[1] Acheneje, G. O., Omale, D., Atokolo, W. and Bolaji, B., Modeling the transmission dynamics
of the co-infection of COVID-19 and monkeypox diseases with optimal control strategies and
cost–benefit analysis, Franklin Open, 8 (2024), 100130.
[2] Acheneje, G. O., Omale, D., Atokolo, W., Celestine, A. B., Abah, E., Dervishi, R., Shyam-
sunder and Bolaji, B., Mathematical modelling on the transmission dynamics of the co-
infection of COVID-19 and monkeypox with treatment as a control strategy, Discov. Public
Health, 22(1) (2025), 535.
[3] Agarwal, P., Jleli, M. and Samet, B., Fixed point theory in metric spaces: recent advances
and applications, Springer, Singapore, 2018.
[4] Ahmad, S., Ullah, A., Shah, K., Salahshour, S., Ahmadian, A. and Ciano, T., Fuzzy
fractional-order model of the novel coronavirus, Adv. Differ. Equ., 1 (2020), 1–17.
[5] Anderson, P. L., Glidden, D. V., Liu, A., Buchbinder, S., Lama, J. R., Guanira, J. V. and
Grant, R. M., Emtricitabine–tenofovir concentrations and pre-exposure prophylaxis efficacy
in men who have sex with men, Sci. Transl. Med., 4(151) (2012), 151ra125.
[6] Anderson, R. M. and May, R. M., Infectious diseases of humans: dynamics and control,
Oxford University Press, Oxford, 1991.
[7] Arafa, A. A. M., Rida, S. Z. and Khalil, M., Fractional modeling dynamics of HIV and
CD4+ T-cells during primary infection, Nonlinear Biomed. Phys., 6(1) (2012), 1–7.
[8] Atangana, A., Fractal–fractional differentiation and integration: connecting fractal calculus
and fractional calculus to predict complex systems, Chaos Solitons Fractals, 102 (2017),
396–406.
[9] Atangana, A., Fractional operators with constant and variable order with application to
geo-hydrology, Academic Press, London, 2017.
[10] Atangana, A. and Baleanu, D., New fractional derivatives with nonlocal and non-singular
kernel: theory and application to heat transfer model, Therm. Sci., 20(2) (2016), 763–769.
[11] Atangana, A. and Qureshi, S., Modeling attractors of chaotic dynamical systems with fractal–
fractional operators, Chaos Solitons Fractals, 123 (2019), 320–337.
[12] Atokolo, W., Aja, R. O., Omale, D., Ahman, Q. O., Acheneje, G. O. and Amos, J., Fractional
mathematical model for the transmission dynamics and control of Lassa fever, Franklin
Open, 7 (2024), 100110.
[13] Baleanu, D., Jajarmi, A., Mohammadi, H. and Rezapour, S., A new study on the mathe-
matical modelling of human liver with Caputo–Fabrizio fractional derivative, Chaos Solitons
Fractals, 134 (2020), 109705.
[14] Bekker, L. G., Johnson, L., Cowan, F., Overs, C., Besada, D., Hillier, S. and Cates Jr.,
W., Combination HIV prevention for female sex workers: what is the evidence?, Lancet,
385(9962) (2015), 72–87.
[15] Bhunu, C. P. and Mushayabasa, S., Modelling the transmission dynamics of HIV/AIDS and
hepatitis B co-infection, HIV AIDS Rev., 11(4) (2012), 118–124.
[16] Blower, S. M. and Dowlatabadi, H., Sensitivity and uncertainty analysis of complex models
of disease transmission: an HIV model as an example, Int. Stat. Rev., 62(2) (1994), 229–243.
[17] Bolaji, B., Ibrahim, A., Ani, F., Omede, B. and Acheneje, G., A model for the control of
transmission dynamics of human monkeypox disease in Sub-Saharan Africa, J. Niger. Soc.
Phys. Sci., 6 (2024), 1800.
[18] Bryant, J., Baxter, L. and Hird, S., Non-occupational post-exposure prophylaxis for HIV: a
systematic review, Health Technol. Assess., 17(20) (2013), 1–195.
[19] Cardo, D. M., Culver, D. H., Ciesielski, C. A., Srivastava, P. U., Marcus, R., Abiteboul, D.
and McKibben, P. S., A case–control study of HIV seroconversion in health care workers
after percutaneous exposure, N. Engl. J. Med., 337(21) (1997), 1485–1490.
[20] Cohen, M. S., Chen, Y. Q., McCauley, M., Gamble, T., Hosseinipour, M. C., Kumarasamy,
N. and Fleming, T. R., Prevention of HIV-1 infection with early antiretroviral therapy, N.
Engl. J. Med., 365(6) (2011), 493–505.
[21] Cohen, M. S., Chen, Y. Q., McCauley, M., Gamble, T., Hosseinipour, M. C., Kumarasamy,
N. and Godbole, S. V., Antiretroviral therapy for the prevention of HIV-1 transmission, N.
Engl. J. Med., 375(9) (2016), 830–839.
[22] Davarian, N., Heydari, A., Hashemi Mehne, S. H. and Heydari, A. A., The investigation of
stability and optimal control of the multigroup COVID-19 epidemic model in Iran, J. Math.
Ext., 19(1) (2025), 1–20.
[23] Deeks, S. G., Overbaugh, J., Phillips, A. and Buchbinder, S., HIV infection, Nat. Rev. Dis.
Primers, 1(1) (2015), 1–22.
