A study of global dynamics and sensitivity analysis of a discrete-time model of the COVID-19 epidemic

Document Type : Research Article


Department of Mathematics and Computer Science, Abdelhamid Ibn Badis Mostaganem University, Algeria.


This study presents a novel approach to understanding the global dynam-ics of COVID-19 transmission with vaccination based on a discrete-time model. We establish biologically meaningful constraints on the model parameters and prove the existence of a disease-free equilibrium and an endemic equilibrium under these constraints, along with their theoretical stabilities. Furthermore, we identify the most sensitive operational param-eters that have a substantial impact on the transmission of the epidemic. Through numerical simulations, we demonstrate the local stability of the two equilibria, depending on the parameter values. Our findings reveal that the discrete-time model is not only dynamically robust but also more realistic than its continuous counterpart under biologically meaningful con-straints. These results provide a foundation for future research in this area and contribute to our understanding of the global dynamics of COVID-19 transmission.


Main Subjects

[1] Alimerina, H., Ablaoui, N.L., Maamar, M.H., Bouzid, L., Tabharit, L. and Bel-hamiti, O. Mathematical modeling of the impact of vaccination on infected linked Covid-19 disease, International Conference on Recent Advances in Mathematics
and Informatics (ICRAMI). IEEE, 2021.
[2] Almatroud, A.O., Djenina, N., Ouannas, A., Grassi, G. and Al-Sawalha, M.M. A novel discrete-time COVID-19 epidemic model including the compartment of vaccinated individuals, Math. Biosci. Eng. 19 (2022), 12387–12404.
[3] Balabdaoui, F. and Mohr, D. Age-stratified discrete compartment model of the COVID-19 epidemic with application to Switzerland, Sci. Rep. 10(1) (2020), 1–12.
[4] Cavanaugh, A.M., Spicer, K.B., Thoroughman, D., Glick, C. and Winter, K. Re-duced risk of reinfection with SARS-CoV-2 after COVID-19 vaccination–Kentucky, May–June 2021, Morb. Mortal. Wkly. Rep. 70(32) (2021), 1081.
[5] Chitnis, N., Hyman, J.M. and Cushing, J.M. Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull. Math. Biol. 70 (2008), 1272–1296. https://doi.org/10.1007/s11538-008-9299-0
[6] Cui, J., Li, F. and Shi, Z.L. Origin and evolution of pathogenic coronaviruses, Nat. Rev. Microbiol. 17(3) (2019), 181–192.
[7] Dammann, O., Gray, P. and Gressens, P., Wolkenhauer, O. and Leviton, A. Systems epidemiology: What’s in a name?, Online J. Public Health Inform. 6(3) (2014).
[8] De la Sen, M., Alonso-Quesada, S. and Ibeas, A. On a discrete SEIR epidemic model with exposed infectivity, feedback vaccination and partial delayed re-susceptibility, Mathematics, 9(5) (2021), 520.
[9] De la Sen, M., Alonso-Quesada, S., Ibeas, A. and Nistal, R. On a discrete SEIR epi-demic model with two-doses delayed feedback vaccination control on the susceptible, Vaccines, 9 (2021), 398.
[10] Diekmann, O., Heesterbeek, J.A.P. and Metz, J.A. On the definition and the com-putation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28 (1990), 365–382.
[11] Imperial College COVID-19 Response Team, Impact of non-pharmaceutical in-terventions (NPIs) to reduce COVID-19 mortality and healthcare demand, 20 (10.25561) (2020), 77482.
[12] Ivorra, B., Ferrández, M.R., Vela-Pérez, M. and Ramos, A.M. Mathematical mod-eling of the spread of the coronavirus disease 2019 (COVID-19) taking into account the undetected infections. The case of China, Commun. Nonlinear Sci. Numer. Simul. 88 (2020), 105303.
