Efficient method for solving nonlinear weakly singular kernel fractional integro-differential equations

Ismail Gad Ameen; Dumitru Baleanu; Hussien Shafei Hussien; Ismail Gad Ameen; Dumitru Baleanu; Hussien Shafei Hussien

doi:10.3934/math.2024764

AIMS Mathematics

2024, Volume 9, Issue 6: 15819-15836. doi: 10.3934/math.2024764

Previous Article Next Article

Research article Special Issues

Efficient method for solving nonlinear weakly singular kernel fractional integro-differential equations

1.
Department of Mathematics, Faculty of Science, South Valley University, Qena 83523, Egypt
2.
Department of Computer Science and Mathematics, Lebanese American University, Beirut, Lebanon

Received: 20 January 2024 Revised: 03 March 2024 Accepted: 25 March 2024 Published: 06 May 2024
MSC : 33E12, 47G20, 65K10, 65G99, 65R20

This paper introduced an efficient method to obtain the solution of linear and nonlinear weakly singular kernel fractional integro-differential equations (WSKFIDEs). It used Riemann-Liouville fractional integration (R-LFI) to remove singularities and approximated the regularized problem with a combined approach using the generalized fractional step-Mittag-Leffler function (GFSMLF) and operational integral fractional Mittag matrix (OIFMM) method. The resulting algebraic equations were turned into an optimization problem. We also proved the method's accuracy in approximating any function, as well as its fractional differentiation and integration within WSKFIDEs. The proposed method was performed on some attractive examples in order to show how their solutions behave at various values of the fractional order $\digamma$ . The paper provided a valuable contribution to the field of fractional calculus (FC) by presenting a novel method for solving WSKFIDEs. Additionally, the accuracy of this method was verified by comparing its results with those obtained using other methods.

Keywords:

singular integro-differential equations,
generalized fractional Mittag-Leffler function,
step-function,
operational integral matrix,
optimization method,
error analysis

Citation: Ismail Gad Ameen, Dumitru Baleanu, Hussien Shafei Hussien. Efficient method for solving nonlinear weakly singular kernel fractional integro-differential equations[J]. AIMS Mathematics, 2024, 9(6): 15819-15836. doi: 10.3934/math.2024764

Related Papers:

[1]	Tyson Loudon, Stephen Pankavich . Mathematical analysis and dynamic active subspaces for a long term model of HIV. Mathematical Biosciences and Engineering, 2017, 14(3): 709-733. doi: 10.3934/mbe.2017040
[2]	A. M. Elaiw, N. H. AlShamrani . Stability of HTLV/HIV dual infection model with mitosis and latency. Mathematical Biosciences and Engineering, 2021, 18(2): 1077-1120. doi: 10.3934/mbe.2021059
[3]	Ting Guo, Zhipeng Qiu . The effects of CTL immune response on HIV infection model with potent therapy, latently infected cells and cell-to-cell viral transmission. Mathematical Biosciences and Engineering, 2019, 16(6): 6822-6841. doi: 10.3934/mbe.2019341
[4]	Yicang Zhou, Yiming Shao, Yuhua Ruan, Jianqing Xu, Zhien Ma, Changlin Mei, Jianhong Wu . Modeling and prediction of HIV in China: transmission rates structured by infection ages. Mathematical Biosciences and Engineering, 2008, 5(2): 403-418. doi: 10.3934/mbe.2008.5.403
[5]	Cameron Browne . Immune response in virus model structured by cell infection-age. Mathematical Biosciences and Engineering, 2016, 13(5): 887-909. doi: 10.3934/mbe.2016022
[6]	Nara Bobko, Jorge P. Zubelli . A singularly perturbed HIV model with treatment and antigenic variation. Mathematical Biosciences and Engineering, 2015, 12(1): 1-21. doi: 10.3934/mbe.2015.12.1
[7]	A. M. Elaiw, N. H. AlShamrani . Analysis of an HTLV/HIV dual infection model with diffusion. Mathematical Biosciences and Engineering, 2021, 18(6): 9430-9473. doi: 10.3934/mbe.2021464
[8]	B. M. Adams, H. T. Banks, Hee-Dae Kwon, Hien T. Tran . Dynamic Multidrug Therapies for HIV: Optimal and STI Control Approaches. Mathematical Biosciences and Engineering, 2004, 1(2): 223-241. doi: 10.3934/mbe.2004.1.223
[9]	Sophia Y. Rong, Ting Guo, J. Tyler Smith, Xia Wang . The role of cell-to-cell transmission in HIV infection: insights from a mathematical modeling approach. Mathematical Biosciences and Engineering, 2023, 20(7): 12093-12117. doi: 10.3934/mbe.2023538
[10]	Tinevimbo Shiri, Winston Garira, Senelani D. Musekwa . A two-strain HIV-1 mathematical model to assess the effects of chemotherapy on disease parameters. Mathematical Biosciences and Engineering, 2005, 2(4): 811-832. doi: 10.3934/mbe.2005.2.811

