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Mathematical Analysis of A Fractional-Order Viral Infection Model with Saturated Infection Rate and Cellular Immune Response

Yıl 2025, Cilt: 13 Sayı: 1, 67 - 77, 30.04.2025

Jaouad Danane , Mehmet Yavuz , Sania Qureshi

Öz

This paper investigates a fractional-order viral infection model with saturated infection rate and cellular immune response. The cellular immunity will be represented by cytotoxic T-lymphocytes (CTL) cells. In order to study mathematically the infection model, we will suggest five fractional differential equations describing the interaction between the uninfected cells, the latently infected cells, the infected cells, the CTL cells and the free viruses. A saturated infection rate will be taken into consideration to represent the viral infection. First, the positivity and boundedness of solutions for non-negative initial data are proved. Next, by constructing suitable Lyapunov functions, the global stability of the disease free equilibrium and the endemic equilibria are established depending on the basic reproduction number $R_0$ and the CTL immune response reproduction number $R_{CTL}$. Finally, numerical simulations are performed in order to show the dynamics behavior of the viral infection and to support the theoretical results.

Anahtar Kelimeler

Global stability viral dynamics CTL immune response viral infection model fractional derivative

Kaynakça

  • [1] World Health Organization, HIV/AIDS key facts.
  • [2] Maziane, M., Lotfi, E. M., Hattaf, K., Yousfi, N. (2015). Dynamics of a class of HIV infection models with cure of infected cells in eclipse stage. Acta Biotheoretica, 63(4), 363-380.
  • [3] Shi, V., Tridane, A., Kuang, Y. (2008). A viral load-based cellular automata approach to modeling HIV dynamics and drug treatment. Journal of theoretical biology, 253(1), 24-35.
  • [4] Wodarz, D. Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses. J. Gen. Virol. 2003, 84, 1743–1750.
  • [5] Li, M.; Zu, J. The review of differential equation models of HBV infection dynamics. J. Virol. Methods 2019, 266, 103–113.
  • [6] Danane, J., Meskaf, A., Allali, K. (2018). Optimal control of a delayed hepatitis B viral infection model with HBV DNA-containing capsids and CTL immune response. Optimal Control Applications and Methods, 39(3), 1262-1272.
  • [7] Burchell, A.N.; Winer, R.L.; de Sanjos´e, S.; Franco, E.L. Epidemiology and transmission dynamics of genital HPV infection. Vaccine 2006, 24, S52–S61.
  • [8] Elbasha, E.H.; Dasbach, E.J.; Insinga, R.P. A multi-type HPV transmission model. Bull. Math. Biol. 2008, 70, 2126–2176.
  • [9] Braaten, K. P., Laufer, M. R. (2008). Human papillomavirus (HPV), HPV-related disease, and the HPV vaccine. Reviews in obstetrics and gynecology, 1(1), 2.
  • [10] Nowak M.A.; Bangham C.R.M. Population dynamics of immune responses to persistent viruses. Science 1996, 272, 74–79.
  • [11] Smith H.L.; De Leenheer P. Virus dynamics: a global analysis. SIAM Journal on Applied Mathematics 2003, 63, 1313-1327.
  • [12] Korobeinikov A. Global properties of basic virus dynamics models. Bulletin of Mathematical Biology 2004, 66, 879–883.
