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"Gonorrhea adalah penyakit menular seksual yang tidak memberikan kekebalan. Proses penyebaran penyakit tersebut dapat dijelaskan oleh model epidemi susceptible infected susceptible (SIS). Skripsi ini membahas model stokastik epidemi SIS. Model stokastik epidemi SIS diturunkan dari model deterministik epidemi SIS dengan memberikan gangguan pada parameter. Simulasi dilakukan menggunakan nilai-nilai parameter dari penelitian klinis penyakit gonorrhea. Metode numerik yang digunakan untuk mengaproksimasi solusi model adalah metode Milstein. Simulasi dilakukan terhadap kondisi kepunahan (extinction), kondisi kebertahanan (persistence), pengaruh intensitas gangguan terhadap jumlah individu terinfeksi pada kesetimbangan endemik model, dan distribusi model. Hasil simulasi menunjukkan semakin besar intensitas gangguan, jumlah individu terinfeksi pada kesetimbangan endemik model semakin berkurang. Kondisi kepunahan dan kondisi kebertahanan untuk model stokastik dipengaruhi oleh bilangan reproduksi dasar stokastik (). Jika, maka penyakit punah dari populasi dengan probabilitas satu. Jika, maka penyakit tetap bertahan di dalam populasi dan model memiliki distribusi stasioner unik.

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Gonorrhea is a sexually transmitted disease which does not confer immunity. The spreading process of the disease can be explained by the susceptible infected susceptible (SIS) epidemic model. This skripsi discusses the stochastic SIS epidemic model. The stochastic SIS epidemic model is derived from deterministic SIS epidemic model by adding a perturbation in the parameter. Simulations are generated based on parameter values from clinical research of gonorrhea disease and focus on conditions of extinction, persistence, and the influence of the noise intensity to the number of infected individuals in endemic equilibrium model as well as distribution of the model. The Milstein method is used to approximate solutions of the model. The simulations show that the greater noise intensity implies the reduction of individual infective number in endemic equilibrium. Extinction and persistence conditions in stochastic model are influenced by the stochastic basic reproduction number (). It shows that the disease dies out from population with probability one if , otherwise the disease persists in population and the model has a unique stationary distribution."

"ABSTRAKModel epidemik SIS (Susceptible Infected Susceptible) diaplikasikan dalampembuatan model matematis penyebaran penyakit influenza. Model penyebaranpenyakit flu dibuat dengan pendekatan stokastik. Model stokastik yang digunakandalam skripsi ini adalah model Continuous Time Markov Chain (CTMC). Padamodel CTMC, dikonstruksi probabilitas transisi, ekspektasi, dan limit distribusidari banyaknya individu yang terinfeksi penyakit flu dengan asumsi banyaknyaindividu terinfeksi hanya dapat bertambah satu, berkurang satu atau tetap dalaminterval waktu yang sangat pendek (
t 􀀀 0). Ekspektasi dari banyaknya individuyang terinfeksi flu tidak dapat diselesaikan secara langsung, tetapi dapat diketahuibahwa rata- rata pada model stokastik lebih kecil dibandingkan dengan solusideterministik. Dari kajian tentang limit distribusi, didapatkan bahwa probabilitastidak ada individu terinfeksi adalah satu saat t 􀀀 ª. Simulasi numerik padapenyebaran penyakit flu diberikan sebagai pendukung untuk interpretasi model

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ABSTRACTMathematical model for the spread of influenza using SIS (Susceptible InfectedSusceptible) Epidemic Model for constant total human population size is discussedin this undergraduate thesis. These influenza model was made with stochasticapproach. Stochastic model that used in this thesis is Continuous Time MarkovChain (CTMC). Transition probability, expectation, and limiting distribution forthe number of infected people were constructed in CTMC with assumption that thenumber of infected people might change by increasing one, decreasing one, or stillin the time interval that tends to zero (
t 􀀀 0). The expectation for the number ofinfected people cannot be solved directly, but we will know that the mean of thestochastic SIS epidemic model is less than the deterministic solution. Fromlimiting distribution analyses, probability that there are no infected people att 􀀀 ª is one. Some numerical simulation for the spread of influenza is given togive a better interpretation and a better understanding about the modelinterpretation"

"Teori katastrofe menjelaskan bahwa perubahan kecil (smooth change) pada suatu parameter akan menyebabkan rusaknya kestabilan dan menimbulkan perubahan perilaku sistem yang drastis secara tiba-tiba. Dengan menggunakan kalkulus stokastik, fungsi delta Dirac, transformasi Fourier terhadap fungsi karakteristik serta persamaan Fokker-Planck, dapat dijelaskan hubungan antara model katastrofe cusp stokastik dengan suatu fungsi densitas probabilitas (FDP) stasioner. Pada tesis ini ditunjukkan bahwa model katastrofe cusp stokastik dapat digunakan untuk menjelaskan peristiwa krisis pasar saham, yaitu krisis Black Monday pada 19 Oktober 1987 di pasar saham Amerika. Estimasi parameter dengan metode momen menunjukkan bahwa terdapat perubahan nilai diskriminan Cardan dari positif ke negatif, sehingga menunjukkan adanya kasus perubahan FDP dari unimodal ke bimodal. Peristiwa katastrofe pada data Black Monday menunjukkan bahwa krisis ini dipengaruhi oleh faktor internal.

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Catastrophe theory explains that a smooth change of parameters can perturb the system stability to a sudden discontinuous state. Using stochastic calculus, Dirac delta function, Fourier transform of characteristic function, and Fokker-Planck equation we show the connection between stochastic cusp catastrophe model to a stationer probability density function (PDF).

This thesis shows that stochastic cusp catastrophe model can explains U.S stock market crash in October 19, 1987 called Black Monday. Parameter estimations using momen method shows change of Cardan discriminant from positive to negative which explain the stationer PDF in unimodal case to bimodal case. Catastrophe in Black Monday data explains that the crisis influenced by internal factor."

"This introductory textbook explains how and why probability models are applied to scientific fields such as medicine, biology, physics, oceanography, economics, and psychology to solve problems about stochastic processes. It does not just show how a problem is solved but explains why by formulating questions and first steps in the solutions. Stochastic Processes is ideal for a course aiming to give examples of the wide variety of empirical phenomena for which stochastic processes provide mathematical models. It introduces the methods of probability model building and provides the reader with mathematically sound techniques as well as the ability to further study the theory of stochastic processes. Originally published in 1962, this was the first comprehensive survey of stochastic processes requiring only a minimal background in introductory probability theory and mathematical analysis. Stochastic Processes continues to be unique, with many topics and examples still not discussed in other textbooks. As new fields of applications (such as finance and DNA analysis) become important, researchers will continue to find the fundamental and accessible topics explained in this book essential background for their research."

"This book develops systematically and rigorously, yet in an expository and lively manner, the evolution of general random processes and their large time properties such as transience, recurrence, and convergence to steady states. The emphasis is on the most important classes of these processes from the viewpoint of theory as well as applications, namely, Markov processes."