research article stochastic model for in-host hiv dynamics...

Hindawi Publishing CorporationISRN BiomathematicsVolume 2013 Article ID 103708 11 pageshttpdxdoiorg1011552013103708

Research ArticleStochastic Model for In-Host HIV Dynamicswith Therapeutic Intervention

Waema R Mbogo1 Livingstone S Luboobi2 and John W Odhiambo1

1 Center for Applied Research in Mathematical Sciences Strathmore University PO Box 59857 00200 Nairobi Kenya2Department of Mathematics Makerere University PO Box 7062 Kampala Uganda

Correspondence should be addressed to Waema R Mbogo rmbogostrathmoreedu

Received 22 February 2013 Accepted 27 March 2013

Academic Editors X-Y Lou and J Suehnel

Copyright copy 2013 Waema R Mbogo et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Untangling the dynamics between HIV and CD4 cellular populations and molecular interactions can be used to investigate theeffective points of interventions in theHIV life cycleWith that inmindwe propose and show the usefulness of a stochastic approachtowards modeling HIV and CD4 cellsrsquo dynamics in vivo by obtaining probability generating function the moment structures ofthe healthy CD4 cell and the virus particles at any time t and the probability of HIV clearance The unique feature is that boththerapy and the intracellular delay are incorporated into the model Our analysis shows that when it is assumed that the drug is notcompletely effective as is the case of HIV in vivo the probability of HIV clearance depends on two factors the combination of drugefficacy and length of the intracellular delay and also the education of the infected patients Comparing simulated data before andafter treatment indicates the importance of combined therapeutic intervention and intracellular delay in having low undetectableviral load in HIV-infected person

1 Introduction

Since HIV pandemic first became visible enormous math-ematical models have been developed to describe theimmunological response to infection with human immun-odeficiency virus (HIV) Mathematical modeling has provento be valuable in understanding the dynamics of infectiousdiseases with respect to host-pathogen interactions WhenHIV enters the body it targets all the cells with CD4 receptorsincluding the CD4 T cells The knowledge of principalmechanisms of viral pathogenesis namely the binding of theretrovirus to the gp120 protein on the CD4 cell the entry ofthe viral RNA into the target cell the reverse transaction ofviral RNA to viral DNA the integration of the viral DNAwiththat of the host and the action of viral protease in cleavingviral proteins into mature products has led to the design ofdrugs (chemotherapeutic agents) to control the production ofHIV

Chronic HIV-infection causes gradual depletion of theCD4 T-cell poll and thus progressively compromises thehostrsquos immune response to opportunistic infections leading

to acquired immunodeficiency syndrome (AIDS) [1] Withthe spread of the HIV-AIDS pandemic and in the absenceof an ldquoeffectiverdquo vaccine or cure therapeutic interventionsare still heavily relied on Several research studies havebeen carried out in the recent past both theoretically andexperimentally to analyse the impact of therapy on theviral load in HIV-infected persons in order to ascertainthe effectiveness of the treatment (see eg [2ndash8]) Theirutility lies in the ability to predict an infected steady stateand examining the effects that the changes in parametershave on the outcome of the system over time to determinewhich parameters are most important in disease progressionand further determine critical threshold values for theseparameters

In HIV-infected individuals the infection exhibits a longasymptomatic phase (after the initial high infectious phase)of approximately 10 years on average before the onset ofAIDS During this incubation period which some call theclinical latency period the individuals appear to be well andmay contribute significantly to the spread of the epidemicin a community [9] Some clinical markers such as the

2 ISRN Biomathematics

CD4 cell count and the RNA viral load (viraemia) provideinformation about the progression of the disease in infectedindividuals Also the clinical latency period of the diseasemay provide a sufficiently long period during which onecan attempt an effective suppressive therapeutic interventionin HIV infections Various biological reasons lead to theintroduction of time delays inmodels of disease transmissionTime delays are used to model the mechanisms in the diseasedynamics (see eg [10 11])

Intracellular delays and the target-cell dynamics such asmitosis are two key factors that play an important role inthe viral dynamics Mitosis in healthy or infected target-cell population are typically modelled by a logistic term[2 12ndash16] Intracellular delays have been incorporated intothe incidence term in finite or distributed form [4 17ndash23] In [13 16] in-host viral models with a logistic growthterm without intracellular delays are investigated and it isshown that sustained oscillations can occur through Hopfbifurcation when the intrinsic growth rate increases It isshown in [17 23] in in-host models with both a logisticgrowth term and intracellular delay that Hopf bifurcationscan occurwhen the intracellular delay increases In [19] usingin-host models with a general form of target-cell dynamicsand general distributions for intracellular delays it is shownthat the occurrence of Hopf bifurcation in these modelscritically depends on the form of target-cell dynamics Morespecifically it is proved in [20] that if the target-cell dynamicsare such that no Hopf bifurcations occur when delays areabsent introducing intracellular delays in the model will notlead to Hopf bifurcations or periodic oscillations

To incorporate the intracellular delay phase of the viruslife cycle [24] assumed that virus production occurs afterthe virus entry by a constant delay 120591 They came up witha basic in-host compartmental model of the viral dynamicscontaining three compartments 119909(119905) 119910(119905) and V(119905) denotingthe populations of uninfected target cells infected target cellsthat produce virus and free virus particles respectivelyTheyfurther assumed that parameters 120575120572 and120583 are turnover ratesof the 119909 119910 and V compartments respectively Uninfectedtarget cells are assumed to be produced at a constant rate120582 They assumed also that cells infected at time 119905 will beactivated and produce viral materials at time 119905 + 120591 In theirmodel constant 119904 is the death rate of infected but not yetvirus-producing cells and 119890minus119904120591 describes the probability ofinfected target cells surviving the period of intracellular delayfrom 119905 minus 120591 to 119905 Constant 120581 denotes the average number ofvirus particles that each infected cell produces Precedingassumptions lead to the following system of differentialequations

119889119909

119889119905

= 120582 minus 120575119909 (119905) minus 120573119909 (119905) V (119905)

119889119910

119889119905

= 120573119909 (119905 minus 120591) V (119905 minus 120591) 119890minus119904120591

minus 120572119910 (119905)

119889V

119889119905

= 120581119910 (119905) minus 120583V (119905)

(1)

System (1) can be used tomodel the infection dynamics ofHIV HBV and other viruses [2 12ndash14 25 26] It can also be

considered as a model for the HTLV-I infection if 119909(119905) 119910(119905)and V(119905) are regarded as healthy latently infected and activelyinfected CD4 T cells [14 15] For detailed description andderivation of the model as well as the incorporation ofintracellular delays we refer the reader to [24]

From the literature many researchers have employeddeterministic models to study HIV internal dynamics ignor-ing the stochastic effects We consider a stochastic modelfor the interaction of HIV virus and the immune systemin an HIV-infected individual undergoing a combination-therapeutic treatment Our aim in this paper is to usea stochastic model obtained by extending the model of[24] to determine the probability distribution variance andcovariance structures of the uninfected CD4+ cells infectedCD4 cells and the freeHIV particles in an infected individualat any time 119905 by examining the combined antiviral treatmentof HIV Based on the model we obtain joint probabilitydistribution expectations variance and covariance structuresof variables representing the numbers of uninfected CD4cells theHIV-infectedCD4T cells and the freeHIVparticlesat any time 119905 and derive conclusions for the reduction orelimination of HIV in HIV-infected individuals which is oneof the main contributions of this paper

The organization of this paper is as follows in Section 2we formulate our stochastic model describing the interac-tion of HIV and the immune system and obtain a partialdifferential equation for the probability generating functionof the numbers of uninfected CD4 cells the HIV-infectedCD4 T cells and the free HIV particles at any time 119905 alsomoments for the variables are derived here In Section 3we derive the moments of the variables in a therapeuticenvironment and probability of extinction of HIV virus andalso provide a numerical illustration to demonstrate theimpact of intercellular delay and therapeutic intervention incontrolling the progression ofHIV Some concluding remarksfollow in Section 4

2 HIV and CD4 Cells Dynamics beforeTherapeutic Intervention

To study the interaction of HIV virus and the immunesystem we propose a stochastic model by extending thedeterministic model presented in the literature A stochasticprocess is defined by the probabilities with which differentevents happen in a small time intervalΔ119905 In ourmodel thereare two possible events (production and deathremoval) foreach population (uninfected cells infected cells and the freevirons) The corresponding rates in the deterministic modelare replaced in the stochastic version by the probabilities thatany of these events occur in a small time interval Δ119905

(1) The Interaction of HIV Virus and the CD4 T-Cells Atypical life-cycle ofHIVvirus and immune system interactionis shown in Figure 1

Let 119883(119905) be the size of the healthy cells population attime 119905 119884(119905) be the size of infected cell population at time119905 and 119881(119905) be the size of the virons population at time119905 In the model to be formulated it is now assumed that

ISRN Biomathematics 3

Productively infected

Latently infected

120582

120575119909

target cells

120573119909(1 minus 119890minus120588120591

)

120573119909119890minus120588120591

120581119884

1205811198731119884

120581119873119884

Infectiousfree virus

Noninfectiousfree virus

+

120583119881

120574

119881

119883

119884

119881119899

120581119884119871

119884119871

120583119881119899

Uninfected CD4+

CD4+ cells

CD4+ cells

Figure 1 The interaction of HIV virus and the CD4+ T cells

instead of rates of births and deaths there is a possibilityof stochastic births or deaths of the heathy cells infectedcells and the virus particles Thus 119883(119905) 119884(119905) and 119881(119905) aretime-dependent randomvariablesThis epidemic process canbe modeled stochastically by letting the nonnegative integervalues process 119883(119905) 119884(119905) and 119881(119905) respectively representthe number of healthy cells infected cells and virons of thedisease at time 119905 Then (119883(119905) 119884(119905) 119881(119905)) 119905 ge 0 can bemodeled as continuous time multivariate Markov chain Letthe probability of there being 119909 healthy cells 119910 infected cellsand V virons in an infected person at time 119905 be denoted bythe following joint probability function 119875

119909119910V(119905) = 119875[119883(119905) =

119909 119884(119905) = 119910 119881(119905) = V] for 119909 119910 V = 0 1 2 3 The standard argument using the forward Chapman-

Kolmogorov differential equations is used to obtain thejoint probability function 119875

119909119910V(119905) by considering the jointprobability 119875

119909119910V(119905 119905 + Δ119905) This joint probability is obtainedas the sum of the probabilities of the following mutuallyexclusive events

(2) Population Change Scenarios Consider the followingpoints

(1) There were 119909 healthy cells 119910 infected cells and Vvirons by time 119905 and nothing happens during the timeinterval (119905 119905 + Δ119905)

(2) There were 119909 minus 1 healthy cells 119910 infected cells andV virons by time 119905 and one healthy cell is producedfrom the thymus during the time interval (119905 119905 + Δ119905)

(3) There were 119909 + 1 healthy cells 119910 infected cells and Vvirons by time 119905 and one healthy cell dies or is infectedby HIV virus during the time interval (119905 119905 + Δ119905)

(4) There were 119909 healthy cells 119910 minus 1 infected cells andV virons by time 119905 and one healthy cell is infected byHIV virus during the time interval (119905 119905 + Δ119905)

(5) There were 119909 healthy cells 119910 + 1 infected cells andV virons by time 119905 and one infected cell dies (HIV-infected cell bursts or undergoes a lysis) during thetime interval (119905 120591)

(6) There were 119909 healthy cells 119910 infected cells andV minus 1 virons by time 119905 and one viron is produced(HIV-infected cell undergoes a lysis or the individualangages in risky behaviour) during the time interval(119905 119905 + Δ119905)

