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.

[2]

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

[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.

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[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.

[10]

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

[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.

[12]

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

[13]

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

[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.

[15]

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

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.

[2]

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

[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.

[4]

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

[5]

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

[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.

[7]

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

[8]

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

[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.

[10]

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

[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.

[12]

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

[13]

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

[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.

[15]

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

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