Mathematical Modeling of COVID-19: Stability Analysis and Numerical Simulation
Keywords:
COVID-19, Reproduction Number, Equilibrium Point, Stability Analysis, Numerical Simulation.Abstract
COVID-19 has highlighted the critical role of population mobility and regional interactions in shaping the dynamics of infectious diseases. This study proposes a nonlinear model depicting the dynamics of the coronavirus, specifically accounting for behavioral heterogeneity between resident and visitor populations. R_0, (the basic reproduction number) is determined as it is the main determining parameter to analyze the dynamics of the disease. Stability analysis reveals that the disease-free- equilibrium is globally asymptotically-stable when R_0 < 1. For R_0 > 1, the global asymptotic stability of the endemic- equilibrium is established using a Lyapunov function. Furthermore, sensitivity analysis is performed to find the key parameters driving the progression of the pandemic. Numerical simulations are presented with available parametric values to validate the theoretical findings, providing insights into the interpretations that can be drawn about the dynamics of the disease.
Downloads
References
[1] The mathematical fractional modeling of tio-2 nanopowder synthesis by sol–gel method at low temperature. 1
[2] Folashade B Agusto, Shamise Easley, Kenneth Freeman, and Madison Thomas, Mathematical model of three age-structured transmission dynamics of chikungunya virus, Computational and mathematical methods in medicine 2016 (2016), no. 1, 4320514. 3.1
[3] W Ahmad, MA Nazir, M Rafiq, AIK Butt, N Ahmad, and M Hussain, Analyzing optimal control techniques in a nonlinear fractional rubella model with the atangana–baleanu derivative, Computers in Biology and Medicine 197 (2025), 110954. 1
[4] W Ahmad, H Ullah, M Rafiq, AIK Butt, and N Ahmad, Analytical and numerical investigations of optimal control techniques for managing ebola virus disease, The European Physical Journal Plus 140 (2025), no. 4, 329. 1
[5] Waheed Ahmad, Muhammad Rafiq, Azhar Iqbal Kashif Butt, Momina Zainab, and Naeed Ahmad, Dynamics of bisusceptibility patterns in covid-19 outbreaks and associated abstain strategies, Modeling Earth Systems and Environment 11 (2025), no. 3, 198. 1
[6] Waheed Ahmad, Muhammad Sultan Aslam, Muhammad Rafiq, Junaid Ahmad, and Ali Raza, Modeling the impact of optimized treatment patterns on dengue transmission dynamics, Chaos: An Interdisciplinary Journal of Nonlinearscience 35 (2025), no. 10. 1
[7] Ruchi Arora, Dharmendra Kumar, Ishita Jhamb, and Avina Kaur Narang, Mathematical modeling of chikungunya dynamics: Stability and simulation, Cubo (Temuco) 22 (2020), no. 2, 177–201. 4.4
[8] C. P. Bhunu, W. Garira, and Z. Mukandavire, Modeling hiv/aids and tuberculosis coinfection, Bulletin of Mathematical Biology 71 (2009), no. 7, 1745–1780. 5.2
[9] Sudhanshu Kumar Biswas, Jayanta Kumar Ghosh, Susmita Sarkar, and Uttam Ghosh, Covid-19 pandemic in india: a mathematical model study, Nonlinear Dynamics 102 (2020), no. 1, 537–553. 4.3
[10] Azhar Iqbal Kashif Butt, Tariq Ismaeel, Sara Khan, Muhammad Imran, Waheed Ahmad, IA Rashid, and Muhammad Sajid Riaz, Investigating the role of antimalarial treatment and mosquito nets in malaria transmission and control through mathematical modeling, Comput. Model. Eng. Sci. 144 (2025), no. 3, 3463–3492. 1
[11] Edmund X DeJesus and Charles Kaufman, Routh-hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations, Physical Review A 35 (1987), no. 12, 5288. 4.3
[12] O Diekmann and JAP Heesterbeek, Mathematical epidemiology of infectious diseases, Model Building, Analysis (1989). 1
[13] Taruna Garg, Madhuchanda Rakshit, et al., Analyzing pneumonia transmission with a fractional sveir model: the role of vaccination in disease eradication, Discover Public Health 23 (2026), no. 1, 36. 1
[14] Taruna Garg, Madhuchanda Rakshit, M Manivel, and Shyamsunder, Modeling of hepatitis b virus transmission with vaccination, treatment, and memory effects, Advanced Theory and Simulations 9 (2026), no. 1, e01358. 1
[15] Delhi Government, Annual report, http://des.delhigovt.nic.in/wps/wcm/connect/3b3bb3804949e070adc3bf26edbf4824/Annual+Report+2017.pdf?MOD=AJPERES&lmod=-1268218958&CACHEID=3b3bb3804949e070adc3bf26edbf4824&Annual%20report%20of%20Registration%20of%20Births%20and%20Deaths%20in%20Delhi%202017.
