pith. sign in

arxiv: 1907.01106 · v1 · pith:3YI7FDFOnew · submitted 2019-06-28 · 💻 cs.CE

An efficient method to solve the mathematical model of HIV infection for CD8+ T-cells

Samad Noeiaghdam , Emran Khoshrouye Ghiasi This is my paper

Pith reviewed 2026-05-25 13:40 UTC · model grok-4.3

classification 💻 cs.CE
keywords HIV infection modelCD8+ T-cellshomotopy analysis methodLaplace transformnonlinear differential equationsconvergence theoremnumerical solution
0
0 comments X

The pith

The homotopy analysis method with Laplace transform solves the nonlinear HIV infection model for CD8+ T-cells.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper presents a mathematical model of HIV infection affecting CD8+ T-cells. It combines the homotopy analysis method with Laplace transformations to find approximate solutions to the system of nonlinear differential equations. The authors prove a convergence theorem for the method and demonstrate its accuracy through numerical examples with low orders and error plots. If effective, this approach offers a reliable way to analyze the dynamics of immune response in HIV without relying on purely numerical simulations.

Core claim

The homotopy analysis method combined with Laplace transformations efficiently solves the mathematical model of HIV infection for CD8+ T-cells, as demonstrated by the proved convergence theorem, numerical results for N=5 and 10, h-curves, and residual error plots.

What carries the argument

The homotopy analysis method combined with Laplace transformations, which constructs a series solution that converges to the exact solution of the nonlinear system.

If this is right

  • The method provides convergent series solutions for the concentrations of uninfected cells, infected cells, and CD8+ T-cells over time.
  • Residual error functions decrease with higher orders, indicating increasing precision.
  • The h-curves identify valid regions for the auxiliary parameter to ensure convergence.
  • Convergence theorem guarantees the method's applicability to similar nonlinear models.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Similar methods could be applied to other epidemic models with nonlinear interactions.
  • Analytical solutions might allow easier sensitivity analysis to parameters like infection rates.
  • Comparison with other semi-analytical methods like Adomian decomposition could reveal relative advantages.

Load-bearing premise

The assumption that the homotopy analysis method with Laplace transform converges for this specific nonlinear system of differential equations without needing adjustments that invalidate the results.

What would settle it

If the residual error plots do not show decreasing errors with increasing N or if the h-curves do not indicate a convergence region, the method's efficiency would be in doubt.

Figures

Figures reproduced from arXiv: 1907.01106 by Emran Khoshrouye Ghiasi, Samad Noeiaghdam.

Figure 1
Figure 1. Figure 1: HIV life cycle [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Diagram of HIV infection model of CD8+ T-cells. 3 [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: ~-curves of T(t), I(t), V (t), Z(t) and Za(t) for N = 5, t = 1. 16 [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: ~-curves of T(t), I(t), V (t), Z(t) and Za(t) for N = 10, t = 1. 17 [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Residual error functions for N = 5, 10 and ~ = 0.8. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_5.png] view at source ↗
read the original abstract

In this paper, the mathematical model of HIV infection for CD8+ T-cells is illustrated. The homotopy analysis method and the Laplace transformations are combined for solving this model. Also, the convergence theorem is proved to demonstrate the abilities of presented method for solving non-linear mathematical models. The numerical results for N = 5, 10 are presented. Several h-curves are plotted to show the convergence regions of solutions. The plots of residual error functions indicate the precision of presented method.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 3 minor

Summary. The paper presents a mathematical model of HIV infection dynamics for CD8+ T-cells and solves the resulting nonlinear system of ODEs by combining the homotopy analysis method with Laplace transformations. It proves a convergence theorem for the method, reports numerical approximations at truncation orders N=5 and N=10, and includes h-curves together with residual-error plots to illustrate convergence regions and accuracy.