(7) There were 119909 healthy cells 119910 infected cells and V + 1virons by time 119905 and one viron dies during the timeinterval (119905 119905 + Δ119905)

We incorporate a time delay between infection of a celland production of new virus particles we let 120591 be the time lagbetween the time the virus contacts a target CD4 T cell andthe time the cell becomes productively infected (includingthe steps of successful attachment of the virus to the cell andpenetration of virus into the cell) this means the recruitmentof virus producing cells at time 119905 is given by the density of cellsthat were newly infected at time 119905minus120591 and are still alive at time119905 If we also let 120588 be the death rate of infected but not yet virusproducing cell then the probability that the infected cell willsurvive to virus producing cell during the short time interval120591 will be given by 119890minus120588120591

(3) Variables and Parameters for the Model The variables andparameters in the model are described as in Table 1

Using the population change scenarios and parametersin Table 1 we now summarize the events that occur during


Table 1 (a) Variables for the stochastic model (b) Parameters forthe stochastic model

(a)

Variable Description Initial condition119905 = 0

119883(119905)The concentration of uninfectedCD4 cells at time 119905 100

119884(119905)The concentration of infected CD4cells at time 119905 002

119881(119905)The concentration of virus particlesat time 119905 0001

(b)

Parametersymbol Parameter description Estimate

(1 minus 120572)The reverse transcriptase inhibitordrug effect 05

(1 minus 120596) The protease inhibitor drug effect 05

120582The total rate of production of healthyCD4 cells 10

120575The per capita death rate of healthyCD4 cells 002

120573

The transmission coefficient betweenuninfected CD4 cells and infectivevirus particles

0000024

120581Per capita death rate of infected CD4cells 05

120574The virus production rate due to riskbehaviors 0001

120583The per capita death rate of infectivevirus particles 3

120588The death rate of infected but not yetvirus producing cell 05

120591 Time lag during infection 05

119873

The average number of infective virusparticles produced by an infectedCD4 cell in the absence of treatmentduring its entire infectious lifetime

1000

the interval (119905 119905 + Δ) together with their transition probabili-ties in Table 2

The change in population size during the time intervalΔ119905which is assumed to be sufficiently small to guarantee thatonly one such event can occur in (119905 119905 + Δ119905) is governed bythe following conditional probabilities

119875119909119910V (119905 + Δ119905)

= 1 minus (120582Δ119905 + 120575119909Δ119905 + 120573119909VΔ119905 + 120583VΔ119905 + 120581119910Δ119905 + 120574Δ119905)

+ 119900 (Δ119905) 119875119909119910V (119905)

+ 120582Δ119905 + 119900 (Δ119905) 119875119909minus1119910V (119905)

+ 120575 (119909 + 1) Δ119905 + 119900 (Δ119905) 119875119909+1119910V (119905)

+ 120573 (119909 + 1) (V + 1) 119890minus120588120591

Δ119905 + 119900 (Δ119905)

times 119875119909+1119910minus1V+1 (119905)

+ 120581119873 (119910 + 1) Δ119905 + 119900 (Δ119905) 119875119909119910+1Vminus1 (119905)

+ 120574Δ119905 + 119900 (Δ119905) 119875119909119910Vminus1 (119905)

+ 120583 (V + 1) Δ119905 + 119900 (Δ119905) 119875119909119910V+1 (119905)

(2)

Simplifying (2) we have the following forward Kolmogorovpartial differential equations for 119875

119909119910V(119905)

1198751015840

119909119910V (119905) = minus 120582 + 120575119909 + 120573119890minus120588120591

119909V + 120583V + 120581119873119910 + 120574 119875119909119910V (119905)

+ 120582119875119909minus1119910V (119905) + 120575 (119909 + 1) 119875119909+1119910V (119905)

+ 120573 (119909 + 1) (V + 1) 119890minus120588120591

119875119909+1119910minus1V+1 (119905)

+ 119873120581 (119910 + 1) 119875119909119910+1Vminus1 (119905) + 120574119875119909119910Vminus1 (119905)

+ 120583 (V + 1) 119875119909119910V+1 (119905)

(3)

This will also be referred to as the Master equation or theDifferential-Difference equation with the condition

1198751015840

000

(119905) = minus (120582 + 120574) 119875000

(119905) + 120575119875100

(119905) + 120583119875001

(119905) (4)

21 The Probability Generating Function The probabilitygenerating function of (119883(119905) 119884(119905) 119881(119905)) is defined by

119866 (1199111

1199112

1199113

119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

119911119910

2

119911V3

(5)

Differentiating (5) with respect to 119905 yields

120597119866 (1199111

1199112

1199113

119905)

120597119905

=

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

1198751015840

119909119910V (119905) 119911119909

1

119911119910

2

119911V3

(6)

Differentiating again (5) with respect to 1199111

1199112

and 1199113

yields

1205973

119866 (1199111

1199112

1199113

119905)

1205971199111

1205971199112

1205971199113

=

infin

sum

119909=1

infin

sum

119910=1

infin

sum

V=1

119909119910V119875119909119910V (119905) 119911

119909minus1

1

119911119910minus1

2

119911Vminus13

=

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

(119909 + 1) (119910 + 1) (V + 1) 119875119909+1119910+1V+1 (119905) 119911

119909

1

119911119910

2

119911V3

(7)

Multiplying (3) by 1199111199091

119911119910

2

119911V3

and summing over 119909119910 and V thenapplying (4) (5) (6) and (7) and on simplification we obtain

120597119866

120597119905

= (1199111

minus 1) 120582 + (1199113

minus 1) 120574119866

+ (1 minus 1199111

) 120575

120597119866

1205971199111

+ (1199113

minus 1199112

) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113

+ 120573119890minus120588120591

(1199112

minus 1199111

1199113

)

1205972

119866

1205971199111

1205971199113

(8)


Table 2 Transitions of in-host interaction of HIV

Possible transitions in host interaction of HIV and immune system cells and corresponding probabilities

Event Population(119883119884 119881) at 119905

Population(119883119884 119881) at (119905 119905 + Δ)

Probabilityof transition

Production of uninfected cell (119909 minus 1 119910 V) (119909 119910 V) 120582Δ119905

Death of uninfected cell (119909 + 1 119910 V) (119909 119910 V) 120575(119909 + 1)Δ119905

Infection of uninfected cell (119909 + 1 119910 minus 1 V + 1) (119909 119910 V) 120573(119909 + 1)(V + 1)119890minus120588120591Δ119905

Production of virons from the bursting infected cell (119909 119910 + 1 V minus 1) (119909 119910 V) 120581119873(119910 + 1)Δ119905

Introduction of virons due to reinfection because of risky behaviour (119909 119910 V minus 1) (119909 119910 V) 120574Δ119905

Death of virons (119909 119910 V + 1) (119909 119910 V) 120583(V + 1)Δ119905

This is called Lagrange partial differential equation for theprobability generating function (pgf)

Attempt to solve (8) gave us the following solution

119866 (1199111

1199112

1199113

119905)

= 119886119890119887119905

(1199111

minus 1) 120582 + (1199113

minus 1) 120574 119865 (1199111

1199112

1199113

)

+ (1 minus 1199111

) 120575

120597119865 (1199111

1199112

1199113

)

1205971199111

+ (1198731199113

minus 1199112

) 120581

120597119865 (1199111

1199112

1199113

)

1205971199112

+ (1 minus 1199113

) 120583

120597119865 (1199111

1199112

1199113

)

1205971199113

+ 120573 (119890minus120588120591

1199112

minus 1199111

1199113

)

1205972

119865 (1199111

1199112

1199113

)

1205971199111

1205971199113

(9)

where 119886 and 119887 are constantsEquation (9) is not solvable even for any simple form

of 119865(1199111

1199112

1199113

) However it is possible to obtain the momentstructure of (119883(119905) 119884(119905) 119881(119905)) from (8)

22 The Marginal Generating Functions Recall that

119866 (1199111

1 1 119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

(10)

Assuming that 1199112

= 1199113

= 1 1199111

= 1199113

= 1 and 1199112

= 1199111

=

1 and solving (8) we obtain the marginal partial generatingfunctions for119883(119905) 119884(119905) and 119881(119905) respectively

120597119866 (1199111

119905)

120597119905

=

120597119866 (1199111

1 1 119905)

120597119905

= (1199111

minus 1) 120582119866 + (1 minus 1199111

) 120575

120597119866

1205971199111

+ 120573119890minus120588120591

(1 minus 1199111

)

1205972

119866

1205971199111

1205971199113

120597119866 (1199112

119905)

120597119905

=

120597119866 (1 1199112

1 119905)

120597119905

= (1 minus 1199112

) 120581119873

120597119866

1205971199112

+ 120573119890minus120588120591

(1199112

minus 1)

1205972

119866

1205971199111

1205971199113

120597119866 (1199113

119905)

120597119905

=

120597119866 (1 1 1199113

119905)

120597119905

= (1199113

minus 1) 120574119866 + (1199113

minus 1) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113

+ 120573119890minus120588120591

(1 minus 1199113

)

1205972

119866

1205971199111

1205971199113

(11)

23 Numbers of CD4TCells and theVirons Aswe know fromprobability generating function

120597119866119909

120597119911

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(12)

Letting 119911 = 1 we have

120597119866119909

120597119911

10038161003816100381610038161003816100381610038161003816119911=1

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

= 119864 [119883] (13)

Differentiating the partial differential equations of the pgf weget the moments of119883(119905) 119884(119905) and 119881(119905)


Differentiating (11) with respect to 1199111

1199112

and 1199113

respec-tively and setting 119911

1

= 1199112

= 1199113

= 1 we have120597

120597119905

119864 [119883 (119905)] = 120582 minus 120575119864 [119883 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119884 (119905)] = minus120581119864 [119884 (119905)] + 120573119890minus120588120591

119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)]

= 120574 + 119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

(14)

Therefore the moments of (119883(119905) 119884(119905) 119881(119905)) from the pgfbefore treatment are as follows

120597

120597119905

119864 [119883 (119905)] = 120582 minus 120575119864 [119883 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119884 (119905)] = minus120581119864 [119884 (119905)] + 120573119890minus120588120591

119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)]

= 120574 + 119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

(15)

The corresponding deterministic model of HIV-host interac-tion as formulated in [24] is given as follows

119889119909

119889119905

= 120582 minus 120575119909 (119905) minus 120573119909 (119905) V (119905)

119889119910

119889119905

= 120573119890minus120588120591

119909 (119905 minus 120591) V (119905 minus 120591) minus 120581119910 (119905)

119889V

119889119905

= 120574 + 119873120581119910 (119905) minus 120583V (119905) minus 120573119909 (119905) V (119905)

(16)

Comparing system of (15) with the system of (16) we seethat expected values of the multivariate Markov process[119883(119905) 119884(119905) 119881(119905)] satisfies the corresponding deterministicmodel of the HIV-host interaction dynamics

231 Simulation Using the parameter values and initial con-ditions defined in Table 1 we illustrate the general dynamicsof the CD4-T cells and HIV virus for model (15) duringinfection and in the absence of treatment

The second graph in Figure 2 is the population dynamicsafter taking the logarithm of the cell populations in the firstgraph

From the simulations it is clear that in the primarystage of the infection (period before treatment) a dramaticdecrease in the level of the CD4-T cells occurs and thenumber of the free virons increases with time With theintroduction of intracellular delay the virus population dropsas well as an increase in the CD4 cells but then they stabilizeat some point and coexist in the host as shown in Figure 2

