SEIQR Model Sensitivity and Bifurcation Analysis of SARS-CoV-2 Dynamics with International in-out Mobility Control in Indonesia

Lasker Pangarapan Sinaga, Dinda Kartika, Nurul Ain Farhana


This study aims to analyze the SEIQR model for the SARS-CoV-2 dynamic by considering in-out mobility. The model construction is based on the COVID-19 response strategy implemented by the Indonesian government, then analyzing the model by determining the equilibrium point and basic reproduction number, analyzing model stability, parameter sensitivity, and bifurcation. The results show that the model has stable disease-free and disease-endemic critical points when the parameter inequality conditions based on the Routh-Hurwitz criteria are satisfied. Numerical simulations show that the system takes a long time to reach equilibrium. Furthermore, the sensitivity analysis of the basic reproduction number shows that the most sensitive parameters are natural birth and death rate susceptible, contact rate of susceptible individuals with infected individuals from local and international subjects, and rate of exposed individuals who have infected. Thus, efforts to handle COVID-19 in Indonesia can be improved by focusing on controlling international in-out mobility, so that the number of exposed individuals who have been infected can be reduced. Moreover, the bifurcation analysis shows that the system undergoes forward or backward bifurcation under disease-free conditions if certain coefficient values are satisfied based on the Castillo-Chavez and Song conditions.


Stability; Sensitivity; Bifurcation; SEIQR Model; SARS-CoV-2; Covid-19

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H. Ouassou et al., “The Pathogenesis of Coronavirus Disease 2019 (COVID-19): Evaluation and Prevention,” J. Immunol. Res., vol. 2020, pp. 1–7, Jul. 2020, doi: 10.1155/2020/1357983.

Kementerian Kesehatan Republik Indonesia, “COVID-19 Situation in Indonesia.” Accessed: Jan. 23, 2023. [Online]. Available:

Gugus Tugas Percepatan Penanganan Covid-19, “Pedoman Pencegahan dan Pengendalian Coronavirus Disease 2019 (COVID-19).” Accessed: Jan. 23, 2023. [Online]. Available:

F. Ndaïrou, I. Area, J. J. Nieto, and D. F. M. Torres, “Mathematical modeling of COVID-19 transmission dynamics with a case study of Wuhan,” Chaos Solitons Fractals, vol. 135, p. 109846, Jun. 2020, doi: 10.1016/j.chaos.2020.109846.

O. J. Peter, A. A. Ayoade, A. I. Abioye, A. A. Victor, and C. E. Akpan, “Sensitivity analysis of the parameters of a cholera model,” J. Appl. Sci. Environ. Manag., vol. 22, no. 4, p. 477, May 2018, doi: 10.4314/jasem.v22i4.6.

H.-F. Huo, P. Yang, and H. Xiang, “Stability and bifurcation for an SEIS epidemic model with the impact of media,” Phys. Stat. Mech. Its Appl., vol. 490, pp. 702–720, Jan. 2018, doi: 10.1016/j.physa.2017.08.139.

S. Annas, Muh. Isbar Pratama, Muh. Rifandi, W. Sanusi, and S. Side, “Stability analysis and numerical simulation of SEIR model for pandemic COVID-19 spread in Indonesia,” Chaos Solitons Fractals, vol. 139, p. 110072, Oct. 2020, doi: 10.1016/j.chaos.2020.110072.

R. Resmawan, A. R. Nuha, and L. Yahya, ”Analisis dinamik model transmisi COVID-19 dengan melibatkan intervensi karantina,” Jambura J. Math, vol. 3, no. 1, pp. 66-79, 2021, doi: 10.34312/jjom.v3i1.8699.

A. Zeb, E. Alzahrani, V. S. Erturk, and G. Zaman, “Mathematical Model for Coronavirus Disease 2019 (COVID-19) Containing Isolation Class,” BioMed Res. Int., vol. 2020, pp. 1–7, Jun. 2020, doi: 10.1155/2020/3452402.