  • [13] Buonomo, B., Vargas-De-Le´on, C.(2012) Global stability for an HIV-1 infection model including an eclipse stage of infected cells, J. Math. Anal. Appl.385(2):709-720.
  • [14] Sun Q.; Min L.; Kuang Y. Global stability of infection-free state and endemic infection state of a modified human immunodeficiency virus infection model. IET systems biology 2015, 9, 95–103.
  • [15] Sun Q.; Min L. Dynamics Analysis and Simulation of a Modified HIV Infection Model with a Saturated Infection Rate. Computational and mathematical methods in medicine 2014, 2014 Article ID 145162, 14 pages.
  • [16] Allali, K., Danane, J., Kuang, Y., 2017. Global Analysis for an HIV Infection Model with CTL Immune Response and Infected Cells in Eclipse Phase. Applied Sciences (2076-3417), 7(8).
  • [17] Raeisi, E., Yavuz, M., Khosravifarsani, M., & Fadaei, Y. (2024). Mathematical modeling of interactions between colon cancer and immune system with a deep learning algorithm. The European Physical Journal Plus, 139(4), 345.
  • [18] Wang X.; Tao Y.; Song X. Global stability of a virus dynamics model with Beddington-DeAngelis incidence rate and CTL immune response. Nonlinear Dynamics 2011, 66, 825–830.
  • [19] Daar, E.S.; Moudgil, T.; Meyer, R.D.; Ho, D.D. Transient high levels of viremia in patients with primary human immunodeficiency virus type 1. N. Engl. J. Med. 1991, 324, 961–964.
  • [20] Naim, M., Zeb, A., Mohsen, A. A., Sabbar, Y., & Yıldız, M. (2024). Local and global stability of a fractional viral infection model with two routes of propagation, cure rate and non-lytic humoral immunity. Mathematical Modelling and Numerical Simulation with Applications, 4(5-Special Issue: ICAME’24), 94-115.
  • [21] Joshi, H., Yavuz, M., Taylan, O., & Alkabaa, A. (2025). Dynamic analysis of fractal–fractional cancer model under chemotherapy drug with generalized Mittag-Leffler kernel. Computer Methods and Programs in Biomedicine, 260, 108565.
  • [22] Kahn, J.O.; Walker, B.D. Acute human immunodeficiency virus type 1 infection. N. Engl. J. Med. 1998, 339, 33–39.
  • [23] Sokolov IM, Chechkin AV, Klafter J. Distributed-order fractional kinetics. Statistical Physics: Fundamentals and Applications 2004; 35:1323–1341.
  • [24] Rida SZ, El-Sayed AMA, Arafa AM. A effect of bacterial memory dependent growth by using fractional derivatives reaction-diffusion chemotactic model. Journal of Statistical Physics 2010; 140:797–811.
  • [25] Arafa AAM, Rida SZ, Khalil M. Fractional modeling dynamics of HIV and CD4CT -cells during primary infection. Nonlinear Biomedical Physics 2012; 6(1).
  • [26] Sweilam NH, AL-Mekhlafi SM. Numerical study for multi-strain tuberculosis (TB) model of variable-order fractional derivatives. Journal of Advanced Research 2016; 7(2):271–283.
  • [27] A. Boukhouima, K. Hattaf, and N. Yousfi, Dynamics of a Fractional Order HIV InfectionModel with Specific Functional Response and Cure Rate, International Journal of Differential Equations, vol. 2017, Article ID 8372140, 8 pages, 2017.
  • [28] Debnath L. Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences 2003; 54:3413–3442.