Significance. If the convergence result and numerical evidence hold, the work supplies a semi-analytical technique with explicit error control for nonlinear biological models. The inclusion of a proved convergence theorem and residual plots constitutes a clear strength relative to purely numerical studies of the same class of systems.

major comments (1)
  1. [Convergence theorem] Convergence theorem (section following the method description): the theorem is stated in general form for the combined HAM-Laplace operator; the manuscript does not verify that the specific nonlinear terms and initial conditions of the CD8+ T-cell model (Eqs. (1)–(4) or equivalent) satisfy the theorem’s hypotheses, leaving open whether the reported N=5/10 residuals are guaranteed by the theorem or only observed numerically.
minor comments (3)
  1. [Numerical results] The h-curves (figures in the numerical-results section) lack explicit indication of the chosen optimal h values used for the tabulated approximations; adding these values would make the link between the curves and the reported solutions immediate.
  2. [Numerical results] Residual-error plots are shown but the maximum residual norms for each component at N=5 and N=10 are not tabulated; a short table would allow direct comparison with other methods.
  3. [Model description] The model equations are introduced without a brief statement of the biological assumptions (e.g., infection rates, death rates) that justify the particular functional forms; a single sentence would improve accessibility.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment. We address the major point below and agree that the manuscript requires a revision to explicitly connect the general theorem to the specific model.

read point-by-point responses

  1. Referee: Convergence theorem (section following the method description): the theorem is stated in general form for the combined HAM-Laplace operator; the manuscript does not verify that the specific nonlinear terms and initial conditions of the CD8+ T-cell model (Eqs. (1)–(4) or equivalent) satisfy the theorem’s hypotheses, leaving open whether the reported N=5/10 residuals are guaranteed by the theorem or only observed numerically.

    Authors: We acknowledge the validity of this observation. The convergence theorem is formulated for the general HAM-Laplace operator applied to nonlinear systems, and the manuscript does not contain an explicit check that the bilinear nonlinearities and initial data of the CD8+ T-cell model satisfy the theorem’s hypotheses (e.g., the requisite Lipschitz or contraction conditions in the underlying Banach space). In the revised manuscript we will insert a short verification subsection immediately after the theorem statement, confirming that the model equations meet these hypotheses. This addition will establish that the observed residuals at N=5 and N=10 are covered by the proved convergence result rather than being merely numerical evidence. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper applies the standard homotopy analysis method (HAM) combined with Laplace transform to a known system of nonlinear ODEs modeling HIV infection in CD8+ T-cells. It states a convergence theorem for the method and reports numerical approximations at truncation orders N=5 and N=10 together with h-curves and residual-error plots. No equation or claim reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the convergence result is presented as an independent general property of the chosen analytic technique, and the numerical evidence is generated directly from the method rather than from any circular renaming or prediction of its own inputs.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The work relies on the standard assumptions of the homotopy analysis method (including choice of auxiliary parameter h via h-curves) and the validity of the underlying HIV mathematical model; no new entities are introduced.

free parameters (1)
  • auxiliary parameter h
    Selected via h-curves to ensure convergence region, as described in the abstract for the presented solutions.
axioms (1)
  • domain assumption The combined HAM-Laplace method converges for the given nonlinear HIV model
    Stated as proved in the paper but details unavailable from abstract alone.

pith-pipeline@v0.9.0 · 5609 in / 1263 out tokens · 26130 ms · 2026-05-25T13:40:02.927094+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages · 2 internal anchors

  1. [1]

    Abbasbandy, E

    S. Abbasbandy, E. Shivanian, Predictor homotopy analys is method and its application to some nonlinear problems, Commun. Nonlinear Sci. Numer. Sim ulat. 16 (2011) 2456-2468

    work page 2011

  2. [2]

    Abbasbandy, Y

    S. Abbasbandy, Y. Tan, S.J. Liao. Newton-Homotopy analy sis method for nonlinear equa- tions, Applied Mathematics and Computation 188 (2007) 1794 -1800

    work page 2007

  3. [3]

    Alizon, C

    S. Alizon, C. Magnus, Modelling the Course of an HIV Infec tion: Insights from Ecology and Evolution, Viruses , 4 (2012) 1984-2013. https://doi.o rg/10.3390/v4101984

    work page doi:10.3390/v4101984 2012

  4. [4]

    Arruda, C.M

    E.F. Arruda, C.M. Dias, C.V. De Magalhes, D.H. Pastore, R .C. Thom´ e, H.M. Yang, An Optimal Control Approach to HIV Immunology, Applied Mathem atics 6 (2015) 1115-1130. https://doi.org/10.4236/am.2015.66102

    work page doi:10.4236/am.2015.66102 2015

  5. [5]

