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Aterosklerosis adalah fenomena penyempitan arteri karena penumpukan plak di dinding arteri sebagai akibat dari suatu pola hidup tidak sehat. Menurut pendekatan ilmu sosial, pola hidup yang salah dapat ditularkan kepada orang lain disekitarnya. Bentuk penanganan pasien aterosklerosis adalah melalui operasi bypass yang harus dilakukan oleh dokter spesialis dan di rumah sakit tertentu, yang keduanya memiliki jumlah yang terbatas. Jika jumlah pasien dengan komplikasi aterosklerosis terus bertambah, itu akan berdampak pada pengobatan yang tidak lagi optimal. Penelitian ini bertujuan untuk membangun model penyebaran aterosklerosis tanpa dan dengan mempertimbangkan pengaruh keterbatasan sumber daya rumah sakit, yang dikenal sebagai efek saturasi. Kedua model dibentuk dengan membagi kompartemen manusia dalam populasi yang rentan, terinfeksi aterosklerosis, dan penderita aterosklerosis yang menjalani pengobatan. Model-model yang telah dibangun kemudian dianalisis secara analitik dan numerik. Studi analitik dilakukan untuk menemukan dan menganalisis titik keseimbangan, menentukan bilangan reproduksi dasar (R0), dan menyelidiki keberadaan bifurkasi dari model yang dibangun. Bifurkasi maju, mundur dan maju dengan hysteresis muncul dari model yang telah terbentuk. Hasil analitik didukung oleh simulasi numerik terkait elastisitas dan sensitivitas R0 serta simulasi autonomous.

 

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Atherosclerosis is a narrowing of the arteries due to the buildup of plaque on the artery walls resulting from an unhealthy lifestyle. According to the social science approach, the wrong lifestyle can be ”infection” to other people. The form of treatment for atherosclerosis patients is through bypass surgery, which must be performed by specialists and in specific hospitals, both of which have a limited number. If the number of patients with atherosclerosis complications continues to increase, it will result in no longer optimal treatment. This study aims to build a model of atherosclerosis spread without and taking into account the effect of limited hospital resources, known as the saturation effect. Both models were formed by dividing the human compartment into populations susceptible, infected with atherosclerosis, and people with atherosclerosis who are undergoing treatment. The models that have built are then analyzed analytically and numerically. Analytical studies carried out to find and analyze the equilibrium point, determine the basic reproduction number (R0), and investigate the existence of a bifurcation of the built model. Forward bifurcation, backward bifurcation, and forward bifurcation with hysteresis appear from the model that has formed. The analytical results supported by numerical simulations related to elasticity and sensitivity of R0 as well as autonomous simulations.

 

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"Dynamical systems arise in all fields of applied mathematics. The author focuses on the description of numerical methods for the detection, computation, and continuation of equilibria and bifurcation points of equilibria of dynamical systems. This subfield has the particular attraction of having links with the geometric theory of differential equations, numerical analysis, and linear algebra.Several features make this book unique. The first is the systematic use of bordered matrix methods in the numerical computation and continuation of various bifurcations. The second is a detailed treatment of bialternate matrix products and their Jordan structure. Govaerts discusses their use in the numerical methods for Hopf and related bifurcations. A third feature is a unified treatment of singularity theory, with and without a distinguished bifurcation parameter, from a numerical point of view. Finally, numerical methods for symmetry-breaking bifurcations are discussed in detail, up to the fundamental cases covered by the equivariant branching lemma."

