Optimal control of smoking-induced asthma and cardiovascular disorders with medical and public health interventions

Muhammad Farman; Noreen Asghar; Muhammad Umer Saleem; Nezihal Gokbulut; Evren Hincal; Aseel Smerat; Mohamed Hafez

doi:10.46481/jnsps.2026.3397

Authors

Muhammad Farman
[email protected]

Faculty of Arts and Sciences, Department of Mathematics, Near East University, Mersin-10, Turkey;

Faculty of Medicine, Department of Biostatistics and Medical Informatics, Karadeniz Technical University, Trabzon, Turkey;

International Center for Interdisciplinary Research in Sciences, The University of Lahore, Lahore, 54600, Pakistan
Noreen Asghar
Department of Mathematics, University of Education, Lahore, 54600, Pakistan
Muhammad Umer Saleem
Department of Mathematics, University of Education, Lahore, 54600, Pakistan
Nezihal Gokbulut
Faculty of Arts and Sciences, Department of Mathematics, Near East University, Mersin-10, Turkey
Evren Hincal
Faculty of Arts and Sciences, Department of Mathematics, Near East University, Mersin-10, Turkey;

Research Center of Applied Mathematics, Khazar University, Baku, AZ1096, Azerbaijan
Aseel Smerat
Faculty of Educational Sciences, Al-Ahliyya Amman University, Amman 19328, Jordan;

Department of Biosciences, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Chennai, 602105, India
Mohamed Hafez
Faculty of Engineering and Quantity Surveying, INTI International University Colleges, 71800, Nilai, Malaysia;

Faculty of Management, Shinawatra University, Pathum Thani, 12000, Thailand

Keywords:

Cardiovascular disease modeling, Optimal control, Nonstandard finite difference (NSFD), Public health interventions

Abstract

Smoking is a major public health concern because of its harmful effects on several organs and its association with respiratory and cardiovascular complications. Cardiovascular disease is more common among smokers, and awareness of smoking cessation is important for reducing smoking prevalence. In this study, a compartmental model is developed to analyze the effect of awareness on the control of smoking and smoking-induced asthma and cardiovascular disorders. The positively invariant region, boundedness, positivity, and other basic dynamical properties of the model are established. Equilibria are obtained, and local and global stability analyses are performed. The normalized forward sensitivity index is used to assess the influence of key parameters on smoking cessation. Numerical simulations are conducted using a reliable nonstandard finite difference (NSFD) method. The simulation results show that smoking substantially increases the risk of respiratory and cardiovascular illnesses and that intervention strategies can reduce disease severity and prevalence. The model is further extended to an optimal-control framework that emphasizes the importance of treatment implementation and public awareness in reducing smoking-related harm. The results highlight the importance of media-based awareness and the need to inform potential smokers about the dangers of smoking. The proposed mathematical framework can support policymakers and health professionals in designing targeted interventions to reduce tobacco and cigarette use.

Dimensions

REFERENCES

[1] I. R. Sofia, S. R. Bandekar & M. Ghosh, ``Mathematical modeling of smoking dynamics in society with impact of media information and awareness'', Results in Control and Optimization 11 (2023) 100233. https://doi.org/10.1016/j.rico.2023.100233.

[2] M. Ghosh, ``Industrial pollution and Asthma: a mathematical model'', Journal of Biological Systems 8 (2000) 347. https://doi.org/10.1142/S0218339000000225.

[3] R. Polosa & N. C. Thomson, ``Smoking and asthma: dangerous liaisons'', European Respiratory Journal 41 (2013) 716. https://doi.org/10.1183/09031936.00073312.

[4] D. P. Strachan, ``The role of environmental factors in asthma'', British Medical Bulletin 56 (2000) 865. https://doi.org/10.1258/0007142001903562.

