A fractional-order model for drinking alcohol behaviour leading to road accidents and violence

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dc.citation.epage518
dc.citation.issue3
dc.citation.journalTitleМатематичне моделювання та комп'ютинг
dc.citation.spage501
dc.contributor.affiliationУніверситет Хасана ІІ Касабланки
dc.contributor.affiliationУніверситет Шуайба Дуккалі
dc.contributor.affiliationHassan II University
dc.contributor.affiliationChouaib Doukkali University
dc.contributor.authorХаджі, Б.
dc.contributor.authorБуджаллал, Л.
dc.contributor.authorЕльхія, М.
dc.contributor.authorБалатіф, О.
dc.contributor.authorРачик, М.
dc.contributor.authorKhajji, B.
dc.contributor.authorBoujallal, L.
dc.contributor.authorElhia, M.
dc.contributor.authorBalatif, O.
dc.contributor.authorRachik, M.
dc.coverage.placenameЛьвів
dc.coverage.placenameLviv
dc.date.accessioned2025-03-04T11:33:05Z
dc.date.created2022-02-28
dc.date.issued2022-02-28
dc.description.abstractУ цій роботі пропонуємо нову модель вживання алкоголю дробового порядку за участю похідної Капуто та шести груп осіб. Вводимо дорожньо-транспортні пригоди та насильство, що пов’язані зі вживанням алкоголю, як окремі класи, щоб підкреслити роль алкоголізму в агресивній та ризикованій поведінці людей, що зловживають алкоголем. Показано існування та єдиність невід’ємних розв’язків і визначено основне число відтворення R0. Проаналізовано чутливість параметрів моделі для характеристики важливих параметрів, які найбільше впливають на число відтворення. Крім того, аналіз стійкості моделі показує, що система локально та глобально асимптотично стійка в рівновазі без пиття E0, якщо R0 < 1, і рівновага з питтям E∗ існує. Система локально та глобально асимптотично стійка для E∗, коли R0 > 1. Насамкінець, проведено чисельне моделювання для ілюстрації теоретичних результатів для різних значень порядку дробової похідної.
dc.description.abstractIn this paper, we propose a new fractional-order model of alcohol drinking involving the Caputo derivative and six groups of individuals. We introduce road accidents and violence related to alcohol consumption as separate classes to highlight the role of alcoholism in the aggressive and risky behaviour of heavy drinkers. We show the existence and uniqueness of the non-negative solutions, and we determine the basic reproduction number R0. The sensitivity analysis of the model parameters is performed to characterize the important parameters that have the most effects on the reproduction number. Furthermore, the stability analysis of the model shows that the system is locally and globally asymptotically stable at drinking-free equilibrium E0 when R0<1, and the drinking present equilibrium E∗exists. The system is locally and globally asymptotically stable at E∗ when R0>1. Finally, numerical simulations are carried out to illustrate the theoretical results for different values of the order of the fractional derivative.
dc.format.extent501-518
dc.format.pages18
dc.identifier.citationA fractional-order model for drinking alcohol behaviour leading to road accidents and violence / B. Khajji, L. Boujallal, M. Elhia, O. Balatif, M. Rachik // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2022. — Vol 9. — No 3. — P. 501–518.
dc.identifier.citationenA fractional-order model for drinking alcohol behaviour leading to road accidents and violence / B. Khajji, L. Boujallal, M. Elhia, O. Balatif, M. Rachik // Mathematical Modeling and Computing. — Lviv : Lviv Politechnic Publishing House, 2022. — Vol 9. — No 3. — P. 501–518.
dc.identifier.doidoi.org/10.23939/mmc2022.03.501
dc.identifier.urihttps://ena.lpnu.ua/handle/ntb/63476
dc.language.isoen
dc.publisherВидавництво Львівської політехніки
dc.publisherLviv Politechnic Publishing House
dc.relation.ispartofМатематичне моделювання та комп'ютинг, 3 (9), 2022
dc.relation.ispartofMathematical Modeling and Computing, 3 (9), 2022
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dc.relation.references[9] Balatif O., Boujallal L., Labzai A., Rachik M. Stability Analysis of a Fractional-Order Model for Abstinence Behavior of Registration on the Electoral Lists. International Journal of Differential Equations. 2020, 4325640 (2020).