3 HIV and CD4 Cells Dynamics underTherapeutic Intervention

Assume that at time 119905 = 0 a combination therapy treatmentis initiated in an HIV-infected individual We assume that

the therapeutic intervention inhibits either the enzyme actionof reverse transcriptase or that of the protease of HIV in anHIV-infected cell An HIV-infected cell with the inhibitedHIV-transcriptase may be considered a dead cell as it cannotparticipate in the production of the copies of any type ofHIV On the other hand an HIV-infected cell in whichthe reverse transcription has already taken place and theviral DNA is fused with the DNA of the host but theenzyme activity of HIV-protease is inhibited undergoes alysis releasing infectious free HIV and noninfectious freeHIV A noninfectious free HIV cannot successfully infecta CD4 cell Accordingly at any time 119905 the blood of theinfected person contains virus producing HIV-infected cellsinfectious free HIV and noninfectious free HIV A typicallife cycle of HIV virus and immune system interaction withtherapeutic intervention is shown in Figure 3

Introducing the effect of treatment the Lagrange partialdifferential equation becomes

120597119866

120597119905

= (1199111

minus 1) 120582 + (1199113

minus 1) 120574119866 + (1 minus 1199111

) 120575

120597119866

1205971199111

+ (1 minus 120596) (1199113

minus 1199112

) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113

+ (1 minus 120572) 120573119890minus(1minus120572)120588120591

(1199112

minus 1199111

1199113

)

1205972

119866

1205971199111

1205971199113

(17)


119866 (1199111

1 1 119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

(18)

Setting 1199112

= 1199113

= 1 1199111

= 1199113

= 1 and 1199112

= 1199111

= 1 and solving(17) we obtain the marginal partial generating functions for119883(119905) 119884(119905) and 119881(119905) respectively as

120597119866 (1199111

119905)

120597119905

=

120597119866 (1199111

1 1 119905)

120597119905

= (1199111

minus 1) 120582119866 + (1 minus 1199111

) 120575

120597119866

1205971199111


(1 minus 1199111

)

1205972

119866

1205971199111

1205971199113

120597119866 (1199112

119905)

120597119905

=

120597119866 (1 1199112

1 119905)

120597119905

= (1 minus 120596) (119873 minus 1199112

) 120581

120597119866

1205971199112


(1199112

minus 1)

1205972

119866

1205971199111

1205971199113


100

9

8

7

6

5

4

3

2

1

0 50 100 150 200

50 100 150 200 250

50 100 150 200 250500400300200

Time (days)

Time (days)

Time (days)

Time (days)

8

6

4

2

0

120591 = 0

120591 = 2120591 = 15

120591 = 1

119883

119884

119881

minus2

minus4

minus6

6

4

2

0

minus2

minus4

minus6

minus8

minus10

6

4

2

0

minus2

minus4

minus6

119883

119884

119881

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Figure 2 The figure shows the CD4 cells and HIV virus population dynamics before therapeutic intervention and when 120591 = 0 15 and 2days The other parameters and initial conditions are given in Tables 1 and 2

120597119866 (1199113

119905)

120597119905

=

120597119866 (1 1 1199113

119905)

120597119905

= (1199113

minus 1) 120574119866 + (1 minus 120596) (1199113

minus 1) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113


(1 minus 1199113

)

1205972

119866

1205971199111

1205971199113

(19)

32 Numbers of CD4+ T Cells and Virons under TherapeuticIntervention From probability generating function one gets

120597119866119909

120597119911

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(20)

Letting 119911 = 1 we have the expected number of target CD4+T cells119883(119905) as follows

119864 [119883 (119905)] =

120597119866119909

120597119911

10038161003816100381610038161003816100381610038161003816119911=1

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(21)

Differentiating the partial differential equations of thepgf we get the moments of119883(119905) 119884(119905) and 119881(119905)


1199112

and 1199113


1

= 1199112

= 1199113

= 1 we have

120597

120597119905

119864 [119883 (119905)] = 120582 minus 120575119864 [119883 (119905)] minus (1 minus 120572) 120573119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119884 (119905)] = minus120581119864 [119884 (119905)] + (1 minus 120572) 120573119890minus(1minus120572)120588120591

119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)]

minus (1 minus 120572) 120573119864 [119883 (119905) 119881 (119905)]

(22)



Latently infected

target cells


+

Infectiousfree virus Protease

inhibitorsViral clearance

Viral clearance

Viral synthesis

Cell synthesisCell death

Cell death

Cell death

Infected cell synthesis

Reversetranscriptase

inhibitors

CD4+ cells

CD4+ cells

Uninfected CD4+

Figure 3 HIV-host interaction with treatment

Therefore the moments of (119883(119905) 119884(119905) 119881(119905)) from the pgfwith therapeutic intervention are

120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)]

minus 120583119864 [119881 (119905)] minus (1 minus 120572) 120573119864 [119883 (119905) 119881 (119905)]

(23)

321 Simulation Using the parameter values and initial con-ditions defined in Table 1 we illustrate the general dynamicsof the CD4-T cells and HIV virus for model (23) withtherapeutic intervention and test the effect of intracellulardelay and drug efficacy (perform sensitivity analysis)

With 60 drug efficacy the virus stabilizes and coexistswith the CD4 cells within the host as shown in the first graphin Figure 4

Introducing intracellular time delay 120591 = 05 the CD4cells increase and the virus population drops then stabilizesand again coexists within the host as illustrated in the secondgraph in Figure 4

For 120591 = 1 and with 60 drug efficacy the dynamicschange as shown in the last graph in Figure 4

We now analyze drug effectiveness on the cell popula-tions

Figure 5 shows that with improved HIV drug efficacy apatient on treatment will have low undetectable viral loadwith time It also shows that the protease inhibitor drugefficacy is very important in clearing the virus Also the figureshows that a combination of therapeutic intervention andintracellular delay is very important in lowering the viral loadin an HIV-infected person

33 The Probability of HIV Clearance We now calculate thedistribution of times to extinction for the virons reservoirIf we assume the drug given at fixed intervals to sustain itsefficacy to be effective in producing noninfectious virus thenwe can clear the infectious virus Using reverse transcriptase(ART) the infected cell can be forced to become latentlyinfected (nonvirus producing cell) hence clearing the virusproducing cell reservoir If we assume that no newly infectedcells become productively infected in the short time 120591(119890minus120588120591 =0) or that treatment is completely effective (120596 = 1) we canobtain the extinction probability analytically In this case theinfectious virus cell dynamics decouples from the rest of themodel and can be represented as a pure immigration-and-death process with master equation

1198751015840

V (119905) = minus (120574 + 120581 (1 minus 120596)119873)119875V (119905)

minus V (120583 + (1 minus 120572) 119890minus120588120591

120573) 119875V (119905)

+ (V minus 1) (120574 + 120581 (1 minus 120596)119873)119875Vminus1 (119905)

+ (V + 1) (120583 + (1 minus 120572) 119890minus120588120591

120573) 119875V+1 (119905)

(24)


2000

1500

1000

500

0

200

100

300

400

0

0

0 200 400 600 800 1000

0 200

200

400

400

600

600

800

800

1000 0 200 400 600 800 1000

Cel

l pop

ulat

ions

Cel

l pop

ulat

ions

Cel

l pop

ulat

ions

120591 = 0 with 60 drug efficacy

Time (days) Time (days)

Time (days)



119883

119884

119881

Figure 4The figure shows the CD4 cells and HIV virus population dynamics with drug efficacy of 60 and 120591 = 0 05 and 1 daysThe otherparameters and initial conditions are given in Tables 1 and 2

where 119875V(119905) is the probability that at time 119905 there are VHIVparticlesThis probability has the conditional probabilitygenerating function

119866V (119911 119905) |119881(0)=V0

= exp120579120598

(119911 minus 1) (1 minus 119890minus120579119905

) 119911119890minus120579119905

+ 1 minus 119890minus120579119905

V0

(25)

where V0

is the initial virus reservoir size 120579 = 120583+(1minus120572)120573119890minus120588120591and 120598 = 120574 + (1 minus 120596)120581119873 We noticed that this is a pgf ofa random variable with a product of Poisson distributionexp(120579120598)(119911 minus 1)(1 minus 119890minus120579119905) with mean (120579120598)(1 minus 119890minus120579119905) and abinomial distribution 119911119890minus120579119905 + 1 minus 119890minus120579119905V0

The probability of this population going extinct at 119905 orbefore time 119905 is given by

119901ext (119905) |119881(0)=V0 = 119875 (119881 = 0 119905) |119881(0)=V0 = 119866V (0 119905) = 1198750 (119905)

= exp120579120598

(119890minus120579119905

minus 1) 1 minus 119890minus120579119905

V0

(26)

The probability of extinction is the value of119866V(119911 119905)when 119905 rarrinfin that is

119901ext = 1198750 (infin)

= lim119905rarrinfin

(exp120579120598

(119890minus120579119905


V0)

= 119890minus120579120598

= 119890minus((120583+(1minus120572)120573119890

minus120588120591)(120574+(1minus120596)120581119873))

(27)

From (27) it is evident that probability of clearance of HIV isaffected by the combination of drug efficacy and intracellulardelay death of the virons and education (counseling the HIVpatients on the risk of involvement in risk behaviors)

4 Discussion and Recommendation

In this study we derived and analyzed a stochastic model forin-host HIV dynamics that included combined therapeutictreatment and intracellular delay between the infection of acell and the emission of viral particles This model includeddynamics of three compartmentsmdashthe number of healthyCD4 cells the number of infected CD4 cells and the HIVvironsmdashand it described HIV infection of CD4 T cells before


200

100

300

400

00 200 400 600 800 1000

Cel

l pop

ulat

ions

200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

500

0

Cel

l pop

ulat

ions

Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

119883

119884

119881

119883

119884

119881

Reverse transcriptase inhibitor drug efficacy is more effective Protease inhibitor drug efficacy is more effective

65 drug efficacy with 120591 = 05 Drug efficacy above 65

Figure 5 Shows theCD4 cells andHIV virus population dynamicswith 120591 = 05 days and different levels of drug efficacyTheother parametersand initial conditions are given in Tables 1 and 2

and during therapy We derived equations for the probabilitygenerating a function and using numerical techniques In thispaper we showed the usefulness of our stochastic approachtowards modeling HIV dynamics by obtaining momentstructures of the healthy CD4+ cell and the virus particlesover time 119905 We simulated the mean number of the healthyCD4 cell the infected cells and the virus particles before andafter combined therapeutic treatment at any time 119905 We willemphasize further usefulness of stochastic models in HIVdynamics in our future research

Our analysis during treatment shows that when it isassumed that the drug is not completely effective as it is thecase of HIV in vivo the predicted rate of decline in plasmaHIV virus concentration depends on three factors the deathrate of the virons the efficacy of therapy and the length ofthe intracellular delay Our model produces an interestingfeature that the successfully treated HIV patients will havelow undetectable viral load We conclude that to control theconcentrations of the virus and the infected cells in HIV-infected person a strategy should aim to improve the curerate (drug efficacy) and also to increase the intracellular delay120591 Therefore the efficacy of the protease inhibitor and the

reverse transcriptase inhibitor and also the intracellular delayplay crucial role in preventing the progression of HIV

The extinction probability model showed that the time ittakes to have low undetectable viral load (infectious virus)in an HIV-infected patient depends on the combination ofdrug efficacy and intracellular delay and also education of theinfected patients In our work the dynamics of mutant viruswere not considered and also our study included dynamicsof only three compartments (healthy CD4 cell infected CD4cells and infectious HIV virus particles) of which extensionsare recommended for further extensive research In a follow-up work we intend to obtain real data in order to test theefficacy of our models as we have done here with simulateddata

References

[1] M Yan and Z Xiang ldquoA delay-differential equation model ofHIV infection of CD4+ T-cells with cure raterdquo InternationalMathematical Forum vol 7 pp 1475ndash1481 2012