Abstract

References

[1]	G. S. Teodoro, J. A. T. Machado, E. C. de Oliveira, A review of definitions of fractional derivatives and other operators, J. Comput. Phys., 388 (2019), 195–208. https://doi.org/10.1016/j.jcp.2019.03.008 doi: 10.1016/j.jcp.2019.03.008
[2]	H. G. Sun, Y. Zhang, D. Baleanu, W. Chen, Y. Q. Chen, A new collection of real world applications of fractional calculus in science and engineering, Commun. Nonlinear Sci. Numer. Simul., 64 (2018), 213–231. https://doi.org/10.1016/j.cnsns.2018.04.019 doi: 10.1016/j.cnsns.2018.04.019
[3]	J. F. Gómez-Aguilar, A. Atangana, Applications of fractional calculus to modeling in dynamics and chaos, Boca Raton: Chapman & Hall/CRC Press, 2022. https://doi.org/10.1201/9781003006244
[4]	J. A. T. Machado, Fractional calculus: fundamentals and applications, In: Acoustics and vibration of mechanical structures—AVMS-2017: Proceedings of the 14th AVMS Conference, 2018, 3–11. https://doi.org/10.1007/978-3-319-69823-6_1
[5]	S. Chakraverty, R. M. Jena, S. K. Jena, Computational fractional dynamical systems: fractional differential equations and applications, John Wiley & Sons, 2023.
[6]	C. Ionescu, A. Lopes, D. Copot, J. A. T. Machado, J. H. T. Bates, The role of fractional calculus in modeling biological phenomena: a review, Commun. Nonlinear Sci. Numer. Simul., 51 (2017), 141–159. https://doi.org/10.1016/j.cnsns.2017.04.001 doi: 10.1016/j.cnsns.2017.04.001
[7]	J. A. T. Machado, M. F. Silva, R. S. Barbosa, I. S. Jesus, C. M. Reis, M. G. Marcos, et al., Some applications of fractional calculus in engineering, Math. Probl. Eng., 2010 (2010), 1–34. https://doi.org/10.1155/2010/639801 doi: 10.1155/2010/639801
[8]	S. Das, Observation of fractional calculus in physical system description, In: Functional fractional calculus, Berlin, Heidelberg: Springer, 2011. https://doi.org/10.1007/978-3-642-20545-3_3
[9]	R. Hilfer, Applications of fractional calculus in physics, World Scientific, 2000.
[10]	H. Jafari, B. Mehdinejadiani, D. Baleanu, Fractional calculus for modeling unconfined groundwater, Berlin, Boston: De Gruyter, 2019. https://doi.org/10.1515/9783110571905-007
[11]	N. Su, Fractional calculus for hydrology, soil science and geomechanics, Boca Raton: CRC Press, 2020. https://doi.org/10.1201/9781351032421
[12]	C. P. Li, Y. Q. Chen, J. Kurths, Fractional calculus and its applications, Phil. Trans. R. Soc. A, 371 (2013), 20130037. http://dx.doi.org/10.1098/rsta.2013.0037 doi: 10.1098/rsta.2013.0037
[13]	Z. J. Meng, L. F. Wang, H. Li, W. Zhang, Legendre wavelets method for solving fractional integro-differential equations, Int. J. Comput. Math., 92 (2015), 1275–1291. https://doi.org/10.1080/00207160.2014.932909 doi: 10.1080/00207160.2014.932909
[14]	K. Kumar, R. K. Pandey, S. Sharma, Comparative study of three numerical schemes for fractional integro-differential equations, J. Comput. Appl. Math., 315 (2017), 287–302. https://doi.org/10.1016/j.cam.2016.11.013 doi: 10.1016/j.cam.2016.11.013
[15]	M. B. Almatrafi, A. R. Alharbi, A. R. Seadawy, Structure of analytical and numerical wave solutions for the Ito integro-differential equation arising in shallow water waves, J. King Saud Univ. Sci., 33 (2021), 101375. https://doi.org/10.1016/j.jksus.2021.101375 doi: 10.1016/j.jksus.2021.101375
[16]	K. Agilan, V. Parthiban, Initial and boundary value problem of fuzzy fractional-order nonlinear Volterra integro-differential equations, J. Appl. Math. Comput., 69 (2023), 1765–1793. https://doi.org/10.1007/s12190-022-01810-2 doi: 10.1007/s12190-022-01810-2