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dc.relation.references[11] Zhang Y., Liu F., Koura Y. H., Wang H. Analysing rumours spreading considering self-purification mechanism. Connection Science. 33 (1), 81–94 (2020).
dc.relation.references[12] Sharma S., Samanta G. Analysis of a drinking epidemic model. International Journal of Dynamics and Control. 3 (3), 288–305 (2015).
dc.relation.references[13] Ma S.-H., Huo H.-F., Meng X.-Y. Modelling alcoholism as a contagious disease: a mathematical model with awareness programs and time delay. Discrete Dynamics in Nature and Society. 2015, 2600195 (2015).
dc.relation.references[14] Wang X.-Y., Hattaf K., Huo H.-F., Xiang H. Stability analysis of a delayed social epidemics model with general contact rate and its optimal control. Journal of Industrial & Management Optimization. 12 (4), 1267–1285 (2016).
dc.relation.references[15] Huo H.-F., Liu Y.-P. The analysis of the SIRS alcoholism models with relapse on weighted networks. SpringerPlus. 5 (1), 722 (2016).
dc.relation.references[16] Xiang H., Song N.-N., Huo H.-F. Modelling effects of public health educational campaigns on drinking dynamics. Journal of Biological Dynamics. 10 (1), 164–178 (2016).
dc.relation.references[17] Giacobbe A., Mulone G., Straughan B., Wang W. Modelling drinking with information. Mathematical Methods in the Applied Sciences. 40 (12), 4400–4411 (2017).
dc.relation.references[18] Adu I. K., Mojeeb A., Yang C. Mathematical model of drinking epidemic. Journal of Advances in Mathematics and Computer Science. 22 (5), 1–10 (2017).
dc.relation.references[19] Bonyah E., Khan M. A., Okosun K. O., G´omez-Aguilar J. F. Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea with optimal control. Mathematical Biosciences. 309, 1–11 (2019).
dc.relation.references[20] Khajji B., Labzai A., Kouidere A., Balatif O., Rachik M. A discrete mathematical modeling of the influence of alcohol treatment centers on the drinking dynamics using optimal control. Journal of Applied Mathematics. 2020, 9284698 (2020).
dc.relation.references[21] Agrawal A., Tenguria A., Modi G. Role of epidemic model to control drinking problem. International Journal of Scientific Research in Mathematical and Statistical Sciences. 5 (4), 324–337 (2018).
dc.relation.references[22] Xiang H., Wang Y., Huo H. Analysis of the binge drinking models with demographics and nonlinear infectivity on networks. Journal of Applied Analysis & Computation. 8 (5), 1535–1554 (2018).
dc.relation.references[23] Agrawal O. P. Formulation of Euler–Lagrange equations for fractional variational problems. Journal of Mathematical Analysis and Applications. 272 (1), 368–379 (2002).
dc.relation.references[24] Jajarmi A., Baleanu D. A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems. Frontiers in Physics. 8, 220 (2020).
dc.relation.references[25] Khan M. A., Atangana A. Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative. Alexandria Engineering Journal. 59 (4), 2379–2389 (2020).
dc.relation.references[26] Pinto C. M. A., Carvalho A. R. M. The HIV/TB coinfection severity in the presence of TB multi-drug resistant strains. Ecological Complexity. 32 (A), 1–20 (2017).
dc.relation.references[27] Fatmawati, Shaiful E. M., Utoyo M. I. A Fractional-Order Model for HIV Dynamics in a Two-Sex Population. International Journal of Mathematics and Mathematical Sciences. 2018, 6801475 (2018).
dc.relation.references[28] Khajji B., Kouidere A., Elhia M., Balatif O., Rachik M. Fractional optimal control problem for an agestructured model of COVID-19 transmission. Chaos, Solitons & Fractals. 143, 110625 (2021).
dc.relation.references[29] Boujallal L. Stability Analysis of Fractional Order Mathematical Model of Leukemia. International Journal of Mathematical Modelling & Computations. 11 (1), 15–27 (2021).
dc.relation.references[30] Veeresha P., Prakasha D. G., Baskonus H. M. Solving smoking epidemic model of fractional order using a modified homotopy analysis transform method. Mathematical Sciences. 13 (2), 115–128 (2019).
dc.relation.references[31] WHO. Global Status Report on Road Safety 2018. WHO: Geneva, Switzerland (2018).
