Birth of Catastrophe and Strange Attractors through Generalized Hopf Bifurcations in Covid-19 Transmission Mathematical Model

Year 2024, Volume: 6 Issue: 3, 159 - 169, 31.07.2024

Ario Wiraya , Yudi Ari Adi Laila Fitriana Triyanto Triyanto Yuvita Andriani Kusumadewi , Azimatus Nur Safitri Aulia Nurmalitasari

https://doi.org/10.51537/chaos.1448912

Cited By: 1

Abstract

Coronavirus can be transmitted through the things that people carry or the things where it sticks to after being spread by the sufferer. Instead, various preventive measures have been carried out. We create a new mathematical model that represents Coronavirus that exists in non-living objects, susceptible, and infected subpopulations interaction by considering the Coronavirus transmission through non-living objects caused by susceptible and infected subpopulations along with its prevention to characterize the dynamics of Coronavirus transmission in the population under those conditions. One disease-free and two infection equilibrium points along with their local stability and coexistence are identified. Global stability of the disease-free equilibria and basic reproduction number are also investigated. Changes in susceptible-Coronavirus interaction rate generate Fold and Hopf bifurcations which represent the emergence of a cycle and the collision of two infection equilibrium points respectively. Catastrophe generated by the collision between an attractor and a repeller is found around a Generalized Hopf bifurcation point by changing susceptible-Coronavirus interaction rate and increasing rate of Coronavirus originating from infected subpopulation. It represents a momentary unpredictable dynamics as the effect of Coronavirus addition and infection. Non-chaotic strange attractors that represent complex but still predictable dynamics are also triggered by Generalized Hopf bifurcation when the susceptible-Coronavirus interaction rate and one of the following parameters, i.e. increasing rate of Coronavirus originating from infected subpopulation or infected subpopulation recovery rate vary.

Keywords

Covid-19, Catastrophe, Generalized Hopf, Strange attractor

Ethical Statement

We confirm that this work is original and has neither been published elsewhere nor is currently under consideration for publication elsewhere.

Supporting Institution

Universitas Sebelas Maret

Project Number

1

Thanks

We would like to express our gratitude to Programme of Study in Mathematics Education, Faculty of Teacher Training and Education, Universitas Sebelas Maret, and Ministry of Research, Technology, and Higher Education of Indonesia for their contribution to this research through "Hibah Penelitian dan Pengabdian kepada Masyarakat (P2M) Penerimaan Negara Bukan Pajak (PNBP) Universitas Sebelas Maret 2023 with Contract Number: 228/UN27.22/PT.01.03/2023".

