{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,28]],"date-time":"2026-07-28T22:27:43Z","timestamp":1785277663913,"version":"3.55.0"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,2,3]],"date-time":"2026-02-03T00:00:00Z","timestamp":1770076800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2026,2,3]],"date-time":"2026-02-03T00:00:00Z","timestamp":1770076800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J. Appl. Math. Comput."],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s12190-026-02770-7","type":"journal-article","created":{"date-parts":[[2026,2,3]],"date-time":"2026-02-03T14:21:53Z","timestamp":1770128513000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Neural network assisted stability exploration of a stochastic Hepatitis B model incorporating dual transmission routes and immune response"],"prefix":"10.1007","volume":"72","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0463-0360","authenticated-orcid":false,"given":"Anwarud","family":"Din","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2026,2,3]]},"reference":[{"key":"2770_CR1","unstructured":"Jinliang, W., Tian, X.: Global stability of a delay differential equation of hepatitis B virus infection with immune response. Electron. J. Differ. Equ. 2013(94), 1\u201311 (2013)"},{"key":"2770_CR2","unstructured":"Nowak, M.A., May, R.M.: Viral Dynamics. Oxford University Press, Oxford (2000)"},{"issue":"9","key":"2770_CR3","doi-asserted-by":"publisher","first-page":"4398","DOI":"10.1073\/pnas.93.9.4398","volume":"93","author":"M.A. Nowak","year":"1996","unstructured":"Nowak, M.A., Bonhoeffer, S., Hill, A.M., Boehme, R., Thomas, H.C., McDade, H.: Viral dynamics in hepatitis B virus infection. Proc. Natl. Acad. Sci. USA 93(9), 4398\u20134402 (1996)","journal-title":"Proc. Natl. Acad. Sci. USA"},{"issue":"1","key":"2770_CR4","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1137\/S0036144598335107","volume":"41","author":"A.S. Perelson","year":"1999","unstructured":"Perelson, A.S., Nelson, P.W.: Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41(1), 3\u201344 (1999)","journal-title":"SIAM Rev."},{"key":"2770_CR5","doi-asserted-by":"publisher","first-page":"281","DOI":"10.1016\/j.jmaa.2006.06.064","volume":"329","author":"X.Q. Song","year":"2007","unstructured":"Song, X.Q., Neuman, A.U.: Global stability and periodic solutions of viral dynamics. J. Math. Anal. Appl. 329, 281\u2013297 (2007)","journal-title":"J. Math. Anal. Appl."},{"key":"2770_CR6","doi-asserted-by":"publisher","first-page":"1573","DOI":"10.1216\/RMJ-2008-38-5-1573","volume":"38","author":"L.Q. Min","year":"2008","unstructured":"Min, L.Q., Su, Y.M., Kuang, Y.: Mathematical analysis of a basic virus infection model with application to HBV infection. Rocky Mt. J. Math. 38, 1573\u20131585 (2008)","journal-title":"Rocky Mt. J. Math."},{"key":"2770_CR7","doi-asserted-by":"publisher","first-page":"1221","DOI":"10.1007\/s11424-010-8467-0","volume":"23","author":"Y. Zheng","year":"2010","unstructured":"Zheng, Y., Min, L.Q., Ji, Y., Su, Y.M., Kuang, Y.: Global stability of endemic equilibrium of a virus infection model with application to HBV infection. J. Syst. Sci. Complex. 23, 1221\u20131230 (2010)","journal-title":"J. Syst. Sci. Complex."