The mathematical model of glioblastoma multiforme brain tumor (GBM) consists of a population of tumor cells that are sensitive ( ) and the pupulation of cells susceptible to tumor ( ). The effect of treatment on sensitive cells is given by Chemoresistant and pleotropic ( ), whereas the effect of treatment on susceptible cells is a precursor to prevention in tumor patients ( ). This article aims to solve the equation of GBM brain tumor model with the effect of treatment using Runge Kutta Fehlberg method. The result of Runge Kutta Fehlberg method has high accuracy and has fulfilled the given error tolerance of 10 . Numerical solutions show that both populations have met the error tolerance when it reaches 200 days with = 1. Based on these results, the numerical solution to the effect of treatment using the Runge Kutta Fehlberg method has a good accuracy in solving nonlinear common differential equations of GBM brain tumor mode.

Model Prey-predator merupakan salah satu model interaksi antara dua spesies yang berbentuk persamaan diferensial biasa nonlinier. Tujuan dari penelitian ini yaitu untuk menganalisis model prey-predator dengan pemanenan konstan pada ikan prey dan melakukan intepretasi pada model tersebut berdasarkan simulasi yang dilakukan. Dengan menggunakan nilai pemanenan 0≤ℎ≤ℎ𝑚𝑎𝑘𝑠,dimana ℎ𝑚𝑎𝑘𝑠 merupakan nilai pemanenan maksimu. Maka didapatkan lima titik kesetimbangan yang terdapat satu titik kesetimbangan yang stabil dengan jenis titik simpul dan jenis kestabilan berupa stabil asimtotik. Dari simulasi yang dilakukan dengan tiga kondisi nilai pemanenan yaitu ketika ℎ<ℎ𝑚𝑎𝑘𝑠, ℎ=ℎ𝑚𝑎𝑘𝑠dan ℎ>ℎ𝑚𝑎𝑘𝑠. Maka dapat disimpulkan bahwa jika nilai pemanenan melebihi nilai pemanenan maksimum maka model tersebut tidak stabil dan populasi ikan prey akan punah dan diikuti oleh populasi ikan predator. Pada penelitian selanjutnya dapat dilakukan analisis pada model prey-predator dengan memberikan perlakuan pemanenan berupa konstan pada kedua spesies dan selain itu juga dengan memberikan perlakuan pemanenan berupa fungsi pemanenan kepada salah satu spesies atau kedua spesies.

ABSTRAKProses produksi sel darah (hematopoiesis) pada kondisi normal diformulasikan dalam bentuk sistem persamaan diferensial nonlinier dengan waktu perlambatan. Waktu perlambatan menunjukkan durasi atau waktu yang diperlukan sel punca berada pada fase proliferasi. Penelitian ini bertujuan untuk menganalisis model matematika pada proses produksi sel darah meliputi analisis titik tetap dan perilaku populasi sel punca hematopoietik. Untuk mempelajari perilaku dinamik model, dilakukan dengan mempelajari persamaan karakteristik dari model tersebut. Hasil simulasi numerik menunjukkan bahwa untuk titik tetap nontrivial model mengalami osilasi. Osilasi pada model matematika proses hematopoiesis mengindikasikan bahwa hematopoiesis yang terjadi tidak stabil sehingga nantinya dapat diimplementasikan pada analisa adanya penyakit-penyakit yang mempengaruhi sel darah.Kata Kunci: Hematopoiesis, osilasi, model matematika, waktu perlambatan ABSTRACT Blood cell production process (hematopoiesis) in normal conditions is formulated in the form of a nonlinear differential equation system with time delay. Time delay indicates the duration or time required for stem cells in the proliferative phase. This study aims to analyze the mathematical model of blood cell production process including fixed point analysis and hematopoietic stem cell population behavior. Studying the characteristics equations of the model was conducted to study the dynamic behavior of the model. The numerical simulation results show that for nontrivial fixed point model experiences oscillations. Oscillations in mathematical models of hematopoiesis process indicate that hematopoiesis occurs unstable so that they can be implemented on an analysis of the presence of diseases that affect blood cells.

Animal feed is one of the most important things for sheep and goat farming. Without a good balance of feed ingredients, sheep and goats will not grow optimally, because the feed given to livestock does not match their needs. Therefore, we need an appropriate way to regulate the nutritional needs of feed required by sheep and goats. This study aims to meet the nutritional needs of sheep and goats from a variety of concentrate and forage feed ingredients. To meet the nutritional needs of feed ingredients, it is necessary to do research on optimization in the manufacture of ration feed formulas. If the nutritional needs have been met, the next goal is to predict the price of the ration economically in order to provide a profit. To solve this problem, an approach is needed to model the relationship between concentrate feed ingredients and forage feedstuff variables. Multivariate linear regression is a regression analysis method that involves more than one response variable.

The spread of hepatitis B virus (HBV) infection has been widely studied using the deterministic SIR model, in which individuals who recover from acute infection have temporary immunity to the virus. However, this deterministic model uses a constant rate of viral infection over time. This is not in accordance with the fact that the infection rate is a random parameter that depends on time. This study discusses the analysis of the stochastic model of hepatitis B virus transmission. The purpose of this study is to construct the SIR stochastic model by dividing the infection rate into two, namely the rate of acute and chronic infection following the Wiener process. The model is then searched for an analytical solution referring to the Ito formula. The analytical solution and the Wiener process are described by substituting parameter values in the form of acute and chronic infection rates (β+α), cure rate (γ), and initial values (S(0) and I(0)) to obtain the mean value (μ). and the standard deviation (σ) of dS(t) and dI(t). The results of the simulation show that the number of infected individuals (I(t)) will decrease rapidly if (γ) is greater but will increase rapidly if (β+α) and (I(0)) are greater.