{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T18:08:33Z","timestamp":1788804513589,"version":"build-2803163510"},"reference-count":44,"publisher":"L and H Scientific Publishing, LLC","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["JAND"],"published-print":{"date-parts":[[2027,3,1]]},"abstract":"<jats:p>A six-compartment fractional model of hepatitis B was developed to analyze the transmission mechanisms of the infection, also taking into account the impact of the media on the population. Its biological validity was confirmed from an epidemiological point of view. The basic reproduction number,  R_0 , was determined using the new generation matrix method. A study of the asymptotic behavior around the equilibrium points showed that the model undergoes a transcritical bifurcation when R_0 = 1, thus establishing R_0 as a critical threshold influencing the evolution of the epidemic. Fractional-order optimal control was applied, incorporating time-evolving prevention and treatment strategies. The results, validated by simulations using realistic parameters, revealed that the fractional approach offers superior performance. The application of prevention and treatment measures led to a progressive reduction in the number of cases and a decrease in the epidemic peak. The analysis was extended to a stochastic framework, incorporating white noise, to study the stochastic stability of the endemic equilibrium point. In addition, sensitivity indices were used to identify key parameters influencing R_0,, facilitating the development of disease control strategies. Numerical simulations corroborated the theoretical results obtained.<\/jats:p>","DOI":"10.5890\/jand.2027.03.005","type":"journal-article","created":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T17:41:02Z","timestamp":1788802862000},"page":"65-112","source":"Crossref","is-referenced-by-count":0,"title":["Global Dynamics of a Fractional Order Hepatitis B Model with Media Effect Incorporating Optimal Control and Stochastic Stability"],"prefix":"10.5890","volume":"16","author":[{"given":"Gallimard Nzinga","family":"Milongo","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Apollinaire Ndondo","family":"Mboma","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Aymard Christbert","family":"Nimi","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Franck Davhys Reval","family":"Langa","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"7015","published-online":{"date-parts":[[2026,9,7]]},"reference":[{"key":"ref1","unstructured":"[1] Zhou, L., Zhang, Z., and He, H. (2017), Modeling of fractional-order infectious diseases and the impact of control strategies, Journal of Mathematical Biology, 75(6), 1379-1402."},{"key":"ref2","unstructured":"[2] Bai, Z. and Xu, W. (2017), Fractional-order models in epidemiology and their applications, Mathematical Models and Methods in Applied Sciences, 27(12), 2215-2238."},{"key":"ref3","doi-asserted-by":"crossref","unstructured":"[3] Badawi, H., Arqub, O., and Shawagfeh, N. (2025), Theoretical study and numerical analysis using the stepwise spectral collocation method for M-fractional stochastic differential models of simplified Brownian motion with constant delay processes, Journal of Applied Mathematics and Computing, 71(1), 1-25. DOI: 10.1007\/s12190-025-02523-y.","DOI":"10.1007\/s12190-025-02523-y"},{"key":"ref4","doi-asserted-by":"crossref","unstructured":"[4] Badawi, H., Abu Arqub, O., and Shawagfeh, N. (2024), Existence, uniqueness, and collocation solutions using the shifted Legendre