Research ArticleInnovative Approximate Analytical Solution for StandardModel of Viral Dynamics Hepatitis C with Direct-ActingAgents as an Implemented Case
Hesham A Elkaranshawy 1 Hossam M Ezzat 1 Yasmine Abouelseoud1
and Nermeen N Ibrahim2
1Department of Engineering Mathematics and Physics Faculty of Engineering Alexandria University Alexandria Egypt2High Institute of Public Health Alexandria University Alexandria Egypt
Correspondence should be addressed to Hesham A Elkaranshawy hesham_elkalexuedueg
Received 15 April 2019 Revised 10 August 2019 Accepted 23 August 2019 Published 16 September 2019
Academic Editor Jose J Muntildeoz
Copyright copy 2019 Hesham A Elkaranshawy et al is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited
In this article a novel approximate analytical solution is presented for solving the standard viral dynamic model Basically thestandard model is used to study viral dynamics in patients for a wide range of viruses like HIV HPV HBV and HCV In thisresearch work the standard model for hepatitis C virus (HCV) is considered in detail however the analysis and results can beapplicable for all other viruses is standard model is used to study viral dynamics in patients treated with direct-acting antiviralagents (DAAs) Power series solution combined with LaplacendashPade resummation method (PSLP) is used to obtain the ap-proximate analytical solutions for the model To test the capability as well as the validity of the proposed method results arecompared with available viral load data published in the literature and with published simulation results Good fits are obtained inthe comparison for all cases considered Given that medical specialists and physicians are more interested in solutions that yielddirect and simple predictions it is expected that the proposed approximate analytical solution would be attractive to them andhelp them to obtain a straightforward and a proper estimation regarding the viral load due to variations in treatment andorpatientrsquos parameters
1 Introduction
Treatment for chronic hepatitis C infection began in theearly 1990s with interferon-alpha [1] is injectable drugworked by improving the immune system rather than byspecifically attacking the virus In 1998 the oral drug ri-bavirin was added to interferon [2] e development of thetreatment occurred in 2002 with the approval of pegylatedinterferon-alpha a process that makes interferon moredurable and effective [3] In 2011 antiviral medications thatstop the virus reproducing which called direct-acting an-tivirals (DAAs) appeared [4] Sofosbuvir commercial nameSovaldi was approved in the United States in December2013 Sovaldi-based oral therapy offers high cure rates forHCV infection with excellent tolerability [5] Currently the
recommended DAA regimens are combinations of multi-drug treatments containing in particular NS5A NS5BNS4A inhibitors such as glecaprevir + pibrentasvir sofos-buvir and ribavirin (SOF+RBV) telaprevir + pegylated in-terferon and ribavirin (TVR+PR) sofosbuvir + simeprevir(SOF+ SIM) and sofosbuvir + velpatasvir For more detailssee [6 7 8] ese regimens offer high cure rates with a rateof curation near 100
e quick step of HCV drug development has led to thehopeful estimate that fully eradication of HCV is possibleAlthough there is still no vaccine for HCV cure anderadication happened but further study for the factors thatincrease this eradication rate is needed e World HealthOrganization (WHO) has formulated the Global HealthSector Strategy on Viral Hepatitis 2016ndash2021 and
HindawiMathematical Problems in EngineeringVolume 2019 Article ID 1454739 8 pageshttpsdoiorg10115520191454739
established service coverage targets to eliminate HCV as apublic health threat by 2030 [9 10] Meanwhile there re-main many barriers that need to be overcome Such barriersinclude the development of simplified and highly effectivedrug regimens improving the rates of detection of infectionand the availability of funds including financial and medicalexpertise
Mathematical modeling is a useful tool in the study ofvirus dynamics for many types of viruses such as HCVHBV HPV and HIV It can be used to predict behaviorunder certain conditions or decide which parameters en-hance the spread of disease It may also be used to calculatethe medications required to eradicate the disease or at leastget it under control ese models can be used also tounderstand the biological mechanisms and interpret theexperimental results [11] Mathematical models in the formof a system of differential equations for the basic dynamics invivo were developed and analyzed for HCV [8 12ndash15] HBV[16 17] HIV [18ndash22] and HPV [23] Models for HCVtreatment with DAAs therapy are considered in [8 14 15]Authors in [24ndash26] developed a multiscale model thatconsidered both intracellular viral RNA replication andextracellular viral infection
In general numerical solutions were obtained for themodels However analytical solutions could be useful for theestimation of parameters and for direct and simple pre-dictions for the viral loads Seldom an analytical or anapproximate analytical solution can be found in the liter-ature A simplified analytical solution for the standardmodelfor HCV viral dynamics was constructed in [12] It wasassumed that the number of target cells is constant hencethe system of ordinary differential equations (ODE) is re-duced linearized and solved It can be noticed that thissystem of ODE contains 7 parameters while the obtainedsolutions contain only 3 parameters An analytical solutionwas obtained for the multiscale model in [24 26] All newinfections during therapy were ignored in that analysis
In this research work the approximate analytical solu-tion is obtained for the standard model Similar standardmodels have been used for HCV HBV and HIV in theliterature [8 12ndash23] Nevertheless the model considered isthe HCV model ere is no reduction for the system ofODE or assumptions to simplify the equations of the modelYet the approximation is in the solution not in the system ofdifferential equations erefore it is more accurate andsatisfactory than the previously mentioned analytical solu-tions Power series solutions are first obtained for the systemof ODE and then the LaplacendashPade resummation method(PSLP) is used to obtain the entailed approximate analyticalsolution e technique used depends on the methodologydeveloped in [27ndash34] for the general system of ODE esolution is used for the analysis of the standard viral dynamicmodel of HCV after any type of DAA treatment initiation
2 Standard Model
Consider the system of nonlinear ODEs for the standardviral dynamic mathematical model for HCV kinetics duringtreatment [11ndash14]
dT
dt s minus dT minus βVT (1)
dI
dt βVT minus δI (2)
dV
dt (1 minus ε)ρI minus cV (3)
where T is the target cells which are produced at a constantrate s and are assumed to die per capita rate d and areinfected by virus V at rate β Infected cells are assumed todie at per capita rate δ Virions are generated at rate ρ perinfected cell and cleared from serum at rate c per virionTreatment is assumed to reduce the average viral productionrate per infected cell from ρ to (1 minus ε)ρ where ε is the in vivoantiviral effectiveness of therapy (0lt εlt 1) At treatmentinitiation (t 0) we used the standard assumption[10 21 22] that the system in the pretreatment is in steadystate given by
dT
dt 0
dI
dt 0
dV
dt 0
(4)
then
T(0) T0 cδβρ
I(0) I0 minus cdδ + βρs
βδρ
V(0) V0 minus cdδ + βρs
βcδ
(5)
3 Solution Method
31 Power Series Solution Assuming the following solutionsof order N
T(t) 1113944N
i0cTi
ti (6)
I(t) 1113944N
i0cIi
ti (7)
V(t) 1113944N
i0cVi
ti (8)
Substituting equations (6)ndash(8) into equations (1)ndash(3) andequating terms having the same powers of t e cTi
cIi and
cVicoefficients can be calculated For instance the power
series solution for V(t) is
2 Mathematical Problems in Engineering
V(t) 1113944N
i0cVi
ti
cV0+ cV1
t + cV2t2
+ cV3t3
+ middot middot middot (9)
where cV0 V0 is the initial condition of V(t) and the
coefficients cV1 cV2
and cV3are as follows
cV1 minus cεc0
cV2 05c
2εc0
cV316c0 minus c
3ε minus c2δε minus cdδ + ρs minus βc0cδ + ε2c2δ1113872
+ cdδε minus ρsβε + cδc0βε)
(10)
32 LaplacendashPade Resummation e Pade approximantfor a series function is a function with the same powerseries expansions as the original series function It issuitable for approximating a divergent series functioneapproximant is derived by expanding the function as aratio of two power series and determining both the nu-merator and denominator coefficients [28ndash34] If we havea function f(u) that can be represented in a power seriesform as
f(u) 1113944infin
i0cui
ui (11)
A Pade approximant is a rational functionL
M1113876 1113877
b0 + b1u + b2u2 + middot middot middot + bLuL
d0 + d1u + d2u2 + middot middot middot + dMuM
(12)
which has a Maclaurin expansion that agrees with the seriesof equation (11) approximately d0 is chosen to be equal to 1So there are L + 1 independent numerator coefficients andM independent denominator coefficients making L + M +
1 unknown coefficients e [LM] should fit the powerseries of equation (11) through the orders 1 u u2 uL+MHence
1113944
infin
i0cui
ui
b0 + b1u + b2u
2 + middot middot middot + bLuL
1 + d1u + d2u2 + middot middot middot + dMuM
+ O uL+M+1
1113872 1113873
(13)
Up to the term of order L + M we can write
cu0+ cu1
u + middot middot middot + cuL+Mu
L+M1113872 1113873 1 + d1u + d2u
2+ middot middot middot + dMu
M1113872 1113873
b0 + b1u + b2u2
+ middot middot middot + bLuL
(14)Denominator coefficients d1 d1 dM can be found
from equation (14) by equating the coefficients ofuL uL+1 uL+M Numerator coefficients b0 + b1 + middot middot middot bL
can be found from equation (14) by equating the coefficientsof u0 u u2 uL Hence we have constructed the [LM]
Pade approximant which agrees with 1113936infini0 cui
ui through theorder uL+M
To extend the convergence zone of the power seriessolution LaplacendashPade resummation can be introduced In
LaplacendashPade resummation the Laplace transform is ap-plied to the given series function then Pade approximate isobtained for the resulted function and consequently theinverse Laplace transform is used to specify the final solutionin the form of exponential functions To apply LaplacendashPaderesummation to the series solution of the viral load V(t) thefollowing procedure is employed
(1) From equation (9) the third-order power seriessolution of V(t) is
V(t) cV0+ cV1
t + cV2t2
+ cV3t3 (15)
(2) Taking Laplace transform to the series solution inequation (15) to obtain
F(s) cV0
s+
cV1
s2+2cV2
s3+6cV3
s4 (16)
(3) 1u is written instead of s in equation (16) to obtain
Z(u) cV0u + cV1
u2
+ 2cV2u3
+ 6cV3u4 (17)
(4) [22] Pade approximant for equation (17) can bewritten as
[22] b0 + b1u + b2u
2
1 + d1u + d2u2 (18)
en
cV0u + cV1
u2
+ 2cV2u3
+ 6cV3u4
b0 + b1u + b2u
2
1 + d1u + d2u2 (19)
en equation (19) can be written as
cV0u + cV1
u2
+ 2cV2u3
+ 6cV3u4
1113872 1113873 1 + d1u + d2u2
1113872 1113873
b0 + b1u + b2u2
(20)
Denominator coefficients d1 and d2 can be found byequating the coefficients of u3 and u4 Numerator coefficientsb0 b1 and b2 can be found by equating the coefficients ofu0 u and u2 Using Mathematica program hence
b0 0
b1 cV0
b2 cV1
3 minus 4cV1cV2
cV0+ 6cV3
cV02
f1113888 1113889
d1 minus 2cV1
cV2minus 3cV3
cV0
cV12 minus 2cV2
cV0
1113888 1113889
d2 minus 2minus 2cV2
2 + 3cV1cV3
cV12 minus 2cV2
cV0
1113888 1113889
(21)
Mathematical Problems in Engineering 3
(5) 1s is written instead of u in [22] hence F(s) isobtained as
F(s) cV0
s1113872 1113873 + b2s2( 1113857
1 + d1s( 1113857 + d2s2( 1113857 (22)
(6) Using inverse Laplace transform for F(s) the viralload V(t) is obtained as
V(t) A1eminus D1lowastt + cV0
minus A11113872 1113873eminus D2lowastt (23)
where
A1 1
2B
radic minus cV1
3+ 3cV1
cV2cV0
minus 3cV3cV0
2+ cV0
B
radic1113872 1113873
D1 minusA minus
B
radic
f
D2 minusA +
B
radic
f
f cV1
2minus 2cV2
cV0
A cV1cV2
minus 3cV3cV0
B minus 3cV1
2cV2
2+ 6cV1
3cV3
+ 8cV2
3cV0
minus 18cV1cV2
cV3cV0
+ 9cV3
2cV0
2
(24)
Equation (23) combined with equations (10) and (24)represents the general approximate analytical solution forthe viral load for the standard dynamic model given inequations (1)ndash(3) It is worth to notice that this solutioncontains the 7 parameters given in the system of equations(1)ndash(3) Hence it reflects properly the biological features
inherited in the system It can be noticed that the simplifiedapproximate solution given in equation (23) contains only 3parameters
4 Study Cases
To illustrate the capabilities and competences of the pro-posed method four cases of study are presented ese casesconsider the analysis of viral kinetics using the standard viraldynamic mathematical model of HCV e proposed PSLPmethod is applied to solve the nonlinear dynamic model ofthe viral kinetics for some patients after the initiation oftreatment with DAAs
For the first two study cases the predictions of the PSLPmethod are compared for each patient with viral load dataavailable in the literature [24 25] To provide best fits of datafor each patient parameters δ ε c and ρ are estimated usingthe relation for initial condition V0 (minus cdδ + βρsβcδ) andequation (23) In the first case patients were infected withHCV genotype 1 and treated with danoprevir e viral loadfor these patients is checked for 13 days after initiation ofdanoprevir the data are available in [24] and the parametersare given in Table 1 In the second case patients were treatedwith daclatasvir and the viral load of patients is checked for2 days after initiation of daclatasvir the data are available in[25] and the parameters are given in Table 2 Figures 1 and 2demonstrate the comparison between the solution of the PSLPmethod and the corresponding viral load data for each patient
Two other cases representing recently used combinationof multidrug DAA treatments are considered e pre-dictions of the PSLP method are compared with the cor-responding viral load obtained by simulation in [7]e usedparameter values are given in Table 3 where δ ε and c havethe same values assigned in [7] and ρ has been chosenaccordingly e PSLP results and the published simulationresults for 25 days after the initiation of treatment withTVR+PR are shown in Figure 3 e PSLP results and thepublished simulation results for 28 days after the initiation oftreatment with SOF+ SIM are shown in Figure 4 In both
