Modelling of immune cells as vectors of HIV spread inside a patient human body

3.7 MB·

by  Marcelo Margon Rossi

Modelling of immune cells as vectors of HIV spread inside
a patient human body
MarceloMargonRossia,∗,LuisFernandezLopezb,c
aLIM01-HCFMUSP,UniversityofSa˜oPaulo,Sa˜oPaulo,SP,01246-903,Brazil
bDept.ofLegalMedicineandMedicalEthics,UniversityofSa˜oPaulo,Sa˜oPaulo,SP,01246-903,Brazil
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cCIARA,FloridaInternationalUniversity,Miami,Fl,33199,USA
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J Abstract

6 ThesearchtounderstandhowtheHIVvirusspreadsinsidethehumanbodyandhow
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theimmuneresponseworkstocontrolithasmotivatedstudiesrelatedtoMathematical

Immunology. Actually,researchesincludetheideaofmathematicalmodelsrepresent-
]
E ingthedynamicsofhealthyandinfectedcellpopulationsandfocusingonmechanisms
P usedbyHIVtoinvadetarget-hostcells,viraldissemination(whichleadstodepletion
o. oftheT-cellpoolandcollapseoftheimmunesystem),andimpairmentofimmunere-
i sponse. In this work, we show the importance of specific cells of immune response,
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asinfectionvectorsinvolvedinthedynamicsofviralproliferationwithinanuntreated
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q patient,byusinganordinarydifferentialequationmodelinwhichweconsideredthat
[ the virus infected target-cells such as macrophages, dendritic cells, and lymphocytes

1 TCD4andTCD8populations. Inconclusion,wedemonstratetheimportanceofeach
v cell-host and the threshold of viral establishment and posterior spread based on the
3 presenceofinfectedmacrophagesanddendriticcellsinantigen-presentingprocessing,
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whichleadstonewinfections(bywildormutantvirions)afterimmuneresponseacti-
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vationepisodes. WepresentedanR expressionthatprovidesthemajorparametersof
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0 HIVinfection.Additionally,wesuggestsomepossibilitiesofnewtargetsforfunctional
. vaccines.
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0 Keywords: MathematicalModeling;HIV/AIDS;ImmuneSystem;ViralInfection;
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Pre-ExposureProphylaxis;BasicReproductionNumber;Vaccine
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r 1. Introduction
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The epidemiology of infectious diseases has been studied for years and has been
consideredthecontrol(anderadication)ofinfectiondependentonthedynamicsofepi-
demicprocessdynamicsandhumanimmunity[1–3]. Thelevelsofimmunizationde-
pendonvaccinationuse,theprobabilityofoccurrenceofnewcasesandthemetabolic
∗Correpondingauthor: MarceloMargonRossi. LIM01-HCFMUSP,UniversityofSa˜oPauloSchoolof
Medicine.AvenidaDoutorArnaldo,455.01246-903.Sa˜oPaulo-SP,Brazil.FAX:[phone].
Emailaddress:[email protected](MarceloMargonRossi)
PreprintsubmittedtoArXiv January19,2016
consequencesofaninfectedhost. Theyaredevelopedaccordingtothelevelofcom-
plexity,theobjectivetobereached,thehost-pathogenrelationshipandthepopulation.
Based on the HIV infection process and dissemination, the range of maximum effi-
ciency of a particular vaccine may not be reached [4, 5] because of the number of
resistantstrainsthatappearaftertreatmentwithantiretroviraldrugs[6,7]orbythefor-
mation of cellular reservoirs (macrophages and lymphocytes TCD4 cells chronically
infected)[8,9]. Becauseviralsecondaryinfectionscanoccurinantigenpresentation
andlymphocyteactivationorbymacrophageactivationininflammationsites,amath-
ematicalindexexpressionconsideringthesefactorscouldhelptheefforttocontroland
eradicatethediseasemoreeffectively.