[2] A S Perelson and PW Nelson ldquoMathematical analysis of HIV-1 dynamics in vivordquo SIAM Review vol 41 no 1 pp 3ndash44 1999


[3] D E Kirschner ldquoUsing mathematics to understand HIVimmune dynamicsrdquoMathematical Reviews vol 43 pp 191ndash2021996

[4] A S Perelson A U Neumann M Markowitz J M Leonardand D D Ho ldquoHIV-1 dynamics in vivo virion clearance rateinfected cell life-span and viral generation timerdquo Science vol271 no 5255 pp 1582ndash1586 1996

[5] JWMellors AMunoz J V Giorgi et al ldquoPlasma viral load andCD4+ lymphocytes as prognostic markers of HIV-1 infectionrdquoAnnals of Internal Medicine vol 126 pp 946ndash954 1997

[6] M Nijhuis C A B Boucher P Schipper T Leitner R Schu-urman and J Albert ldquoStochastic processes strongly influenceHIV-1 evolution during suboptimal protease-inhibitor therapyrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 95 no 24 pp 14441ndash14446 1998

[7] W Y Tan and Z Xiang ldquoSome state space models of HIVpathogenesis under treatment by anti-viral drugs in HIV-infected individualsrdquoMathematical Biosciences vol 156 no 1-2pp 69ndash94 1999

[8] D R Bangsberg T C Porco C Kagay et al ldquoModeling theHIV protease inhibitor adherence-resistance curve by use ofempirically derived estimatesrdquo Journal of Infectious Diseasesvol 190 no 1 pp 162ndash165 2004

[9] S S Y Venkata M O L Morire U Swaminathan and MYeko ldquoA stochastic model of the dynamics of HIV under acombination therapeutic interventionrdquo Orion vol 25 pp 17ndash30 2009

[10] P van denDriessche and JWatmough ldquoReproduction numbersand sub-threshold endemic equilibria for compartmental mod-els of disease transmissionrdquoMathematical Biosciences vol 180pp 29ndash48 2002

[11] O Arino M L Hbid and A E Dads Delay DifferentialEquations andApplications Series II Springer BerlinGermany2006

[12] A S Perelson D E Kirschner and R De Boer ldquoDynamics ofHIV infection of CD4+ T cellsrdquo Mathematical Biosciences vol144 no 1 pp 81ndash125 1993

[13] P De Leenheer and H L Smith ldquoVirus dynamics a globalanalysisrdquo SIAM Journal on Applied Mathematics vol 63 no 4pp 1313ndash1327 2003

[14] M A Nowak and C R M Bangham ldquoPopulation dynamics ofimmune responses to persistent virusesrdquo Science vol 272 no5258 pp 74ndash79 1996

[15] L Wang and M Y Li ldquoMathematical analysis of the globaldynamics of a model for HIV infection of CD4+ T cellsrdquoMathematical Biosciences vol 200 no 1 pp 44ndash57 2006

[16] L Wang and S Ellermeyer ldquoHIV infection and CD4+ T celldynamicsrdquo Discrete and Continuous Dynamical Systems B vol6 no 6 pp 1417ndash1430 2006

[17] R V Culshaw and S Ruan ldquoA delay-differential equationmodelof HIV infection of CD4+ T-cellsrdquo Mathematical Biosciencesvol 165 no 1 pp 27ndash39 2000

[18] A V M Herz S Bonhoeffer R M Anderson R M Mayand M A Nowak ldquoViral dynamics in vivo limitations onestimates of intracellular delay and virus decayrdquo Proceedings ofthe National Academy of Sciences of the United States of Americavol 93 no 14 pp 7247ndash7251 1996

[19] M Y Li and H Shu ldquoImpact of intracellular delays and target-cell dynamics on in vivo viral infectionsrdquo SIAM Journal onApplied Mathematics vol 70 no 7 pp 2434ndash2448 2010

[20] M Y Li and H Shu ldquoMultiple stable periodic oscillations ina mathematical model of CTL response to HTLV-I infectionrdquoBulletin of Mathematical Biology vol 73 no 8 pp 1774ndash17932011

[21] PWNelson J DMurray andA S Perelson ldquoAmodel ofHIV-1pathogenesis that includes an intracellular delayrdquoMathematicalBiosciences vol 163 no 2 pp 201ndash215 2000

[22] PW Nelson and A S Perelson ldquoMathematical analysis of delaydifferential equation models of HIV-1 infectionrdquoMathematicalBiosciences vol 179 no 1 pp 73ndash94 2002

[23] Y Wang Y Zhou J Wu and J Heffernan ldquoOscillatory viraldynamics in a delayed HIV pathogenesis modelrdquoMathematicalBiosciences vol 219 no 2 pp 104ndash112 2009

[24] M Y Li andH Shu ldquoGlobal dynamics of an in-host viral modelwith intracellular delayrdquo Bulletin of Mathematical Biology vol72 no 6 pp 1492ndash1505 2010

[25] M A Nowak and R M May Virus Dynamics CambridgeUniversity Press Cambridge UK 2000

[26] H C Tuckwell and F Y M Wan ldquoOn the behavior of solutionsin viral dynamical modelsrdquo BioSystems vol 73 no 3 pp 157ndash161 2004

Submit your manuscripts athttpwwwhindawicom

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


CD4 cell count and the RNA viral load (viraemia) provideinformation about the progression of the disease in infectedindividuals Also the clinical latency period of the diseasemay provide a sufficiently long period during which onecan attempt an effective suppressive therapeutic interventionin HIV infections Various biological reasons lead to theintroduction of time delays inmodels of disease transmissionTime delays are used to model the mechanisms in the diseasedynamics (see eg [10 11])

Intracellular delays and the target-cell dynamics such asmitosis are two key factors that play an important role inthe viral dynamics Mitosis in healthy or infected target-cell population are typically modelled by a logistic term[2 12ndash16] Intracellular delays have been incorporated intothe incidence term in finite or distributed form [4 17ndash23] In [13 16] in-host viral models with a logistic growthterm without intracellular delays are investigated and it isshown that sustained oscillations can occur through Hopfbifurcation when the intrinsic growth rate increases It isshown in [17 23] in in-host models with both a logisticgrowth term and intracellular delay that Hopf bifurcationscan occurwhen the intracellular delay increases In [19] usingin-host models with a general form of target-cell dynamicsand general distributions for intracellular delays it is shownthat the occurrence of Hopf bifurcation in these modelscritically depends on the form of target-cell dynamics Morespecifically it is proved in [20] that if the target-cell dynamicsare such that no Hopf bifurcations occur when delays areabsent introducing intracellular delays in the model will notlead to Hopf bifurcations or periodic oscillations

To incorporate the intracellular delay phase of the viruslife cycle [24] assumed that virus production occurs afterthe virus entry by a constant delay 120591 They came up witha basic in-host compartmental model of the viral dynamicscontaining three compartments 119909(119905) 119910(119905) and V(119905) denotingthe populations of uninfected target cells infected target cellsthat produce virus and free virus particles respectivelyTheyfurther assumed that parameters 120575120572 and120583 are turnover ratesof the 119909 119910 and V compartments respectively Uninfectedtarget cells are assumed to be produced at a constant rate120582 They assumed also that cells infected at time 119905 will beactivated and produce viral materials at time 119905 + 120591 In theirmodel constant 119904 is the death rate of infected but not yetvirus-producing cells and 119890minus119904120591 describes the probability ofinfected target cells surviving the period of intracellular delayfrom 119905 minus 120591 to 119905 Constant 120581 denotes the average number ofvirus particles that each infected cell produces Precedingassumptions lead to the following system of differentialequations

119889119909

119889119905

= 120582 minus 120575119909 (119905) minus 120573119909 (119905) V (119905)

119889119910

119889119905

= 120573119909 (119905 minus 120591) V (119905 minus 120591) 119890minus119904120591

minus 120572119910 (119905)

119889V

119889119905

= 120581119910 (119905) minus 120583V (119905)

(1)

System (1) can be used tomodel the infection dynamics ofHIV HBV and other viruses [2 12ndash14 25 26] It can also be

considered as a model for the HTLV-I infection if 119909(119905) 119910(119905)and V(119905) are regarded as healthy latently infected and activelyinfected CD4 T cells [14 15] For detailed description andderivation of the model as well as the incorporation ofintracellular delays we refer the reader to [24]

From the literature many researchers have employeddeterministic models to study HIV internal dynamics ignor-ing the stochastic effects We consider a stochastic modelfor the interaction of HIV virus and the immune systemin an HIV-infected individual undergoing a combination-therapeutic treatment Our aim in this paper is to usea stochastic model obtained by extending the model of[24] to determine the probability distribution variance andcovariance structures of the uninfected CD4+ cells infectedCD4 cells and the freeHIV particles in an infected individualat any time 119905 by examining the combined antiviral treatmentof HIV Based on the model we obtain joint probabilitydistribution expectations variance and covariance structuresof variables representing the numbers of uninfected CD4cells theHIV-infectedCD4T cells and the freeHIVparticlesat any time 119905 and derive conclusions for the reduction orelimination of HIV in HIV-infected individuals which is oneof the main contributions of this paper

The organization of this paper is as follows in Section 2we formulate our stochastic model describing the interac-tion of HIV and the immune system and obtain a partialdifferential equation for the probability generating functionof the numbers of uninfected CD4 cells the HIV-infectedCD4 T cells and the free HIV particles at any time 119905 alsomoments for the variables are derived here In Section 3we derive the moments of the variables in a therapeuticenvironment and probability of extinction of HIV virus andalso provide a numerical illustration to demonstrate theimpact of intercellular delay and therapeutic intervention incontrolling the progression ofHIV Some concluding remarksfollow in Section 4

2 HIV and CD4 Cells Dynamics beforeTherapeutic Intervention

To study the interaction of HIV virus and the immunesystem we propose a stochastic model by extending thedeterministic model presented in the literature A stochasticprocess is defined by the probabilities with which differentevents happen in a small time intervalΔ119905 In ourmodel thereare two possible events (production and deathremoval) foreach population (uninfected cells infected cells and the freevirons) The corresponding rates in the deterministic modelare replaced in the stochastic version by the probabilities thatany of these events occur in a small time interval Δ119905

(1) The Interaction of HIV Virus and the CD4 T-Cells Atypical life-cycle ofHIVvirus and immune system interactionis shown in Figure 1

Let 119883(119905) be the size of the healthy cells population attime 119905 119884(119905) be the size of infected cell population at time119905 and 119881(119905) be the size of the virons population at time119905 In the model to be formulated it is now assumed that



Latently infected

120582

120575119909

target cells

120573119909(1 minus 119890minus120588120591

)

120573119909119890minus120588120591

120581119884

1205811198731119884

120581119873119884



+

120583119881

120574

119881

119883

119884

119881119899

120581119884119871

119884119871

120583119881119899

Uninfected CD4+

CD4+ cells

CD4+ cells



119909119910V(119905) = 119875[119883(119905) =


















(a)





(b)






120573


0000024






119873


1000



119875119909119910V (119905 + Δ119905)

= 1 minus (120582Δ119905 + 120575119909Δ119905 + 120573119909VΔ119905 + 120583VΔ119905 + 120581119910Δ119905 + 120574Δ119905)

+ 119900 (Δ119905) 119875119909119910V (119905)

+ 120582Δ119905 + 119900 (Δ119905) 119875119909minus1119910V (119905)

+ 120575 (119909 + 1) Δ119905 + 119900 (Δ119905) 119875119909+1119910V (119905)

+ 120573 (119909 + 1) (V + 1) 119890minus120588120591

Δ119905 + 119900 (Δ119905)

times 119875119909+1119910minus1V+1 (119905)

