{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,6]],"date-time":"2026-10-06T21:24:14Z","timestamp":1791321854097,"version":"4.3.1"},"reference-count":51,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"10","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["MBE"],"published-print":{"date-parts":[[2025]]},"DOI":"10.3934\/mbe.2025100","type":"journal-article","created":{"date-parts":[[2025,8,28]],"date-time":"2025-08-28T08:43:43Z","timestamp":1756370623000},"page":"2720-2760","source":"Crossref","is-referenced-by-count":4,"title":["Backward bifurcation and periodic dynamics in a tuberculosis model with integrated control strategies"],"prefix":"10.3934","volume":"22","author":[{"given":"Dipo","family":"Aldila","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"name":"Department of Mathematics, Faculty of Mathematics and Natural Sciences, Universitas Indonesia, Depok 16424, Indonesia","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Chidozie Williams","family":"Chukwu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Eka D. A.","family":"Ginting","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"F.","family":"Fatmawati","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Faishal Farrel","family":"Herdicho","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mohammad Ivan","family":"Azis","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"S.","family":"Sutrisno","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"name":"Innovative Mathematics and Predictive Analytics for Complex System and Technology Laboratory (IMPACT Lab), Universitas Indonesia, Depok 16424, Indonesia","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"name":"Department of Mathematical Sciences, Georgia Southern University, 65 Georgia Ave., P.O. Box 8093, Statesboro, Georgia 30460, USA","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"name":"Department of Mathematics, Faculty of Science and Technology, Universitas Airlangga, Surabaya, Indonesia","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"name":"Department of Mathematics, Hasanuddin University, Makassar, Indonesia","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"name":"Department of Mathematics, Diponegoro University, Tembalang, Semarang 50275, Indonesia","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"2321","reference":[{"key":"key-10.3934\/mbe.2025100-1","doi-asserted-by":"publisher","unstructured":"D. Aldila, B. L. Fardian, C. W. Chukwu, M. H. N. Aziz, P. Z. Kamalia, Improving tuberculosis control: Assessing the value of medical masks and case detection\u2014a multi-country study with cost-effectiveness analysis, R. Soc. Open Sci.<\/i>, 11<\/b> (2024), 231715. https:\/\/doi.org\/10.1098\/rsos.231715","DOI":"10.1098\/rsos.231715"},{"key":"key-10.3934\/mbe.2025100-2","unstructured":"Centers for Disease Control and Prevention, Signs and Symptoms of Tuberculosis, 2024, [Online; accessed 10-May-2024]. Available from: https:\/\/www.cdc.gov\/tb\/signs-symptoms\/?CDC_AAref_Val = https:\/\/www.cdc.gov\/tb\/topic\/basics\/signsandsymptoms.htm<\/ext-link>."},{"key":"key-10.3934\/mbe.2025100-3","doi-asserted-by":"publisher","unstructured":"C. W. Chukwu, E. Bonyah, M. L. Juga, F. Fatmawati, On mathematical modeling of fractional-order stochastic for tuberculosis transmission dynamics, Results Control Optim.