Nonlocal diffusion model of HIV/AIDS transmission dynamics among MSM populations with ART in heterogeneous environments

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Abstract

In this paper, we develop a spatial heterogeneous compartment model to investigate the combined impacts of nonlocal diffusion and antiretroviral therapy (ART) on the transmission dynamics of human immunodeficiency virus/acquired immunodeficiency syndrome (HIV/AIDS) within men who have sex with men (MSM). We have overcome the difficulty caused by the lack of compactness of the nonlocal operator. With the updated equation, we obtain the function formulation of the next generation operator R. Then, the basic reproduction number R₀ is obtained, which is the spectral radius of R. Concerning analytical investigation, according to Lipschitz continuity assumption of the parameters, the linearized model under the assumption of no disease at steady state has a dominant eigenvalue associated with a non-negative eigenfunction with positive values. By specifying the eigenfunction to be the integral kernel of the Lyapunov function, we demonstrate that, for R₀ < 1, as time progresses, all solutions approach the disease-free steady state globally. Based on the uniform persistence theory applicable to dissipative systems, it is proved that when R₀ > 1, uniform persistence is maintained in the model, which implies the presence of at least one positive equilibrium. In the numerical simulation part, the theoretical results are verified, and these findings indicate that the MSM population experiences significantly reduced HIV transmission rates through effective ART implementation.

Keywords

nonlocal dispersal / HIV/AIDS / the basic reproduction number / ART / uniform persistent

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References

[1] UNAIDS, New UNAIDS report shows AIDS pandemic can be ended by 2030, but only if leaders boost resources and protect human rights now, 2024. Available from: https://www.unaids.org/en/resources/presscentre/pressreleaseandstatementarchive/2024/july/20240722global-aids-update.

[2] S. Wang, W. Kang, J. Hu, D. Zhang, J. Xu, H. Tang, et al., Recollection: Synergizing digital and physical approaches: experience summary of the HIV PrEP promotion project, China CDC Wkly., 7 (2025), 57-62. https://doi.org/10.46234/ccdcw2025.012

[3] 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

[4] N. He, Review: Research progress in the epidemiology of HIV/AIDS in China, China CDC Wkly., 3 (2021), 1022-1030. https://doi.org/10.46234/ccdcw2021.249

[5] World Health Organization, HIV and AIDS, 2025. Available from: https://who.int/news-room/fact-sheets/detail/hiv-aids#Treatment.

[6] United Nations, Resolution adopted by the General Assembly on 8 June 2016, 2016. Available from: https://dpnsee.org/wp-content/uploads/2019/04/UN-Political-Declaration-on-HIV-and-AIDS-2016-70-266.pdf.

[7] G. Akudibillah, A. Pandey, J. Medlock, Maximizing the benefits of ART and PrEP in resource-limited settings, Epidemiol. Infect., 145 (2017), 942-956. https://doi.org/10.1017/s0950268816002958

[8] H. Zhao, P. Wu, S. Ruan, Dynamic analysis and optimal control of a three-age-class HIV/AIDS epidemic model in China, Discrete Contin. Dyn. Syst. B, 25 (2020), 3491-3521. https://doi.org/10.3934/dcdsb.2020070

[9] I. Ghosh, P. K. Tiwari, S. Samanta, I. M. Elmojtaba, N. Al-Salti, J. Chattopadhyay, A simple SI-type model for HIV/AIDS with media and self-imposed psychological fear, Math. Biosci., 306 (2018), 160-169. https://doi.org/10.1016/j.mbs.2018.09.014

[10] L. Wang, A. Din, P. Wu, Dynamics and optimal control of a spatial diffusion HIV/AIDS model with antiretrovial therapy and pre-exposure prophylaxis treatments, Math. Methods Appl. Sci., 45 (2022), 10136-10161. https://doi.org/10.1002/mma.8359

[11] P. Wu, H. Zhao, Mathematical analysis of an age-structured HIV/AIDS epidemic model with HAART and spatial diffusion, Nonlinear Anal. Real World Appl., 60 (2021), 103289. https://doi.org/10.1016/j.nonrwa.2021.103289

[12] N. Bacaer, X. Abdurahman, J. Ye, Modeling the HIV/AIDS epidemic among injecting drug users and sex workers in Kunming, China, Bull. Math. Biol., 68 (2006), 525-550. https://doi.org/10.1007/s11538-005-9051-y

