Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment
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. Specifical...
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| Format: | Article |
| Language: | English |
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MDPI AG
2023-06-01
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| Series: | Axioms |
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| Online Access: | https://www.mdpi.com/2075-1680/12/6/608 |
| _version_ | 1797596091626029056 |
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| author | Fahad Al Basir Biru Rajak Bootan Rahman Khalid Hattaf |
| author_facet | Fahad Al Basir Biru Rajak Bootan Rahman Khalid Hattaf |
| author_sort | Fahad Al Basir |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-11T02:46:43Z |
| format | Article |
| id | doaj.art-5a8997ed564a4d2abdca7ee129fb0318 |
| institution | Directory Open Access Journal |
| issn | 2075-1680 |
| language | English |
| last_indexed | 2024-03-11T02:46:43Z |
| publishDate | 2023-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj.art-5a8997ed564a4d2abdca7ee129fb03182023-11-18T09:17:25ZengMDPI AGAxioms2075-16802023-06-0112660810.3390/axioms12060608Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and TreatmentFahad Al Basir0Biru Rajak1Bootan Rahman2Khalid Hattaf3Department of Mathematics, Asansol Girls’ College, Asansol-4, West Bengal 713304, IndiaDepartment of Computer Science, Asansol Girls’ College, Asansol-4, West Bengal 713304, IndiaMathematics Unit, School of Science and Engineering, University of Kurdistan Hewler (UKH), Erbil 44001, IraqEquipe de Recherche en Modélisation et Enseignement des Mathématiques (ERMEM), Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Derb Ghalef, Casablanca 20340, MoroccoInfectious 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.https://www.mdpi.com/2075-1680/12/6/608mathematical modelbasic reproduction numberstability theoryforward bifurcationminimum principlenumerical simulations |
| spellingShingle | Fahad Al Basir Biru Rajak Bootan Rahman Khalid Hattaf Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment Axioms mathematical model basic reproduction number stability theory forward bifurcation minimum principle numerical simulations |
| title | Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment |
| title_full | Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment |
| title_fullStr | Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment |
| title_full_unstemmed | Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment |
| title_short | Hopf Bifurcation Analysis and Optimal Control of an Infectious Disease with Awareness Campaign and Treatment |
| title_sort | hopf bifurcation analysis and optimal control of an infectious disease with awareness campaign and treatment |
| topic | mathematical model basic reproduction number stability theory forward bifurcation minimum principle numerical simulations |
| url | https://www.mdpi.com/2075-1680/12/6/608 |
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