[17]	M. Derakhshan, M. Jahanshahi, H. K. demneh, Investigation the boundary and initial value problems including fractional integro-differential equations with singular kernels, J. Adv. Math. Model., 11 (2021), 97–108. https://doi.org/10.22055/JAMM.2021.34670.1848 doi: 10.22055/JAMM.2021.34670.1848
[18]	X. H. Yang, Z. M. Zhang, On conservative, positivity preserving, nonlinear FV scheme on distorted meshes for the multi-term nonlocal Nagumo-type equations, Appl. Math. Lett., 150 (2024), 108972. https://doi.org/10.1016/j.aml.2023.108972 doi: 10.1016/j.aml.2023.108972
[19]	J. W. Wang, X. X. Jiang, X. H. Yang, H. X. Zhang, A nonlinear compact method based on double reduction order scheme for the nonlocal fourth-order PDEs with Burgers' type nonlinearity, J. Appl. Math. Comput., 70 (2024), 489–511. https://doi.org/10.1007/s12190-023-01975-4 doi: 10.1007/s12190-023-01975-4
[20]	J. W. Wang, X. X. Jiang, H. X. Zhang, A BDF3 and new nonlinear fourth-order difference scheme for the generalized viscous Burgers' equation, Appl. Math. Lett., 151 (2024), 109002. https://doi.org/10.1016/j.aml.2024.109002 doi: 10.1016/j.aml.2024.109002
[21]	L. J. Wu, H. X. Zhang, X. H. Yang, F. R. Wang, A second-order finite difference method for the multi-term fourth-order integral-differential equations on graded meshes, Comput. Appl. Math., 41 (2022), 313. https://doi.org/10.1007/s40314-022-02026-7 doi: 10.1007/s40314-022-02026-7
[22]	X. H. Yang, W. L. Qiu, H. F. Chen, H. X. Zhang, Second-order BDF ADI Galerkin finite element method for the evolutionary equation with a nonlocal term in three-dimensional space, Appl. Numer. Math., 172 (2022), 497–513. https://doi.org/10.1016/j.apnum.2021.11.004 doi: 10.1016/j.apnum.2021.11.004
[23]	F. R. Wang, X. H. Yang, H. X. Zhang, L. J. Wu, A time two-grid algorithm for the two dimensional nonlinear fractional PIDE with a weakly singular kernel, Math. Comput. Simul., 199 (2022), 38–59. https://doi.org/10.1016/j.matcom.2022.03.004 doi: 10.1016/j.matcom.2022.03.004
[24]	H. X. Zhang, X. X. Jiang, F. R. Wang, X. H. Yang, The time two-grid algorithm combined with difference scheme for 2D nonlocal nonlinear wave equation, J. Appl. Math. Comput., 2024, 1–25. https://doi.org/10.1007/s12190-024-02000-y
[25]	F. Safari, An accurate RBF-based meshless technique for the inverse multi-term time-fractional integro-differential equation, Eng. Anal. Bound. Elem., 153 (2023), 116–125. https://doi.org/10.1016/j.enganabound.2023.05.015 doi: 10.1016/j.enganabound.2023.05.015
[26]	S. Z. Rida, H. S. Hussien, Efficient Mittag-Leffler collocation method for solving linear and nonlinear fractional differential equations, Mediterr. J. Math., 15 (2018), 1–15. https://doi.org/10.1007/s00009-018-1174-0 doi: 10.1007/s00009-018-1174-0
[27]	S. Z. Rida, H. S. Hussien, A. H. Noreldeen, M. M. Farag, Effective fractional technical for some fractional initial value problems, Int. J. Appl. Comput. Math., 8 (2022), 149. https://doi.org/10.1007/s40819-022-01346-w doi: 10.1007/s40819-022-01346-w
[28]	M. S. Akel, H. S. Hussein, Numerical treatment of solving singular integral equations by using Sinc approximations, Appl. Math. Comput., 218 (2011), 3565–3573. https://doi.org/10.1016/j.amc.2011.08.102 doi: 10.1016/j.amc.2011.08.102
[29]	S. Behera, S. S. Ray, On a wavelet-based numerical method for linear and nonlinear fractional Volterra integro-differential equations with weakly singular kernels, Comput. Appl. Math., 41 (2022), 211. https://doi.org/10.1007/s40314-022-01897-0 doi: 10.1007/s40314-022-01897-0