A concept of great interest in epidemiology is the Basic Reproduction Number, R ,
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which is the expected number of secondary cases per primary cases of infection in a
completely susceptible population. This concept was initially proposed by MacDon-
ald in the 1950s [10], when it was shown that for cases in which R >1, the disease
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spreads into the population, and if R < 1, the disease will not progress. However,
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inacomplexsystem,thedeductionofsimpleexpressionsofR iscomplicateddueto
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thepersistenceofinfectionphenomena,suchasvariousconditionsandcharacteristics,
thecomplexitylevel, theobjectivetobereached, andthenatureofthehost-pathogen
relationship[3]. Forthesecomplexcases, thedefinitiondevelopedbyDiekmannand
Heesterbeek[11,12],definedasBasicReproductionRatio,hasbeenusedtostudythe
thresholdcriteriainheterogeneoussystems.
In this study, we adapted the concept of R to represent the phenomena of the HIV-1
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infectionandthedynamicsoftheimmunesystemtocontrolandeliminateitasanes-
sentialtooltotestahypothesisthatresultsinmoreaccurateoutcomes. Weappliedthe
NextGenerationMatrixOperatormethodologytoestablishthethresholdofviralinfec-
tionbasedonimmunologicalmechanisms,whichisanimportantindicatorofeffortsto
eliminatetheinfection.
2. Descriptionofthemodel
Themodeldescriptionweproposeisarealizationoftheimmunesysteminwhich
thevariouscellpopulationsdeterminethesetofequations[39,40]. Thecellpopula-
tionswereT,thenaveandnot-infectedTCD4+lymphocytepopulation,I,theproduc-
tively infected, and L, the latently (or chronically) TCD4+ infected lymphocyte sub-
populations. Thefreeviralloadisrepresentedbythevequation. Theseequationscon-
sideradditionalcellpopulationsthatcontributetoinfectionestablishment. Thesecells
areMp,activenot-infectedmacrophages;Mpi,activeinfectedmacrophages;iDC,im-
mature not-infected dendritic cells and mDC the infected and mature dendritic cells.
TheCTL compartment of the model represents the cytotoxic action of TCD8+ lym-
phocytes. The effectors and the memory cytotoxic lymphocyte subpopulations were
consideredintheCTLflux(Fig.1). Allkineticparametersrepresentingtheglobalflux
areshowninTable1.
Themacrophagepopulationispresentedby(1),
dMp vn
= s +l Mp−δ Mp−β Mp (1)
dt m 1 f 1vn+K
v
2
OncetheHIVviruspenetratesthetissuebarriers,macrophagesentinelsfinditandsig-
nal other immune cells. In (1), macrophages are recruited by a s rate and activated
m
accordingtotheimmune responselevelbyl rate. δ isthenaturaldeathrate. Once
1 f
activemacrophagescontactviralparticles,theparticlesareabsorbed,andtheseabsorp-
tionprocessesarerepresentedbyasaturationterm(HollingIItype),withamaximum
rate parameter β to lead to production of antigens. The exponent n is a Hill coeffi-
1
cientandrepresentsthepossiblecooperationamongtheHIV-1molecularinteractions
withactivemacrophages. Theviralparticleingestioninitiatestheactiveandinfected
macrophagepopulationaccordingto(2),
dMpi vn
=β Mp−(δ +α(cid:48))Mpi−β MpiT −k MpiCTL. (2)
dt 1vn+K f mf 1
v
These population dynamics show that infected macrophages do not suffer from the
HIV-1cytolyticeffectsasstronglyasTcells,andtheycanremainintheviralreservoir
foralongperiod. AccordingtoPernoetal. [13]andAquaroetal. [14],virionsmay
occurinmacrophageswithgeneticsequencesanddifferentalternativesplicingofthose
foundininfectedlymphocyteTCD4+cells,showingalessergeneticvariance. These
infectedmacrophagesaregivenanadditionaldeathrate,α(cid:48),thatreflectspossibleHIV
biochemicaleffectsincytosol,leadingtoanapoptosisstate.Theinfectedmacrophages
carry trapped viral particles to TCD4+ lymphocyte activation in the so-called ”infec-
tious synapse” process. These phenomena were quantified in the third term of (2),
according to the Mass Action Law. β is the parameter encapsulating the contact
mf
and complex formation rate. The last term in (2) shows the cell-cell encounter rate
displayedbytheimmuneresponse. Thisimmuneaction, relatedtothedestructionof
theseinfectiouscellsbycytotoxicTcelllymphocytes(CTL),isproportionaltotherate
k .
