Analisis Dinamik Pada Model Penyebaran Penyakit Campak dengan Pengaruh Vaksin Permanen


Dani Suandi(1*)

(1) Politeknik PIKSI Ganesha, Indonesia
(*) Corresponding Author

Abstract


Penyakit campak merupakan penyakit menular yang disebabkan oleh virus golongan Paramixovirus. Kasus campak di Indonesia sering terjadi meskipun telah berhasil direduksi dari angka kejadian 180.000 kasus pada tahun 1990 menjadi sekitar 20.000 kasus pada tahun 2010. Pemberian vaksin campak kepada balita dan anak usia sekolah dasar merupakan salah satu program pemerintah dalam mencegah dan menanggulangi kenaikan angka kejadian penyakit campak. Pada paper ini dikembangkan model matematika untuk penyebaran penyakit campak. Model merupakan sistem dinamik non linear empat dimensi yang menggambarkan pengaruh vaksin permanen terhadap penyebaran penyakit campak. Metode Routh Hurwith digunakan untuk menganalisis kestabilan dari titik ekulibrium endemik. Kita menggunakan basic roproduction number untuk menganlisis keendemikan penyakit yang diperoleh dengan metode next generation matrix. Hasil Analisis dan Simulasi numerik memberikan informasi bahwa laju vaksinasi permanen berpengaruh sangat significant terhadap penurunan populasi manusia yang terinveksi penyakit campak.


Keywords


Model dinamik; Kestabilan titik ekuilibrium endemik; Basic reproduction number; next generation matrix; penyakit campak

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References


D. Didik Budijanto, B. Hardhana, M. Yudianto, and Dkk, “Propil Kesehatan Indonesia 2016,” 2017.

Depkes RI, “Kementerian kesehatan republik indonesia,” 2016.

Y. Zhou, W. Zhang, S. Yuan, and H. Hu, “Persistence and extinction in stochastic sirs models with

general nonlinear incidence rate,” Electron. J. Differ. Equations, vol. 2014, 2014.

X. Wang, Y. Tao, and X. Song, “Analysis of pulse vaccination strategy in SIRVS epidemic model,”

Commun. Nonlinear Sci. Numer. Simul., vol. 14, no. 6, pp. 2747–2756, 2009.

J. M. Heffernan and M. J. Keeling, “An in-host model of acute infection: Measles as a case study,”

Theor. Popul. Biol., vol. 73, no. 1, pp. 134–147, 2008.

G. Zaman, Y. Han Kang, and I. H. Jung, “Stability analysis and optimal vaccination of an SIR

epidemic model,” BioSystems, vol. 93, no. 3, pp. 240–249, 2008.

G. Zaman, Y. H. Kang, G. Cho, and I. H. Jung, “Optimal strategy of vaccination & treatment in an

SIR epidemic model,” Math. Comput. Simul., vol. 136, pp. 63–77, 2017.

T. K. Kar and A. Batabyal, “Stability analysis and optimal control of an SIR epidemic model with

vaccination,” BioSystems, vol. 104, no. 2–3, pp. 127–135, 2011.

A. A. Lashari, “Optimal control of an SIR epidemic model with a saturated treatment,” Appl. Math.

Inf. Sci., vol. 10, no. 1, pp. 185–191, 2016.

H. Laarabi, A. Abta, M. Rachik, J. Bouyaghroumni, and E. H. Labriji, “Stability Analysis and Optimal

Vaccination Strategies for an SIR Epidemic Model with a Nonlinear Incidence Rate,” ISSN Int. J.

Nonlinear Sci., vol. 16, no. 4, pp. 1749–3889, 2013.

Jurnal Kubik, Volume 2 No. 2 ISSN : 2338-0896

Y. Zhao and D. Jiang, “The threshold of a stochastic SIRS epidemic model with saturated

incidence,” Appl. Math. Lett., vol. 34, no. 1, pp. 90–93, 2014.

Y. Cai, X. Wang, W. Wang, and M. Zhao, “Stochastic dynamics of an SIRS epidemic model with

ratio-dependent incidence rate,” Abstr. Appl. Anal., vol. 2013, 2013.

Q. Liu and Q. Chen, “Analysis of the deterministic and stochastic SIRS epidemic models with

nonlinear incidence,” Phys. A Stat. Mech. its Appl., vol. 428, pp. 140–153, 2015.

O. Diekmann, J. a P. Heesterbeek, and M. G. Roberts, “The construction of next-generation

matrices for compartmental epidemic models.,” J. R. Soc. Interface, vol. 7, no. 47, pp. 873–885,

J. A. P. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis

and Interpretation, vol. 26 Suppl 4. 2000.

P. Van Den Driessche and J. Watmough, “Reproduction numbers and sub-threshold endemic

equilibria for compartmental models of disease transmission,” Math. Biosci., vol. 180, pp. 29–48,

J. J. Anagnost and C. A. Desoer, “An elementary proof of the Routh-Hurwitz stability criterion,”

Circuits Syst. Signal Process., vol. 10, no. 1, pp. 101–114, 1991.

X. Yang, “Generalized form of Hurwitz-Routh criterion and Hopf bifurcation of higher order,” Appl.

Math. Lett., vol. 15, no. 5, pp. 615–621, 2002.




DOI: https://doi.org/10.15575/kubik.v2i2.1854

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