<\/i>, 11<\/b> (2023), 100238. https:\/\/doi.org\/10.1016\/j.rico.2023.100238","DOI":"10.1016\/j.rico.2023.100238"},{"key":"key-10.3934\/mbe.2025100-4","unstructured":"WHO, Global Tuberculosis Report 2022, 2022, [Online; accessed 22-May-2024]. Available from: https:\/\/cdn.who.int\/media\/docs\/default-source\/hq-tuberculosis\/global-tuberculosis-report-2022\/global-tb-report-2022-factsheet.pdf?sfvrsn = 88f8d76_8 & download = true<\/ext-link>."},{"key":"key-10.3934\/mbe.2025100-5","unstructured":"WHO, Global Tuberculosis Report 2023, 2023, [Online; accessed 25-May-2024]. Available from: https:\/\/www.who.int\/publications\/i\/item\/9789240061729<\/ext-link>."},{"key":"key-10.3934\/mbe.2025100-6","unstructured":"WHO, Global Tuberculosis Report 2020, 2020, [Online; accessed 15-May-2024]. Available from: https:\/\/www.who.int\/publications\/i\/item\/9789240013131<\/ext-link>."},{"key":"key-10.3934\/mbe.2025100-7","unstructured":"WHO, Global Tuberculosis Report 2019, 2019, [Online; accessed 2-May-2024]. Available from: https:\/\/www.who.int\/publications\/i\/item\/9789241565714<\/ext-link>."},{"key":"key-10.3934\/mbe.2025100-8","unstructured":"S. D. Lawn, A. I. Zumla, Tuberculosis, Lancet<\/i>, 378<\/b> (2011), 57\u201372. https:\/\/doi.org\/10.1016\/S0140-6736(10)62173-3<\/ext-link>"},{"key":"key-10.3934\/mbe.2025100-9","doi-asserted-by":"publisher","unstructured":"B. Song, C. Castillo-Chavez, J. P. Aparicio, Tuberculosis models with fast and slow dynamics: The role of close and casual contacts, Math. Biosci.<\/i>, 180<\/b> (2002), 187\u2013205. https:\/\/doi.org\/10.1016\/S0025-5564(02)00112-8","DOI":"10.1016\/S0025-5564(02)00112-8"},{"key":"key-10.3934\/mbe.2025100-10","doi-asserted-by":"publisher","unstructured":"R. Xu, J. Yang, X. Tian, J. Lin, Global dynamics of a tuberculosis model with fast and slow progression and age-dependent latency and infection, J. Biol. Dyn.<\/i>, 13<\/b> (2019), 675\u2013705. https:\/\/doi.org\/10.1080\/17513758.2019.1683628","DOI":"10.1080\/17513758.2019.1683628"},{"key":"key-10.3934\/mbe.2025100-11","unstructured":"Stop TB Partnership, Stop TB Partnership: Global Plan to End TB 2023\u20132030, [Online; accessed 22-May-2024]. Available from: https:\/\/www.stoptb.org\/news\/launched-global-plan-end-tb-2023-2030<\/ext-link>."},{"key":"key-10.3934\/mbe.2025100-12","unstructured":"Centers for Disease Control and Prevention. TB Disease Treatment: 6- or 9-Month Ripe tb Treatment Regimen, 2023, [Online; accessed 20-May-2024]. Available from: https:\/\/www.cdc.gov\/tb\/webcourses\/tb101\/page16413.html<\/ext-link>."},{"key":"key-10.3934\/mbe.2025100-13","unstructured":"A. S. Weichel, Antimycobacterial therapy for tuberculosis, Am. Fam. Physician<\/i>, 61<\/b> (2000), 861\u2013862. Available from: https:\/\/www.aafp.org\/pubs\/afp\/issues\/2000\/0201\/p861.html<\/ext-link>."},{"key":"key-10.3934\/mbe.2025100-14","unstructured":"International Union Against Tuberculosis and Lung Disease, Guide for the Clinical and Operational Management of Drug-Resistant Tuberculosis, 2013, [Online; accessed 23-July-2025]. Available from: https:\/\/theunion.org\/sites\/default\/files\/2020-08\/mdr-tbguide_6-19-13_web.pdf<\/ext-link>."},{"key":"key-10.3934\/mbe.2025100-15","doi-asserted-by":"publisher","unstructured":"M. Majumder, P. K. Tiwari, S. Pal, Impact of saturated treatments on HIV-TB dual epidemic as a consequence of COVID-19: Optimal control with awareness and treatment, Nonlinear Dyn.