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All Journal JURNAL MATEMATIKA STATISTIKA DAN KOMPUTASI Jurnal Daya Matematis Prosiding SI MaNIs (Seminar Nasional Integrasi Matematika dan Nilai-Nilai Islami) Desimal: Jurnal Matematika JTAM (Jurnal Teori dan Aplikasi Matematika) Advances in Food Science, Sustainable Agriculture and Agroindustrial Engineering (AFSSAAE) Proximal: Jurnal Penelitian Matematika dan Pendidikan Matematika JATI EMAS (Jurnal Aplikasi Teknik dan Pengabdian Masyarakat) InPrime: Indonesian Journal Of Pure And Applied Mathematics Pedagogy : Jurnal Pendidikan Matematika Journal of Fundamental Mathematics and Applications (JFMA)
Kasbawati Kasbawati
Industrial & Financial Mathematics Research Group, Institut Teknologi Bandung, Jl. Ganesha 10, Bandung 40132, Indonesia Department Of Mathematics, Hasanuddin University, Jl. P. Kemerdekaan Km. 10 Makassar 90245, Indonesia
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A Mathematical Model Analysis of COVID-19 Transmission with Vaccination in Caputo Fractional Derivatives: Case Study in Indonesia Nisardi, Muhammad Rifki; Kasbawati, Kasbawati; Khaeruddin, Khaeruddin
JTAM (Jurnal Teori dan Aplikasi Matematika) Vol 8, No 4 (2024): October
Publisher : Universitas Muhammadiyah Mataram

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.31764/jtam.v8i4.24711

Abstract

This study aims to investigate a fractional-order mathematical model of COVID-19 transmission using the Caputo derivative definition which suitable to epidemiological cases by its advantage to explain memory effects. The model incorporates compartments for asymptomatic infections and includes a vaccination strategy aimed at mitigating the spread of COVID-19. We derived the disease-free and endemic equilibrium points for the fractional model and computed the basic reproduction number (R_0 )  using the Next-generation Matrix method. Additionally, we conducted sensitivity analyses of parameters affecting R_0. The stability of the fractional model requires specific conditions to be met by the model parameters. To approximate active COVID-19 cases in Indonesia, we utilized the Explicit Grunwald-Letnikov method which well fit with Caputo fractional differential system. Simulation results demonstrate that the fractional-order model offers a flexible approach for modelling active COVID-19 cases in these regions. We found that fractional order for active cases COVID-19 in Indonesia is α=0.9856. The simulation showed that decreasing the vaccination rate and the efficacy of the vaccine would affect the reduction of COVID-19 transmission.
Analytical Study of the Existence of a Hopf Bifurcation in the Tumor Cell Growth Model with Time Delay Yusnaeni, A.; Kasbawati, Kasbawati; Syamsuddin, Toaha
InPrime: Indonesian Journal of Pure and Applied Mathematics Vol. 3 No. 1 (2021)
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.v3i1.19515

Abstract

AbstractIn this paper, we study a mathematical model of an immune response system consisting of a number of immune cells that work together to protect the human body from invading tumor cells. The delay differential equation is used to model the immune system caused by a natural delay in the activation process of immune cells. Analytical studies are focused on finding conditions in which the system undergoes changes in stability near a tumor-free steady-state solution. We found that the existence of a tumor-free steady-state solution was warranted when the number of activated effector cells was sufficiently high. By considering the lag of stimulation of helper cell production as the bifurcation parameter, a critical lag is obtained that determines the threshold of the stability change of the tumor-free steady state. It is also leading the system undergoes a Hopf bifurcation to periodic solutions at the tumor-free steady-state solution.Keywords: tumor–immune system; delay differential equation; transcendental function; Hopf bifurcation. AbstrakDalam makalah ini, dikaji model matematika dari sistem respon imun yang terdiri dari sejumlah sel imun yang bekerja sama untuk melindungi tubuh manusia dari invasi sel tumor. Persamaan diferensial tunda digunakan untuk memodelkan sistem kekebalan yang disebabkan oleh keterlambatan alami dalam proses aktivasi sel-sel imun. Studi analitik difokuskan untuk menemukan kondisi di mana sistem mengalami perubahan stabilitas di sekitar solusi kesetimbangan bebas tumor. Diperoleh bahwa solusi kesetimbangan bebas tumor dijamin ada ketika jumlah sel efektor yang diaktifkan cukup tinggi. Dengan mempertimbangkan tundaan stimulasi produksi sel helper sebagai parameter bifurkasi, didapatkan lag kritis yang menentukan ambang batas perubahan stabilitas dari solusi kesetimbangan bebas tumor. Parameter tersebut juga mengakibatkan sistem mengalami percabangan Hopf untuk solusi periodik pada solusi kesetimbangan bebas tumor.Kata kunci: sistem tumor–imun; persamaan differensial tundaan; fungsi transedental; bifurkasi Hopf.
Fractional Mathematical Model of Covid-19 with Quarantine Nisardi, Muhammad Rifki; Kasbawati, Kasbawati; Khaeruddin, Khaeruddin; Robinet, Antonin; Chetehouna, Khaled
InPrime: Indonesian Journal of Pure and Applied Mathematics Vol. 4 No. 1 (2022)
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.v4i1.23719