1
DuringtheHIVparticlebypassingthetissuebarriers,therecognizingprocessisacti-
vated, and the adaptive immune response initiates the presenting process by the den-
dritic cells. The s parameter shows the cell influx into a specific tissue, and the
id
increase of the active cell concentration (by adaptive immune response) is shown by
thesecondtermin(3),withthel parameterrepresentingtheactivationrate.
2
diDC v
= s +l iDC−δ iDC−K iDC. (3)
dt id 2 id 1v+κ
v
inwhichdendriticcellsdieanaturaldeathattherateδ . Itwasdemonstratedthatthe
id
relationship between the number of viral particles and mono-covalent connections to
DC-membrane receptors follow a saturation process over time, according to Vanham
etal. [15]andHlavaceketal. [16,17]. Thesimplermathematicalexpressionusedto
represent these phenomena resembles the enzymatic rate by Michaelis-Menten. The
process in which the virus has been holding onto the receptor is encapsulated in the
third term of (3), where K is the maximum rate of the connected virus to cellular
1
membrane receptors, κ is the viral particle half-maximum concentration of dendritic
v
cells.ThisformulationwaschoseninsteadofMassActionLawtoexhibitthebiochem-
icalmechanismsdescribedbyStebbingandBower[18]andWuandKewalramari[19].
Duringthisprocess,thesecellsmature(mDC)andprocessantigensintoMHCclassII
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complex,activatingTCD4+lymphocytes(4). Therefore,themDCexpressionis
dmDC v
= K iDC−δ mDC−β mDCT −k mDCCTL. (4)
dt 1v+κ dc dc 2
v
Thenumberofcontactsbetweenthesematuredendriticcells(withfragmentsorentire
virustrapped)andnaveTCD4+lymphocytesisproportionaltothesetwopopulations
andhastheprobabilityβ [20]. Thesecellsdieatarateδ duetothedeathcellac-
dc dc
tivationinducedprocess[21]andCTLlymphocytecytotoxicaction,ataratek . The
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infecteddendriticcellpopulationdirectlymigratestotheinteriorofthelymphoidor-
ganstomeetnaveTCD4+lymphocytes,activatingandinfectingit. Theefficiencyof
infection in these lymphocytes is intimately associated with the viral integrity, an ef-
fectivetransfectionprocessandadequatetrappinginviralandTcellbinding[22–24].
TheTCD4+lymphocytepopulationhasanavecell s influx. Theselymphocytesare
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activatedaftercontactwithHIV-infectedmacrophagesordendriticcells,whichoccurs
during the antigen-presenting process and is increased by cloning expansion (5). In
activeHIV-infectedmacrophages,thisviraltransportcanoccurinlymphorgansorat
inflammationsites[25]. Theβ andβ parametersarecontactratesthatquantifythe
dc mf
intensity of the adaptive immune response and the T-cell activation level by antigen
processing [26]. Additionally, ξ and φ parameters define, respectively, the infectious
contact parameters of macrophages and dendritic cells in the antigen-presenting pro-
cess. ThisantigenpresentationwasmodeledbyQueuingTheory,consideringthatthe
arriving process is close to a Poisson distribution’s mean 1/c [27]. The c parameter
showstheexponentialdecaylifeexpectancyoffreevirusinblood. Thesurvivorvirus
fromimmunologicalattackreachesthenaveoractiveCD4moleculeswithprobability
[1−exp(−ct)]/c, with the parameter a representing the probability of the encounters
tobeinfectious. Thus,themathematicalexpressionΨ(a,c;t)=a[1−exp(−ct)]/crep-
resentstheefficiencyofinfectiouscontactbetweendendriticcellsormacrophages,in
theantigenpresentation,andTCD4+andCTLlymphocyteactivation. Thedynamics
ofnaveTCD4+lymphocytesfollowtheexpression,
dT (cid:16) (cid:17)
= s + β Mpi+β mDC T −Ψ(a,c;t)(ξMpi+φmDC)T −δT. (5)
dt 4 mf dc
Thismathematicalformulationofaninfectiousencounterisverysimilartothemodel
usedbyMcDonaldinvector-hostdiseases[10,27].