<\/i>, 109<\/b> (2022), 143\u2013176. https:\/\/doi.org\/10.1007\/s11071-022-07395-6","DOI":"10.1007\/s11071-022-07395-6"},{"key":"key-10.3934\/mbe.2025100-16","doi-asserted-by":"publisher","unstructured":"S. Athithan, M. Ghosh, Mathematical modelling of TB with the effects of case detection and treatment, Int. J. Dyn. Control<\/i>, 1<\/b> (2013), 223\u2013230. https:\/\/doi.org\/10.1007\/s40435-013-0020-2","DOI":"10.1007\/s40435-013-0020-2"},{"key":"key-10.3934\/mbe.2025100-17","doi-asserted-by":"publisher","unstructured":"D. Aldila, J. P. Ch\u00e1vez, K. P. Wijaya, N. C. Ganegoda, G. M. Simorangkir, H. Tasman, et al., A tuberculosis epidemic model as a proxy for the assessment of the novel M72\/AS01<\/i>E<\/i><\/sub> vaccine, Commun. Nonlinear Sci. Numer. Simul.<\/i>, 120<\/b> (2023), 107162. https:\/\/doi.org\/10.1016\/j.cnsns.2023.107162","DOI":"10.1016\/j.cnsns.2023.107162"},{"key":"key-10.3934\/mbe.2025100-18","doi-asserted-by":"publisher","unstructured":"Y. D. Zhang, H. F. Huo, H. Xiang, Dynamics of tuberculosis with fast and slow progression and media coverage, Math. Biosci. Eng.<\/i>, 16<\/b> (2019), 1150\u20131170. https:\/\/doi.org\/10.3934\/mbe.2019055","DOI":"10.3934\/mbe.2019055"},{"key":"key-10.3934\/mbe.2025100-19","doi-asserted-by":"publisher","unstructured":"K. K. Avilov, A. A. Romanyukha, E. M. Belilovsky, S. E. Borisov, Mathematical modelling of the progression of active tuberculosis: Insights from fluorography data, Infect. Dis. Modell.<\/i>, 7<\/b> (2022), 374\u2013386. https:\/\/doi.org\/10.1016\/j.idm.2022.06.007","DOI":"10.1016\/j.idm.2022.06.007"},{"key":"key-10.3934\/mbe.2025100-20","doi-asserted-by":"publisher","unstructured":"T. Sumner, R. G. White, The predicted impact of tuberculosis preventive therapy: The importance of disease progression assumptions, BMC Infect. Dis.<\/i>, 20<\/b> (2020), 880. https:\/\/doi.org\/10.1186\/s12879-020-05592-5","DOI":"10.1186\/s12879-020-05592-5"},{"key":"key-10.3934\/mbe.2025100-21","doi-asserted-by":"publisher","unstructured":"K. C. Chong, C. C. Leung, W. W. Yew, B. C. Y. Zee, G. C. H. Tam, M. H. Wang, et al., Mathematical modelling of the impact of treating latent tuberculosis infection in the elderly in a city with intermediate tuberculosis burden, Sci. Rep.<\/i>, 9<\/b> (2019), 4869. https:\/\/doi.org\/10.1038\/s41598-019-41256-4","DOI":"10.1038\/s41598-019-41256-4"},{"key":"key-10.3934\/mbe.2025100-22","doi-asserted-by":"publisher","unstructured":"S. Bowong, J. J. Tewa, Mathematical analysis of a tuberculosis model with differential infectivity, Commun. Nonlinear Sci. Numer. Simul.<\/i>, 14<\/b> (2009), 4010\u20134021. https:\/\/doi.org\/10.1016\/j.cnsns.2009.02.017","DOI":"10.1016\/j.cnsns.2009.02.017"},{"key":"key-10.3934\/mbe.2025100-23","doi-asserted-by":"publisher","unstructured":"L. Liu, Y. Wang, A mathematical study of a tb model with treatment interruptions and two latent periods, Comput. Math. Methods Med.<\/i>, 2014<\/b> (2014), 932186. https:\/\/doi.org\/10.1155\/2014\/932186","DOI":"10.1155\/2014\/932186"},{"key":"key-10.3934\/mbe.2025100-24","doi-asserted-by":"publisher","unstructured":"D. Okuonghae, B. O. Ikhimwin, Dynamics of a mathematical model for tuberculosis with variability in susceptibility and disease progressions due to difference in awareness level, Front. Microbiol.<\/i>, 6<\/b> (2016), 1530. https:\/\/doi.org\/10.3389\/fmicb.2015.01530","DOI":"10.3389\/fmicb.2015.01530"},{"key":"key-10.3934\/mbe.2025100-25","doi-asserted-by":"publisher","unstructured":"D. C. Maulana, M. I. Utoyo, U. D. Purwati, C. W. Chukw, Parameter estimation and analysis on sis-seis types model of tuberculosis transmission in east java indonesia, Commun. Math. Biol. Neurosci.