[5] A. J. Driscoll, S. H. Arshad, L. Bont, S. M. Brunwasser, T. Cherian, J. A. Englund, D. B. Fell, L. L. Hammitt, T. V. Hartert, B. L. Innis, R. A. Karron, G. E. Langley, E. K. Mulholland, P. K. Munywoki, H. Nair, J. R. Ortiz, D. A. Savitz, N. M. Scheltema, E. A. F. Simões, P. G. Smith & D. R. Feikin., ``Does respiratory syncytial virus lower respiratory illness in early life cause recurrent wheeze of early childhood and asthma? Critical review of the evidence and guidance for future studies from a World Health Organization-sponsored meeting'', Vaccine 38 (2020) 2435. https://doi.org/10.1016/j.vaccine.2020.01.020.

[6] Y. Riza, D. Adam, Y. Christina, A. Z. Anwary, Netty, E. Handayani, H. K. Inayah, A. Malik, Y. Oktaviani & A. Widyarni, ``Public health policy in action: the impact of enforcing Local Regulation No. 7 of 2013 on smoke-free areas'', Universal Journal of Public Health 13 (2025) 250. https://doi.org/10.13189/ujph.2025.130125.

[7] N. E. Collishaw, J. Kirkbride & D. T. Wigle, ``Tobacco smoke in the workplace: an occupational health hazard'', Canadian Medical Association Journal 131 (1984) 1199. Available online: https://pmc.ncbi.nlm.nih.gov/articles/PMC1483688/.

[8] D. W. Dockery, C. A. Pope, X. Xu, J. D. Spengler, J. H. Ware, M. E. Fay, B. G. Ferris Jr. & F. E. Speizer, ``An association between air pollution and mortality in six US cities'', New England Journal of Medicine 329 (1993) 1753. https://doi.org/10.1056/NEJM199312093292401.

[9] M. J. Utell & R. J. Looney, ``Environmentally induced asthma'', Toxicology Letters 82 (1995) 47. https://doi.org/10.1016/0378-4274(95)03467-6.

[10] R. Jan & Y. Xiao, ``Effect of partial immunity on transmission dynamics of dengue disease with optimal control'', Mathematical Methods in the Applied Sciences 42 (2019) 1967. https://doi.org/10.1002/mma.5491.

[11] Z. Hammouch & T. Mekkaoui, ``Control of a new chaotic fractional-order system using Mittag-Leffler stability'', Nonlinear Studies 22 (2015) 565. Available online: https://nonlinearstudies.com/index.php/nonlinear/article/view/1245.

[12] M. Sohaib, ``Mathematical modeling and numerical simulation of HIV infection model'', Results in Applied Mathematics 7 (2020) 100118. https://doi.org/10.1016/j.rinam.2020.100118.

[13] R. J. Attaullah & A. Jabeen, ``Solution of the HIV infection model with full logistic proliferation and variable source term using Galerkin scheme'', Matrix Science Mathematic 4 (2020) 37. https://www.researchgate.net/publication/349945043_Solution_of_The_Hiv_Infection_Model_With_Full_Logistic_Proliferation_and_Variable_Source_Term_Using_Galerkin_Scheme.

[14] R. Jan & Y. Xiao, ``Effect of pulse vaccination on dynamics of dengue with periodic transmission functions'', Advances in Difference Equations 2019 (2019) 368. https://doi.org/10.1186/s13662-019-2314-y.

[15] M. R. Mahmoudi, D. Baleanu, S. S. Band & A. Mosavi, ``Factor analysis approach to classify COVID-19 datasets in several regions'', Results in Physics 25 (2021) 104071. https://doi.org/10.1016/j.rinp.2021.104071.

[16] H. M. Baskonus, Z. Hammouch, T. Mekkaoui & H. Bulut, ``Chaos in the fractional order logistic delay system: circuit realization and synchronization'', in AIP Conference Proceedings, 1738 (2016) 290005. https://doi.org/10.1063/1.4952077.

[17] M. Slowikowska, J. Bajzert, J. Miller, T. Stefaniak & A. Niedźwiedź, ``The dynamics of circulating immune complexes in horses with severe equine asthma'', Animals 11 (2021) 1001. https://doi.org/10.3390/ani11041001.