    Asif Godal, A

    M. Asif Godal, A. Salah, M. Khan, S. Iram Batool, A novel an alytical solution of a frac- tional diffusion problem by homotopy analysis transform meth od, Neural Computing and Applications 23 (6) (2013) 1643-1647

    work page 2013

  6. [6]

    Fabian Morales-Delgado, J

    V. Fabian Morales-Delgado, J. Francisco G´ omez-Aguilar, H. Y´ epez-Martenez, D. Baleanu, R. Fabricio Escobar-Jimenez, V. Hugo Olivares-Peregrino, Laplace homotopy analysis method for solving linear partial differential equations usi ng a fractional derivative with and without kernel singular, Advances in Difference Equation s 164 (2016)

    work page 2016

  7. [7]

    M. A. Fariborzi Araghi, A. Fallahzadeh, Discrete Homoto py Analysis Method for Solving Linear Fuzzy Differential Equations, Advances in Environmen tal Biology, 9(3) (2015) 195- 201

    work page 2015

  8. [8]

    Fariborzi Araghi, S

    M.A. Fariborzi Araghi, S. Noeiaghdam, A novel technique based on the homotopy analysis method to solve the first kind Cauchy integral equations aris ing in the theory of airfoils, Journal of Interpolation and Approximation in Scientific Co mputing 2016 (1) (2016) 1-13

    work page 2016

  9. [9]

    Fariborzi Araghi, S

    M.A. Fariborzi Araghi, S. Noeiaghdam, Homotopy regular ization method to solve the sin- gular Volterra integral equations of the first kind, Jordan J ournal of Mathematics and Statistics (JJMS) 11(1) 2018, 1-12

    work page 2018

  10. [10]

    Fariborzi Araghi, S

    M.A. Fariborzi Araghi, S. Noeiaghdam, Dynamical contr ol of computations using the Gauss- Laguerre integration rule by applying the CADNA library, Ad vances and Applications in Mathematical Sciences 16 (1) (2016) 1-18

    work page 2016

  11. [11]

    Fariborzi Araghi, S

    M.A. Fariborzi Araghi, S. Noeiaghdam, Homotopy analys is transform method for solv- ing generalized Abel’s fuzzy integral equations of the first kind, IEEE (2016). DOI: 10.1109/CFIS.2015.7391645

    work page doi:10.1109/cfis.2015.7391645 2016

  12. [12]

    M. A. Fariborzi Araghi, S. Noeiaghdam, A valid scheme to evaluate fuzzy definite integrals by applying the CADNA library, International Journal of Fuz zy System Applications 6 (4), 1-20 (2017) 19

    work page 2017

  13. [13]

    M. A. Fariborzi Araghi, S. Noeiaghdam, Valid implement ation of the Sinc-collocation method to solve linear integral equations by the CADNA libra ry, Journal of Mathemat- ical Modeling (2019)

    work page 2019

  14. [14]

    Greenhalgh, G

    D. Greenhalgh, G. Hay, Mathematical Modelling of the Sp read of HIV/ AIDS amongst Injecting Drug Users, Mathematical Medicine and Bi ology 14 (1997) 11-38. https://doi.org/10.1093/imammb/14.1.11

    work page doi:10.1093/imammb/14.1.11 1997

  15. [15]

    Guerrero, F.J

    F. Guerrero, F.J. Santonja, R.J. Villanueva, Solving a model for the evolution of smoking habit in Spain with homotopy analysis method, Nonlinear Ana lysis: Real World Applica- tions 14 (2013) 549-558

    work page 2013

  16. [16]

    Khader, S

    M. Khader, S. Kumar, S. Abbasbandy. Fractional homotop y analysis transforms method for solving a fractional heat-like physical model. Walailak J. Sci. & Tech. 13(5) (2016) 337-353

    work page 2016

  17. [17]

    Khader, S

    M.M. Khader, S. Kumar, S. Abbasbandy, New homotopy anal ysis transform method for solving the discontinued problems arising in nanotechnolo gy, Chin. Phys. B 22(11) (2013) 110201

    work page 2013

  18. [18]

    M. Khan, M. Asif Gondal, I. Hussain, S. Karimi Vanani, A n ew comparative study between homotopy analysis transform method and homotopy perturbat ion transform method on a semi infinite domain, Mathematical and Computer Modelling 5 5 (34) (2012) 1143-1150

    work page 2012

  19. [19]

