Fractional-Order Modeling and Neural Network Simulation of ‎NK Cell-Oncolytic Virus Interactions

  • Authors

    • M. Sharmila Department of Mathematics, Kunthavai Naacchiyaar Government Arts College for Women(Autonomous), ‎ ‎(“Affiliated to Bharathidasan University” Tiruchirappalli, 620024), Thanjavur - 613007, Tamil Nadu, India‎
    • S. Indrakala Department of Mathematics, Kunthavai Naacchiyaar Government Arts College for Women(Autonomous), ‎ ‎(“Affiliated to Bharathidasan University” Tiruchirappalli, 620024), Thanjavur - 613007, Tamil Nadu, India‎
    https://doi.org/10.14419/9pjcye33

    Received date: October 16, 2025

    Accepted date: November 22, 2025

    Published date: December 4, 2025

  • Oncolytic Virotherapy; Mathematical Modeling; Discrete Fractional-Order Model; Existence Theory; Stability Analysis.

  • Abstract

    Oncolytic virotherapy (OV) uses viruses to selectively destroy cancer cells, but the role of natural killer (NK) cells ‎in this process remains unclear. This study develops a discrete fractional-order model to examine NK cell activity ‎and identify activation conditions. Existence of the model solution is established via fixed-point theorems, and ‎the basic reproduction number (R_0) is derived using the next-generation matrix method with sensitivity analysis ‎to determine key parameters. Stability of equilibria is analyzed, and numerical simulations reveal how NK cell ‎activation affects interactions among cancer cells, infected cells, and viruses. Additionally, an artificial neural ‎network (ANN) trained with the Levenberg-Marquardt algorithm efficiently approximates the model's dynamics, ‎showing high accuracy (low MSE) and providing a rapid, reliable tool for predicting complex biological behaviors.