+ 120581119873 (119910 + 1) Δ119905 + 119900 (Δ119905) 119875119909119910+1Vminus1 (119905)

+ 120574Δ119905 + 119900 (Δ119905) 119875119909119910Vminus1 (119905)

+ 120583 (V + 1) Δ119905 + 119900 (Δ119905) 119875119909119910V+1 (119905)

(2)


119909119910V(119905)

1198751015840

119909119910V (119905) = minus 120582 + 120575119909 + 120573119890minus120588120591

119909V + 120583V + 120581119873119910 + 120574 119875119909119910V (119905)

+ 120582119875119909minus1119910V (119905) + 120575 (119909 + 1) 119875119909+1119910V (119905)

+ 120573 (119909 + 1) (V + 1) 119890minus120588120591

119875119909+1119910minus1V+1 (119905)

+ 119873120581 (119910 + 1) 119875119909119910+1Vminus1 (119905) + 120574119875119909119910Vminus1 (119905)

+ 120583 (V + 1) 119875119909119910V+1 (119905)

(3)


1198751015840

000

(119905) = minus (120582 + 120574) 119875000

(119905) + 120575119875100

(119905) + 120583119875001

(119905) (4)


119866 (1199111

1199112

1199113

119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

119911119910

2

119911V3

(5)


120597119866 (1199111

1199112

1199113

119905)

120597119905

=

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

1198751015840

119909119910V (119905) 119911119909

1

119911119910

2

119911V3

(6)


1199112

and 1199113

yields

1205973

119866 (1199111

1199112

1199113

119905)

1205971199111

1205971199112

1205971199113

=

infin

sum

119909=1

infin

sum

119910=1

infin

sum

V=1

119909119910V119875119909119910V (119905) 119911

119909minus1

1

119911119910minus1

2

119911Vminus13

=

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

(119909 + 1) (119910 + 1) (V + 1) 119875119909+1119910+1V+1 (119905) 119911

119909

1

119911119910

2

119911V3

(7)


119911119910

2

119911V3


120597119866

120597119905

= (1199111

minus 1) 120582 + (1199113

minus 1) 120574119866

+ (1 minus 1199111

) 120575

120597119866

1205971199111

+ (1199113

minus 1199112

) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113

+ 120573119890minus120588120591

(1199112

minus 1199111

1199113

)

1205972

119866

1205971199111

1205971199113

(8)





Population(119883119884 119881) at (119905 119905 + Δ)










119866 (1199111

1199112

1199113

119905)

= 119886119890119887119905

(1199111

minus 1) 120582 + (1199113

minus 1) 120574 119865 (1199111

1199112

1199113

)

+ (1 minus 1199111

) 120575

120597119865 (1199111

1199112

1199113

)

1205971199111

+ (1198731199113

minus 1199112

) 120581

120597119865 (1199111

1199112

1199113

)

1205971199112

+ (1 minus 1199113

) 120583

120597119865 (1199111

1199112

1199113

)

1205971199113

+ 120573 (119890minus120588120591

1199112

minus 1199111

1199113

)

1205972

119865 (1199111

1199112

1199113

)

1205971199111

1205971199113

(9)


of 119865(1199111

1199112

1199113



119866 (1199111

1 1 119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

(10)


= 1199113

= 1 1199111

= 1199113

= 1 and 1199112

= 1199111

=


120597119866 (1199111

119905)

120597119905

=

120597119866 (1199111

1 1 119905)

120597119905

= (1199111

minus 1) 120582119866 + (1 minus 1199111

) 120575

120597119866

1205971199111

+ 120573119890minus120588120591

(1 minus 1199111

)

1205972

119866

1205971199111

1205971199113

120597119866 (1199112

119905)

120597119905

=

120597119866 (1 1199112

1 119905)

120597119905

= (1 minus 1199112

) 120581119873

120597119866

1205971199112

+ 120573119890minus120588120591

(1199112

minus 1)

1205972

119866

1205971199111

1205971199113

120597119866 (1199113

119905)

120597119905

=

120597119866 (1 1 1199113

119905)

120597119905

= (1199113

minus 1) 120574119866 + (1199113

minus 1) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113

+ 120573119890minus120588120591

(1 minus 1199113

)

1205972

119866

1205971199111

1205971199113

(11)


120597119866119909

120597119911

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(12)


120597119866119909

120597119911

10038161003816100381610038161003816100381610038161003816119911=1

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

= 119864 [119883] (13)




1199112

and 1199113


1

= 1199112

= 1199113

= 1 we have120597

120597119905

119864 [119883 (119905)] = 120582 minus 120575119864 [119883 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119884 (119905)] = minus120581119864 [119884 (119905)] + 120573119890minus120588120591

119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)]

= 120574 + 119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

(14)


120597

120597119905

119864 [119883 (119905)] = 120582 minus 120575119864 [119883 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119884 (119905)] = minus120581119864 [119884 (119905)] + 120573119890minus120588120591

119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)]

= 120574 + 119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

(15)


119889119909

119889119905

= 120582 minus 120575119909 (119905) minus 120573119909 (119905) V (119905)

119889119910

119889119905

= 120573119890minus120588120591


119889V

119889119905

= 120574 + 119873120581119910 (119905) minus 120583V (119905) minus 120573119909 (119905) V (119905)

(16)









120597119866

120597119905

= (1199111

minus 1) 120582 + (1199113

minus 1) 120574119866 + (1 minus 1199111

) 120575

120597119866

1205971199111

+ (1 minus 120596) (1199113

minus 1199112

) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113


(1199112

minus 1199111

1199113

)

1205972

119866

1205971199111

1205971199113

(17)


119866 (1199111

1 1 119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

(18)

Setting 1199112

= 1199113

= 1 1199111

= 1199113

= 1 and 1199112

= 1199111


120597119866 (1199111

119905)

120597119905

=

120597119866 (1199111

1 1 119905)

120597119905

= (1199111

minus 1) 120582119866 + (1 minus 1199111

) 120575

120597119866

1205971199111


(1 minus 1199111

)

1205972

119866

1205971199111

1205971199113

120597119866 (1199112

119905)

120597119905

=

120597119866 (1 1199112

1 119905)

120597119905

= (1 minus 120596) (119873 minus 1199112

) 120581

120597119866

1205971199112


(1199112

minus 1)

1205972

119866

1205971199111

1205971199113


100

9

8

7

6

5

4

3

2

1

0 50 100 150 200

50 100 150 200 250

50 100 150 200 250500400300200

Time (days)

Time (days)

Time (days)

Time (days)

8

6

4

2

0

120591 = 0

120591 = 2120591 = 15

120591 = 1

119883

119884

119881

minus2

minus4

minus6

6

4

2

0

minus2

minus4

minus6

minus8

minus10

6

4

2

0

minus2

minus4

minus6

119883

119884

119881

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)


120597119866 (1199113

119905)

120597119905

=

120597119866 (1 1 1199113

119905)

120597119905

= (1199113

minus 1) 120574119866 + (1 minus 120596) (1199113

minus 1) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113


(1 minus 1199113

)

1205972

119866

1205971199111

1205971199113

(19)


120597119866119909

120597119911

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(20)


119864 [119883 (119905)] =

120597119866119909

120597119911

10038161003816100381610038161003816100381610038161003816119911=1

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(21)



1199112

and 1199113


1

= 1199112

= 1199113

= 1 we have

120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)]

minus (1 minus 120572) 120573119864 [119883 (119905) 119881 (119905)]

(22)



Latently infected

target cells


+



Viral clearance

Viral synthesis


Cell death

Cell death



inhibitors

CD4+ cells

CD4+ cells

Uninfected CD4+



120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)]


(23)








1198751015840

V (119905) = minus (120574 + 120581 (1 minus 120596)119873)119875V (119905)


120573) 119875V (119905)


+ (V + 1) (120583 + (1 minus 120572) 119890minus120588120591

120573) 119875V+1 (119905)

(24)


2000

1500

1000

500

0

200

100

300

400

0

0

0 200 400 600 800 1000

0 200

200

400

400

600

600

800

800

1000 0 200 400 600 800 1000

Cel

l pop

ulat

ions

Cel

l pop

ulat

ions

Cel

l pop

ulat

ions



Time (days)



119883

119884

119881



119866V (119911 119905) |119881(0)=V0

= exp120579120598


) 119911119890minus120579119905

+ 1 minus 119890minus120579119905

V0

(25)

where V0



119901ext (119905) |119881(0)=V0 = 119875 (119881 = 0 119905) |119881(0)=V0 = 119866V (0 119905) = 1198750 (119905)

= exp120579120598

(119890minus120579119905


V0

(26)


119901ext = 1198750 (infin)


(exp120579120598

(119890minus120579119905


V0)

= 119890minus120579120598

= 119890minus((120583+(1minus120572)120573119890

minus120588120591)(120574+(1minus120596)120581119873))

(27)





200

100

300

400

00 200 400 600 800 1000

Cel

l pop

ulat

ions

200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

500

0

Cel

l pop

ulat

ions

Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

119883

119884

119881

119883

119884

119881








References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Latently infected

120582

120575119909

target cells

120573119909(1 minus 119890minus120588120591

)

120573119909119890minus120588120591

120581119884

1205811198731119884

120581119873119884



+

120583119881

120574

119881

119883

119884

119881119899

120581119884119871

119884119871

120583119881119899

Uninfected CD4+

CD4+ cells

CD4+ cells



119909119910V(119905) = 119875[119883(119905) =


















(a)





(b)






120573


0000024






119873


1000



119875119909119910V (119905 + Δ119905)

= 1 minus (120582Δ119905 + 120575119909Δ119905 + 120573119909VΔ119905 + 120583VΔ119905 + 120581119910Δ119905 + 120574Δ119905)

+ 119900 (Δ119905) 119875119909119910V (119905)

+ 120582Δ119905 + 119900 (Δ119905) 119875119909minus1119910V (119905)

+ 120575 (119909 + 1) Δ119905 + 119900 (Δ119905) 119875119909+1119910V (119905)

+ 120573 (119909 + 1) (V + 1) 119890minus120588120591

Δ119905 + 119900 (Δ119905)

times 119875119909+1119910minus1V+1 (119905)

+ 120581119873 (119910 + 1) Δ119905 + 119900 (Δ119905) 119875119909119910+1Vminus1 (119905)

+ 120574Δ119905 + 119900 (Δ119905) 119875119909119910Vminus1 (119905)

+ 120583 (V + 1) Δ119905 + 119900 (Δ119905) 119875119909119910V+1 (119905)

(2)


119909119910V(119905)

1198751015840

119909119910V (119905) = minus 120582 + 120575119909 + 120573119890minus120588120591

119909V + 120583V + 120581119873119910 + 120574 119875119909119910V (119905)

+ 120582119875119909minus1119910V (119905) + 120575 (119909 + 1) 119875119909+1119910V (119905)

+ 120573 (119909 + 1) (V + 1) 119890minus120588120591

119875119909+1119910minus1V+1 (119905)

+ 119873120581 (119910 + 1) 119875119909119910+1Vminus1 (119905) + 120574119875119909119910Vminus1 (119905)

+ 120583 (V + 1) 119875119909119910V+1 (119905)

(3)


1198751015840

000

(119905) = minus (120582 + 120574) 119875000

(119905) + 120575119875100

(119905) + 120583119875001

(119905) (4)


119866 (1199111

1199112

1199113

119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

119911119910

2

119911V3

(5)


120597119866 (1199111

1199112

1199113

119905)

120597119905

=

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

1198751015840

119909119910V (119905) 119911119909

1

119911119910

2

119911V3

(6)