Table 2 Parameter values used for the patients treated with daclatasvir
cases the PSLP solution and the simulation results are al-most identical e comparison shows that the proposedPSLP method provides adequate approximate analytical
solutions using the standard viral dynamic model for all theconsidered cases It is worth to mention that in these twocases the rate of curation is near 100
PSLPViral data
0 2 4 6 8 10 12Days
V (t) = 173 times 107endash738t + 905309endash0149t
10
1000
105
107H
CV R
NA
(a)
PSLPViral data
0
V (t) = 523707251endash1244t + 1100209endash028t
Days2 4 6 8 10 12
10
1000
105
107
HCV
RN
A
(b)
V (t) = 61532097endash1050t + 127402endash016t
PSLPViral data
0 2 4 6 8 10 12Days
10
1000
105
107
HCV
RN
A
(c)
V (t) = 934420474endash940t + 20572111endash034t
PSLPViral data
0Days
2 4 6 8 10 12
10
1000
105
107
HCV
RN
A
(d)
V (t) = 426351825endash1026t + 227693endash033t
PSLPViral data
0 2 4 6 8 10 12Days
10
1000
105
107
HCV
RN
A
(e)
Figure 1 Comparison between the approximate analytical solution using the PSLP method and viral data for patients treated withdanoprevir s 13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1 and the rest of the parameter values are given in Table 1(a) 01ndash94GK (b) 03ndash94HD (c) 03ndash94EA (d) 03ndash94KG and (e) 04ndash94XD
Mathematical Problems in Engineering 5
Since it is highly desirable to predict the patientrsquos re-sponse to a specific treatment regimen prior to the starting ofthe treatment itself the PSLP solution offers a simple and
powerful tool for medical specialists and physicians toperform this task ey can use the patientrsquos parameters tocalculate the constants in equations (10) and (24) and
00 05 10 15 20
HCV
RN
A
10
1000
105
107
PSLPViral data
Days
V (t) = 43453925endash3130t + 197657endash188t
(a)
00 05 10 15 20
HCV
RN
A
10
1000
105
107
PSLPViral data
Days
V (t) = 44575617endash21t + 92743endash039t
(b)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
Days
V (t) = 140 times 107endash2020t + 3215905endash127t
(c)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
Days
V (t) = 199312388endash22t + 213843endash075t
(d)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
DaysV (t) = 28059926endash2180t + 123903endasht
(e)
Figure 2 Comparison between the approximate analytical solution using the PSLP method and viral data for patients treated withdaclatasvir s 13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1 and the rest of the parameter values are given in Table 2(a) PAT 8 (b) PAT 42 (c) PAT 68 (d) PAT 69 and (e) PAT 83
6 Mathematical Problems in Engineering
substitute them in equation (23) to get a closed-form so-lution for the viral load Hence the viral load can be plottedversus time or simply the viral load can be estimated at anyinstant by direct substitution in the closed-form solution
5 Conclusions
Power series solution combined with the LaplacendashPaderesummation method (PSLP) has been used to obtain a
general approximate analytical solution for the nonlinearstandard viral dynamic model of HCV for patients treatedwith DAAs However the solution is not limited to HCVmodel and it can be applied to other viruses like HIV andHBV for example given that proper parameters are used Totest the applicability and accuracy of the proposed methodresults have been compared with viral load data and withpublished simulated results Satisfactory agreement betweenthe PSLP solution and the corresponding viral load data hasbeen found for all the considered casese PSLP results andthe published simulated results are almost identical ecomparison proves that this innovative PSLP solution can beused with confidence for solving the nonlinear standard viraldynamic model is solution can conveniently be used to fitpatient data and estimate parameter values So it wouldfacilitate for physicians to monitor the changes in the viralload due to changes in treatment and to deal with changes inpatientrsquos parameters
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D W Powell B Z Abramson J A Balint et al ldquoNationalinstitutes of health consensus development conference panelstatement management of hepatitis Crdquo Hepatology vol 26no 3 pp 2Sndash10S 1997
[2] J G McHutchison S C Gordon E R Schiff et al ldquoInterferonalfa-2b alone or in combination with ribavirin as initialtreatment for chronic hepatitis Crdquo New England Journal ofMedicine vol 339 no 21 pp 1485ndash1492 1998
[3] T Poynard J McHutchison M Manns et al ldquoImpact ofpegylated interferon alfa-2b and ribavirin on liver fibrosis inpatients with chronic hepatitis Crdquo Gastroenterology vol 122no 5 pp 1303ndash1313 2002
[4] T Asselah and P Marcellin ldquoNew direct-acting antiviralsrsquocombination for the treatment of chronic hepatitis Crdquo LiverInternational vol 31 pp 68ndash77 2011
[5] B Lam L Henry and Z Younossi ldquoSofosbuvir (Sovaldi) forthe treatment of hepatitis Crdquo Expert Review of ClinicalPharmacology vol 7 no 5 pp 555ndash566 2014
[6] J Chhatwal Q Chen T Ayer et al ldquoHepatitis C virus re-treatment in the era of direct-acting antivirals projections inthe USArdquo Alimentary Pharmacology amp 3erapeutics vol 47no 7 pp 1023ndash1031 2018
[7] V Cento T H T Nguyen D Di Carlo et al ldquoImprovementof ALT decay kinetics by all-oral HCV treatment role of
Table 3 Parameter values used with SOF+ SIM and TVR+PR
Figure 3 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with TVR+PR s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
PSLPRef [7]
HCV
RN
A
0 5 10 15 20 25Days
10
1000
105
107
V (t) = 95287079endash528t + 212179endash027t
Figure 4 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with SOF+SIM s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
Mathematical Problems in Engineering 7
NS5A inhibitors and differences with IFN-based regimensrdquoPLoS One vol 12 no 5 Article ID e0177352 2017
[8] T H T Nguyen J Guedj S L Uprichard A KohliS Kottilil and A S Perelson ldquoe paradox of highly ef-fective sofosbuvir-based combination therapy despite slowviral decline can we still rely on viral kineticsrdquo ScientificReports vol 7 no 1 Article ID 10233 2017
[9] World Health Organization Global Hepatitis Report 2017Global Hepatitis Programme Department of HIVAIDSWHO Geneva Switzerland 2017
[10] World Health Organization Combating Hepatitis B and C toReach Elimination by 2030 World Health OrganizationGeneva Switzerland 2016
[11] D S Jones and B D Sleeman ldquoDifferential equations andmathematical biologyrdquo inMathematical Biology andMedicineSeries Chapman amp HallCRC Boca Raton FL USA 2003
[12] A U Neumann N P Lam H Dahari et al ldquoHepatitis C viraldynamics in vivo and the antiviral efficacy of interferon-therapyrdquo Science vol 282 no 5386 pp 103ndash107 1998
[13] J Guedj and A U Neumann ldquoUnderstanding hepatitis Cviral dynamics with direct-acting antiviral agents due to theinterplay between intracellular replication and cellular in-fection dynamicsrdquo Journal of 3eoretical Biology vol 267no 3 pp 330ndash340 2010
[14] H Dahari J Guedj A S Perelson and T J LaydenldquoHepatitis C viral kinetics in the era of direct acting antiviralagents and interleukin-28Brdquo Current Hepatitis Reportsvol 10 no 3 pp 214ndash227 2011
[15] A Chaterjee J Guedj and A S Perelson ldquoMathematicalmodeling of HCV infection what can it teach us in the era ofdirect antiviral agentsrdquo Antiviral 3erapy vol 17 no 6pp 1171ndash1182 2012
[16] R J Payne M A Nowak and B S Blumberg ldquoe dynamicsof hepatitis B virus infectionrdquo Proceedings of the NationalAcademy of Sciences vol 93 no 13 pp 6542ndash6546 1996
[17] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[18] D Wodarz and M A Nowak ldquoMathematical models of HIVpathogenesis and treatmentrdquo BioEssays vol 24 no 12pp 1178ndash1187 2002
[19] W H Ho and A L F Chan ldquoHybrid taguchi-differentialevolution algorithm for parameter estimation of differentialequation models with application to HIV dynamicsrdquoMathematical Problems in Engineering vol 2011 Article ID514756 14 pages 2011
[20] 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 Sciencevol 271 no 5255 pp 1582ndash1586 1996
[21] Q Li and Y Xiao ldquoGlobal dynamics of a virus-immunesystem with virus-guided therapy and saturation growth ofvirusrdquo Mathematical Problems in Engineering vol 2018Article ID 4710586 18 pages 2018
[22] H Zarei A V Kamyad and S Effati ldquoMultiobjective optimalcontrol of HIV dynamicsrdquo Mathematical Problems in Engi-neering vol 2010 Article ID 568315 29 pages 2010
[23] T S N Asih S Lenhart S Wise et al ldquoe dynamics of HPVinfection and cervical cancer cellsrdquo Bulletin of MathematicalBiology vol 78 no 1 pp 4ndash20 2016
[24] L Rong J Guedj H Dahari et al ldquoAnalysis of hepatitis Cvirus decline during treatment with the protease inhibitor
danoprevir using a multiscale modelrdquo PLoS ComputationalBiology vol 9 no 3 Article ID e1002959 2013
[25] J Guedj H Dahari L Rong et al ldquoModeling shows that theNS5A inhibitor daclatasvir has two modes of action and yieldsa shorter estimate of the hepatitis C virus half-liferdquo Pro-ceedings of the National Academy of Sciences vol 110 no 10pp 3991ndash3996 2013
[26] L Rong and A S Perelson ldquoMathematical analysis of mul-tiscale models for hepatitis C virus dynamics under therapywith direct-acting antiviral agentsrdquoMathematical Biosciencesvol 245 no 1 pp 22ndash30 2013
[27] H A Elkaranshawy A M Abdelrazek and H M EzzatldquoPower series solution to sliding velocity in three-dimensionalmultibody systems with impact and frictionrdquo InternationalJournal of Mathematical Computational Physical Electricaland Computer Engineering vol 9 no 10 2015
[28] I M Abdelrazik and H A Elkaranshawy ldquoExtended Parker-Sochacki method for Michaelis-Menten enzymatic reactionmodelrdquo Analytical Biochemistry vol 496 pp 50ndash54 2016
[29] I M Abdelrazik H A Elkaranshawy and A M AbdelrazekldquoModified Parker-Sochacki method for solving nonlinearoscillatorsrdquo Mechanics Based Design of Structures and Ma-chines vol 45 no 2 pp 239ndash252 2016
[30] I M Abdelrazik Improving the solutions of parker sochackimethod for nonlinear ordinary differential equations PhDthesis Department of Engineering Mathematics and PhysicsFaculty of Engineering Alexandria University AlexandriaEgypt 2016
[31] G A Baker and P R Graves-Morris Pade Approximants vol59 Cambridge University Press Cambridge UK 1996
[32] B Raftari and A Yildirim ldquoSeries solution of a nonlinearODE arising in magnetohydrodynamic by HPM-Pade tech-niquerdquo Computers amp Mathematics with Applications vol 61no 6 pp 1676ndash1681 2011
[33] J Aubard P Levoir A Denis and P Claverie ldquoDirectanalysis of chemical relaxation signals by a method based onthe combination of Laplace transform and Pade approx-imantsrdquo Computers amp Chemistry vol 11 no 3 pp 163ndash1781987
[34] H Vazquez-Leal and F Guerrero ldquoApplication of seriesmethod with Pade and Laplace-Pade resummation methodsto solve a model for the evolution of smoking habit in SpainrdquoComputational and Applied Mathematics vol 33 no 1pp 181ndash192 2014
8 Mathematical Problems in Engineering
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Stochastic AnalysisInternational Journal of
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established service coverage targets to eliminate HCV as apublic health threat by 2030 [9 10] Meanwhile there re-main many barriers that need to be overcome Such barriersinclude the development of simplified and highly effectivedrug regimens improving the rates of detection of infectionand the availability of funds including financial and medicalexpertise
Mathematical modeling is a useful tool in the study ofvirus dynamics for many types of viruses such as HCVHBV HPV and HIV It can be used to predict behaviorunder certain conditions or decide which parameters en-hance the spread of disease It may also be used to calculatethe medications required to eradicate the disease or at leastget it under control ese models can be used also tounderstand the biological mechanisms and interpret theexperimental results [11] Mathematical models in the formof a system of differential equations for the basic dynamics invivo were developed and analyzed for HCV [8 12ndash15] HBV[16 17] HIV [18ndash22] and HPV [23] Models for HCVtreatment with DAAs therapy are considered in [8 14 15]Authors in [24ndash26] developed a multiscale model thatconsidered both intracellular viral RNA replication andextracellular viral infection
In general numerical solutions were obtained for themodels However analytical solutions could be useful for theestimation of parameters and for direct and simple pre-dictions for the viral loads Seldom an analytical or anapproximate analytical solution can be found in the liter-ature A simplified analytical solution for the standardmodelfor HCV viral dynamics was constructed in [12] It wasassumed that the number of target cells is constant hencethe system of ordinary differential equations (ODE) is re-duced linearized and solved It can be noticed that thissystem of ODE contains 7 parameters while the obtainedsolutions contain only 3 parameters An analytical solutionwas obtained for the multiscale model in [24 26] All newinfections during therapy were ignored in that analysis
In this research work the approximate analytical solu-tion is obtained for the standard model Similar standardmodels have been used for HCV HBV and HIV in theliterature [8 12ndash23] Nevertheless the model considered isthe HCV model ere is no reduction for the system ofODE or assumptions to simplify the equations of the modelYet the approximation is in the solution not in the system ofdifferential equations erefore it is more accurate andsatisfactory than the previously mentioned analytical solu-tions Power series solutions are first obtained for the systemof ODE and then the LaplacendashPade resummation method(PSLP) is used to obtain the entailed approximate analyticalsolution e technique used depends on the methodologydeveloped in [27ndash34] for the general system of ODE esolution is used for the analysis of the standard viral dynamicmodel of HCV after any type of DAA treatment initiation
2 Standard Model
Consider the system of nonlinear ODEs for the standardviral dynamic mathematical model for HCV kinetics duringtreatment [11ndash14]
dT
dt s minus dT minus βVT (1)
dI
dt βVT minus δI (2)
dV
dt (1 minus ε)ρI minus cV (3)
