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All Journal BAREKENG: Jurnal Ilmu Matematika dan Terapan Communication in Biomathematical Sciences PengabdianMu: Jurnal Ilmiah Pengabdian kepada Masyarakat Contemporary Mathematics and Applications (ConMathA) Limits: Journal of Mathematics and Its Applications
Windarto
Departemen Matematika, Universitas Airlangga
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Pendekatan Numerik pada Model Penyebaran SARS dengan Method of Lines Patria Arif Bijaksana; Windarto; Fatmawati
Limits: Journal of Mathematics and Its Applications Vol. 15 No. 1 (2018): Limits: Journal of Mathematics and Its Applications Volume 15 Nomor1 Edisi Mar
Publisher : Pusat Publikasi Ilmiah LPPM Institut Teknologi Sepuluh Nopember

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Abstract

Pada paper ini dikaji pendekatan numerik model matematika penyebaran SARS dengan adanya suku difusi. Suku difusi pada model tersebut mengilustrasikan penyebaran SARS berdasarkan lokasi. Solusi numerik dilakukan dengan menggunakan Method of Lines. Selanjutnya dibandingkan hasil simulasi numerik antara model penyebaran SARS tanpa suku difusi dan dengan adanya suku difusi. Hasil simulasi dari model penyebaran penyakit SARS tanpa suku difusi hanya menunjukkan terjadinya penyebaran SARS secara periodik waktu. Berdasarkan hasil simulasi pada model SARS dengan adanya suku difusi dapat diketahui bahwa penyebaran SARS dapat ditinjau dari titik awal penyebaran SARS secara spasial dan juga perodik waktu. Lebih lanjut, dari hasil simulasi menunjukkan bahwa semakin jauh dari pusat penyebaran SARS, laju penyebaran penyakit SARS akan semakin kecil
Model Predator Prey Leslie Gower dengan Fungsional Respon Crowley Martin dan Adanya Prey Terinfeksi Serta Faktor Ketakutan Miswanto; Nunik Suroiyah; Windarto
Limits: Journal of Mathematics and Its Applications Vol. 20 No. 3 (2023): Limits: Journal of Mathematics and Its Applications Volume 20 Nomor 3 Edisi No
Publisher : Pusat Publikasi Ilmiah LPPM Institut Teknologi Sepuluh Nopember

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Abstract

Ekosistem adalah suatu sistem yang terdiri dari organisme hidup dan lingkungannya yang seringkali terjadi interaksi antara makhluk hidup dengan lingkungannya, atau makhluk hidup yang satu dengan mahkluk hidup yang lain. Sebagai contoh interaksi antara predator dengan prey, yaitu singa dengan rusa, ular dengan tikus, burung elang dengan ular dan lain-lain. Pada interaksi antara predator prey seringkali nampak adanya prey yang terinfeksi. Hal ini berdampak adanya penyebaran penyakit oleh prey terinfeksi. Penelitian ini mengkaji model predator prey Leslie Gower fungsi respon Crowley Martin dengan prey terinfeksi serta adanya faktor ketakutan. Metode yang digunakan dalam penelitian ini adalah metode analitik dan simulasi numerik. Metode analitik digunakan untuk mengkaji eksistensi titik setimbang dan analisis kestabilan titik setimbang model, sedangkan metode simulasi numerik digunakan untuk mendukung hasil metode analitik. Berdasarkan hasil analitik diperoleh dua titik setimbang yang cenderung stabil asimtotis, yaitu titik setimbang kepunahan prey terinfeksi dan predator (E1) dan titik setimbang kepunahan predator (E2) Sedangkan titik setimbang kepunahan prey terinfeksi (E3) dan titik setimbang koeksistensi (E4) tidak diperoleh secara analitik. Oleh karena analisis kestabilannya menggunakan bidang fase. Hasil simulasi numerik menunjukkan bahwa keempat titik setimbang, yaitu E1,E2, E3, dan E4 cenderung stabil asimtotis.
Analisis Kestabilan dan Kontrol Optimal pada Model Matematika Penyebaran Penyakit Mumps Miswanto; Farah Biba; Windarto
Limits: Journal of Mathematics and Its Applications Vol. 21 No. 1 (2024): Limits: Journal of Mathematics and Its Applications Volume 21 Nomor 1 Edisi Ma
Publisher : Pusat Publikasi Ilmiah LPPM Institut Teknologi Sepuluh Nopember

