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MSEICR Fractional Order Mathematical Model of The Spread Hepatitis B Suriani Suriani; Syamsuddin Toaha; Kasbawati Kasbawati
Jurnal Matematika, Statistika dan Komputasi Vol. 17 No. 2 (2021): JANUARY 2021
Publisher : Department of Mathematics, Hasanuddin University

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.20956/jmsk.v17i2.10994

This research aims to develop the MSEICR model by reviewing fractional orders on the spread of Hepatitis B by administering vaccinations and treatment, and analyzing fractional effects by numerical simulations of the MSEICR mathematical model using the method Grunwald Letnikov. Researchers use qualitative methods to achieve the object of research. The steps are to determine the MSEICR model by reviewing the fractional order, looking for endemic equilibrium points for each non-endemic and endemic equilibrium point, determining the equality of characteristics and eigenvalues of the Jacobian matrix. Next, look for values (Basic Reproductive Numbers), analyze stability around non-endemic and endemic equilibrium points and complete numerical simulations. From the simulation provided, it is known that by giving a fractional alpha value of and , the greater the value of the fractional order parameters used, the movement of the solution graphs is getting closer to the equilibrium point. If given and still endemic, whereas if and the value is increased to non-endemic, then the number of hepatitis B sufferers will disappear.

ANALISIS KESTABILAN DAN SOLUSI APROKSIMASI PADA MODEL MATEMATIKA KECANDUAN MEDIA SOSIAL MENGGUNAKAN METODE PERTURBASI HOMOTOPI Musdalifa Pagga; Syamsuddin Toaha; Kasbawati Kasbawati
Pedagogy: Jurnal Pendidikan Matematika Vol. 7 No. 2 (2022): Pedagogy : Jurnal Pendidikan Matematika
Publisher : Universitas Cokroaminoto Palopo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30605/pedagogy.v7i2.1932

Kecanduan adalah perilaku adiktif yang mudah menjadi kebiasaan, kecanduan internet sebagai media sosial dapat menyebabkan gangguan jiwa seiring dengan bertambahnya jumlah penggunanya dan dampak yang terjadi juga sangan berbahaya.. Tujuan dari penelitian ini adalah untuk melihat bagaimana pengembangan model matematika kecanduan media sosial dengan sensitivitas untuk mengetahui parameter yang berpengaruh pada bilangan reproduksi dasar , dan analisis kestabilan. Dari hasil analisis sensitivitas ditemukan hubungan parameter dengan yang dapat meningkatkan dan menurunkan nilai , dan analisis kestabilan menunjukkan pengaruh perubahan kestabilan titik kesetimbangan akibat perubahan nilai parameter yang berhubungan dengan . Simulasi model diperoleh artinya masih terdapat individu yang kecanduan media sosial di dalam populasi. Selanjutnya dicari solusi numerik model matematika kecanduan penggunaaan media sosial menggunakan metode perturbasi homotopi.

Stability Analysis of the Development Mathematical Models for the Spread of Covid-19 Effects of Vaccination and Campaigns as Processes in Controlling the Spread of Disease Curex, Nola; Toaha, Syamsuddin; Kasbawati
Proximal: Jurnal Penelitian Matematika dan Pendidikan Matematika Vol. 5 No. 2 (2022): Volume 5 Nomor 2 Tahun 2022
Publisher : Universitas Cokroaminoto Palopo

Novel Coronavirus or corona virus is a type of virus that was first discovered in 2003, until now this virus has mutated to form a new type of corona virus (SARS-CoV-2) and causes the emergence of a disease called Coronavirus Disease-19 (COVID-19). The purpose of this study was to see how the influence of vaccination and campaigns in the disease control process with showed sensitivity analysis to determine the parameters that affect the basic reproduction number , and stability analysis. The results obtained from the sensitivity analysis, which found a parameter relationship with which could increase and decrease the value of , and the stability analysis showed the effect of changes in the stability of the equilibrium point due to changes in the values of the parameters , and . The model simulation shows that vaccination and campaigning can control the spread of COVID-19 disease.

Analisis Kestabilan Model Matematika Penyebaran Penyakit HIV Dengan Klasifikasi Gejala Pada Penderita Faisah, Faisah; Toaha, Syamsuddin; Kasbawati, Kasbawati
Proximal: Jurnal Penelitian Matematika dan Pendidikan Matematika Vol. 5 No. 2 (2022): Volume 5 Nomor 2 Tahun 2022
Publisher : Universitas Cokroaminoto Palopo

