A SIR model and the Routh-Hurwitz test

Authors

Keywords:

Epidemiology. Compartmental models. Equilibrium points. Routh-Hurwitz test.

Abstract

Compartmental models are used in epidemiology to describe the dynamics of disease transmission. The objective of this study is to analyze the stability of equilibrium points in a compartmental model where birth and death rates are not necessarily equal, and birth rate depends on the total population size. To analyze the stability of equilibrium points, we applied the so-called Routh-Hurwitz test. For completeness, we also provide a proof of the Routh-Hurwitz test. Through the application of the test, we conclude that the equilibrium points, both disease-free and endemic, in the studied compartmental model are asymptotically stable.

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Author Biographies

  • Natanael de Jesus Oliveira, Rio de Janeiro State University

    Completed high school at CIEP MARIO QUINTANA (2011). Currently, he is studying for a bachelor's degree in Mathematics at the State University of Rio de Janeiro.

  • Patrícia Nunes da Silva, Rio de Janeiro State University

    She has a Bachelor's degree in Mathematics (1996), a Master's degree in Applied Mathematics (1999) and a PhD in Applied Mathematics from the State University of Campinas (2003). She is currently an Associate Professor at the State University of Rio de Janeiro, member of the editorial board of Cadernos do IME. Série Matemática, reviewer for several periodicals. He works in the postgraduate and Computational Sciences programs and in the master's program in mathematics on a national network (PROFMAT) at the Institute of Mathematics and Statistics at UERJ. She has experience in the area of ​​Mathematics and Mathematics Teaching.

References

BODSON, Marc. Explaining the Routh–Hurwitz Criterion: A Tutorial Presentation, IEEE Control Systems Magazine, Piscataway, v. 40, p. 45-51, 2020.

BRAUER, F.; CASTILLO-CHAVEZ, C.; FENG, Z. Mathematical Models in Epidemiology, Nova Iorque, Springer, 2019.

CRONIN, Jane. Some Mathematics of Biological Oscillations. SIAM Review, Filadélfia, v. 19, p. 100-138, 1977.

DIETZ, K.; HEESTERBEEK, J. A. P. Daniel Bernoulli’s epidemiological model revisited,

Mathematical Biosciences, Amsterdã, v. 180, p. 1-21, 2002.

HETHCOTE, Herbert. W. Qualitative analyses of communicable disease models, Mathematical Biosciences, Amsterdã, v. 28, p. 335–356, 1976.

HETHCOTE, Herbert. W. The Mathematics of Infectious Diseases. SIAM Review, Filadélfia, v. 42, p. 599-653, 2000.

KANG, Chul-Goo. Origin of Stability Analysis: “On Governors” by J.C. Maxwell. IEEE Control Systems Magazine, Piscataway, v. 36, p. 77-88, 2016.

KERMACK, W. O.; MCKENDRICK, A. G. Contributions to the mathematical theory of epidemics II. the problem of endemicity, Bulletin of Mathematical Biology, Nova Iorque, v. 53, p. 57-87, 1991.

MEINSMA, Gjerrit. Elementary proof of the Routh-Hurwitz test, Systems & Control Letters, Amsterdã, v. 25, p. 237-242, 1995.

RODRIGUES, Helena Sofia. Application of SIR Epidemiological Model: New Trends. International Journal of Applied Mathematics and Informatics, Bridgewater, v. 10, p. 92-97, 2016.

ROUTH, Edward John. A Treatise on the Stability of a Given State of Motion: Particularly Steady Motion. Londres, Macmillan and Company, 1877.

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Published

2024-09-24

Issue

Volume 11, 2024 – publicação Contínua

Section

Artigos

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Copyright (c) 2024 Revista Brasileira de Iniciação Científica

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How to Cite

A SIR model and the Routh-Hurwitz test. (2024). Revista Brasileira De Iniciação Científica, 024041. https://periodicoscientificos.itp.ifsp.edu.br/index.php/rbic/article/view/1436