dc.relation.references[32] Lipsey M. W., Wilson D. B., Cohen M. A., Derzon J. H. Is there a causal relationship between alcohol use and violence? Recent Developments in Alcoholism. 13, 245–282 (1997).
dc.relation.references[33] Khajji B., Moumine E. M., Ferjouchia H., Balatif O., Rachik M. Optimal control and discrete-time modelling of alcohol model with physical and psychological complications. Journal of Mathematical and Computational Science. 10 (5), 1969–1986 (2020).
dc.relation.references[34] P´erez E. Mathematical modeling of the spread of alcoholism among Colombian College Students. Ingenieria y Ciencia. 16 (32), 195–223 (2020).
dc.relation.references[35] Sharma S., Samanta G. Drinking as an epidemic: a mathematical model with dynamic behaviour. Journal of applied mathematics & informatics. 31 (1_2), 1–25 (2013).
dc.relation.references[36] Global Status Report on Alcohol and Health. Available at http://www.who.int/.
dc.relation.references[37] Diethelm K. The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer Science & Business Media (2010).
dc.relation.references[38] Huo H. F., Song N. N. Global stability for a binge drinking model with two stages. Discrete Dynamics in Nature and Society. 2012, 829386 (2012).
dc.relation.references[39] Hu Z., Teng Z., Jiang H. Stability analysis in a class of discrete SIRS epidemic models. Nonlinear Analysis: Real World Applications. 13 (5), 2017–2033 (2012).
dc.relation.references[40] Matignon D. Stability results for fractional differential equations with applications to control processing. Computational engineering in systems applications. 2, 963–968 (1996).
dc.relation.references[41] Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier (1998).
dc.relation.references[42] Lin W. Global existence theory and chaos control of fractional differential equations. Journal of Mathematical Analysis and Applications. 332 (1), 709–726 (2007).
dc.relation.references[43] Diethelm K. Monotonicity of functions and sign changes of their Caputo derivatives. Fractional Calculus and Applied Analysis. 19 (2), 561–566 (2016).
dc.relation.references[44] Van den Driessche P., Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences. 180 (1–2), 29–48 (2002).
dc.relation.references[45] La Salle J. P. The stability of dynamical systems. SIAM (1976).
dc.relation.references[46] Chitnis N., Hyman J. M., Cushing J. M. Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bulletin of Mathematical Biology. 70 (5), 1272 (2008).
dc.relation.references[47] Odibat Z., Momani S. An algorithm for the numerical solution of differential equations of fractional order. Journal of Applied Mathematics & Informatics. 26 (1–2), 15–27 (2008).
dc.relation.referencesen[1] Organization W. H. Global status report on alcohol and health 2018: Executive summary. Technical report, World Health Organization (2018).
dc.relation.referencesen[2] Elhia M., Boujallal L., Alkama M., Balatif O., Rachik M. Set-valued control approach applied to a COVID19 model with screening and saturated treatment function. Complexity. 2020, 9501028 (2020).
dc.relation.referencesen[3] Elhia M., Balatif O., Boujallal L., Rachik M. Optimal control problem for a tuberculosis model with multiple infectious compartments and time delays. An International Journal of Optimization and Control: Theories & Applications. 11 (1), 75–91 (2021).
dc.relation.referencesen[4] Boujallal L., Balatif O., Elhia M. A set-valued approach applied to a control problem of tuberculosis with treatment. IMA Journal of Mathematical Control and Information. 38 (3), 1010–1027 (2021).
dc.relation.referencesen[5] Djilali S., Touaoula T. M., Miri S. E. H. A heroin epidemic model: very general non linear incidence, treatage, and global stability. Acta Applicandae Mathematicae. 152 (1), 171–194 (2017).
dc.relation.referencesen[6] Liu S., Zhang L., Xing Y. Dynamics of a stochastic heroin epidemic model. Journal of Computational and Applied Mathematics. 351, 260–269 (2019).
dc.relation.referencesen[7] Singh J., Kumar D., Al Qurashi M., Baleanu D. A new fractional model for giving up smoking dynamics. Advances in Difference Equations. 2017 (1), 1–16 (2017).
dc.relation.referencesen[8] Ba˜nuelos S., Danet T., Flores C., Ramos A. An epidemiological math model approach to a political system with three parties. CODEE Journal. 12 (1), 8 (2019).
