Analisis Model Matematika Penyebaran Penyakit COVID-19 Tipe SEIT1T2R

Muhammad Reza Subakti, Dr.rer.nat. Lina Aryati, M.S.

2026 | Tesis | S2 Matematika

Pandemi COVID-19 yang disebabkan oleh SARS-CoV-2 terus menjadi ancaman kesehatan masyarakat global, meskipun status darurat telah dicabut oleh WHO (Mei 2023), fluktuasi kasus masih berlanjut hingga pertengahan 2025. Untuk memahami dinamika ini, dalam penelitian ini dikembangkan model deterministik SEIT1T2R sebagai modifikasi model SEIQR dengan membedakan dua jenis treatment: karantina mandiri (T_1) dan perawatan secara medis (T_2). Pada model ini juga dipertimbangkan faktor vaksinasi dan reinfeksi. Analisis yang dilakukan meliputi penentuan daerah invarian positif, titik ekuilibrium bebas penyakit (E0) yang selalu eksis, dan titik ekuilibrium endemik (E1). Eksistensi E1 dijamin apabila bilangan reproduksi dasar memenuhi kondisi R0 > 1 dan rasio R0E/R0I > k2/k1. Kestabilan ekuilibrium dianalisis menggunakan metode linearisasi, kriteria Routh–Hurwitz, kriteria Castillo–Chavez (berbasis sifat M-matriks), serta konstruksi fungsi Lyapunov.

Secara analitik ditunjukkan bahwa E0 bersifat stabil asimtotik lokal maupun global ketika R0<1>1, termasuk yang tidak memenuhi syarat cukup pada pembuktian Lyapunov. Selain itu, penurunan jumlah individu terinfeksi diperoleh ketika vaksinasi dan intervensi medis ditingkatkan.

The COVID-19 pandemic, caused by SARS-CoV-2, remains a global public health threat. Although the World Health Organization (WHO) lifted the Public Health Emergency of International Concern (PHEIC) status in May 2023, case fluctuations persisted through mid-2025. To capture these evolving dynamics, we develop a deterministic SEIT1T2R model, modifying the classical SEIQR framework by distinguishing two types of treatment: self-quarantine (T1) and medical care (T2). The model also accounts for vaccination and reinfection. Our analysis identifies a positively invariant region and establishes the existence of a disease-free equilibrium (E0) and an endemic equilibrium (E1). The endemic equilibrium E1 exists uniquely provided that the basic reproduction number satisfies R0 > 1 and the ratio condition R0E/R0I > k2/k1 holds. We investigate the stability of these equilibria using linearization, the Routh-Hurwitz criterion, and Lyapunov stability theory.

Analytically, we show that E0 is locally and globally asymptotically stable whenever R0 < 1> 1, E1 is locally asymptotically stable under the Routh-Hurwitz conditions. Furthermore, we derive sufficient conditions for the global asymptotic stability of E1 by constructing an appropriate Lyapunov function. Sensitivity analysis of R0 reveals that contact rates have the most significant positive impact, while the self-quarantine rate has the most significant negative impact. Numerical simulations corroborate the theoretical results, demonstrating convergence toward E1 for various initial conditions when R0 > 1. Finally, the results indicate that increasing vaccination coverage and medical intervention effectively reduce the infected population.

Kata Kunci : COVID-19, SEIT1T2R, Bilangan Reproduksi Dasar, analisis kestabilan, analisis sensitivitas.