# American Institute of Mathematical Sciences

August  2012, 32(8): 2675-2699. doi: 10.3934/dcds.2012.32.2675

## Numerical recipes for investigating endemic equilibria of age-structured SIR epidemics

 1 Department of Mathematics and Computer Science, University of Udine, via delle Scienze 206, I33100 Udine, Italy 2 Department of Mathematics and Computer Science, University of Trieste, via Valerio 12, I34127 Trieste, Italy

Received  May 2011 Revised  July 2011 Published  March 2012

The subject of this paper is the analysis of the equibria of a SIR type epidemic model, which is taken as a case study among the wide family of dynamical systems of infinite dimension. For this class of systems both the determination of the stationary solutions and the analysis of their local asymptotic stability are often unattainable theoretically, thus requiring the application of existing numerical tools and/or the development of new ones. Therefore, rather than devoting our attention to the SIR model's features, its biological and physical interpretation or its theoretical mathematical analysis, the main purpose here is to discuss how to study its equilibria numerically, especially as far as their stability is concerned. To this end, we briefly analyze the construction and solution of the system of nonlinear algebraic equations leading to the stationary solutions, and then concentrate on two numerical recipes for approximating the stability determining values known as the characteristic roots. An algorithm for the purpose is given in full detail. Two applications are presented and discussed in order to show the kind of results that can be obtained with these tools.
Citation: Dimitri Breda, Stefano Maset, Rossana Vermiglio. Numerical recipes for investigating endemic equilibria of age-structured SIR epidemics. Discrete and Continuous Dynamical Systems, 2012, 32 (8) : 2675-2699. doi: 10.3934/dcds.2012.32.2675
##### References:
 [1] E. L. Allgower and K. Georg, "Introduction to Numerical Continuation Methods," Reprint of the 1990 edition [Springer-Verlag, Berlin; MR1059455 (92a:65165)], Classics in Applied Mathematics, 45, SIAM, Philadelphia, PA, 2003. [2] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, New York, Tokyo, 1991. [3] E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period, Math. Biosci. Eng., 8 (2011), 931-952. doi: 10.3934/mbe.2011.8.931. [4] D. Breda, "Numerical Computation of Characteristic Roots for Delay Differential Equations," Ph.D thesis, Ph.D in Computational Mathematics, Università di Padova, 2004. [5] D. Breda, Nonautonomous delay differential equations in Hilbert spaces and Lyapunov exponents, Diff. Int. Equations, 23 (2010), 935-956. [6] D. Breda, C. Cusulin, M. Iannelli, S. Maset and R. Vermiglio, Stability analysis of age-structured population equations by pseudospectral differencing methods, J. Math. Biol., 54 (2007), 701-720. doi: 10.1007/s00285-006-0064-4. [7] 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. [8] D. Breda, S. Maset and R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27 (2005), 482-495. doi: 10.1137/030601600. [9] D. Breda, S. Maset and R. Vermiglio, Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions, Appl. Numer. Math., 56 (2006), 318-331. doi: 10.1016/j.apnum.2005.04.011. [10] D. Breda, S. Maset and R. Vermiglio, Numerical approximation of characteristic values of partial retarded functional differential equations, Numer. Math., 113 (2009), 181-242. doi: 10.1007/s00211-009-0233-7. [11] D. Breda, S. Maset and R. Vermiglio, Approximation of eigenvalues of evolution operators for linear retarded functional differential equations,, 2012, (). [12] D. Breda, S. Maset and R. Vermiglio, Computing eigenvalues of Gurtin-MacCamy models with diffusion, IMA J. Numer. Anal., published online, 2011. doi: 10.1093/imanum/drr004. [13] 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. [14] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Mathematical Surveys and Monographs, 70, American Mathematical Society, Providence, RI, 1999. [15] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, "Delay Equations. Functional, Complex, and Nonlinear Analysis," American Mathematical Sciences, 110, Springer-Verlag, New York, 1995. [16] K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 1999. [17] A. Franceschetti, A. Pugliese and D. Breda, Multiple endemic states in age-structured SIR epidemic models,, 2012, (). [18] W. J. F. Govaerts, "Numerical Methods for Bifurcations of Dynamical Equilibria," SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719543. [19] D. Greenhalgh, Threshold and stability results for an epidemic model with an age-structured meeting rate, IMA J. Math. Appl. Med. Biol., 5 (1988), 81-100. doi: 10.1093/imammb/5.2.81. [20] M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent population dynamics, Archiv. Rat. Mech. Anal., 54 (1974), 281-300. doi: 10.1007/BF00250793. [21] M. Iannelli, "Mathematical Theory of Age-Structured Population Dynamics," Applied Mathematics Monographs (C.N.R.), Giardini Editori e Stampatori, Pisa, Italy, 1994. [22] H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434. doi: 10.1007/BF00178326. [23] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Second edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998. [24] S. Liu, E. Beretta and D. Breda, Predator-prey model of Beddington-DeAngelis type with maturation and gestation delays, Nonlinear Anal. Real World Appl., 11 (2010), 4072-4091. doi: 10.1016/j.nonrwa.2010.03.013. [25] A. Lyapunov, "Problém Géneral de la Stabilité du Mouvement," Annals of Mathematics Studies, 17, Princeton University Press, 1949. [26] R. M. May and R. M. Anderson, Endemic infections in growing populations, Mathematical Biosciences, 77 (1985), 141-156. doi: 10.1016/0025-5564(85)90093-8. [27] I. Mazzer, "Un Modello per la Dinamica di Più Popolazioni: Esistenza, Unicitàe Approssimazione Numerica della Soluzione," Master's thesis, University of Udine, (in italian), 2009. [28] L. N. Trefethen, "Spectral Methods in MATLAB," Software, Environment, and Tools, 10, SIAM, Philadelphia, PA, 2000. [29] J. H. Wilkinson, The perfidious polynomial, in "Studies in Numerical Analysis" (ed. G. H. Golub), Studies in Mathematics, 24, Mathematical Association of America, Washington, DC, (1984), 1-28.