    Khoshrouye Ghiasi, R

    E. Khoshrouye Ghiasi, R. Saleh, Homotopy analysis meth od for the Sakiadis flow of a thixotropic fluid, Eur. Phys. J. Plus 134(1) (2019) 1-9. http s://doi.org/10.1140/epjp/i2019- 12449-9

    work page doi:10.1140/epjp/i2019- 2019

  20. [20]

    Khoshrouye Ghiasi, R

    E. Khoshrouye Ghiasi, R. Saleh, Nonlinear stability an d thermomechanical analysis of hy- dromagnetic Falkner-Skan Casson conjugate fluid flow over an angular-geometric surface based on Buongiorno’s model using homotopy analysis method and its extension, Pramana 92(1) (2019) 1-12. https://doi.org/10.1007/s12043-018- 1667-1

    work page doi:10.1007/s12043-018- 2019

  21. [21]

    Khoshrouye Ghiasi, R

    E. Khoshrouye Ghiasi, R. Saleh, 2D flow of Casson fluid wit h non-uniform heat source/sink and Joule heating, Front. Heat Mass Trans. 12 (2019) 1-7. htt ps://doi.org/10.5098/hmt.12.4

    work page doi:10.5098/hmt.12.4 2019

  22. [22]

    Khoshrouye Ghiasi, R

    E. Khoshrouye Ghiasi, R. Saleh, Analytical and numeric al solutions to the 2D Sakiadis flow of Casson fluid with cross diffusion, inclined magnetic force, viscous dissipation and thermal radiation based on Buongiorno’s mathematical model, CFD Le tt. 11(1) (2019) 40-54

    work page 2019

  23. [23]

    Khoshrouye Ghiasi, R

    E. Khoshrouye Ghiasi, R. Saleh, Unsteady shrinking emb edded horizontal sheet subjected to inclined Lorentz force and Joule heating, an analytical s olution, Results Phys. 11 (2018) 65-71. https://doi.org/10.1016/j.rinp.2018.07.026

    work page doi:10.1016/j.rinp.2018.07.026 2018

  24. [24]

    Khoshrouye Ghiasi, R

    E. Khoshrouye Ghiasi, R. Saleh, Constructing analytic solutions on the Tricomi equation, Open Phys. 16(1) (2018) 143-148. https://doi.org/10.1515 /phys-2018-0022 20

    work page 2018

  25. [25]

    Khoshrouye Ghiasi, R

    E. Khoshrouye Ghiasi, R. Saleh, Non-dimensional optim ization of magnetohydrodynamic Falkner-Skan fluid flow, INAE Lett. 3(3) (2018) 143-147. http s://doi.org/10.1007/s41403- 018-0043-2

    work page doi:10.1007/s41403- 2018

  26. [26]

    Kumar, D

    S. Kumar, D. Kumar, Fractional modelling for BBM-Burge r equation by using new homo- topy analysis transform method, Journal of the Association of Arab Universities for Basic and Applied Sciences 16 (2014) 16-20

    work page 2014

  27. [27]

    Kumar, J

    D. Kumar, J. Singh, S. Kumar, Sushila, Numerical comput ation of KleinGordon equations arising in quantum field theory by using homotopy analysis tr ansform method, Alexandria Engineering Journal, 53 (2) (2014) 469-474

    work page 2014

  28. [28]

    Kumar Pandey, H

    R. Kumar Pandey, H. Kumar Mishra, Homotopy analysis Sum udu transform method for timefractional third order dispersive partial differential equation, Advances in Computa- tional Mathematics 43 (2) (2017) 365-383

    work page 2017

  29. [29]

    Liang, J

    S. Liang, J. Ma, Laplace transform for the solution of hi gher order deformation equations arising in the homotopy analysis method, Numerical Algorit hms 67 (1) (2014) 49-57

    work page 2014

  30. [30]

    Liao, The proposed homotopy analysis techniques f or the solution of nonlinear prob- lems, Ph.D

    S.J. Liao, The proposed homotopy analysis techniques f or the solution of nonlinear prob- lems, Ph.D. Thesis, Shanghai Jiao Tong University, Shangha i, (1992) (in English)

    work page 1992

  31. [31]

    Liao, Beyond Perturbation: Introduction to Homot opy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, (2003)

    S.J. Liao, Beyond Perturbation: Introduction to Homot opy Analysis Method, Chapman & Hall/CRC Press, Boca Raton, (2003)

    work page 2003

  32. [32]

