Dynamic analysis of a fractional-order HIV/AIDS model with generalized nonlinear incidence rate

Mi Yang; Da-peng Gao; Shi-qiang Feng; Jin-dong Li; Mi Yang; Da-peng Gao; Shi-qiang Feng; Jin-dong Li

doi:10.3934/math.20251110

AIMS Mathematics

2025, Volume 10, Issue 10: 25049-25084. doi: 10.3934/math.20251110

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Research article

Dynamic analysis of a fractional-order HIV/AIDS model with generalized nonlinear incidence rate

1.
School of Mathematical Sciences, China West Normal University, Nanchong, Sichuan 637009, China
2.
Sichuan Colleges and Universities Key Laboratory of Optimization Theory and Applications, China West Normal University, Nanchong, Sichuan 637009, China
3.
Institute of Nonlinear Analysis and Applications, China West Normal University, Nanchong, Sichuan 637009, China
4.
School of Mathematical Sciences, Chengdu University of Technology, Chengdu, Sichuan 610059, China

Received: 13 June 2025 Revised: 06 October 2025 Accepted: 15 October 2025 Published: 31 October 2025
MSC : 26A33, 34A08, 37N25

This paper investigates a fractional-order human immunodeficiency virus (HIV)/acquired immune deficiency syndrome (AIDS) model with generalized nonlinear incidence rates $ f(S, I) $ and $ g(S, E) $. First, the existence, uniqueness, non-negativity, and boundedness of the model solutions are proven, and the basic reproduction number $ R_0^\alpha $ is derived. The analysis indicates that the model exhibits two equilibrium points: A disease-free equilibrium and an endemic equilibrium. The local asymptotic stability of each equilibrium point is examined using the Routh-Hurwitz criterion. Additionally, by employing Lyapunov functionals and applying LaSalle's invariance principle, the global stability of the equilibrium points is demonstrated. The main conclusion is that under appropriate conditions, if $ R_0^\alpha < 1 $, then the disease will eventually disappear, whereas if $ R_0^\alpha > 1 $, then it will persist. Finally, the model is utilized to predict and control of HIV/AIDS transmission in Mexico, thereby highlighting the role of mutural preventive measures adopted by susceptible individuals and HIV-infected individuals in reducing disease spread. Simulations are performed to confirm the theoretical validity and practical significance of the model.
- caputo fractional derivative,
- reproduction number,
- local stability,
- global stability
Citation: Mi Yang, Da-peng Gao, Shi-qiang Feng, Jin-dong Li. Dynamic analysis of a fractional-order HIV/AIDS model with generalized nonlinear incidence rate[J]. AIMS Mathematics, 2025, 10(10): 25049-25084. doi: 10.3934/math.20251110

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Abstract

This paper investigates a fractional-order human immunodeficiency virus (HIV)/acquired immune deficiency syndrome (AIDS) model with generalized nonlinear incidence rates $ f(S, I) $ and $ g(S, E) $. First, the existence, uniqueness, non-negativity, and boundedness of the model solutions are proven, and the basic reproduction number $ R_0^\alpha $ is derived. The analysis indicates that the model exhibits two equilibrium points: A disease-free equilibrium and an endemic equilibrium. The local asymptotic stability of each equilibrium point is examined using the Routh-Hurwitz criterion. Additionally, by employing Lyapunov functionals and applying LaSalle's invariance principle, the global stability of the equilibrium points is demonstrated. The main conclusion is that under appropriate conditions, if $ R_0^\alpha < 1 $, then the disease will eventually disappear, whereas if $ R_0^\alpha > 1 $, then it will persist. Finally, the model is utilized to predict and control of HIV/AIDS transmission in Mexico, thereby highlighting the role of mutural preventive measures adopted by susceptible individuals and HIV-infected individuals in reducing disease spread. Simulations are performed to confirm the theoretical validity and practical significance of the model.

