Eminugroho Ratna Sari
Department of Mathematics Education, Mathematics Study Program, Faculty of Mathematics and Natural Sciences, Universitas Negeri Yogyakarta, Indonesia

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A Long-Term Dynamics of Chikungunya Transmission with Chronic Infection: Global Stability Analysis of An SICR-SI Model Ghifa Attaya Ulhaq; Eminugroho Ratna Sari
InPrime: Indonesian Journal of Pure and Applied Mathematics Vol. 8 No. 1 (2026)
Publisher : Department of Mathematics, Faculty of Sciences and Technology, UIN Syarif Hidayatullah

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.15408/e8792k67

Abstract

Climate change and environmental factors have increased the risk of Chikungunya transmission in tropical regions, highlighting the need for mathematical models that capture its long-term epidemiological dynamics. This study proposes a modified SICR-SI compartmental model incorporating a chronic infection stage, where infected individuals may either recover directly or progress to a non-infectious chronic compartment before recovery. The model extends previous Chikungunya transmission frameworks by repositioning the chronic class before recovery and by establishing a new global stability analysis. Qualitative analysis is performed through positivity and invariant-region investigations, derivation of the basic reproduction number using the Next Generation Matrix approach, local stability analysis via eigenvalue and Routh–Hurwitz criteria, and global stability analysis using a Lyapunov function. The analysis shows that the disease-free equilibrium exists and is locally as well as globally asymptotically stable when the basic reproduction number satisfies R_0 < 1, whereas an endemic equilibrium exists and is locally asymptotically stable when R_0 > 1. Numerical simulations based on Chikungunya data from West Java Province, Indonesia, confirm the theoretical findings and indicate a disease-free scenario with R_0 < 1. Sensitivity exploration further demonstrates that increasing transmission probabilities between humans and mosquitoes significantly elevates R_0, potentially driving the system toward endemic persistence. These results emphasize the importance of reducing vector–host transmission intensity as an effective strategy for controlling Chikungunya outbreaks and provide a theoretical foundation for future epidemiological intervention studies.   Abstrak Perubahan iklim dan faktor lingkungan telah meningkatkan risiko penularan Chikungunya di daerah tropis, menyoroti perlunya model matematika yang menangkap dinamika epidemiologi jangka panjangnya. Studi ini mengusulkan model kompartemen SICR-SI yang dimodifikasi yang menggabungkan tahap infeksi kronis, di mana individu yang terinfeksi dapat langsung pulih atau berkembang ke kompartemen kronis non-infeksius sebelum pulih. Model ini memperluas kerangka kerja penularan Chikungunya sebelumnya dengan memposisikan ulang kelas kronis sebelum pemulihan dan dengan menetapkan analisis stabilitas global baru. Analisis kualitatif dilakukan melalui investigasi positif dan wilayah invarian, derivasi angka reproduksi dasar menggunakan pendekatan Matriks Generasi Berikutnya, analisis stabilitas lokal melalui kriteria nilai eigen dan Routh–Hurwitz, dan analisis stabilitas global menggunakan fungsi Lyapunov. Analisis menunjukkan bahwa keseimbangan bebas penyakit ada dan stabil secara asimtotik lokal maupun global ketika angka reproduksi dasar memenuhi R_0 < 1, sedangkan keseimbangan endemik ada dan stabil secara asimtotik lokal ketika R_0 > 1. Simulasi numerik berdasarkan data Chikungunya dari Provinsi Jawa Barat, Indonesia, mengkonfirmasi temuan teoritis dan menunjukkan skenario bebas penyakit dengan R_0 < 1. Eksplorasi sensitivitas lebih lanjut menunjukkan bahwa peningkatan probabilitas penularan antara manusia dan nyamuk secara signifikan meningkatkan R_0, berpotensi mendorong sistem menuju persistensi endemik. Hasil ini menekankan pentingnya mengurangi intensitas penularan vektor-inang sebagai strategi efektif untuk mengendalikan wabah Chikungunya dan memberikan landasan teoritis untuk studi intervensi epidemiologi di masa mendatang. Kata Kunci: Chikungunya; model SICR-SI; Infeksi kronis; Angka reproduksi dasar; Fungsi Lyapunov; Stabilitas global.   2020MSC: 92D30, 37N25.