where T is the target cells which are produced at a constantrate s and are assumed to die per capita rate d and areinfected by virus V at rate β Infected cells are assumed todie at per capita rate δ Virions are generated at rate ρ perinfected cell and cleared from serum at rate c per virionTreatment is assumed to reduce the average viral productionrate per infected cell from ρ to (1 minus ε)ρ where ε is the in vivoantiviral effectiveness of therapy (0lt εlt 1) At treatmentinitiation (t 0) we used the standard assumption[10 21 22] that the system in the pretreatment is in steadystate given by
dT
dt 0
dI
dt 0
dV
dt 0
(4)
then
T(0) T0 cδβρ
I(0) I0 minus cdδ + βρs
βδρ
V(0) V0 minus cdδ + βρs
βcδ
(5)
3 Solution Method
31 Power Series Solution Assuming the following solutionsof order N
T(t) 1113944N
i0cTi
ti (6)
I(t) 1113944N
i0cIi
ti (7)
V(t) 1113944N
i0cVi
ti (8)
Substituting equations (6)ndash(8) into equations (1)ndash(3) andequating terms having the same powers of t e cTi
cIi and
cVicoefficients can be calculated For instance the power
series solution for V(t) is
2 Mathematical Problems in Engineering
V(t) 1113944N
i0cVi
ti
cV0+ cV1
t + cV2t2
+ cV3t3
+ middot middot middot (9)
where cV0 V0 is the initial condition of V(t) and the
coefficients cV1 cV2
and cV3are as follows
cV1 minus cεc0
cV2 05c
2εc0
cV316c0 minus c
3ε minus c2δε minus cdδ + ρs minus βc0cδ + ε2c2δ1113872
+ cdδε minus ρsβε + cδc0βε)
(10)
32 LaplacendashPade Resummation e Pade approximantfor a series function is a function with the same powerseries expansions as the original series function It issuitable for approximating a divergent series functioneapproximant is derived by expanding the function as aratio of two power series and determining both the nu-merator and denominator coefficients [28ndash34] If we havea function f(u) that can be represented in a power seriesform as
f(u) 1113944infin
i0cui
ui (11)
A Pade approximant is a rational functionL
M1113876 1113877
b0 + b1u + b2u2 + middot middot middot + bLuL
d0 + d1u + d2u2 + middot middot middot + dMuM
(12)
which has a Maclaurin expansion that agrees with the seriesof equation (11) approximately d0 is chosen to be equal to 1So there are L + 1 independent numerator coefficients andM independent denominator coefficients making L + M +
1 unknown coefficients e [LM] should fit the powerseries of equation (11) through the orders 1 u u2 uL+MHence
1113944
infin
i0cui
ui
b0 + b1u + b2u
2 + middot middot middot + bLuL
1 + d1u + d2u2 + middot middot middot + dMuM
+ O uL+M+1
1113872 1113873
(13)
Up to the term of order L + M we can write
cu0+ cu1
u + middot middot middot + cuL+Mu
L+M1113872 1113873 1 + d1u + d2u
2+ middot middot middot + dMu
M1113872 1113873
b0 + b1u + b2u2
+ middot middot middot + bLuL
(14)Denominator coefficients d1 d1 dM can be found
from equation (14) by equating the coefficients ofuL uL+1 uL+M Numerator coefficients b0 + b1 + middot middot middot bL
can be found from equation (14) by equating the coefficientsof u0 u u2 uL Hence we have constructed the [LM]
Pade approximant which agrees with 1113936infini0 cui
ui through theorder uL+M
To extend the convergence zone of the power seriessolution LaplacendashPade resummation can be introduced In
LaplacendashPade resummation the Laplace transform is ap-plied to the given series function then Pade approximate isobtained for the resulted function and consequently theinverse Laplace transform is used to specify the final solutionin the form of exponential functions To apply LaplacendashPaderesummation to the series solution of the viral load V(t) thefollowing procedure is employed
(1) From equation (9) the third-order power seriessolution of V(t) is
V(t) cV0+ cV1
t + cV2t2
+ cV3t3 (15)
(2) Taking Laplace transform to the series solution inequation (15) to obtain
F(s) cV0
s+
cV1
s2+2cV2
s3+6cV3
s4 (16)
(3) 1u is written instead of s in equation (16) to obtain
Z(u) cV0u + cV1
u2
+ 2cV2u3
+ 6cV3u4 (17)
(4) [22] Pade approximant for equation (17) can bewritten as
[22] b0 + b1u + b2u
2
1 + d1u + d2u2 (18)
en
cV0u + cV1
u2
+ 2cV2u3
+ 6cV3u4
b0 + b1u + b2u
2
1 + d1u + d2u2 (19)
en equation (19) can be written as
cV0u + cV1
u2
+ 2cV2u3
+ 6cV3u4
1113872 1113873 1 + d1u + d2u2
1113872 1113873
b0 + b1u + b2u2
(20)
Denominator coefficients d1 and d2 can be found byequating the coefficients of u3 and u4 Numerator coefficientsb0 b1 and b2 can be found by equating the coefficients ofu0 u and u2 Using Mathematica program hence
b0 0
b1 cV0
b2 cV1
3 minus 4cV1cV2
cV0+ 6cV3
cV02
f1113888 1113889
d1 minus 2cV1
cV2minus 3cV3
cV0
cV12 minus 2cV2
cV0
1113888 1113889
d2 minus 2minus 2cV2
2 + 3cV1cV3
cV12 minus 2cV2
cV0
1113888 1113889
(21)
Mathematical Problems in Engineering 3
(5) 1s is written instead of u in [22] hence F(s) isobtained as
F(s) cV0
s1113872 1113873 + b2s2( 1113857
1 + d1s( 1113857 + d2s2( 1113857 (22)
(6) Using inverse Laplace transform for F(s) the viralload V(t) is obtained as
V(t) A1eminus D1lowastt + cV0
minus A11113872 1113873eminus D2lowastt (23)
where
A1 1
2B
radic minus cV1
3+ 3cV1
cV2cV0
minus 3cV3cV0
2+ cV0
B
radic1113872 1113873
D1 minusA minus
B
radic
f
D2 minusA +
B
radic
f
f cV1
2minus 2cV2
cV0
A cV1cV2
minus 3cV3cV0
B minus 3cV1
2cV2
2+ 6cV1
3cV3
+ 8cV2
3cV0
minus 18cV1cV2
cV3cV0
+ 9cV3
2cV0
2
(24)
Equation (23) combined with equations (10) and (24)represents the general approximate analytical solution forthe viral load for the standard dynamic model given inequations (1)ndash(3) It is worth to notice that this solutioncontains the 7 parameters given in the system of equations(1)ndash(3) Hence it reflects properly the biological features
inherited in the system It can be noticed that the simplifiedapproximate solution given in equation (23) contains only 3parameters
4 Study Cases
To illustrate the capabilities and competences of the pro-posed method four cases of study are presented ese casesconsider the analysis of viral kinetics using the standard viraldynamic mathematical model of HCV e proposed PSLPmethod is applied to solve the nonlinear dynamic model ofthe viral kinetics for some patients after the initiation oftreatment with DAAs
For the first two study cases the predictions of the PSLPmethod are compared for each patient with viral load dataavailable in the literature [24 25] To provide best fits of datafor each patient parameters δ ε c and ρ are estimated usingthe relation for initial condition V0 (minus cdδ + βρsβcδ) andequation (23) In the first case patients were infected withHCV genotype 1 and treated with danoprevir e viral loadfor these patients is checked for 13 days after initiation ofdanoprevir the data are available in [24] and the parametersare given in Table 1 In the second case patients were treatedwith daclatasvir and the viral load of patients is checked for2 days after initiation of daclatasvir the data are available in[25] and the parameters are given in Table 2 Figures 1 and 2demonstrate the comparison between the solution of the PSLPmethod and the corresponding viral load data for each patient
Two other cases representing recently used combinationof multidrug DAA treatments are considered e pre-dictions of the PSLP method are compared with the cor-responding viral load obtained by simulation in [7]e usedparameter values are given in Table 3 where δ ε and c havethe same values assigned in [7] and ρ has been chosenaccordingly e PSLP results and the published simulationresults for 25 days after the initiation of treatment withTVR+PR are shown in Figure 3 e PSLP results and thepublished simulation results for 28 days after the initiation oftreatment with SOF+ SIM are shown in Figure 4 In both
Table 2 Parameter values used for the patients treated with daclatasvir
cases the PSLP solution and the simulation results are al-most identical e comparison shows that the proposedPSLP method provides adequate approximate analytical
solutions using the standard viral dynamic model for all theconsidered cases It is worth to mention that in these twocases the rate of curation is near 100
PSLPViral data
0 2 4 6 8 10 12Days
V (t) = 173 times 107endash738t + 905309endash0149t
10
1000
105
107H
CV R
NA
(a)
PSLPViral data
0
V (t) = 523707251endash1244t + 1100209endash028t
Days2 4 6 8 10 12
10
1000
105
107
HCV
RN
A
(b)
V (t) = 61532097endash1050t + 127402endash016t
PSLPViral data
0 2 4 6 8 10 12Days
10
1000
105
107
HCV
RN
A
(c)
V (t) = 934420474endash940t + 20572111endash034t
PSLPViral data
0Days
2 4 6 8 10 12
10
1000
105
107
HCV
RN
A
(d)
V (t) = 426351825endash1026t + 227693endash033t
PSLPViral data
0 2 4 6 8 10 12Days
10
1000
105
107
HCV
RN
A
(e)
Figure 1 Comparison between the approximate analytical solution using the PSLP method and viral data for patients treated withdanoprevir s 13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1 and the rest of the parameter values are given in Table 1(a) 01ndash94GK (b) 03ndash94HD (c) 03ndash94EA (d) 03ndash94KG and (e) 04ndash94XD
Mathematical Problems in Engineering 5
Since it is highly desirable to predict the patientrsquos re-sponse to a specific treatment regimen prior to the starting ofthe treatment itself the PSLP solution offers a simple and
powerful tool for medical specialists and physicians toperform this task ey can use the patientrsquos parameters tocalculate the constants in equations (10) and (24) and
00 05 10 15 20
HCV
RN
A
10
1000
105
107
PSLPViral data
Days
V (t) = 43453925endash3130t + 197657endash188t
(a)
00 05 10 15 20
HCV
RN
A
10
1000
105
107
PSLPViral data
Days
V (t) = 44575617endash21t + 92743endash039t
(b)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
Days
V (t) = 140 times 107endash2020t + 3215905endash127t
(c)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
Days
V (t) = 199312388endash22t + 213843endash075t
(d)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
DaysV (t) = 28059926endash2180t + 123903endasht
(e)
Figure 2 Comparison between the approximate analytical solution using the PSLP method and viral data for patients treated withdaclatasvir s 13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1 and the rest of the parameter values are given in Table 2(a) PAT 8 (b) PAT 42 (c) PAT 68 (d) PAT 69 and (e) PAT 83
6 Mathematical Problems in Engineering
substitute them in equation (23) to get a closed-form so-lution for the viral load Hence the viral load can be plottedversus time or simply the viral load can be estimated at anyinstant by direct substitution in the closed-form solution
5 Conclusions
Power series solution combined with the LaplacendashPaderesummation method (PSLP) has been used to obtain a
general approximate analytical solution for the nonlinearstandard viral dynamic model of HCV for patients treatedwith DAAs However the solution is not limited to HCVmodel and it can be applied to other viruses like HIV andHBV for example given that proper parameters are used Totest the applicability and accuracy of the proposed methodresults have been compared with viral load data and withpublished simulated results Satisfactory agreement betweenthe PSLP solution and the corresponding viral load data hasbeen found for all the considered casese PSLP results andthe published simulated results are almost identical ecomparison proves that this innovative PSLP solution can beused with confidence for solving the nonlinear standard viraldynamic model is solution can conveniently be used to fitpatient data and estimate parameter values So it wouldfacilitate for physicians to monitor the changes in the viralload due to changes in treatment and to deal with changes inpatientrsquos parameters
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D W Powell B Z Abramson J A Balint et al ldquoNationalinstitutes of health consensus development conference panelstatement management of hepatitis Crdquo Hepatology vol 26no 3 pp 2Sndash10S 1997
[2] J G McHutchison S C Gordon E R Schiff et al ldquoInterferonalfa-2b alone or in combination with ribavirin as initialtreatment for chronic hepatitis Crdquo New England Journal ofMedicine vol 339 no 21 pp 1485ndash1492 1998
[3] T Poynard J McHutchison M Manns et al ldquoImpact ofpegylated interferon alfa-2b and ribavirin on liver fibrosis inpatients with chronic hepatitis Crdquo Gastroenterology vol 122no 5 pp 1303ndash1313 2002
[4] T Asselah and P Marcellin ldquoNew direct-acting antiviralsrsquocombination for the treatment of chronic hepatitis Crdquo LiverInternational vol 31 pp 68ndash77 2011
[5] B Lam L Henry and Z Younossi ldquoSofosbuvir (Sovaldi) forthe treatment of hepatitis Crdquo Expert Review of ClinicalPharmacology vol 7 no 5 pp 555ndash566 2014
[6] J Chhatwal Q Chen T Ayer et al ldquoHepatitis C virus re-treatment in the era of direct-acting antivirals projections inthe USArdquo Alimentary Pharmacology amp 3erapeutics vol 47no 7 pp 1023ndash1031 2018
[7] V Cento T H T Nguyen D Di Carlo et al ldquoImprovementof ALT decay kinetics by all-oral HCV treatment role of
Table 3 Parameter values used with SOF+ SIM and TVR+PR
Figure 3 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with TVR+PR s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
PSLPRef [7]
HCV
RN
A
0 5 10 15 20 25Days
10
1000
105
107
V (t) = 95287079endash528t + 212179endash027t
Figure 4 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with SOF+SIM s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
Mathematical Problems in Engineering 7
NS5A inhibitors and differences with IFN-based regimensrdquoPLoS One vol 12 no 5 Article ID e0177352 2017
[8] T H T Nguyen J Guedj S L Uprichard A KohliS Kottilil and A S Perelson ldquoe paradox of highly ef-fective sofosbuvir-based combination therapy despite slowviral decline can we still rely on viral kineticsrdquo ScientificReports vol 7 no 1 Article ID 10233 2017
[9] World Health Organization Global Hepatitis Report 2017Global Hepatitis Programme Department of HIVAIDSWHO Geneva Switzerland 2017
[10] World Health Organization Combating Hepatitis B and C toReach Elimination by 2030 World Health OrganizationGeneva Switzerland 2016
[11] D S Jones and B D Sleeman ldquoDifferential equations andmathematical biologyrdquo inMathematical Biology andMedicineSeries Chapman amp HallCRC Boca Raton FL USA 2003
[12] A U Neumann N P Lam H Dahari et al ldquoHepatitis C viraldynamics in vivo and the antiviral efficacy of interferon-therapyrdquo Science vol 282 no 5386 pp 103ndash107 1998
[13] J Guedj and A U Neumann ldquoUnderstanding hepatitis Cviral dynamics with direct-acting antiviral agents due to theinterplay between intracellular replication and cellular in-fection dynamicsrdquo Journal of 3eoretical Biology vol 267no 3 pp 330ndash340 2010