References

[1]	Centers for Disease Control and Prevention. HIV and AIDS-United States, 1981-2000, Morbidity Mortality Weekly Rep., 50 (2001), 430–434.
[2]	World Health Organization, HIV, 2022. Available from: https://www.who.int/news-room/fact-sheets/detail/hiv-aids.
[3]	G. Assembly, Resolution adopted by the general assembly on 3 2015, New York, United Nations, 2016. https://doi.org/10.18356/9789210026253c001
[4]	H. F. Huo, R. Chen, X. Y. Wang, Modelling and stability of HIV/AIDS epidemic model with treatment, Appl. Math. Model., 40 (2016), 6550–6559. https://doi.org/10.1016/j.apm.2016.01.054 doi: 10.1016/j.apm.2016.01.054
[5]	J. W. Jia, G. L. Qin, Stability analysis of HIV/AIDS epidemic model with nonlinear incidence and treatment, Adv. Differ. Equ., 2017 (2017), 1–13. https://doi.org/10.1186/s13662-017-1175-5 doi: 10.1186/s13662-017-1175-5
[6]	S. Mangal, O. P. Misra, J. Dhar, Fractional-order deterministic epidemic model for the spread and control of HIV/AIDS with special reference to Mexico and India, Math. Comput. Simul., 210 (2023), 82–102. https://doi.org/10.1016/j.matcom.2023.03.008 doi: 10.1016/j.matcom.2023.03.008
[7]	R. M. Aderson, R. M. May, Infections diseases of humans: Dynamics and control, Oxford: Oxford Univ. Press, 1991.
[8]	H. W. Hethcote, The mathematics of infections diseases, SIAM Rev., 42 (2000), 599–653. https://doi.org/10.1137/S0036144500371907 doi: 10.1137/S0036144500371907
[9]	C. Liu, J. Gao, J. Kanesan, Dynamics analysis and optimal control of delayed SEIR model in COVID-19 epidemic, J. Inequal. Appl., 2024 (2024), 1–37. https://doi.org/10.1186/s13660-024-03140-2 doi: 10.1186/s13660-024-03140-2
[10]	C. Vargas-De-León, On the global stability of SIS, SIR and SIRS epdemic models with standard incidence, Chaos Soliton. Fract., 44 (2011), 1106–1110. https://doi.org/10.1016/j.chaos.2011.09.002 doi: 10.1016/j.chaos.2011.09.002
[11]	V. Capasso, G. Serio, A generalization of the Kermack-Mckendrick determinstic epidemic model, Math. Biosci., 42 (1978), 43–61. https://doi.org/10.1016/0025-5564(78)90006-8 doi: 10.1016/0025-5564(78)90006-8
[12]	L. W. Wang, X. G. Zhang, Z. J. Liu, An SEIR epidemic model with relapse and general nonlinear incidence rate with application to media impact, Qual. Theory Dyn. Syst., 17 (2018), 309–329. https://doi.org/10.1007/s12346-017-0231-6 doi: 10.1007/s12346-017-0231-6
[13]	Q. Tang, Z. D. Teng, X. Abdurahman, A new Lyapunov function for SIRS epidemic models, Bull. Malays. Math. Sci. Soc., 40 (2017), 237–258. https://doi.org/10.1007/s40840-016-0315-5 doi: 10.1007/s40840-016-0315-5
[14]	F. M. Al-Askar, C. Cesarano, W. W. Mohammed, The analytical solutions of stochastic-fractional drinfel'd-Sokolov-Wilson equations via (G'/G) -expansion method, Symmetry, 14 (2022), 2105. https://doi.org/10.3390/sym14102105 doi: 10.3390/sym14102105
[15]	H. Ahmad, M. Tariq, S. K. Sahoo, New estimations of Hermite–Hadamard type integral inequalities for special functions, Fractal Fract., 5 (2021), 144. https://doi.org/10.3390/fractalfract5040144 doi: 10.3390/fractalfract5040144
[16]	C. J. Sun, Y. P. Lin, S. P. Tang, Global stability for an special SEIR epidemic model with nonlinear incidence rates, Chaos Soliton. Fract., 33 (2017), 290–297. https://doi.org/10.1016/j.chaos.2005.12.028 doi: 10.1016/j.chaos.2005.12.028
[17]	C. Liu, X. Yi, Z. Gong, The control parametrization technique for numerically solving fractal–fractional optimal control problems involving Caputo–Fabrizio derivatives, J. Comput. Appl. Math., 472 (2026), 116814. https://doi.org/10.1016/j.cam.2025.116814 doi: 10.1016/j.cam.2025.116814
[18]	I. Yi, C. Liu, H. T. Cheong, A third-order numerical method for solving fractional ordinary differential equations, AIMS Math., 9 (2024), 21125–21143. https://doi.org/10.3934/math.20241026 doi: 10.3934/math.20241026
[19]	C. Liu, Z. Gong, C. Yu, S. Wang, Optimal control computation for nonlinear fractional time-delay systems with state inequality constraints, J. Optim. Theory Appl., 191 (2021), 83–117. https://doi.org/10.1007/s10957-021-01926-8 doi: 10.1007/s10957-021-01926-8