Penelitian ini bertujuan untuk membahas tentang model matematika dinamika penyabaran penyakit HIV/AIDS. Model ini terdiri dari enam kelas populasi yaitu individu yang rentan terinfeksi atau Susceptible (S), individu yang terpapar HIV atau Eksposed (E), individu yang terinfeksi HIV tanpa gejala atau Infected (I), individu yang terinfeksi dengan gejala ringan ( ), individu yang terinfeksi denga gejala berat ( ) dan individu yang terinfeksi AIDS (A). Tujuan dari penelitian ini adalah untuk mencegah semakin meluasnya penyebaran penyakit HIV/AIDS menjadi wabah di masyarakat dengan analisis sensitivitas terhadap . Dari hasil sensitivitas yang dilakukan, diperoleh nilai negatif untuk pengaruh parameter pembentukan komisi penanggulangan HIV/AIDS (p) sebagai bentuk pencegahan individu terinfeksi penyakit ini. Makna dari hubungan negatif antara parameter p dan yaitu semakin besar pemberian nilai untuk parameter (p) maka nilai bilangan reproduksi dasar semakin kecil. Kemudian untuk parameter interaksi antara individu terinfeksi dengan individu lain (β), memiliki hubungan positif dengan bilangan reproduksi dasar ( ). Maknanya, semakin besar nilai parameter β maka semakin besar pula nilai namun semakin kecil nilai parameter ini maka nilai bilangan reproduksi dasar juga akan mengecil.

ANALISIS KESTABILAN DAN BIFURKASI PADA MODEL MATEMATIKA TINGKAT PENGANGGURAN PADA MASA PANDEMI COVID-19 Bohari, Nurul Aulia; Toaha, Syamsuddin; Kasbawati, Kasbawati
Proximal: Jurnal Penelitian Matematika dan Pendidikan Matematika Vol. 5 No. 2 (2022): Volume 5 Nomor 2 Tahun 2022
Publisher : Universitas Cokroaminoto Palopo

Covid-19 is a type of virus from the Coronaviridae which has implications for infectious and deadly diseases that attack mammals such as humans in the respiratory tract to the lungs. In the conditions of the outbreak of the Covid-19 pandemic, it turns out that there are many impacts, mainly in the Indonesian economic sector, for example unemployment, with the spread of the Covid-19 virus in Indonesia to date, it is possible that the unemployment rate in Indonesia will increase. The purpose of this study is to see how the unemployment rate during the Covid-19 pandemic is taking into account several compartments, namely susceptible, unemployed, employed, reduction, and lead, showing a sensitivity analysis to determine the parameters that affect the basic reproduction number , and bifurcation analysis. The results obtained from the sensitivity analysis, which found a parameter relationship with which could increase and decrease the value of , and the bifurcation analysis showed the effect of changes in the stability of the equilibrium point due to changes in the value of the parameter . model simulation shows the unemployment rate during the Covid 19 pandemic and to show the effect of the governments policy rate ( on unemployment during the Covid 19 pandemic.

Kontrol Optimal Model Matematika Dinamika Korupsi dengan Pemberian Edukasi dan Kampanye, Perbaikan Sistem, dan Represif Wahid, Amira; Toaha, Syamsuddin; Kasbawati, Kasbawati
Proximal: Jurnal Penelitian Matematika dan Pendidikan Matematika Vol. 6 No. 1 (2023): Volume 6 Nomor 1 tahun 2023
Publisher : Universitas Cokroaminoto Palopo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30605/proximal.v6i1.1973

Salah satu masalah yang menarik untuk dikaji melalui pendekatan model matematika yaitu perilaku korupsi yang mengancam kehidupan masyarakat. Sektor pelayanan publik merupakan salah satu contoh lahan basah terkait korupsi birokrasi. Selain itu, adapula korupsi yang lebih besar karena mencakup pembuatan kebijakan politik. Pengembangan model dalam artikel ini dilakukan berdasarkan model matematika korupsi yang telah dikembangkan oleh (Fantaye dan Birhanu, 2021) dengan membagi populasi menjadi lima kompartemen yaitu susceptible (S), exposed (E), corrupt (C), jailed (J) dan honest (H). Penelitian ini bertujuan untuk menganalisis titik kesetimbangan pada model matematika dinamika korupsi serta memberikan penerapan kontrol optimal pada dinamika korupsi melalui strategi yang telah diusung oleh KPK yaitu edukasi dan kampanye, perbaikan sistem, dan strategi represif diharapkan mampu menangani kasus korupsi secara efektif. Dari hasil analisis model diperoleh dua titik kesetimbangan yaitu titik kesetimbangan tanpa korupsi dan titik kesetimbangan adanya korupsi. Titik kesetimbangan tersebut akan stabil jika memenuhi syarat yang ditetapkan oleh aturan Routh-Hurwitz. Berdasarkan hasil simulasi numerik, menunjukkan bahwa peran KPK dalam memberantas korupsi dengan edukasi dan kampanye, perbaikan sistem, dan strategi represif memberikan hasil yang efektif.