1199112

and 1199113

yields

1205973

119866 (1199111

1199112

1199113

119905)

1205971199111

1205971199112

1205971199113

=

infin

sum

119909=1

infin

sum

119910=1

infin

sum

V=1

119909119910V119875119909119910V (119905) 119911

119909minus1

1

119911119910minus1

2

119911Vminus13

=

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

(119909 + 1) (119910 + 1) (V + 1) 119875119909+1119910+1V+1 (119905) 119911

119909

1

119911119910

2

119911V3

(7)


119911119910

2

119911V3


120597119866

120597119905

= (1199111

minus 1) 120582 + (1199113

minus 1) 120574119866

+ (1 minus 1199111

) 120575

120597119866

1205971199111

+ (1199113

minus 1199112

) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113

+ 120573119890minus120588120591

(1199112

minus 1199111

1199113

)

1205972

119866

1205971199111

1205971199113

(8)





Population(119883119884 119881) at (119905 119905 + Δ)










119866 (1199111

1199112

1199113

119905)

= 119886119890119887119905

(1199111

minus 1) 120582 + (1199113

minus 1) 120574 119865 (1199111

1199112

1199113

)

+ (1 minus 1199111

) 120575

120597119865 (1199111

1199112

1199113

)

1205971199111

+ (1198731199113

minus 1199112

) 120581

120597119865 (1199111

1199112

1199113

)

1205971199112

+ (1 minus 1199113

) 120583

120597119865 (1199111

1199112

1199113

)

1205971199113

+ 120573 (119890minus120588120591

1199112

minus 1199111

1199113

)

1205972

119865 (1199111

1199112

1199113

)

1205971199111

1205971199113

(9)


of 119865(1199111

1199112

1199113



119866 (1199111

1 1 119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

(10)


= 1199113

= 1 1199111

= 1199113

= 1 and 1199112

= 1199111

=


120597119866 (1199111

119905)

120597119905

=

120597119866 (1199111

1 1 119905)

120597119905

= (1199111

minus 1) 120582119866 + (1 minus 1199111

) 120575

120597119866

1205971199111

+ 120573119890minus120588120591

(1 minus 1199111

)

1205972

119866

1205971199111

1205971199113

120597119866 (1199112

119905)

120597119905

=

120597119866 (1 1199112

1 119905)

120597119905

= (1 minus 1199112

) 120581119873

120597119866

1205971199112

+ 120573119890minus120588120591

(1199112

minus 1)

1205972

119866

1205971199111

1205971199113

120597119866 (1199113

119905)

120597119905

=

120597119866 (1 1 1199113

119905)

120597119905

= (1199113

minus 1) 120574119866 + (1199113

minus 1) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113

+ 120573119890minus120588120591

(1 minus 1199113

)

1205972

119866

1205971199111

1205971199113

(11)


120597119866119909

120597119911

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(12)


120597119866119909

120597119911

10038161003816100381610038161003816100381610038161003816119911=1

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

= 119864 [119883] (13)




1199112

and 1199113


1

= 1199112

= 1199113

= 1 we have120597

120597119905

119864 [119883 (119905)] = 120582 minus 120575119864 [119883 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119884 (119905)] = minus120581119864 [119884 (119905)] + 120573119890minus120588120591

119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)]

= 120574 + 119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

(14)


120597

120597119905

119864 [119883 (119905)] = 120582 minus 120575119864 [119883 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119884 (119905)] = minus120581119864 [119884 (119905)] + 120573119890minus120588120591

119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)]

= 120574 + 119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

(15)


119889119909

119889119905

= 120582 minus 120575119909 (119905) minus 120573119909 (119905) V (119905)

119889119910

119889119905

= 120573119890minus120588120591


119889V

119889119905

= 120574 + 119873120581119910 (119905) minus 120583V (119905) minus 120573119909 (119905) V (119905)

(16)









120597119866

120597119905

= (1199111

minus 1) 120582 + (1199113

minus 1) 120574119866 + (1 minus 1199111

) 120575

120597119866

1205971199111

+ (1 minus 120596) (1199113

minus 1199112

) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113


(1199112

minus 1199111

1199113

)

1205972

119866

1205971199111

1205971199113

(17)


119866 (1199111

1 1 119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

(18)

Setting 1199112

= 1199113

= 1 1199111

= 1199113

= 1 and 1199112

= 1199111


120597119866 (1199111

119905)

120597119905

=

120597119866 (1199111

1 1 119905)

120597119905

= (1199111

minus 1) 120582119866 + (1 minus 1199111

) 120575

120597119866

1205971199111


(1 minus 1199111

)

1205972

119866

1205971199111

1205971199113

120597119866 (1199112

119905)

120597119905

=

120597119866 (1 1199112

1 119905)

120597119905

= (1 minus 120596) (119873 minus 1199112

) 120581

120597119866

1205971199112


(1199112

minus 1)

1205972

119866

1205971199111

1205971199113


100

9

8

7

6

5

4

3

2

1

0 50 100 150 200

50 100 150 200 250

50 100 150 200 250500400300200

Time (days)

Time (days)

Time (days)

Time (days)

8

6

4

2

0

120591 = 0

120591 = 2120591 = 15

120591 = 1

119883

119884

119881

minus2

minus4

minus6

6

4

2

0

minus2

minus4

minus6

minus8

minus10

6

4

2

0

minus2

minus4

minus6

119883

119884

119881

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)


120597119866 (1199113

119905)

120597119905

=

120597119866 (1 1 1199113

119905)

120597119905

= (1199113

minus 1) 120574119866 + (1 minus 120596) (1199113

minus 1) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113


(1 minus 1199113

)

1205972

119866

1205971199111

1205971199113

(19)


120597119866119909

120597119911

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(20)


119864 [119883 (119905)] =

120597119866119909

120597119911

10038161003816100381610038161003816100381610038161003816119911=1

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(21)



1199112

and 1199113


1

= 1199112

= 1199113

= 1 we have

120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)]

minus (1 minus 120572) 120573119864 [119883 (119905) 119881 (119905)]

(22)



Latently infected

target cells


+



Viral clearance

Viral synthesis


Cell death

Cell death



inhibitors

CD4+ cells

CD4+ cells

Uninfected CD4+



120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)]


(23)








1198751015840

V (119905) = minus (120574 + 120581 (1 minus 120596)119873)119875V (119905)


120573) 119875V (119905)


+ (V + 1) (120583 + (1 minus 120572) 119890minus120588120591

120573) 119875V+1 (119905)

(24)


2000

1500

1000

500

0

200

100

300

400

0

0

0 200 400 600 800 1000

0 200

200

400

400

600

600

800

800

1000 0 200 400 600 800 1000

Cel

l pop

ulat

ions

Cel

l pop

ulat

ions

Cel

l pop

ulat

ions



Time (days)



119883

119884

119881



119866V (119911 119905) |119881(0)=V0

= exp120579120598


) 119911119890minus120579119905

+ 1 minus 119890minus120579119905

V0

(25)

where V0



119901ext (119905) |119881(0)=V0 = 119875 (119881 = 0 119905) |119881(0)=V0 = 119866V (0 119905) = 1198750 (119905)

= exp120579120598

(119890minus120579119905


V0

(26)


119901ext = 1198750 (infin)


(exp120579120598

(119890minus120579119905


V0)

= 119890minus120579120598

= 119890minus((120583+(1minus120572)120573119890

minus120588120591)(120574+(1minus120596)120581119873))

(27)





200

100

300

400

00 200 400 600 800 1000

Cel

l pop

ulat

ions

200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

500

0

Cel

l pop

ulat

ions

Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

119883

119884

119881

119883

119884

119881








References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











(a)





(b)






120573


0000024






119873


1000



119875119909119910V (119905 + Δ119905)

= 1 minus (120582Δ119905 + 120575119909Δ119905 + 120573119909VΔ119905 + 120583VΔ119905 + 120581119910Δ119905 + 120574Δ119905)

+ 119900 (Δ119905) 119875119909119910V (119905)

+ 120582Δ119905 + 119900 (Δ119905) 119875119909minus1119910V (119905)

+ 120575 (119909 + 1) Δ119905 + 119900 (Δ119905) 119875119909+1119910V (119905)

+ 120573 (119909 + 1) (V + 1) 119890minus120588120591

Δ119905 + 119900 (Δ119905)

times 119875119909+1119910minus1V+1 (119905)

+ 120581119873 (119910 + 1) Δ119905 + 119900 (Δ119905) 119875119909119910+1Vminus1 (119905)

+ 120574Δ119905 + 119900 (Δ119905) 119875119909119910Vminus1 (119905)

+ 120583 (V + 1) Δ119905 + 119900 (Δ119905) 119875119909119910V+1 (119905)

(2)


119909119910V(119905)

1198751015840

119909119910V (119905) = minus 120582 + 120575119909 + 120573119890minus120588120591

119909V + 120583V + 120581119873119910 + 120574 119875119909119910V (119905)

+ 120582119875119909minus1119910V (119905) + 120575 (119909 + 1) 119875119909+1119910V (119905)

+ 120573 (119909 + 1) (V + 1) 119890minus120588120591

119875119909+1119910minus1V+1 (119905)

+ 119873120581 (119910 + 1) 119875119909119910+1Vminus1 (119905) + 120574119875119909119910Vminus1 (119905)

+ 120583 (V + 1) 119875119909119910V+1 (119905)

(3)


1198751015840

000

(119905) = minus (120582 + 120574) 119875000

(119905) + 120575119875100

(119905) + 120583119875001

(119905) (4)


119866 (1199111

1199112

1199113

119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

119911119910

2

119911V3

(5)


120597119866 (1199111

1199112

1199113

119905)

120597119905

=

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

1198751015840

119909119910V (119905) 119911119909

1

119911119910

2

119911V3

(6)


1199112

and 1199113

yields

1205973

119866 (1199111

1199112

1199113

119905)

1205971199111

1205971199112

1205971199113

=

infin

sum

119909=1

infin

sum

119910=1

infin

sum

V=1

119909119910V119875119909119910V (119905) 119911

119909minus1

1

119911119910minus1

2

119911Vminus13

=

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

(119909 + 1) (119910 + 1) (V + 1) 119875119909+1119910+1V+1 (119905) 119911

119909

1

119911119910

2

119911V3

(7)


119911119910

2

119911V3


120597119866

120597119905

= (1199111

minus 1) 120582 + (1199113

minus 1) 120574119866

+ (1 minus 1199111

) 120575

120597119866

1205971199111

+ (1199113

minus 1199112

) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113

+ 120573119890minus120588120591

(1199112

minus 1199111

1199113

)

1205972

119866

1205971199111

1205971199113

(8)





Population(119883119884 119881) at (119905 119905 + Δ)










119866 (1199111

1199112

1199113

119905)

= 119886119890119887119905

(1199111

minus 1) 120582 + (1199113

minus 1) 120574 119865 (1199111

1199112

1199113

)

+ (1 minus 1199111

) 120575

120597119865 (1199111

1199112

1199113

)

1205971199111

+ (1198731199113

minus 1199112

) 120581

120597119865 (1199111

1199112

1199113

)

1205971199112

+ (1 minus 1199113

) 120583

120597119865 (1199111

1199112

1199113

)

1205971199113

+ 120573 (119890minus120588120591

1199112

minus 1199111

1199113

)

1205972

119865 (1199111

1199112

1199113

)

1205971199111

1205971199113

(9)


of 119865(1199111

1199112

1199113



119866 (1199111

1 1 119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

(10)


= 1199113

= 1 1199111

= 1199113

= 1 and 1199112

= 1199111

=


120597119866 (1199111

119905)

120597119905

=

120597119866 (1199111

1 1 119905)

