American Institute of Mathematical Sciences

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

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

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

Received  June 2011 Revised  February 2012 Published  July 2012

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

show all references

References:
 [1] Dimitri Breda, Stefano Maset, Rossana Vermiglio. Numerical recipes for investigating endemic equilibria of age-structured SIR epidemics. Discrete & Continuous Dynamical Systems, 2012, 32 (8) : 2675-2699. doi: 10.3934/dcds.2012.32.2675 [2] Hao Kang, Qimin Huang, Shigui Ruan. Periodic solutions of an age-structured epidemic model with periodic infection rate. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4955-4972. doi: 10.3934/cpaa.2020220 [3] Hisashi Inaba. Mathematical analysis of an age-structured SIR epidemic model with vertical transmission. Discrete & Continuous Dynamical Systems - B, 2006, 6 (1) : 69-96. doi: 10.3934/dcdsb.2006.6.69 [4] Hassan Tahir, Asaf Khan, Anwarud Din, Amir Khan, Gul Zaman. Optimal control strategy for an age-structured SIR endemic model. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2535-2555. doi: 10.3934/dcdss.2021054 [5] Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643 [6] Shengqin Xu, Chuncheng Wang, Dejun Fan. Stability and bifurcation in an age-structured model with stocking rate and time delays. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2535-2549. doi: 10.3934/dcdsb.2018264 [7] Hossein Mohebbi, Azim Aminataei, Cameron J. Browne, Mohammad Reza Razvan. Hopf bifurcation of an age-structured virus infection model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 861-885. doi: 10.3934/dcdsb.2018046 [8] Xianlong Fu, Zhihua Liu, Pierre Magal. Hopf bifurcation in an age-structured population model with two delays. Communications on Pure & Applied Analysis, 2015, 14 (2) : 657-676. doi: 10.3934/cpaa.2015.14.657 [9] Zhihua Liu, Yayun Wu, Xiangming Zhang. Existence of periodic wave trains for an age-structured model with diffusion. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021009 [10] Cameron J. Browne, Sergei S. Pilyugin. Global analysis of age-structured within-host virus model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 1999-2017. doi: 10.3934/dcdsb.2013.18.1999 [11] Yuan Yuan, Xianlong Fu. Mathematical analysis of an age-structured HIV model with intracellular delay. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021123 [12] Abdennasser Chekroun, Mohammed Nor Frioui, Toshikazu Kuniya, Tarik Mohammed Touaoula. Mathematical analysis of an age structured heroin-cocaine epidemic model. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4449-4477. doi: 10.3934/dcdsb.2020107 [13] Xue-Zhi Li, Ji-Xuan Liu, Maia Martcheva. An age-structured two-strain epidemic model with super-infection. Mathematical Biosciences & Engineering, 2010, 7 (1) : 123-147. doi: 10.3934/mbe.2010.7.123 [14] Toshikazu Kuniya, Mimmo Iannelli. $R_0$ and the global behavior of an age-structured SIS epidemic model with periodicity and vertical transmission. Mathematical Biosciences & Engineering, 2014, 11 (4) : 929-945. doi: 10.3934/mbe.2014.11.929 [15] Shaoli Wang, Jianhong Wu, Libin Rong. A note on the global properties of an age-structured viral dynamic model with multiple target cell populations. Mathematical Biosciences & Engineering, 2017, 14 (3) : 805-820. doi: 10.3934/mbe.2017044 [16] Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641 [17] Ryszard Rudnicki, Radosław Wieczorek. On a nonlinear age-structured model of semelparous species. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2641-2656. doi: 10.3934/dcdsb.2014.19.2641 [18] Mohammed Nor Frioui, Tarik Mohammed Touaoula, Bedreddine Ainseba. Global dynamics of an age-structured model with relapse. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2245-2270. doi: 10.3934/dcdsb.2019226 [19] Zhong-Kai Guo, Hai-Feng Huo, Hong Xiang. Analysis of an age-structured model for HIV-TB co-infection. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021037 [20] Linlin Li, Bedreddine Ainseba. Large-time behavior of matured population in an age-structured model. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2561-2580. doi: 10.3934/dcdsb.2020195

2018 Impact Factor: 1.313