{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,15]],"date-time":"2026-05-15T03:17:43Z","timestamp":1778815063969,"version":"3.51.4"},"reference-count":45,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,13]],"date-time":"2022-12-13T00:00:00Z","timestamp":1670889600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Ministry of Higher Education (MoHE)","award":["FRGS\/1\/2021\/STG06\/USM\/02\/9"],"award-info":[{"award-number":["FRGS\/1\/2021\/STG06\/USM\/02\/9"]}]},{"name":"Fundamental Research Grant Scheme (FRGS)","award":["FRGS\/1\/2021\/STG06\/USM\/02\/9"],"award-info":[{"award-number":["FRGS\/1\/2021\/STG06\/USM\/02\/9"]}]},{"name":"Research Creativity and Management Office (RCMO), Universiti Sains Malaysia","award":["FRGS\/1\/2021\/STG06\/USM\/02\/9"],"award-info":[{"award-number":["FRGS\/1\/2021\/STG06\/USM\/02\/9"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Tuberculosis (TB), caused by Mycobacterium tuberculosis is one of the treacherous infectious diseases of global concern. In this paper, we consider a deterministic model of TB infection with the public health education and hospital treatment impact. The effective reproductive number, Rph, that measures the potential spread of TB is presented by employing the next generation matrix approach. We investigate local and global stability of the TB-free equilibrium point, endemic equilibrium point, and sensitivity analysis. The analyses of the proposed model show that the model undergoes the phenomenon of backward bifurcation when the effective reproduction number (Rph) is less than one, where two stable equilibria, namely, the DFE and an EEP coexist. Further, we compute the sensitivity of the impact of each parameter on the effective reproductive number of the model by employing a normalized sensitivity index formula. Numerical simulation of the proposed model was conducted using Maple 2016 and MatLab R2020b software and compared with the theoretical results for illustration purposes. The investigation results can be useful in providing information to policy makers and public health authorities in mitigating the spread of TB infection by public health education and hospital treatment.<\/jats:p>","DOI":"10.3390\/axioms11120723","type":"journal-article","created":{"date-parts":[[2022,12,13]],"date-time":"2022-12-13T03:32:32Z","timestamp":1670902352000},"page":"723","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":19,"title":["Dynamical Behaviour of a Modified Tuberculosis Model with Impact of Public Health Education and Hospital Treatment"],"prefix":"10.3390","volume":"11","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6164-414X","authenticated-orcid":false,"given":"Fatima","family":"Sulayman","sequence":"first","affiliation":[{"name":"School of Mathematical Sciences, Universiti Sains Malaysia (USM), Penang 11800, Malaysia"},{"name":"Department of Mathematical Sciences, Ibrahim Badamasi Babangida University, Lapai 911101, Nigeria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5215-9617","authenticated-orcid":false,"given":"Farah Aini","family":"Abdullah","sequence":"additional","affiliation":[{"name":"School of Mathematical Sciences, Universiti Sains Malaysia (USM), Penang 11800, Malaysia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,13]]},"reference":[{"key":"ref_1","unstructured":"World Health Organization (2019). Global Tuberculosis Report, WHO."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"181","DOI":"10.1016\/j.matcom.2019.03.012","article-title":"Modeling and analysis of tuberculosis (TB) in Khyber Pakhtunkhwa, Pakistan","volume":"165","author":"Ullah","year":"2019","journal-title":"Math. Comput. Simul."},{"key":"ref_3","first-page":"149","article-title":"Tuberculosis: A global overview of the situation today","volume":"70","author":"Sudre","year":"1992","journal-title":"Bull. World Health Organ."},{"key":"ref_4","first-page":"213","article-title":"Global tuberculosis incidence and mortality during","volume":"72","author":"Dolin","year":"1994","journal-title":"Bull. World Health Organ."},{"key":"ref_5","unstructured":"World Health Organization (2020). Global Tuberculosis Report 2020, WHO."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"361","DOI":"10.3934\/mbe.2004.1.361","article-title":"Dynamical models of tuberculosis and their applications","volume":"1","author":"Song","year":"2004","journal-title":"Math. Biosci. Eng."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"642","DOI":"10.1186\/s13662-020-03091-0","article-title":"Analysis and simulation of a mathematical model of tuberculosis transmission in Democratic Republic of the Congo","volume":"2020","year":"2020","journal-title":"Adv. Differ. Equ."},{"key":"ref_8","unstructured":"Adebiyi, A.O. (2016). Mathematical Modeling of the Population Dynamics of Tuberculosis. [Master\u2019s Thesis, University of the Western Cape]."