[20]	Y. Luo, Y. Yuan, X. Wang, Z. Teng, Dynamic analysis of a degenerated temporal-spatial acquired immun-odeficiency syndrome model with voluntary counseling and testing awareness and antiretroviral therapy treatment, J. Math. Phys., 66 (2025), 072704. https://doi.org/doi:10.1063/5.0223690 doi: 10.1063/5.0223690
[21]	Y. Hao, Y. Luo, J. Huang, L. Zhang, Z. Teng, Analysis of a stochastic HIV/AIDS model with commercial heterosexual activity and Ornstein-Uhlenbeck process, Math. Comput. Simul., 234 (2025), 50–72. https://doi.org/10.1016/j.matcom.2025.02.020 doi: 10.1016/j.matcom.2025.02.020
[22]	Y. Luo, J. Huang, Z. Teng, Q. Liu, Role of ART and PrEP treatments in a stochastic HIV/AIDS epidemic model, Math. Comput. Simul., 221 (2024), 337–357. https://doi.org/10.1016/j.matcom.2024.03.010 doi: 10.1016/j.matcom.2024.03.010
[23]	M. Naim, F. Lahmidi, A. Namir, A. Kouidere, Dynamics of an fractional SEIR epidemic model with infectivity in latent period and general nonlinear incidence rate, Chaos Soliton. Fract., 152 (2021), 111456. https://doi.org/10.1016/j.chaos.2021.111456 doi: 10.1016/j.chaos.2021.111456
[24]	I. Podlubny, Fractional differential equations, Academic Press, New York, 1999.
[25]	Z. M. Odibat, N. T. Shawagfeh, Generalized Taylor's formula, Appl. Math. Comput., 186 (2007), 286–293. https://doi.org/10.1016/j.amc.2006.07.102
[26]	H. L. Li, L. Zhang, C. Hu, Y. L. Jiang, Z. D. Teng, Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, J. Appl. Math. Comput., 54 (2017), 435–449. https://doi.org/10.1007/s12190-016-1017-8 doi: 10.1007/s12190-016-1017-8
[27]	S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional integral and derivatives, 1 (1993), Gordon and Breach science publishers, Yverdon, Yverdon-Les-Bains, Switzerland.
[28]	K. Diethelm, The analysis of fractional differential equations, an application-oriented exposition using operators of Caputo type, Springer, Berlin, 2004. https://doi.org/10.1006/JMAA.2000.7194
[29]	L. W. Wang, Z. J. Liu, X. G. Zhang, Global dynamics for an age-structed epidemic model with media impact and incomplete vaccination, Nonlinear Anal. Real World Appl., 32 (2016), 136–158. https://doi.org/10.1016/j.nonrwa.2016.04.009 doi: 10.1016/j.nonrwa.2016.04.009
[30]	M. Naim, F. Lahmidi, A. Namir, Global stability of a fractional order SIR epidemic model with double epidemic hypothesis and nonlinear incidence rate, Commun. Math. Biol. Neurosci., 2020 (2020), 38.
[31]	Z. M. Li, Z. D. Teng, H. Miao, Modeling and control for HIV/AIDS transmission in China based on data from 2004 to 2016, Comput. Math. Method Med., 2017 (2017), 8935314. https://doi.org/10.1155/2017/8935314 doi: 10.1155/2017/8935314
[32]	A. Lahrouz, L. Omari, D. Kiouach, A. Belmaâti, Complete global stability for an SIRS epidemic model with generalized non-linear incidence and vaccination, Appl. Math. Comput., 218 (2012), 6519–6525. https://doi.org/10.1016/j.amc.2011.12.024 doi: 10.1016/j.amc.2011.12.024
[33]	A. Mouaouine, A. Boukhouima, K. Hattaf, N. Yousfi, A fractional order SIR epidemic model with nonlinear incidence rate, Adv. Differ. Equ., 2018 (2018), 1–9. https://doi.org/10.1186/s13662-018-1613-z doi: 10.1186/s13662-018-1613-z
[34]	M. R. S. Ammi, M. Tahiri, D. F. M. Torres, Global stability of a Caputo fractional SIRS model with general incidence rate, Math. Comput. Sci., 15 (2020), 91–105. https://doi.org/10.1007/s11786-020-00467-z doi: 10.1007/s11786-020-00467-z
[35]	A. Boukhouima, K. Hattaf, N. Yousfi, Dynamics of a fractional order HIV infection model with specific functional response and cure rate, Int. J. Differ. Equat., 2017 (2017), 8372140. https://doi.org/10.1155/2017/8372140 doi: 10.1155/2017/8372140
[36]	P. V. D. Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29–48. https://doi.org/10.1016/s0025-5564(02)00108-6 doi: 10.1016/s0025-5564(02)00108-6
[37]	A. E. Matouk, Stability conditions, hyperchans and control in a novel fractional order hyperchaotic system, Phys. Lett. A., 373 (2009), 2166–2173. https://doi.org/10.1016/j.physleta.2009.04.032 doi: 10.1016/j.physleta.2009.04.032
[38]	https://ourworldindate.org/hiv-aids.

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