  • References

    1. O. W. Abdulwahhab and N. H. Abbas. A new method to tune a fractional-order pid controller for a twin rotor aerodynamic system. Arabian Journal for Science and Engineering, 42:5179-5189, 2017. https://doi.org/10.1007/s13369-017-2629-5
    2. A. Al-Khedhairi, A. A. Elsadany, and A. Elsonbaty. On the dynamics of a discrete fractional-order cournot-bertrand competition duopoly game. Mathematical Problems in Engineering, page Article ID 8249215, 2022. https://doi.org/10.1155/2022/8249215.
    3. S. M. Al-Tuwairqi, N. O. Al-Johani, and E. A. Simbawa. Modeling dynamics of cancer virotherapy with immune
    4. response. Advances in Difference Equations, 2020:438, 2020.
    5. Z. Ali. Theoretical and Computational Study of Fractional-order Mathematical Models for Infectious Diseases. PhD thesis, Monash University, 2023.
    6. Z. Ali, S. N. Nia, F. Rabiei, K. Shah, and M. K. Tan. A semi-analytical approach for the solution of time-fractional navier-stokes equation. Advances in Mathematical Physics, page Article ID 5547804, 2021. 13 pages. https://doi.org/10.1155/2021/5547804.
    7. Z. Ali, F. Rabiei, K. Shah, and Z. A. Majid. Dynamics of sir mathematical model for covid-19 outbreak in pakistan under fractal-fractional derivative. Fractals, 29(5):2150120, 2021. https://doi.org/10.1142/S0218348X21501206.
    8. A. M. Alqahtani, S. Akram, J. Ahmad, K. A. Aldwoah, and M. ur Rahman. Stochastic wave solutions of fractional radhakrishnan-kundu-lakshmanan equation arising in optical fibers with their sensitivity analysis. Journal of Optics, 2024. https://doi.org/10.1007/s12596-024-01850-w.
    9. J. Altomonte, L. Wu, M. Meseck, L. Chen, O. Ebert, A. Garcia-Sastre, J. Fallon, J. Mandeli, and S. L. Woo. Enhanced oncolytic potency of vesicular stomatitis virus through vector-mediated inhibition of nk and nkt cells. Cancer Gene Therapy, 16:266-278, 2009. https://doi.org/10.1038/cgt.2008.74.
    10. C. A. Alvarez-Breckenridge et al. Nk cells impede glioblastoma virotherapy through nkp30 and nkp46 natural cytotoxicity receptors. Nature Medicine, 18:1827-1834, 2012. https://doi.org/10.1038/nm.3013
    11. S. C-Vázquez et al. Applications of fractional operators in robotics: A review. Journal of Intelligent and Robotic Systems, 104:63, 2022. https://doi.org/10.1007/s10846-022-01597-1.
    12. N. H. Can, H. Jafari, and M. N. Ncube. Fractional calculus in data fitting. Alexandria Engineering Journal, 59(5):3269-3274, 2020. https://doi.org/10.1016/j.aej.2020.09.002.
    13. C. Coll, A. Herrero, D. Ginestar, and E. Sánchez. The discrete fractional order difference applied to an epidemicmodel with indirect transmission. Applied Mathematical Modelling, 103:636-648, 2022. https://doi.org/10.1007/s10846-022-01597-1.
    14. M. F. Danca. Fractional order logistic map: Numerical approach. Chaos, Solitons and Fractals, 157:111851, 2022. https://doi.org/10.1016/j.chaos.2022.111851.
    15. S. Das. Application of generalized fractional calculus in electrical circuit analysis and electromagnetics. In Functional Fractional Calculus. Springer, Berlin, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-20545-3_8.
    16. E. C. de Oliveira and J. A. T. Machado. A review of definitions for fractional derivatives and integral. Mathematical Problems in Engineering, 2014:238459, 2014. 6 pages. https://doi.org/10.1155/2014/238459.
    17. A. Desjardins et al. Recurrent glioblastoma treated with recombinant poliovirus. New England Journal of Medicine, 379:150-161, 2018. https://doi.org/10.1056/NEJMoa1716435
    18. N. Djenina, A. Ouannas, I. M. Batiha, G. Grassi, T.-E. Oussaeif, and S. Momani. A novel fractional-order discrete sir model for predicting covid-19 behavior. Mathematics, 10(13):2224, 2022. https://doi.org/10.3390/math10132224