<\/i>, 2022<\/b> (2022), 114. https:\/\/doi.org\/10.28919\/cmbn\/7739","DOI":"10.28919\/cmbn\/7739"},{"key":"key-10.3934\/mbe.2025100-26","doi-asserted-by":"crossref","unstructured":"E. P. Hafidh, N. Aulida, B. D. Handari, D. Aldila, Optimal control problem from tuberculosis and multidrug resistant tuberculosis transmission model, in AIP Conference Proceedings 2018<\/i>, AIP Publishing, 2023<\/b> (2018), 020223. https:\/\/doi.org\/10.1063\/1.5064220<\/ext-link>","DOI":"10.1063\/1.5064220"},{"key":"key-10.3934\/mbe.2025100-27","doi-asserted-by":"publisher","unstructured":"O. Diekmann, J. A. P. Heesterbeek, M. G. Roberts, The construction of next-generation matrices for compartmental epidemic models, J. R. Soc. Interface<\/i>, 7<\/b> (2010), 873\u2013885. https:\/\/doi.org\/10.1098\/rsif.2009.0386","DOI":"10.1098\/rsif.2009.0386"},{"key":"key-10.3934\/mbe.2025100-28","doi-asserted-by":"publisher","unstructured":"C. Castillo-Chavez, B. Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng.<\/i>, 1<\/b> (2004), 361\u2013404. https:\/\/doi.org\/10.3934\/mbe.2004.1.361","DOI":"10.3934\/mbe.2004.1.361"},{"key":"key-10.3934\/mbe.2025100-29","doi-asserted-by":"publisher","unstructured":"D. Aldila, A. H. Hassan, M. H. N. Aziz, P. Z. Kamalia, An analytical transmission model for evaluating pneumonia vaccination and control strategies, Healthcare Anal.<\/i>, 7<\/b> (2025), 100394. https:\/\/doi.org\/10.1016\/j.health.2025.100394","DOI":"10.1016\/j.health.2025.100394"},{"key":"key-10.3934\/mbe.2025100-30","doi-asserted-by":"publisher","unstructured":"D. Aldila, A. H. Hassan, C. W. Chukwu, S. Y. Tchoumi, M. H. N. Aziz, Evaluating vaccination and quarantine for measles intervention strategy in jakarta, indonesia through mathematical modeling, Partial Differ. Equations Appl. Math.<\/i>, 14<\/b> (2025), 101191. https:\/\/doi.org\/10.1016\/j.padiff.2025.101191","DOI":"10.1016\/j.padiff.2025.101191"},{"key":"key-10.3934\/mbe.2025100-31","doi-asserted-by":"publisher","unstructured":"F. A. Oguntolu, O. J. Peter, D. Aldila, G. B. Balogun, O. P. Ogunmola, B. I. Omede, Mathematical analysis of the transmission dynamics of hepatitis B virus, Braz. J. Phys.<\/i>, 55<\/b> (2025), 160. https:\/\/doi.org\/10.1007\/s13538-025-01772-3","DOI":"10.1007\/s13538-025-01772-3"},{"key":"key-10.3934\/mbe.2025100-32","doi-asserted-by":"publisher","unstructured":"O. J. Peter, D. Aldila, T. A. Ayoola, G. B. Balogun, F. A. Oguntolu, Modeling tuberculosis dynamics with vaccination and treatment strategies, Sci. Afr.<\/i>, 28<\/b> (2025), e02647. https:\/\/doi.org\/10.1016\/j.sciaf.2025.e02647","DOI":"10.1016\/j.sciaf.2025.e02647"},{"key":"key-10.3934\/mbe.2025100-33","doi-asserted-by":"publisher","unstructured":"D. Aldila, Change in stability direction induced by temporal interventions: A case study of a tuberculosis transmission model with relapse and reinfection, Front. Appl. Math. Stat.<\/i>, 11<\/b> (2025), 1541981. https:\/\/doi.org\/10.3389\/fams.2025.1541981","DOI":"10.3389\/fams.2025.1541981"},{"key":"key-10.3934\/mbe.2025100-34","doi-asserted-by":"publisher","unstructured":"H. A. Fatahillah, D. Aldila, Forward and backward bifurcation analysis from an imperfect vaccine efficacy model with saturated treatment and saturated infection, Jambura J. Biomath.