dc.relation.referencesen[9] Balatif O., Boujallal L., Labzai A., Rachik M. Stability Analysis of a Fractional-Order Model for Abstinence Behavior of Registration on the Electoral Lists. International Journal of Differential Equations. 2020, 4325640 (2020).
dc.relation.referencesen[10] Balatif O., Elhia M., Rachik M. Optimal control problem for an electoral behavior model. Differential Equations and Dynamical Systems. 1–18 (2020).
dc.relation.referencesen[11] Zhang Y., Liu F., Koura Y. H., Wang H. Analysing rumours spreading considering self-purification mechanism. Connection Science. 33 (1), 81–94 (2020).
dc.relation.referencesen[12] Sharma S., Samanta G. Analysis of a drinking epidemic model. International Journal of Dynamics and Control. 3 (3), 288–305 (2015).
dc.relation.referencesen[13] Ma S.-H., Huo H.-F., Meng X.-Y. Modelling alcoholism as a contagious disease: a mathematical model with awareness programs and time delay. Discrete Dynamics in Nature and Society. 2015, 2600195 (2015).
dc.relation.referencesen[14] Wang X.-Y., Hattaf K., Huo H.-F., Xiang H. Stability analysis of a delayed social epidemics model with general contact rate and its optimal control. Journal of Industrial & Management Optimization. 12 (4), 1267–1285 (2016).
dc.relation.referencesen[15] Huo H.-F., Liu Y.-P. The analysis of the SIRS alcoholism models with relapse on weighted networks. SpringerPlus. 5 (1), 722 (2016).
dc.relation.referencesen[16] Xiang H., Song N.-N., Huo H.-F. Modelling effects of public health educational campaigns on drinking dynamics. Journal of Biological Dynamics. 10 (1), 164–178 (2016).
dc.relation.referencesen[17] Giacobbe A., Mulone G., Straughan B., Wang W. Modelling drinking with information. Mathematical Methods in the Applied Sciences. 40 (12), 4400–4411 (2017).
dc.relation.referencesen[18] Adu I. K., Mojeeb A., Yang C. Mathematical model of drinking epidemic. Journal of Advances in Mathematics and Computer Science. 22 (5), 1–10 (2017).
dc.relation.referencesen[19] Bonyah E., Khan M. A., Okosun K. O., G´omez-Aguilar J. F. Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea with optimal control. Mathematical Biosciences. 309, 1–11 (2019).
dc.relation.referencesen[20] Khajji B., Labzai A., Kouidere A., Balatif O., Rachik M. A discrete mathematical modeling of the influence of alcohol treatment centers on the drinking dynamics using optimal control. Journal of Applied Mathematics. 2020, 9284698 (2020).
dc.relation.referencesen[21] Agrawal A., Tenguria A., Modi G. Role of epidemic model to control drinking problem. International Journal of Scientific Research in Mathematical and Statistical Sciences. 5 (4), 324–337 (2018).
dc.relation.referencesen[22] Xiang H., Wang Y., Huo H. Analysis of the binge drinking models with demographics and nonlinear infectivity on networks. Journal of Applied Analysis & Computation. 8 (5), 1535–1554 (2018).
dc.relation.referencesen[23] Agrawal O. P. Formulation of Euler–Lagrange equations for fractional variational problems. Journal of Mathematical Analysis and Applications. 272 (1), 368–379 (2002).
dc.relation.referencesen[24] Jajarmi A., Baleanu D. A new iterative method for the numerical solution of high-order non-linear fractional boundary value problems. Frontiers in Physics. 8, 220 (2020).
dc.relation.referencesen[25] Khan M. A., Atangana A. Modeling the dynamics of novel coronavirus (2019-nCov) with fractional derivative. Alexandria Engineering Journal. 59 (4), 2379–2389 (2020).
dc.relation.referencesen[26] Pinto C. M. A., Carvalho A. R. M. The HIV/TB coinfection severity in the presence of TB multi-drug resistant strains. Ecological Complexity. 32 (A), 1–20 (2017).
dc.relation.referencesen[27] Fatmawati, Shaiful E. M., Utoyo M. I. A Fractional-Order Model for HIV Dynamics in a Two-Sex Population. International Journal of Mathematics and Mathematical Sciences. 2018, 6801475 (2018).
dc.relation.referencesen[28] Khajji B., Kouidere A., Elhia M., Balatif O., Rachik M. Fractional optimal control problem for an agestructured model of COVID-19 transmission. Chaos, Solitons & Fractals. 143, 110625 (2021).