120597119905

= (1199111

minus 1) 120582119866 + (1 minus 1199111

) 120575

120597119866

1205971199111

+ 120573119890minus120588120591

(1 minus 1199111

)

1205972

119866

1205971199111

1205971199113

120597119866 (1199112

119905)

120597119905

=

120597119866 (1 1199112

1 119905)

120597119905

= (1 minus 1199112

) 120581119873

120597119866

1205971199112

+ 120573119890minus120588120591

(1199112

minus 1)

1205972

119866

1205971199111

1205971199113

120597119866 (1199113

119905)

120597119905

=

120597119866 (1 1 1199113

119905)

120597119905

= (1199113

minus 1) 120574119866 + (1199113

minus 1) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113

+ 120573119890minus120588120591

(1 minus 1199113

)

1205972

119866

1205971199111

1205971199113

(11)


120597119866119909

120597119911

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(12)


120597119866119909

120597119911

10038161003816100381610038161003816100381610038161003816119911=1

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

= 119864 [119883] (13)




1199112

and 1199113


1

= 1199112

= 1199113

= 1 we have120597

120597119905

119864 [119883 (119905)] = 120582 minus 120575119864 [119883 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119884 (119905)] = minus120581119864 [119884 (119905)] + 120573119890minus120588120591

119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)]

= 120574 + 119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

(14)


120597

120597119905

119864 [119883 (119905)] = 120582 minus 120575119864 [119883 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119884 (119905)] = minus120581119864 [119884 (119905)] + 120573119890minus120588120591

119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)]

= 120574 + 119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

(15)


119889119909

119889119905

= 120582 minus 120575119909 (119905) minus 120573119909 (119905) V (119905)

119889119910

119889119905

= 120573119890minus120588120591


119889V

119889119905

= 120574 + 119873120581119910 (119905) minus 120583V (119905) minus 120573119909 (119905) V (119905)

(16)









120597119866

120597119905

= (1199111

minus 1) 120582 + (1199113

minus 1) 120574119866 + (1 minus 1199111

) 120575

120597119866

1205971199111

+ (1 minus 120596) (1199113

minus 1199112

) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113


(1199112

minus 1199111

1199113

)

1205972

119866

1205971199111

1205971199113

(17)


119866 (1199111

1 1 119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

(18)

Setting 1199112

= 1199113

= 1 1199111

= 1199113

= 1 and 1199112

= 1199111


120597119866 (1199111

119905)

120597119905

=

120597119866 (1199111

1 1 119905)

120597119905

= (1199111

minus 1) 120582119866 + (1 minus 1199111

) 120575

120597119866

1205971199111


(1 minus 1199111

)

1205972

119866

1205971199111

1205971199113

120597119866 (1199112

119905)

120597119905

=

120597119866 (1 1199112

1 119905)

120597119905

= (1 minus 120596) (119873 minus 1199112

) 120581

120597119866

1205971199112


(1199112

minus 1)

1205972

119866

1205971199111

1205971199113


100

9

8

7

6

5

4

3

2

1

0 50 100 150 200

50 100 150 200 250

50 100 150 200 250500400300200

Time (days)

Time (days)

Time (days)

Time (days)

8

6

4

2

0

120591 = 0

120591 = 2120591 = 15

120591 = 1

119883

119884

119881

minus2

minus4

minus6

6

4

2

0

minus2

minus4

minus6

minus8

minus10

6

4

2

0

minus2

minus4

minus6

119883

119884

119881

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)


120597119866 (1199113

119905)

120597119905

=

120597119866 (1 1 1199113

119905)

120597119905

= (1199113

minus 1) 120574119866 + (1 minus 120596) (1199113

minus 1) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113


(1 minus 1199113

)

1205972

119866

1205971199111

1205971199113

(19)


120597119866119909

120597119911

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(20)


119864 [119883 (119905)] =

120597119866119909

120597119911

10038161003816100381610038161003816100381610038161003816119911=1

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(21)



1199112

and 1199113


1

= 1199112

= 1199113

= 1 we have

120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)]

minus (1 minus 120572) 120573119864 [119883 (119905) 119881 (119905)]

(22)



Latently infected

target cells


+



Viral clearance

Viral synthesis


Cell death

Cell death



inhibitors

CD4+ cells

CD4+ cells

Uninfected CD4+



120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)]


(23)








1198751015840

V (119905) = minus (120574 + 120581 (1 minus 120596)119873)119875V (119905)


120573) 119875V (119905)


+ (V + 1) (120583 + (1 minus 120572) 119890minus120588120591

120573) 119875V+1 (119905)

(24)


2000

1500

1000

500

0

200

100

300

400

0

0

0 200 400 600 800 1000

0 200

200

400

400

600

600

800

800

1000 0 200 400 600 800 1000

Cel

l pop

ulat

ions

Cel

l pop

ulat

ions

Cel

l pop

ulat

ions



Time (days)



119883

119884

119881



119866V (119911 119905) |119881(0)=V0

= exp120579120598


) 119911119890minus120579119905

+ 1 minus 119890minus120579119905

V0

(25)

where V0



119901ext (119905) |119881(0)=V0 = 119875 (119881 = 0 119905) |119881(0)=V0 = 119866V (0 119905) = 1198750 (119905)

= exp120579120598

(119890minus120579119905


V0

(26)


119901ext = 1198750 (infin)


(exp120579120598

(119890minus120579119905


V0)

= 119890minus120579120598

= 119890minus((120583+(1minus120572)120573119890

minus120588120591)(120574+(1minus120596)120581119873))

(27)





200

100

300

400

00 200 400 600 800 1000

Cel

l pop

ulat

ions

200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

500

0

Cel

l pop

ulat

ions

Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

119883

119884

119881

119883

119884

119881








References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Population(119883119884 119881) at (119905 119905 + Δ)










119866 (1199111

1199112

1199113

119905)

= 119886119890119887119905

(1199111

minus 1) 120582 + (1199113

minus 1) 120574 119865 (1199111

1199112

1199113

)

+ (1 minus 1199111

) 120575

120597119865 (1199111

1199112

1199113

)

1205971199111

+ (1198731199113

minus 1199112

) 120581

120597119865 (1199111

1199112

1199113

)

1205971199112

+ (1 minus 1199113

) 120583

120597119865 (1199111

1199112

1199113

)

1205971199113

+ 120573 (119890minus120588120591

1199112

minus 1199111

1199113

)

1205972

119865 (1199111

1199112

1199113

)

1205971199111

1205971199113

(9)


of 119865(1199111

1199112

1199113



119866 (1199111

1 1 119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

(10)


= 1199113

= 1 1199111

= 1199113

= 1 and 1199112

= 1199111

=


120597119866 (1199111

119905)

120597119905

=

120597119866 (1199111

1 1 119905)

120597119905

= (1199111

minus 1) 120582119866 + (1 minus 1199111

) 120575

120597119866

1205971199111

+ 120573119890minus120588120591

(1 minus 1199111

)

1205972

119866

1205971199111

1205971199113

120597119866 (1199112

119905)

120597119905

=

120597119866 (1 1199112

1 119905)

120597119905

= (1 minus 1199112

) 120581119873

120597119866

1205971199112

+ 120573119890minus120588120591

(1199112

minus 1)

1205972

119866

1205971199111

1205971199113

120597119866 (1199113

119905)

120597119905

=

120597119866 (1 1 1199113

119905)

120597119905

= (1199113

minus 1) 120574119866 + (1199113

minus 1) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113

+ 120573119890minus120588120591

(1 minus 1199113

)

1205972

119866

1205971199111

1205971199113

(11)


120597119866119909

120597119911

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(12)


120597119866119909

120597119911

10038161003816100381610038161003816100381610038161003816119911=1

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

= 119864 [119883] (13)




1199112

and 1199113


1

= 1199112

= 1199113

= 1 we have120597

120597119905

119864 [119883 (119905)] = 120582 minus 120575119864 [119883 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119884 (119905)] = minus120581119864 [119884 (119905)] + 120573119890minus120588120591

119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)]

= 120574 + 119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

(14)


120597

120597119905

119864 [119883 (119905)] = 120582 minus 120575119864 [119883 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119884 (119905)] = minus120581119864 [119884 (119905)] + 120573119890minus120588120591

119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)]

= 120574 + 119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

(15)


119889119909

119889119905

= 120582 minus 120575119909 (119905) minus 120573119909 (119905) V (119905)

119889119910

119889119905

= 120573119890minus120588120591


119889V

119889119905

= 120574 + 119873120581119910 (119905) minus 120583V (119905) minus 120573119909 (119905) V (119905)

(16)









120597119866

120597119905

= (1199111

minus 1) 120582 + (1199113

minus 1) 120574119866 + (1 minus 1199111

) 120575

120597119866

1205971199111

+ (1 minus 120596) (1199113

minus 1199112

) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113


(1199112

minus 1199111

1199113

)

1205972

119866

1205971199111

1205971199113

(17)


119866 (1199111

1 1 119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

(18)

Setting 1199112

= 1199113

= 1 1199111

= 1199113

= 1 and 1199112

= 1199111


120597119866 (1199111

119905)

120597119905

=

120597119866 (1199111

1 1 119905)

120597119905

= (1199111

minus 1) 120582119866 + (1 minus 1199111

) 120575

120597119866

1205971199111


(1 minus 1199111

)

1205972

119866

1205971199111

1205971199113

120597119866 (1199112

119905)

120597119905

=

120597119866 (1 1199112

1 119905)

120597119905

= (1 minus 120596) (119873 minus 1199112

) 120581

120597119866

1205971199112


(1199112

minus 1)

1205972

119866

1205971199111

1205971199113


100

9

8

7

6

5

4

3

2

1

0 50 100 150 200

50 100 150 200 250

50 100 150 200 250500400300200

Time (days)

Time (days)

Time (days)

Time (days)

8

6

4

2

0

120591 = 0

120591 = 2120591 = 15

120591 = 1

119883

119884

119881

minus2

minus4

minus6

6

4

2

0

minus2

minus4

minus6

minus8

minus10

6

4

2

0

minus2

minus4

minus6

119883

119884

119881

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)


120597119866 (1199113

119905)

120597119905

=

120597119866 (1 1 1199113

119905)

120597119905

= (1199113

minus 1) 120574119866 + (1 minus 120596) (1199113

minus 1) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113


(1 minus 1199113

)

1205972

119866

1205971199111

1205971199113

(19)


120597119866119909

120597119911

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(20)


119864 [119883 (119905)] =

120597119866119909

120597119911

10038161003816100381610038161003816100381610038161003816119911=1

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(21)



1199112

and 1199113


1

= 1199112

= 1199113

= 1 we have

120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)]

minus (1 minus 120572) 120573119864 [119883 (119905) 119881 (119905)]

(22)



Latently infected

target cells


+



Viral clearance

Viral synthesis


Cell death

Cell death



inhibitors

CD4+ cells

CD4+ cells

Uninfected CD4+



120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)]


(23)








1198751015840

V (119905) = minus (120574 + 120581 (1 minus 120596)119873)119875V (119905)


120573) 119875V (119905)


+ (V + 1) (120583 + (1 minus 120572) 119890minus120588120591

120573) 119875V+1 (119905)

(24)


2000

1500

1000

500

0

200

100

300

400

0

0

0 200 400 600 800 1000

0 200

200

400

400

600

600

800

800

1000 0 200 400 600 800 1000

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ions

Cel

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ions



Time (days)



119883

119884

119881



119866V (119911 119905) |119881(0)=V0

= exp120579120598


) 119911119890minus120579119905

+ 1 minus 119890minus120579119905

V0

(25)

where V0



119901ext (119905) |119881(0)=V0 = 119875 (119881 = 0 119905) |119881(0)=V0 = 119866V (0 119905) = 1198750 (119905)

