Mathematics Model of COVID-19 with Two-Stage Vaccination, Symptomatic, Asymptomatic, and Quarantine Individuals

Nur Inayah, Muhammad Manaqib, Muhammad Febry Fadillah


This research developed a model of COVID-19 based on the SEIR model which was further developed by dividing the infected subpopulation into symptomatic and asymptomatic, adding quarantine of infected individuals and vaccination in two steps. Making this model begins with making a compartment diagram of the disease and then forming a system of differential equations. After the model is formed, the disease-free equilibrium point, endemic equilibrium point, and basic reproduction number (R0) are obtained. Analysis of the stability of the disease-free equilibrium point was locally asymptotically stable if R0<1 and an endemic equilibrium point existed if R0>1. Numerical simulation for the model that has been made is in line with the analysis. Furthermore, the sensitivity analysis obtained that the parameters that have a significant effect on the spread of COVID-19 are the rate of the first dose vaccination, the rate of contact with symptomatic or asymptomatic individuals, and the rate of quarantine of symptomatic infected individuals.


COVID-19, Equilibrium Point, Basic Reproduction Number, Sensitivity Analysis

Full Text:



D. O. Wiebers and V. L. Feigin, “What the COVID-19 crisis is telling humanity,” Neuroepidemiology, p. 1, 2020.

B. K. Mishra et al., “COVID-19 created chaos across the globe: Three novel quarantine epidemic models,” Chaos, Solitons & Fractals, vol. 138, p. 109928, 2020.

Worldometer, “COVID-19 Coronavirus Pandemic,” 2021. [Online]. Available: [Accessed: 20-Jul-2021].

R. Tosepu, J. Gunawan, D. S. Effendy, H. Lestari, H. Bahar, and P. Asfian, “Correlation between weather and Covid-19 pandemic in Jakarta, Indonesia,” Sci. Total Environ., vol. 725, p. 138436, 2020.

SATGAS-COVID19, “Pasien Sembuh Terus Meningkat Mencapai 1.890.287 Orang,” 2021. [Online]. Available:

J. B. Aguilar, J. S. Faust, L. M. Westafer, and J. B. Gutierrez, “Investigating the impact of asymptomatic carriers on COVID-19 transmission,” MedRxiv, 2020.

A. A. Sayampanathan, C. S. Heng, P. H. Pin, J. Pang, T. Y. Leong, and V. J. Lee, “Infectivity of asymptomatic versus symptomatic COVID-19,” Lancet, vol. 397, no. 10269, pp. 93–94, 2021.

T. T. Le et al., “The COVID-19 vaccine development landscape,” Nat Rev Drug Discov, vol. 19, no. 5, pp. 305–306, 2020.

WHO, “WHO validates Sinovac COVID-19 vaccine for emergency use and issues interim policy recommendations,” 2021. [Online]. Available:

G. O. Fosu, J. M. Opong, and J. K. Appati, “Construction of compartmental models for COVID-19 with quarantine, lockdown and vaccine interventions,” Lockdown Vaccine Interv. (April 12, 2020), 2020.

I. Ahmed, G. U. Modu, A. Yusuf, P. Kumam, and I. Yusuf, “A mathematical model of Coronavirus Disease (COVID-19) containing asymptomatic and symptomatic classes,” Results Phys., vol. 21, p. 103776, 2021.

E. A. Iboi, C. N. Ngonghala, and A. B. Gumel, “Will an imperfect vaccine curtail the COVID-19 pandemic in the US?,” Infect. Dis. Model., vol. 5, pp. 510–524, 2020.

M. Martcheva, An introduction to mathematical epidemiology, vol. 61. Springer, 2015.

B. Anderson, J. Jackson, and M. Sitharam, “Descartes’ rule of signs revisited,” Am. Math. Mon., vol. 105, no. 5, pp. 447–451, 1998.

P. Lawrence, “Differential equations and dynamical systems.” Springer-Verlag, New York, 1991.

BPS, “Jumlah Penduduk Hasil SP2020 Provinsi DKI Jakarta sebesar 10.56 juta jiwa,” 2021. [Online]. Available:

BPOM, “Badan POM Terbitkan EUA, Vaksin CoronaVac Sinovac Siap Disuntikkan,” 2021. [Online]. Available:

WHO, “The Sinovac COVID-19 vaccine: What you need to know,” 2021. [Online]. Available:

G. Perkasa, “Berapa lama tubuh akan kebal dari virus corona setelah disuntik vaksin,” 2020. [Online]. Available:

G. S. Putri, “Antibodi Baru Terbentuk Hari Ke-28 Setelah Vaksin, Ini Prosesnya,” 2021. [Online]. Available:

R. Resmawan and L. Yahya, “Sensitivity analysis of mathematical model of coronavirus disease (COVID-19) transmission,” Cauchy, vol. 6, no. 2, pp. 91–99, 2020.



  • There are currently no refbacks.

Copyright (c) 2022 Nur Inayah, Muhammad Manaqib, Muhammad Febry Fadillah

Creative Commons License
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Editorial Office
Mathematics Department,
Universitas Islam Negeri Maulana Malik Ibrahim Malang
Gajayana Street 50 Malang, East Java, Indonesia 65144
Faximile (+62) 341 558933

Creative Commons License
CAUCHY: Jurnal Matematika Murni dan Aplikasi is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.