Aftersomenavelymphocytesareactivatedandinfected,aproportion f becomespro-
ductivelyinfectedTCD4+lymphocytesandaproportion(1− f)becomeschronically
infected. ItisassumedthatthislastT-cellgroupisactivatedat0.0005cell(µl.day)−1
[28],dependingontheneedtostrugglewithopportunisticinfections,anditproduces
avirusinthisprocess,asshownby(6and7),
dI
= f Ψ(a,c;t)(ξMpi(t−τ)+φmDC(t−τ))T(t−τ)e−(δ+α∗)τ
dt (6)
−(δ+α∗)I+0.0005L−k ICTL,
3
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and,
dL
dt =(1− f)Ψ(a,c;t)(ξMpi(t−τlat)+φmDC(t−τlat))T(t−τlat)e−(δ+α∗lat)τlat (7)
−(δ+α∗ )L−0,0005L.
lat
Within these equations, τ and τ parameters represent the period that HIV needs to
lat
completeitslifecycleinahostcell. TheinfectedT-cellsdieduetotheHIVcytolytic
actionortheCTLsimmuneresponseintensity. Thisoccurrenceisdescribedbyα∗and
α∗ parametersandthecelllifeexpectancyby(δ+α∗)and(δ+α∗ ),whicharethedeath
lat lat
rateparametersassociatedwithproductiveandchroniccellpopulations,respectively.
TheCTLcellsplayanimportantroleintheimmuneprocess, signalinginfectedcells
and destroying them by apoptosis mechanisms. Equation (8) shows the dynamics of
theCTLlymphocytepopulationontheimmunesystemtocontrolthespreadofinfec-
tion. Thefirstterm, s ,representsthecellinflux,andδ representsthenaturaldeath
8 ctl
rate. ξ(cid:48) and φ(cid:48) are parameters related to the activation of induced cell death of CTL
by macrophages and dendritic cells, respectively. Each population cited previously
contributes to immune response level through the rates kctl (i=1,2,3), cooperating to
i
increasetheactiveeffectorsintheCTLpopulation.
dCTL = s −δ CTL−(cid:0)ξ(cid:48)Mpi+φ(cid:48)mDC(cid:1)CTL
dt 8 ctl (8)
(cid:16) (cid:17)
+ kctlMpi+kctlmDC+kctlI CTL
1 2 3
Viralbalance(9)showstheformationoffreeHIV-1virusparticlesintheblood. The
dynamicsfollowtheexpression,
dv vn
= Q (δ+α∗)I+Q (δ+α∗ )L+Q Mpi−K iDCv−σ Mp−cv. (9)
dt 1 2 lat 3 2 vn+κ
v
wherethetwofirsttermsrepresentthecontributionsfromproductivelyandchronically
infected lymphocyte subpopulations, after cell death by HIV-1 cytolytic effects. The
thirdtermindicatestheinfectedmacrophagepopulationeffectonbasalviralload. Pa-
rametersQ ,Q andQ aretheratesofparticipationofeachinfectedcellpopulationin
1 2 3
HIVdisseminationandviralpathogenesis. Theothertermsdescribedasthefreeviral
particlesarewithdrawnfromthebloodbydendriticcells(atrateK )andthenumbers
2
of HIV-1 virions that are encountered (and are captured) by active macrophages (at
rateσ). cistheviralclearance. Theseparametersevaluatetheefficacyoftheimmuno-
logicalactions. Allkineticparameterswerebasedonimmunologicalandbiochemical
mechanismsoftheimmuneresponsetotheHIVinfectionprocessandweregathered
fromcitedliterature. ThevaluesareshowninTable1.