Kontrol Optimal Model Matematika Dinamika Korupsi dengan Pemberian Edukasi dan Kampanye, Perbaikan Sistem, dan Represif Wahid, Amira; Toaha, Syamsuddin; Kasbawati, Kasbawati
Proximal: Jurnal Penelitian Matematika dan Pendidikan Matematika Vol. 6 No. 1 (2023): Inovasi Teknologi, Psikologi Belajar, dan Adaptasi Pembelajaran Matematika di E
Publisher : Universitas Cokroaminoto Palopo

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30605/proximal.v6i1.1973

Salah satu masalah yang menarik untuk dikaji melalui pendekatan model matematika yaitu perilaku korupsi yang mengancam kehidupan masyarakat. Sektor pelayanan publik merupakan salah satu contoh lahan basah terkait korupsi birokrasi. Selain itu, adapula korupsi yang lebih besar karena mencakup pembuatan kebijakan politik. Pengembangan model dalam artikel ini dilakukan berdasarkan model matematika korupsi yang telah dikembangkan oleh (Fantaye dan Birhanu, 2021) dengan membagi populasi menjadi lima kompartemen yaitu susceptible (S), exposed (E), corrupt (C), jailed (J) dan honest (H). Penelitian ini bertujuan untuk menganalisis titik kesetimbangan pada model matematika dinamika korupsi serta memberikan penerapan kontrol optimal pada dinamika korupsi melalui strategi yang telah diusung oleh KPK yaitu edukasi dan kampanye, perbaikan sistem, dan strategi represif diharapkan mampu menangani kasus korupsi secara efektif. Dari hasil analisis model diperoleh dua titik kesetimbangan yaitu titik kesetimbangan tanpa korupsi dan titik kesetimbangan adanya korupsi. Titik kesetimbangan tersebut akan stabil jika memenuhi syarat yang ditetapkan oleh aturan Routh-Hurwitz. Berdasarkan hasil simulasi numerik, menunjukkan bahwa peran KPK dalam memberantas korupsi dengan edukasi dan kampanye, perbaikan sistem, dan strategi represif memberikan hasil yang efektif.

COMPARISON OF PROJECTED UNIT CREDIT AND ENTRY AGE NORMAL METHODS IN PENSION FUND VASICEK AND COX-INGERSOLL-ROSS MODELS Sulma, Sulma; Widana, I Nyoman; Toaha, Syamsuddin; Fitria, Ita
BAREKENG: Jurnal Ilmu Matematika dan Terapan Vol 17 No 4 (2023): BAREKENG: Journal of Mathematics and Its Applications
Publisher : PATTIMURA UNIVERSITY

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30598/barekengvol17iss4pp2421-2432

The pension program is one of the pension fund's efforts to anticipate the risks that will be experienced by participants in old age. Actuarial calculations help to determine the benefits that participants will receive by considering life chances, interest rates, age when becoming a participant, and normal retirement age. This study aims to determine normal contributions and actuarial liabilities with the Projected Unit Credit and Entry Age Normal methods using stochastic interest rates, namely Vasicek and Cox-Ingersoll-Ross (CIR). The data used in this study are civil servants who work at the Natural Resources Management Office, Bulukumba Regency. The results of the calculation analysis showed that normal cost using the Projected Unit Credit (PUC) method with the Vasicek and Cox-Ingersoll-Ross (CIR) model interest rates increased as the length of service increased, and at the end of the working period the Cox-Ingersoll-Ross (CIR) model interest rate reached Rp14.773.176,- which was higher than Vasicek by Rp3.849.898,-. The results of the calculation of normal cost using the Entry Age Normal (EAN) method with the Vasicek model increase in the period 0-20 years of service, then decrease towards the contribution value at the beginning of the service period of Rp1.499.725,-. At the beginning of the working year, the normal cost using the Entry Age Normal method with the Cox-Ingersoll-Ross (CIR) model interest rate is Rp7.581.593,- then decreases for 24 years of service to Rp5.849.854,- after which it increases again towards the initial contribution value of the working year. The results of the calculation of actuarial liabilities show an increase as the length of service increases, for the Entry Age Normal (EAN) and Projected Unit Credit (PUC) methods with the Cox-Ingersoll-Ross (CIR) interest rate model at the end of the service period, it is found that both are the same value, namely Rp443.195.285,-. By using the Vasicek interest rate model for both methods, the same result is obtained at the end of the service period of Rp115.496.951,-. This shows that the actuarial liabilities for both methods used are affected by interest rates, and the Cox-Ingersoll-Ross (CIR) model is higher than Vasicek

Analysis Dynamics Two Prey of a Predator-Prey Model with Crowley–Martin Response Function Pratama, Rian Ade; Toaha, Syamsuddin
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 7, No 3 (2023): July
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v7i3.14506