"ABSTRAKPenyakit Ebola disebabkan oleh virus Ebola yang termasuk dalam keluarga virus floviridae. Penyakit ini menyebar melalui kontak langsung dengan cairan tubuh individu terinfeksi. Dalam penelitian ini, dibahas mengenai model matematika transmisi penyakit ebola dengan relapse dan reinfeksi. Penyakit tersebut dimodelkan dengan menggunakan sistem persamaan diferensial biasa berdimensi tujuh. Model ini menunjukkan adanya fenomena backwards bifurcation yang dianalisa dengan memperhatikan perubahan arah pada titik keseimbangan endemik. Eksistensi backward bifurcation pada model penyakit Ebola dikarenakan adanya relapse dan reinfeksi sehingga terdapat titik keseimbangan endemik saat basic reproduction number R0 kurang dari satu. Jumlah total kasus baru individu terinfeksi Ebola meningkat dengan meningkatnya nilai parameter relapse dan reinfeksi.

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ABSTRACTEbola disease is caused by the Ebola virus which belongs to the floviridae virus family.This disease spreads through direct contact with the body fluids of infected individuals.In this undergraduate thesis, we discussed the mathematical model of Ebola diseasetransmission with relapse and reinfection. This infection is modeled using systemof seven dimensions ordinary differential equation. This model shows the backwardbifurcation phenomenon that is analyzed by considering the direction change in theendemic equilibrium point. The existence of backward bifurcation in the Ebola diseasemodel is due to relapse and reinfection so there is an endemic equilibrium point whenbasic reproduction number R0 is less than one. The total number of new cases ofindividuals infected with Ebola increases with increasing values of the parameters relapseand reinfection."

"Tuberkulosis (TB) adalah penyakit yang sangat menular yang disebabkan oleh bakteri. Pada skripsi ini sebuah model epidemi SEIR dibentuk pada penyebaran penyakit tuberkulosis dengan memperhatikan infeksi lambat dan infeksi cepat. Model ini menggunakan sistem persamaaan diferensial biasa nonlinear berdimensi 4. Dilakukan kajian mengenai Basic Reproduction Number (R0), titik keseimbangan bebas penyakit atau Disease Free Equilibrium (DFE), serta analisa kestabilan lokal dan analisa bifurkasi yang dilakukan secara analitik dan numerik pada model. Metode yang digunakan untuk melakukan analisa bifurkasi yakni menggunakan Teorema yang telah dibuktikan oleh Castillo-Chavez&Song. Model yang dibentuk menunjukkan adanya kemungkinan terjadi bifurkasi mundur yang ditandai dengan munculnya dua titik ekuilibrium endemik saat nilai R0 < 1.

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Tuberculosis (TB) is a highly contagious disease caused by bacteria. In this paper, a SEIR epidemic model was formed in the spread of tuberculosis with regard to slow infection and fast infection. This model uses a dimensionless nonlinear ordinary differential equation system. A study is conducted on Basic Reproduction Number (R0), Disease Free Equilibrium (DFE), and local stability analysis and analytical and numerical bifurcation analysis on the model. The method used to carry out bifurcation analysis is using the theorem that has been proven by Castillo-Chavez & Song. The model formed shows the possibility of a backward bifurcation which is indicated by the appearance of two endemic equilibrium points when the value of Ro < 1."

"Kemajuan teknologi dalam bidang epidemi tak luput dari pemodelan matematika. Banyak peneliti yang memprediksikan mengenai wabah melalui model matematika dalam berbagai bentuk yang beragam. Model epidemic sederhana menggunakan pendekatan eksponensial. Karena pertumbuhan suatu populasi memiliki sumber daya yang terbatas, maka perlu menggunakan pendekatan logistic. Selain itu, tingkat saturasi dalam epidemiolgi diperlukan untuk mengukur dampak psikologis pada suatu populasi. Dalam penelitian, ini akan dibahas bifurkasi model SI tanpa saturasi dan model yang memiliki saturasi. Penelitian ini juga akan membahas kapan bifurkasi Hopf dapat muncul yang berupa solusi periodik.

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Technological advances in the field of epidemics cannot be separated from mathematical modeling. Many researchers predict outbreaks through mathematical models in various forms. The simple epidemic models use an exponential approach. Because a growing population has limited resources, it is necessary to use a logistical approach. In addition, the saturation level in epidemiology is needed to measure the psychological impact on a population. In this research, the bifurcation of SI models without saturation and models with saturation will be discussed. This research will also discuss when Hopf bifurcations can appear in the form of periodic solutions."