},{"key":"2770_CR8","doi-asserted-by":"publisher","first-page":"e00049","DOI":"10.7554\/eLife.00049","volume":"1","author":"H. Yan","year":"2012","unstructured":"Yan, H., Zhong, G., Xu, G., He, W., Jing, Z., Gao, Z., et al.: Sodium taurocholate cotransporting polypeptide is a functional receptor for human hepatitis B and D virus. eLife 1, e00049 (2012). https:\/\/doi.org\/10.7554\/eLife.00049","journal-title":"eLife"},{"issue":"1","key":"2770_CR9","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1172\/JCI73943","volume":"125","author":"A. Bertoletti","year":"2015","unstructured":"Bertoletti, A., Kennedy, P.T.: HBV infection: immunological dynamics and control. J. Clin. Invest. 125(1), 55\u201364 (2015)","journal-title":"J. Clin. Invest."},{"key":"2770_CR10","doi-asserted-by":"crossref","first-page":"775","DOI":"10.3389\/fmicb.2020.00775","volume":"11","author":"Y. Zhu","year":"2020","unstructured":"Zhu, Y., Yamamoto, T., Cullen, J.M., Liu, L.: Cell-to-cell transmission of hepatitis B virus: a potential mechanism for viral persistence. Front. Microbiol. 11, 775 (2020)","journal-title":"Front. Microbiol."},{"issue":"6","key":"2770_CR11","first-page":"1258","volume":"36","author":"Y. Yin","year":"2021","unstructured":"Yin, Y., He, J., Ji, C.: Cell-to-cell transmission and cccDNA maintenance: two keystones of HBV chronic infection. Virol. Sin. 36(6), 1258\u20131267 (2021)","journal-title":"Virol. Sin."},{"issue":"3","key":"2770_CR12","doi-asserted-by":"publisher","first-page":"393","DOI":"10.1016\/j.immuni.2007.08.007","volume":"27","author":"S.M. Kaech","year":"2007","unstructured":"Kaech, S.M., Wherry, E.J.: Heterogeneity and cell-fate decisions in effector and memory CD8 T cell differentiation during viral infection. Immunity 27(3), 393\u2013405 (2007)","journal-title":"Immunity"},{"issue":"3","key":"2770_CR13","doi-asserted-by":"publisher","first-page":"215","DOI":"10.1038\/nri1573","volume":"5","author":"B. Rehermann","year":"2005","unstructured":"Rehermann, B., Nascimbeni, M.: Immunology of hepatitis B virus and hepatitis C virus infection. Nat. Rev. Immunol. 5(3), 215\u2013229 (2005)","journal-title":"Nat. Rev. Immunol."},{"issue":"8","key":"2770_CR14","doi-asserted-by":"publisher","first-page":"486","DOI":"10.1038\/nri3862","volume":"15","author":"E.J. Wherry","year":"2015","unstructured":"Wherry, E.J., Kurachi, M.: Molecular and cellular insights into T cell exhaustion. Nat. Rev. Immunol. 15(8), 486\u2013499 (2015)","journal-title":"Nat. Rev. Immunol."},{"issue":"3","key":"2770_CR15","doi-asserted-by":"publisher","first-page":"399","DOI":"10.1142\/S0218339019500177","volume":"27","author":"H. Hui","year":"2019","unstructured":"Hui, H., Nie, L.-F.: Analysis of a stochastic HBV infection model with nonlinear incidence rate. J. Biol. Syst. 27(3), 399\u2013421 (2019)","journal-title":"J. Biol. Syst."},{"key":"2770_CR16","doi-asserted-by":"publisher","first-page":"110839","DOI":"10.1016\/j.chaos.2021.110839","volume":"146","author":"A. Din","year":"2021","unstructured":"Din, A., Li, Y., Yusuf, A.: Delayed hepatitis B epidemic model with stochastic analysis. Chaos Solitons Fractals 146, 110839 (2021)","journal-title":"Chaos Solitons Fractals"},{"key":"2770_CR17","doi-asserted-by":"publisher","first-page":"4759","DOI":"10.1007\/s12190-022-01721-2","volume":"68","author":"B. Karaman","year":"2022","unstructured":"Karaman, B.: The global stability investigation of a fractional-order HBV infection model. J. Appl. Math. Comput. 68, 4759\u20134775 (2022)","journal-title":"J. Appl. Math. Comput."},{"issue":"5","key":"2770_CR18","doi-asserted-by":"publisher","first-page":"318","DOI":"10.1007\/s40808-025-02468-y","volume":"11","author":"V. Singh","year":"2025","unstructured":"Singh, V., Thakur, P., Srivastava, S.K.: A nonlinear time-fractional-order epidemic model for tobacco smoking with sensitivity analysis. Model. Earth Syst. Environ. 