spectral method for the Hilfer fractional stochastic integro-differential equations regarding stochastic Brownian motion, Results in Applied Mathematics, 24, 100504. DOI: 10.1016\/j.rinam.2024.100504.","DOI":"10.1016\/j.rinam.2024.100504"},{"key":"ref5","doi-asserted-by":"crossref","unstructured":"[5] Omar Abou, A. and Nabil, S. (2021), Optimal control problems with Fredholm constraints solved via the reproducing kernel Hilbert space method: error estimates and convergence analysis, Mathematical Methods in the Applied Sciences, 44(10), 8123-8138. DOI: 10.1002\/mma.5530.","DOI":"10.1002\/mma.5530"},{"key":"ref6","unstructured":"[6] Liu, X., Zhang, Z., and Wang, L. (2020), Stochastic stability analysis of fractional-order epidemic models, Nonlinear Dynamics, 100(3), 1751-1766."},{"key":"ref7","doi-asserted-by":"crossref","unstructured":"[7] Arqub, O.A. (2019), Computational algorithm for solving singular Fredholm time-fractional partial integrodifferential equations with error estimates, Journal of Applied Mathematics and Computing, 59, 227\u2013243.","DOI":"10.1007\/s12190-018-1176-x"},{"key":"ref8","doi-asserted-by":"crossref","unstructured":"[8] Duarte, F.B. and Machado, J.T. (2002), Chaotic phenomena and fractional-order dynamics in the trajectory control of redundant manipulators, Nonlinear Dynamics, 29(1\u20134), 315\u2013342.","DOI":"10.1023\/A:1016559314798"},{"key":"ref9","doi-asserted-by":"crossref","unstructured":"[9] Laskin, N. (2000), Fractional market dynamics, Physica A: Statistical Mechanics and its Applications, 287, 482\u2013492. DOI: 10.1016\/S0378-4371(00)00387-3.","DOI":"10.1016\/S0378-4371(00)00387-3"},{"key":"ref10","doi-asserted-by":"crossref","unstructured":"[10] Huang, C., Cai, L., and Cao, J. (2018), Linear control for synchronization of a fractional-order time delayed chaotic financial system, Chaos, Solitons and Fractals, 113, 326-332.","DOI":"10.1016\/j.chaos.2018.05.022"},{"key":"ref11","doi-asserted-by":"crossref","unstructured":"[11] Huo, J., Zhao, H., and Zhu, L. (2015), The effect of vaccines on backward bifurcation in a fractional order HIV model, Nonlinear Analysis: Real World Applications, 26, 289\u2013305.","DOI":"10.1016\/j.nonrwa.2015.05.014"},{"key":"ref12","doi-asserted-by":"crossref","unstructured":"[12] Rakkiyappan, R., Velmurugan, G., and Cao, J. (2015), Stability analysis of fractional-order complex-valued neural networks with time delays, Chaos, Solitons and Fractals, 78, 297\u2013316.","DOI":"10.1016\/j.chaos.2015.08.003"},{"key":"ref13","doi-asserted-by":"crossref","unstructured":"[13] Li, H.L., Zhang, L., Hu, C., Jiang, Y.L., and Teng, Z. (2016), Dynamical analysis of a fractional-order predator-prey model incorporating a prey refuge, Journal of Applied Mathematics and Computing, 54(1\u20132), 435\u2013449. DOI: 10.1007\/s12190-016-1017-8.","DOI":"10.1007\/s12190-016-1017-8"},{"key":"ref14","doi-asserted-by":"crossref","unstructured":"[14] Diethelm, K. (2010), The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type, Springer, 2010.","DOI":"10.1007\/978-3-642-14574-2"},{"key":"ref15","unstructured":"[15] Ali, M. and Podlubny, I. (1999), Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Academic Press, 1999."},{"key":"ref16","unstructured":"[16] Chueshov, I. (2002), Introduction to the theory of infinite-dimensional dissipative systems, Acta, 2002."