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##### References:
 [1] E. L. Allgower and K. Georg, "Introduction to Numerical Continuation Methods," Reprint of the 1990 edition [Springer-Verlag, Berlin; MR1059455 (92a:65165)], Classics in Applied Mathematics, 45, SIAM, Philadelphia, PA, 2003. [2] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford University Press, New York, Tokyo, 1991. [3] E. Beretta and D. Breda, An SEIR epidemic model with constant latency time and infectious period, Math. Biosci. Eng., 8 (2011), 931-952. doi: 10.3934/mbe.2011.8.931. [4] D. Breda, "Numerical Computation of Characteristic Roots for Delay Differential Equations," Ph.D thesis, Ph.D in Computational Mathematics, Università di Padova, 2004. [5] D. Breda, Nonautonomous delay differential equations in Hilbert spaces and Lyapunov exponents, Diff. Int. Equations, 23 (2010), 935-956. [6] D. Breda, C. Cusulin, M. Iannelli, S. Maset and R. Vermiglio, Stability analysis of age-structured population equations by pseudospectral differencing methods, J. Math. Biol., 54 (2007), 701-720. doi: 10.1007/s00285-006-0064-4. [7] 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. [8] D. Breda, S. Maset and R. Vermiglio, Pseudospectral differencing methods for characteristic roots of delay differential equations, SIAM J. Sci. Comput., 27 (2005), 482-495. doi: 10.1137/030601600. [9] D. Breda, S. Maset and R. Vermiglio, Pseudospectral approximation of eigenvalues of derivative operators with non-local boundary conditions, Appl. Numer. Math., 56 (2006), 318-331. doi: 10.1016/j.apnum.2005.04.011. [10] D. Breda, S. Maset and R. Vermiglio, Numerical approximation of characteristic values of partial retarded functional differential equations, Numer. Math., 113 (2009), 181-242. doi: 10.1007/s00211-009-0233-7. [11] D. Breda, S. Maset and R. Vermiglio, Approximation of eigenvalues of evolution operators for linear retarded functional differential equations,, 2012, (). [12] D. Breda, S. Maset and R. Vermiglio, Computing eigenvalues of Gurtin-MacCamy models with diffusion, IMA J. Numer. Anal., published online, 2011. doi: 10.1093/imanum/drr004. [13] 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. [14] C. Chicone and Y. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Mathematical Surveys and Monographs, 70, American Mathematical Society, Providence, RI, 1999. [15] O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, "Delay Equations. Functional, Complex, and Nonlinear Analysis," American Mathematical Sciences, 110, Springer-Verlag, New York, 1995. [16] K. Engel and R. Nagel, "One-Parameter Semigroups for Linear Evolution Equations," Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 1999. [17] A. Franceschetti, A. Pugliese and D. Breda, Multiple endemic states in age-structured SIR epidemic models,, 2012, (). [18] W. J. F. Govaerts, "Numerical Methods for Bifurcations of Dynamical Equilibria," SIAM, Philadelphia, PA, 2000. doi: 10.1137/1.9780898719543. [19] D. Greenhalgh, Threshold and stability results for an epidemic model with an age-structured meeting rate, IMA J. Math. Appl. Med. Biol., 5 (1988), 81-100. doi: 10.1093/imammb/5.2.81. [20] M. E. Gurtin and R. C. MacCamy, Non-linear age-dependent population dynamics, Archiv. Rat. Mech. Anal., 54 (1974), 281-300. doi: 10.1007/BF00250793. [21] M. Iannelli, "Mathematical Theory of Age-Structured Population Dynamics," Applied Mathematics Monographs (C.N.R.), Giardini Editori e Stampatori, Pisa, Italy, 1994. [22] H. Inaba, Threshold and stability results for an age-structured epidemic model, J. Math. Biol., 28 (1990), 411-434. doi: 10.1007/BF00178326. [23] Y. A. Kuznetsov, "Elements of Applied Bifurcation Theory," Second edition, Applied Mathematical Sciences, 112, Springer-Verlag, New York, 1998. [24] S. Liu, E. Beretta and D. Breda, Predator-prey model of Beddington-DeAngelis type with maturation and gestation delays, Nonlinear Anal. Real World Appl., 11 (2010), 4072-4091. doi: 10.1016/j.nonrwa.2010.03.013. [25] A. Lyapunov, "Problém Géneral de la Stabilité du Mouvement," Annals of Mathematics Studies, 17, Princeton University Press, 1949. [26] R. M. May and R. M. Anderson, Endemic infections in growing populations, Mathematical Biosciences, 77 (1985), 141-156. doi: 10.1016/0025-5564(85)90093-8. [27] I. Mazzer, "Un Modello per la Dinamica di Più Popolazioni: Esistenza, Unicitàe Approssimazione Numerica della Soluzione," Master's thesis, University of Udine, (in italian), 2009. [28] L. N. Trefethen, "Spectral Methods in MATLAB," Software, Environment, and Tools, 10, SIAM, Philadelphia, PA, 2000. [29] J. H. Wilkinson, The perfidious polynomial, in "Studies in Numerical Analysis" (ed. G. H. Golub), Studies in Mathematics, 24, Mathematical Association of America, Washington, DC, (1984), 1-28.
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