THE DEVELOPMENT OF COVID-19 USING OUTBREAK THE SUSCEPTIBLE, INFECTED, AND RECOVERED (SIR) MODEL WITH VACCINATION

  • Dorrah Azis Mathematics Department, Faculty of Mathematic and Natural Sciences, University of Lampung, Indonesia
  • La Zakaria Mathematics Department, Faculty of Mathematic and Natural Sciences, University of Lampung, Indonesia
  • Tiryono Ruby Mathematics Department, Faculty of Mathematic and Natural Sciences, University of Lampung, Indonesia
  • Muhammad Is’ad Arifaldi Mathematics Department, Faculty of Mathematic and Natural Sciences, University of Lampung, Indonesia
DOI: https://doi.org/10.30598/barekengvol17iss3pp1325-1340
Keywords: SIR Model, Covid-19, Basic Reproduction Number, Routh-Hurwitz Criterion

Abstract

The Covid-19 pandemic in 2020 has caused severe problems in Indonesia. The Covid-19 virus epidemic can be modeled using the Susceptible, Infected, and Recovered (SIR) model. This modeling aims to look at the dynamics of Covid-19 to predict when disease-free and endemic disease occurs and to find the basic reproduction number ( ) for policy making in suppressing the spread of Covid-19. In this article, we describe and solve a research result on the SIR model with an assumption. The assumption in the model is that there is vaccination for the population. There are live stages of research conducted. The first is creating the SIR model and determining the equilibrium points on disease-free and disease-endemic. The Second is getting the basic reproduction number. The third is determining the stability around the equilibrium points using the Routh-Hurwitz criteria. Fourth, create a diagram for the subpopulations state at a specific time using Wolfram Mathematica software. As an implementation of the model created, COVID-19 data at the Batanghari Community Health Center Inpatient UPTD was used. Finally, determine the model error percentage with MAPE. The SIR Covid-19 model was made using eight parameters, namely , which are all positive. The results showed that the disease-free and disease-endemic equilibrium points were locally asymptotically stable after being analyzed using the Routh-Hurwitz stability criteria. The model trial using data from UPTD Puskesmas Batanghari obtained a stable condition for up to 100 months with a MAPE of 2.8%. From this study, obtained an . This means that if you want to reduce the rate of spread, then reduce the number of people who are easily infected ( ) and reduce contacts ( ), and increase the healing rate ( ).

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References

E. Callaway, “The race for coronavirus vaccines: a graphical guide,” Nature, vol. 580, no. 7805, pp. 576–577, 2020, doi: doi:10.1038/d41586-020-01221-y.

A. Ajbar, R. T. Alqahtani, and B. Mourad, “Dynamics of an SIR-Based COVID-19 Model With Linear Incidence Rate, Nonlinear Removal Rate, and Public Awareness,” Front. Phys., vol. 9, no. 634251, pp. 1–13, 2021, doi: https://doi.org/10.3389/fphy.2021.634251.

B. Wacker and J. Schlüter, “Time-continuous and time-discrete SIR models revisited: theory and applications,” Adv. Differ. Equations, vol. 2020, no. 556, pp. 1–44, 2020, doi: doi.org/10.1186/s13662-020-02995-1.

Z. Liao, P. Lan, Z. Liao, Y. Zhang, and S. Liu, “TW-SIR: time-window based SIR for COVID-19 forecasts.,” Sci. Rep., vol. 10, no. 1, 2020, doi: https://doi.org/10.1038/s41598-020-80007-8.

A. Mortellaro and P. Ricciardi-Castagnoli, “From vaccine practice to vaccine science: The contribution of human immunology to the prevention of infectious disease,” Immunol. Cell Biol., vol. 89, no. 3, pp. 332–339, 2011, doi: 10.1038/icb.2010.152.

P. Thapa, “Predicating COVID19 Epidemic in Nepal Using the SIR Model,” Stud. Syst. Decis. Control, vol. 358, no. September, pp. 229–237, 2021, doi: 10.1007/978-3-030-69744-0_14.

O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, “On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations,” J. Math. Biol., vol. 28, no. 4, pp. 365–382, 1990, doi: 10.1007/BF00178324.

