{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T16:09:28Z","timestamp":1774368568978,"version":"3.50.1"},"reference-count":33,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2023,6,19]],"date-time":"2023-06-19T00:00:00Z","timestamp":1687132800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Infectious diseases continue to be a significant threat to human health and civilization, and finding effective methods to combat them is crucial. In this paper, we investigate the impact of awareness campaigns and optimal control techniques on infectious diseases without proper vaccines. Specifically, we develop an SIRS-type mathematical model that incorporates awareness campaigns through media and treatment for disease transmission dynamics and control. The model displays two equilibria, a disease-free equilibrium and an endemic equilibrium, and exhibits Hopf bifurcation when the bifurcation parameter exceeds its critical value, causing a switch in the stability of the system. We also propose an optimal control problem that minimizes the cost of control measures while achieving a desired level of disease control. By applying the minimum principle to the optimal control problem, we obtain analytical and numerical results that show how the infection rate of the disease affects the stability of the system and how awareness campaigns and treatment can maintain the stability of the system. This study highlights the importance of awareness campaigns in controlling infectious diseases and demonstrates the effectiveness of optimal control theory in achieving disease control with minimal cost.<\/jats:p>","DOI":"10.3390\/axioms12060608","type":"journal-article","created":{"date-parts":[[2023,6,20]],"date-time":"2023-06-20T03:02:13Z","timestamp":1687230133000},"page":"608","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment"],"prefix":"10.3390","volume":"12","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3744-5524","authenticated-orcid":false,"given":"Fahad","family":"Al Basir","sequence":"first","affiliation":[{"name":"Department of Mathematics, Asansol Girls\u2019 College, Asansol-4, West Bengal 713304, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7360-062X","authenticated-orcid":false,"given":"Biru","family":"Rajak","sequence":"additional","affiliation":[{"name":"Department of Computer Science, Asansol Girls\u2019 College, Asansol-4, West Bengal 713304, India"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6695-155X","authenticated-orcid":false,"given":"Bootan","family":"Rahman","sequence":"additional","affiliation":[{"name":"Mathematics Unit, School of Science and Engineering, University of Kurdistan Hewler (UKH), Erbil 44001, Iraq"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5032-3639","authenticated-orcid":false,"given":"Khalid","family":"Hattaf","sequence":"additional","affiliation":[{"name":"Equipe de Recherche en Mod\u00e9lisation et Enseignement des Math\u00e9matiques (ERMEM), Centre R\u00e9gional des M\u00e9tiers de l\u2019Education et de la Formation (CRMEF), Derb Ghalef,  Casablanca 20340, Morocco"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2023,6,19]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2123","DOI":"10.1016\/S0140-6736(12)60907-6","article-title":"Tackling pneumonia and diarrhoea: The deadliest diseases for the world\u2019s poorest children","volume":"379","author":"Gupta","year":"2012","journal-title":"Lancet"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1016\/S0140-6736(10)60356-X","article-title":"Progress and barriers for the control of diarrhoeal disease","volume":"376","author":"Santosham","year":"2010","journal-title":"Lancet"},{"key":"ref_3","unstructured":"World Health Organization (WHO) (2021). Global Tuberculosis Report 2021, WHO. Available online: https:\/\/www.who.int\/publications\/i\/item\/9789240037021."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Al Basir, F., and Abraha, T. (2023). Mathematical Modelling and Optimal Control of Malaria Using Awareness-Based Interventions. Mathematics, 11.","DOI":"10.20944\/preprints202302.0491.v1"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Liu, H., Li, M., Jin, M., Jing, F., Wang, H., and Chen, K. (2013). Public awareness of three major infectious diseases in rural Zhejiang province, China: A cross-sectional study. BMC Infect. Dis., 13.","DOI":"10.1186\/1471-2334-13-192"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"1261","DOI":"10.1016\/S0140-6736(10)60809-4","article-title":"Use of mass media campaigns to change health behaviour","volume":"376","author":"Wakefield","year":"2010","journal-title":"Lancet"},{"key":"ref_7","first-page":"665","article-title":"Exploring the effects of awareness and time delay in controlling malaria disease propagation","volume":"22","author":"Banerjee","year":"2020","journal-title":"Int. J. Nonlin. Sci. Numer. Simul."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"318","DOI":"10.1007\/s10729-014-9310-2","article-title":"Effect of individual protective behaviors on influenza transmission: An agent-based model","volume":"18","author":"Karimi","year":"2015","journal-title":"Health Care Manag. Sci."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"5364","DOI":"10.3934\/mbe.2021272","article-title":"Farming awareness based optimum interventions for crop pest control","volume":"18","author":"Abraha","year":"2021","journal-title":"Math. Biosci. Eng."},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Jones, J.H., and Salathe, M. (2009). Early assessment of anxiety and behavioral response to novel swine origin influenza A (H1N1). PLoS ONE, 4.","DOI":"10.1371\/journal.pone.0008032"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Ahorsu, D.K., and Lin, C.Y. (2023). The Effect of COVID-19 Vaccine Acceptance, Intention, and\/or Hesitancy and Its Association with Our Health and\/or Important Areas of Functioning. Vaccines, 11.","DOI":"10.3390\/vaccines11020368"},{"key":"ref_12","first-page":"6965","article-title":"Effect of awareness programs by media on the epidemic outbreaks: A mathematical model","volume":"219","author":"Samanta","year":"2013","journal-title":"Appl. Math. Comput."