"Malaria adalah penyakit yang ditularkan dari individu ke individu lainnya melalui perantara nyamuk Anopheles betina. Hingga saat ini, beberapa upaya pengendalian yang dilakukan oleh WHO untuk menekan angka kejadian dan kematian akibat malaria antara lain insecticide-treated mosquito nets (ITN), indoor spraying with residual insecticides (IRS), obat anti malaria, serta vaksinasi. Namun dalam proses penanganan di berbagai negara endemik, beberapa hal seringkali terabaikan sehingga dapat menyebabkan malaria akan terus mewabah. Kurangnya informasi dan pengetahuan tentang malaria, tingkat kesadaran yang buruk, sumber daya yang tidak memadai serta kurangnya keterampilan dalam mengendalikan penyakit malaria menyebabkan terjadinya kekambuhan (recurrence). Kekambuhan (recurrence) pada malaria terbagi dalam tiga jenis yaitu, relapse yang disebabkan oleh melemahnya imun seseorang sehingga terjadi reaktivasi parasit dalam sel hati, reinfection yang disebabkan oleh individu yang telah terinfeksi dan dalam status dormant menerima kembali gigitan nyamuk Anopheles betina terinfeksi, dan recrudescence yang disebabkan oleh gagalnya pengobatan. Sebuah model matematika penyebaran penyakit malaria dengan mempertimbangkan proses kekambuhan (relapse, reinfection, dan recrudescence) dibahas dalam penelitian ini. Kebaruan terletak pada konstruksi model yang melibatkan setiap tahapan infeksi yang terjadi dalam tubuh, sehingga perubahan bentuk parasit akan menentukan status individu tersebut. Model matematika yang terbentuk didasarkan pada model SIR-UV dengan penambahan empat kompartemen terinfeksi lainnya yaitu, Exposed (E), Dormant (D), Latent (L), Under-treatment (T). Analisa kestabilan lokal dari titik keseimbangan dan basic reproduction number (R0) akan ditampilkan secara analitik. Hasil numerik dari beberapa skenario berbeda akan dilakukan untuk menunjukkan situasi yang mungkin ditemukan di lapangan.

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Malaria is a disease that is transmitted from individuals to other individuals through intermediaries female Anopheles mosquito. Until now, the government has made several control efforts to suppress the incidence and mortality rates due to malaria such as insecticide-treated mosquito nets (ITN), indoor spraying with residuals insecticides (IRS), anti-malaria drugs, and vaccinations. But in the process of handling, some things are often overlooked so that it can cause malaria continue to plague. Lack of information and knowledge about the disease, poor level of awareness, inadequate resources and lack of control skills malaria causes recurrence. Recurrence in malaria is divided into three types namely, relapse caused by weakening one’s immune resulting in reactivation of parasites in liver cells, reinfection caused by individuals who have been infected and are in dormant status receive back the bite of an infected female Anopheles mosquito, and the recrudescence caused by treatment failure. A mathematical model of the spread of malaria by considering the recurrence process (relapse, reinfection, and recrudescence) are discussed in this thesis. The novelty lies in the construction of models that involve each stage of infection that occurs in the body, so the change in the shape of the parasite will determine the status of the individual. The mathematical model formed is based on the SIR-UV model with additions four other infected compartments namely, Exposed (E), Dormant (D), Latent (L), Under-treatment (T). Analysis of local stability from the equilibrium point and basic reproduction number (R0) will be displayed analytically. Numerical results from several different scenarios will be done to show situations that might be found in the field."

