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All Journal Square : Journal of Mathematics and Mathematics Education
Abdur Rahman Hakim
Universitas Islam Negeri Sultan Syarif Kasim Riau
Education Mathematics

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Model SIRS Nonlinear Incidence Rate dengan Faktor Imigrasi Terinfeksi pada Penyebaran Penyakit COVID-19 Abdur Rahman Hakim; Oktarisa Ramadhanti; Granita
Square : Journal of Mathematics and Mathematics Education Vol. 8 No. 1 (2026)
Publisher : UIN Walisongo Semarang

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Abstract

ABSTRACT COVID-19 is a rapidly spreading disease driven by high human mobility, intense social interactions, and the emergence of new variants that enable reinfection. Classical SIR models generally assume permanent immunity, utilize a linear transmission rate, and do not account for individual immigration, making them inadequate for representing waning immunity, changes in community behavior, and more complex disease transmission dynamics. Therefore, this study extends the SIR model with a nonlinear incidence rate into an SIRS model with a nonlinear incidence rate, incorporating an infected immigration factor. The mathematical analysis includes determining the disease-free and endemic equilibrium points, calculating the basic reproduction number using the Direct Method, and conducting stability analysis through the Jacobian matrix and the Routh–Hurwitz criterion. The analysis yields the disease-free equilibrium point , the basic reproduction number and the endemic equilibrium point . Stability analysis shows that when , the disease-free equilibrium is locally asymptotically stable, indicating that the disease will not spread widely and will eventually disappear from the population. Conversely, when , the endemic equilibrium is locally asymptotically stable, implying that the infection will persist and continue to spread within the population, eventually becoming endemic. Keywords: SIRS model, nonlinear incidence rate, infected immigration, COVID-19.
Co-Authors Granita Oktarisa Ramadhanti