[30]	G. D. Shi, Y. L. Gong, M. X. Yi, Alternative Legendre polynomials method for nonlinear fractional integro-differential equations with weakly singular kernel, J. Math., 2021 (2021), 1–13. https://doi.org/10.1155/2021/9968237 doi: 10.1155/2021/9968237
[31]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006.
[32]	D. Baleanu, Z. B. Guvenc, J. A. T. Machado, New trends in nanotechnology and fractional calculus applications, Dordrecht: Springer, 2010. https://doi.org/10.1007/978-90-481-3293-5
[33]	M. Bahmanpour, M. T. Kajani, M. Maleki, Solving Fredholm integral equations of the first kind using Muntz wavelets, Appl. Numer. Math., 143 (2019), 159–171. https://doi.org/10.1016/j.apnum.2019.04.007 doi: 10.1016/j.apnum.2019.04.007
[34]	S. C. Shiralashetti, S. Kumbinarasaiah, Laguerre wavelets exact Parseval frame-based numerical method for the solution of system of differential equations, Int. J. Comput. Math., 6 (2020), 101. https://doi.org/10.1007/s40819-020-00848-9 doi: 10.1007/s40819-020-00848-9
[35]	B. B. Tavasani, A. H. R. Sheikhani, H. Aminikhah, Numerical scheme to solve a class of variable-order Hilfer-Prabhakar fractional differential equations with Jacobi wavelets polynomials, Appl. Math. J. Chinese Univ., 37 (2022), 35–51. https://doi.org/10.1007/s11766-022-4241-z doi: 10.1007/s11766-022-4241-z
[36]	D. Hong, J. Z. Wang, R. Gardner, Real analysis with an introduction to wavelets and applications, Elsevier, 2005.
[37]	A. M. Mathai, H. J. Haubold, Special functions for applied scientists, New York: Springer, 2008. https://doi.org/10.1007/978-0-387-75894-7
[38]	J. Shahni, R. Singh, Laguerre wavelet method for solving Thomas-Fermi type equations, Eng. Comput., 38 (2022), 2925–2935. https://doi.org/10.1007/s00366-021-01309-7 doi: 10.1007/s00366-021-01309-7
[39]	B. Q. Tang, X. F. Li, Solution of a class of Volterra integral equations with singular and weakly singular kernels, Appl. Math. Comput., 199 (2008), 406–413. https://doi.org/10.1016/j.amc.2007.09.058 doi: 10.1016/j.amc.2007.09.058
[40]	P. K. Kythe, P. Puri, Computational methods for linear integral equations, Boston: Birkhauser, 2002.
[41]	M. X. Yi, J. Huang, CAS wavelet method for solving the fractional integro-differential equation with a weakly singular kernel, Int. J. Comput. Math., 92 (2015), 1715–1728. https://doi.org/10.1080/00207160.2014.964692 doi: 10.1080/00207160.2014.964692
[42]	V. V. Zozulya, P. I. Gonzalez-Chi, Weakly singular, singular and hypersingular integrals in 3-D elasticity and fracture mechanics, J. Chin. Inst. Eng., 22 (1999), 763–775. https://doi.org/10.1080/02533839.1999.9670512 doi: 10.1080/02533839.1999.9670512
[43]	S. Nemati, S. Sedaghat, I. Mohammadi, A fast numerical algorithm based on the second kind Chebyshev polynomials for fractional integro-differential equations with weakly singular kernels, J. Comput. Appl. Math., 308 (2016), 231–242. https://doi.org/10.1016/j.cam.2016.06.012 doi: 10.1016/j.cam.2016.06.012
[44]	Y. X. Wang, L. Zhu, Z. Wang, Fractional-order Euler functions for solving fractional integro-differential equations with weakly singular kernel, Adv. Differ. Equ., 2018 (2018), 1–13. https://doi.org/10.1186/s13662-018-1699-3 doi: 10.1186/s13662-018-1699-3
[45]	S. Nemati, P. M. Lima, Numerical solution of nonlinear fractional integro-differential equations with weakly singular kernels via a modification of hat functions, Appl. Math. Comput., 327 (2018), 79–92. https://doi.org/10.1016/j.amc.2018.01.030 doi: 10.1016/j.amc.2018.01.030

