7)
2.4. Drug treatment
With the advances in medicine, there are many drugsavailable to treat patients with HIV. Currently, two typesof basic inhibitors are used—RT inhibitors and proteaseinhibitors. The RT inhibitors block the RT process of thevirus after it enters a healthy cell, preventing the cell frombeing infected. The protease inhibitors cause infected cellsto produce non-infectious virions. Standard mono-therapyuses RT inhibitors alone. Combination therapy of RTinhibitors with later-developed protease inhibitors usuallyyields better treatment. The latest development in drugtherapy is the highly active antiretroviral therapy(HAART), which typically combines three or more drugsof both the RT and protease inhibitors.To simulate the effect of the RT inhibitor, a new cellstate, the exposed but not infected state (An), is added tothe model to represent the state where a cell is invaded bythe virus but stays healthy because of the treatment. Theefﬁcacy of the drug is simulated by a probability
P
rt
thatdecides whether a healthy cell, normally to be infectedunder the rules when no treatment is given, will be infected(A1) or stay in an exposed but not infected (An) state dueto RT inhibition. The An cells will remain uninfected for ashort period of time before they become susceptible toinfection again, either due to viral RNA degradation ornew strains of viral mutation.Assume the efﬁcacy of the protease inhibitor isrepresented by the numerical value of the percentage
P
pi
of virions that are non-infectious during burst. Then thesimulation of the protease inhibitor is a simple change of the viral load equation (1):
V
t
þ
1
¼ ð
1
À
P
pi
Þ Á
d
T
Ã
t
þ ð
1
À
c
Þ Á
V
t
. (8)Because the viral load affects the infection rate throughthe change of the number of neighbors, the CA model willautonomously adjust the dynamics of the infection toreﬂect the effect of the protease inhibitor therapy.Combination therapy and HAART can be modeled bysetting both
P
rt
and
P
pi
to levels that represent the efﬁcacyof the individual RT and protease inhibitors or thecombined efﬁcacy of two or more RT or proteaseinhibitors.Drug resistance is modeled by using a variable diminish-ing drug efﬁcacy
P
rt
or
P
pi
. One possible form of the declining efﬁcacy is to assume a decay model of
P
0
rt
¼ À
lP
rt
. Solving the differential equation gives
P
rt
¼
P
rt
0
Á
e
À
l
ðð
t
À
t
s
Þ
=
t
s
Þ
, where
t
s
is the drug therapy starttime.
3. Methods
A square lattice with 700
Â
700 cells was used tosimulate the lymphoid tissues. The lattice was initializedwith infected cells randomly distributed among healthycells. Each time step represented 1 week. At each time step,the state of each cell in the lattice was updated based on therules and its neighbors’ states. After all the cell states wereupdated, the viral load was calculated using Eq. (1), andthe number of cells in the neighborhood that would affectthe state of the center cell was updated for the next timestep using Eq. (7).The CA model was implemented in MATLAB. Conﬁg-urations were set up for different simulation purposes,such as no treatment, mono or combination therapy, and
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HAART treatment. For each conﬁguration, multiplesimulation runs were conducted and the data averaged toproduce the ﬁnal results. The following sections outline thedetails of the CA model, including cell states, rules, andparameters used.In this model, there are six cell states as opposed to thefour states thatZorzenon dos Santos and Coutinho (2001)andSloot et al. (2002)used. In addition to the healthy (T),infected stage 1 (A1), infected stage 2 (A2), and dead (D)states, the latently infected (A0) and exposed but notinfected (An) states are incorporated to better simulate thebiological processes of HIV infection. A state transitiondiagram is presented inFig. 1to illustrate the relationshipof these states and the conditions for each state transition.The states of all cells are updated using the rules listed inTable 2at each time step. Viral load and the number of cells in the neighborhood are updated after all the cells areupdated. Each time step represents 1 week.Table 3lists all the parameters used in the model. Theunit for time is in weeks.
4. Results
The CA model developed here is a comprehensive toolthat can simulate different drug therapy regimens. Bysetting the parameters
P
pi
and
P
rt
to different values toform various conﬁgurations, different types of treatmentcan be evaluated. Because each simulation run represents aparticular individual case, the simulation was run 20 timesfor each conﬁguration and the results averaged over theruns. The following ﬁgures show the results from thesimulation runs of selected conﬁgurations.When
P
pi
and
P
rt
are set to zero, the model is simulatingthe effect of no drug treatment.Fig. 2shows the dynamicsof the HIV infection without treatment. The three phasesof the infection, acute, chronic, and AIDS, are evident inthe graph, which shows that the model output qualitativelymatches clinical data. Note the different time scales of weeks and years in the graphs. It is also worth mentioningthat the probability that determines the number of initialinfected cells
P
HIV
is set to 0.005 as opposed to 0.05 in thedos Santos’ model. The value of 0.05, according to critics,was too high compared to clinical experiment data, and thedos Santos’ model failed to show the acute phase dynamicswhen
P
HIV
was set to 0.005 (Strain and Levine, 2002). Byincorporating the viral load and varying the number of cells in the neighborhood, the infection rate changes basedon viral load, and the model is able to maintain the acutephase characteristics even when
P
HIV
equals 0.005. Theresults presented here prove the robustness of this model.The graphs also show that the number of cells in theneighborhood truthfully reﬂects the viral load and effec-tively changes the infection rate. Both the viral load andM-cell neighbors exhibit the three-phase dynamics. Theresults also show that the chronic phase is sustained by theactivation of the latently infected cells.In addition to simulating the dynamics of the HIVinfection without treatment, the model is very ﬂexible insimulating drug therapy. Different types of drug therapy
ARTICLE IN PRESS
TA1A2A0DAn
Atleast 1 A1 or4 A2 neighborst>
τ
3
t>
τ
1
t>
τ
2 &
P<P
act
NexttimestepT - HealthyA0-Latently infectedA1- Inf ectedstage1A1- Inf ectedstage2D- DeadAn - Exposed but not infectedP<P
rt
P>P
inf
P>P
rt
P<P
inf
P<P
repl (simulatesreplenishment byimmunesystem)
Fig. 1. State transition diagram of cell states.Table 2Model rulesRule no. Rule descriptionRule 1 Update of healthy cellsIf a healthy cell has at least one A1 neighbor or
R
A2 neighbors within the M-cell neighborhood, then it becomes an An cell withprobability
P
rt
. Otherwise it becomes an infected A1 cell with probability
P
inf
or a latently infected A0 with probability (1
À
P
inf
).Otherwise it stays healthy.Rule 2 Update of infected-A1 cellsAn infected-A1 cell becomes infected-A2 cell after
t
1 time steps.Rule 3 Update of infected-A2 cellsInfected-A2 cells become dead cells in the next time step.Rule 4 Update of dead cellsDead cells can be replaced by healthy cells with probability
P
repl
in the next time step (or remain dead with probability 1
À
P
repl
). Thissimulates the new cells that are replenished by the immune system.Rule 5 Update of latently infected A0 cellsAfter a long time delay (
t
2), the latently infected-A0 cell becomes actively infected (A1) with a probability
P
act
, otherwise it stays A0Rule 6 Update of exposed but not infected An cellsAfter a short time delay (
t
3), the An cells become healthy (T) cells.
V. Shi et al. / Journal of Theoretical Biology 253 (2008) 24–35
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