ANALISIS KESTABILAN DAN SENSITIVITAS PADA MODEL MATEMATIKA SEIRD DARI PENYEBARAN COVID-19: STUDI KASUS DI KALIMANTAN SELATAN

The cases of Covid-19 that occurred in South Kalimantan were classified into 5 groups, namely suspected, treated, recovered, deaths, and healthy population who were susceptible for being infected with Covid-19. The dynamics of changes in the number of cases in each group can be studied mathematically through epidemiological mathematical modeling. In this study, the SEIRD Model (Susceptible, Exposed, Infected, Recovered, and Deaths) was formed to describe the dynamics of changing the number of Covid-19 cases in South Kalimantan. In this model, stability analysis and formulation of indicators for the controllability of the spread of Covid-19 are given, known as the Basic Reproduction Number. Furthermore, a sensitivity analysis of the parameters contained in the Basic Reproduction Number is given to determine the priority efforts that can be made to suppress the spread of Covid-19 in South Kalimantan

MODEL MATEMATIKA SEIRD (SUSCEPTIBLE, EXPOSED, INFECTED, RECOVERED, DAN DEATH) UNTUK PENYEBARAN PENYAKIT ISPA

In the last few decades, the Upper Respiratory Tract Infection has become one of the three leading causes of death and disability in the world, both in developing countries and in developed countries. In Indonesia, the trend of this disease continues to increase throughout 2016 - 2019 and in children it has caused 1 - 4 children under five to die every hour. In this study, the spread of this disease was modeled mathematically by using the SEIRD Model (Susceptible, Exposed, Infected, Recovered, and Death). Then, the equilibrium points of the model are determined, stability analysis is performed, and the model solution is determined using the Runge Kutta Metho

ESTIMASI PARAMETER PADA PERSAMAAN OSILATOR HARMONIK FUZZY: PERBANDINGAN SOLUSI HUKUHARA DIFERENSIAL DAN INKLUSI DIFERENSIAL FUZZY DENGAN MENGGUNAKAN METODE RUNGE-KUTTA DIPERLUAS

Most of the real systems in the world may contain uncertainties, which are possibly due to the limitations of available data, complexity of the network of systems, and environmental or demographic changes at the time of observation. One of the system behaviors that often appears in mathematical modeling is periodic behavior, which often shows complex dynamic behavior, depending on initial values and parameters. By accommodating the uncertainties in the model, in-depth studies are needed to describe mathematical structure, methodology for determining solution, and procedure for estimating parameters. Among the mathematical models that describe periodic behavior is harmonic oscillator equation. In this paper, the model is assumed to have uncertainty in the initial values in the form of fuzzy numbers, which is then called by fuzzy harmonic oscillator equation. The model is examined by comparing three fuzzy differential approaches, namely Hukuhara differential, generalized Hukuhara differential and fuzzy differential inclusions. Applications of fuzzy arithmetic concepts to the models lead to a deterministic alpha-cut systems, which are solved using extended Runge-Kutta method. In contrast to the standard Runge-Kutta method, the extended Runge-Kutta method using the first derivative approximation of the evaluation function to increase the accuracy of the solution. Among the three fuzzy approaches, the fuzzy differential inclusion type is the most appropriate approach to capture the periodic behavior of the equation. Next, it is shown how to estimate the parameters of solution of the fuzzy differential inclusion type and simulation of fuzzy data using lsqnonlin method

ANALISIS KESTABILAN MODEL SEIR UNTUK PENYEBARAN COVID-19 DENGAN PARAMETER VAKSINASI

Covid-19 adalah penyakit menular yang disebabkan oleh coronavirus disease jenis baru, yaitu SARS-CoV-2. Oleh WHO, penyebaran Covid-19 telah ditetapkan sebagai pandemi global sejak 11 Maret 2020. Pada penelitian ini, penyebaran Covid-19 dimodelkan dengan menggunakan model matematika epidemik, yaitu model SEIR (Susceptible, Exposed, Infected, and Recovered) dengan memperhatikan faktor vaksinasi sebagai parameter. Selanjutnya, ditentukan titik ekuilibrium dan bilangan reproduksi dasar, serta diberikan analisis kestabilan pada model