"ABSTRACTThe objective of the present study is to investigate the influence of external magnetic field on unsteady incompressible flow of water based nanofluid through a successively expanding or contracting channel with porous walls. the basic governing equations with boundary conditions are non-dimensionalized using appropriate transformation to ordinary differential equations, which are then solved using power series with the help of Hermite-Padè approximation method. the instability of the flow is shown using bifurcation graph and the dominating singularity behavior numerically. the regular effects of the different governing physical parameters specifically Hartmann number, volume friction of nanoparticles, non-dimensional shear stress and permeation Reynolds number on velocity profiles are depicted graphically."

"This book gives a unified presentation in an abstract setting of the main theorems in bifurcation theory, as well as more recent and lesser known results. It covers both the local and global theory of one-parameter bifurcations for operators acting in infinite-dimensional Banach spaces, and shows how to apply the theory to problems involving partial differential equations. In addition to existence, qualitative properties such as stability and nodal structure of bifurcating solutions are treated in depth. "

"A discussion of developments in the field of bifurcation theory, with emphasis on symmetry breaking and its interrelationship with singularity theory. The notions of universal solutions, symmetry breaking, and unfolding of singularities are discussed in detail. The book not only reviews recent mathematical developments but also provides a stimulus for further research in the field."

"ABSTRAKMeskipun belakangan ini ada tindakan pencegahan sanitasi yang buruk, penyakit Penyakit tifus masih banyak ditemukan di beberapa negara berkembang. Demam tifoid atau umum Disebut tifus adalah penyakit yang disebabkan oleh bakteri Salmonella Thypii. Gejala yang dialami termasuk demam tinggi yang berlangsung lama pendarahan internal dan bahkan kematian. Infeksi tifus dapat ditularkan melalui kontak langsung dengan orang yang terinfeksi atau mengonsumsi makanan dan / atau minuman yang telah terkontaminasi bakteri. Selain sanitasi yang buruk, ada keterbatasan sumber daya Tenaga pelayanan kesehatan juga dapat berperan dalam penyebaran penyakit tifus. Dalam tesis ini dibahas model distribusi tifus dengan menambahkan batasan-batasan sumber daya layanan kesehatan. Model dibangun untuk melihat efeknya dari sumber daya pelayanan kesehatan yang terbatas hingga penyebaran tifus.Model yang telah dibangun kemudian dianalisis secara analitik dan numerik. Belajar Analisis dilakukan untuk mengetahui keberadaan dan kestabilan titik kesetimbangan titik ekuilibrium bebas penyakit dan endemik dalam model, serta menentukan reproduksi dasar numberR0. Selain itu, analisis sensitivitas R0 terhadap parameter juga dilakukantingkat infeksi dan parameter tingkat maksimum pengobatan serta melawan parameter tingkat kesembuhan alami pada individu yang terinfeksi dan parameter tingkat maksimum pengobatan numerik. Akhirnya, simulasi otonom dilakukan untuk melihat pengaruh tingkat pengobatan maksimum terhadap penyebaran penyakit tipus.ABSTRACTTyphus disease is still found in many developing countries. Typhoid fever or commonly called typhus is a disease caused by Salmonella Thypii bacteria. Symptoms include high fever, prolonged internal bleeding and even death. Typhoid infection can be transmitted through direct contact with an infected person or by consuming food and / or drink that has been contaminated with bacteria. Apart from poor sanitation, there are limited resources. Health care workers can also play a role in the spread of typhoid. This thesis discusses the typhus distribution model by adding the limitations of health care resources. The model was built to see the effects ranging from limited health care resources to the spread of typhus. The model that has been built is then analyzed analytically and numerically. Learning analysis is carried out to determine the presence and stability of disease-free and endemic equilibrium points in the model, and to determine the basic reproduction number R0. In addition, a sensitivity analysis of R0 to parameters was also carried out infection rate and maximum treatment rate parameter as well as against natural cure rate parameter in infected individuals and numerical maximum treatment rate parameter. Finally, autonomous simulations were carried out to see the effect of maximum treatment rates on the spread of typhoi"