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Abstract

Mumps is an acute disease in children and adults, caused by paramyxovirus. In this thesis, a mathematical model analysis of the spread of mumps disease was carried out and the application of optimal control, namely prevention by giving vaccinations and treatment. Based on the analysis of the model without control obtained a non-endemic equilibrium point and an endemic equilibrium point. The non-endemic equilibrium point is local asymptotic stable if the basic reproduction number is less than one while the endemic equilibrium point tends to be local asymptotic stable if the basic reproduction number is more than one . Optimal control on the mathematical model of the spread of mumps disease was carried out using the Pontryagin’s Maximum Principle. The results of numerical simulations show that the provision of control, namely prevention and treatment, is simultaneously considered the most effective and efficient to minimize the number of individual populations infected with mumps disease with minimum cost.
A MATHEMATICAL MODEL OF DIPHTHERIA TRANSMISSION DYNAMICS WITH HETEROGENEOUS SUSCEPTIBILITY Mohamad Tafrikan; Fatmawati Fatmawati; Windarto Windarto; Chinwendu E. Madubueze
BAREKENG: Jurnal Ilmu Matematika dan Terapan Vol 20 No 3 (2026): BAREKENG: Journal of Mathematics and Its Application
Publisher : PATTIMURA UNIVERSITY

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.30598/barekengvol20iss3pp1967-1984

Abstract

Despite the availability of vaccines, diphtheria continues to pose a public health risk in Indonesia due to uneven vaccination coverage across regions. Previous models have not distinguished between highly susceptible (unvaccinated) and susceptible (vaccinated) populations, nor have they been calibrated with actual Indonesian epidemiological data. To address this gap, this study develops a five-compartment diphtheria transmission model: Highly susceptible (unvaccinated)-Susceptible (vaccinated)-Exposed-Infectious-Recovered (S_1 S_2 EIR), which incorporates two levels of susceptibility based on vaccination status, using empirical diphtheria case data in Indonesia from 2012 to 2023. The analysis begins by proving the positivity, boundedness, and uniqueness of solutions, followed by the calculation of the basic reproduction number using the Next-Generation Matrix method. The analysis shows that the disease-free equilibrium (DFE) is locally and globally asymptotically stable when R₀<1, while the endemic equilibrium (EE) is globally stable when R₀>1. Simulations indicate that the interaction parameter for the unvaccinated group η₁, strongly accelerates epidemic growth, leading to a higher and earlier infection peak, whereas increased vaccination coverage and recovery rates effectively suppress transmission. This model can be used because the Mean Absolute Percentage Error (MAPE) between the data and the model solution for diphtheria cases in Indonesia is 8.77%. These results highlight the importance of interventions focused on highly susceptible groups to prevent more severe outbreaks. Therefore, this study is significant in strengthening the theoretical understanding of diphtheria transmission, while also providing data-driven insights as recommendations for policymakers to implement effective and efficient outbreak control measures.
Co-Authors Abdul Faliq Anwar Adiluhung Setya Pambudi Alfiniyah, Cicik Bertha Aurellia Pamudya Fajar Biba, Farah Chinwendu E. Madubueze Dinda Ariska Putri Eridani Farah Biba Fatmawati, Fatmawati Inna Kuswandari Jaelani, Abdulloh M. Miswanto Miswanto Mohamad Tafrikan Nanda Amalia Rahma Nisrina Firsta Ammara Nunik Suroiyah Patria Arif Bijaksana Patria Arif Bijaksana Riris Nur Patria Putri Utami Dyah Purwati