[18] C. Castillo-Garsow, G. Jordan-Salivia & A. Rodriguez-Herrera, ``Mathematical models for the dynamics of tobacco use, recovery and relapse'', Biometrics Unit Technical Report BU-1505-M, Cornell University, Ithaca, USA, 1997. https://mcmsc.asu.edu/sites/g/files/litvpz576/files/2024-09/MTBI%201997%20Tobacco%20Use%20Report.pdf

[19] O. Sharomi & A. B. Gumel, ``Curtailing smoking dynamics: a mathematical modeling approach'', Applied Mathematics and Computation 195 (2008) 475. https://doi.org/10.1016/j.amc.2007.05.012.

[20] V. Verma, ``Optimal control analysis of a mathematical model on smoking'', Modeling Earth Systems and Environment 6 (2020) 2535. https://doi.org/10.1007/s40808-020-00847-1.

[21] G. Zaman, Y. H. Kang & I. H. Jung, ``Optimal treatment of an SIR epidemic model with time delay'', BioSystems 98 (2009) 43. https://doi.org/10.1016/j.biosystems.2009.05.006.

[22] S. Lee, E. Jung & C. Castillo-Chavez, ``Optimal control intervention strategies in low and high risk problem drinking populations'', Socio-Economic Planning Sciences 44 (2010) 258. https://doi.org/10.1016/j.seps.2010.07.006.

[23] A. S. Devi, P. A. Naik, S. Boulaaras, N. Sene & Z. Huang. ``Understanding the transmission mechanism of HIV/TB co-infection using fractional framework with optimal control'', International Journal of Numerical Modelling: Electronic Networks, Devices and Fields 38 (2025) e70097. https://doi.org/10.1002/jnm.70097.

[24] P. A. Naik, B. M. Yeolekar, S. Qureshi, M. Yeolekar & A. Madzvamuse. ``Modeling and analysis of the fractional-order epidemic model to investigate mutual influence in HIV/HCV co-infection'', Nonlinear Dynamics 112 (2024) 11679. https://doi.org/10.1007/s11071-024-09653-1.

[25] O. Zakary, M. Rachik & I. Elmouki, ``On the analysis of a multi-regions discrete SIR epidemic model: an optimal control approach'', International Journal of Dynamics and Control 5 (2017) 917. https://doi.org/10.1007/s40435-016-0233-2.

[26] B. Khajji, A. Kouidere, O. Balatif & M. Rachik. ``Mathematical modeling, analysis and optimal control of an alcohol drinking model with liver complication'', Communications in Mathematical Biology and Neuroscience 2020 (2020) 32. https://doi.org/10.28919/cmbn/4553.

[27] B. Khajji, A. Labzai, A. Kouidere, O. Balatif & M. Rachik. ``A discrete mathematical modeling of the influence of alcohol treatment centers on the drinking dynamics using optimal control'', Journal of Applied Mathematics 2020 (2020) 9284698. https://doi.org/10.1155/2020/9284698.

[28] A. Omame & M. Abbas, ``The stability analysis of a co-circulation model for COVID-19, dengue, and zika with nonlinear incidence rates and vaccination strategies'', Healthcare Analytics 3 (2023) 100151. https://doi.org/10.1016/j.health.2023.100151.

[29] A. Omame & M. Abbas, ``Modeling SARS-CoV-2 and HBV co-dynamics with optimal control'', Physica A: Statistical Mechanics and its Applications 615 (2023) 128607. https://doi.org/10.1016/j.physa.2023.128607.

[30] A. Omame, M. Abbas & C. P. Onyenegecha. ``Backward bifurcation and optimal control in a co-infection model for SARS-CoV-2 and ZIKV'', Results in Physics 37 (2022) 105481. https://doi.org/10.1016/j.rinp.2022.105481.

[31] T. Faniran, A. Ali, M. O. Adewole, B. Adebo & O. O. Akanni. ``Asymptotic behavior of tuberculosis between smokers and non-smokers'', Partial Differential Equations in Applied Mathematics 5 (2022) 100244. https://doi.org/10.1016/j.padiff.2021.100244.

[32] A. Sharma & A. K. Misra, ``Backward bifurcation in a smoking cessation model with media campaigns'', Applied Mathematical Modelling 39 (2015) 1087. https://doi.org/10.1016/j.apm.2014.07.022.