W. F. Putra, “Analisis Efikasi dan Efektivitas Vaksin COVID-19 terhadap Varian SARS-CoV-2: Sebuah Tinjauan Literatur,” J. Kedokt. Meditek, vol. 28, no. 1, pp. 107–119, 2022, doi: 10.36452/jkdoktmeditek.v28i1.2243.

C. van Oosterhout, N. Hall, H. Ly, and K. M. Tyler, “COVID-19 evolution during the pandemic–Implications of new SARS-CoV-2 variants on disease control and public health policies,” Virulence, vol. 12, no. 1, pp. 507–508, 2021, doi: 10.1080/21505594.2021.1877066.

J. Li et al., “Epidemiology of COVID-19 : A Systematic Review and Meta-analysis of Clinical Epidemiology of COVID-19 : A systematic review and meta-analysis of clinical characteristics, risk factors , and outcomes,” Med. Virol., pp. 1–10, 2020, doi: 10.1002/jmv.26424.

WHO, “Coronavirus disease ( COVID-19 ),” 2020. [Online]. Available: https://www.who.int/docs/default-source/coronaviruse/situation-reports/20201012-weekly-epi-update-9.pdf

J. Sun et al., “COVID-19 : Epidemiology , Evolution , and Cross-Disciplinary Perspectives,” no. January, 2020.

C. Van Oosterhout, N. Hall, H. Ly, and K. M. Tyler, “COVID-19 evolution during the pandemic – Implications of new SARS-CoV-2 variants on disease control and public health policies,” Virulence, vol. 12, no. 1, pp. 507–508, 2021, doi: 10.1080/21505594.2021.1877066.

S. Setiati and M. K. Azwar, “Dilemma of Prioritising Health and the Economy During COVID-19 Pandemic in Indonesia,” Acta Med Indones, vol. 52, no. 3, pp. 196–198, 2020.

A. Mortellaro and P. Ricciardi-Castagnoli, “From vaccine practice to vaccine science : the contribution of human immunology to the prevention of infectious disease,” Immunol. Cell Biol., vol. 89, pp. 332–339, 2011, doi: 10.1038/icb.2010.152.

WHO, “Tanya Jawab : Lockdown dan herd immunity,” 2021. who.int/indonesia/news/novel-coronavirus/qa/qa-lockdown-and-herd-immunity (accessed Jun. 10, 2020).

N. M. Nasir, I. S. Joyosemito, B. Boerman, and I. Ismaniah, “Kebijakan Vaksinasi COVID-19 : Pendekatan Pemodelan Matematika Dinamis Pada Efektivitas Dan Dampak Vaksin Di Indonesia,” J. Abdimas UBJ, vol. 4, no. 2, pp. 191–204, 2021.

W. F. Putra, “Analisis Efikasi dan Efektivitas Vaksin COVID-19 terhadap Varian SARS-CoV-2 : Sebuah Tinjauan Literatur Analysis the Efficacy and Effectivity of COVID-19 Vaccines to the SARS-CoV-2 Variants : A Literature Review,” Meditek, vol. 28, no. 1, pp. 107–119, 2022, doi: https://doi.org/10.36452/jkdoktmeditek.v28i1.2243.

W. O. Kermarck and A. G. McKendrick, “A Contribution to the Mathematical Theory o f Epidemics.,” in The Royal Society London A, Royal Society, 1927, pp. 700–721. doi: 10.1098/rspa.1927.0118.

P. Thapa, “Predicating COVID19 epidemic in Nepal using the SIR model,” Artif. Intell. COVID-19., 2021, doi: 10.1007/978-3-030-69744-0_14.

M. A. Shereen, S. Khan, A. Kazmi, N. Bashir, and R. Siddique, “COVID-19 infection: origin, transmission, and characteristics of human coronaviruses.,” J. Adv. Res., vol. 24, no. 2020, pp. 91–98, 2020, doi: 10.1016/j.jare.2020.03.005.

Y. F. Lin et al., “Spread and Impact of COVID-19 in China: A Systematic Review and Synthesis of Predictions From Transmission-Dynamic Models. Frontiers in Medicine, 7. doi:10.3389/fmed.2020.00321,” Front. Med., vol. 7, no. 321, pp. 1–11, 2020, doi: doi:10.3389/fmed.2020.00321.