This article has been cited by:

1.	Marios M. Hadjiandreou, Raúl Conejeros, D. Ian Wilson, Planning of patient-specific drug-specific optimal HIV treatment strategies, 2009, 64, 00092509, 4024, 10.1016/j.ces.2009.06.009
2.	Hassan Zarei, Ali Vahidian Kamyad, Sohrab Effati, Maximizing of Asymptomatic Stage of Fast Progressive HIV Infected Patient Using Embedding Method, 2010, 01, 2153-0653, 48, 10.4236/ica.2010.11006
3.	Xinqi Xie, Junling Ma, P. van den Driessche, Backward bifurcation in within-host HIV models, 2021, 00255564, 108569, 10.1016/j.mbs.2021.108569
4.	Esteban A. Hernandez-Vargas, Dhagash Mehta, Richard H. Middleton, Towards Modeling HIV Long Term Behavior, 2011, 44, 14746670, 581, 10.3182/20110828-6-IT-1002.00685
5.	I Hosseini, F Mac Gabhann, Mechanistic Models Predict Efficacy of CCR5‐Deficient Stem Cell Transplants in HIV Patient Populations, 2016, 5, 2163-8306, 82, 10.1002/psp4.12059
6.	Esteban A. Hernandez-Vargas, Richard H. Middleton, Modeling the three stages in HIV infection, 2013, 320, 00225193, 33, 10.1016/j.jtbi.2012.11.028
7.	M. M. Hadjiandreou, R. Conejeros, D. Ian Wilson, 2012, Controlling AIDS progression in patients with rapid HIV dynamics, 978-1-4577-1096-4, 4078, 10.1109/ACC.2012.6314737
8.	Esteban A. Hernandez-Vargas, Richard H. Middleton, Patrizio Colaneri, Franco Blanchini, 2010, Dynamic optimization algorithms to mitigate HIV escape, 978-1-4244-7745-6, 827, 10.1109/CDC.2010.5717251
9.	Robert J. Smith, B. D. Aggarwala, Can the viral reservoir of latently infected CD4+ T cells be eradicated with antiretroviral HIV drugs?, 2009, 59, 0303-6812, 697, 10.1007/s00285-008-0245-4
10.	YUEPING DONG, WANBIAO MA, GLOBAL PROPERTIES FOR A CLASS OF LATENT HIV INFECTION DYNAMICS MODEL WITH CTL IMMUNE RESPONSE, 2012, 10, 0219-6913, 1250045, 10.1142/S0219691312500452
11.	Hassan Zarei, Ali Vahidian Kamyad, Sohrab Effati, Multiobjective Optimal Control of HIV Dynamics, 2010, 2010, 1024-123X, 1, 10.1155/2010/568315
12.	M. Arantxa Colchero, Yanink N. Caro-Vega, Gilberto Sánchez-González, Sergio Bautista-Arredondo, A literature review of reporting standards of HIV progression models, 2012, 26, 0269-9370, 1335, 10.1097/QAD.0b013e3283533ae2
13.	G. Bocharov, V. Chereshnev, I. Gainova, S. Bazhan, B. Bachmetyev, J. Argilaguet, J. Martinez, A. Meyerhans, Human Immunodeficiency Virus Infection : from Biological Observations to Mechanistic Mathematical Modelling, 2012, 7, 0973-5348, 78, 10.1051/mmnp/20127507
14.	Elizabeth Gross, Brent Davis, Kenneth L. Ho, Daniel J. Bates, Heather A. Harrington, Numerical algebraic geometry for model selection and its application to the life sciences, 2016, 13, 1742-5689, 20160256, 10.1098/rsif.2016.0256