[33] I. R. Sofia & M. Ghosh, ``Mathematical modeling of smoking habits in the society'', Stochastic Analysis and Applications 41 (2023) 918. https://doi.org/10.1080/07362994.2022.2093223.

[34] A. Labzai, O. Balatif & M. Rachik. ``Optimal control strategy for a discrete time smoking model with specific saturated incidence rate'', Discrete Dynamics in Nature and Society 2018 (2018) 5949303. https://doi.org/10.1155/2018/5949303.

[35] S. Musa, S. M. Ahmad & H. B. Aliyu, ``Impact of health education campaign on the dynamics of cigarette smoking in a varying population'', Dutse Journal of Pure and Applied Sciences 5 (2019) 141.

[36] S. S. Alzaid & B. S. T. Alkahtani, ``Asymptotic analysis of a giving up smoking model with relapse and harmonic mean type incidence rate'', Results in Physics 28 (2021) 104437. https://doi.org/10.1016/j.rinp.2021.104437.

[37] O. Khyar, J. Danane & K. Allali. ``Mathematical analysis and optimal control of giving up the smoking model'', International Journal of Differential Equations 2021 (2021) 8673020. https://doi.org/10.1155/2021/8673020.

[38] M. Farman, N. Asghar, M. U. Saleem, K. S. Nisar, K. Hosseini & M. Hafez. ``Predictive and global effect of active smoker in asthma dynamics with Caputo fractional derivative'', Computer Modeling in Engineering & Sciences 145 (2025) 721. https://doi.org/10.32604/cmes.2025.069541.

[39] M. Farman, N. Asghar, M. U. Saleem, S. Salahshour, A. Smerat & M. Hafez. ``Optimal control and dynamical transmission of asthma due to smoking populations: incorporating medical and public health measures'', Results in Control and Optimization 22 (2026) 100674. https://doi.org/10.1016/j.rico.2026.100674.

[40] R. Anguelov & J. M. S. Lubuma, ``Contributions to the mathematics of the nonstandard finite difference method and applications'', Numerical Methods for Partial Differential Equations 17 (2001) 518. https://doi.org/10.1002/num.1025.

[41] R. E. Mickens (Ed.), Nonstandard finite difference models of differential equations, World Scientific, Singapore, 1993, pp. 68-92. https://doi.org/10.1142/9789814440882_0003.

[42] J. M. S. Lubuma & K. C. Patidar, ``Non-standard methods for singularly perturbed problems possessing oscillatory/layer solutions'', Applied Mathematics and Computation 187 (2007) 1147. https://doi.org/10.1016/j.amc.2006.09.011.

[43] A. B. Gumel (Ed.), Mathematics of Continuous and Discrete Dynamical Systems, Contemporary Mathematics Series, Vol. 618, American Mathematical Society, Providence, Rhode Island, USA, 2014. Available online: https://bookstore.ams.org/conm-618.

[44] M. McAsey, L. Mou & W. Han ``Convergence of the forward-backward sweep method in optimal control'', Computational Optimization and Applications 53 (2012) 207. https://doi.org/10.1007/s10589-011-9454-7.

[45] C. L. Hwang & L. T. Fan, ``A discrete version of Pontryagin's maximum principle'', Operations Research 15 (1967) 139. https://doi.org/10.1287/opre.15.1.139.

[46] V. Guibout & A. Bloch, ``A discrete maximum principle for solving optimal control problems'', in Proceedings of the 43rd IEEE Conference on Decision and Control (CDC), IEEE, Bahamas (2004), pp. 1806--1811. https://ieeexplore.ieee.org/document/1430309.

[47] L. S. Pontryagin, Mathematical theory of optimal processes, Routledge, London, 2018. https://www.taylorfrancis.com/books/mono/10.1201/9780203749319/mathematical-theory-optimal-processes-pontryagin.

[48] E. Todorov & M. I. Jordan, ``Optimal feedback control as a theory of motor coordination'', Nature Neuroscience 5 (2002) 1226. https://doi.org/10.1038/nn963.