Optimal Control of an SEIRS Epidemic Model for Online Shopping Addiction Dynamics among University Students

Muhammad Isbar Pratama, Wahidah Sanusi, Rhida Anggita Dasri

Abstract


Online shopping addiction is a growing behavioral problem among university students, driven by the rapid development of e-commerce platforms. This study constructs, analyzes, and applies optimal control to an SEIRS (susceptible--exposed--infected--recovered--susceptible) mathematical model of online shopping addiction dynamics among students of the Faculty of Mathematics and Natural Sciences, Universitas Negeri Makassar. The population is divided into susceptible, exposed, addicted, and recovered compartments, and the model incorporates the possibility of relapse from the recovered to the susceptible compartment. The analysis determines the addiction-free and endemic equilibria, examines local stability through Jacobian linearization, and computes the basic reproduction number using the next-generation matrix method. Optimal control is formulated using Pontryagin's minimum principle, with an educational and counseling intervention as the control applied to the addicted compartment. Primary data were obtained from questionnaires distributed to ninety-eight active students, and numerical simulations were carried out using the fourth-order Runge--Kutta and forward--backward sweep methods in Python. The basic reproduction number is approximately 1.03, which is greater than one, indicating that online shopping addiction can persist and spread within the population; consistently, the addiction-free equilibrium is unstable and the endemic equilibrium is locally asymptotically stable. Applying the optimal control suppresses the peak addicted population by 43.1 percent, accelerates the movement of addicted individuals toward recovery during the intervention window, and preserves a substantially larger never-addicted susceptible population. These results demonstrate that educational intervention is mathematically effective in controlling the spread of online shopping addiction.

Keywords


Basic reproduction number; Online shopping addiction; Optimal control; Pontryagin's minimum principle; SEIRS epidemic model.

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References


Ahfa, L. (2023). Kontrol optimal model SEIPQ pada pecandu game online dengan mempertimbangkan edukasi dan treatment [Undergraduate thesis, Universitas Negeri Makassar]. UNM Repository.

Alemneh, H. T., & Makinde, O. D. (2020). Optimal control analysis for co-infection of dengue and leptospirosis disease. Advances in Difference Equations, 2020, 526. https://doi.org/10.1186/s13662-020-02878-z

Anwar, A. (2020). Pemodelan matematika SEIRS terhadap kecanduan game online pada mahasiswa Jurusan Matematika FMIPA Universitas Negeri Makassar [Undergraduate thesis, Universitas Negeri Makassar]. UNM Repository.

Indonesian Internet Service Providers Association (APJII). (2023). Indonesia internet penetration survey 2023. APJII.

Brauer, F., & Castillo-Chavez, C. (2012). Mathematical models in population biology and epidemiology (2nd ed.). Springer. https://doi.org/10.1007/978-1-4614-1686-9

Castillo-Chavez, C., & Song, B. (2004). Dynamical models of tuberculosis and their applications. Mathematical Biosciences and Engineering, 1(2), 361–404. https://doi.org/10.3934/mbe.2004.1.361

Christakis, N. A., & Fowler, J. H. (2007). The spread of obesity in a large social network over 32 years. New England Journal of Medicine, 357(4), 370–379. https://doi.org/10.1056/NEJMsa066082

Christakis, N. A., & Fowler, J. H. (2008a). The collective dynamics of smoking in a large social network. New England Journal of Medicine, 358(21), 2249–2258. https://doi.org/10.1056/NEJMsa0706154

Christakis, N. A., & Fowler, J. H. (2008b). Dynamic spread of happiness in a large social network. BMJ, 337, a2338. https://doi.org/10.1136/bmj.a2338

Diekmann, O., Heesterbeek, J. A. P., & Roberts, M. G. (2010). The construction of next-generation matrices for compartmental epidemic models. Journal of the Royal Society Interface, 7(47), 873–885. https://doi.org/10.1098/rsif.2009.0386

Girsang, M. K., Yulindra, D., Nuryaman, A., & Yanto, B. (2024). Pemodelan matematika kecanduan masyarakat terhadap perilaku belanja online di Shopee. Jurnal Riset dan Aplikasi Matematika, 8(2), 168–177.

Guo, Y., & Li, T. (2022). Dynamics and optimal control of an online game addiction model with considering family education. AIMS Mathematics, 7(2), 2710–2735. https://doi.org/10.3934/math.2022152

Hafiza, N., Shoffah, S. N. A., & Nursaptini, N. (2024). E-commerce memicu maraknya perilaku konsumtif di kalangan mahasiswa Universitas Mataram. Jurnal Dinamika Sosial Budaya, 26(1), 24–35.