2.1. Thresholdcalculation
The necessity of considering other cellular reservoirs in addition to the TCD4+
lymphocytes (such as macrophages and dendritic cells) is very important for HIV
spreadcontainmentanditscompletecontrolintothepatientsbody, aswellastheex-
istence of a remarkable high efficiency of HIV vaccines. The model (1-9) represent,
from a mathematical point of view, the immunological aspects of HIV infection, the
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Table1:ParametervaluesandbiochemicalmeaningsforparametersetfromEquations1-9
Parameter Value(unit) Biochemicalmeaning References
s 16.3cell(µl.day)−1 Activemacrophageinfluxrate [36]
m
l 0.002cell(µl)−1 Activatedmacrophagegrowthrate [36]
1
β 1.10 Infectedmacrophagewithviralparticles [38,39]
1
n 0.55 Hillcoefficient Adopted
K 100virions(ml)−1 Saturationcoefficient Adopted
v
δ 0.0285(day)−1 Activatedmacrophagenaturaldeathrate [36]
f
β 4.50×10−04(day)−1 MacrophageandCD4+lymphocytecontactactivationrate [15]
mf
α(cid:48) 0.02(day)−1 InfectedmacrophagedeathrateduetoHIVvirus [38]
s 37.6cell(µl.day)−1 Immaturedendriticcellinfluxrate [24,40]
id
l 0.085cell(µl)−1 Dendriticcellgrowthrate [33]
2
K 145 Immaturedendriticcellwithviralparticles [33]
1
κ 4.5 ComplexDendriticcell HIVsaturationconstant Adopted
v
δ 0.005(day)−1 Immaturedendriticcellnaturaldeathrate [15]
id
β 4.17×10−03(day)−1 DendriticcellandCD4+lymphocytecontactactivationrate [32]
dc
δ 0.025(day)−1 Infectedmaturedendriticcelldeathrate [38]
dc
s 0.025cell(µl.day)−1 LymphocyteCD4+influxrate [24,40]
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a [0..1] Effectivecontactfractionamongactivatedandinfectedcells
ξ 0.0028(day)−1 InfectedmacrophageandTCD4+contactrate Estimated
φ 8.62×10−05(day)−1 InfecteddendriticcellandTCD4+contactrate Estimated
δ 0.015(day)−1 ActivatedlymphocyteCD4+naturaldeathrate [32]
f [0..1] ProductivelyinfectedlymphocyteCD4+fraction
τ 0.13day LymphocyteCD4+intracellularviralproductiondelay [1]
α∗ 0.332(day)−1 ProductivelyinfectedlymphocyteCD4+additionaldeathrate [1,40]
τ 20day ViralproductioninlatentlymphocyteCD4+delay Adopted
lat
α∗ 0.132(day)−1 LatentlymphocyteCD4+additionaldeathrate [32]
lat
s 0.0023cell(µl.day)−1 CD8+lymphocyteinfluxrate [40]
8
δ 0.136(day)−1 CD8+lymphocytenaturaldeathrate [32]
ctl
k 1.50×10−03(day)−1 InfectedmacrophageandeffectorCTLcontactrate [37]
1
k 2.00×10−04(day)−1 Dendriticcell-HIVcomplexandeffectorCTLcontactrate [37]
2
k 2.50×10−04(day)−1 InfectedCD4+lymphocyteandeffectorCTLcontactrate [37]
3
Q 750virions(cell)−1 ViralparticlenumberperactivatedinfectedCD4+lymphocyte [32]
1
Q 80virions(cell)−1 ViralparticlenumberperlatentlyinfectedCD4+lymphocyte [32]
2
Q 250virions(cell)−1 Viralparticlenumberperinfectedmacrophage [3]
3
K 0.15ml(cell)−1 AbsorptionrateofHIVparticlesbyimmaturedendriticcells [15]
2
σ 0.82ml(cell)−1 AbsorptionrateofHIVparticlesbyactivatedmacrophages [15]
ξ(cid:48) 4.80×10−04(day)−1 InfectedmacrophageinducedCTLdeathrate Estimated
φ(cid:48) 2.10×10−02(day)−1 MaturedendriticcellinducedCTLdeathrate Estimated
kctl 1.50×10−03cell(µl.day)−1 CTLgrowthraterelatedtoinfectedmacrophage [40]
1
kctl 1.50×10−04cell(µl.day)−1 CTLgrowthraterelatedtomaturedendriticcells [40]
2
kctl 6.1755×10−03cell(µl.day)−1 CTLgrowthraterelatedtoinfectedCD4+lymphocyte [40]
3
c 2.3(day)−1 Viralclearancerate [3]
growthanddeathrates,thenumberofcontagiouscontacts(astheforceofviralspread)
and the ”control” exerted by CTL populations on viral dissemination. This approach
is very interesting to understand which factors in HIV infection mechanisms are im-
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portanttoinitiatecellinvasion,proliferationanddissemination,andtodepleteTCD4+
populations,whichcauseimpairmentofthehumanimmunesystem.
Equations(1-9)wereconductedusingBerkeleyMadonnasoftware8.0.1,andsimula-
tion runs were performed with fourth order Runge-Kutta integration algorithm. The
mainobjectivewasreachingtheexactpredictionoftheHIVinfectionanddiseaseevo-
lutioninanuntreatedpatient.