[14] H Dahari J Guedj A S Perelson and T J LaydenldquoHepatitis C viral kinetics in the era of direct acting antiviralagents and interleukin-28Brdquo Current Hepatitis Reportsvol 10 no 3 pp 214ndash227 2011
[15] A Chaterjee J Guedj and A S Perelson ldquoMathematicalmodeling of HCV infection what can it teach us in the era ofdirect antiviral agentsrdquo Antiviral 3erapy vol 17 no 6pp 1171ndash1182 2012
[16] R J Payne M A Nowak and B S Blumberg ldquoe dynamicsof hepatitis B virus infectionrdquo Proceedings of the NationalAcademy of Sciences vol 93 no 13 pp 6542ndash6546 1996
[17] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[18] D Wodarz and M A Nowak ldquoMathematical models of HIVpathogenesis and treatmentrdquo BioEssays vol 24 no 12pp 1178ndash1187 2002
[19] W H Ho and A L F Chan ldquoHybrid taguchi-differentialevolution algorithm for parameter estimation of differentialequation models with application to HIV dynamicsrdquoMathematical Problems in Engineering vol 2011 Article ID514756 14 pages 2011
[20] 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 Sciencevol 271 no 5255 pp 1582ndash1586 1996
[21] Q Li and Y Xiao ldquoGlobal dynamics of a virus-immunesystem with virus-guided therapy and saturation growth ofvirusrdquo Mathematical Problems in Engineering vol 2018Article ID 4710586 18 pages 2018
[22] H Zarei A V Kamyad and S Effati ldquoMultiobjective optimalcontrol of HIV dynamicsrdquo Mathematical Problems in Engi-neering vol 2010 Article ID 568315 29 pages 2010
[23] T S N Asih S Lenhart S Wise et al ldquoe dynamics of HPVinfection and cervical cancer cellsrdquo Bulletin of MathematicalBiology vol 78 no 1 pp 4ndash20 2016
[24] L Rong J Guedj H Dahari et al ldquoAnalysis of hepatitis Cvirus decline during treatment with the protease inhibitor
danoprevir using a multiscale modelrdquo PLoS ComputationalBiology vol 9 no 3 Article ID e1002959 2013
[25] J Guedj H Dahari L Rong et al ldquoModeling shows that theNS5A inhibitor daclatasvir has two modes of action and yieldsa shorter estimate of the hepatitis C virus half-liferdquo Pro-ceedings of the National Academy of Sciences vol 110 no 10pp 3991ndash3996 2013
[26] L Rong and A S Perelson ldquoMathematical analysis of mul-tiscale models for hepatitis C virus dynamics under therapywith direct-acting antiviral agentsrdquoMathematical Biosciencesvol 245 no 1 pp 22ndash30 2013
[27] H A Elkaranshawy A M Abdelrazek and H M EzzatldquoPower series solution to sliding velocity in three-dimensionalmultibody systems with impact and frictionrdquo InternationalJournal of Mathematical Computational Physical Electricaland Computer Engineering vol 9 no 10 2015
[28] I M Abdelrazik and H A Elkaranshawy ldquoExtended Parker-Sochacki method for Michaelis-Menten enzymatic reactionmodelrdquo Analytical Biochemistry vol 496 pp 50ndash54 2016
[29] I M Abdelrazik H A Elkaranshawy and A M AbdelrazekldquoModified Parker-Sochacki method for solving nonlinearoscillatorsrdquo Mechanics Based Design of Structures and Ma-chines vol 45 no 2 pp 239ndash252 2016
[30] I M Abdelrazik Improving the solutions of parker sochackimethod for nonlinear ordinary differential equations PhDthesis Department of Engineering Mathematics and PhysicsFaculty of Engineering Alexandria University AlexandriaEgypt 2016
[31] G A Baker and P R Graves-Morris Pade Approximants vol59 Cambridge University Press Cambridge UK 1996
[32] B Raftari and A Yildirim ldquoSeries solution of a nonlinearODE arising in magnetohydrodynamic by HPM-Pade tech-niquerdquo Computers amp Mathematics with Applications vol 61no 6 pp 1676ndash1681 2011
[33] J Aubard P Levoir A Denis and P Claverie ldquoDirectanalysis of chemical relaxation signals by a method based onthe combination of Laplace transform and Pade approx-imantsrdquo Computers amp Chemistry vol 11 no 3 pp 163ndash1781987
[34] H Vazquez-Leal and F Guerrero ldquoApplication of seriesmethod with Pade and Laplace-Pade resummation methodsto solve a model for the evolution of smoking habit in SpainrdquoComputational and Applied Mathematics vol 33 no 1pp 181ndash192 2014
8 Mathematical Problems in Engineering
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AnalysisInternational Journal of
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Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
V(t) 1113944N
i0cVi
ti
cV0+ cV1
t + cV2t2
+ cV3t3
+ middot middot middot (9)
where cV0 V0 is the initial condition of V(t) and the
coefficients cV1 cV2
and cV3are as follows
cV1 minus cεc0
cV2 05c
2εc0
cV316c0 minus c
3ε minus c2δε minus cdδ + ρs minus βc0cδ + ε2c2δ1113872
+ cdδε minus ρsβε + cδc0βε)
(10)
32 LaplacendashPade Resummation e Pade approximantfor a series function is a function with the same powerseries expansions as the original series function It issuitable for approximating a divergent series functioneapproximant is derived by expanding the function as aratio of two power series and determining both the nu-merator and denominator coefficients [28ndash34] If we havea function f(u) that can be represented in a power seriesform as
f(u) 1113944infin
i0cui
ui (11)
A Pade approximant is a rational functionL
M1113876 1113877
b0 + b1u + b2u2 + middot middot middot + bLuL
d0 + d1u + d2u2 + middot middot middot + dMuM
(12)
which has a Maclaurin expansion that agrees with the seriesof equation (11) approximately d0 is chosen to be equal to 1So there are L + 1 independent numerator coefficients andM independent denominator coefficients making L + M +
1 unknown coefficients e [LM] should fit the powerseries of equation (11) through the orders 1 u u2 uL+MHence
1113944
infin
i0cui
ui
b0 + b1u + b2u
2 + middot middot middot + bLuL
1 + d1u + d2u2 + middot middot middot + dMuM
+ O uL+M+1
1113872 1113873
(13)
Up to the term of order L + M we can write
cu0+ cu1
u + middot middot middot + cuL+Mu
L+M1113872 1113873 1 + d1u + d2u
2+ middot middot middot + dMu
M1113872 1113873
b0 + b1u + b2u2
+ middot middot middot + bLuL
(14)Denominator coefficients d1 d1 dM can be found
from equation (14) by equating the coefficients ofuL uL+1 uL+M Numerator coefficients b0 + b1 + middot middot middot bL
can be found from equation (14) by equating the coefficientsof u0 u u2 uL Hence we have constructed the [LM]
Pade approximant which agrees with 1113936infini0 cui
ui through theorder uL+M
To extend the convergence zone of the power seriessolution LaplacendashPade resummation can be introduced In
LaplacendashPade resummation the Laplace transform is ap-plied to the given series function then Pade approximate isobtained for the resulted function and consequently theinverse Laplace transform is used to specify the final solutionin the form of exponential functions To apply LaplacendashPaderesummation to the series solution of the viral load V(t) thefollowing procedure is employed
(1) From equation (9) the third-order power seriessolution of V(t) is
V(t) cV0+ cV1
t + cV2t2
+ cV3t3 (15)
(2) Taking Laplace transform to the series solution inequation (15) to obtain
F(s) cV0
s+
cV1
s2+2cV2
s3+6cV3
s4 (16)
(3) 1u is written instead of s in equation (16) to obtain
Z(u) cV0u + cV1
u2
+ 2cV2u3
+ 6cV3u4 (17)
(4) [22] Pade approximant for equation (17) can bewritten as
[22] b0 + b1u + b2u
2
1 + d1u + d2u2 (18)
en
cV0u + cV1
u2
+ 2cV2u3
+ 6cV3u4
b0 + b1u + b2u
2
1 + d1u + d2u2 (19)
en equation (19) can be written as
cV0u + cV1
u2
+ 2cV2u3
+ 6cV3u4
1113872 1113873 1 + d1u + d2u2
1113872 1113873
b0 + b1u + b2u2
(20)
Denominator coefficients d1 and d2 can be found byequating the coefficients of u3 and u4 Numerator coefficientsb0 b1 and b2 can be found by equating the coefficients ofu0 u and u2 Using Mathematica program hence
b0 0
b1 cV0
b2 cV1
3 minus 4cV1cV2
cV0+ 6cV3
cV02
f1113888 1113889
d1 minus 2cV1
cV2minus 3cV3
cV0
cV12 minus 2cV2
cV0
1113888 1113889
d2 minus 2minus 2cV2
2 + 3cV1cV3
cV12 minus 2cV2
cV0
1113888 1113889
(21)
Mathematical Problems in Engineering 3
(5) 1s is written instead of u in [22] hence F(s) isobtained as
F(s) cV0
s1113872 1113873 + b2s2( 1113857
1 + d1s( 1113857 + d2s2( 1113857 (22)
(6) Using inverse Laplace transform for F(s) the viralload V(t) is obtained as
V(t) A1eminus D1lowastt + cV0
minus A11113872 1113873eminus D2lowastt (23)
where
A1 1
2B
radic minus cV1
3+ 3cV1
cV2cV0
minus 3cV3cV0
2+ cV0
B
radic1113872 1113873
D1 minusA minus
B
radic
f
D2 minusA +
B
radic
f
f cV1
2minus 2cV2
cV0
A cV1cV2
minus 3cV3cV0
B minus 3cV1
2cV2
2+ 6cV1
3cV3
+ 8cV2
3cV0
minus 18cV1cV2
cV3cV0
+ 9cV3
2cV0
2
(24)
Equation (23) combined with equations (10) and (24)represents the general approximate analytical solution forthe viral load for the standard dynamic model given inequations (1)ndash(3) It is worth to notice that this solutioncontains the 7 parameters given in the system of equations(1)ndash(3) Hence it reflects properly the biological features
inherited in the system It can be noticed that the simplifiedapproximate solution given in equation (23) contains only 3parameters
4 Study Cases
To illustrate the capabilities and competences of the pro-posed method four cases of study are presented ese casesconsider the analysis of viral kinetics using the standard viraldynamic mathematical model of HCV e proposed PSLPmethod is applied to solve the nonlinear dynamic model ofthe viral kinetics for some patients after the initiation oftreatment with DAAs
For the first two study cases the predictions of the PSLPmethod are compared for each patient with viral load dataavailable in the literature [24 25] To provide best fits of datafor each patient parameters δ ε c and ρ are estimated usingthe relation for initial condition V0 (minus cdδ + βρsβcδ) andequation (23) In the first case patients were infected withHCV genotype 1 and treated with danoprevir e viral loadfor these patients is checked for 13 days after initiation ofdanoprevir the data are available in [24] and the parametersare given in Table 1 In the second case patients were treatedwith daclatasvir and the viral load of patients is checked for2 days after initiation of daclatasvir the data are available in[25] and the parameters are given in Table 2 Figures 1 and 2demonstrate the comparison between the solution of the PSLPmethod and the corresponding viral load data for each patient
Two other cases representing recently used combinationof multidrug DAA treatments are considered e pre-dictions of the PSLP method are compared with the cor-responding viral load obtained by simulation in [7]e usedparameter values are given in Table 3 where δ ε and c havethe same values assigned in [7] and ρ has been chosenaccordingly e PSLP results and the published simulationresults for 25 days after the initiation of treatment withTVR+PR are shown in Figure 3 e PSLP results and thepublished simulation results for 28 days after the initiation oftreatment with SOF+ SIM are shown in Figure 4 In both
Table 2 Parameter values used for the patients treated with daclatasvir
cases the PSLP solution and the simulation results are al-most identical e comparison shows that the proposedPSLP method provides adequate approximate analytical
solutions using the standard viral dynamic model for all theconsidered cases It is worth to mention that in these twocases the rate of curation is near 100
PSLPViral data
0 2 4 6 8 10 12Days
V (t) = 173 times 107endash738t + 905309endash0149t
10
1000
105
107H
CV R
NA
(a)
PSLPViral data
0
V (t) = 523707251endash1244t + 1100209endash028t
Days2 4 6 8 10 12
10
1000
105
107
HCV
RN
A
(b)
V (t) = 61532097endash1050t + 127402endash016t
PSLPViral data
0 2 4 6 8 10 12Days
10
1000
105
107
HCV
RN
A
(c)
V (t) = 934420474endash940t + 20572111endash034t
PSLPViral data
0Days
2 4 6 8 10 12
10
1000
105
107
HCV
RN
A
(d)
V (t) = 426351825endash1026t + 227693endash033t
PSLPViral data
0 2 4 6 8 10 12Days
10
1000
105
107
HCV
RN
A
(e)
Figure 1 Comparison between the approximate analytical solution using the PSLP method and viral data for patients treated withdanoprevir s 13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1 and the rest of the parameter values are given in Table 1(a) 01ndash94GK (b) 03ndash94HD (c) 03ndash94EA (d) 03ndash94KG and (e) 04ndash94XD
Mathematical Problems in Engineering 5
Since it is highly desirable to predict the patientrsquos re-sponse to a specific treatment regimen prior to the starting ofthe treatment itself the PSLP solution offers a simple and
powerful tool for medical specialists and physicians toperform this task ey can use the patientrsquos parameters tocalculate the constants in equations (10) and (24) and
00 05 10 15 20
HCV
RN
A
10
1000
105
107
PSLPViral data
Days
V (t) = 43453925endash3130t + 197657endash188t
(a)
00 05 10 15 20
HCV
RN
A
10
1000
105
107
PSLPViral data
Days
V (t) = 44575617endash21t + 92743endash039t
(b)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
Days
V (t) = 140 times 107endash2020t + 3215905endash127t
(c)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
Days
V (t) = 199312388endash22t + 213843endash075t
(d)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
DaysV (t) = 28059926endash2180t + 123903endasht
(e)
Figure 2 Comparison between the approximate analytical solution using the PSLP method and viral data for patients treated withdaclatasvir s 13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1 and the rest of the parameter values are given in Table 2(a) PAT 8 (b) PAT 42 (c) PAT 68 (d) PAT 69 and (e) PAT 83
6 Mathematical Problems in Engineering
substitute them in equation (23) to get a closed-form so-lution for the viral load Hence the viral load can be plottedversus time or simply the viral load can be estimated at anyinstant by direct substitution in the closed-form solution
5 Conclusions
Power series solution combined with the LaplacendashPaderesummation method (PSLP) has been used to obtain a
general approximate analytical solution for the nonlinearstandard viral dynamic model of HCV for patients treatedwith DAAs However the solution is not limited to HCVmodel and it can be applied to other viruses like HIV andHBV for example given that proper parameters are used Totest the applicability and accuracy of the proposed methodresults have been compared with viral load data and withpublished simulated results Satisfactory agreement betweenthe PSLP solution and the corresponding viral load data hasbeen found for all the considered casese PSLP results andthe published simulated results are almost identical ecomparison proves that this innovative PSLP solution can beused with confidence for solving the nonlinear standard viraldynamic model is solution can conveniently be used to fitpatient data and estimate parameter values So it wouldfacilitate for physicians to monitor the changes in the viralload due to changes in treatment and to deal with changes inpatientrsquos parameters
