Pemodelan Matematika Penyebaran COVID-19 di Gresik Dengan Pengaruh Tindakan Rawat Inap di Rumah Sakit

Mathematical Modeling of the Spread of COVID-19 in Gresik with Inpatient Measures at the Hospital

COVID-19 merupakan suatu jenis penyakit menular yang telah menyebar ke banyak wilayah dengan cepat dan menginfeksi secara global tanpa terkecuali di Kabupaten Gresik. Dalam upaya pengendalian penyebaran COVID-19 di Kabupaten Gresik individu yang terinfeksi diberikan tindakan rawat inap di rumah sakit. Tindakan rawat inap diharapkan dapat menekan angka kasus terinfeksi COVID-19. Berdasarkan karakteristik penyebaran COVID-19 di kabupaten Gresik dapat dikontruksi model SIR yang dimodifikasi dengan menambahkan kompartemen rawat inap di rumah sakit (H) dan memperhitungkan individu terinfeksi yang tidak terdeteksi terdeteksi dan tanpa menjalani tindakan rawat inap di rumah sakit. Penelitian ini bertujuan untuk mengontruksi model matematika penyebaran COVID-19 di kabupaten Gresik dengan pengaruh tindakan rawat inap di rumah sakit. Model matematika yang telah dikontruksi dari hasil modifikasi kedalam bentuk persamaan non-linier kemudian dicari titik kesetimbangan sistem, bilangan reproduksi dasar, analisis kestabilan titik kesetimbangan beserta mensimulasi hasil menggunakan software Matlab. Berdasarkan hasil analisis model secara analitik diperoleh titik kesetimbangan bebas penyakit E₀ = ( s, i, u, h, r_u, r_h) dan titik kesetimbangan endemik E₁ = ( s*, i*, u*, h*, r*_u, r*_h) , beserta analisis kestabilan sistem di sekitar titik kesetimbangan. Nilai parameter diperoleh dari hasil penurunan data sebenarnya di Kabupaten Gresik. Hasil substitusi nilai parameter kedalam bilangan reproduksi dasar diperoleh R₀ = 3.3536 > 1 , artinya COVID-19 di kabupaten Gresik masih mewabah. Berdasarkan hasil simulasi numerik saat  R₀ = 3.3536 > 1 grafik setiap kompartemen stabil disekitar titik kesetimbangan endemik sedangkan saat  R₀ = 0.7703 < 1  grafik setiap kompartemen stabil disekitar titik kesetimbangan bebas penyakit.

COVID-19 is a type of infectious disease that has spread to many regions quickly and infected globally without exception in Gresik Regency. In an effort to control the spread of COVID-19 in Gresik Regency, infected individuals are given hospitalized. Inpatient measures are expected to reduce the number of cases of COVID-19 infection. Based on the characteristics of the spread of COVID-19 in Gresik district, a modified SIR model can be constructed by adding the inpatient compartment at the hospital (H) and taking into account infected individuals who are undetected and without undergoing inpatient treatment at the hospital . This study aims to construct a mathematical model of the spread of COVID-19 in Gresik district with the effect of inpatient treatment at the hospital. Mathematical models that have been constructed from the modified results into non-linear equations are then searched for the system's equilibrium point, the basic reproduction number, the stability analysis of the equilibrium point and simulate the results using Matlab software. Based on the analysis of the model analytically, the disease-free equilibrium point is E₀ = ( s, i, u, h, r_u, r_h)  and the endemic equilibrium point E₁ = ( s*, i*, u*, h*, r*_u, r*_h) , along with an analysis of the stability of the system around the equilibrium point. The parameter values ​​are obtained from the actual data reduction in Gresik Regency. The result of substitution of parameter values ​​into the basic reproduction number obtained R₀ = 3.3536 > 1, meaning that COVID-19 in Gresik district is still endemic. Based on the numerical simulation results when R₀ = 3.3536 > 1  graphs of each compartment are stable around the endemic equilibrium point, while when R₀ = 0.7703 < 1  graphs of each compartment are stable around the disease-free equilibrium point.