11(5), 318 (2025)","journal-title":"Model. Earth Syst. Environ."},{"key":"2770_CR19","doi-asserted-by":"publisher","first-page":"64","DOI":"10.3389\/fphy.2020.00064","volume":"8","author":"A. Alshabanat","year":"2020","unstructured":"Alshabanat, A., Jleli, M., Kumar, S., Samet, B.: Generalization of Caputo\u2013Fabrizio fractional derivative and applications to electrical circuits. Front. Phys. 8, 64 (2020)","journal-title":"Front. Phys."},{"key":"2770_CR20","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1007\/978-981-96-3094-3_14","volume-title":"Biology and Sustainable Development Goals: applications of Mathematical Methods","author":"M. Shams","year":"2025","unstructured":"Shams, M., Kausar, N., Agarwal, P.: Artificial neural network-based single-step method for solving biomedical engineering applications. In: Biology and Sustainable Development Goals: applications of Mathematical Methods, pp. 317\u2013334. Springer Nature Singapore, Singapore (2025)"},{"key":"2770_CR21","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1007\/978-981-16-6534-9_11","volume-title":"Stability Analysis of Neural Networks","author":"G. Rajchakit","year":"2021","unstructured":"Rajchakit, G., Agarwal, P., Ramalingam, S.: Robust finite-time passivity of Markovian jump discrete-time BAM neural networks. In: Stability Analysis of Neural Networks, pp. 341\u2013371. Springer, Singapore (2021)"},{"issue":"7","key":"2770_CR22","doi-asserted-by":"publisher","first-page":"7671","DOI":"10.1002\/mma.7065","volume":"46","author":"S. Kumar","year":"2023","unstructured":"Kumar, S., Kumar, R., Momani, S., Hadid, S.: A study on fractional COVID-19 disease model using Hermite wavelets. Math. Methods Appl. Sci. 46(7), 7671\u20137687 (2023)","journal-title":"Math. Methods Appl. Sci."},{"issue":"12","key":"2770_CR23","doi-asserted-by":"publisher","first-page":"123104","DOI":"10.1063\/5.0063050","volume":"31","author":"A. Din","year":"2021","unstructured":"Din, A.: The stochastic bifurcation analysis and stochastic delayed optimal control for an epidemic model with general incidence function. Chaos 31(12), 123104 (2021)","journal-title":"Chaos"},{"issue":"5","key":"2770_CR24","doi-asserted-by":"publisher","first-page":"312","DOI":"10.7763\/IJAPM.2012.V2.121","volume":"2","author":"D.K. Saini","year":"2012","unstructured":"Saini, D.K.: Cyber defense: mathematical modeling and simulation. Int. J. Appl. Phys. Math. 2(5), 312\u2013315 (2012)","journal-title":"Int. J. Appl. Phys. Math."},{"key":"2770_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4899-7612-3","volume-title":"An Introduction to Mathematical Epidemiology","author":"M. Martcheva","year":"2015","unstructured":"Martcheva, M.: An Introduction to Mathematical Epidemiology. Springer, New York (2015)"},{"key":"2770_CR26","doi-asserted-by":"publisher","first-page":"786","DOI":"10.1109\/TIFS.2020.3023277","volume":"16","author":"K. Cao","year":"2020","unstructured":"Cao, K., Wang, B., Ding, H., Lv, L., Dong, R., Cheng, T., Gong, F.: Improving physical layer security of uplink NOMA via energy harvesting jammers. IEEE Trans. Inf. Forensics Secur. 16, 786\u2013799 (2020)","journal-title":"IEEE Trans. Inf. Forensics Secur."