<\/i>, 5<\/b> (2024), 132\u2013143. https:\/\/doi.org\/10.37905\/jjbm.v5i2.28810","DOI":"10.37905\/jjbm.v5i2.28810"},{"key":"key-10.3934\/mbe.2025100-35","doi-asserted-by":"publisher","unstructured":"D. Aldila, J. P. Ch\u00e1vez, C. W. Chukwu, A. Y. Fathiyah, J. W. Puspita, K. A. D. Setio, et al., Unraveling dengue dynamics with data calibration from palu and jakarta: Optimizing active surveillance and fogging interventions, Chaos, Solitons Fractals<\/i>, 189<\/b> (2024), 115729. https:\/\/doi.org\/10.1016\/j.chaos.2024.115729","DOI":"10.1016\/j.chaos.2024.115729"},{"key":"key-10.3934\/mbe.2025100-36","doi-asserted-by":"publisher","unstructured":"W. M. Liu, Criterion of hopf bifurcations without using eigenvalues, J. Math. Anal. Appl.<\/i>, 182<\/b> (1994), 250\u2013256. https:\/\/doi.org\/10.1006\/jmaa.1994.1079","DOI":"10.1006\/jmaa.1994.1079"},{"key":"key-10.3934\/mbe.2025100-37","unstructured":"World Bank, Incidence of Tuberculosis (per 100,000 people), 2024, [Online; accessed 08-March-2024]. Available from: https:\/\/data.worldbank.org\/indicator\/SH.TBS.INCD<\/ext-link>."},{"key":"key-10.3934\/mbe.2025100-38","doi-asserted-by":"publisher","unstructured":"S. Marino, I. B. Hogue, C. J. Ray, D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology, J. Theor. Biol.<\/i>, 254<\/b> (2008), 178\u2013196. https:\/\/doi.org\/10.1016\/j.jtbi.2008.04.011","DOI":"10.1016\/j.jtbi.2008.04.011"},{"key":"key-10.3934\/mbe.2025100-39","doi-asserted-by":"publisher","unstructured":"H. Tasman, D. Aldila, P. A. Dumbela, M. Z. Ndii, F. Fatmawati, F. F. Herdicho, et al., Assessing the impact of relapse, reinfection and recrudescence on malaria eradication policy: A bifurcation and optimal control analysis, Trop. Med. Infect. Dis.<\/i>, 7<\/b> (2022), 263. https:\/\/doi.org\/10.3390\/tropicalmed7100263","DOI":"10.3390\/tropicalmed7100263"},{"key":"key-10.3934\/mbe.2025100-40","doi-asserted-by":"publisher","unstructured":"B. D. Handari, R. A. Ramadhani, C. W. Chukwu, S. H. A. Khoshnaw, D. Aldila, An optimal control model to understand the potential impact of the new vaccine and transmission-blocking drugs for malaria: A case study in papua and west papua, Indonesia, Vaccines<\/i>, 10<\/b> (2022), 1174. https:\/\/doi.org\/10.3390\/vaccines10081174","DOI":"10.3390\/vaccines10081174"},{"key":"key-10.3934\/mbe.2025100-41","doi-asserted-by":"publisher","unstructured":"F. Fatmawati, C. W. Chukwu, R. T. Alqahtani, C. Alfiniyah, F. F. Herdicho, Tasmi, A Pontryagin's maximum principle and optimal control model with cost-effectiveness analysis of the COVID-19 epidemic, Decis. Anal. J.<\/i>, 8<\/b> (2023), 100273. https:\/\/doi.org\/10.1016\/j.dajour.2023.100273","DOI":"10.1016\/j.dajour.2023.100273"},{"key":"key-10.3934\/mbe.2025100-42","doi-asserted-by":"publisher","unstructured":"F. Fatmawati, F. F. Herdicho, W. Windarto, C. W. Chukwu, H. Tasman, An optimal control of malaria transmission model with mosquito seasonal factor, Results Phys.<\/i>, 25<\/b> (2021), 104238. https:\/\/doi.org\/10.1016\/j.rinp.2021.104238","DOI":"10.1016\/j.rinp.2021.104238"},{"key":"key-10.3934\/mbe.2025100-43","doi-asserted-by":"publisher","unstructured":"D. Aldila, N. Awdinda, F. Fatmawati, F. F. Herdicho, M. Z. Ndii, C. W. Chukwu, Optimal control of pneumonia transmission model with seasonal factor: Learning from jakarta incidence data, Heliyon<\/i>, 9<\/b> (2023), e18096. https:\/\/doi.org\/10.1016\/j.heliyon.2023.e18096","DOI":"10.1016\/j.heliyon.2023.e18096"},{"key":"key-10.3934\/mbe.2025100-44","doi-asserted-by":"publisher","unstructured":"C. W. Chukwu, Z. Chazuka, S. Safdar, I. H. Febriana, D. Aldila, Assessing Syphilis transmission among msm population incorporating low and high-risk infection: A modeling study, Comput. Appl. Math.