No abstract

This article examines the mathematical model of divorce. This model consists of four population classes, namely the Married class (M), the population class who experiences separation of separated beds (S), the population class who is divorced by Divorce (D), and the population class who experiences depression or stress due to divorce Hardship (H). This study focuses on the stability analysis of divorce-free and endemic equilibrium points. Local stability was analyzed using linearization and eigenvalues ​​methods. In addition, the basic reproduction number  is provided via the next generation matrix method. The existence and stability of the equilibrium point are determined from . The results showed that the rate of interaction between population M and populations other than H is very influential on efforts to minimize divorce. Divorce can be minimized when the transmission rate is reduced to . Reducing the transmission rate and increasing the rate of transfer from split bed class to married class can turn divorce endemic cases into non-endemic cases. A numerical simulation is given to confirm the analysis results.

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.

Oil palm empty fruit bunches (OPEFBs) are highly abundant in Indonesia and have been highlighted as a potential feedstock for bioethanol and xylitol production. However, the efficacy of the fermentation technology to convert OPEFBs to bioethanol and xylitol, either in partial (i.e. mono-production) or integrated (i.e. co-production) process, still needs further improvement. This study aimed to evaluate the partial and integrated process for xylitol and bioethanol production from OPEFBs. In the integrated process, the remaining solid residues after xylitol extraction are used as feedstock for bioethanol due to their high cellulose compounds. This solid residue is more susceptible to be degraded by cellulase enzymes into glucose and further transformed into bioethanol. In the partial process of xylitol production, xylanase enzyme was used to hydrolyze xylan into xylose, which was then converted into xylitol using Debaryomyces hansenii. While in the partial process of bioethanol production, the hydrolysis of cellulose in the OPEFB into glucose was carried out using cellulase enzymes, followed by fermentation using Saccharomyces cerevisiae. The results show that the partial process produced xylitol yield (Yp/s) of 0.10 g-xylitol/g-xylose, while bioethanol at yield (Yp/s) of 0.32 g-bioethanol/g-glucose, respectively. The integrated process generates xylitol yield (Yp/s)of 0.298 g-xylitol/g-xylose, with bioethanol yield from the remaining solid at 0.051 g-bioethanol/g-OPEFB (or 0.078 g-bioethanol/g-glucose). These findings, therefore, confirmed that the integrated process of xylitol with bioethanol production might offer higher efficacy of OPEFB utilization into high value-added products.

Abstrak Penelitian ini mengkaji model satu mangsa dan satu pemangsa yang saling berkompetisi. Fungsi predasi dari pemangsa diasumsikan menggunakan fungs1 respon Holling tipe II. Dengan asumsi bahwa adanya kompetisi intraspesifik pada popuasi pemangsa serta dilakukan pemanenan ambang batas pada popuasi pemangsa. Pada model tersebut dilakukan analisis tentang syarat kewujudan dan kestabilan titik keseimbangan interior. Analisis kestabilan titik keseimbangan interior dilakukan dengan metode linearisasi dan dengan memperhatikan nilai eigen dari matriks Jacobi yang diperoleh. Terdapat sepuluh titik kesetimbangan yang diperoleh pada model, satu diantaranya dapat dinterpretasikan. Titik tersebut dinyatakan stabil asimtotik. Berdasarkan hasil anasis menggunakan beberapa parameter, diketahui bahwa ada suatu waktu pemanenan ambang batas harus dihentikan karna sudah tidak memenuhi syarat kriteria ambang batas yang telah ditentukan.Kata kunci : Model mangsa pemangsa, Pemanenan ambang batas, Titik kesetimbanganAbstract This study examines the model of one prey and one predator who mutates each other. The predation function of predators is assumed to use the Holling type II response function. Assuming that the existence of intraspecific competition in predatory population and theshold harvesting for predatory population is carried out. In this model, an analysis of the actual conditions and stability of the interior balance point is carried out. Analysis of the interior stability balance points was carried out by linearization method and by taking into account the eigenvalues of the Jacobian matrix obtained. There are ten equilibrium points of engagement obtained on the model, one of which can be interpreted. This point is stated as asymptotically stable. Based on the results of analysis using several parameters, it is known that there is a time when harvesting the threshold must be stopped because it has not fulfill the specified criteria for threshold.Keyword : Prey-predator model threshold harvesting, equibrium point