Heffernan, J. M., Smith, R. J., & Wahl, L. M. (2005). Perspectives on the basic reproductive ratio. Journal of the Royal Society Interface, 2(4), 281–293. https://doi.org/10.1098/rsif.2005.0042

Hethcote, H. W. (1989). Three basic epidemiological models. In S. A. Levin, T. G. Hallam, & L. J. Gross (Eds.), Applied mathematical ecology (pp. 119–144). Springer. https://doi.org/10.1007/978-3-642-61317-3_5

Hethcote, H. W. (2000). The mathematics of infectious diseases. SIAM Review, 42(4), 599–653. https://doi.org/10.1137/S0036144500371907

Huo, H. F., & Yokoyama, T. (2021). Properties of a dynamic SAIR model of smoking with optimal control. Discrete and Continuous Dynamical Systems – Series B, 27(1), 413–438. https://doi.org/10.3934/dcdsb.2021050

Ilmayasinta, N., & Purnawan, H. (2021). Optimal control in a mathematical model of smoking. Journal of Mathematical and Fundamental Sciences, 53(3), 380–394. https://doi.org/10.5614/j.math.fund.sci.2021.53.3.4

Kermack, W. O., & McKendrick, A. G. (1927). A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London. Series A, 115(772), 700–721. https://doi.org/10.1098/rspa.1927.0118

Lenhart, S., & Workman, J. T. (2007). Optimal control applied to biological models. Chapman & Hall/CRC.

Lestari, S. A. C., Nurul Izzah, A., & Putri Agustin, N. (2023). Online shopping habit sebagai budaya masyarakat modern. Jurnal Sosiologi dan Humaniora, 6(1), 129.

Li, T., & Guo, Y. (2019). Stability and optimal control in a mathematical model of online game addiction. Filomat, 33(17), 5691–5711. https://doi.org/10.2298/FIL1917691L

Li, J., & Ma, Z. (2002). Qualitative analyses of SIS epidemic model with vaccination and varying total population size. Mathematical and Computer Modelling, 35(11–12), 1235–1243. https://doi.org/10.1016/S0895-7177(02)00082-1

Li, J., & Ma, Z. (2004). Global analysis of SIS epidemic models with variable total population size. Mathematical and Computer Modelling, 39(11–12), 1231–1242. https://doi.org/10.1016/j.mcm.2004.06.004

Müller, A., Mitchell, J. E., & de Zwaan, M. (2015). Compulsive buying. The American Journal on Addictions, 24(2), 132–137. https://doi.org/10.1111/ajad.12111

Müller, A., Brand, M., Claes, L., Demetrovics, Z., de Zwaan, M., Fernández-Aranda, F., … Kyrios, M. (2019). Buying-shopping disorder—Is there enough evidence to support its inclusion in ICD-11? CNS Spectrums, 24(4), 374–379. https://doi.org/10.1017/S1092852918001323

Müller, A., Trotzke, P., Mitchell, J. E., de Zwaan, M., & Brand, M. (2022). Excessive shopping on the internet: Recent trends in compulsive buying-shopping disorder. Current Opinion in Behavioral Sciences, 44, 101116. https://doi.org/10.1016/j.cobeha.2022.101116

Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., & Mishchenko, E. F. (1962). The mathematical theory of optimal processes. Interscience Publishers.

Putri, A. M., Fitri, W., Susanti, M., & Agustin, L. T. (2022). Kecanduan belanja online: Analisis perilaku kaum rebahan di lingkungan mahasiswa UIN Imam Bonjol Padang. Jurnal Dakwah dan Ilmu Komunikasi, 2, 2685–1881.

Saha, S., & Samanta, G. P. (2023). Analysis of an SEIRS model for two diseases with optimal control. Results in Control and Optimization, 12, 100267. https://doi.org/10.1016/j.rico.2023.100267

Septiansari, D., & Handayani, T. (2021). Pengaruh belanja online terhadap perilaku konsumtif pada mahasiswa di masa pandemi Covid-19. Jurnal Ekonomi dan Manajemen Teknologi, 5(1), 53–65.

Sharomi, O., & Gumel, A. B. (2008). Curtailing smoking dynamics: A mathematical modeling approach. Applied Mathematics and Computation, 195(2), 475–499. https://doi.org/10.1016/j.amc.2007.05.012

Siregar, A. F. P., & Panjaitan, D. J. (2024). Pemodelan matematika terhadap kecanduan game online berdasarkan model SEIRS. Methoda, 14(1), 87–92.

Ulya, A., Putri, O. N., & Naylawati, W. A. (2023). Budaya konsumtif belanja online di kalangan mahasiswa. Prosiding Seminar Nasional, 1300–1308.

van den Driessche, P., & Watmough, J. (2002). Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 180(1–2), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6

Weinstein, A., Mezig, H., Mizrachi, S., & Lejoyeux, M. (2015). A study investigating the association between compulsive buying with measures of anxiety and obsessive–compulsive behavior among internet shoppers. Comprehensive Psychiatry, 57, 46–50. https://doi.org/10.1016/j.comppsych.2014.11.003

Widad, N. Al. (2023). Analisis dan simulasi model matematika kecanduan game online menggunakan metode beda hingga nonstandar [Undergraduate thesis, Universitas Negeri Makassar]. UNM Repository.

Zhao, H., Tian, W., & Xin, T. (2017). The development and validation of the Online Shopping Addiction Scale. Frontiers in Psychology, 8, 735. https://doi.org/10.3389/fpsyg.2017.00735




DOI: https://doi.org/10.18860/cauchy.v11i2.45099

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