  • [29] Hilfer ER. Applications of Fractional Calculus in Physics. World Scientific: River Edge, NJ, USA, 2000.
  • [30] Yuste SB, Acedo L, Lindenberg K. Reaction front in a reaction-subdiffusion process. Physical Review E 2004; 69:036126-1–036126-10.
  • [31] Lin W. Global existence theory and chaos control of fractional differential equations. Journal of Mathematical Analysis and Applications 2007; 332:709–726.
  • [32] Sheng H, Chen YQ, Qiu TS. Fractional Processes and Fractional-order Signal Processing. Springer: New York, NY, USA, 2012.
  • [33] Ferdi Y. Some applications of fractional order calculus to design digital filters for biomedical signal processing. Journal of Mechanics in Medicine and Biology 2012; 12:1–13.
  • [34] Chen W-C. Nonlinear dynamics and chaos in a fractional-order financial system. Journal of Chaos, Solitons and Fractals 2008; 36:1305–1314
  • [35] Naik, P. A., Yeolekar, B. M., Qureshi, S., Yavuz, M., Huang, Z., & Yeolekar, M. (2025). Fractional insights in tumor modeling: an interactive study between tumor carcinogenesis and macrophage activation. Advanced Theory and Simulations, 2401477.
  • [36] Manivel, M., Venkatesh, A., & Kumawat, S. (2025). A comprehensive study of monkeypox disease through fractional mathematical modeling. Mathematical Modelling and Numerical Simulation with Applications, 5(1), 65-96.
  • [37] Naik, P. A., Yavuz, M., Qureshi, S., Owolabi, K. M., Soomro, A., Ganie, A. H. (2024). Memory impacts in hepatitis C: A global analysis of a fractional-order model with an effective treatment. Computer Methods and Programs in Biomedicine, 254, 108306.
  • [38] Jha, B. K., Vora, H., & Singh, T. P. (2024). The effect of amyloid beta, membrane, and ER pathways on the fractional behavior of neuronal calcium. Bulletin of Biomathematics, 2(2), 198-217.
  • [39] Yavuz, M., Akman, M., Usta, F., & O¨ zdemir, N. (2022, November). Effect of the awareness parameter on a fractional-order tuberculosis model. In AIP Conference Proceedings (Vol. 2483, No. 1). AIP Publishing.
  • [40] Evirgen, F., O¨ zko¨se, F., Yavuz, M., & O¨ zdemir, N. (2023). Real data-based optimal control strategies for assessing the impact of the Omicron variant on heart attacks. AIMS Bioengineering, 10(3).
  • [41] Joshi, H., & Yavuz, M. (2025). Chaotic dynamics of a cancer model with singular and non-singular kernel. Discrete and Continuous Dynamical Systems-S, 18(5), 1416-1439.
  • [42] Nguefack, D. J. T. (2024). Mathematical modeling of schistosomiasis transmission using reaction-diffusion equations. Fundamental Journal of Mathematics and Applications, 7(2), 118-136.
  • [43] E˘gilmez H˙I, Haspolat E. (2024) Temperature-Dependent Parameters in Enzyme Kinetics: Impacts on Enzyme Denaturation. Fundamental Journal of Mathematics and Applications, 7(4), 226-235.
  • [44] Yavuz, M., Özdemir, F. N., Akyu¨z, K., & Bayraktar, N. B. The relationship between colon cancer and immune system: a fractional order modelling approach. Balıkesir U¨ niversitesi Fen Bilimleri Enstitu¨su¨ Dergisi, 27(1), 126-144.
  • [45] Iwa, L. L., Omame, A., & Inyama, S. C. (2024). A fractional-order model of COVID-19 and Malaria co-infection. Bulletin of Biomathematics, 2(2), 133-161.