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"71","DOI":"10.1177\/135910539900400101","article-title":"Tuberculosis and stigma: Predictors of prejudice against people with tuberculosis","volume":"4","author":"Jaramillo","year":"1999","journal-title":"J. Health Psychol."},{"key":"ref_10","unstructured":"Daniel, T.M., Bates, J.H., and Downes, K.A. (1994). History of tuberculosis. Tuberculosis: Pathogenesis, Protection, and Control, Wiley Online Library."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1687814019854835","DOI":"10.1177\/1687814019854835","article-title":"Modeling the transmission dynamics of tuberculosis in Khyber Pakhtunkhwa Pakistan","volume":"11","author":"Khan","year":"2019","journal-title":"Adv. Mech. Eng."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"52","DOI":"10.1016\/j.physa.2018.01.014","article-title":"Dynamics of tuberculosis transmission with exogenous reinfections and endogenous reactivation","volume":"497","author":"Khajanchi","year":"2018","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"159","DOI":"10.1016\/j.jtbi.2014.10.019","article-title":"Mathematical modeling of tuberculosis data of China","volume":"365","author":"Zhang","year":"2015","journal-title":"J. Theor. Biol."},{"key":"ref_14","unstructured":"Bar, B. (1922). Tuberculose et grossesse. Bull. Acad. M\u00e9d., 219."},{"key":"ref_15","unstructured":"World Health Organization (2020). WHO Consolidated Guidelines on Tuberculosis: Tuberculosis Preventive Treatment, WHO."},{"key":"ref_16","first-page":"245","article-title":"Modelling and stability analysis for a tuberculosis model with healthy education and treatment","volume":"32","author":"Zhou","year":"2015","journal-title":"Comput. Appl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"164","DOI":"10.1080\/17513758.2015.1115562","article-title":"Modelling effects of public health educational campaigns on drinking dynamics","volume":"10","author":"Xiang","year":"2016","journal-title":"J. Biol. Dyn."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"228","DOI":"10.1016\/j.mbs.2005.03.006","article-title":"Effects of behavioural changes in a smallpox attack model","volume":"195","author":"Hethcote","year":"2005","journal-title":"Math. Biosci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"243","DOI":"10.1515\/ijnsns-2016-0084","article-title":"Modeling the effects of health education and early therapy on tuberculosis transmission dynamics","volume":"20","author":"Xiang","year":"2019","journal-title":"Int. J. Nonlinear Sci. Numer."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"9474","DOI":"10.1016\/j.apm.2016.06.029","article-title":"Modelling effects of treatment at home on tuberculosis transmission dynamics","volume":"40","author":"Huo","year":"2016","journal-title":"Appl. Math. Model."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2349","DOI":"10.1002\/oca.2658","article-title":"Optimal control analysis of a tuberculosis model with exogenous reinfection and incomplete treatment","volume":"41","author":"Abimbade","year":"2020","journal-title":"Optim. Control Appl. Methods"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Ojo, M.M., Peter, O.J., Goufo, E.F.D., Panigoro, H.S., and Oguntolu, F.A. (2022). Mathematical model for control of tuberculosis epidemiology. J. Appl. Math. Comput., 1\u201319.","DOI":"10.1007\/s12190-022-01734-x"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"129","DOI":"10.1007\/s12190-018-1172-1","article-title":"Analysis of a mathematical model for tuberculosis with diagnosis","volume":"59","author":"Egonmwan","year":"2019","journal-title":"J. Appl. Math Comput."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"55","DOI":"10.17512\/jamcm.2022.1.05","article-title":"Mathematical modeling and analysis of mycobacterium tuberculosis transmission in humans with hospitalization and reinfection","volume":"21","author":"Mustapha","year":"2022","journal-title":"J. Appl. Math. Comput. Mech."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"433","DOI":"10.1007\/s10852-012-9210-8","article-title":"Global dynamics of a tuberculosis epidemic model and the influence of backward bifurcation","volume":"11","author":"Kar","year":"2012","journal-title":"J. Math. Model. Algorithms"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.jtbi.2010.09.044","article-title":"Analysis of a mathematical model for tuberculosis: What could be done to increase case detection","volume":"269","author":"Okuonghae","year":"2011","journal-title":"J. Theor. Biol."