Model Matematika Makrofag dan Sitokin pasca Infark Miokard

ABSTRAK

Model matematika makrofag dan sitokin pasca Infark Miokard yang telah dirumuskan oleh Yunji Wang, dkk (2012) menjelaskan tentang aktivasi makrofag yang mensekresikan sitokin. Model matematika makrofag dan sitokin pasca Infark Miokard merupakan sistem persamaan diferensial nonlinier yang terdiri dari 6 persamaan sehingga perlu menentukan metode khusus dalam menentukan solusinya. Penyelesaian sistem persamaan diferensial nonlinier umumnya sulit diselesaikan secara analitik. Metode Runge Kutta orde-4 dan ODE45 merupakan salah satu metode numerik untuk menyelesaikan persamaan diferensial nonlinier.

Tujuan penelitian ini adalah untuk menginterpretasikan model dan menyelesaikan sistem persamaan diferensial secara numerik dengan menggunakan metode Runge Kutta orde-4 dan ODE45. Hasil penelitian menunjukkan pada saat t=90 dan h=0.01 diperoleh solusi yakni pada metode Runge-Kutta M_u (t)=15554.763527729, M_1 (t)=21735.245246907, M_2 (t)=12709.990494324, I_10 (t)=2.480470719, T_α (t)=0.251660894, dan I_1 (t)= 0.941588383. Solusi pada ODE45 diperoleh M_u (t)=15554.764065455, M_1 (t)=21735.245051221, M_2 (t)=12709.990152284,I_10 (t)=2.480470654, T_α (t)=0.25167585240, I_1 (t)=0.941588583. Berdasarkan hasil penyelesaian model matematika makrofag dan sitokin pasca Infark Miokard dengan metode numerik Runge Kutta orde-4 dan ODE45, dapat ditarik kesimpulan bahwa solusi dari kedua metode saling berdekatan dan hampir sama. Hal tersebut dapat ditunjukkan dengan grafik solusi dari kedua metode tersebut.

ABSTRACT

Mathematical model of macrophages and cytokines post Myocardial Infarction by Yunji Wang, et al (2012) describes the activation of macrophages that secretion cytokines. Mathematical model of macrophages and cytokines post Myocardial Infarction is a system of nonlinear differential equations consist of 6 equations so that it is necessary to determine a specific method in determining the solution. Completion of systems of nonlinear differential equations is generally difficult to solve with analytical method. Fourth order Runge Kutta method and ODE45 is one of the numerical methods to solve nonlinear differential equations.

The purpose of this study is to interpret the model and solve the system of differential equations with numerical methods using Fourth order Runge Kutta methods and ODE45. The results show that when t = 90 and h = 0.01 the solution is obtained in the Fourth order Runge Kutta methods M_u (t)=15554.763527729, M_1 (t)=21735.245246907, M_2 (t)=12709.990494324, I_10 (t)=2.480470719, T_α (t)=0.251660894, and I_1 (t)= 0.941588383. The solution on ODE45 is obtained M_u (t)=15554.764065455, M_1 (t)=21735.245051221, M_2 (t)=12709.990152284,I_10 (t)=2.480470654, T_α (t)=0.25167585240, I_1 (t)=0.941588583. Based on the results of solving mathematical model of macrophages and cytokines post Myocardial Infarction with the numerical Fourth order Runge Kutta method and ODE45, it can be concluded that the solutions of the two methods are close together and almost be the same. It can be indicated by a graph of the solutions of the two methods.