<\/i>, 43<\/b> (2024), 205. https:\/\/doi.org\/10.1007\/s40314-024-02669-8","DOI":"10.1007\/s40314-024-02669-8"},{"key":"key-10.3934\/mbe.2025100-45","doi-asserted-by":"publisher","unstructured":"M. McAsey, L. Mou, W. Han, Convergence of the forward-backward sweep method in optimal control, Comput. Optim. Appl.<\/i>, 53<\/b> (2012) 207\u2013226. https:\/\/doi.org\/10.1007\/s10589-011-9454-7","DOI":"10.1007\/s10589-011-9454-7"},{"key":"key-10.3934\/mbe.2025100-46","doi-asserted-by":"publisher","unstructured":"D. Aldila, R. P. Dhanendra, S. H. A. Khoshnaw, J. W. Puspita, P. Z. Kamalia, M. Shahzad, Understanding HIV\/AIDS dynamics: Insights from CD4+T cells, antiretroviral treatment, and country-specific analysis, Front. Public Health<\/i>, 12<\/b> (2024), 1324858. https:\/\/doi.org\/10.3389\/fpubh.2024.1324858","DOI":"10.3389\/fpubh.2024.1324858"},{"key":"key-10.3934\/mbe.2025100-47","doi-asserted-by":"publisher","unstructured":"I. H. Febiriana, A. H. Hassan, D. Aldila, Enhancing Malaria control strategy: Optimal control and cost-effectiveness analysis on the impact of vector bias on the efficacy of mosquito repellent and hospitalization, J. Appl. Math.<\/i>, 2024<\/b> (2024), 9943698. https:\/\/doi.org\/10.1155\/2024\/9943698","DOI":"10.1155\/2024\/9943698"},{"key":"key-10.3934\/mbe.2025100-48","doi-asserted-by":"publisher","unstructured":"C. K. Mahadhika, D. Aldila, A deterministic transmission model for analytics-driven optimization of COVID-19 post-pandemic vaccination and quarantine strategies, Math. Biosci. Eng.<\/i>, 21<\/b> (2024), 4956\u20134988. https:\/\/doi.org\/10.3934\/mbe.2024219","DOI":"10.3934\/mbe.2024219"},{"key":"key-10.3934\/mbe.2025100-49","doi-asserted-by":"publisher","unstructured":"A. H. Hassan, D. Aldila, M. H. N. Aziz, Optimal control and stability analysis of monkeypox transmission dynamics with the impact of contaminated surfaces, Front. Appl. Math. Stat.<\/i>, 10<\/b> (2024), 1372579. https:\/\/doi.org\/10.3389\/fams.2024.1372579","DOI":"10.3389\/fams.2024.1372579"},{"key":"key-10.3934\/mbe.2025100-50","doi-asserted-by":"publisher","unstructured":"W. Li, Y. Wang, J. Cao, M. Abdel-Aty, Dynamics and backward bifurcations of sei tuberculosis models in homogeneous and heterogeneous populations, J. Math. Anal. Appl.<\/i>, 543<\/b> (2025), 128924. https:\/\/doi.org\/10.1016\/j.jmaa.2024.128924","DOI":"10.1016\/j.jmaa.2024.128924"},{"key":"key-10.3934\/mbe.2025100-51","doi-asserted-by":"publisher","unstructured":"R. Pastor-Satorras, C. Castellano, P. V. Mieghem, A. Vespignani, Epidemic processes in complex networks, Rev. Mod. Phys.<\/i>, 87<\/b> (2015), 925\u2013979. https:\/\/doi.org\/10.1103\/RevModPhys.87.925","DOI":"10.1103\/RevModPhys.87.925"}],"container-title":["Mathematical Biosciences and Engineering"],"original-title":[],"link":[{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/mbe.2025100?viewType=html","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,8,28]],"date-time":"2025-08-28T08:43:46Z","timestamp":1756370626000},"score":1,"resource":{"primary":{"URL":"http:\/\/www.aimspress.com\/article\/doi\/10.3934\/mbe.2025100"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025]]},"references-count":51,"journal-issue":{"issue":"10","published-print":{"date-parts":[[2025]]}},"URL":"https:\/\/doi.org\/10.3934\/mbe.2025100","relation":{},"ISSN":["1551-0018"],"issn-type":[{"value":"1551-0018","type":"print"}],"subject":[],"published":{"date-parts":[[2025]]}}}