},{"key":"ref17","doi-asserted-by":"crossref","unstructured":"[17] Van den Driessche, P. and Watmough, J. (2002), Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180, 29\u201348.","DOI":"10.1016\/S0025-5564(02)00108-6"},{"key":"ref18","doi-asserted-by":"crossref","unstructured":"[18] Guckenheimer, G. and Holmes, P. (1983), Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Springer Verlag, 1983.","DOI":"10.1007\/978-1-4612-1140-2"},{"key":"ref19","doi-asserted-by":"crossref","unstructured":"[19] Fleming, W.H. and Rishel, R.W. (1975), Deterministic and stochastic optimal control, Springer Verlag, 1975.","DOI":"10.1007\/978-1-4612-6380-7"},{"key":"ref20","unstructured":"[20] Lukes, D.L. (1982), Differential equations: classical to control, mathematics in science and engineering, Academic Press, 1982."},{"key":"ref21","unstructured":"[21] Pontryagin, L.S., Boltyanskii, V.G., Gamkrelidze, R.V., and Mishchenko, E.F. (1962), The mathematical theory of optimal processes, Wiley, 1962."},{"key":"ref22","doi-asserted-by":"crossref","unstructured":"[22] Shi, R. and Lu, T. (2020), Dynamic analysis and optimal control of a fractional order model for hand-foot-mouth disease, Journal of Applied Mathematics and Computing, 64, 565\u201390.","DOI":"10.1007\/s12190-020-01369-w"},{"key":"ref23","doi-asserted-by":"crossref","unstructured":"[23] Lahrouz, A., Omari, L., and Kiouach, D. (2011), Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Analysis: Modelling and Control, 16(1), 69\u201376.","DOI":"10.15388\/NA.16.1.14115"},{"key":"ref24","doi-asserted-by":"crossref","unstructured":"[24] Huang, Z., Yang, Q., and Cao, J. (2011), Complex dynamics in a stochastic internal HIV model, Chaos, Solitons and Fractals, 44, 954\u2013963.","DOI":"10.1016\/j.chaos.2011.07.017"},{"key":"ref25","doi-asserted-by":"crossref","unstructured":"[25] Beretta, E., Kolmanovskii, V., and Shaikhet, L. (1998), Stability of epidemic model with time delays influenced by stochastic perturbations, Mathematical and Computers in Simulation, 45, 269\u2013277.","DOI":"10.1016\/S0378-4754(97)00106-7"},{"key":"ref26","doi-asserted-by":"crossref","unstructured":"[26] Jana, S., Haldar, P., and Kar, T.K. (2016), Complex dynamics of an epidemic model with vaccination and treatment controls, International Journal of Dynamics and Control, 4, 318\u2013329.","DOI":"10.1007\/s40435-015-0189-7"},{"key":"ref27","unstructured":"[27] Arnold, L. (1972), Stochastic differential equations: theory and applications, Wiley, 1972."},{"key":"ref28","unstructured":"[28] Mao, X. (1997), Stochastic differential equations and applications, Horwood, 1997."},{"key":"ref29","doi-asserted-by":"crossref","unstructured":"[29] Dalal, N., Greenhalgh, D., and Mao, X. (2007), A stochastic model of AIDS and condom use, Journal of Mathematical Analysis and Applications, 325, 36\u201353.","DOI":"10.1016\/j.jmaa.2006.01.055"},{"key":"ref30","doi-asserted-by":"crossref","unstructured":"[30] Mao, X. (2011), Stochastic differential equations and applications, second edition, Woodhead Publishing, 2011.","DOI":"10.1533\/9780857099402.47"},{"key":"ref31","doi-asserted-by":"crossref","unstructured":"[31] Afanas'ev, V.N., Kolmanowskii, V.B., and Nosov, V.R. (1996), Mathematical theory of control systems design, Kluwer Academic, 1996.","DOI":"10.1007\/978-94-017-2203-2"},{"key":"ref32","doi-asserted-by":"crossref","unstructured":"[32] Cai, L. and Li, X. (2008), A note on global stability of an SEI epidemic model with acute and chronic stages, Applied Mathematics and Computation, 196(2), 923\u201330.","DOI":"10.1016\/j.amc.2007.07.024"},{"key":"ref33","doi-asserted-by":"crossref","unstructured":"[33] Din, A., Li, Y., and Yusuf, A. (2021), Delayed hepatitis B epidemic model with stochastic