3. ResultsandDiscussion
The TCD4+ lymphocyte dynamics (Fig. 2) are similar to the classic HIV values
found in the literature, and differences that appear could be due to the delays in the
model. If we consider the biochemical events relative to the viral life cycle and the
cellhosthalf-life,this”timedilation”betweenthearrivalandthecompletionofcom-
plete virions could be a type of immune system adaptation process to HIV-1. In the
early phase of the disease (0 to 60 days) (Fig. 2), approximately 6.0 % of TCD4+
lymphocyteswereinfected. Intheasymptomaticphase,thiscontentwasalmost8.4%.
In chronically infected TCD4+ lymphocytes the content was 2.0 % of the total lym-
phocyte population in the early phase and 0.15 % in asymptomatic phase. However,
ifweconsiderthedelayedinput,thechronicallyinfectedTCD4+lymphocytesplaya
specific (and unclear) role and act as a possible reservoir to HIV pathogenesis. The
formationoftheproductivelyinfectedTCD4+cells(Fig. 3)followstheprofilescited
in specific HIV literature. In this figure, the value reaches 55 cells µl−1 (in the early
phase) and 65 cells µl−1 during the asymptomatic phase and constant cytotoxic lym-
phocytepressure. Inevaluatingthebeginningofinfectionmorecarefully,thepeakof
infectedcellsreaches100cellsµl−1inatimeintervalthatisdependentorindependent
ofdelay. Theinfectedlymphocytesshowadeathrateofapproximately0.0067day−1
duetoCTLaction.TherelationshipbetweentheinfectedTCD4+andtheCTLcontent
isapproximately18%,similarto[34].
3.1. Macrophagecells
When active macrophages meet viral particles, there is an increase in these cells
atinflammationsitesduringearlyinfectionstages,andasmallproportionsufferfrom
infection.Themathematicalsimulationsshowedthat34%oftotalactivemacrophages
becomeinfected(almosttwo-foldthevaluefoundin[29]duringtheinstallationphase
ofHIVpathogenesis),andthislevelreaches43%duringtheasymptomaticphase.The
kineticsofmacrophageproliferationshowapseudo-steadystateininfectedmacrophage
dynamics(Fig.4),reachinganaveragecontentof18cellsµl−1.Consideringthatnearly
one-third of active macrophages become infected, its importance as a viral prolifera-
tionreservoirandinfectionmaintenanceissignificant.
Thedynamicalequilibrium,asdescribedin(8),ismodifiedbyCTLlymphocytepop-
ulationinterference. Theparameterk ,representingthecytotoxiclymphocyteactions
1
intheinfectedmacrophagecellcontrol,transfersalargesensibilitytosystemequilib-
rium,butthiscontrolisnotaspronouncedindendriticcells. Allthesecharacteristics
reflectthecontrolofinfectiondynamics.
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3.2. Dendriticcells
Thedendriticcellsshowasimilarprofiletothemacrophagepopulation.Duringthe
first60days(earlyinfectionphase),approximately95%ofmaturedendriticcellshave
some viral particles trapped on its receptors. After an initial period, we can imagine
thatalldendriticcellsderivedfromrecognizedHIVvirionsareinfected. Anequilib-
riumbaselineisestablished(correspondingtotheasymptomaticphase)atanaverage
concentrationof15cellsµl−1. Inthelastinfectionphase(Fig.5),thereisanexponen-
tialincreaseofinfectedcellsandviralproliferationrate[30]. Inanalyzingthekinetics
parametervaluesandthesimulationoutcomes,theamountofvirionsabsorbedpercell
is80virions.cell−1. Thisvalueisinagreementwiththerangeof100to180particles
percellshownbyMorisetal. [31]andHaaseetal. [32]. Thecontactraterangebe-
tweendendriticcellstrappingvirusandTCD4+lymphocytesis0.088[15]and0.0109
[31]. Thevalueofsaturationconstantκ wasobtainedfromsimulationoutcomes.
v
3.3. CTLlymphocytes
Themathematicalsimulationsshowthethreestagesofviralinfection(Fig.4): (a)
the initial equilibrium perturbation as the viral cycle and infection is established; (b)
theasymptomaticorchronicphase,inwhichapseudo-steadystateoccursinallcellular
compartments and the virus spreads posteriorly, and (c) the final phase, in which the
viral proliferation leads to depletion of the immune system. The CTL population is
not sufficient to eradicate the HIV presence or the infection propagation because of
either viral escape mechanisms or signaling cytokine depletion in the immunological
response.