Yıl 2025, Cilt: 13 Sayı: 1, 67 - 77, 30.04.2025

Jaouad Danane , Mehmet Yavuz , Sania Qureshi

Öz

Kaynakça

  • [1] World Health Organization, HIV/AIDS key facts.
  • [2] Maziane, M., Lotfi, E. M., Hattaf, K., Yousfi, N. (2015). Dynamics of a class of HIV infection models with cure of infected cells in eclipse stage. Acta Biotheoretica, 63(4), 363-380.
  • [3] Shi, V., Tridane, A., Kuang, Y. (2008). A viral load-based cellular automata approach to modeling HIV dynamics and drug treatment. Journal of theoretical biology, 253(1), 24-35.
  • [4] Wodarz, D. Hepatitis C virus dynamics and pathology: The role of CTL and antibody responses. J. Gen. Virol. 2003, 84, 1743–1750.
  • [5] Li, M.; Zu, J. The review of differential equation models of HBV infection dynamics. J. Virol. Methods 2019, 266, 103–113.
  • [6] Danane, J., Meskaf, A., Allali, K. (2018). Optimal control of a delayed hepatitis B viral infection model with HBV DNA-containing capsids and CTL immune response. Optimal Control Applications and Methods, 39(3), 1262-1272.
  • [7] Burchell, A.N.; Winer, R.L.; de Sanjos´e, S.; Franco, E.L. Epidemiology and transmission dynamics of genital HPV infection. Vaccine 2006, 24, S52–S61.
  • [8] Elbasha, E.H.; Dasbach, E.J.; Insinga, R.P. A multi-type HPV transmission model. Bull. Math. Biol. 2008, 70, 2126–2176.
  • [9] Braaten, K. P., Laufer, M. R. (2008). Human papillomavirus (HPV), HPV-related disease, and the HPV vaccine. Reviews in obstetrics and gynecology, 1(1), 2.
  • [10] Nowak M.A.; Bangham C.R.M. Population dynamics of immune responses to persistent viruses. Science 1996, 272, 74–79.
  • [11] Smith H.L.; De Leenheer P. Virus dynamics: a global analysis. SIAM Journal on Applied Mathematics 2003, 63, 1313-1327.
  • [12] Korobeinikov A. Global properties of basic virus dynamics models. Bulletin of Mathematical Biology 2004, 66, 879–883.
  • [13] Buonomo, B., Vargas-De-Le´on, C.(2012) Global stability for an HIV-1 infection model including an eclipse stage of infected cells, J. Math. Anal. Appl.385(2):709-720.
  • [14] Sun Q.; Min L.; Kuang Y. Global stability of infection-free state and endemic infection state of a modified human immunodeficiency virus infection model. IET systems biology 2015, 9, 95–103.
  • [15] Sun Q.; Min L. Dynamics Analysis and Simulation of a Modified HIV Infection Model with a Saturated Infection Rate. Computational and mathematical methods in medicine 2014, 2014 Article ID 145162, 14 pages.
  • [16] Allali, K., Danane, J., Kuang, Y., 2017. Global Analysis for an HIV Infection Model with CTL Immune Response and Infected Cells in Eclipse Phase. Applied Sciences (2076-3417), 7(8).
  • [17] Raeisi, E., Yavuz, M., Khosravifarsani, M., & Fadaei, Y. (2024). Mathematical modeling of interactions between colon cancer and immune system with a deep learning algorithm. The European Physical Journal Plus, 139(4), 345.
  • [18] Wang X.; Tao Y.; Song X. Global stability of a virus dynamics model with Beddington-DeAngelis incidence rate and CTL immune response. Nonlinear Dynamics 2011, 66, 825–830.
  • [19] Daar, E.S.; Moudgil, T.; Meyer, R.D.; Ho, D.D. Transient high levels of viremia in patients with primary human immunodeficiency virus type 1. N. Engl. J. Med. 1991, 324, 961–964.
  • [20] Naim, M., Zeb, A., Mohsen, A. A., Sabbar, Y., & Yıldız, M. (2024). Local and global stability of a fractional viral infection model with two routes of propagation, cure rate and non-lytic humoral immunity. Mathematical Modelling and Numerical Simulation with Applications, 4(5-Special Issue: ICAME’24), 94-115.
  • [21] Joshi, H., Yavuz, M., Taylan, O., & Alkabaa, A. (2025). Dynamic analysis of fractal–fractional cancer model under chemotherapy drug with generalized Mittag-Leffler kernel. Computer Methods and Programs in Biomedicine, 260, 108565.
  • [22] Kahn, J.O.; Walker, B.D. Acute human immunodeficiency virus type 1 infection. N. Engl. J. Med. 1998, 339, 33–39.
  • [23] Sokolov IM, Chechkin AV, Klafter J. Distributed-order fractional kinetics. Statistical Physics: Fundamentals and Applications 2004; 35:1323–1341.