[16] Herbert W. Hethcote, The mathematics of infectious diseases, SIAM Review 42 (2000), no. 4, 599–653. 1
[17] Manh Tuan Hoang, A simple approach for studying stability properties of an seirs epidemic model, Journal of Applied Analysis 31 (2025), no. 1, 143–156. 5.3
[18] Ruchi Kaur, Prabhanshi, Ishita Jhamb, and Pratibha Verma, Transmission dynamics of covid-19 across a region: A mathematical model, Proceedings of the National Academy of Sciences, India, Section A: Physical Sciences (2025), In press. 1, 2
[19] Ruchi Kaur, Naman Taggar, Vaishnavi Rajagopalan, and Jaspreet Kaur, Mathematical modelling in epidemiology, RESONANCE (2026), 45. 1
[20] Subhas Khajanchi, Kankan Sarkar, and Jayanta Mondal, Dynamics of the covid-19 pandemic in india, (2021).
[21] Gwenan M Knight, Thi Mui Pham, James Stimson, Sebastian Funk, Yalda Jafari, Diane Pople, Stephanie Evans, Mo Yin, Colin S Brown, Alex Bhattacharya, et al., The contribution of hospital-acquired infections to the covid-19 epidemic in england in the first half of 2020, BMC infectious diseases 22 (2022), no. 1, 556. 2
[22] Kok Yew Ng and Meei Mei Gui, Covid-19: Development of a robust mathematical model and simulation package with consideration for ageing population and time delay for control action and resusceptibility, Physica D: Nonlinear Phenomena 411 (2020), 132599. 3.2
[23] SD Purohit et al., A novel study of the impact of vaccination on pneumonia via fractional approach, Partial Differential Equations in Applied Mathematics 10 (2024), 100698. 1
[24] O Sadek, L Sadek, S Touhtouh, and A Hajjaji, The mathematical fractional modeling of tio-2 nanopowder synthesis by sol–gel method at low temperature, Math. Model. Comput 9 (2022), no. 3, 616–626. 1
[25] Piu Samui, Jayanta Mondal, and Subhas Khajanchi, A mathematical model for covid-19 transmission dynamics with a case study of india, Chaos, Solitons & Fractals 140 (2020), 110173. 5.1
[26] Nita H. Shah, Nisha Sheoran, Ekta Jayswal, Dhairya Shukla, Nehal Shukla, Jagdish Shukla, and Yash Shah, Modelling covid-19 transmission in the united states through interstate and foreign travels and evaluating impact of governmental public health interventions, Journal of Mathematical Analysis and Applications (2020), 124896. 1
[27] Shyamsunder and Manisha Meena, A comparative analysis of vector-borne disease: monkeypox transmission outbreak, Journal of Applied Mathematics and Computing 71 (2025), no. 4, 6119–6155. 1
[28] G.F. Simmons, Differential equations with applications and historical notes, Textbooks in Mathematics, CRC Press, 2016. 5.2
[29] Shakir Ullah, Imtiaz Ahmad, Mohammad Idrees, Saeed Islam, and Ishtiaq Ali, Analysis of tuberculosis transmission dynamics with environmental and reinfection factors: a mathematical approach, Journal of Applied Mathematics and Computing 72 (2026), no. 2, 86. 3.2
[30] WHO, Covid-19 vaccines, https://www.who.int/emergencies/diseases/novel-coronavirus-2019/covid-19-vaccines.
1
Downloads
Published
Issue
Section
License
Copyright (c) 2026 Ruchi Kaur, Prabhanshi, Ishita Jhamb
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
