Mathematical Biosciences and Engineering, 2012, 9(3): 577-599. doi: 10.3934/mbe.2012.9.577.

Primary: 92D30; Secondary: 45G15, 65P30.

Export file:


  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text


  • Citation Only
  • Citation and Abstract

Multiple endemic states in age-structured $SIR$ epidemic models

1. Dept. Mathematics, Università di Trento, Via Sommarive 14, 38123 Povo (TN)
2. Department of Mathematics and Computer Science, University of Udine, via delle Scienze 206, I33100 Udine

$SIR$ age-structured models are very often used as a basic model of epidemic spread. Yet, their behaviour, under generic assumptions on contact rates between different age classes, is not completely known, and, in the most detailed analysis so far, Inaba (1990) was able to prove uniqueness of the endemic equilibrium only under a rather restrictive condition.
    Here, we show an example in the form of a $3 \times 3$ contact matrix in which multiple non-trivial steady states exist. This instance of non-uniqueness of positive equilibria differs from most existing ones for epidemic models, since it arises not from a backward transcritical bifurcation at the disease free equilibrium, but through two saddle-node bifurcations of the positive equilibrium. The dynamical behaviour of the model is analysed numerically around the range where multiple endemic equilibria exist; many other features are shown to occur, from coexistence of multiple attractive periodic solutions, some with extremely long period, to quasi-periodic and chaotic attractors.
    It is also shown that, if the contact rates are in the form of a $2 \times 2$ WAIFW matrix, uniqueness of non-trivial steady states always holds, so that 3 is the minimum dimension of the contact matrix to allow for multiple endemic equilibria.
  Article Metrics

Keywords periodic and chaotic solutions; Age-structured epidemic model; multiple endemic equilibria; fixed point index.; numerical bifurcation analysis

Citation: Andrea Franceschetti, Andrea Pugliese, Dimitri Breda. Multiple endemic states in age-structured $SIR$ epidemic models. Mathematical Biosciences and Engineering, 2012, 9(3): 577-599. doi: 10.3934/mbe.2012.9.577


This article has been cited by

  • 1. Dimitri Breda, Odo Diekmann, Stefano Maset, Rossana Vermiglio, A numerical approach for investigating the stability of equilibria for structured population models, Journal of Biological Dynamics, 2013, 7, sup1, 4, 10.1080/17513758.2013.789562
  • 2. Toshikazu Kuniya, Jinliang Wang, Hisashi Inaba, A multi-group SIR epidemic model with age structure, Discrete and Continuous Dynamical Systems - Series B, 2016, 21, 10, 3515, 10.3934/dcdsb.2016109
  • 3. Mimmo Iannelli, Fabio Milner, , The Basic Approach to Age-Structured Population Dynamics, 2017, Chapter 10, 277, 10.1007/978-94-024-1146-1_10
  • 4. Hisashi Inaba, , Age-Structured Population Dynamics in Demography and Epidemiology, 2017, Chapter 6, 287, 10.1007/978-981-10-0188-8_6
  • 5. Toshikazu Kuniya, Existence of a nontrivial periodic solution in an age-structured SIR epidemic model with time periodic coefficients, Applied Mathematics Letters, 2014, 27, 15, 10.1016/j.aml.2013.08.008
  • 6. D.H. Knipl, G. Röst, Large number of endemic equilibria for disease transmission models in patchy environment, Mathematical Biosciences, 2014, 258, 201, 10.1016/j.mbs.2014.08.012
  • 7. Jinliang Wang, Ran Zhang, Toshikazu Kuniya, The dynamics of an SVIR epidemiological model with infection age: Table 1., IMA Journal of Applied Mathematics, 2016, 81, 2, 321, 10.1093/imamat/hxv039
  • 8. Toshikazu Kuniya, Stability Analysis of an Age-Structured SIR Epidemic Model with a Reduction Method to ODEs, Mathematics, 2018, 6, 9, 147, 10.3390/math6090147
  • 9. Toshikazu Kuniya, Hopf bifurcation in an age-structured SIR epidemic model, Applied Mathematics Letters, 2019, 92, 22, 10.1016/j.aml.2018.12.010
  • 10. Toshikazu Kuniya, Global Behavior of a Multi-Group SIR Epidemic Model with Age Structure and an Application to the Chlamydia Epidemic in Japan, SIAM Journal on Applied Mathematics, 2019, 79, 1, 321, 10.1137/18M1205947
  • 11. Xue-Zhi Li, Junyuan Yang, Maia Martcheva, , Age Structured Epidemic Modeling, 2020, Chapter 1, 1, 10.1007/978-3-030-42496-1_1

Reader Comments

your name: *   your email: *  

Copyright Info: 2012, , licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (

Download full text in PDF

Export Citation

Copyright © AIMS Press All Rights Reserved