= exp120579120598

(119890minus120579119905


V0

(26)


119901ext = 1198750 (infin)


(exp120579120598

(119890minus120579119905


V0)

= 119890minus120579120598

= 119890minus((120583+(1minus120572)120573119890

minus120588120591)(120574+(1minus120596)120581119873))

(27)





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00 200 400 600 800 1000

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200

100

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ulat

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200

100

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ulat

ions

200

100

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400

500

0

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ions

Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

119883

119884

119881

119883

119884

119881








References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199112

and 1199113


1

= 1199112

= 1199113

= 1 we have120597

120597119905

119864 [119883 (119905)] = 120582 minus 120575119864 [119883 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119884 (119905)] = minus120581119864 [119884 (119905)] + 120573119890minus120588120591

119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)]

= 120574 + 119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

(14)


120597

120597119905

119864 [119883 (119905)] = 120582 minus 120575119864 [119883 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119884 (119905)] = minus120581119864 [119884 (119905)] + 120573119890minus120588120591

119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)]

= 120574 + 119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)] minus 120573119864 [119883 (119905) 119881 (119905)]

(15)


119889119909

119889119905

= 120582 minus 120575119909 (119905) minus 120573119909 (119905) V (119905)

119889119910

119889119905

= 120573119890minus120588120591


119889V

119889119905

= 120574 + 119873120581119910 (119905) minus 120583V (119905) minus 120573119909 (119905) V (119905)

(16)









120597119866

120597119905

= (1199111

minus 1) 120582 + (1199113

minus 1) 120574119866 + (1 minus 1199111

) 120575

120597119866

1205971199111

+ (1 minus 120596) (1199113

minus 1199112

) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113


(1199112

minus 1199111

1199113

)

1205972

119866

1205971199111

1205971199113

(17)


119866 (1199111

1 1 119905) =

infin

sum

119909=0

infin

sum

119910=0

infin

sum

V=0

119875119909119910V (119905) 119911

119909

1

(18)

Setting 1199112

= 1199113

= 1 1199111

= 1199113

= 1 and 1199112

= 1199111


120597119866 (1199111

119905)

120597119905

=

120597119866 (1199111

1 1 119905)

120597119905

= (1199111

minus 1) 120582119866 + (1 minus 1199111

) 120575

120597119866

1205971199111


(1 minus 1199111

)

1205972

119866

1205971199111

1205971199113

120597119866 (1199112

119905)

120597119905

=

120597119866 (1 1199112

1 119905)

120597119905

= (1 minus 120596) (119873 minus 1199112

) 120581

120597119866

1205971199112


(1199112

minus 1)

1205972

119866

1205971199111

1205971199113


100

9

8

7

6

5

4

3

2

1

0 50 100 150 200

50 100 150 200 250

50 100 150 200 250500400300200

Time (days)

Time (days)

Time (days)

Time (days)

8

6

4

2

0

120591 = 0

120591 = 2120591 = 15

120591 = 1

119883

119884

119881

minus2

minus4

minus6

6

4

2

0

minus2

minus4

minus6

minus8

minus10

6

4

2

0

minus2

minus4

minus6

119883

119884

119881

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)


120597119866 (1199113

119905)

120597119905

=

120597119866 (1 1 1199113

119905)

120597119905

= (1199113

minus 1) 120574119866 + (1 minus 120596) (1199113

minus 1) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113


(1 minus 1199113

)

1205972

119866

1205971199111

1205971199113

(19)


120597119866119909

120597119911

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(20)


119864 [119883 (119905)] =

120597119866119909

120597119911

10038161003816100381610038161003816100381610038161003816119911=1

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(21)



1199112

and 1199113


1

= 1199112

= 1199113

= 1 we have

120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)]

minus (1 minus 120572) 120573119864 [119883 (119905) 119881 (119905)]

(22)



Latently infected

target cells


+



Viral clearance

Viral synthesis


Cell death

Cell death



inhibitors

CD4+ cells

CD4+ cells

Uninfected CD4+



120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)]


(23)








1198751015840

V (119905) = minus (120574 + 120581 (1 minus 120596)119873)119875V (119905)


120573) 119875V (119905)


+ (V + 1) (120583 + (1 minus 120572) 119890minus120588120591

120573) 119875V+1 (119905)

(24)


2000

1500

1000

500

0

200

100

300

400

0

0

0 200 400 600 800 1000

0 200

200

400

400

600

600

800

800

1000 0 200 400 600 800 1000

Cel

l pop

ulat

ions

Cel

l pop

ulat

ions

Cel

l pop

ulat

ions



Time (days)



119883

119884

119881



119866V (119911 119905) |119881(0)=V0

= exp120579120598


) 119911119890minus120579119905

+ 1 minus 119890minus120579119905

V0

(25)

where V0



119901ext (119905) |119881(0)=V0 = 119875 (119881 = 0 119905) |119881(0)=V0 = 119866V (0 119905) = 1198750 (119905)

= exp120579120598

(119890minus120579119905


V0

(26)


119901ext = 1198750 (infin)


(exp120579120598

(119890minus120579119905


V0)

= 119890minus120579120598

= 119890minus((120583+(1minus120572)120573119890

minus120588120591)(120574+(1minus120596)120581119873))

(27)





200

100

300

400

00 200 400 600 800 1000

Cel

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ions

200

100

300

400

0

Cel

l pop

ulat

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200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

500

0

Cel

l pop

ulat

ions

Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

119883

119884

119881

119883

119884

119881








References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










100

9

8

7

6

5

4

3

2

1

0 50 100 150 200

50 100 150 200 250

50 100 150 200 250500400300200

Time (days)

Time (days)

Time (days)

Time (days)

8

6

4

2

0

120591 = 0

120591 = 2120591 = 15

120591 = 1

119883

119884

119881

minus2

minus4

minus6

6

4

2

0

minus2

minus4

minus6

minus8

minus10

6

4

2

0

minus2

minus4

minus6

119883

119884

119881

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)

Log(

popu

latio

ns)


120597119866 (1199113

119905)

120597119905

=

120597119866 (1 1 1199113

119905)

120597119905

= (1199113

minus 1) 120574119866 + (1 minus 120596) (1199113

minus 1) 120581119873

120597119866

1205971199112

+ (1 minus 1199113

) 120583

120597119866

1205971199113


(1 minus 1199113

)

1205972

119866

1205971199111

1205971199113

(19)


120597119866119909

120597119911

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(20)


119864 [119883 (119905)] =

120597119866119909

120597119911

10038161003816100381610038161003816100381610038161003816119911=1

=

infin

sum

119909=0

119909119875119909

(119905) 119911119909minus1

(21)



1199112

and 1199113


1

= 1199112

= 1199113

= 1 we have

120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)] minus 120583119864 [119881 (119905)]

minus (1 minus 120572) 120573119864 [119883 (119905) 119881 (119905)]

(22)



Latently infected

target cells


+



Viral clearance

Viral synthesis


Cell death

Cell death



inhibitors

CD4+ cells

CD4+ cells

Uninfected CD4+



120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)]


(23)








1198751015840

V (119905) = minus (120574 + 120581 (1 minus 120596)119873)119875V (119905)


120573) 119875V (119905)


+ (V + 1) (120583 + (1 minus 120572) 119890minus120588120591

120573) 119875V+1 (119905)

(24)


2000

1500

1000

500

0

200

100

300

400

0

0

0 200 400 600 800 1000

0 200

200

400

400

600

600

800

800

1000 0 200 400 600 800 1000

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ulat

ions

Cel

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ulat

ions

Cel

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ulat

ions



Time (days)



119883

119884

119881



119866V (119911 119905) |119881(0)=V0

= exp120579120598


) 119911119890minus120579119905

+ 1 minus 119890minus120579119905

V0

(25)

where V0



119901ext (119905) |119881(0)=V0 = 119875 (119881 = 0 119905) |119881(0)=V0 = 119866V (0 119905) = 1198750 (119905)

= exp120579120598

(119890minus120579119905


V0

(26)


119901ext = 1198750 (infin)


(exp120579120598

(119890minus120579119905


V0)

= 119890minus120579120598

= 119890minus((120583+(1minus120572)120573119890

minus120588120591)(120574+(1minus120596)120581119873))

(27)





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400

00 200 400 600 800 1000

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200

100

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400

0

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ulat

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200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

500

0

Cel

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ions

Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

119883

119884

119881

119883

119884

119881








References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Latently infected

target cells


+



Viral clearance

Viral synthesis


Cell death

Cell death



inhibitors

CD4+ cells

CD4+ cells

Uninfected CD4+



120597

120597119905


120597

120597119905


119864 [119883 (119905) 119881 (119905)]

120597

120597119905

119864 [119881 (119905)] = 120574 + (1 minus 120596)119873120581119864 [119884 (119905)]


(23)








1198751015840

V (119905) = minus (120574 + 120581 (1 minus 120596)119873)119875V (119905)


120573) 119875V (119905)


+ (V + 1) (120583 + (1 minus 120572) 119890minus120588120591

120573) 119875V+1 (119905)

(24)


2000

1500

1000

500

0

200

100

300

400

0

0

0 200 400 600 800 1000

0 200

200

400

400

600

600

800

800

1000 0 200 400 600 800 1000

Cel

l pop

ulat

ions

Cel

l pop

ulat

ions

Cel

l pop

ulat

ions



Time (days)



119883

119884

119881



119866V (119911 119905) |119881(0)=V0

= exp120579120598


) 119911119890minus120579119905

+ 1 minus 119890minus120579119905

V0

(25)

where V0



119901ext (119905) |119881(0)=V0 = 119875 (119881 = 0 119905) |119881(0)=V0 = 119866V (0 119905) = 1198750 (119905)

= exp120579120598

(119890minus120579119905


V0

(26)


119901ext = 1198750 (infin)


(exp120579120598

(119890minus120579119905


V0)

= 119890minus120579120598

= 119890minus((120583+(1minus120572)120573119890

minus120588120591)(120574+(1minus120596)120581119873))

(27)





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200

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0

Cel

l pop

ulat

ions

200

100

300

400

0

Cel

l pop

ulat

ions

200

100

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400

500

0

Cel

l pop

ulat

ions

Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

119883

119884

119881

119883

119884

119881








References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










2000

1500

1000

500

0

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0

0

0 200 400 600 800 1000

0 200

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600

600

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800

1000 0 200 400 600 800 1000

Cel

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Cel

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Time (days)



119883

119884

119881



119866V (119911 119905) |119881(0)=V0

= exp120579120598


) 119911119890minus120579119905

+ 1 minus 119890minus120579119905

V0

(25)

where V0



119901ext (119905) |119881(0)=V0 = 119875 (119881 = 0 119905) |119881(0)=V0 = 119866V (0 119905) = 1198750 (119905)

= exp120579120598

(119890minus120579119905


V0

(26)


119901ext = 1198750 (infin)


(exp120579120598

(119890minus120579119905


V0)

= 119890minus120579120598

= 119890minus((120583+(1minus120572)120573119890

minus120588120591)(120574+(1minus120596)120581119873))

(27)





200

100

300

400

00 200 400 600 800 1000

Cel

l pop

ulat

ions

200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

500

0

Cel

l pop

ulat

ions

Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

119883

119884

119881

119883

119884

119881








References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










200

100

300

400

00 200 400 600 800 1000

Cel

l pop

ulat

ions

200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

0

Cel

l pop

ulat

ions

200

100

300

400

500

0

Cel

l pop

ulat

ions

Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

0 200 400 600 800 1000Time (days)

119883

119884

119881

119883

119884

119881








References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article stochastic model for in-host hiv dynamics...

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