AccordingtoRouzineetal.[33],therecouldbetwopossiblemechanismstocontrolthe
HIVspread,andbothrelatetotheviralloadlevel. Iftheviralloadislower,theeffort
of CTL lymphocytes is TCD4+ active concentration dependent. A higher viral load
isTCD4-independentbutantigen-presentingdependent. Thiscontrolmechanismcan
beverifiedin(8)andresultsfromamathematicalsimulation,inwhichtheTCD4/CTL
ratioreachesvaluesbetween2.67and1.85.Theviralescapephenomena,relatedtothe
inefficiencyofCTLcontrol,canbeobservedinFigs.1to5.
3.4. Infectionlevel
This study adapts the ”force of infection” and ”basic reproduction number” epi-
demiologicalmethodologies,whicharewellestablishedinepidemiologicalstudies,as
parametersofdiseasepropagationinasusceptiblepopulation. Inourstudy,theparam-
eters represent the intensity of the antigen-presenting process, produced by infected
macrophages and dendritic cells at infectious synapses, to nave TCD4+ lymphocytes
intheadaptiveimmuneresponse. Afterthepathogen-hostcomplexesareformed,HIV
is transferredto TCD4+ lymphocytes, whereits genetic materialis injected intohost
DNA, destroying it and infecting other susceptible cells. The progression of the dis-
easeisquantifiedbyanindexcalledBasicReproductionNumber,whichcorresponds
to the average number of secondary cases that a unique infected case will cause in a
susceptiblepopulationwithouttreatmentoranycontroloftheinfectiousprocess[12].
8
3.5. Basicreproductivenumbercalculation
AfterHIVpenetratesthetissuebarrier,itstartsacascadeofcellularandbiochem-
ical mechanisms represented by the model (1-9). The efficiency of the humoral and
adaptive response is represented in the matrix Σ (A.1, Appendix A), which shows
the probability of virus elimination (the first to fifth row) and the life expectancy of
HIV in human blood (last vector in the form of the clearance rate c). The matrix Λ
(A.2,AppendixA)representsthedeathratesofinfectedcellpopulations, suchasin-
fectedmacrophagesormaturedendriticcellsandproductivelyinfectedTCD4lympho-
cytes. Thelasttwovectorsarezerobecausetheyarerepresentativesofpopulationsof
CTLlymphocytesandviralload. Conceptually,thespectralradiusρ(K)isthelargest
absolutevalueofaneigenvalueofK(K=-Σ(s)Λ−1 whereρ(K))=sup|λ|;λ ∈ σ(K))
andisassociatedwithR ,accordingto(10).Mathematically,thespectralradiusrelated
0
tosomepositivematrixconsiderstheexpectednumbersofoffspringthatanaveragein-
dividualwouldproduceduringitslifetimeifthepopulationlingersatlowdensity.Each
(i, j)entryoftheNextGenerationMatrixK(A.4,AppendixA)isanexpectednumber
ofsecondaryinfectionsproducedincompartmentibyacaseinitiallyin j.
This concept is related to the balance between the production of new infections by
the group of infected or uninfected macrophages and dendritic cells on TCD4+ lym-
phocytes. It is also related to the control of viral proliferation (with their removal)
generated by HIV-specific CTL response. Herein, the constant antigen presentations
by HIV-infected macrophages impact new generations of viral infections and remain
importantreservoirstothecontinuityofHIVproliferationandT-celldepletion.
Byapplyingthenextgenerationmatrixmethodologyonmodel(1-9),theoutcomeis
f ξ (1− f)φ
R (τ)= Mpi(τ)+ mDC(τ). (10)
0 δ+α∗−0.0005 δ+α∗ +0.0005
lat
Equation(10)showsthatapossibletargettothedrug’sactionisfocusedontheτterm.
The sensitivity is seen in 11, showing that the depletion profile of TCD4+ lympho-
cytes depends on the viral infection in macrophages and dendritic cells and the level
ofimmunization. Additionally,inanalyzingthedestructionofallinfectedcells,such
asHIV-infectedmacrophages,thevariationofR hasalineardependenceonthedeath
0
rate of these cells. Thus, although the target cells used by drug therapy are TCD4+
lymphocytes, the presence of dendritic cells and macrophages working like antigen-
presenting cells still causes new infections, which maintain low levels of HIV virus.