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D W Powell B Z Abramson J A Balint et al ldquoNationalinstitutes of health consensus development conference panelstatement management of hepatitis Crdquo Hepatology vol 26no 3 pp 2Sndash10S 1997
[2] J G McHutchison S C Gordon E R Schiff et al ldquoInterferonalfa-2b alone or in combination with ribavirin as initialtreatment for chronic hepatitis Crdquo New England Journal ofMedicine vol 339 no 21 pp 1485ndash1492 1998
[3] T Poynard J McHutchison M Manns et al ldquoImpact ofpegylated interferon alfa-2b and ribavirin on liver fibrosis inpatients with chronic hepatitis Crdquo Gastroenterology vol 122no 5 pp 1303ndash1313 2002
[4] T Asselah and P Marcellin ldquoNew direct-acting antiviralsrsquocombination for the treatment of chronic hepatitis Crdquo LiverInternational vol 31 pp 68ndash77 2011
[5] B Lam L Henry and Z Younossi ldquoSofosbuvir (Sovaldi) forthe treatment of hepatitis Crdquo Expert Review of ClinicalPharmacology vol 7 no 5 pp 555ndash566 2014
[6] J Chhatwal Q Chen T Ayer et al ldquoHepatitis C virus re-treatment in the era of direct-acting antivirals projections inthe USArdquo Alimentary Pharmacology amp 3erapeutics vol 47no 7 pp 1023ndash1031 2018
[7] V Cento T H T Nguyen D Di Carlo et al ldquoImprovementof ALT decay kinetics by all-oral HCV treatment role of
Table 3 Parameter values used with SOF+ SIM and TVR+PR
Figure 3 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with TVR+PR s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
PSLPRef [7]
HCV
RN
A
0 5 10 15 20 25Days
10
1000
105
107
V (t) = 95287079endash528t + 212179endash027t
Figure 4 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with SOF+SIM s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
Mathematical Problems in Engineering 7
NS5A inhibitors and differences with IFN-based regimensrdquoPLoS One vol 12 no 5 Article ID e0177352 2017
[8] T H T Nguyen J Guedj S L Uprichard A KohliS Kottilil and A S Perelson ldquoe paradox of highly ef-fective sofosbuvir-based combination therapy despite slowviral decline can we still rely on viral kineticsrdquo ScientificReports vol 7 no 1 Article ID 10233 2017
[9] World Health Organization Global Hepatitis Report 2017Global Hepatitis Programme Department of HIVAIDSWHO Geneva Switzerland 2017
[10] World Health Organization Combating Hepatitis B and C toReach Elimination by 2030 World Health OrganizationGeneva Switzerland 2016
[11] D S Jones and B D Sleeman ldquoDifferential equations andmathematical biologyrdquo inMathematical Biology andMedicineSeries Chapman amp HallCRC Boca Raton FL USA 2003
[12] A U Neumann N P Lam H Dahari et al ldquoHepatitis C viraldynamics in vivo and the antiviral efficacy of interferon-therapyrdquo Science vol 282 no 5386 pp 103ndash107 1998
[13] J Guedj and A U Neumann ldquoUnderstanding hepatitis Cviral dynamics with direct-acting antiviral agents due to theinterplay between intracellular replication and cellular in-fection dynamicsrdquo Journal of 3eoretical Biology vol 267no 3 pp 330ndash340 2010
[14] H Dahari J Guedj A S Perelson and T J LaydenldquoHepatitis C viral kinetics in the era of direct acting antiviralagents and interleukin-28Brdquo Current Hepatitis Reportsvol 10 no 3 pp 214ndash227 2011
[15] A Chaterjee J Guedj and A S Perelson ldquoMathematicalmodeling of HCV infection what can it teach us in the era ofdirect antiviral agentsrdquo Antiviral 3erapy vol 17 no 6pp 1171ndash1182 2012
[16] R J Payne M A Nowak and B S Blumberg ldquoe dynamicsof hepatitis B virus infectionrdquo Proceedings of the NationalAcademy of Sciences vol 93 no 13 pp 6542ndash6546 1996
[17] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[18] D Wodarz and M A Nowak ldquoMathematical models of HIVpathogenesis and treatmentrdquo BioEssays vol 24 no 12pp 1178ndash1187 2002
[19] W H Ho and A L F Chan ldquoHybrid taguchi-differentialevolution algorithm for parameter estimation of differentialequation models with application to HIV dynamicsrdquoMathematical Problems in Engineering vol 2011 Article ID514756 14 pages 2011
[20] 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 Sciencevol 271 no 5255 pp 1582ndash1586 1996
[21] Q Li and Y Xiao ldquoGlobal dynamics of a virus-immunesystem with virus-guided therapy and saturation growth ofvirusrdquo Mathematical Problems in Engineering vol 2018Article ID 4710586 18 pages 2018
[22] H Zarei A V Kamyad and S Effati ldquoMultiobjective optimalcontrol of HIV dynamicsrdquo Mathematical Problems in Engi-neering vol 2010 Article ID 568315 29 pages 2010
[23] T S N Asih S Lenhart S Wise et al ldquoe dynamics of HPVinfection and cervical cancer cellsrdquo Bulletin of MathematicalBiology vol 78 no 1 pp 4ndash20 2016
[24] L Rong J Guedj H Dahari et al ldquoAnalysis of hepatitis Cvirus decline during treatment with the protease inhibitor
danoprevir using a multiscale modelrdquo PLoS ComputationalBiology vol 9 no 3 Article ID e1002959 2013
[25] J Guedj H Dahari L Rong et al ldquoModeling shows that theNS5A inhibitor daclatasvir has two modes of action and yieldsa shorter estimate of the hepatitis C virus half-liferdquo Pro-ceedings of the National Academy of Sciences vol 110 no 10pp 3991ndash3996 2013
[26] L Rong and A S Perelson ldquoMathematical analysis of mul-tiscale models for hepatitis C virus dynamics under therapywith direct-acting antiviral agentsrdquoMathematical Biosciencesvol 245 no 1 pp 22ndash30 2013
[27] H A Elkaranshawy A M Abdelrazek and H M EzzatldquoPower series solution to sliding velocity in three-dimensionalmultibody systems with impact and frictionrdquo InternationalJournal of Mathematical Computational Physical Electricaland Computer Engineering vol 9 no 10 2015
[28] I M Abdelrazik and H A Elkaranshawy ldquoExtended Parker-Sochacki method for Michaelis-Menten enzymatic reactionmodelrdquo Analytical Biochemistry vol 496 pp 50ndash54 2016
[29] I M Abdelrazik H A Elkaranshawy and A M AbdelrazekldquoModified Parker-Sochacki method for solving nonlinearoscillatorsrdquo Mechanics Based Design of Structures and Ma-chines vol 45 no 2 pp 239ndash252 2016
[30] I M Abdelrazik Improving the solutions of parker sochackimethod for nonlinear ordinary differential equations PhDthesis Department of Engineering Mathematics and PhysicsFaculty of Engineering Alexandria University AlexandriaEgypt 2016
[31] G A Baker and P R Graves-Morris Pade Approximants vol59 Cambridge University Press Cambridge UK 1996
[32] B Raftari and A Yildirim ldquoSeries solution of a nonlinearODE arising in magnetohydrodynamic by HPM-Pade tech-niquerdquo Computers amp Mathematics with Applications vol 61no 6 pp 1676ndash1681 2011
[33] J Aubard P Levoir A Denis and P Claverie ldquoDirectanalysis of chemical relaxation signals by a method based onthe combination of Laplace transform and Pade approx-imantsrdquo Computers amp Chemistry vol 11 no 3 pp 163ndash1781987
[34] H Vazquez-Leal and F Guerrero ldquoApplication of seriesmethod with Pade and Laplace-Pade resummation methodsto solve a model for the evolution of smoking habit in SpainrdquoComputational and Applied Mathematics vol 33 no 1pp 181ndash192 2014
8 Mathematical Problems in Engineering
Hindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
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Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
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AnalysisInternational Journal of
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Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
(5) 1s is written instead of u in [22] hence F(s) isobtained as
F(s) cV0
s1113872 1113873 + b2s2( 1113857
1 + d1s( 1113857 + d2s2( 1113857 (22)
(6) Using inverse Laplace transform for F(s) the viralload V(t) is obtained as
V(t) A1eminus D1lowastt + cV0
minus A11113872 1113873eminus D2lowastt (23)
where
A1 1
2B
radic minus cV1
3+ 3cV1
cV2cV0
minus 3cV3cV0
2+ cV0
B
radic1113872 1113873
D1 minusA minus
B
radic
f
D2 minusA +
B
radic
f
f cV1
2minus 2cV2
cV0
A cV1cV2
minus 3cV3cV0
B minus 3cV1
2cV2
2+ 6cV1
3cV3
+ 8cV2
3cV0
minus 18cV1cV2
cV3cV0
+ 9cV3
2cV0
2
(24)
Equation (23) combined with equations (10) and (24)represents the general approximate analytical solution forthe viral load for the standard dynamic model given inequations (1)ndash(3) It is worth to notice that this solutioncontains the 7 parameters given in the system of equations(1)ndash(3) Hence it reflects properly the biological features
inherited in the system It can be noticed that the simplifiedapproximate solution given in equation (23) contains only 3parameters
4 Study Cases
To illustrate the capabilities and competences of the pro-posed method four cases of study are presented ese casesconsider the analysis of viral kinetics using the standard viraldynamic mathematical model of HCV e proposed PSLPmethod is applied to solve the nonlinear dynamic model ofthe viral kinetics for some patients after the initiation oftreatment with DAAs
For the first two study cases the predictions of the PSLPmethod are compared for each patient with viral load dataavailable in the literature [24 25] To provide best fits of datafor each patient parameters δ ε c and ρ are estimated usingthe relation for initial condition V0 (minus cdδ + βρsβcδ) andequation (23) In the first case patients were infected withHCV genotype 1 and treated with danoprevir e viral loadfor these patients is checked for 13 days after initiation ofdanoprevir the data are available in [24] and the parametersare given in Table 1 In the second case patients were treatedwith daclatasvir and the viral load of patients is checked for2 days after initiation of daclatasvir the data are available in[25] and the parameters are given in Table 2 Figures 1 and 2demonstrate the comparison between the solution of the PSLPmethod and the corresponding viral load data for each patient
Two other cases representing recently used combinationof multidrug DAA treatments are considered e pre-dictions of the PSLP method are compared with the cor-responding viral load obtained by simulation in [7]e usedparameter values are given in Table 3 where δ ε and c havethe same values assigned in [7] and ρ has been chosenaccordingly e PSLP results and the published simulationresults for 25 days after the initiation of treatment withTVR+PR are shown in Figure 3 e PSLP results and thepublished simulation results for 28 days after the initiation oftreatment with SOF+ SIM are shown in Figure 4 In both
Table 2 Parameter values used for the patients treated with daclatasvir
cases the PSLP solution and the simulation results are al-most identical e comparison shows that the proposedPSLP method provides adequate approximate analytical
solutions using the standard viral dynamic model for all theconsidered cases It is worth to mention that in these twocases the rate of curation is near 100
PSLPViral data
0 2 4 6 8 10 12Days
V (t) = 173 times 107endash738t + 905309endash0149t
10
1000
105
107H
CV R
NA
(a)
PSLPViral data
0
V (t) = 523707251endash1244t + 1100209endash028t
Days2 4 6 8 10 12
10
1000
105
107
HCV
RN
A
(b)
V (t) = 61532097endash1050t + 127402endash016t
PSLPViral data
0 2 4 6 8 10 12Days
10
1000
105
107
HCV
RN
A
(c)
V (t) = 934420474endash940t + 20572111endash034t
PSLPViral data
0Days
2 4 6 8 10 12
10
1000
105
107
HCV
RN
A
(d)
V (t) = 426351825endash1026t + 227693endash033t
PSLPViral data
0 2 4 6 8 10 12Days
10
1000
105
107
HCV
RN
A
(e)
Figure 1 Comparison between the approximate analytical solution using the PSLP method and viral data for patients treated withdanoprevir s 13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1 and the rest of the parameter values are given in Table 1(a) 01ndash94GK (b) 03ndash94HD (c) 03ndash94EA (d) 03ndash94KG and (e) 04ndash94XD
Mathematical Problems in Engineering 5
Since it is highly desirable to predict the patientrsquos re-sponse to a specific treatment regimen prior to the starting ofthe treatment itself the PSLP solution offers a simple and
powerful tool for medical specialists and physicians toperform this task ey can use the patientrsquos parameters tocalculate the constants in equations (10) and (24) and
00 05 10 15 20
HCV
RN
A
10
1000
105
107
PSLPViral data
Days
V (t) = 43453925endash3130t + 197657endash188t
(a)
00 05 10 15 20
HCV
RN
A
10
1000
105
107
PSLPViral data
Days
V (t) = 44575617endash21t + 92743endash039t
(b)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
Days
V (t) = 140 times 107endash2020t + 3215905endash127t
(c)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
Days
V (t) = 199312388endash22t + 213843endash075t
(d)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
DaysV (t) = 28059926endash2180t + 123903endasht
(e)
Figure 2 Comparison between the approximate analytical solution using the PSLP method and viral data for patients treated withdaclatasvir s 13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1 and the rest of the parameter values are given in Table 2(a) PAT 8 (b) PAT 42 (c) PAT 68 (d) PAT 69 and (e) PAT 83
6 Mathematical Problems in Engineering
substitute them in equation (23) to get a closed-form so-lution for the viral load Hence the viral load can be plottedversus time or simply the viral load can be estimated at anyinstant by direct substitution in the closed-form solution
5 Conclusions
Power series solution combined with the LaplacendashPaderesummation method (PSLP) has been used to obtain a
general approximate analytical solution for the nonlinearstandard viral dynamic model of HCV for patients treatedwith DAAs However the solution is not limited to HCVmodel and it can be applied to other viruses like HIV andHBV for example given that proper parameters are used Totest the applicability and accuracy of the proposed methodresults have been compared with viral load data and withpublished simulated results Satisfactory agreement betweenthe PSLP solution and the corresponding viral load data hasbeen found for all the considered casese PSLP results andthe published simulated results are almost identical ecomparison proves that this innovative PSLP solution can beused with confidence for solving the nonlinear standard viraldynamic model is solution can conveniently be used to fitpatient data and estimate parameter values So it wouldfacilitate for physicians to monitor the changes in the viralload due to changes in treatment and to deal with changes inpatientrsquos parameters