ANALISA PELAKSANAAN NEW NORMAL DI KALIMANTAN SELATAN MELALUI MODEL MATEMATIKA SIRD

Mathematical models of epidemiology are very useful in studying the interrelationships among various epidemiological cases, conducting evaluations of efforts to deal with these cases, and preparing preventive actions and control of health problems in a population. One of the most popular models is SIR Model (Susceptible, Infectious, Recovered). Along with the rapid development in the field of epidemiology, the SIR Model has also undergone many modifications, one of which is the SIRD Model. The SIRD Model is modified for cases that explicitly separate Recovered and Deaths subpopulations. Since the positive case of Coronavirus Disease (Covid-19) was first confirmed in the Province of South Borneo on March 22, 2020, this outbreak has continued to increase significantly until the end of May 2020, exactly where the Large-scale Social Restrictions simultaneously ended throughout the region. The end of this restriction is the starting point for the start of 'New Normal' in South Kalimantan, which is called the New Life Order in the midst of the Covid-19 outbreak. In this study, an analysis was conducted to measure the implementation of the New Normal in South Borneo, as part of the evaluation material for the community and the local government on the implementation of the New Normal. Analysis was conducted using the SIRD Model and the data of Covid-19 in South Borneo in the period June 16 to July 17, 2020. The data showed an increase in the Attack Rate, which illustrates that the positive cases of Covid-19 in South Borneo are still experiencing an increase. The data also shows an increase in the Case Recovery Rate and a decrease in the Case Fatality Rate, which indicates that efforts to accelerate the handling of Covid-19 cases in South Borneo have given positive results. On the other hand, the parameter estimation process of the SIRD Model produces a Basic Reproduction Number of 2 and an Effective Reproductive Number of 1.82. Both of these numbers indicate that the transmission of Covid-19 in South Borneo is still out of control and it is estimated that the high transmission will still occur until the end of August 202

PREDIKSI JUMLAH PENDUDUK KALIMANTAN SELATAN MENGGUNAKAN METODE NONLINEAR LEAST-SQUARES

Dalam tulisan ini disajikan prediksi jumlah penduduk Kalimantan Selatan menggunakan model pertumbuhan logistik. Untuk memprediksi jumlah penduduk Kalimantan Selatan tersebut digunakan Metode Nonlinear Least-Squares untuk mengestimasi parameter-parameter yang mempengaruhi model. Pada model pertumbuhan logistik terdapat dua parameter yang mempengaruhi yaitu tingkat pertumbuhan dan daya tampung (Carrying Capacity). Penelitian ini dilakukan dalam tiga tahapan metode. Pertama, menentukan solusi model, kedua mengestimasi parameter tingkat pertumbuhan penduduk dan daya tampung penduduk Kalimantan Selatan dengan cara meminimumkan fungsi error yaitu antara data jumlah penduduk dan solusi model menggunakan Metode Nonlinear Least Squares. Ketiga melakukan prediksi jumlah penduduk Kalimantan Selatan untuk tahun-tahun mendatang. Berdasarkan hasil penelitian ini diperoleh  parameter hasil estimasi yaitu tingkat pertumbuhan penduduk Kalimantan Selatan sebesar 0,14055 per tahun dan daya tampung penduduk Kalimantan Selatan adalah 8.521.817 jiwa. Selanjutnya, disajikan prediksi jumlah penduduk Kalimantan Selatan untuk tahun-tahun mendatang menggunakan hasil estimasi parameter-parameter yang telah diperoleh. Hasil prediksi menunjukkan setiap tahun terjadi peningkatan jumlah penduduk dan peningkatan tersebut dari waktu kewaktu mendekati  daya tampung penduduk Kalimantan Selatan

Analisa Kestabilan dan Solusi Pendekatan Pada Persamaan Van der Pol

Abstrak: Di dalam tulisan ini disajikan analisa kestabilan, diselidiki eksistensi dan kestabilan limit cycle, dan ditentukan solusi pendekatan dengan menggunakan metode multiple scale dari persamaan Van der Pol. Penelitian ini dilakukan dalam tiga tahapan metode. Pertama, menganalisa perilaku dinamik persamaan Van der Pol di sekitar ekuilibrium, meliputi transformasi persamaan ke sistem persamaan, analisa kestabilan persamaan melalui linearisasi, dan analisa kemungkinan terjadinya bifukasi pada persamaan. Kedua, membuktikan eksistensi dan kestabilan limit cycle dari persamaan Van der Pol dengan menggunakan teorema Lienard. Ketiga, menentukan solusi pendekatan dari persamaan Van der Pol dengan menggunakan metode multiple scale. Hasil penelitian adalah, berdasarkan variasi nilai parameter kekuatan redaman, daerah kestabilan dari persamaan Van der Pol terbagi menjadi tiga. Untuk parameter kekuatan redaman bernilai positif mengakibatkan ekuilibrium tidak stabil, dan sebaliknya, untuk parameter kekuatan redaman bernilai negatif mengakibatkan ekuilibrium stabil asimtotik, serta tanpa kekuatan redaman mengakibatkan ekuilibrium stabil. Pada kondisi tanpa kekuatan redaman, persamaan Van der Pol memiliki solusi periodik dan mengalami bifurkasi hopf. Selain itu, dengan menggunakan teorema Lienard dapat dibuktikan bahwa solusi periodik dari persamaan Van der Pol berupa limit cycle yang stabil. Pada akhirnya, dengan menggunakan metode multiple scale dan memberikan variasi nilai amplitudo awal dapat ditunjukkan bahwa solusi persamaan Van der Pol konvergen ke solusi periodik dengan periode dua. Abstract: In this paper, the stability analysis is given, the existence and stability of the limit cycle are investigated, and the approach solution is determined using the multiple scale method of the Van der Pol equation. This research was conducted in three stages of method. First, analyzing the dynamic behavior of the equation around the equilibrium, including the transformation of equations into a system of equations, analysis of the stability of equations through linearization, and analysis of the possibility of bifurcation of the equations. Second, the existence and stability of the limit cycle of the equation are proved using the Lienard theorem. Third, the approach solution of the Van der Pol equation is determined using the multiple scale method. Our results, based on variations in the values of the damping strength parameters, the stability region of the Van der Pol equation is divided into three types. For the positive value, it is resulting in unstable equilibrium, and contrary, for the negative value, it is resulting in asymptotic stable equilibrium, and without the damping force, it is resulting in stable equilibrium. In conditions without damping force, the Van der Pol equation has a periodic solution and has hopf bifurcation. In addition, by using the Lienard theorem, it is proven that the periodic solution is a stable limit cycle. Finally, by using the multiple scale method with varying the initial amplitude values, it is shown that the solution of the Van der Pol equation is converge to a periodic solution with a period of two