[13] J. Lou, J. Wu, L. Chen, Y. Ruan, Y. Shao, A sex-role-preference model for HIV transmission among men who have sex with men in China, BMC Public Health, 9 (2009), 10. https://doi.org/10.1186/1471-2458-9-s1-s10

[14] S. M. A. Rahman, N. K. Vaidya, X. Zou, Impact of Tenofovir gel as a PrEP on HIV infection: A mathematical model, J. Theor. Biol., 347 (2014), 151-159. https://doi.org/10.1016/j.jtbi.2013.12.021

[15] M. Shen, Y. Xiao, L. Rong, L. A. Meyers, S. E. Bellan, The cost-effectiveness of oral HIV pre-exposure prophylaxis and early antiretroviral therapy in the presence of drug resistance among men who have sex with men in San Francisco, BMC Med., 16 (2018), 58. https://doi.org/10.1186/s12916-018-1047-1

[16] P. Wu, X. Wang, H. Wang, Threshold dynamics of a nonlocal dispersal HIV/AIDS epidemic model with spatial heterogeneity and antiretroviral therapy, Commun. Nonlinear Sci. Numer. Simul., 115 (2022), 106728. https://doi.org/10.1016/j.cnsns.2022.106728

[17] L. Lu, J. Wang, X. Zhao, Spatiotemporal dynamics of impulsive nonlocal diffusive systems in heterogeneous shifting environments, Sci. China Math., (2025), 1-28.

[18] W. Zeng, P. Sun, R. Wang, W. Li, A nonlocal diffusion model with free boundaries in a spatially heterogeneous environment, Discrete Contin. Dyn. Syst. B, 30 (2025), 1249-1279. https://doi.org/10.3934/dcdsb.2024128

[19] P. Wu, C. Fang, Spatiotemporal dynamics of syphilis in Xinjiang via a demographic-geographic data-validated reaction diffusion model, J. Math. Phys., 66 (2025), 062704. https://doi.org/10.1063/5.0273893

[20] P. Wu, T. Chen, S. Ruan, Spatio-temporal modeling and analysis of two HIV strain infections via demographic-geographic data, Math. Biosci., 389 (2025), 109539. https://doi.org/10.1016/j.mbs.2025.109539

[21] J. García-Melián, J. D. Rossi, On the principal eigenvalue of some nonlocal diffusion problems, J. Differ. Equations, 246 (2009), 21-38. https://doi.org/10.1016/j.jde.2008.04.015

[22] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations , 1^st edition, Springer, 1983. https://doi.org/10.1007/978-1-4612-5561-1

[23] G. F. Webb, Theory of Nonlinear Age-dependent Population Dynamics , Marcel Dekker, 1985.

[24] K. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations , Springer, 2000. https://doi.org/10.1007/b97696

[25] X. Wang, J. Yang, Dynamics of a nonlocal dispersal foot-and-mouth disease model in a spatially heterogeneous environment, Acta Math. Sci., 41 (2021), 552-572. https://doi.org/10.1007/s10473-021-0217-y

[26] J. K. Hale, Asymptotic Behavior of Dissipative Systems , American Mathematical Society, 1988. https://doi.org/10.1090/surv/025

[27] P. Magal, G. F. Webb, Y. Wu, On the basic reproduction number of reaction-diffusion epidemic models, SIAM J. Appl. Math., 79 (2019), 284-304. https://doi.org/10.1137/18M1182243

[28] N. D. Alikakos, G. Fusco, A dynamical systems proof of the Krein-Rutman Theorem and an extension of the Perron Theorem, in Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 117 (1991), 209-214. https://doi.org/10.1017/S0308210500024689

[29] D. Gerbet, K. Röbenack, Proving asymptotic stability with LaSalle’s invariance principle: On the automatic computation of invariant sets using quantifier elimination, in 2020 7th International Conference on Control, Decision and Information Technologies (CoDIT), (2020), 306-311. https://doi.org/10.1109/CoDIT49905.2020.9263958

[30] H. L. Smith, H. R. Thieme, Dynamical Systems and Population Persistence , American Mathematical Society, 2011.

[31] H. L. Smith, X. Zhao, Robust persistence for semidynamical systems, Nonlinear Anal. Theory Methods Appl., 47 (2001), 6169-6179. https://doi.org/10.1016/s0362-546x(01)00678-2

[32] J. Yang, Z. Jin, F. Xu, Threshold dynamics of an age-space structured SIR model on heterogeneous environment, Appl. Math. Lett., 96 (2019), 69-74. https://doi.org/10.1016/j.aml.2019.03.009