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"86","DOI":"10.1111\/j.1467-8268.2009.00205.x","article-title":"HIV\/AIDS knowledge and behaviour: Have information campaigns reduced HIV infection? The case of Kenya","volume":"21","author":"Frolich","year":"2009","journal-title":"Afr. Dev. Rev."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"12091","DOI":"10.3934\/math.2022672","article-title":"COVID-19 propagation and the usefulness of awareness-based control measures: A mathematical model with delay","volume":"7","author":"Maji","year":"2022","journal-title":"AIMS Math."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1142\/S0218339011004020","article-title":"Effect of awareness programs in controlling the prevalence of an epidemic with time delay","volume":"19","author":"Misra","year":"2011","journal-title":"J. Biol. Syst."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"680743","DOI":"10.1155\/2014\/680743","article-title":"An SIRS Epidemic Model Incorporating Media Coverage with Time Delay","volume":"2014","author":"Zhao","year":"2014","journal-title":"Comput. Math. Methods Med."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1007\/s10884-007-9075-0","article-title":"The impact of media on the control of infectious diseases","volume":"20","author":"Cui","year":"2007","journal-title":"J. Dyn. Differ. Equ."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1142\/S0218339014400051","article-title":"Modeling the impact of awareness created by media campaigns on vaccination coverage in a variable population","volume":"22","author":"Sharma","year":"2014","journal-title":"J. Biol. Syst."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.mbs.2009.11.009","article-title":"The impact of information transmission on epidemic outbreaks","volume":"225","author":"Kiss","year":"2010","journal-title":"Math. Biosci."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1186\/s13662-015-0549-9","article-title":"Effect of awareness programs in controlling the disease HIV\/AIDS: An optimal control theoretic approach","volume":"2015","author":"Roy","year":"2015","journal-title":"Adv. Differ. Equ."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.mbs.2017.09.007","article-title":"Dynamics of vaccination in a time-delayed epidemic model with awareness","volume":"294","author":"Agaba","year":"2017","journal-title":"Math. Biosci."},{"key":"ref_22","first-page":"372","article-title":"Role of media coverage and delay in controlling infectious diseases: A mathematical model","volume":"337","author":"Basir","year":"2018","journal-title":"Appl. Math. Comput."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.biosystems.2015.11.002","article-title":"Stability analysis and optimal control of an epidemic model with awareness programs by media","volume":"138","author":"Misra","year":"2015","journal-title":"Biosystems"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"1123","DOI":"10.1080\/09720502.2019.1706839","article-title":"Modeling of media impact with stability analysis and optimal solution of SEIRS epidemic model","volume":"22","author":"Sharma","year":"2019","journal-title":"J. Interdiscip. Math."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"127804","DOI":"10.1016\/j.physa.2022.127804","article-title":"Optimal control and cost-effectiveness analysis for a COVID-19 model with individual protection awareness","volume":"603","author":"Yuan","year":"2022","journal-title":"Phys. A Stat. Mech. Appl."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Hale, J.K. (1977). Theory of Functional Differential Equations, Springer.","DOI":"10.1007\/978-1-4612-9892-2"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1098\/rsif.2005.0042","article-title":"Perspectives on the basic reproductive ratio","volume":"2","author":"Heffernan","year":"2005","journal-title":"J. R. Soc. Interface"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"92","DOI":"10.1016\/j.ecocom.2018.07.004","article-title":"Modelling the effects of awareness-based interventions to control the mosaic disease of Jatropha curcas","volume":"36","author":"Blyuss","year":"2018","journal-title":"Ecol. Complex."},{"key":"ref_29","first-page":"278234","article-title":"Complete coefficient criteria for five-dimensional Hopf bifurcations, with an application to economic dynamics","volume":"2015","author":"Douskos","year":"2015","journal-title":"J. Nonlinear Dyn."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Fleming, W.H., and Rishel, R.W. (1975). Deterministic and Stochastic Optimal Control, Springer.","DOI":"10.1007\/978-1-4612-6380-7"},{"key":"ref_31","first-page":"539","article-title":"Awareness programs control infectious disease-Multiple delay induced mathematical model","volume":"251","author":"Greenhalgh","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"287","DOI":"10.1080\/17486700802653917","article-title":"The dynamics of an HIV\/AIDS model with screened disease carriers","volume":"10","author":"Nyabadza","year":"2009","journal-title":"Comput. Math. Methods Med."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Lenhart, S., and Workman, J.T. (2007). Optimal Control Applied to Biological Models, Chapman & Hall\/CRC.","DOI":"10.1201\/9781420011418"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/6\/608\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T19:56:39Z","timestamp":1760126199000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/12\/6\/608"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,6,19]]},"references-count":33,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2023,6]]}},"alternative-id":["axioms12060608"],"URL":"https:\/\/doi.org\/10.3390\/axioms12060608","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,6,19]]}}}