},{"key":"ref_27","doi-asserted-by":"crossref","unstructured":"Tang, H., Li, M., Yan, X., Lu, Z., and Jia, Z. (2021). Modeling the dynamics of drug spreading in China. Int. J. Environ. Res. Public Health, 18.","DOI":"10.3390\/ijerph18010288"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Olaniyi, S., Falowo, O.D., Okosun, K.O., Mukamuri, M., Obabiyi, O.S., and Adepoju, O.A. Effect of saturated treatment on malaria spread with optimal intervention. Alex. Eng. J., 2022. in press.","DOI":"10.1016\/j.aej.2022.09.024"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"938","DOI":"10.1140\/epjp\/s13360-020-00954-z","article-title":"Mathematical modelling and optimal cost-effective control of COVID-19 transmission dynamics","volume":"135","author":"Olaniyi","year":"2020","journal-title":"Eur. Phys. J. Plus."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"240","DOI":"10.1007\/s002850050051","article-title":"Analysis of an SEIRS epidemic model with two delays","volume":"35","author":"Cooke","year":"1996","journal-title":"J. Math. Biol."},{"key":"ref_31","first-page":"135","article-title":"On the stability analysis of a mathematical model of Lassa fever disease dynamics","volume":"2","author":"Obasi","year":"2019","journal-title":"J. Nig. Soc. Math. Biol."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1140\/epjp\/s13360-022-03201-9","article-title":"Nonlinear dynamics of a SIRI model incorporating the impact of information and saturated treatment with optimal control","volume":"137","author":"Srivastava","year":"2022","journal-title":"Eur. Phys. J. Plus."},{"key":"ref_33","unstructured":"Rota, G.C., and Birkhoff, G. (1969). Ordinary Differential Equations, Blaisdell Publishing Company."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"599","DOI":"10.1137\/S0036144500371907","article-title":"The mathematics of infectious diseases","volume":"42","author":"Hethcote","year":"2000","journal-title":"SIAM Rev."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/S0025-5564(02)00108-6","article-title":"Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission","volume":"180","author":"Watmough","year":"2002","journal-title":"Math. Biosci."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Falowo, O.D., Olaniyi, S., and Oladipo, A.T. (2022). Optimal control assessment of Rift Valley fever model with vaccination and environmental sanitation in the presence of treatment delay. Model. Earth Syst. Environ., 1\u201315.","DOI":"10.1007\/s40808-022-01508-1"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Keeling, M.J., and Rohani, P. (2011). Temporally forced models. Modeling Infectious Diseases in Humans and Animals, Princeton University Press.","DOI":"10.2307\/j.ctvcm4gk0"},{"key":"ref_38","first-page":"229","article-title":"On the computation of R0 and its role on","volume":"Volume 1","author":"Feng","year":"2002","journal-title":"Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction"},{"key":"ref_39","unstructured":"Islam, M.S., Ira, J.I., Kabir, K.A., and Kamrujjaman, M. (2020). COVID-19 Epidemic compartments model and Bangladesh. Preprint, 10."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"969","DOI":"10.18576\/amis\/120510","article-title":"Dynamics of Zika virus model with nonlinear incidence and optimal control strategies","volume":"12","author":"Olaniyi","year":"2018","journal-title":"Appl. Math. Inf. Sci."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"1272","DOI":"10.1007\/s11538-008-9299-0","article-title":"Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model","volume":"70","author":"Chitnis","year":"2008","journal-title":"Bull. Math. Biol."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"187","DOI":"10.1016\/S0025-5564(02)00112-8","article-title":"Tuberculosis models with fast and slow dynamics: The role of close and casual contacts","volume":"180","author":"Song","year":"2002","journal-title":"Math. Biosci."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"2473","DOI":"10.1098\/rspb.2011.2712","article-title":"How host heterogeneity governs tuberculosis reinfection?","volume":"279","author":"Gomes","year":"2012","journal-title":"Proc. R. Soc. B Biol. Sci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1016\/j.tpb.2006.10.004","article-title":"Drug resistance in tuberculosis\u2014A reinfection model","volume":"71","author":"Rodrigues","year":"2007","journal-title":"Theor. Popul. Biol."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"3621","DOI":"10.1016\/j.cnsns.2010.01.007","article-title":"Global analysis of a dynamical model for transmission of tuberculosis with a general contact rate","volume":"15","author":"Bowong","year":"2010","journal-title":"Commun. Nonlinear. Sci. Numer. Simul. Commun Nonlinear Sci."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/723\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:40:10Z","timestamp":1760146810000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/723"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,13]]},"references-count":45,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["axioms11120723"],"URL":"https:\/\/doi.org\/10.3390\/axioms11120723","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,12,13]]}}}