However,anyinflammationprocesscouldtriggernewHIVinfectioncasesinlympho-
cytesresultinginviralloadelevationanddisseminationofthevirus.
Asensitivityanalysisin(10)ofeachparameterisshownby(andAppendixB),
∂R ξMpi(τ)
0 =−f (11a)
∂(δ+α∗) (δ+α∗)2
∂R ξMpi(τ)
0 =−f (11b)
∂(δ+α∗ ) (δ+α∗ )2
lat lat
∂R ξMpi(τ)
0 =−f (11c)
∂ξ (δ+α∗)−0.0005
9
∂R ξmDC(τ)
0 =−f (11d)
∂φ (δ+α∗ )+0.0005
lat
(cid:34) (cid:35)
∂R dτ ξ ∂Mpi(τ) φ ∂mDC(τ)
0 = f +(1− f) (11e)
∂τ dt δ+α∗ ∂τ δ+α∗ ∂τ
lat
Equations (11a) to (11e) show the relationship between the R index and the lym-
0
phocyte TCD4+ life expectancy, which notes that higher productively or chronically
infectedlymphocytedeathratesareassociatedwithlowerratesofinfectionduringthe
adaptiveimmuneresponse. Duringtheacute(orearly)stage,cellularlysesinducedby
HIVoccurandinfectedcellsdiebyCTLaction(11aand11b). Thus, ifthecell-host
dies,virussurvivorsarekilledbytheimmunesystem. However,thisdynamicequilib-
rium does not lower viral proliferation, and HIV-1 continues to infect and proliferate
(under mutant particles form) escaping the immune system and infecting new target-
cells. Thus,theviciouscyclerestarts.
After the acute phase, the viral load is defined by the HIV replication rate and its
clearance. Blipsofintenseviralreplicationarefoundduringthisperiodandaremost
likely caused by the inflammation process or opportunistic infections triggered by
macrophages (or dendritic cells) and lymphocyte cell interaction. Hence, if HIV-
infected macrophages (or dendritic cells) interact with nave TCD4+ and infect them,
startinganew viraldisseminationwave, theviralproliferation triggersotherinfected
TCD4+ lymphocytes. This process is presented in (11c) and (11d). Specific studies
[9, 14] cite that the viral population within the infected macrophages has a different
mutantprofilecomparedtothatfoundintheinfectedlymphocytesubpopulation,prob-
ablyduetodifferentselectivepressurestowhichHIV-1isexposed. Thisinformation
couldexplaintherapiddominanceofwild-typevirusaftertherapycessation(Equations
10and11).
4. Conclusion
Equation11erelatestheimpactofviraldissemination(e.g.,theinfectionlevel)and
the HIV replication cycle on infected macrophages and dendritic cells. It concludes
that the interaction between the HIV virus and these host cells seems to have a pro-
foundinfluenceontheexhaustionrateoftheimmunesystem.
Therelations f Ψ(a,c;t)ξand f Ψ(a,c;t)φshowtheefficiencyofeachcontactbetween
HIV-infected macrophages and dendritic cells, respectively, to lymphocyte T-cells in
theantigen-presentingprocess. Astheaξ andaφtermsrepresentthecontentofviral
particles that infect the host cell and f is the fraction of lymphocytes that are pro-
ductivelyinfected(theexpressioninparenthesesrepresentsthevirusthatsurvivesthe
immunesystemattack), pre-exposureprophylaxistreatmenttherapieswithfusion-or
ligand-inhibitordrugs,forexample,decreasethecontacttermsaξandaφ.Thisprocess
couldbeabarrierofviralentranceintotheintracellularhostenvironment. Infectious
contacts caused by HIV-infected macrophages and dendritic cells lead to new infec-
tionsinnaveTCD4+lymphocytecells,influencingnewimmuneresponsestoavoidvi-
ralproliferation,whichcouldresultinimmunereconstitutioninflammatorysyndrome.
TheR equationshowsthemechanismofimmuneresponsethatmayrelatetothetar-
0
getsofnewdrugsornewformsofpre-exposureprophylaxistreatment. Theimpactof
10