Analisis Kestabilan Global dengan Menggunakan Fungsi Lyapunov pada Model Dinamik Epidemik SIR

Lisna Nurjanah; Fadilah Ilahi; Dani Suandi

doi:10.15575/kubik.v3i1.2733

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Lisna Nurjanah
UIN Sunan Gunung Djati
Indonesia

Fadilah Ilahi
UIN Sunan Gunung Djati
Indonesia

Dani Suandi
Politeknik PIKSI Ganesha
Indonesia

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Analisis Kestabilan Global dengan Menggunakan Fungsi Lyapunov pada Model Dinamik Epidemik SIR

Lisna Nurjanah^(1*), Fadilah Ilahi⁽²⁾, Dani Suandi⁽³⁾

(1) UIN Sunan Gunung Djati, Indonesia
(2) UIN Sunan Gunung Djati, Indonesia
(3) Politeknik PIKSI Ganesha, Indonesia
(*) Corresponding Author

Abstract

Artikel ini menganalisis kestabilan global dengan menggunakan fungsi Lyapunov pada model dinamik epidemik SIR. Populasi manusia diasumsikan menjadi tiga bagian yaitu individu rentan (susceptible), terinfeksi (infected) dan kebal (recovered). Titik tetap terdiri dari titik tetap bebas penyakit dan endemik. Kestabilan yang dikaji berupa kestabilan global dari titik tetap bebas penyakit dan endemik menggunakan fungsi Lyapunov. Berdasarkan hasil analisis, pada titik tetap bebas penyakit dapat disimpulkan bahwa titik tersebut bersifat stabil asimtot global jika . Sedangkan pada titik tetap endemik dapat disimpulkan bersifat stabil global karena definit positif dan turunan fungsi tersebut semi definit negatif.

Keywords

fungsi Lyapunov; kestabilan global; model SIR; epidemik; endemik

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References

D. G. Zill, A First Course in Differential Equations with Modeling Applications. 2009.

O. O. Rufai, O. S. Obabiyi, and O. Y. Saka-balogun, “Psycho-Social Effects on Sir Epidemic Model,” vol. 30, no. 1, p. 6107, 2015.

A. Korobeinikov and G. C. Wake, “Lyapunov functions and global stability for SIR, SIRS, and SIS epidemiological models,” Appl. Math. Lett., vol. 15, no. 8, pp. 955–960, 2002.

B. S. Goh, “Global stability in many-species systems,” Am. Nat., vol. 111, no. 977, pp. 135–143, 1977.

A. Korobeinikov and P. K. Maini, “A Lyapunov function and global properties for SIR and SEIR epidemiological models with nonlinear incidence,” Math. Biosci. Eng., vol. 1, no. 1, pp. 57–60, 2004.

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C. C. McCluskey, “Complete global stability for an SIR epidemic model with delay—distributed or discrete,” Nonlinear Anal. Real World Appl., vol. 11, no. 1, pp. 55–59, 2010.

DOI: https://doi.org/10.15575/kubik.v3i1.2733

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Analisis Kestabilan Global dengan Menggunakan Fungsi Lyapunov pada Model Dinamik Epidemik SIR

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