References

• Adi, Y., N. Irsalinda, A. Wiraya, S. Sugiyarto, and Z. Rafsanjani, 2023 Mathematical Modeling and Computing 10: 311–325, DOI: 10.23939/mmc2023.02.311.
• AlQadi, H. and M. Bani-Yaghoub, 2022 Incorporating global dynamics to improve the accuracy of disease models: Example of a covid-19 sir model. PloS one 17: e0265815, DOI: 10.1371/journal. pone.0265815.
• Bosi, S. and D. Desmarchelier, 2019 Local bifurcations of three and four-dimensional systems: A tractable characterization with economic applications. Mathematical Social Sciences 97: 1–1, DOI: 10.1016/j.mathsocsci.2018.11.001.
• Carraturo, F., C. Giudice, M. Morelli, V. Cerullo, G. Libralato, et al., 2020 Persistence of sars-cov-2 in the environment and covid-19 transmission risk from environmental matrices and surfaces. Environmental Pollution 265, DOI: 10.1016/j.envpol.2020.
• Castillo-Garsow, C. and C. Castillo-Chavez, 2020 A tour of the basic reproductive number and the next generation of researchers. Springer International Publishing, Berlin/Heidelberg, Germany.
• Cencini, M., F. Cecconi, and A. Vulpiani, 2009 Chaos: from Simple Models to Complex Systems. World Scientific, Singapore. Dieci, L., R. Russell, and E. Van Vleck, 1997 SIAM Journal on Numerical Analysis 34: 402–423,
• https://api.semanticscholar.org/CorpusID:18204582. Din, R. and E. Algehyne, 2021 Mathematical analysis of covid-19 by using sir model with convex incidence rate. Results in Physics 23, DOI: 10.1016/j.rinp.2021.103970.
• for Disease Prevention, E. C. and Control, 2020 Using face masks in the community reducing covid-19 transmission from potentially asymptomatic or pre-symptomatic people through the use of face masks. Technical report, Stockholm.
• Gandhi, M., D. Yokoe, and D. Havlir, 2020 Asymptomatic transmission, the achilles’ heel of current strategies to control covid-19. The New England Journal of Medicine 382: 2158–2160, DOI: 10.1056/NEJMe2009758.
• He, F., Y. Deng, and W. Li, 2020 Coronavirus disease 2019: What we know? Journal of Medical Virology 92: 719–725, DOI: 10.1002/jmv.25766.
• Kuznetsov, Y., 1998 Element of Applied Bifurcation Theory. Springer- Verlag, Inc., New York.
• LaSalle, J. and S. Lefschetz, 1961 Stability by Lyapunov’s Direct Method with Applications. Academic Press, New York.
• Mondal, B., A. Thirthar, N. Sk, M. Alqudah, and T. Abdeljawad, 2024 Complex dynamics in a two species system with crowley– martin response function: Role of cooperation, additional food and seasonal perturbations. Mathematics and Computers in Simulation 221: 415–434, DOI: 10.1016/j.matcom.2024.03.015.
• Obi, O. and D. Odoh, 2021 Transmission of coronavirus (sarscov- 2) by presymptomatic and asymptomatic covid-19 carriers? European Journal of Medical and Educational Technologies 14, DOI: 10.30935/ejmets/11060.
• Pakhira, R., B. Mondal, A. Thirthar, M. Alqudah, and T. Abdeljawad, 2024 Developing a fuzzy logic-based carbon emission cost-incorporated inventory model with memory effects. Ain Shams Engineering Journal p. 102746, DOI: 10.1016/j.asej.2024.102746.
• Pedersen, S. and Y. Ho, 2020 Sars-cov-2: a storm is raging. The Journal of Clinical Investigation 130: 2202–2205, DOI: 10.1172/JCI137647.
• Perko, L., 2001 Differential Equations and Dynamical Systems. Springer-Verlag, Inc., New York, NY.
• Ramesh, N., A. Siddaiah, and B. Joseph, 2020 Tackling coronavirus disease 2019 (covid 19) in workplaces. Indian Journal of Occupational and Environmental Medicine 24: 16–18.
• Sender, R., Y. Bar-On, S. Gleizer, B. Bernshtein, A. Flamholz, et al., 2021 The total number and mass of sars-cov-2 virions. Proceedings of the National Academy of Sciences of the United States of America 118, DOI: 10.1073/pnas.2024815118.
• Sk, N., B. Mondal, A. Thirthar, M. Alqudah, and T. Abdeljawad, 2023 Bistability and tristability in a deterministic prey–predator model: Transitions and emergent patterns in its stochastic counterpart. Chaos, Solitons and Fractals 176: 114073, DOI: 10.1016/j.matcom.2024.03.015.
• Thirthar, A., 2023 A mathematical modelling of a plantherbivore community with additional effects of food on the environment. Iraqi Journal of Science 64: 3551–3566, DOI: 10.24996/ijs.2023.64.7.34.
• Thirthar, A., N. Sk, B. Mondal, M. Alqudah, and T. Abdeljawad, 2023 Utilizing memory effects to enhance resilience in diseasedriven prey-predator systems under the influence of global warming. Journal of Applied Mathematics and Computing 69: 4617–4643, DOI: 10.1007/s12190-023-01936-x.
• van Doremalen, N., T. Bushmaker, D. Morris, M. Holbrook, A. Gamble, et al., 2020 Aerosol and surface stability of sarscov- 2 as compared with sars-cov-1. The New England Journal of Medicine 382: 1564–1567, DOI: 10.1056/NEJMc2004973.
• Verhulst, F., 1996 Nonlinear differential equation and dynamical systems. Springer-Verlag, Inc., New York.
• Vermund, S. and V. Pitzer, 2021 Asymptomatic transmission and the infection fatality risk for covid-19: Implications for school reopening. Clinical Infectious Diseases 7: 1493–1496, DOI: 10.1093/cid/ciaa855.
• WHO, 2020 Website of the who coronavirus (covid-19) dashboard. Technical report, World Health Organization.
• Wiggins, S., 2003 Introduction To Applied Nonlinear Dynamical Systems And Chaos. Springer-Verlag, Inc., New York.
• Wiraya, A., Y. Adi, L. Fitriana, Triyanto, and S. Khoirunnisa, 2022 Global stability of latency equilibria on mathematical model for human inflammatory response to coronavirus infection. In Internationa Conference of Mathematics and Mathematics Education (I-CMME) 2021, I-CMME 2021, Surakarta, Indonesia, pp. 030009– 1–030009–9.
• Wiraya, A. and F. Adi-Kusumo, 2023 Torus and homoclinic bifurcations on a cells repair regulations model of the metastatic nasopharyngeal carcinoma. Journal of Nonlinear Science 33: 1– 21, DOI: 10.1007/s00332-023-09925-x.
• Wiraya, A., L. Fitriana, Triyanto, Y. Adi, Y. Kusumadewi, et al., 2024 Bifurcation analysis of the dynamics in covid-19 transmission through living and nonliving media. Journal of Applied Mathematics 2024: 1–15, DOI: 10.1155/2024/5669308.
• Yang, C. and J. Wang, 2020 A mathematical model for the novel coronavirus epidemic in wuhan, china. Mathematical Biosciences and Engineering 17: 2708–2724, DOI: 10.3934/mbe.2020148.
• Zu, Z., M. Jiang, P. Xu, W. Chen, Q. Ni, et al., 2020 Coronavirus disease 2019 (covid-19): A perspective from china. Radiology 296: E15–E25, DOI: 10.1148/radiol.2020200490.