# 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:
 [1] R.M. Anderson and R.M. May, Vaccination against rubella and measles: Quantitative investigations of different policies, J. Hyg. Camb., 90 (1983), 259-325. doi: 10.1017/S002217240002893X. [2] R.M. Anderson and R.M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford New York Tokyo, 1991. [3] V.Andreasen, The effect of age-dependent host mortality on the dynamics of an endemic disease, Math.Biosci., 114 (1993), 5-29. doi: 10.1016/0025-5564(93)90041-8. [4] V.Andreasen, Instability in an {SIR}-model with age dependent susceptibility, in "Theory of Epidemics" (eds. O. Arino, D. Axelrod, M. Kimmel and M. Langlais), Mathematical Population Dynamics, 1, Wuerz Publ., Winnipeg, (1995), 3-14. [5] D.Breda, M.Iannelli, S.Maset and R.Vermiglio, Stability analysis of the Gurtin-MacCamy model, SIAM J. Numer. Anal., 46 (2008), 980-995. doi: 10.1137/070685658. [6] D.Breda and D.Visetti, Existence, multiplicity and stability of endemic states for an age-structured S-I epidemic model, Math. Biosci., 235 (2012), 19-31. doi: 10.1016/j.mbs.2011.10.004. [7] S.Busenberg, K.Cooke and M.Iannelli, Endemic thresholds and stability in a class of age-structured epidemics, SIAM J. Appl. Math., 48 (1988), 1379-1395. doi: 10.1137/0148085. [8] J.M. Cushing, Robert Costantino, Brian Dennis, Robert Desharnais and S.Henson, "Chaos in Ecology: Experimental Nonlinear Dynamics," Academic Press, 2002. [9] K.Deimling, "Nonlinear Functional Analysis," Springer Verlag, Berlin, 1985. [10] A.Franceschetti and A.Pugliese, Threshold behaviour of a SIR epidemic model with age structure and immigration, J. Math. Biol., 57 (2008), 1-27. doi: 10.1007/s00285-007-0143-1. [11] D.Greenhalgh, Threshold and stability results for an epidemic model with an age structured meeting-rate, IMA Journal of Mathematics applied in Medicine and Biology, 5 (1988), 81-100. doi: 10.1093/imammb/5.2.81. [12] H.Guo, M.Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canadian Appl. Math. Quart., 14 (2006), 259-284. [13] H. W. Hethcote and H. R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci., 75 (1985), 205-227. doi: 10.1016/0025-5564(85)90038-0. [14] M.Iannelli, "Mathematical Theory of Age-Structured Population Dynamics," Giardini editori e stampatori in Pisa, Pisa, 1994. [15] H.Inaba, Threshold and stability results for an age-structured epidemic model, Journal of Mathematical Biology, 28 (1990), 411-434. doi: 10.1007/BF00178326. [16] T.Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. [17] J.Mossong, N.Hens, M.Jit, P.Beutels, K.Auranen, R.Mikolajczyk, M.Massari, S.Salmaso, G.Scalia Tomba, J.Wallinga, J.Heijne, M.Sadkowska-Todys, M.Rosinska and W.J. Edmunds, Social contacts and mixing patterns relevant to the spread of infectious diseases, PlOS Medicine, 5 (2008), 381-391. doi: 10.1371/journal.pmed.0050074. [18] H. R. Thieme, Stability change of the endemic equilibrium in age-structured models for the spread of SIR type infectious diseases, in "Differential Equations Models in Biology, Epidemiology and Ecology" (eds. S. Busenberg and M. Martelli) (Claremont, CA, 1990), Lecture Notes in Biomathematics, 92, Springer, Berlin, (1991), 139-158. [19] E.Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed Point Theorems," Springer-Verlag, New York, 1986.

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##### References:
 [1] R.M. Anderson and R.M. May, Vaccination against rubella and measles: Quantitative investigations of different policies, J. Hyg. Camb., 90 (1983), 259-325. doi: 10.1017/S002217240002893X. [2] R.M. Anderson and R.M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, Oxford New York Tokyo, 1991. [3] V.Andreasen, The effect of age-dependent host mortality on the dynamics of an endemic disease, Math.Biosci., 114 (1993), 5-29. doi: 10.1016/0025-5564(93)90041-8. [4] V.Andreasen, Instability in an {SIR}-model with age dependent susceptibility, in "Theory of Epidemics" (eds. O. Arino, D. Axelrod, M. Kimmel and M. Langlais), Mathematical Population Dynamics, 1, Wuerz Publ., Winnipeg, (1995), 3-14. [5] D.Breda, M.Iannelli, S.Maset and R.Vermiglio, Stability analysis of the Gurtin-MacCamy model, SIAM J. Numer. Anal., 46 (2008), 980-995. doi: 10.1137/070685658. [6] D.Breda and D.Visetti, Existence, multiplicity and stability of endemic states for an age-structured S-I epidemic model, Math. Biosci., 235 (2012), 19-31. doi: 10.1016/j.mbs.2011.10.004. [7] S.Busenberg, K.Cooke and M.Iannelli, Endemic thresholds and stability in a class of age-structured epidemics, SIAM J. Appl. Math., 48 (1988), 1379-1395. doi: 10.1137/0148085. [8] J.M. Cushing, Robert Costantino, Brian Dennis, Robert Desharnais and S.Henson, "Chaos in Ecology: Experimental Nonlinear Dynamics," Academic Press, 2002. [9] K.Deimling, "Nonlinear Functional Analysis," Springer Verlag, Berlin, 1985. [10] A.Franceschetti and A.Pugliese, Threshold behaviour of a SIR epidemic model with age structure and immigration, J. Math. Biol., 57 (2008), 1-27. doi: 10.1007/s00285-007-0143-1. [11] D.Greenhalgh, Threshold and stability results for an epidemic model with an age structured meeting-rate, IMA Journal of Mathematics applied in Medicine and Biology, 5 (1988), 81-100. doi: 10.1093/imammb/5.2.81. [12] H.Guo, M.Y. Li and Z. Shuai, Global stability of the endemic equilibrium of multigroup SIR epidemic models, Canadian Appl. Math. Quart., 14 (2006), 259-284. [13] H. W. Hethcote and H. R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci., 75 (1985), 205-227. doi: 10.1016/0025-5564(85)90038-0. [14] M.Iannelli, "Mathematical Theory of Age-Structured Population Dynamics," Giardini editori e stampatori in Pisa, Pisa, 1994. [15] H.Inaba, Threshold and stability results for an age-structured epidemic model, Journal of Mathematical Biology, 28 (1990), 411-434. doi: 10.1007/BF00178326. [16] T.Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. [17] J.Mossong, N.Hens, M.Jit, P.Beutels, K.Auranen, R.Mikolajczyk, M.Massari, S.Salmaso, G.Scalia Tomba, J.Wallinga, J.Heijne, M.Sadkowska-Todys, M.Rosinska and W.J. Edmunds, Social contacts and mixing patterns relevant to the spread of infectious diseases, PlOS Medicine, 5 (2008), 381-391. doi: 10.1371/journal.pmed.0050074. [18] H. R. Thieme, Stability change of the endemic equilibrium in age-structured models for the spread of SIR type infectious diseases, in "Differential Equations Models in Biology, Epidemiology and Ecology" (eds. S. Busenberg and M. Martelli) (Claremont, CA, 1990), Lecture Notes in Biomathematics, 92, Springer, Berlin, (1991), 139-158. [19] E.Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed Point Theorems," Springer-Verlag, New York, 1986.
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