dc.relation.referencesen[29] Boujallal L. Stability Analysis of Fractional Order Mathematical Model of Leukemia. International Journal of Mathematical Modelling & Computations. 11 (1), 15–27 (2021).
dc.relation.referencesen[30] Veeresha P., Prakasha D. G., Baskonus H. M. Solving smoking epidemic model of fractional order using a modified homotopy analysis transform method. Mathematical Sciences. 13 (2), 115–128 (2019).
dc.relation.referencesen[31] WHO. Global Status Report on Road Safety 2018. WHO: Geneva, Switzerland (2018).
dc.relation.referencesen[32] Lipsey M. W., Wilson D. B., Cohen M. A., Derzon J. H. Is there a causal relationship between alcohol use and violence? Recent Developments in Alcoholism. 13, 245–282 (1997).
dc.relation.referencesen[33] Khajji B., Moumine E. M., Ferjouchia H., Balatif O., Rachik M. Optimal control and discrete-time modelling of alcohol model with physical and psychological complications. Journal of Mathematical and Computational Science. 10 (5), 1969–1986 (2020).
dc.relation.referencesen[34] P´erez E. Mathematical modeling of the spread of alcoholism among Colombian College Students. Ingenieria y Ciencia. 16 (32), 195–223 (2020).
dc.relation.referencesen[35] Sharma S., Samanta G. Drinking as an epidemic: a mathematical model with dynamic behaviour. Journal of applied mathematics & informatics. 31 (1_2), 1–25 (2013).
dc.relation.referencesen[36] Global Status Report on Alcohol and Health. Available at http://www.who.int/.
dc.relation.referencesen[37] Diethelm K. The analysis of fractional differential equations: An application-oriented exposition using differential operators of Caputo type. Springer Science & Business Media (2010).
dc.relation.referencesen[38] Huo H. F., Song N. N. Global stability for a binge drinking model with two stages. Discrete Dynamics in Nature and Society. 2012, 829386 (2012).
dc.relation.referencesen[39] Hu Z., Teng Z., Jiang H. Stability analysis in a class of discrete SIRS epidemic models. Nonlinear Analysis: Real World Applications. 13 (5), 2017–2033 (2012).
dc.relation.referencesen[40] Matignon D. Stability results for fractional differential equations with applications to control processing. Computational engineering in systems applications. 2, 963–968 (1996).
dc.relation.referencesen[41] Podlubny I. Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier (1998).
dc.relation.referencesen[42] Lin W. Global existence theory and chaos control of fractional differential equations. Journal of Mathematical Analysis and Applications. 332 (1), 709–726 (2007).
dc.relation.referencesen[43] Diethelm K. Monotonicity of functions and sign changes of their Caputo derivatives. Fractional Calculus and Applied Analysis. 19 (2), 561–566 (2016).
dc.relation.referencesen[44] Van den Driessche P., Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences. 180 (1–2), 29–48 (2002).
dc.relation.referencesen[45] La Salle J. P. The stability of dynamical systems. SIAM (1976).
dc.relation.referencesen[46] Chitnis N., Hyman J. M., Cushing J. M. Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bulletin of Mathematical Biology. 70 (5), 1272 (2008).
dc.relation.referencesen[47] Odibat Z., Momani S. An algorithm for the numerical solution of differential equations of fractional order. Journal of Applied Mathematics & Informatics. 26 (1–2), 15–27 (2008).
dc.relation.urihttp://www.who.int/
dc.rights.holder© Національний університет “Львівська політехніка”, 2022
dc.subjectмодель дробового порядку
dc.subjectповедінка при вживанні алкоголю
dc.subjectдорожньо-транспортна пригода
dc.subjectепідеміологічний підхід
dc.subjectаналіз стійкості
dc.subjectаналіз чутливості
dc.subjectfraction-order model
dc.subjectdrinking alcohol behaviour
dc.subjectroad accident
dc.subjectepidemiological approach
dc.subjectstability analysis
dc.subjectsensitivity analysis
dc.titleA fractional-order model for drinking alcohol behaviour leading to road accidents and violence
dc.title.alternativeМодель дробового порядку поведінки в стані алкогольного сп’яніння, що призводить до дорожньо-транспортних пригод і насильства
dc.typeArticle

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Mathematical Modeling And Computing. – 2022. – Vol. 9, No. 3