[24] Ding, Y., Wang, Z. and Ye, H., Optimal control of a fractional-order HIV-immune system
with memory, IEEE Trans. Control Syst. Technol., 20(3) (2012), 763–769.
[25] French, P. P., Latka, M., Gollub, E. L., Rogers, C., Hoover, D. R. and Stein, Z. A., Use-
effectiveness of the female versus male condom in preventing sexually transmitted disease
in women, Sex. Transm. Dis., 30(5) (2003), 433–439.
[26] Grant, R. M. and Glidden, D. V., HIV moments and pre-exposure prophylaxis, Lancet,
387(10027) (2016), 1507–1508.
[27] Grant, R. M., Lama, J. R., Anderson, P. L., McMahan, V., Liu, A. Y., Vargas, L. and
Glidden, D. V., Preexposure chemoprophylaxis for HIV prevention in men who have sex
with men, N. Engl. J. Med., 363(27) (2010), 2587–2599.
[28] Khan, M. A., Ullah, S. and Farhan, M., The dynamics of Zika virus with Caputo fractional
derivative, AIMS Math., 4(1) (2020), 134–154.
[29] Kilbas, A. A., Srivastava, H. M. and Trujillo, J. J., Theory and applications of fractional
differential equations, Elsevier, Amsterdam, 2006.
[30] Kuhar, D. T., Henderson, D. K., Struble, K. A., Heneine, W., Thomas, V., Cheever, L. W.
and Panlilio, A. L., Updated US public health service guidelines for the management of
occupational exposures to HIV and recommendations for postexposure prophylaxis, Infect.
Control Hosp. Epidemiol., 34(9) (2013), 875–892.
[31] Kumar, S., Kumar, R., Cattani, C. and Samet, B., Chaotic behaviour of fractional predator–
prey dynamical system, Chaos Solitons Fractals, 135 (2020), 109811.
[32] La Salle, J. P., The stability of dynamical systems, SIAM, Philadelphia, 1976.
[33] Odiba, P., Acheneje, G. O. and Bolaji, B., A compartmental deterministic epidemiological
model with nonlinear differential equations for analyzing the co-infection dynamics between
COVID-19, HIV and monkeypox diseases, Healthc. Anal., 5 (2024), 100311.
[34] Omame, A., Han, Q., Iyaniwura, S. A., Ebenezer, A., Bragazzi, N. L., Wang, X. and
Woldegerima, W. A., Understanding the impact of HIV on mpox transmission in the MSM
population: a mathematical modeling study, Infect. Dis. Model., 9(4) (2024), 1117–1137.
[35] Owolabi, K. M. and Atangana, A., Mathematical analysis and computational experiments
for an epidemic system with nonlocal and nonsingular derivative, Chaos Solitons Fractals,
126 (2019), 41–49.
[36] Owolabi, K. M. and Atangana, A., Fractal fractional-order derivative for HIV/AIDS model
with Mittag–Leffler kernel, Alexandria Eng. J., 61 (2022), 10965–10980.
[37] Palella Jr., F. J., Delaney, K. M., Moorman, A. C., Loveless, M. O., Fuhrer, J., Satten,
G. A. and Holmberg, S. D., Declining morbidity and mortality among patients with advanced
human immunodeficiency virus infection, N. Engl. J. Med., 338(13) (1998), 853–860.
[38] Perelson, A. S., Modelling viral and immune system dynamics, Nat. Rev. Immunol., 2(1)
(2002), 28–36.
[39] Potts, M., Halperin, D. T., Kirby, D., Swidler, A., Marseille, E., Klausner, J. D. and Gayle,
H., Reassessing HIV prevention, Science, 320(5877) (2008), 749–750.
[40] Rodger, A. J., Cambiano, V., Bruun, T., Vernazza, P., Collins, S., Degen, O. and Lundgren,
J., Sexual activity without condoms and risk of HIV transmission in serodifferent couples
when the HIV-positive partner is using suppressive antiretroviral therapy, JAMA, 316(2)
(2016), 171–181.
[41] Shah, K., Alqudah, M. A., Jarad, F. and Abdeljawad, T., Semi-analytical study of Pine Wilt
Disease model with convex rate under Caputo–Fabrizio fractional order derivative, Chaos
Solitons Fractals, 135 (2020), 109754.
[42] Smart, D. R., Fixed point theorems, Cambridge University Press, Cambridge, 1974.
[43] Sharp, P. M. and Hahn, B. H., Origins of HIV and the AIDS pandemic, Cold Spring Harb.
Perspect. Med., 1(1) (2011), a006841.
[44] Toufik, M. and Atangana, A., New numerical approximation of fractional derivative with
non-local and non-singular kernel: application to chaotic models, Eur. Phys. J. Plus, 132(10)
(2017), 444.
[45] UNAIDS, Global HIV & AIDS statistics: fact sheet, Joint United Nations Programme on
HIV/AIDS, 2021.
[46] UNAIDS, 2024 UNAIDS global AIDS update summary for South Africa, Joint United Na-
tions Programme on HIV/AIDS, 2024.
[47] Van Den Driessche, P. and Watmough, J., Reproduction numbers and sub-threshold endemic
equilibria for compartmental models of disease transmission, Math. Biosci., 180(1–2) (2002),
29–48.
[48] Weller, S. and Davis, K., Condom effectiveness in reducing heterosexual HIV transmission,
Cochrane Database Syst. Rev., 1 (2002), CD003255.
[49] World Health Organization, Guideline on when to start antiretroviral therapy and on pre-
exposure prophylaxis for HIV, World Health Organization, Geneva, 2015.