},{"key":"2770_CR27","doi-asserted-by":"publisher","first-page":"108256","DOI":"10.1016\/j.compbiolchem.2024.108256","volume":"113","author":"M. Farhan","year":"2024","unstructured":"Farhan, M., Ling, Z., Shah, Z., Islam, S., Alshehri, M.H., Antonescu, E.: A multi-layer neural network approach for the stability analysis of the hepatitis B model. Comput. Biol. Chem. 113, 108256 (2024)","journal-title":"Comput. Biol. Chem."},{"issue":"2","key":"2770_CR28","first-page":"278","volume":"28","author":"X. Chen","year":"2013","unstructured":"Chen, X., Min, L.Q., Sun, Q.L.: Dynamics analysis and numerical simulation of an amended HBV infection model. J. Biomath. 28(2), 278\u2013284 (2013)","journal-title":"J. Biomath."},{"key":"2770_CR29","doi-asserted-by":"publisher","first-page":"161","DOI":"10.1016\/j.nonrwa.2015.05.007","volume":"26","author":"A.M. Elaiw","year":"2015","unstructured":"Elaiw, A.M., AlShamrani, N.H.: Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal. Nonlinear Anal. Real World Appl. 26, 161\u2013190 (2015)","journal-title":"Nonlinear Anal. Real World Appl."},{"key":"2770_CR30","doi-asserted-by":"publisher","first-page":"130604","DOI":"10.1016\/j.neucom.2025.130604","volume":"562","author":"R. Sriraman","year":"2025","unstructured":"Sriraman, R., Manoj, N., Agarwal, P., Vigo-Aguiar, J., Jain, S.: Event-triggered control for exponential synchronization of reaction\u2013diffusion fractional-order Clifford-valued delayed neural networks and its application to image encryption. Neurocomputing 562, 130604 (2025)","journal-title":"Neurocomputing"},{"issue":"2","key":"2770_CR31","doi-asserted-by":"publisher","first-page":"1673","DOI":"10.1002\/num.22603","volume":"37","author":"S. Kumar","year":"2021","unstructured":"Kumar, S., Kumar, R., Samet, B., Dutta, H.: A study on fractional host\u2013parasitoid population dynamical model to describe insect species. Numer. Methods Partial Differ. Equ. 37(2), 1673\u20131692 (2021)","journal-title":"Numer. Methods Partial Differ. Equ."},{"issue":"1","key":"2770_CR32","doi-asserted-by":"publisher","first-page":"e22707","DOI":"10.1002\/num.22707","volume":"40","author":"S. Kumar","year":"2024","unstructured":"Kumar, S., Chauhan, R.P., Momani, S., Hadid, S.: Numerical investigations on COVID-19 models through singular and non-singular fractional operators. Numer. Methods Partial Differ. Equ. 40(1), e22707 (2024)","journal-title":"Numer. Methods Partial Differ. Equ."},{"issue":"1","key":"2770_CR33","doi-asserted-by":"publisher","first-page":"e22689","DOI":"10.1002\/num.22689","volume":"40","author":"B. Ghanbari","year":"2024","unstructured":"Ghanbari, B., Kumar, S.: A study on a fractional predator\u2013prey\u2013pathogen model with Mittag\u2013Leffler kernel-based operators. Numer. Methods Partial Differ. Equ. 40(1), e22689 (2024)","journal-title":"Numer. Methods Partial Differ. Equ."},{"issue":"1","key":"2770_CR34","doi-asserted-by":"publisher","first-page":"234","DOI":"10.1016\/j.chaos.2005.12.029","volume":"33","author":"C. Sun","year":"2007","unstructured":"Sun, C., Cao, Z., Lin, Y.: Analysis of stability and Hopf bifurcation for a viral infectious model with delay. Chaos Solitons Fractals 33(1), 234\u2013245 (2007)","journal-title":"Chaos Solitons Fractals"},{"issue":"1","key":"2770_CR35","doi-asserted-by":"publisher","first-page":"26","DOI":"10.1007\/s40819-023-01664-7","volume":"10","author":"S. Thakur","year":"2024","unstructured":"Thakur, S., Singh, V., Kumar, A., Singh, A.K., Srivastava, S.K.: A time-fractional-order HIV\/AIDS epidemic model with q-HATM. Int. J. Appl. Comput. Math. 10(1), 26 (2024)","journal-title":"Int. J. Appl. Comput. Math."