  • [24] Rida SZ, El-Sayed AMA, Arafa AM. A effect of bacterial memory dependent growth by using fractional derivatives reaction-diffusion chemotactic model. Journal of Statistical Physics 2010; 140:797–811.
  • [25] Arafa AAM, Rida SZ, Khalil M. Fractional modeling dynamics of HIV and CD4CT -cells during primary infection. Nonlinear Biomedical Physics 2012; 6(1).
  • [26] Sweilam NH, AL-Mekhlafi SM. Numerical study for multi-strain tuberculosis (TB) model of variable-order fractional derivatives. Journal of Advanced Research 2016; 7(2):271–283.
  • [27] A. Boukhouima, K. Hattaf, and N. Yousfi, Dynamics of a Fractional Order HIV InfectionModel with Specific Functional Response and Cure Rate, International Journal of Differential Equations, vol. 2017, Article ID 8372140, 8 pages, 2017.
  • [28] Debnath L. Recent applications of fractional calculus to science and engineering. International Journal of Mathematics and Mathematical Sciences 2003; 54:3413–3442.
  • [29] Hilfer ER. Applications of Fractional Calculus in Physics. World Scientific: River Edge, NJ, USA, 2000.
  • [30] Yuste SB, Acedo L, Lindenberg K. Reaction front in a reaction-subdiffusion process. Physical Review E 2004; 69:036126-1–036126-10.
  • [31] Lin W. Global existence theory and chaos control of fractional differential equations. Journal of Mathematical Analysis and Applications 2007; 332:709–726.
  • [32] Sheng H, Chen YQ, Qiu TS. Fractional Processes and Fractional-order Signal Processing. Springer: New York, NY, USA, 2012.
  • [33] Ferdi Y. Some applications of fractional order calculus to design digital filters for biomedical signal processing. Journal of Mechanics in Medicine and Biology 2012; 12:1–13.
  • [34] Chen W-C. Nonlinear dynamics and chaos in a fractional-order financial system. Journal of Chaos, Solitons and Fractals 2008; 36:1305–1314
  • [35] Naik, P. A., Yeolekar, B. M., Qureshi, S., Yavuz, M., Huang, Z., & Yeolekar, M. (2025). Fractional insights in tumor modeling: an interactive study between tumor carcinogenesis and macrophage activation. Advanced Theory and Simulations, 2401477.
  • [36] Manivel, M., Venkatesh, A., & Kumawat, S. (2025). A comprehensive study of monkeypox disease through fractional mathematical modeling. Mathematical Modelling and Numerical Simulation with Applications, 5(1), 65-96.
  • [37] Naik, P. A., Yavuz, M., Qureshi, S., Owolabi, K. M., Soomro, A., Ganie, A. H. (2024). Memory impacts in hepatitis C: A global analysis of a fractional-order model with an effective treatment. Computer Methods and Programs in Biomedicine, 254, 108306.
  • [38] Jha, B. K., Vora, H., & Singh, T. P. (2024). The effect of amyloid beta, membrane, and ER pathways on the fractional behavior of neuronal calcium. Bulletin of Biomathematics, 2(2), 198-217.
  • [39] Yavuz, M., Akman, M., Usta, F., & O¨ zdemir, N. (2022, November). Effect of the awareness parameter on a fractional-order tuberculosis model. In AIP Conference Proceedings (Vol. 2483, No. 1). AIP Publishing.
  • [40] Evirgen, F., O¨ zko¨se, F., Yavuz, M., & O¨ zdemir, N. (2023). Real data-based optimal control strategies for assessing the impact of the Omicron variant on heart attacks. AIMS Bioengineering, 10(3).
  • [41] Joshi, H., & Yavuz, M. (2025). Chaotic dynamics of a cancer model with singular and non-singular kernel. Discrete and Continuous Dynamical Systems-S, 18(5), 1416-1439.
  • [42] Nguefack, D. J. T. (2024). Mathematical modeling of schistosomiasis transmission using reaction-diffusion equations. Fundamental Journal of Mathematics and Applications, 7(2), 118-136.
  • [43] E˘gilmez H˙I, Haspolat E. (2024) Temperature-Dependent Parameters in Enzyme Kinetics: Impacts on Enzyme Denaturation. Fundamental Journal of Mathematics and Applications, 7(4), 226-235.
  • [44] Yavuz, M., Özdemir, F. N., Akyu¨z, K., & Bayraktar, N. B. The relationship between colon cancer and immune system: a fractional order modelling approach. Balıkesir U¨ niversitesi Fen Bilimleri Enstitu¨su¨ Dergisi, 27(1), 126-144.
  • [45] Iwa, L. L., Omame, A., & Inyama, S. C. (2024). A fractional-order model of COVID-19 and Malaria co-infection. Bulletin of Biomathematics, 2(2), 133-161.
Toplam 45 adet kaynakça vardır.