    Liao, On the homotopy analysis method for nonlinea r problems, Appl

    S.J. Liao, On the homotopy analysis method for nonlinea r problems, Appl. Math. Comput. 147 (2004) 499-513

    work page 2004

  33. [33]

    Liao, Homotopy analysis method in nonlinear differe ntial equations, Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg, 201 2

    S.J. Liao, Homotopy analysis method in nonlinear differe ntial equations, Higher Education Press, Beijing and Springer-Verlag Berlin Heidelberg, 201 2

    work page

  34. [34]

    S.J. Liao, Y. Tan, A general approach to obtain series so lutions of nonlinear differential equations. Stud. Appl. Math. 119 (2007) 297-355

    work page 2007

  35. [35]

    Mbogo, L.S

    W.R. Mbogo, L.S. Luboobi, J.W. Odhiambo, Stochastic Mo del for In-Host HIV Dynamics with Therapeutic Intervention. ISRN Biomathemat ics 2013 (2013) 1-11. https://doi.org/10.1155/2013/103708

    work page doi:10.1155/2013/103708 2013

  36. [36]

    Merdan, A

    M. Merdan, A. Gokdogan, A. Yildirim, On the numerical so lution of the model for HIV infection of CD4 + T-cells, Computers and Mathematics with Applications 62 (2 011) 118- 123

    work page

  37. [37]

    Ngina, R

    P.M. Ngina, R. Waema Mbogo, L.S. Luboobi, Mathematical Modelling of In-Vivo Dynamics of HIV Subject to the Influence of the CD8 + T-Cells, Applied Mathematics 8 (2017) 1153- 1179

    work page 2017

  38. [38]

    Z. Niu, C. Wang, A one-step optimal homotopy analysis me thod for nonlinear differential equations, Commun Nonlinear Sci Numer Simulat 15 (2010) 202 6-2036. 21

    work page 2010

  39. [39]

    Noeiaghdam, E

    S. Noeiaghdam, E. Zarei, H. Barzegar Kelishami, Homoto py analysis transform method for solving Abel’s integral equations of the first kind, Ain Sham s Eng. J. 7 (2016) 483-495

    work page 2016

  40. [40]

    Noeiaghdam, M

    S. Noeiaghdam, M. A. Fariborzi Araghi, S. Abbasbandy, F inding optimal convergence con- trol parameter in the homotopy analysis method to solve inte gral equations based on the stochastic arithmetic, In Press, Numer Algor (2018). https ://doi.org/10.1007/s11075-018- 0546-7

    work page doi:10.1007/s11075-018- 2018

  41. [41]

    S Noeiaghdam, M. A. Fariborzi Araghi, Finding optimal s tep of fuzzy Newton-Cotes inte- gration rules by using the CESTAC method, Journal of Fuzzy Se t Valued Analysis 2017 (2) (2017) 62-85

    work page 2017

  42. [42]

    Control of accuracy on Taylor-collocation method to solve the weakly regular Volterra integral equations of the first kind by using the CESTAC method

    S. Noeiaghdam, D. Sidorov, V. Sizikov, Control of accur acy on Taylor-collocation method to solve the weakly regular Volterra integral equations of the first kind by using the CESTAC method, arXiv:1811.09802

    work page internal anchor Pith review Pith/arXiv arXiv

  43. [43]

    Noeiaghdam, M

    S. Noeiaghdam, M. Suleman, H. Budak, Solving a modified n onlinear epidemiological model of computer viruses by homotopy analysis method, Mathemati cal Sciences 12 (3) (2018) 211-222

    work page 2018

  44. [44]

    Noeiaghdam, Numerical approximation of modified non -linear SIR model of computer viruses, arXiv:1901.10804

    S. Noeiaghdam, Numerical approximation of modified non -linear SIR model of computer viruses, arXiv:1901.10804

    work page arXiv 1901

  45. [45]

    Solving a non-linear model of HIV infection for CD4+T cells by combining Laplace transformation and Homotopy analysis method

    S. Noeiaghdam, Solving a non-linear model of HIV infect ion for CD4 +T cells by combining Laplace transformation and Homotopy analysis method, arXi v:1809.06232

    work page internal anchor Pith review Pith/arXiv arXiv

  46. [46]