},{"issue":"1","key":"2770_CR36","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1007\/s12190-018-01219-w","volume":"60","author":"H. Kheiri","year":"2019","unstructured":"Kheiri, H., Jafari, M.: Fractional optimal control of an HIV\/AIDS epidemic model with random testing and contact tracing. J. Appl. Math. Comput. 60(1), 387\u2013411 (2019)","journal-title":"J. Appl. Math. Comput."},{"issue":"1","key":"2770_CR37","first-page":"331","volume":"20","author":"S. Ullah","year":"2020","unstructured":"Ullah, S., Khan, M.A., Farooq, M.: Mathematical modeling of hepatitis B infection with vaccination and optimal control interventions. J. Comput. Methods Sci. Eng. 20(1), 331\u2013349 (2020)","journal-title":"J. Comput. Methods Sci. Eng."},{"key":"2770_CR38","doi-asserted-by":"publisher","first-page":"112855","DOI":"10.1016\/j.chaos.2022.112855","volume":"165","author":"Z. Yaagoub","year":"2022","unstructured":"Yaagoub, Z., Allali, K.: Fractional HBV infection model with both cell-to-cell and virus-to-cell transmissions and adaptive immunity. Chaos. Solitons Fractals 165, 112855 (2022)","journal-title":"Chaos. Solitons Fractals"},{"key":"2770_CR39","doi-asserted-by":"publisher","first-page":"121548","DOI":"10.1016\/j.physa.2019.121548","volume":"531","author":"X.-B. Zhang","year":"2019","unstructured":"Zhang, X.-B., Wang, X.-D., Huo, H.-F.: Extinction and stationary distribution of a stochastic SIRS epidemic model with standard incidence rate and partial immunity. Physica A 531, 121548 (2019)","journal-title":"Physica A"},{"key":"2770_CR40","volume-title":"Stochastic Stability of Differential Equations","author":"R. Khasminskii","year":"2011","unstructured":"Khasminskii, R.: Stochastic Stability of Differential Equations. Springer, Berlin (2011)"},{"key":"2770_CR41","doi-asserted-by":"crossref","unstructured":"El Koufi, A., Edraoui, M.: Stochastic analysis of a generalized epidemic model with two coexisting diseases. J. Appl. Math. Comput. 1\u201325 (2025)","DOI":"10.1007\/s12190-025-02447-7"},{"key":"2770_CR42","doi-asserted-by":"crossref","unstructured":"Wang, Z., Zhang, C.: Stability analysis of the Milstein method for stochastic differential equations with delay. Comput. Math. Appl. 51(9\u201310), 1445\u20131452 (2006)","DOI":"10.1016\/j.camwa.2006.01.004"}],"container-title":["Journal of Applied Mathematics and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02770-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s12190-026-02770-7","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s12190-026-02770-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,27]],"date-time":"2026-03-27T14:08:10Z","timestamp":1774620490000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s12190-026-02770-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,3]]},"references-count":42,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["2770"],"URL":"https:\/\/doi.org\/10.1007\/s12190-026-02770-7","relation":{},"ISSN":["1598-5865","1865-2085"],"issn-type":[{"value":"1598-5865","type":"print"},{"value":"1865-2085","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,3]]},"assertion":[{"value":"25 September 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 January 2026","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"6 January 2026","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"3 February 2026","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"There is no conflict of interest among the authors.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing interests"}}],"article-number":"116"}}