15.	2019, 9780128130520, 221, 10.1016/B978-0-12-813052-0.00023-3
16.	Tyson Loudon, Stephen Pankavich, Mathematical analysis and dynamic active subspaces for a long term model of HIV, 2017, 14, 1551-0018, 709, 10.3934/mbe.2017040
17.	Marcos A. Capistrán, A study of latency, reactivation and apoptosis throughout HIV pathogenesis, 2010, 52, 08957177, 1011, 10.1016/j.mcm.2010.03.022
18.	Marios M. Hadjiandreou, Raul Conejeros, D. Ian Wilson, Long-term HIV dynamics subject to continuous therapy and structured treatment interruptions, 2009, 64, 00092509, 1600, 10.1016/j.ces.2008.12.010
19.	2019, 9780128130520, 105, 10.1016/B978-0-12-813052-0.00017-8
20.	Ali Heydari, S.N. Balakrishnan, Optimal multi-therapeutic HIV treatment using a global optimal switching scheme, 2013, 219, 00963003, 7872, 10.1016/j.amc.2013.01.070
21.	Esteban A. Hernandez-Vargas, Modeling Kick-Kill Strategies toward HIV Cure, 2017, 8, 1664-3224, 10.3389/fimmu.2017.00995
22.	Emiliano Mancini, Rick Quax, Andrea De Luca, Sarah Fidler, Wolfgang Stohr, Peter M. A. Sloot, Siddappa N. Byrareddy, A study on the dynamics of temporary HIV treatment to assess the controversial outcomes of clinical trials: An in-silico approach, 2018, 13, 1932-6203, e0200892, 10.1371/journal.pone.0200892
23.	2019, 9780128130520, 129, 10.1016/B978-0-12-813052-0.00018-X
24.	Esteban Hernandez-Vargas, Patrizio Colaneri, Richard Middleton, Franco Blanchini, Discrete-time control for switched positive systems with application to mitigating viral escape, 2011, 21, 10498923, 1093, 10.1002/rnc.1628
25.	H. Zarei, A. V. Kamyad, M. H. Farahi, Optimal Control of HIV Dynamic Using Embedding Method, 2011, 2011, 1748-670X, 1, 10.1155/2011/674318
26.	Matthias Haering, Andreas Hördt, Michael Meyer-Hermann, Esteban A. Hernandez-Vargas, Computational Study to Determine When to Initiate and Alternate Therapy in HIV Infection, 2014, 2014, 2314-6133, 1, 10.1155/2014/472869
27.	Esteban A. Hernandez-Vargas, Richard H. Middleton, Patrizio Colaneri, Optimal and MPC Switching Strategies for Mitigating Viral Mutation and Escape, 2011, 44, 14746670, 14857, 10.3182/20110828-6-IT-1002.01137
28.	Charlotte Lew, 2022, Neural Network Modeling of HIV Acute and Chronic Phases With and Without Antiretroviral Intervention, 978-1-6654-7184-8, 47, 10.1109/TransAI54797.2022.00014
29.	Qiang-hui Xu, Ji-cai Huang, Yue-ping Dong, Yasuhiro Takeuchi, A Delayed HIV Infection Model with the Homeostatic Proliferation of CD4+ T Cells, 2022, 38, 0168-9673, 441, 10.1007/s10255-022-1088-2
30.	Konstantin E. Starkov, Anatoly N. Kanatnikov, Eradication Conditions of Infected Cell Populations in the 7-Order HIV Model with Viral Mutations and Related Results, 2021, 9, 2227-7390, 1862, 10.3390/math9161862