Ayrıntılar

Birincil Dil İngilizce
Konular Biyolojik Matematik
Bölüm Articles
Yazarlar

Jaouad Danane

Mehmet Yavuz 0000-0002-3966-6518

Sania Qureshi 0000-0002-7225-2309

Erken Görünüm Tarihi 28 Nisan 2025
Yayımlanma Tarihi 30 Nisan 2025
Gönderilme Tarihi 21 Eylül 2024
Kabul Tarihi 7 Nisan 2025
Yayımlandığı Sayı Yıl 2025 Cilt: 13 Sayı: 1

Kaynak Göster

APA Danane, J., Yavuz, M., & Qureshi, S. (2025). Mathematical Analysis of A Fractional-Order Viral Infection Model with Saturated Infection Rate and Cellular Immune Response. Konuralp Journal of Mathematics, 13(1), 67-77.
AMA Danane J, Yavuz M, Qureshi S. Mathematical Analysis of A Fractional-Order Viral Infection Model with Saturated Infection Rate and Cellular Immune Response. Konuralp J. Math. Nisan 2025;13(1):67-77.
Chicago Danane, Jaouad, Mehmet Yavuz, ve Sania Qureshi. “Mathematical Analysis of A Fractional-Order Viral Infection Model With Saturated Infection Rate and Cellular Immune Response”. Konuralp Journal of Mathematics 13, sy. 1 (Nisan 2025): 67-77.
EndNote Danane J, Yavuz M, Qureshi S (01 Nisan 2025) Mathematical Analysis of A Fractional-Order Viral Infection Model with Saturated Infection Rate and Cellular Immune Response. Konuralp Journal of Mathematics 13 1 67–77.
IEEE J. Danane, M. Yavuz, ve S. Qureshi, “Mathematical Analysis of A Fractional-Order Viral Infection Model with Saturated Infection Rate and Cellular Immune Response”, Konuralp J. Math., c. 13, sy. 1, ss. 67–77, 2025.
ISNAD Danane, Jaouad vd. “Mathematical Analysis of A Fractional-Order Viral Infection Model With Saturated Infection Rate and Cellular Immune Response”. Konuralp Journal of Mathematics 13/1 (Nisan 2025), 67-77.
JAMA Danane J, Yavuz M, Qureshi S. Mathematical Analysis of A Fractional-Order Viral Infection Model with Saturated Infection Rate and Cellular Immune Response. Konuralp J. Math. 2025;13:67–77.
MLA Danane, Jaouad vd. “Mathematical Analysis of A Fractional-Order Viral Infection Model With Saturated Infection Rate and Cellular Immune Response”. Konuralp Journal of Mathematics, c. 13, sy. 1, 2025, ss. 67-77.
Vancouver Danane J, Yavuz M, Qureshi S. Mathematical Analysis of A Fractional-Order Viral Infection Model with Saturated Infection Rate and Cellular Immune Response. Konuralp J. Math. 2025;13(1):67-7.
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