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D W Powell B Z Abramson J A Balint et al ldquoNationalinstitutes of health consensus development conference panelstatement management of hepatitis Crdquo Hepatology vol 26no 3 pp 2Sndash10S 1997
[2] J G McHutchison S C Gordon E R Schiff et al ldquoInterferonalfa-2b alone or in combination with ribavirin as initialtreatment for chronic hepatitis Crdquo New England Journal ofMedicine vol 339 no 21 pp 1485ndash1492 1998
[3] T Poynard J McHutchison M Manns et al ldquoImpact ofpegylated interferon alfa-2b and ribavirin on liver fibrosis inpatients with chronic hepatitis Crdquo Gastroenterology vol 122no 5 pp 1303ndash1313 2002
[4] T Asselah and P Marcellin ldquoNew direct-acting antiviralsrsquocombination for the treatment of chronic hepatitis Crdquo LiverInternational vol 31 pp 68ndash77 2011
[5] B Lam L Henry and Z Younossi ldquoSofosbuvir (Sovaldi) forthe treatment of hepatitis Crdquo Expert Review of ClinicalPharmacology vol 7 no 5 pp 555ndash566 2014
[6] J Chhatwal Q Chen T Ayer et al ldquoHepatitis C virus re-treatment in the era of direct-acting antivirals projections inthe USArdquo Alimentary Pharmacology amp 3erapeutics vol 47no 7 pp 1023ndash1031 2018
[7] V Cento T H T Nguyen D Di Carlo et al ldquoImprovementof ALT decay kinetics by all-oral HCV treatment role of
Table 3 Parameter values used with SOF+ SIM and TVR+PR
Figure 3 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with TVR+PR s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
PSLPRef [7]
HCV
RN
A
0 5 10 15 20 25Days
10
1000
105
107
V (t) = 95287079endash528t + 212179endash027t
Figure 4 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with SOF+SIM s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
Mathematical Problems in Engineering 7
NS5A inhibitors and differences with IFN-based regimensrdquoPLoS One vol 12 no 5 Article ID e0177352 2017
[8] T H T Nguyen J Guedj S L Uprichard A KohliS Kottilil and A S Perelson ldquoe paradox of highly ef-fective sofosbuvir-based combination therapy despite slowviral decline can we still rely on viral kineticsrdquo ScientificReports vol 7 no 1 Article ID 10233 2017
[9] World Health Organization Global Hepatitis Report 2017Global Hepatitis Programme Department of HIVAIDSWHO Geneva Switzerland 2017
[10] World Health Organization Combating Hepatitis B and C toReach Elimination by 2030 World Health OrganizationGeneva Switzerland 2016
[11] D S Jones and B D Sleeman ldquoDifferential equations andmathematical biologyrdquo inMathematical Biology andMedicineSeries Chapman amp HallCRC Boca Raton FL USA 2003
[12] A U Neumann N P Lam H Dahari et al ldquoHepatitis C viraldynamics in vivo and the antiviral efficacy of interferon-therapyrdquo Science vol 282 no 5386 pp 103ndash107 1998
[13] J Guedj and A U Neumann ldquoUnderstanding hepatitis Cviral dynamics with direct-acting antiviral agents due to theinterplay between intracellular replication and cellular in-fection dynamicsrdquo Journal of 3eoretical Biology vol 267no 3 pp 330ndash340 2010
[14] H Dahari J Guedj A S Perelson and T J LaydenldquoHepatitis C viral kinetics in the era of direct acting antiviralagents and interleukin-28Brdquo Current Hepatitis Reportsvol 10 no 3 pp 214ndash227 2011
[15] A Chaterjee J Guedj and A S Perelson ldquoMathematicalmodeling of HCV infection what can it teach us in the era ofdirect antiviral agentsrdquo Antiviral 3erapy vol 17 no 6pp 1171ndash1182 2012
[16] R J Payne M A Nowak and B S Blumberg ldquoe dynamicsof hepatitis B virus infectionrdquo Proceedings of the NationalAcademy of Sciences vol 93 no 13 pp 6542ndash6546 1996
[17] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[18] D Wodarz and M A Nowak ldquoMathematical models of HIVpathogenesis and treatmentrdquo BioEssays vol 24 no 12pp 1178ndash1187 2002
[19] W H Ho and A L F Chan ldquoHybrid taguchi-differentialevolution algorithm for parameter estimation of differentialequation models with application to HIV dynamicsrdquoMathematical Problems in Engineering vol 2011 Article ID514756 14 pages 2011
[20] 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 Sciencevol 271 no 5255 pp 1582ndash1586 1996
[21] Q Li and Y Xiao ldquoGlobal dynamics of a virus-immunesystem with virus-guided therapy and saturation growth ofvirusrdquo Mathematical Problems in Engineering vol 2018Article ID 4710586 18 pages 2018
[22] H Zarei A V Kamyad and S Effati ldquoMultiobjective optimalcontrol of HIV dynamicsrdquo Mathematical Problems in Engi-neering vol 2010 Article ID 568315 29 pages 2010
[23] T S N Asih S Lenhart S Wise et al ldquoe dynamics of HPVinfection and cervical cancer cellsrdquo Bulletin of MathematicalBiology vol 78 no 1 pp 4ndash20 2016
[24] L Rong J Guedj H Dahari et al ldquoAnalysis of hepatitis Cvirus decline during treatment with the protease inhibitor
danoprevir using a multiscale modelrdquo PLoS ComputationalBiology vol 9 no 3 Article ID e1002959 2013
[25] J Guedj H Dahari L Rong et al ldquoModeling shows that theNS5A inhibitor daclatasvir has two modes of action and yieldsa shorter estimate of the hepatitis C virus half-liferdquo Pro-ceedings of the National Academy of Sciences vol 110 no 10pp 3991ndash3996 2013
[26] L Rong and A S Perelson ldquoMathematical analysis of mul-tiscale models for hepatitis C virus dynamics under therapywith direct-acting antiviral agentsrdquoMathematical Biosciencesvol 245 no 1 pp 22ndash30 2013
[27] H A Elkaranshawy A M Abdelrazek and H M EzzatldquoPower series solution to sliding velocity in three-dimensionalmultibody systems with impact and frictionrdquo InternationalJournal of Mathematical Computational Physical Electricaland Computer Engineering vol 9 no 10 2015
[28] I M Abdelrazik and H A Elkaranshawy ldquoExtended Parker-Sochacki method for Michaelis-Menten enzymatic reactionmodelrdquo Analytical Biochemistry vol 496 pp 50ndash54 2016
[29] I M Abdelrazik H A Elkaranshawy and A M AbdelrazekldquoModified Parker-Sochacki method for solving nonlinearoscillatorsrdquo Mechanics Based Design of Structures and Ma-chines vol 45 no 2 pp 239ndash252 2016
[30] I M Abdelrazik Improving the solutions of parker sochackimethod for nonlinear ordinary differential equations PhDthesis Department of Engineering Mathematics and PhysicsFaculty of Engineering Alexandria University AlexandriaEgypt 2016
[31] G A Baker and P R Graves-Morris Pade Approximants vol59 Cambridge University Press Cambridge UK 1996
[32] B Raftari and A Yildirim ldquoSeries solution of a nonlinearODE arising in magnetohydrodynamic by HPM-Pade tech-niquerdquo Computers amp Mathematics with Applications vol 61no 6 pp 1676ndash1681 2011
[33] J Aubard P Levoir A Denis and P Claverie ldquoDirectanalysis of chemical relaxation signals by a method based onthe combination of Laplace transform and Pade approx-imantsrdquo Computers amp Chemistry vol 11 no 3 pp 163ndash1781987
[34] H Vazquez-Leal and F Guerrero ldquoApplication of seriesmethod with Pade and Laplace-Pade resummation methodsto solve a model for the evolution of smoking habit in SpainrdquoComputational and Applied Mathematics vol 33 no 1pp 181ndash192 2014
8 Mathematical Problems in Engineering
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
cases the PSLP solution and the simulation results are al-most identical e comparison shows that the proposedPSLP method provides adequate approximate analytical
solutions using the standard viral dynamic model for all theconsidered cases It is worth to mention that in these twocases the rate of curation is near 100
PSLPViral data
0 2 4 6 8 10 12Days
V (t) = 173 times 107endash738t + 905309endash0149t
10
1000
105
107H
CV R
NA
(a)
PSLPViral data
0
V (t) = 523707251endash1244t + 1100209endash028t
Days2 4 6 8 10 12
10
1000
105
107
HCV
RN
A
(b)
V (t) = 61532097endash1050t + 127402endash016t
PSLPViral data
0 2 4 6 8 10 12Days
10
1000
105
107
HCV
RN
A
(c)
V (t) = 934420474endash940t + 20572111endash034t
PSLPViral data
0Days
2 4 6 8 10 12
10
1000
105
107
HCV
RN
A
(d)
V (t) = 426351825endash1026t + 227693endash033t
PSLPViral data
0 2 4 6 8 10 12Days
10
1000
105
107
HCV
RN
A
(e)
Figure 1 Comparison between the approximate analytical solution using the PSLP method and viral data for patients treated withdanoprevir s 13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1 and the rest of the parameter values are given in Table 1(a) 01ndash94GK (b) 03ndash94HD (c) 03ndash94EA (d) 03ndash94KG and (e) 04ndash94XD
Mathematical Problems in Engineering 5
Since it is highly desirable to predict the patientrsquos re-sponse to a specific treatment regimen prior to the starting ofthe treatment itself the PSLP solution offers a simple and
powerful tool for medical specialists and physicians toperform this task ey can use the patientrsquos parameters tocalculate the constants in equations (10) and (24) and
00 05 10 15 20
HCV
RN
A
10
1000
105
107
PSLPViral data
Days
V (t) = 43453925endash3130t + 197657endash188t
(a)
00 05 10 15 20
HCV
RN
A
10
1000
105
107
PSLPViral data
Days
V (t) = 44575617endash21t + 92743endash039t
(b)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
Days
V (t) = 140 times 107endash2020t + 3215905endash127t
(c)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
Days
V (t) = 199312388endash22t + 213843endash075t
(d)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
DaysV (t) = 28059926endash2180t + 123903endasht
(e)
Figure 2 Comparison between the approximate analytical solution using the PSLP method and viral data for patients treated withdaclatasvir s 13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1 and the rest of the parameter values are given in Table 2(a) PAT 8 (b) PAT 42 (c) PAT 68 (d) PAT 69 and (e) PAT 83
6 Mathematical Problems in Engineering
substitute them in equation (23) to get a closed-form so-lution for the viral load Hence the viral load can be plottedversus time or simply the viral load can be estimated at anyinstant by direct substitution in the closed-form solution
5 Conclusions
Power series solution combined with the LaplacendashPaderesummation method (PSLP) has been used to obtain a
general approximate analytical solution for the nonlinearstandard viral dynamic model of HCV for patients treatedwith DAAs However the solution is not limited to HCVmodel and it can be applied to other viruses like HIV andHBV for example given that proper parameters are used Totest the applicability and accuracy of the proposed methodresults have been compared with viral load data and withpublished simulated results Satisfactory agreement betweenthe PSLP solution and the corresponding viral load data hasbeen found for all the considered casese PSLP results andthe published simulated results are almost identical ecomparison proves that this innovative PSLP solution can beused with confidence for solving the nonlinear standard viraldynamic model is solution can conveniently be used to fitpatient data and estimate parameter values So it wouldfacilitate for physicians to monitor the changes in the viralload due to changes in treatment and to deal with changes inpatientrsquos parameters
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D W Powell B Z Abramson J A Balint et al ldquoNationalinstitutes of health consensus development conference panelstatement management of hepatitis Crdquo Hepatology vol 26no 3 pp 2Sndash10S 1997
[2] J G McHutchison S C Gordon E R Schiff et al ldquoInterferonalfa-2b alone or in combination with ribavirin as initialtreatment for chronic hepatitis Crdquo New England Journal ofMedicine vol 339 no 21 pp 1485ndash1492 1998
[3] T Poynard J McHutchison M Manns et al ldquoImpact ofpegylated interferon alfa-2b and ribavirin on liver fibrosis inpatients with chronic hepatitis Crdquo Gastroenterology vol 122no 5 pp 1303ndash1313 2002
[4] T Asselah and P Marcellin ldquoNew direct-acting antiviralsrsquocombination for the treatment of chronic hepatitis Crdquo LiverInternational vol 31 pp 68ndash77 2011
[5] B Lam L Henry and Z Younossi ldquoSofosbuvir (Sovaldi) forthe treatment of hepatitis Crdquo Expert Review of ClinicalPharmacology vol 7 no 5 pp 555ndash566 2014
[6] J Chhatwal Q Chen T Ayer et al ldquoHepatitis C virus re-treatment in the era of direct-acting antivirals projections inthe USArdquo Alimentary Pharmacology amp 3erapeutics vol 47no 7 pp 1023ndash1031 2018
[7] V Cento T H T Nguyen D Di Carlo et al ldquoImprovementof ALT decay kinetics by all-oral HCV treatment role of
Table 3 Parameter values used with SOF+ SIM and TVR+PR
Figure 3 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with TVR+PR s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
PSLPRef [7]
HCV
RN
A
0 5 10 15 20 25Days
10
1000
105
107
V (t) = 95287079endash528t + 212179endash027t
Figure 4 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with SOF+SIM s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
Mathematical Problems in Engineering 7
NS5A inhibitors and differences with IFN-based regimensrdquoPLoS One vol 12 no 5 Article ID e0177352 2017
[8] T H T Nguyen J Guedj S L Uprichard A KohliS Kottilil and A S Perelson ldquoe paradox of highly ef-fective sofosbuvir-based combination therapy despite slowviral decline can we still rely on viral kineticsrdquo ScientificReports vol 7 no 1 Article ID 10233 2017
[9] World Health Organization Global Hepatitis Report 2017Global Hepatitis Programme Department of HIVAIDSWHO Geneva Switzerland 2017
[10] World Health Organization Combating Hepatitis B and C toReach Elimination by 2030 World Health OrganizationGeneva Switzerland 2016
[11] D S Jones and B D Sleeman ldquoDifferential equations andmathematical biologyrdquo inMathematical Biology andMedicineSeries Chapman amp HallCRC Boca Raton FL USA 2003
[12] A U Neumann N P Lam H Dahari et al ldquoHepatitis C viraldynamics in vivo and the antiviral efficacy of interferon-therapyrdquo Science vol 282 no 5386 pp 103ndash107 1998
[13] J Guedj and A U Neumann ldquoUnderstanding hepatitis Cviral dynamics with direct-acting antiviral agents due to theinterplay between intracellular replication and cellular in-fection dynamicsrdquo Journal of 3eoretical Biology vol 267no 3 pp 330ndash340 2010