I. Darti, Trisilowati, M. Rayungsari, R. R. Musafir, and A. Suryanto, “A SEIQRD epidemic model to study the dynamics of COVID-19 disease,” Commun. Math. Biol. Neurosci., 2023, doi: 10.28919/cmbn/7822.

O. E. Deeb and M. Jalloul, “The Dynamics of COVID-19 spread: Evidence from Lebanon,” Math. Biosci. Eng., vol. 17, no. 5, pp. 5618–5632, 2020, doi: 10.3934/mbe.2020302.

C. E. Madubueze, S. Dachollom, and I. O. Onwubuya, “Controlling the Spread of COVID-19: Optimal Control Analysis,” Comput. Math. Methods Med., vol. 2020, pp. 1–14, Sep. 2020, doi: 10.1155/2020/6862516.

R. Resmawan and L. Yahya, ”Sensitivity analysis of mathematical model of coronavirus disease (COVID-19) transmission,” CAUCHY: Jurnal Matematika Murni dan Aplikasi, vol. 6, no. 2. pp. 91-99, 2020, doi: 10.18860/ca.v6i2.9165.

T. Hussain, M. Ozair, F. Ali, S. U. Rehman, T. A. Assiri, and E. E. Mahmoud, “Sensitivity analysis and optimal control of COVID-19 dynamics based on SEIQR model,” Results Phys., vol. 22, p. 103956, Mar. 2021, doi: 10.1016/j.rinp.2021.103956.

M. Parsamanesh and M. Erfanian, “Stability and bifurcations in a discretetime SIVS model with saturated incidence rate,” Chaos Solitons Fractals, vol. 150, p. 111178, Sep. 2021, doi: 10.1016/j.chaos.2021.111178.

L. P. Sinaga, D. Kartika, A. D. R. Sianipar, and R. L. Purba, “Stability analysis of modified SEIQR on dynamics of SARS-CoV-2 with control on vaccination, quarantine and international in-out mobility in Indonesia,” in The 9th Annual International Seminar on Trends in Science and Science Education (AISTSSE) 2022, Sciendo, 2023, pp. 534–544. doi: 10.2478/9788367405195-069.

Kementerian Kesehatan Republik Indonesia, “Pedoman Pencegahan dan Pengendalian Coronavirus Disease (COVID-19).” Kementerian Kesehatan Republik Indonesia, Jul. 13, 2020. Accessed: Jan. 23, 2023. [Online]. Available:

L. Perko, Differential equations and dynamical systems, 3rd ed. in Texts in applied mathematics, no. 7. New York: Springer, 2001.

R. C. Dorf and R. H. Bishop, Modern control systems, Thirteenth edition. Hoboken: Pearson, 2016.

P. Van Den Driessche and J. Watmough, “Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission,” Math. Biosci., vol. 180, no. 1–2, pp. 29–48, Nov. 2002, doi: 10.1016/S0025-5564(02)00108-6.

N. Chitnis, J. M. Hyman, and J. M. Cushing, “Determining Important Parameters in the Spread of Malaria Through the Sensitivity Analysis of a Mathematical Model,” Bull. Math. Biol., vol. 70, no. 5, pp. 1272–1296, Jul. 2008, doi: 10.1007/s11538-008-9299-0.

C. Castillo-Chavez, B. Song, 1. Mathematics, Computational and Modeling Sciences Center, Arizona State University, PO Box 871904, Tempe, AZ 85287, and 2. Department of Mathematical Sciences, Montclair State University, Upper Montclair, NJ 07043, “Dynamical Models of Tuberculosis and Their Applications,” Math. Biosci. Eng., vol. 1, no. 2, pp. 361–404, 2004, doi: 10.3934/mbe.2004.1.361.

C. Elik, “Bifurcation Analysis and Its Applications,” in Numerical Simulation - From Theory to Industry, M. Andriychuk, Ed., InTech, 2012. doi: 10.5772/50075.


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