Abstract

This study aims to observe the dynamics of the spread of COVID-19 with the SIR-Model by considering the quarantine (Q) scheme. We also involve a fractional order in the model. Then the basic reproduction numbers were calculated using the generation matrix method, analyzed the local stability of the fractional model for each equilibrium point, and observed its relation to the basic reproduction numbers. We perform the sensitivity analysis to see the effect of parameters on changes in the basic reproduction numbers. We applied the Grunwald-Letnikov method for numerical simulations. Estimation for parameters was also carried out on the existing parameters in the model to obtain parameter values that could represent the actual conditions. Furthermore, with a fractional model, we approximated the model to the data of COVID-19 in West Sulawesi, Indonesia, so that we could obtain a fractional order since it could describe the data more accurately.Keywords: SIR-Q Model; COVID-19; basic reproduction number; Fractional Mathematical Model; Grunwald Letnikov Method. AbstrakPenelitian ini bertujuan untuk mengkaji dinamika penyebaran COVID-19 dengan model matematika orde fraksional penyebaran penyakit SIR-Q dengan mempertimbangkan skema karantina (Q) untuk mengendalikan penyebaran COVID-19. Bilangan reproduksi dasar dihitung menggunakan metode matriks generasi. Kemudian, dianalisa kestabilan lokal model fraksional untuk titik kesetimbangan dan lalu dianalisa kaitannya dengan bilangan reproduksi dasar. Analisis sensitivitas dilakukan untuk mengamati pengaruh parameter terhadap perubahan bilangan reproduksi dasar. Simulasi numerik dilakukan dengan menggunakan metode eksplisit Grunwald-Letnikov. Estimasi juga dilakukan terhadap parameter yang ada pada model untuk memperoleh nilai parameter yang merepresentasikan kondisi aktual penyebaran COVID-19 di Sulawesi Barat. Selanjutnya dengan model fraksional dilakukan pendekatan terhadap data kasus aktif COVID-19 di Sulawesi Barat sehingga diperoleh orde fraksional tertentu yang menghasilkan pendekatan nilai kasus aktif COVID-19 yang lebih akurat terhadap real data.Kata Kunci: Model SIR-Q; COVID-19; bilangan Reproduksi Dasar; Model Matematika Fraksional; Metode Grunwald-Letnikov.
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

Abstract

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 
Co-Authors A. Muh. Amil Siddik A. Yusnaeni Afifah Afifah Agus Try Hartono Aidawayati, Aidawayati Ainun Mawaddah Abdal Amil Siddik, A. Muh. Amir Kamal Amir Andi Utari Samsir Anggriani, Nita Anisa Anisa Anna Islamiyati Anting Wulandari Antonin Robinet Anwar, Andi M. Aprilyani, Lilis Dwi Sapta B., Hasmawati Bohari, Nurul Aulia Budi Mandra Harahap Chetehouna, Khaled Dadi, Oswaldus Dwi Meldya Lestari Edy Saputra Rusdi Efri Mardawati Emilda Ayu Febrianti Faisah, Faisah Febriyanti, Himmatul Ulya Febryanti Febryanti Haryanto, Loeky Herlina Ahmad I Gusti Agung Komang Diafari Djuni Hartawan Irma Andriani Irwan Akib Jaya, A. Kresna Jusmawati Jusmawati, Jusmawati Jusmawati Massalesse Khaeruddin Khaeruddin Khaeruddin Khaeruddin Khaled Chetehouna Mahie, Andi Galsan Muhammad Rifki Nisardi Muralia Hustim Musdalifa Pagga Mutmainnah Syamsul Naimah Aris Naimah Aris Najhah Aris Natasha Putri Siahaan Nur Erawaty Nur Erawaty Nur Suci Ramadhani Nurdin Nurdin Okta Nofri Oswaldus Dadi Pratama, Rian Ade Putra, Restu Ananda Rabiatul Adawiyah Ramadhan, Muh. Nur Bahri Rangkuti, Aidawayati Rian Ade Pratama Robinet, Antonin Sahriman, Sitti Siduppa, Muh. Nursyam Silvia Yudiastuti Sirajang, Nasrah Sitti Sahriman Sri Astuti Thamrin Sri Suhartini, PhD Sulma Sulma Suriani Suriani Syamsir Muaraf syamsuddin Toaha Syamsuddin Toaha Syamsuddin Toaha Syamsuddin, Toaha Syamsul Agus Thoha, Syamsuddin Toaha Syamsuddin Toaha Toaha Wahda, Wahda Wahid, Amira Yusnaeni, A. Yusrianto Yusrianto