[14] H Dahari J Guedj A S Perelson and T J LaydenldquoHepatitis C viral kinetics in the era of direct acting antiviralagents and interleukin-28Brdquo Current Hepatitis Reportsvol 10 no 3 pp 214ndash227 2011
[15] A Chaterjee J Guedj and A S Perelson ldquoMathematicalmodeling of HCV infection what can it teach us in the era ofdirect antiviral agentsrdquo Antiviral 3erapy vol 17 no 6pp 1171ndash1182 2012
[16] R J Payne M A Nowak and B S Blumberg ldquoe dynamicsof hepatitis B virus infectionrdquo Proceedings of the NationalAcademy of Sciences vol 93 no 13 pp 6542ndash6546 1996
[17] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[18] D Wodarz and M A Nowak ldquoMathematical models of HIVpathogenesis and treatmentrdquo BioEssays vol 24 no 12pp 1178ndash1187 2002
[19] W H Ho and A L F Chan ldquoHybrid taguchi-differentialevolution algorithm for parameter estimation of differentialequation models with application to HIV dynamicsrdquoMathematical Problems in Engineering vol 2011 Article ID514756 14 pages 2011
[20] 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 Sciencevol 271 no 5255 pp 1582ndash1586 1996
[21] Q Li and Y Xiao ldquoGlobal dynamics of a virus-immunesystem with virus-guided therapy and saturation growth ofvirusrdquo Mathematical Problems in Engineering vol 2018Article ID 4710586 18 pages 2018
[22] H Zarei A V Kamyad and S Effati ldquoMultiobjective optimalcontrol of HIV dynamicsrdquo Mathematical Problems in Engi-neering vol 2010 Article ID 568315 29 pages 2010
[23] T S N Asih S Lenhart S Wise et al ldquoe dynamics of HPVinfection and cervical cancer cellsrdquo Bulletin of MathematicalBiology vol 78 no 1 pp 4ndash20 2016
[24] L Rong J Guedj H Dahari et al ldquoAnalysis of hepatitis Cvirus decline during treatment with the protease inhibitor
danoprevir using a multiscale modelrdquo PLoS ComputationalBiology vol 9 no 3 Article ID e1002959 2013
[25] J Guedj H Dahari L Rong et al ldquoModeling shows that theNS5A inhibitor daclatasvir has two modes of action and yieldsa shorter estimate of the hepatitis C virus half-liferdquo Pro-ceedings of the National Academy of Sciences vol 110 no 10pp 3991ndash3996 2013
[26] L Rong and A S Perelson ldquoMathematical analysis of mul-tiscale models for hepatitis C virus dynamics under therapywith direct-acting antiviral agentsrdquoMathematical Biosciencesvol 245 no 1 pp 22ndash30 2013
[27] H A Elkaranshawy A M Abdelrazek and H M EzzatldquoPower series solution to sliding velocity in three-dimensionalmultibody systems with impact and frictionrdquo InternationalJournal of Mathematical Computational Physical Electricaland Computer Engineering vol 9 no 10 2015
[28] I M Abdelrazik and H A Elkaranshawy ldquoExtended Parker-Sochacki method for Michaelis-Menten enzymatic reactionmodelrdquo Analytical Biochemistry vol 496 pp 50ndash54 2016
[29] I M Abdelrazik H A Elkaranshawy and A M AbdelrazekldquoModified Parker-Sochacki method for solving nonlinearoscillatorsrdquo Mechanics Based Design of Structures and Ma-chines vol 45 no 2 pp 239ndash252 2016
[30] I M Abdelrazik Improving the solutions of parker sochackimethod for nonlinear ordinary differential equations PhDthesis Department of Engineering Mathematics and PhysicsFaculty of Engineering Alexandria University AlexandriaEgypt 2016
[31] G A Baker and P R Graves-Morris Pade Approximants vol59 Cambridge University Press Cambridge UK 1996
[32] B Raftari and A Yildirim ldquoSeries solution of a nonlinearODE arising in magnetohydrodynamic by HPM-Pade tech-niquerdquo Computers amp Mathematics with Applications vol 61no 6 pp 1676ndash1681 2011
[33] J Aubard P Levoir A Denis and P Claverie ldquoDirectanalysis of chemical relaxation signals by a method based onthe combination of Laplace transform and Pade approx-imantsrdquo Computers amp Chemistry vol 11 no 3 pp 163ndash1781987
[34] H Vazquez-Leal and F Guerrero ldquoApplication of seriesmethod with Pade and Laplace-Pade resummation methodsto solve a model for the evolution of smoking habit in SpainrdquoComputational and Applied Mathematics vol 33 no 1pp 181ndash192 2014
8 Mathematical Problems in Engineering
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Since it is highly desirable to predict the patientrsquos re-sponse to a specific treatment regimen prior to the starting ofthe treatment itself the PSLP solution offers a simple and
powerful tool for medical specialists and physicians toperform this task ey can use the patientrsquos parameters tocalculate the constants in equations (10) and (24) and
00 05 10 15 20
HCV
RN
A
10
1000
105
107
PSLPViral data
Days
V (t) = 43453925endash3130t + 197657endash188t
(a)
00 05 10 15 20
HCV
RN
A
10
1000
105
107
PSLPViral data
Days
V (t) = 44575617endash21t + 92743endash039t
(b)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
Days
V (t) = 140 times 107endash2020t + 3215905endash127t
(c)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
Days
V (t) = 199312388endash22t + 213843endash075t
(d)
HCV
RN
A
10
1000
105
107
00 05 10 15 20
PSLPViral data
DaysV (t) = 28059926endash2180t + 123903endasht
(e)
Figure 2 Comparison between the approximate analytical solution using the PSLP method and viral data for patients treated withdaclatasvir s 13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1 and the rest of the parameter values are given in Table 2(a) PAT 8 (b) PAT 42 (c) PAT 68 (d) PAT 69 and (e) PAT 83
6 Mathematical Problems in Engineering
substitute them in equation (23) to get a closed-form so-lution for the viral load Hence the viral load can be plottedversus time or simply the viral load can be estimated at anyinstant by direct substitution in the closed-form solution
5 Conclusions
Power series solution combined with the LaplacendashPaderesummation method (PSLP) has been used to obtain a
general approximate analytical solution for the nonlinearstandard viral dynamic model of HCV for patients treatedwith DAAs However the solution is not limited to HCVmodel and it can be applied to other viruses like HIV andHBV for example given that proper parameters are used Totest the applicability and accuracy of the proposed methodresults have been compared with viral load data and withpublished simulated results Satisfactory agreement betweenthe PSLP solution and the corresponding viral load data hasbeen found for all the considered casese PSLP results andthe published simulated results are almost identical ecomparison proves that this innovative PSLP solution can beused with confidence for solving the nonlinear standard viraldynamic model is solution can conveniently be used to fitpatient data and estimate parameter values So it wouldfacilitate for physicians to monitor the changes in the viralload due to changes in treatment and to deal with changes inpatientrsquos parameters
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D W Powell B Z Abramson J A Balint et al ldquoNationalinstitutes of health consensus development conference panelstatement management of hepatitis Crdquo Hepatology vol 26no 3 pp 2Sndash10S 1997
[2] J G McHutchison S C Gordon E R Schiff et al ldquoInterferonalfa-2b alone or in combination with ribavirin as initialtreatment for chronic hepatitis Crdquo New England Journal ofMedicine vol 339 no 21 pp 1485ndash1492 1998
[3] T Poynard J McHutchison M Manns et al ldquoImpact ofpegylated interferon alfa-2b and ribavirin on liver fibrosis inpatients with chronic hepatitis Crdquo Gastroenterology vol 122no 5 pp 1303ndash1313 2002
[4] T Asselah and P Marcellin ldquoNew direct-acting antiviralsrsquocombination for the treatment of chronic hepatitis Crdquo LiverInternational vol 31 pp 68ndash77 2011
[5] B Lam L Henry and Z Younossi ldquoSofosbuvir (Sovaldi) forthe treatment of hepatitis Crdquo Expert Review of ClinicalPharmacology vol 7 no 5 pp 555ndash566 2014
[6] J Chhatwal Q Chen T Ayer et al ldquoHepatitis C virus re-treatment in the era of direct-acting antivirals projections inthe USArdquo Alimentary Pharmacology amp 3erapeutics vol 47no 7 pp 1023ndash1031 2018
[7] V Cento T H T Nguyen D Di Carlo et al ldquoImprovementof ALT decay kinetics by all-oral HCV treatment role of
Table 3 Parameter values used with SOF+ SIM and TVR+PR
Figure 3 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with TVR+PR s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
PSLPRef [7]
HCV
RN
A
0 5 10 15 20 25Days
10
1000
105
107
V (t) = 95287079endash528t + 212179endash027t
Figure 4 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with SOF+SIM s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
Mathematical Problems in Engineering 7
NS5A inhibitors and differences with IFN-based regimensrdquoPLoS One vol 12 no 5 Article ID e0177352 2017
[8] T H T Nguyen J Guedj S L Uprichard A KohliS Kottilil and A S Perelson ldquoe paradox of highly ef-fective sofosbuvir-based combination therapy despite slowviral decline can we still rely on viral kineticsrdquo ScientificReports vol 7 no 1 Article ID 10233 2017
[9] World Health Organization Global Hepatitis Report 2017Global Hepatitis Programme Department of HIVAIDSWHO Geneva Switzerland 2017
[10] World Health Organization Combating Hepatitis B and C toReach Elimination by 2030 World Health OrganizationGeneva Switzerland 2016
[11] D S Jones and B D Sleeman ldquoDifferential equations andmathematical biologyrdquo inMathematical Biology andMedicineSeries Chapman amp HallCRC Boca Raton FL USA 2003
[12] A U Neumann N P Lam H Dahari et al ldquoHepatitis C viraldynamics in vivo and the antiviral efficacy of interferon-therapyrdquo Science vol 282 no 5386 pp 103ndash107 1998
[13] J Guedj and A U Neumann ldquoUnderstanding hepatitis Cviral dynamics with direct-acting antiviral agents due to theinterplay between intracellular replication and cellular in-fection dynamicsrdquo Journal of 3eoretical Biology vol 267no 3 pp 330ndash340 2010
[14] H Dahari J Guedj A S Perelson and T J LaydenldquoHepatitis C viral kinetics in the era of direct acting antiviralagents and interleukin-28Brdquo Current Hepatitis Reportsvol 10 no 3 pp 214ndash227 2011
[15] A Chaterjee J Guedj and A S Perelson ldquoMathematicalmodeling of HCV infection what can it teach us in the era ofdirect antiviral agentsrdquo Antiviral 3erapy vol 17 no 6pp 1171ndash1182 2012
[16] R J Payne M A Nowak and B S Blumberg ldquoe dynamicsof hepatitis B virus infectionrdquo Proceedings of the NationalAcademy of Sciences vol 93 no 13 pp 6542ndash6546 1996
[17] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[18] D Wodarz and M A Nowak ldquoMathematical models of HIVpathogenesis and treatmentrdquo BioEssays vol 24 no 12pp 1178ndash1187 2002
[19] W H Ho and A L F Chan ldquoHybrid taguchi-differentialevolution algorithm for parameter estimation of differentialequation models with application to HIV dynamicsrdquoMathematical Problems in Engineering vol 2011 Article ID514756 14 pages 2011
[20] 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 Sciencevol 271 no 5255 pp 1582ndash1586 1996
[21] Q Li and Y Xiao ldquoGlobal dynamics of a virus-immunesystem with virus-guided therapy and saturation growth ofvirusrdquo Mathematical Problems in Engineering vol 2018Article ID 4710586 18 pages 2018
[22] H Zarei A V Kamyad and S Effati ldquoMultiobjective optimalcontrol of HIV dynamicsrdquo Mathematical Problems in Engi-neering vol 2010 Article ID 568315 29 pages 2010
[23] T S N Asih S Lenhart S Wise et al ldquoe dynamics of HPVinfection and cervical cancer cellsrdquo Bulletin of MathematicalBiology vol 78 no 1 pp 4ndash20 2016
[24] L Rong J Guedj H Dahari et al ldquoAnalysis of hepatitis Cvirus decline during treatment with the protease inhibitor
danoprevir using a multiscale modelrdquo PLoS ComputationalBiology vol 9 no 3 Article ID e1002959 2013
[25] J Guedj H Dahari L Rong et al ldquoModeling shows that theNS5A inhibitor daclatasvir has two modes of action and yieldsa shorter estimate of the hepatitis C virus half-liferdquo Pro-ceedings of the National Academy of Sciences vol 110 no 10pp 3991ndash3996 2013
[26] L Rong and A S Perelson ldquoMathematical analysis of mul-tiscale models for hepatitis C virus dynamics under therapywith direct-acting antiviral agentsrdquoMathematical Biosciencesvol 245 no 1 pp 22ndash30 2013
[27] H A Elkaranshawy A M Abdelrazek and H M EzzatldquoPower series solution to sliding velocity in three-dimensionalmultibody systems with impact and frictionrdquo InternationalJournal of Mathematical Computational Physical Electricaland Computer Engineering vol 9 no 10 2015
[28] I M Abdelrazik and H A Elkaranshawy ldquoExtended Parker-Sochacki method for Michaelis-Menten enzymatic reactionmodelrdquo Analytical Biochemistry vol 496 pp 50ndash54 2016
[29] I M Abdelrazik H A Elkaranshawy and A M AbdelrazekldquoModified Parker-Sochacki method for solving nonlinearoscillatorsrdquo Mechanics Based Design of Structures and Ma-chines vol 45 no 2 pp 239ndash252 2016
[30] I M Abdelrazik Improving the solutions of parker sochackimethod for nonlinear ordinary differential equations PhDthesis Department of Engineering Mathematics and PhysicsFaculty of Engineering Alexandria University AlexandriaEgypt 2016
[31] G A Baker and P R Graves-Morris Pade Approximants vol59 Cambridge University Press Cambridge UK 1996
[32] B Raftari and A Yildirim ldquoSeries solution of a nonlinearODE arising in magnetohydrodynamic by HPM-Pade tech-niquerdquo Computers amp Mathematics with Applications vol 61no 6 pp 1676ndash1681 2011
[33] J Aubard P Levoir A Denis and P Claverie ldquoDirectanalysis of chemical relaxation signals by a method based onthe combination of Laplace transform and Pade approx-imantsrdquo Computers amp Chemistry vol 11 no 3 pp 163ndash1781987
[34] H Vazquez-Leal and F Guerrero ldquoApplication of seriesmethod with Pade and Laplace-Pade resummation methodsto solve a model for the evolution of smoking habit in SpainrdquoComputational and Applied Mathematics vol 33 no 1pp 181ndash192 2014
8 Mathematical Problems in Engineering
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
substitute them in equation (23) to get a closed-form so-lution for the viral load Hence the viral load can be plottedversus time or simply the viral load can be estimated at anyinstant by direct substitution in the closed-form solution
5 Conclusions
Power series solution combined with the LaplacendashPaderesummation method (PSLP) has been used to obtain a