PEMODELAN MATEMATIKA PENYEBARAN COVID-19 DI PROVINSI KALIMANTAN SELATAN

Mathematical modeling in epidemiology has a very important role in the study of the dynamics of an epidemic. The outbreak of Covid-19, which is currently being spread widely in the world requires in-depth study, starting from the search for sources, prediction of spread patterns, to strategies for handling this virus outbreak. Mathematical modeling can be applied to support various fields of the study. In this paper, we discuss mathematical modeling of the spread of Covid-19 by providing analysis and predictions based on data from the case of Covid-19 in South Kalimantan Province. This study was conducted by estimating parameters of the SIR Model, which is accommodates the death cases in the data, supported by several methods, namely Runge Kutta Method and Nonlinear Least Squares Method. Our analysis to the data and the model yields a Basic Reproduction Number , which means that one individual infected by Covid-19 can produce three new infected individuals. Whereas our prediction shows that infected cases can reach to 37.82% and cases of death can reach to 0.49% of the population who remained in normal activities during the PSBB. The peaks of this case are estimated to occur in the 2nd week of August to the 1st week of October 2020. The fewer people who have normal activities, then the spread of Covid-19 is predicted to pass faster with smaller cases of infection and death. Conversely, the more people who have normal activities, then the spread of Covid-19 in South Kalimantan can take longer and take a higher number of victims

Parameter Estimations of Fuzzy Forced Duffing Equation: Numerical Performances by the Extended Runge-Kutta Method

In this work, the forced Duffing equation with secondary resonance will be considered our subject by assuming that the initial values has uncertainty in terms of a fuzzy number. The resulted fuzzy models will be studied by three fuzzy differential approaches, namely, Hukuhara differential and its generalization and fuzzy differential inclusion. Applications of fuzzy arithmetics to the models lead to a set of alpha-cut deterministic systems with some additional equations. These systems are then solved by the extended Runge-Kutta method. The extended Runge-Kutta method is chosen as our numerical approach in order to enhance the order of accuracy of the solutions by including both function and its first derivative values in calculations. Among our fuzzy approaches, our simulations show that the fuzzy differential inclusion is the most appropriate approach to capture oscillation behaviors of the model. Using the aforementioned fuzzy approach, we then demonstrate how to estimate parameters to our generated fuzzy simulation data

Analysis of stability and bifurcation in logistics models with harvesting in the form of the holling type III functional response

The logistic model can be applied in the field of biological studies to investigate population growth problems and some important aspects of the ecological situation. This model is a growth model with a limited population growth rate, and ecologists describe this rate as carrying capacity. Carrying capacity can be interpreted as the ideal population size, where individuals in the population can live properly in their environment. The growth rate of a population can be influenced by the harvesting factor, in this case, it is assumed that harvesting is not constant. The effect of the harvest on the growth rate can be analyzed mathematically by using the Holling type III functional response. In this paper, describe the formation of a logistic model taking into account the effects of harvesting, using the Holling type III functional response. Then,  perform a nondimensional process in the model, namely simplifying a model that has four parameters to a model that only has two parameters. Next, determine the equilibrium point of the model, perform a stability analysis at that equilibrium point, and investigate the possibility of bifurcation. As result, first obtained a logistic model which has two non-dimensional parameters, where one of the equilibrium points is zero and is unstable. Next, determine another equilibrium point through an implicit equation and investigate its stability through simulation. Finally, obtained two equilibrium points, which are fold bifurcation