31.	Huseyin Tunc, Murat Sari, Seyfullah Kotil, Machine learning aided multiscale modelling of the HIV-1 infection in the presence of NRTI therapy, 2023, 11, 2167-8359, e15033, 10.7717/peerj.15033
32.	A. N. Kanatnikov, O. S. Tkacheva, Behavior of Trajectories of a Four-Dimensional Model of HIV Infection, 2023, 59, 0012-2661, 1451, 10.1134/S00122661230110022
33.	Cameron Clarke, Stephen Pankavich, Three-stage modeling of HIV infection and implications for antiretroviral therapy, 2024, 88, 0303-6812, 10.1007/s00285-024-02056-1
34.	A. N. Kanatnikov, O. S. Tkacheva, Behavior of Trajectories of a Four-Dimensional Model of HIV Infection, 2023, 59, 0374-0641, 1451, 10.31857/S037406412311002X
35.	A. M. Elaiw, E. A. Almohaimeed, A. D. Hobiny, Analysis of HHV-8/HIV-1 co-dynamics model with latency, 2024, 139, 2190-5444, 10.1140/epjp/s13360-024-05202-2
36.	A. M. Elaiw, E. A. Almohaimeed, A. D. Hobiny, Stability of HHV-8 and HIV-1 co-infection model with latent reservoirs and multiple distributed delays, 2024, 9, 2473-6988, 19195, 10.3934/math.2024936
37.	Yueping Dong, Jicai Huang, Yasuhiro Takeuchi, Qianghui Xu, 2024, Chapter 2, 978-981-97-7849-2, 11, 10.1007/978-981-97-7850-8_2
38.	A. M. Elaiw, E. A. Almohaimeed, A. D. Hobiny, Modeling the co-infection of HTLV-2 and HIV-1 in vivo, 2024, 32, 2688-1594, 6032, 10.3934/era.2024280
39.	Xia Wang, Yue Wang, Yueping Dong, Libin Rong, Dynamics of a delayed HIV infection model with cell-to-cell transmission and homeostatic proliferation, 2024, 139, 2190-5444, 10.1140/epjp/s13360-024-05845-1
40.	A.M. Elaiw, E.A. Almohaimeed, A.D. Hobiny, Stability analysis of a diffusive HTLV-2 and HIV-1 co-infection model, 2025, 116, 11100168, 232, 10.1016/j.aej.2024.11.074
41.	Jing Cai, Jun Zhang, Kai Wang, Zhixiang Dai, Zhiliang Hu, Yueping Dong, Zhihang Peng, Evaluating the long-term effects of combination antiretroviral therapy of HIV infection: a modeling study, 2025, 90, 0303-6812, 10.1007/s00285-025-02196-y
42.	Jun Zhang, Jing Cai, Yasuhiro Takeuchi, Yueping Dong, Zhihang Peng, Rich dynamics induced by homeostatic proliferation of both CD4+ T cells and macrophages in HIV infection and persistence, 2025, 2025, 2731-4235, 10.1186/s13662-025-03941-9
43.	A. M. Elaiw, E. A. Almohaimeed, Within-host dynamics of HTLV-2 and HIV-1 co-infection with delay, 2025, 19, 1751-3758, 10.1080/17513758.2025.2506536
44.	A. M. Elaiw, E. Dahy, H. Z. Zidan, A. A. Abdellatif, Stability of an HIV-1 abortive infection model with antibody immunity and delayed inflammatory cytokine production, 2025, 140, 2190-5444, 10.1140/epjp/s13360-025-06475-x

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)