    19. A. Elsonbaty and A. A. Elsadany. On discrete fractional-order lotka-volterra model based on the caputo difference discrete operator. Mathematical Sciences, 17(1):67-79, 2023. https://doi.org/10.1007/s40096-021-00442-0
    20. P. F. Ferrucci, L. Pala, F. Conforti, and E. Cocorocchio. Talimogene laherparepvec (t-vec): An intralesional cancer immunotherapy for advanced melanoma. Cancers, 13:1383, 2021. https://doi.org/10.3390/cancers13061383.
    21. B. Gesundheit et al. Effective treatment of glioblastoma multiforme with oncolytic virotherapy: A case-series. Frontiers in Oncology, 10:702, 2020. https://doi.org/10.3389/fonc.2020.00702
    22. J. Han, C. A. Alvarez-Breckenridge, Q. E. Wang, and J. Yu. Tgf-beta signaling and its targeting for glioma treatment. American Journal of Cancer Research, 5:945-955, 2015.
    23. R. Hilfer. Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000. https://doi.org/10.1142/3779.
    24. L.-L. Huang, J. H. Park, G.-C. Wu, and Z.-W. Mo. Variable-order fractional discrete-time recurrent neural networks. Journal of Computational and Applied Mathematics, 370:112633, 2020. https://doi.org/10.1016/j.cam.2019.112633.
    25. Y. Jiang et al. Fractional-order autonomous circuits with order larger than one. Journal of Advanced Research, 25:217-225, 2020. https://doi.org/10.1016/j.jare.2020.05.005.
    26. D. Kim, D.-H. Shin, and C. K. Sung. The optimal balance between oncolytic viruses and natural killer cells: A mathematical approach. Mathematics, 10:3370, 2022. https://doi.org/10.3390/math10183370.
    27. Y. Kim, J. Y. Yoo, T. J. Lee, J. Liu, J. Yu, M. A. Caligiuri, B. Kaur, and A. Friedman. Complex role of nk cells in regulation of oncolytic virus-bortezomib therapy. Proceedings of the National Academy of Sciences of the USA, 115:4927-4932, 2018. https://doi.org/10.1073/pnas.1715295115
    28. E. Y. L. Leung et al. Nk cells augment oncolytic adenovirus cytotoxicity in ovarian cancer. Molecular Therapy Oncolytics, 16:289-301, 2020. https://doi.org/10.1016/j.omto.2020.02.001.
    29. X. Li, P. Wang, H. Li, X. Du, M. Liu, Q. Huang, Y. Wang, and S. Wang. The efficacy of oncolytic adenovirus is mediated by t -cell responses against virus and tumor in syrian hamster model. Clinical Cancer Research, 23:239-249, 2017. https://doi.org/10.1158/1078-0432.CCR-16-0477
    30. Y. A. Madani, Z. Ali, M. Rabih, A. Alsulami, N. H. E. Eljaneid, K. Aldwoah, and B. Muflh. Discrete fractional-order modeling of recurrent childhood diseases using the caputo difference operator. Fractal and Fractional, 9(1):55, 2025. https://doi.org/10.3390/fractalfract9010055
    31. Y. A. Madani, M. N. A. Rabih, F. A. Alqarni, Z. Ali, K. A. Aldwoah, and M. Hleili. Existence, uniqueness, and stability of a nonlinear tripled fractional order differential system. Fractal and Fractional, 8(7):416, 2024. https://doi.org/10.3390/fractalfract8070416.
    32. R. L. Magin. Fractional calculus in bioengineering. Critical Reviews in Biomedical Engineering, 32(1):1-104, 2004. https://doi.org/10.1615/CritRevBiomedEng.v32.10.
    33. R. Matušu. Application of fractional order calculus to control theory. International Journal of Mathematical Models and Methods in Applied Sciences, 5(7):1162-1169, 2011.
    34. R. P. Meilanov and R. A. Magomedov. Thermodynamics in fractional calculus. Journal of Engineering Physics and Thermophysics, 87:1521-1531, 2014. https://doi.org/10.1007/s10891-014-1158-2.
    35. F. Meral, T. Royston, and R. Magin. Fractional calculus in viscoelasticity: an experimental study. Communications in Nonlinear Science and Numerical Simulation, 15:939-945, 2010. https://doi.org/10.1016/j.cnsns.2009.05.004.
    36. M. Mondal, J. Guo, P. He, and D. Zhou. Recent advances of oncolytic virus in cancer therapy. Human Vaccines ^6 Immunotherapeutics, 16:2389-2402, 2020. https://doi.org/10.1080/21645515.2020.1723363.