general approximate analytical solution for the nonlinearstandard viral dynamic model of HCV for patients treatedwith DAAs However the solution is not limited to HCVmodel and it can be applied to other viruses like HIV andHBV for example given that proper parameters are used Totest the applicability and accuracy of the proposed methodresults have been compared with viral load data and withpublished simulated results Satisfactory agreement betweenthe PSLP solution and the corresponding viral load data hasbeen found for all the considered casese PSLP results andthe published simulated results are almost identical ecomparison proves that this innovative PSLP solution can beused with confidence for solving the nonlinear standard viraldynamic model is solution can conveniently be used to fitpatient data and estimate parameter values So it wouldfacilitate for physicians to monitor the changes in the viralload due to changes in treatment and to deal with changes inpatientrsquos parameters
Data Availability
e data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
e authors declare that there are no conflicts of interestregarding the publication of this paper
References
[1] D W Powell B Z Abramson J A Balint et al ldquoNationalinstitutes of health consensus development conference panelstatement management of hepatitis Crdquo Hepatology vol 26no 3 pp 2Sndash10S 1997
[2] J G McHutchison S C Gordon E R Schiff et al ldquoInterferonalfa-2b alone or in combination with ribavirin as initialtreatment for chronic hepatitis Crdquo New England Journal ofMedicine vol 339 no 21 pp 1485ndash1492 1998
[3] T Poynard J McHutchison M Manns et al ldquoImpact ofpegylated interferon alfa-2b and ribavirin on liver fibrosis inpatients with chronic hepatitis Crdquo Gastroenterology vol 122no 5 pp 1303ndash1313 2002
[4] T Asselah and P Marcellin ldquoNew direct-acting antiviralsrsquocombination for the treatment of chronic hepatitis Crdquo LiverInternational vol 31 pp 68ndash77 2011
[5] B Lam L Henry and Z Younossi ldquoSofosbuvir (Sovaldi) forthe treatment of hepatitis Crdquo Expert Review of ClinicalPharmacology vol 7 no 5 pp 555ndash566 2014
[6] J Chhatwal Q Chen T Ayer et al ldquoHepatitis C virus re-treatment in the era of direct-acting antivirals projections inthe USArdquo Alimentary Pharmacology amp 3erapeutics vol 47no 7 pp 1023ndash1031 2018
[7] V Cento T H T Nguyen D Di Carlo et al ldquoImprovementof ALT decay kinetics by all-oral HCV treatment role of
Table 3 Parameter values used with SOF+ SIM and TVR+PR
Figure 3 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with TVR+PR s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8 ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
PSLPRef [7]
HCV
RN
A
0 5 10 15 20 25Days
10
1000
105
107
V (t) = 95287079endash528t + 212179endash027t
Figure 4 Comparison between the results obtained by the PSLPmethod and by simulation in [7] for treatment with SOF+SIM s
13lowast 105 cellsml d 001 dayminus 1 and β 5lowast 10minus 8ml dayminus 1virionminus 1
and the rest of the parameter values are given in Table 3
Mathematical Problems in Engineering 7
NS5A inhibitors and differences with IFN-based regimensrdquoPLoS One vol 12 no 5 Article ID e0177352 2017
[8] T H T Nguyen J Guedj S L Uprichard A KohliS Kottilil and A S Perelson ldquoe paradox of highly ef-fective sofosbuvir-based combination therapy despite slowviral decline can we still rely on viral kineticsrdquo ScientificReports vol 7 no 1 Article ID 10233 2017
[9] World Health Organization Global Hepatitis Report 2017Global Hepatitis Programme Department of HIVAIDSWHO Geneva Switzerland 2017
[10] World Health Organization Combating Hepatitis B and C toReach Elimination by 2030 World Health OrganizationGeneva Switzerland 2016
[11] D S Jones and B D Sleeman ldquoDifferential equations andmathematical biologyrdquo inMathematical Biology andMedicineSeries Chapman amp HallCRC Boca Raton FL USA 2003
[12] A U Neumann N P Lam H Dahari et al ldquoHepatitis C viraldynamics in vivo and the antiviral efficacy of interferon-therapyrdquo Science vol 282 no 5386 pp 103ndash107 1998
[13] J Guedj and A U Neumann ldquoUnderstanding hepatitis Cviral dynamics with direct-acting antiviral agents due to theinterplay between intracellular replication and cellular in-fection dynamicsrdquo Journal of 3eoretical Biology vol 267no 3 pp 330ndash340 2010
[14] H Dahari J Guedj A S Perelson and T J LaydenldquoHepatitis C viral kinetics in the era of direct acting antiviralagents and interleukin-28Brdquo Current Hepatitis Reportsvol 10 no 3 pp 214ndash227 2011
[15] A Chaterjee J Guedj and A S Perelson ldquoMathematicalmodeling of HCV infection what can it teach us in the era ofdirect antiviral agentsrdquo Antiviral 3erapy vol 17 no 6pp 1171ndash1182 2012
[16] R J Payne M A Nowak and B S Blumberg ldquoe dynamicsof hepatitis B virus infectionrdquo Proceedings of the NationalAcademy of Sciences vol 93 no 13 pp 6542ndash6546 1996
[17] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[18] D Wodarz and M A Nowak ldquoMathematical models of HIVpathogenesis and treatmentrdquo BioEssays vol 24 no 12pp 1178ndash1187 2002
[19] W H Ho and A L F Chan ldquoHybrid taguchi-differentialevolution algorithm for parameter estimation of differentialequation models with application to HIV dynamicsrdquoMathematical Problems in Engineering vol 2011 Article ID514756 14 pages 2011
[20] 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 Sciencevol 271 no 5255 pp 1582ndash1586 1996
[21] Q Li and Y Xiao ldquoGlobal dynamics of a virus-immunesystem with virus-guided therapy and saturation growth ofvirusrdquo Mathematical Problems in Engineering vol 2018Article ID 4710586 18 pages 2018
[22] H Zarei A V Kamyad and S Effati ldquoMultiobjective optimalcontrol of HIV dynamicsrdquo Mathematical Problems in Engi-neering vol 2010 Article ID 568315 29 pages 2010
[23] T S N Asih S Lenhart S Wise et al ldquoe dynamics of HPVinfection and cervical cancer cellsrdquo Bulletin of MathematicalBiology vol 78 no 1 pp 4ndash20 2016
[24] L Rong J Guedj H Dahari et al ldquoAnalysis of hepatitis Cvirus decline during treatment with the protease inhibitor
danoprevir using a multiscale modelrdquo PLoS ComputationalBiology vol 9 no 3 Article ID e1002959 2013
[25] J Guedj H Dahari L Rong et al ldquoModeling shows that theNS5A inhibitor daclatasvir has two modes of action and yieldsa shorter estimate of the hepatitis C virus half-liferdquo Pro-ceedings of the National Academy of Sciences vol 110 no 10pp 3991ndash3996 2013
[26] L Rong and A S Perelson ldquoMathematical analysis of mul-tiscale models for hepatitis C virus dynamics under therapywith direct-acting antiviral agentsrdquoMathematical Biosciencesvol 245 no 1 pp 22ndash30 2013
[27] H A Elkaranshawy A M Abdelrazek and H M EzzatldquoPower series solution to sliding velocity in three-dimensionalmultibody systems with impact and frictionrdquo InternationalJournal of Mathematical Computational Physical Electricaland Computer Engineering vol 9 no 10 2015
[28] I M Abdelrazik and H A Elkaranshawy ldquoExtended Parker-Sochacki method for Michaelis-Menten enzymatic reactionmodelrdquo Analytical Biochemistry vol 496 pp 50ndash54 2016
[29] I M Abdelrazik H A Elkaranshawy and A M AbdelrazekldquoModified Parker-Sochacki method for solving nonlinearoscillatorsrdquo Mechanics Based Design of Structures and Ma-chines vol 45 no 2 pp 239ndash252 2016
[30] I M Abdelrazik Improving the solutions of parker sochackimethod for nonlinear ordinary differential equations PhDthesis Department of Engineering Mathematics and PhysicsFaculty of Engineering Alexandria University AlexandriaEgypt 2016
[31] G A Baker and P R Graves-Morris Pade Approximants vol59 Cambridge University Press Cambridge UK 1996
[32] B Raftari and A Yildirim ldquoSeries solution of a nonlinearODE arising in magnetohydrodynamic by HPM-Pade tech-niquerdquo Computers amp Mathematics with Applications vol 61no 6 pp 1676ndash1681 2011
[33] J Aubard P Levoir A Denis and P Claverie ldquoDirectanalysis of chemical relaxation signals by a method based onthe combination of Laplace transform and Pade approx-imantsrdquo Computers amp Chemistry vol 11 no 3 pp 163ndash1781987
[34] H Vazquez-Leal and F Guerrero ldquoApplication of seriesmethod with Pade and Laplace-Pade resummation methodsto solve a model for the evolution of smoking habit in SpainrdquoComputational and Applied Mathematics vol 33 no 1pp 181ndash192 2014
8 Mathematical Problems in Engineering
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
NS5A inhibitors and differences with IFN-based regimensrdquoPLoS One vol 12 no 5 Article ID e0177352 2017
[8] T H T Nguyen J Guedj S L Uprichard A KohliS Kottilil and A S Perelson ldquoe paradox of highly ef-fective sofosbuvir-based combination therapy despite slowviral decline can we still rely on viral kineticsrdquo ScientificReports vol 7 no 1 Article ID 10233 2017
[9] World Health Organization Global Hepatitis Report 2017Global Hepatitis Programme Department of HIVAIDSWHO Geneva Switzerland 2017
[10] World Health Organization Combating Hepatitis B and C toReach Elimination by 2030 World Health OrganizationGeneva Switzerland 2016
[11] D S Jones and B D Sleeman ldquoDifferential equations andmathematical biologyrdquo inMathematical Biology andMedicineSeries Chapman amp HallCRC Boca Raton FL USA 2003
[12] A U Neumann N P Lam H Dahari et al ldquoHepatitis C viraldynamics in vivo and the antiviral efficacy of interferon-therapyrdquo Science vol 282 no 5386 pp 103ndash107 1998
[13] J Guedj and A U Neumann ldquoUnderstanding hepatitis Cviral dynamics with direct-acting antiviral agents due to theinterplay between intracellular replication and cellular in-fection dynamicsrdquo Journal of 3eoretical Biology vol 267no 3 pp 330ndash340 2010
[14] H Dahari J Guedj A S Perelson and T J LaydenldquoHepatitis C viral kinetics in the era of direct acting antiviralagents and interleukin-28Brdquo Current Hepatitis Reportsvol 10 no 3 pp 214ndash227 2011
[15] A Chaterjee J Guedj and A S Perelson ldquoMathematicalmodeling of HCV infection what can it teach us in the era ofdirect antiviral agentsrdquo Antiviral 3erapy vol 17 no 6pp 1171ndash1182 2012
[16] R J Payne M A Nowak and B S Blumberg ldquoe dynamicsof hepatitis B virus infectionrdquo Proceedings of the NationalAcademy of Sciences vol 93 no 13 pp 6542ndash6546 1996
[17] L Min Y Su and Y Kuang ldquoMathematical analysis of a basicvirus infection model with application to HBV infectionrdquoRocky Mountain Journal of Mathematics vol 38 no 5pp 1573ndash1585 2008
[18] D Wodarz and M A Nowak ldquoMathematical models of HIVpathogenesis and treatmentrdquo BioEssays vol 24 no 12pp 1178ndash1187 2002
[19] W H Ho and A L F Chan ldquoHybrid taguchi-differentialevolution algorithm for parameter estimation of differentialequation models with application to HIV dynamicsrdquoMathematical Problems in Engineering vol 2011 Article ID514756 14 pages 2011
[20] 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 Sciencevol 271 no 5255 pp 1582ndash1586 1996
[21] Q Li and Y Xiao ldquoGlobal dynamics of a virus-immunesystem with virus-guided therapy and saturation growth ofvirusrdquo Mathematical Problems in Engineering vol 2018Article ID 4710586 18 pages 2018
[22] H Zarei A V Kamyad and S Effati ldquoMultiobjective optimalcontrol of HIV dynamicsrdquo Mathematical Problems in Engi-neering vol 2010 Article ID 568315 29 pages 2010
[23] T S N Asih S Lenhart S Wise et al ldquoe dynamics of HPVinfection and cervical cancer cellsrdquo Bulletin of MathematicalBiology vol 78 no 1 pp 4ndash20 2016
[24] L Rong J Guedj H Dahari et al ldquoAnalysis of hepatitis Cvirus decline during treatment with the protease inhibitor
danoprevir using a multiscale modelrdquo PLoS ComputationalBiology vol 9 no 3 Article ID e1002959 2013
[25] J Guedj H Dahari L Rong et al ldquoModeling shows that theNS5A inhibitor daclatasvir has two modes of action and yieldsa shorter estimate of the hepatitis C virus half-liferdquo Pro-ceedings of the National Academy of Sciences vol 110 no 10pp 3991ndash3996 2013
[26] L Rong and A S Perelson ldquoMathematical analysis of mul-tiscale models for hepatitis C virus dynamics under therapywith direct-acting antiviral agentsrdquoMathematical Biosciencesvol 245 no 1 pp 22ndash30 2013
[27] H A Elkaranshawy A M Abdelrazek and H M EzzatldquoPower series solution to sliding velocity in three-dimensionalmultibody systems with impact and frictionrdquo InternationalJournal of Mathematical Computational Physical Electricaland Computer Engineering vol 9 no 10 2015
[28] I M Abdelrazik and H A Elkaranshawy ldquoExtended Parker-Sochacki method for Michaelis-Menten enzymatic reactionmodelrdquo Analytical Biochemistry vol 496 pp 50ndash54 2016
[29] I M Abdelrazik H A Elkaranshawy and A M AbdelrazekldquoModified Parker-Sochacki method for solving nonlinearoscillatorsrdquo Mechanics Based Design of Structures and Ma-chines vol 45 no 2 pp 239ndash252 2016
[30] I M Abdelrazik Improving the solutions of parker sochackimethod for nonlinear ordinary differential equations PhDthesis Department of Engineering Mathematics and PhysicsFaculty of Engineering Alexandria University AlexandriaEgypt 2016
[31] G A Baker and P R Graves-Morris Pade Approximants vol59 Cambridge University Press Cambridge UK 1996
[32] B Raftari and A Yildirim ldquoSeries solution of a nonlinearODE arising in magnetohydrodynamic by HPM-Pade tech-niquerdquo Computers amp Mathematics with Applications vol 61no 6 pp 1676ndash1681 2011
[33] J Aubard P Levoir A Denis and P Claverie ldquoDirectanalysis of chemical relaxation signals by a method based onthe combination of Laplace transform and Pade approx-imantsrdquo Computers amp Chemistry vol 11 no 3 pp 163ndash1781987
[34] H Vazquez-Leal and F Guerrero ldquoApplication of seriesmethod with Pade and Laplace-Pade resummation methodsto solve a model for the evolution of smoking habit in SpainrdquoComputational and Applied Mathematics vol 33 no 1pp 181ndash192 2014
8 Mathematical Problems in Engineering
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
![For the cyclic process shown, W is:D A] 0, because it’s a loop B] p 0 V 0 C] - p 0 V 0 D] 2 p 0 V 0 E] 6 p 0 V 0 For the cyclic process shown, U is:](https://static.documents.pub/doc/80x56/56649d7e5503460f94a609a5/for-the-cyclic-process-shown-w-isd-a-0-because-its-a-loop-b-p-0-v-0.jpg)
![Y C%& «#.0 M*ñ >+ ¥8Q b Ê ] v | &k &É M&É i Y C ~ b …...1 Î'v>*'¨ è4 >&'¨>/'v>*'¨>0'v>' %Ê'2 c>*%& «#.0 b0i!l ? }&k &É _ > E \7 #.0 M*ñ>& è W &k &É \7 #.0 M*ñ>'](https://static.documents.pub/doc/80x56/5f0ebf1c7e708231d440bd28/y-c-0-m-8q-b-v-k-m-i-y-c-b-.jpg)
