Akurasi Metode Euler Dan Heun Pada Simulasi Numerik Model SIR Dan Efektivitas Intervensi

Nayla Octamandhani; Sopi Amelia Salsabila; Naufal Hafizh Wahdiyat; Taufik Rahmat Kurniawan

doi:10.69693/ijim.v4i3.1732

Authors

Nayla Octamandhani Universitas Muhammadiyah Bandung
Sopi Amelia Salsabila Universitas Muhammadiyah Bandung
Naufal Hafizh Wahdiyat Universitas Muhammadiyah Bandung
Taufik Rahmat Kurniawan Universitas Muhammadiyah Bandung

DOI:

https://doi.org/10.69693/ijim.v4i3.1732

Keywords:

Model SIR, Metode Euler, Metode Heun, Metode Numerik, Epidemiologi

Abstract

Model Susceptible-Infected-Recovered (SIR) adalah kerangka epidemiologi klasik untuk menggambarkan dinamika penyebaran penyakit menular pada populasi tertutup. Karena sistem persamaan diferensial biasa (PDB) SIR bersifat nonlinear, solusi analitik eksaknya sulit diperoleh sehingga metode numerik menjadi pendekatan yang esensial. Penelitian ini mengimplementasikan Metode Euler dan Metode Heun secara mandiri (from scratch) menggunakan Python untuk menyelesaikan sistem PDB SIR, dengan parameter dasar β = 0,30/hari, γ = 0,05/hari (R0 = 6,0) pada populasi N = 10.000 individu. Akurasi kedua metode diverifikasi terhadap solusi rujukan scipy.integrate.odeint dan dievaluasi melalui analisis sensitivitas ukuran langkah h ∈ {0,25; 0,5; 1; 2; 5; 10} hari. Hasil menunjukkan Metode Heun menghasilkan galat rata-rata absolut (MAE) sekitar 13 kali lebih kecil dibanding Metode Euler pada h = 1 hari (11,27 berbanding 147,76), dengan orde konvergensi empiris ≈1 untuk Euler dan ≈2 untuk Heun, konsisten dengan teori. Simulasi lebih lanjut dengan menurunkan laju transmisi β menunjukkan efek pelandaian kurva epidemi (flattening the curve): attack rate akhir turun dari 99,7% menjadi 31,8% ketika R0 diturunkan dari 6,0 menjadi 1,2. Temuan ini menegaskan bahwa Metode Heun lebih unggul baik dari sisi akurasi maupun efisiensi komputasi, serta bahwa penurunan laju transmisi merupakan jalur intervensi yang efektif dalam menekan penyebaran epidemi.

Downloads

Download data is not yet available.

References

Ashgi, R., Pratama, M.A.A. & Purwani, S. 2021. Comparison of Numerical Simulation of Epidemiological Model between Euler Method with 4th Order Runge Kutta Method. International Journal of Global Operations Research, 2(1): 37-44.

Arif, M.S., Abodayeh, K. & Nawaz, Y. 2024. A Third-Order Two Stage Numerical Scheme and Neural Network Simulations for SEIR Epidemic Model: A Numerical Study. Emerging Science Journal, 8(1): 326-340.

Chapra, S.C. & Canale, R.P. 2015. Numerical Methods for Engineers (7th ed.). New York: McGraw-Hill Education.

Diekmann, O., Heesterbeek, J.A.P. & Metz, J.A.J. 1990. On the Definition and the Computation of the Basic Reproduction Ratio R0 in Models for Infectious Diseases in Heterogeneous Populations. Journal of Mathematical Biology, 28(4): 365-382.

Hairer, E., Nørsett, S.P. & Wanner, G. 1993. Solving Ordinary Differential Equations I: Nonstiff Problems (2nd ed.). Berlin: Springer-Verlag.

Harris, C.R., Millman, K.J., van der Walt, S.J. et al. 2020. Array Programming with NumPy. Nature, 585(7825): 357-362.

Hethcote, H.W. 2000. The Mathematics of Infectious Diseases. SIAM Review, 42(4): 599-653.

Hossain, M.T., Miah, M.M. & Hossain, M.B. 2017. Numerical Study of Kermack-McKendrick SIR Model to Predict the Outbreak of Ebola Virus Diseases Using Euler and Fourth Order Runge-Kutta Methods. American Scientific Research Journal for Engineering, Technology, and Sciences, 37(1): 1-21.

Hunter, J.D. 2007. Matplotlib: A 2D Graphics Environment. Computing in Science & Engineering, 9(3): 90-95.

Irwan, M., Pathuddin, H., Jalil, E. & Mariani, A. 2020. Model Epidemik SIR pada Kasus COVID-19 di Indonesia. Jurnal MSA (Matematika dan Statistika serta Aplikasinya), 8(2): 83-87.

Iskandar, D. & Tiong, O.C. 2022. The Application of Runge Kutta Fourth Order Method in SIR Model for Simulation of COVID-19 Cases. Proceedings of Science and Mathematics, 10: 61-70.

Kantner, M. & Koprucki, T. 2020. Beyond Just “Flattening the Curve”: Optimal Control of Epidemics with Purely Non-Pharmaceutical Interventions. Journal of Mathematics in Industry, 10: art. 23.

Kermack, W.O. & McKendrick, A.G. 1927. A Contribution to the Mathematical Theory of Epidemics. Proceedings of the Royal Society of London A, 115(772): 700-721.

Mehdizadeh Khalsaraei, I., Shokri, A., Ramos, H., Yao, S.-W. & Molayi, M. 2022. Efficient Numerical Solutions to a SIR Epidemic Model. Mathematics, 10(18): art. 3299.

Mohd Idris, N.A., Mohtar, S.K., Md Ali, Z. & Abdul Hamid, K. 2022. A Dynamic SIR Model for the Spread of Novel Coronavirus Disease 2019 (COVID-19) in Malaysia. Malaysian Journal of Computing, 7(2): 1108-1119.

Murray, J.D. 2002. Mathematical Biology: I. An Introduction (3rd ed.). New York: Springer.

Nawaz, Y., Arif, M.S., Abodayeh, K. & Shatanawi, W. 2021. An Explicit Unconditionally Stable Scheme: Application to Diffusive COVID-19 Epidemic Model. Advances in Continuous and Discrete Models, 2021: art. 63.

Valdez Martinez, J.S., Guevara Lopez, P., Audelo Gonzalez, J. & Delgado Reyes, G. 2015. Numerical Approaching of SIR Epidemic Model for Propagation of Computer Worms. IEEE Latin America Transactions, 13(10): 3452-3460.

Van den Driessche, P. & Watmough, J. 2002. Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission. Mathematical Biosciences, 180(1-2): 29-48.

Virtanen, P. et al. 2020. SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python. Nature Methods, 17: 261-272.