    37. K. Oldham. Fractional differential equations in electrochemistry. Advances in Engineering Software, 41:9-12, 2010. https://doi.org/10.1016/j.advengsoft.2008.12.012
    38. P. Ostalczyk. Discrete Fractional Calculus: Applications in Control and Image Processing, volume 4. World Scientific, 2015. https://doi.org/10.1142/9833.
    39. Y. Peng, S. He, and K. Sun. Chaos in the discrete memristor-based system with fractional-order difference. Results in Physics, 24:104106, 2021. https://doi.org/10.1016/j.rinp.2021.104106
    40. T. A. Phan and J. P. Tian. The role of the innate immune system in oncolytic virotherapy. Computational and Mathematical Methods in Medicine, 2017:6587258, 2017. https://doi.org/10.1155/2017/6587258.
    41. N. S. Senekal, K. J. Mahasa, A. Eladdadi, L. de Pillis, and R. Ouifki. Natural killer cells recruitment in oncolytic virotherapy: A mathematical model. Bulletin of Mathematical Biology, 83:75, 2021. https://doi.org/10.1007/s11538-021-00903-6
    42. N. A. Shah, D. Vieru, and C. Fetecau. Effects of the fractional order and magnetic field on the blood flow in cylindrical domains. Journal of Magnetism and Magnetic Materials, 409:10-19, 2016. https://doi.org/10.1016/j.jmmm.2016.02.013.
    43. V. E. Tarasov. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer Science and Business Media, 2011. https://doi.org/10.1007/978-3-642-14003-7
    44. V. E. Tarasov and V. V. Tarasova. Long and short memory in economics: fractional-order difference and differentiation. arXiv preprint arXiv:1612.07903, 2016. https://doi.org/10.21013/jmss.v5.n2.p10
    45. S. H. Thorne, T. H. Hwang, et al. Rational strain selection and engineering creates a broad-spectrum, systemically effective oncolytic poxvirus, jx-963. Journal of Clinical Investigation, 117:3350-3358, 2007. https://doi.org/10.1172/JCI32727.
    46. O. W. Abdulwahhab and N. H. Abbas. A new method to tune a fractional-order pid controller for a twin rotor aerodynamic system. Arabian Journal for Science and Engineering, 42:5179-5189, 2017. https://doi.org/10.1007/s13369-017-2629-5
    47. A. Al-Khedhairi, A. A. Elsadany, and A. Elsonbaty. On the dynamics of a discrete fractional-order cournot-bertrand competition duopoly game. Mathematical Problems in Engineering, page Article ID 8249215, 2022. https://doi.org/10.1155/2022/8249215.
    48. S. M. Al-Tuwairqi, N. O. Al-Johani, and E. A. Simbawa. Modeling dynamics of cancer virotherapy with immune
    49. response. Advances in Difference Equations, 2020:438, 2020.
    50. Z. Ali. Theoretical and Computational Study of Fractional-order Mathematical Models for Infectious Diseases. PhD thesis, Monash University, 2023.
    51. Z. Ali, S. N. Nia, F. Rabiei, K. Shah, and M. K. Tan. A semi-analytical approach for the solution of time-fractional navier-stokes equation. Advances in Mathematical Physics, page Article ID 5547804, 2021. 13 pages. https://doi.org/10.1155/2021/5547804.
    52. Z. Ali, F. Rabiei, K. Shah, and Z. A. Majid. Dynamics of sir mathematical model for covid-19 outbreak in pakistan under fractal-fractional derivative. Fractals, 29(5):2150120, 2021. https://doi.org/10.1142/S0218348X21501206.
    53. A. M. Alqahtani, S. Akram, J. Ahmad, K. A. Aldwoah, and M. ur Rahman. Stochastic wave solutions of fractional radhakrishnan-kundu-lakshmanan equation arising in optical fibers with their sensitivity analysis. Journal of Optics, 2024. https://doi.org/10.1007/s12596-024-01850-w.
    54. J. Altomonte, L. Wu, M. Meseck, L. Chen, O. Ebert, A. Garcia-Sastre, J. Fallon, J. Mandeli, and S. L. Woo. Enhanced oncolytic potency of vesicular stomatitis virus through vector-mediated inhibition of nk and nkt cells. Cancer Gene Therapy, 16:266-278, 2009. https://doi.org/10.1038/cgt.2008.74.
    55. C. A. Alvarez-Breckenridge et al. Nk cells impede glioblastoma virotherapy through nkp30 and nkp46 natural cytotoxicity receptors. Nature Medicine, 18:1827-1834, 2012. https://doi.org/10.1038/nm.3013
