ANALISIS KESTABILAN LOKAL DAN BIFURKASI TITIK EKUILIBRIUM MODEL SEIRS DENGAN WAKTU TUNDAAN DAN LAJU INSIDENSI TAK LINEAR

R Setiawan

Abstract


Dalam paper ini dianalisa sifat kualitatif secara lokal dari model epidemi SEIRS (Susceptible Exposed Infected Recovered Susceptible) dengan waktu tundaan dan laju insidensi nonlinear (strong nonlinear incidence). Waktu tundaan diskrit dalam model ini merupakan waktu yang dibutuhkan individu rentan penyakit ketika mulai terinfeksi penyakit sampai masuk ke dalam kelas exposed. Dapat dibuktikan bahwa terdapat dua titik ekuilibrium yaitu titik ekuilibrium penyakit dan bebas penyakit yang eksistensinya bergantung pada angka reproduksi dasar yang telah didefinisikan sebelumnya. Analisa kestabilan lokal dilakukan untuk masing masing titik ekuilibrium serta pengaruh adanya waktu tundaan terhadap perubahan kestabilan dari masing masing titik ekuilibrium yang memungkinkan terjadinya proses bifurkasi.

Susceptible Exposed Infected Recovered Susceptible was analyzed qualitatively bu using delay time and strong nonlinear incidence. The discrete delay time in this model is the time needed by the person that is vurnerable to disease from the infection phase to reach the exposed class. It can be proved that there were two equilibirum points, they are disease equilibrium points and disease free point that their existence depend on their the reproduction rate that has been defined previously, The local stability anylisis was dore to each equilibrium point and the influence of delay time towards the stability change from each equilibrium point that enables the bifurcation process.


Keywords


Model SEIRS; nonlinear incidence; Hopf bifurcation, time delays

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References


Cappaso V. 1993. Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics Vol.97. Springer Verlag, Berlin Heidelberg, Germany.

Jiao J, Chen L, & Cai S. 2008. An SEIRS Epidemic Model With Two Delays and Pulse Vaccination. J Syst Sci Complex. 21: 217 225.

Kuang Y. 1993. Delay Differential Equations, with Applications in Population Dynamics, Academic Press, Inc. San Diego, USA.

Kuznetsov YA. 1998. Elements of Applied Bifurcation Theory Second Edition, Applied Mathematical Sciences Vol. 112, Springer Verlag, New York, USA.

Setiawan R, Widodo, & Aryati L. 2010. Stability Analysis of Delayed SIR Epidemic Model With Saturated Incidence. Presented on International Conference on Algebra (ICA) UGM Yogyakarta, Yogyakarta, Indonesia.

Setiawan R. 2012. Analisis Model SEIRS dengan Waktu Tundaan dan Laju Insidensi Jenuh. Prosiding Seminar Nasional Matematika FMIPA UNS Surakarta.: 79-83

Setiawan R. 2009. Bifurkasi Hopf dalam Model Penyebaran Penyakit. J PYTHAGORAS. 5 (1)

Verhulst F. 1990. Nonlinear Differential Equations and Dynamical Systems. SpringerVerlag, New-York, USA.


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