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Parameter estimation and uncertainty quantification for an epidemic model
Multiple endemic states in agestructured $SIR$ epidemic models
1.  Dept. Mathematics, Università 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 
Here, we show an example in the form of a $3 \times 3$ contact matrix in which multiple nontrivial steady states exist. This instance of nonuniqueness 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 saddlenode 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 quasiperiodic 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 nontrivial steady states always holds, so that 3 is the minimum dimension of the contact matrix to allow for multiple endemic equilibria.
References:
[1] 
R.M. Anderson and R.M. May, Vaccination against rubella and measles: Quantitative investigations of different policies, J. Hyg. Camb., 90 (1983), 259325. 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 agedependent host mortality on the dynamics of an endemic disease, Math.Biosci., 114 (1993), 529. doi: 10.1016/00255564(93)900418. 
[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), 314. 
[5] 
D.Breda, M.Iannelli, S.Maset and R.Vermiglio, Stability analysis of the GurtinMacCamy model, SIAM J. Numer. Anal., 46 (2008), 980995. doi: 10.1137/070685658. 
[6] 
D.Breda and D.Visetti, Existence, multiplicity and stability of endemic states for an agestructured SI epidemic model, Math. Biosci., 235 (2012), 1931. doi: 10.1016/j.mbs.2011.10.004. 
[7] 
S.Busenberg, K.Cooke and M.Iannelli, Endemic thresholds and stability in a class of agestructured epidemics, SIAM J. Appl. Math., 48 (1988), 13791395. 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), 127. doi: 10.1007/s0028500701431. 
[11] 
D.Greenhalgh, Threshold and stability results for an epidemic model with an age structured meetingrate, IMA Journal of Mathematics applied in Medicine and Biology, 5 (1988), 81100. 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), 259284. 
[13] 
H. W. Hethcote and H. R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci., 75 (1985), 205227. doi: 10.1016/00255564(85)900380. 
[14] 
M.Iannelli, "Mathematical Theory of AgeStructured Population Dynamics," Giardini editori e stampatori in Pisa, Pisa, 1994. 
[15] 
H.Inaba, Threshold and stability results for an agestructured epidemic model, Journal of Mathematical Biology, 28 (1990), 411434. doi: 10.1007/BF00178326. 
[16] 
T.Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der mathematischen Wissenschaften, Band 132, SpringerVerlag 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.SadkowskaTodys, M.Rosinska and W.J. Edmunds, Social contacts and mixing patterns relevant to the spread of infectious diseases, PlOS Medicine, 5 (2008), 381391. doi: 10.1371/journal.pmed.0050074. 
[18] 
H. R. Thieme, Stability change of the endemic equilibrium in agestructured 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), 139158. 
[19] 
E.Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed Point Theorems," SpringerVerlag, New York, 1986. 
show all references
References:
[1] 
R.M. Anderson and R.M. May, Vaccination against rubella and measles: Quantitative investigations of different policies, J. Hyg. Camb., 90 (1983), 259325. 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 agedependent host mortality on the dynamics of an endemic disease, Math.Biosci., 114 (1993), 529. doi: 10.1016/00255564(93)900418. 
[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), 314. 
[5] 
D.Breda, M.Iannelli, S.Maset and R.Vermiglio, Stability analysis of the GurtinMacCamy model, SIAM J. Numer. Anal., 46 (2008), 980995. doi: 10.1137/070685658. 
[6] 
D.Breda and D.Visetti, Existence, multiplicity and stability of endemic states for an agestructured SI epidemic model, Math. Biosci., 235 (2012), 1931. doi: 10.1016/j.mbs.2011.10.004. 
[7] 
S.Busenberg, K.Cooke and M.Iannelli, Endemic thresholds and stability in a class of agestructured epidemics, SIAM J. Appl. Math., 48 (1988), 13791395. 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), 127. doi: 10.1007/s0028500701431. 
[11] 
D.Greenhalgh, Threshold and stability results for an epidemic model with an age structured meetingrate, IMA Journal of Mathematics applied in Medicine and Biology, 5 (1988), 81100. 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), 259284. 
[13] 
H. W. Hethcote and H. R. Thieme, Stability of the endemic equilibrium in epidemic models with subpopulations, Math. Biosci., 75 (1985), 205227. doi: 10.1016/00255564(85)900380. 
[14] 
M.Iannelli, "Mathematical Theory of AgeStructured Population Dynamics," Giardini editori e stampatori in Pisa, Pisa, 1994. 
[15] 
H.Inaba, Threshold and stability results for an agestructured epidemic model, Journal of Mathematical Biology, 28 (1990), 411434. doi: 10.1007/BF00178326. 
[16] 
T.Kato, "Perturbation Theory for Linear Operators," Die Grundlehren der mathematischen Wissenschaften, Band 132, SpringerVerlag 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.SadkowskaTodys, M.Rosinska and W.J. Edmunds, Social contacts and mixing patterns relevant to the spread of infectious diseases, PlOS Medicine, 5 (2008), 381391. doi: 10.1371/journal.pmed.0050074. 
[18] 
H. R. Thieme, Stability change of the endemic equilibrium in agestructured 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), 139158. 
[19] 
E.Zeidler, "Nonlinear Functional Analysis and its Applications. I. Fixed Point Theorems," SpringerVerlag, New York, 1986. 
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