The predator-prey model has been extensively developed in recent research. This research is an applied literature study with a proposed dynamics model using the Crowley–Martin response function, namely the development of the Beddington-DeAngelis response function. The aim of this research is to construct a mathematical model of the predator-prey model, equilibrium analysis and population trajectories analysis. The results showed that the predator-prey model produced seven non-negative equilibrium points, but only one equilibrium point was tested for stability. Stability analysis produces negative eigenvalues indicating fulfillment of the Routh-Hurwitz criteria so that the equilibrium point is locally asymtotically stable. Analysis of the stability of the equilibrium point indicates stable population growth over a long period of time. Numerical simulation is also given to see the trajectories of the population growth movement. The population growth of first prey and second prey is not much different, significant growth occurs at the beginning of the growth period, while after reaching the peak the trajectory growth slopes towards a stable point. Different growth is shown by the predator population, which grows linearly with time. The growth of predators is very significant because predators have the freedom to eat resources. Various types of trajectory patterns in ecological parameters show good results for population growth with the given assumptions.

Stability Analysis and Optimal Control of Mathematical Model of Thypoid Fever Spread Siduppa, Muh. Nursyam; Toaha, Syamsuddin; Kasbawati, Kasbawati
InPrime: Indonesian Journal of Pure and Applied Mathematics Vol. 5 No. 1 (2023)
Publisher : Department of Mathematics, Faculty of Sciences and Technology, UIN Syarif Hidayatullah

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.15408/inprime.v5i1.27205

AbstractTyphoid fever is an endemic disease caused by infection with Salmonella Typhi. The transmission of typhoid fever is through food and drink contaminated with Salmonella Typhi bacteria, which is excreted through the feces or urine of an infected person. The problem of typhoid fever is increasingly complex because of the increase in carrier cases, making it difficult for treatment and prevention efforts. This study develops a mathematical model for the control of typhoid fever, which consists of two equilibrium points, namely endemic and non-endemic equilibrium points. The endemic and non-endemic equilibrium point is asymptotically stable if it satisfies the condition given by the Routh-Hurwitz criterion. Optimal control theory is applied to the mathematical model by providing control through health campaigns, screening, and treatment to minimize the number of asymptomatic individuals, symptomatic individuals, and chronic carriers. The Pontryagin Minimum principle is used to determine the optimal control form. Numerical simulations are performed using the Forward-Backward Sweep Runge-Kutta method of order 4. The simulation results indicate a decrease in each infected subpopulation after applying optimal control for ten months. It is found that control in health campaigns has a more significant impact than control in screening and treatment in decreasing the number of asymptomatic and symptomatic individuals. The control of treatment effectively reduces infected individuals with symptoms of becoming chronic carriers. In conclusion, the most effective strategy in controlling the spread of typhoid fever is to simultaneously apply controls in the form of health campaigns, screening, and treatment.Keywords: health campaign; screening; treatment; optimal control; Pontryagin minimum principle; forward-backward sweep. AbstrakDemam tifoid merupakan penyakit endemik yang disebabkan oleh infeksi bakteri Salmonella Typhi. Proses penularan demam tifoid melalui makanan dan minuman yang telah terkontaminasi bakteri Salmonella Typhi yang dikeluarkan melalui tinja maupun urin dari orang yang telah terinfeksi. Permasalahan tentang demam tifoid semakin kompleks karena meningkatnya kasus - kasus carrier, sehingga menyulitkan upaya pengobatan dan pencegahan. Model matematika yang dikembangkan memiliki dua titik kesetimbangan yaitu titik setimbang nonendemik dan titik setimbang endemik. Titik setimbang nonendemik dan endemik akan stabil asimtotik jika memenuhi kondisi yang diberikan oleh aturan Routh-Hurwitz. Teori kontrol optimal diterapkan pada model matematika dengan pemberian kontrol berupa kampanye kesehatan, screening dan pengobatan untuk meminimumkan jumlah individu asymptomatic, individu symptomatic dan carrier chronic. Penentuan bentuk kontrol optimal menggunakan prinsip Minimum Pontryagin. Simulasi numerik dilakukan dengan menggunakan metode Forward-Backward Sweep Runge-Kutta orde 4. Berdasarkan hasil simulasi, terjadi penurunan disetiap subpopulasi terinfeksi setelah penerapan kontrol optimal selama 10 bulan. Kontrol berupa kampanye kesehatan memiliki pengaruh yang besar dibandingkan kontrol berupa screening dan pengobatan dalam menekan meningkatnya individu asymptomatic dan individu symptomatic. Penerapan kontrol berupa pengobatan sangat efektif dalam menekan individu terinfeksi dengan gejala menjadi individu carrier chronic.Kata Kunci: kampanye kesehatan; screening; pengobatan; kontrol optimal; prinsip minimum Pontryagin; forward-backward sweep. 2020MSC: 00A71, 92B05

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