    56. S. C-Vázquez et al. Applications of fractional operators in robotics: A review. Journal of Intelligent and Robotic Systems, 104:63, 2022. https://doi.org/10.1007/s10846-022-01597-1.
    57. N. H. Can, H. Jafari, and M. N. Ncube. Fractional calculus in data fitting. Alexandria Engineering Journal, 59(5):3269-3274, 2020. https://doi.org/10.1016/j.aej.2020.09.002.
    58. C. Coll, A. Herrero, D. Ginestar, and E. Sánchez. The discrete fractional order difference applied to an epidemicmodel with indirect transmission. Applied Mathematical Modelling, 103:636-648, 2022. https://doi.org/10.1007/s10846-022-01597-1.
    59. M. F. Danca. Fractional order logistic map: Numerical approach. Chaos, Solitons and Fractals, 157:111851, 2022. https://doi.org/10.1016/j.chaos.2022.111851.
    60. S. Das. Application of generalized fractional calculus in electrical circuit analysis and electromagnetics. In Functional Fractional Calculus. Springer, Berlin, Heidelberg, 2011. https://doi.org/10.1007/978-3-642-20545-3_8.
    61. E. C. de Oliveira and J. A. T. Machado. A review of definitions for fractional derivatives and integral. Mathematical Problems in Engineering, 2014:238459, 2014. 6 pages. https://doi.org/10.1155/2014/238459.
    62. A. Desjardins et al. Recurrent glioblastoma treated with recombinant poliovirus. New England Journal of Medicine, 379:150-161, 2018. https://doi.org/10.1056/NEJMoa1716435
    63. N. Djenina, A. Ouannas, I. M. Batiha, G. Grassi, T.-E. Oussaeif, and S. Momani. A novel fractional-order discrete sir model for predicting covid-19 behavior. Mathematics, 10(13):2224, 2022. https://doi.org/10.3390/math10132224
    64. A. Elsonbaty and A. A. Elsadany. On discrete fractional-order lotka-volterra model based on the caputo difference discrete operator. Mathematical Sciences, 17(1):67-79, 2023. https://doi.org/10.1007/s40096-021-00442-0
    65. P. F. Ferrucci, L. Pala, F. Conforti, and E. Cocorocchio. Talimogene laherparepvec (t-vec): An intralesional cancer immunotherapy for advanced melanoma. Cancers, 13:1383, 2021. https://doi.org/10.3390/cancers13061383.
    66. B. Gesundheit et al. Effective treatment of glioblastoma multiforme with oncolytic virotherapy: A case-series. Frontiers in Oncology, 10:702, 2020. https://doi.org/10.3389/fonc.2020.00702
    67. J. Han, C. A. Alvarez-Breckenridge, Q. E. Wang, and J. Yu. Tgf-beta signaling and its targeting for glioma treatment. American Journal of Cancer Research, 5:945-955, 2015.
    68. R. Hilfer. Applications of Fractional Calculus in Physics. World Scientific, Singapore, 2000. https://doi.org/10.1142/3779.
    69. L.-L. Huang, J. H. Park, G.-C. Wu, and Z.-W. Mo. Variable-order fractional discrete-time recurrent neural networks. Journal of Computational and Applied Mathematics, 370:112633, 2020. https://doi.org/10.1016/j.cam.2019.112633.
    70. Y. Jiang et al. Fractional-order autonomous circuits with order larger than one. Journal of Advanced Research, 25:217-225, 2020. https://doi.org/10.1016/j.jare.2020.05.005.
    71. D. Kim, D.-H. Shin, and C. K. Sung. The optimal balance between oncolytic viruses and natural killer cells: A mathematical approach. Mathematics, 10:3370, 2022. https://doi.org/10.3390/math10183370.
    72. Y. Kim, J. Y. Yoo, T. J. Lee, J. Liu, J. Yu, M. A. Caligiuri, B. Kaur, and A. Friedman. Complex role of nk cells in regulation of oncolytic virus-bortezomib therapy. Proceedings of the National Academy of Sciences of the USA, 115:4927-4932, 2018. https://doi.org/10.1073/pnas.1715295115
    73. E. Y. L. Leung et al. Nk cells augment oncolytic adenovirus cytotoxicity in ovarian cancer. Molecular Therapy Oncolytics, 16:289-301, 2020. https://doi.org/10.1016/j.omto.2020.02.001.