[13] Killerby, M.E., Biggs, H.M., Midgley, C.M., Gerber, S.I. and Watson, J.T. Mid-dle East respiratory syndrome coronavirus transmission, Emerg. Infect. Dis. 26(2) (2020), 191.
[14] Kranz, J. and Hau, B. Systems analysis in epidemiology, Annu. Rev. Phytopathol. 18(1) (1980), 67–83.
[15] Krishna, M.V. Mathematical modelling on diffusion and control of COVID–19, Infect. Dis. Model., 5 (2020), 588–597.
[16] Ksiazek, T.G., Erdman, D., Goldsmith, C.S., Zaki, S.R., Peret, T., Emery, S., Tong, S., Urbani, C., Comer, J.A., Lim, W. and Rollin, P.E. A novel coronavirus associated with severe acute respiratory syndrome, N. Engl. J. Med., 348(20) (2003),
[17] Liu, Y., Yan, L.M., Wan, L., Xiang, T.X., Le, A., Liu, J.M. and Zhang, W. Viral dynamics in mild and severe cases of COVID-19, Lancet Infect. Dis. 20(6) (2020), 656–657.
[18] Morgan, G., de Azambuja, E., Punie, K., Ades, F., Heinrich, K., Personeni, N., Rahme, R., Ferrara, R., Pels, K., Garassino, M. and von Bergwelt-Baildon, M. OncoAlert round table discussions: The global COVID-19 experience, JCO Global
Oncology, 7 (2021), 455–463.
[19] Peiris, J.S., Guan, Y. and Yuen, K. Severe acute respiratory syndrome, Nat. Med. 10(12) (2004), S88–S97.
[20] Rabitz, H., Kramer, M. and Dacol, D. Sensitivity analysis in chemical kinetics, Annu. Rev. Phys. Chem. 34 (1983), 419–461.
[21] Ramosa, A.M., Ferrández, M.R., Vela-Pérez, M. and Ivorra, B. A simple but com-plex enough θ-SIR type model to be used with COVID-19 real data. Application to the case of Italy, Phys. D: Nonlinear Phenom. 421 (2021), 132839.
[22] Van den Driessche, P. and Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), 29–48. https://doi.org/10.1016/S0025-5564(02)00108-6.
[23] World Health Organization, Naming the coronavirus disease (COVID-19) and the virus that causes it, Braz. J. Implantol. Health Sci. 2(3) (2020).
[24] World Health Organization, Novel Coronavirus (2019-nCoV): situation report 1, https://www.who.int/docs/default-source/coronaviruse/ situation-reports/20200121-sitrep-1-2019-ncov.pdf (accessed on 20 January 2023).
[25] World Health Organization, Novel Coronavirus (2019-nCoV):situation report 51, https://www.who.int/docs/default-source/coronaviruse/ situation-reports/20200311-sitrep-51-covid-19.pdf (accessed on 20 January 2023).
[26] World Health Organization, Novel Coronavirus (2019-nCoV):situation report 75, https://www.who.int/docs/default-source/coronaviruse/ situation-reports/20200404-sitrep-75-covid-19.pdf (accessed on 20 January 2023).
[27] World Health Organization, Weekly epidemiological update on COVID-19 - 21 December 2021, https://www.who.int/publications/m/item/ weekly-epidemiological-update-on-covid-19---21-december-2021 (accessed on 20 January 2023).
[28] World Health Organization, Weekly epidemiological update on COVID-19 09 November 2022, https://www.who.int/publications/m/item/ weekly-epidemiological-update-on-covid-19---9-november-2022 (accessed on 20 January 2023)
[29] Xu, M., Wang, D., Wang, H., Zhang, X., Liang, T., Dai, J., Li, M., Zhang, J., Zhang, K., Xu, D. and Yu, X. COVID-19 diagnostic testing: technology perspective, Clin. Transl. Med. 10(4) (2020), e158.
[30] Yang, B., Yu, Z. and Cai, Y. The impact of vaccination on the spread of COVID-19: Studying by a mathematical model, Phys. A: Stat. Mech. 590 (2022), 126717.
[31] Yavuz, M.Ö., Co┼čar, F., Günay, F. and Özdemir, F.N. A new mathematical modeling of the COVID-19 pandemic including the vaccination campaign, Open Journal of Modelling and Simulation, 9(3) (2021), 299–321.
[32] Zaki, A.M., Van Boheemen, S. , Bestebroer, T.M., Osterhaus, A.D. and Fouchier, R.A. Isolation of a novel coronavirus from a man with pneumonia in Saudi Arabia, New Engl. J. Med. 367(19) (2012), 1814–1820.