Q. Griette and P. Magal, “Clarifying predictions for COVID-19 from testing data: The example of New York State.,” Infect. Dis. Model., vol. 6, pp. 273–283, 2021, doi: https://doi.org/10.1016/j.idm.2020.12.011.

Z. Liu, P. Magal, and G. Webb, “Predicting the number of reported and unreported cases for the COVID- 19 epidemics in China, South Korea, Italy, France, Germany and United Kingdom,” J. Theor. Biol. 509, 110501., vol. 509, no. 110501, 2021, doi: doi:10.1016/j.jtbi.2020.110501.

A. J. Kucharski, T. W. Russell, J. Diamond, C., Liu, Y., Edmunds, S. Funk, and R. M. Eggo, “Early dynamics of transmission and control of COVID-19: a mathematical modelling study.,” Lancet Infect. Dis., vol. 20, pp. 1–7, 2020, doi: https://doi.org/10.1016/S1473-3099(20)30144-4.

E. S. Kurkina and E. M. Koltsova, “Mathematical Modeling of the Propagation of Covid-19 Pandemic Waves in the World.,” Comput. Math. Model., vol. 32, no. 2, pp. 147–170, 2021, doi: doi:10.1007/s10598-021-09523-0.

A. Lobo et al., “COVID-19 epidemic in Brazil: where we at?,” Int. J. Infect. Dis., vol. 97, pp. 382–385, 2020, doi: doi:10.1016/j.ijid.2020.06.044.

K. Roosa et al., “Real-time forecasts of the COVID-19 epidemic in China from February 5th to February 24th, 2020.,” Infect. Dis. Model., vol. 5, no. 2020, pp. 256–263, 2020, doi: 10.1016/j.idm.2020.02.002.

H. W. Hethcote, “The mathematics of infectious diseases,” SIAM Rev., vol. 42, no. 4, pp. 599–653, 2000, [Online]. Available: http://www.siam.org/journals/sirev/42-4/37190.html

K. R. Meyer, “Normal forms for the general equilibrium,” Funkc. Ekvacioj, vol. 27, pp. 261–271, 1984.

G. J. Olsder and J. W. van der Woude, Mathematical Systems Theory, Second. Delft, The Netherland: Delft Univrsity Press, 1997.

O. Diekmann, J. A. P. Heesterbeek, and J. A. J. Metz, “On the definition and the computation of the basic reproduction ratio Ro in models for infectious diseases in heterogeneous populations,” J. Math. Biol, vol. 28, pp. 365–382, 1990.

A. Faruk, “Model Epidemik Tuberkulosis Seir dengan Terapi pada Individu Terinfeksi,” J. Penelit. Sains, vol. 18, no. 3, pp. 99–104, 2016, doi: 10.56064/jps.v18i3.16.

J. Giesecke, Modern Infectious Disease Epidemiology, Third. CRC Press Taylor & Francis Group, 2017.

P. Van Den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,” Math. Biosci. 180, vol. 180, pp. 29–48, 2002, [Online]. Available: http://www.math.unb.ca/?watmough.

F. R. Gantmacher, The theory of matrices. Chelsea Publishing Company, 1959.

C. D. Lewis, Demand Forecasting and Inventory Control: A computer aided learning approach, vol. 2. Woodhead Publishing Ltd, 1997.

WHO, “Evaluation of COVID-19 vaccine effectiveness,” 2021. https://www.who.int/publications/i/item/WHO-2019-nCoV-vaccine_effectiveness- measurement-2021

Published
2023-09-30
How to Cite
[1]
D. Azis, L. Zakaria, T. Ruby, and M. Arifaldi, “THE DEVELOPMENT OF COVID-19 USING OUTBREAK THE SUSCEPTIBLE, INFECTED, AND RECOVERED (SIR) MODEL WITH VACCINATION”, BAREKENG: J. Math. & App., vol. 17, no. 3, pp. 1325-1340, Sep. 2023.
Issue
Vol 17 No 3 (2023): BAREKENG: Journal of Mathematics and Its Applications
Section
Articles

Copyright (c) 2023 Dorrah Azis, La Zakaria, Tiryono Ruby, Muhammad Is’ad Arifaldi

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