    74. X. Li, P. Wang, H. Li, X. Du, M. Liu, Q. Huang, Y. Wang, and S. Wang. The efficacy of oncolytic adenovirus is mediated by t -cell responses against virus and tumor in syrian hamster model. Clinical Cancer Research, 23:239-249, 2017. https://doi.org/10.1158/1078-0432.CCR-16-0477
    75. Y. A. Madani, Z. Ali, M. Rabih, A. Alsulami, N. H. E. Eljaneid, K. Aldwoah, and B. Muflh. Discrete fractional-order modeling of recurrent childhood diseases using the caputo difference operator. Fractal and Fractional, 9(1):55, 2025. https://doi.org/10.3390/fractalfract9010055
    76. Y. A. Madani, M. N. A. Rabih, F. A. Alqarni, Z. Ali, K. A. Aldwoah, and M. Hleili. Existence, uniqueness, and stability of a nonlinear tripled fractional order differential system. Fractal and Fractional, 8(7):416, 2024. https://doi.org/10.3390/fractalfract8070416.
    77. R. L. Magin. Fractional calculus in bioengineering. Critical Reviews in Biomedical Engineering, 32(1):1-104, 2004. https://doi.org/10.1615/CritRevBiomedEng.v32.10.
    78. R. Matušu. Application of fractional order calculus to control theory. International Journal of Mathematical Models and Methods in Applied Sciences, 5(7):1162-1169, 2011.
    79. R. P. Meilanov and R. A. Magomedov. Thermodynamics in fractional calculus. Journal of Engineering Physics and Thermophysics, 87:1521-1531, 2014. https://doi.org/10.1007/s10891-014-1158-2.
    80. F. Meral, T. Royston, and R. Magin. Fractional calculus in viscoelasticity: an experimental study. Communications in Nonlinear Science and Numerical Simulation, 15:939-945, 2010. https://doi.org/10.1016/j.cnsns.2009.05.004.
    81. M. Mondal, J. Guo, P. He, and D. Zhou. Recent advances of oncolytic virus in cancer therapy. Human Vaccines ^6 Immunotherapeutics, 16:2389-2402, 2020. https://doi.org/10.1080/21645515.2020.1723363.
    82. K. Oldham. Fractional differential equations in electrochemistry. Advances in Engineering Software, 41:9-12, 2010. https://doi.org/10.1016/j.advengsoft.2008.12.012
    83. P. Ostalczyk. Discrete Fractional Calculus: Applications in Control and Image Processing, volume 4. World Scientific, 2015. https://doi.org/10.1142/9833.
    84. Y. Peng, S. He, and K. Sun. Chaos in the discrete memristor-based system with fractional-order difference. Results in Physics, 24:104106, 2021. https://doi.org/10.1016/j.rinp.2021.104106
    85. T. A. Phan and J. P. Tian. The role of the innate immune system in oncolytic virotherapy. Computational and Mathematical Methods in Medicine, 2017:6587258, 2017. https://doi.org/10.1155/2017/6587258.
    86. N. S. Senekal, K. J. Mahasa, A. Eladdadi, L. de Pillis, and R. Ouifki. Natural killer cells recruitment in oncolytic virotherapy: A mathematical model. Bulletin of Mathematical Biology, 83:75, 2021. https://doi.org/10.1007/s11538-021-00903-6
    87. N. A. Shah, D. Vieru, and C. Fetecau. Effects of the fractional order and magnetic field on the blood flow in cylindrical domains. Journal of Magnetism and Magnetic Materials, 409:10-19, 2016. https://doi.org/10.1016/j.jmmm.2016.02.013.
    88. V. E. Tarasov. Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer Science and Business Media, 2011. https://doi.org/10.1007/978-3-642-14003-7
    89. V. E. Tarasov and V. V. Tarasova. Long and short memory in economics: fractional-order difference and differentiation. arXiv preprint arXiv:1612.07903, 2016. https://doi.org/10.21013/jmss.v5.n2.p10
    90. S. H. Thorne, T. H. Hwang, et al. Rational strain selection and engineering creates a broad-spectrum, systemically effective oncolytic poxvirus, jx-963. Journal of Clinical Investigation, 117:3350-3358, 2007. https://doi.org/10.1172/JCI32727.

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    Sharmila, M., & Indrakala, S. (2025). Fractional-Order Modeling and Neural Network Simulation of ‎NK Cell-Oncolytic Virus Interactions. International Journal of Basic and Applied Sciences, 14(8), 106-120. https://doi.org/10.14419/9pjcye33