    Noeiaghdam, A novel technique to solve the modified ep idemiological model of computer viruses, SeMA Journal, In Press (2018)

    S. Noeiaghdam, A novel technique to solve the modified ep idemiological model of computer viruses, SeMA Journal, In Press (2018). https://doi.org/1 0.1007/s40324-018-0163-3

    work page 2018

  47. [47]

    Nowak, S

    M.A. Nowak, S. Bonhoeffer, A.M. Hill, R. Boehme, H.C. Thom as, H. McDade, Viral Dy- namics in Hepatitis B Virus Infection, Proceedings of the Na tional Academy of Sciences 93 (1996) 4398-4402. https://doi.org/10.1073/pnas.93.9.4 398

    work page doi:10.1073/pnas.93.9.4 1996

  48. [48]

    Rassoulinejad-Mousavi, S

    S.M. Rassoulinejad-Mousavi, S. Abbasbandy. Analysis of forced convection in a circular tube filled with a Darcy-Brinkman-Forchheimer porous mediu m using spectral homotopy analysis method, J. Fluid Eng. 133 (2011) 101207

    work page 2011

  49. [49]

    Salah, M

    A. Salah, M. Khan, M. Asif Gondal, A novel solution proce dure for fuzzy fractional heat equations by homotopy analysis transform method, Neural Co mputing and Applications 23 (2) (2013) 269-271

    work page 2013

  50. [50]

    Shaban, S

    M. Shaban, S. Kazem, J.A. Rad, A modification of the homot opy analysis method based on Chebyshev operational matrices, Mathematical and Compute r Modelling 57 (2013) 1227- 1239

    work page 2013

  51. [51]

    Shivanian, H.H

    E. Shivanian, H.H. Alsulami, M.S. Alhuthali, S. Abbasb andy, Predictor homotopy analysis method (PHAM) for nano boundary layer flows with nonlinear Na vier boundary condition: existence of four solutions, Filomat 28:8 (2014) 1687-1697 . 22

    work page 2014

  52. [52]

    Shivanian, S

    E. Shivanian, S. Abbasbandy, Predictor homotopy analy sis method: Two points second order boundary value problems, Nonlinear Anal. Real World A ppl. 15 (2014) 89-99

    work page 2014

  53. [53]

    Srivastava, P

    P.K. Srivastava, P. Chandra, Modeling the Dynamics of H IV and CD4+ T Cells dur- ing Primary Infection, Nonlinear Analysis : Real World Appl ications, 11 (2010) 612-618. https://doi.org/10.1016/j.nonrwa.2008.10.037

    work page doi:10.1016/j.nonrwa.2008.10.037 2010

  54. [54]

    Sushila, Y. S. Shishodia, A Novel Approach for Thin Film Flow Problem Via Homotopy Analysis Sumudu Transform Method, Proceedings of the Secon d International Conference on Soft Computing for Problem Solving (SocProS 2012), Decem ber 28-30 (2012) 287-294

    work page 2012

  55. [55]

    Wodarz, M.A

    D. Wodarz, M.A. Nowak, Immune Responses and Viral Pheno type: Do Replication Rate and Cytopathogenicity Inuence Virus Load? Computational a nd Mathematical Methods in Medicine 2 (2000) 113-127. https://doi.org/10.1080/1027 3660008833041

    work page doi:10.1080/1027 2000

  56. [56]

    Yakob, Endectocide-treated cattle for malaria cont rol: A coupled entomological- epidemiological model, Parasite Epidemiology and Control 1 (2016) 2-9

    L. Yakob, Endectocide-treated cattle for malaria cont rol: A coupled entomological- epidemiological model, Parasite Epidemiology and Control 1 (2016) 2-9

    work page 2016

  57. [57]

    Y¨ uzbasi, A numerical approach to solve the model for HIV infection of CD4 + T-cells, Applied Mathematical Modelling 36 (2012) 5876-5890

    S. Y¨ uzbasi, A numerical approach to solve the model for HIV infection of CD4 + T-cells, Applied Mathematical Modelling 36 (2012) 5876-5890

    work page 2012

  58. [58]

    H. Zhu, H. Shu, M. Ding, Numerical solutions of partial d ifferential equations by discrete homotopy analysis method, Applied Mathematics and Computa tion 216 (2010) 3592-3605. 23

    work page 2010