American Institute of Mathematical Sciences

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2015, 2015(special): 85-93. doi: 10.3934/proc.2015.0085

Canard-type solutions in epidemiological models

Jacek Banasiak 1, and  Eddy Kimba Phongi 2,

1. 

School of Mathematical Sciences, UKZN, Durban
2. 

School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4041

Received  September 2014 Revised  January 2015 Published  November 2015

The paper concerns an epidemiological model with an age structure and with two time scales: the slow one refers to the demographical processes and the fast describes the dynamics of a fast disease such as flu or common cold. The model in the singular limit corresponding to the infinite disease-related rates has two intersecting quasi-stationary steady states. We investigate the asymptotic behaviour of solutions passing close to the intersection manifold and show that in the models with increasing total populations there is a delay in switching between the quasi-stationary states which resembles the so-called canard solutions.
Keywords: canard solutions, population dynamics, Tikhonov theorem, age structure, stability switch., multiple time scales, singularly perturbed dynamical systems, SIS system.
Mathematics Subject Classification: Primary: 34E15, 92A15; Secondary: 34E1.
Citation: Jacek Banasiak, Eddy Kimba Phongi. Canard-type solutions in epidemiological models. Conference Publications, 2015, 2015 (special) : 85-93. doi: 10.3934/proc.2015.0085
References:
[1]

J. Banasiak, E. Kimba Phongi and M. Lachowicz, A singularly perturbed SIS model with age structure, Math. Biosci. Eng., 10 (3), (2013), 499-521. Google Scholar

[2]

J. Banasiak and M. Lachowicz, "Methods of Small Parameter in Mathematical Biology'', Birkhäuser Springer, Heidelberg, New York, 2014 Google Scholar

[3]

J. Banasiak and R.Y. M'pika Massoukou, A singularly perturbed age structured SIRS model with fast recovery, Discrete Cont. Dyn. Sys. Ser B, 19 (8), (2014), 2383-2399. Google Scholar

[4]

J. Banasiak and M.S. Seuneu Tchamga, Delayed stability switches in singularly perturbed predator-prey models,, in preparation., ().   Google Scholar

[5]

E. Benoît, J. F. Callot, F. Diener, M. Diener, Chasse au canard, Collectanea Mathematica, 31-32 (1-3), (1981), 37-119. Google Scholar

[6]

V.F. Butuzov, N.N. Nefedov and K.R. Schneider, Singularly perturbed problems in case of exchange of stabilities, J. Math. Sci. (N. Y.), 121 (2004), no. 1, 1973-2079. Google Scholar

[7]

J. Cronin, Electrically active cells and singular perturbation problems, Math. Intelligencer, 12 (4), (1990), 57-64. Google Scholar

[8]

M. Diener, The canard unchained or how fast/slow dynamical systems bifurcate, Math. Intelligencer, 6 (3), (1984), 38-49. Google Scholar

[9]

W. Eckhaus, Relaxation oscillations including a standard chase on French ducks, in Asymptotic analysis II (ed. F. Verhulst), Lecture Notes in Math., 985, Springer, Berlin, (1983), 449-494. Google Scholar

[10]

G. Hek, Geometrical singular perturbation theory in biological practice, J. Math. Biol., 60, (2010), 347-386. Google Scholar

[11]

M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points - fold and canard points in two dimensions, SIAM J. Math. Anal., 33(2), (2001), 286-314. Google Scholar

[12]

A.I. Neĭshtadt, Persistence of stability loss for dynamical bifurcations I, Diff. Eq. 23, (1987), 1385-1391 Google Scholar

[13]

H. L. Smith and P. Waltman, "The Theory of the Chemostat'', Cambridge University Press, Cambridge, 2008. Google Scholar

[14]

N. Siewe, The Tikhonov Theorem in Multiscale Modelling: An Application To The SIRS Epidemic Model, African Institute of Mathematical Sciences Postgraduate Diploma Essay 2011/12, http://archive.aims.ac.za/2011/nourridine.pdf. Google Scholar

[15]

G. Wallet, Entrée-sortie dans un tourbillon, Annales de l'Institut Fourier, 36 (4), (1986), 157-184. Google Scholar

show all references

References:
[1]

J. Banasiak, E. Kimba Phongi and M. Lachowicz, A singularly perturbed SIS model with age structure, Math. Biosci. Eng., 10 (3), (2013), 499-521. Google Scholar

[2]

J. Banasiak and M. Lachowicz, "Methods of Small Parameter in Mathematical Biology'', Birkhäuser Springer, Heidelberg, New York, 2014 Google Scholar

[3]

J. Banasiak and R.Y. M'pika Massoukou, A singularly perturbed age structured SIRS model with fast recovery, Discrete Cont. Dyn. Sys. Ser B, 19 (8), (2014), 2383-2399. Google Scholar

[4]

J. Banasiak and M.S. Seuneu Tchamga, Delayed stability switches in singularly perturbed predator-prey models,, in preparation., ().   Google Scholar

[5]

E. Benoît, J. F. Callot, F. Diener, M. Diener, Chasse au canard, Collectanea Mathematica, 31-32 (1-3), (1981), 37-119. Google Scholar

[6]

V.F. Butuzov, N.N. Nefedov and K.R. Schneider, Singularly perturbed problems in case of exchange of stabilities, J. Math. Sci. (N. Y.), 121 (2004), no. 1, 1973-2079. Google Scholar

[7]

J. Cronin, Electrically active cells and singular perturbation problems, Math. Intelligencer, 12 (4), (1990), 57-64. Google Scholar

[8]

M. Diener, The canard unchained or how fast/slow dynamical systems bifurcate, Math. Intelligencer, 6 (3), (1984), 38-49. Google Scholar

[9]

W. Eckhaus, Relaxation oscillations including a standard chase on French ducks, in Asymptotic analysis II (ed. F. Verhulst), Lecture Notes in Math., 985, Springer, Berlin, (1983), 449-494. Google Scholar

[10]

G. Hek, Geometrical singular perturbation theory in biological practice, J. Math. Biol., 60, (2010), 347-386. Google Scholar

[11]

M. Krupa and P. Szmolyan, Extending geometric singular perturbation theory to nonhyperbolic points - fold and canard points in two dimensions, SIAM J. Math. Anal., 33(2), (2001), 286-314. Google Scholar

[12]

A.I. Neĭshtadt, Persistence of stability loss for dynamical bifurcations I, Diff. Eq. 23, (1987), 1385-1391 Google Scholar

[13]

H. L. Smith and P. Waltman, "The Theory of the Chemostat'', Cambridge University Press, Cambridge, 2008. Google Scholar

[14]

N. Siewe, The Tikhonov Theorem in Multiscale Modelling: An Application To The SIRS Epidemic Model, African Institute of Mathematical Sciences Postgraduate Diploma Essay 2011/12, http://archive.aims.ac.za/2011/nourridine.pdf. Google Scholar

[15]

G. Wallet, Entrée-sortie dans un tourbillon, Annales de l'Institut Fourier, 36 (4), (1986), 157-184. Google Scholar

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