analysis, Chaos, Solitons and Fractals, 146, 110839.","DOI":"10.1016\/j.chaos.2021.110839"},{"key":"ref34","doi-asserted-by":"crossref","unstructured":"[34] Kar, T.K., Nandi, S.K., Jana, S., and Mandal, M. (2019), Stability and bifurcation analysis of an epidemic model with the effect of media, Chaos, Solitons and Fractals, 120, 188-199.","DOI":"10.1016\/j.chaos.2019.01.025"},{"key":"ref35","doi-asserted-by":"crossref","unstructured":"[35] M.S., Jana, S., Das, D.K., and Kar, T.K. (2022), Global dynamics of a fractional-order HFMD model incorporating optimal treatment and stochastic stability, Chaos, Solitons and Fractals, 161, 112291.","DOI":"10.1016\/j.chaos.2022.112291"},{"key":"ref36","unstructured":"[36] Zinihi, A., Sidi Ammi, M.R., and Ehrhardt, M. (2024), Optimal control of a diffusive epidemiological model involving the Caputo-Fabrizio fractional time-derivative, Journal of Mathematical Analysis and Applications, 530(1), 127-145."},{"key":"ref37","unstructured":"[37] Bouvenot, G., Le Coz, P., and Juillet, Y. (2022), Public perception of drug-related risks and the role of the media (Report 22-09), Economic Committee for Health Products, 2022."},{"key":"ref38","unstructured":"[38] Abahour, R. and Boumzaid, Y. (2022), Stochastic modeling of epidemic spread, Master's thesis, University of Bejaia, 2022."},{"key":"ref39","unstructured":"[39] Zinihi, A., Sidi Ammi, M.R., and Torres, D.F.M. (2024), Optimal control analysis of a Caputo-type fractional epidemic model with diffusion, Computational and Applied Mathematics, 43(2), 88-105."},{"key":"ref40","doi-asserted-by":"crossref","unstructured":"[40] Moneim, I.A. and Khali, H.A.l. (2015), Modeling and simulation of the spread of HBV disease with infectious latent, Applied Mathematics, 6, 745--753.","DOI":"10.4236\/am.2015.65070"},{"key":"ref41","doi-asserted-by":"crossref","unstructured":"[41] Ali, A. and Munir, M. (2020), Studying the transmission of hepatitis B virus through sensitivity analysis, Educational Assessment Review, 20(4), 999--1010.","DOI":"10.46939\/J.Sci.Arts-20.4-c01"},{"key":"ref42","doi-asserted-by":"crossref","unstructured":"[42] Diethelm, K., Ford, N.J., and Freed, A.D. (2002), A predictor\u2013corrector approach for the numerical solution of fractional differential equations, Nonlinear Dynamics, 29, 3\u201322.","DOI":"10.1023\/A:1016592219341"},{"key":"ref43","doi-asserted-by":"crossref","unstructured":"[43] Sara, H. and Herrmann, R. (2014), Fractional calculus: an introduction for physicists, World Scientific, 2014.","DOI":"10.1142\/8934"},{"key":"ref44","doi-asserted-by":"crossref","unstructured":"[44] Raissi, M., Perdikaris, P., and Karniadakis, G.E. (2019), Physics-informed neural networks: a deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations, Journal of Computational Physics, 378, 686\u2013707.","DOI":"10.1016\/j.jcp.2018.10.045"}],"container-title":["Journal of Applied Nonlinear Dynamics"],"original-title":[],"language":"en","deposited":{"date-parts":[[2026,9,7]],"date-time":"2026-09-07T17:42:14Z","timestamp":1788802934000},"score":1,"resource":{"primary":{"URL":"https:\/\/lhscientificpublishing.com\/index.php\/jand\/article\/view\/2487"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,9,7]]},"references-count":44,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2027,3,1]]}},"URL":"https:\/\/doi.org\/10.5890\/jand.2027.03.005","relation":{},"ISSN":["2164-6457","2164-6473"],"issn-type":[{"value":"2164-6457","type":"print"},{"value":"2164-6473","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,9,7]]}}}