American Institute of Mathematical Sciences

Advanced
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 (2013), 499.   Google Scholar

[2]

J. Banasiak and M. Lachowicz, "Methods of Small Parameter in Mathematical Biology'',, Birkhäuser Springer, (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 (2014), 2383.   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 (1981), 31.   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), 1973.   Google Scholar

[7]

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

[8]

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

[9]

W. Eckhaus, Relaxation oscillations including a standard chase on French ducks,, in Asymptotic analysis II (ed. F. Verhulst), (1983), 449.   Google Scholar

[10]

G. Hek, Geometrical singular perturbation theory in biological practice,, J. Math. Biol., 60 (2010), 347.   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 (2001), 286.   Google Scholar

[12]

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

[13]

H. L. Smith and P. Waltman, "The Theory of the Chemostat'',, Cambridge University Press, (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, (2011).   Google Scholar

[15]

G. Wallet, Entrée-sortie dans un tourbillon,, Annales de l'Institut Fourier, 36 (1986), 157.   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 (2013), 499.   Google Scholar

[2]

J. Banasiak and M. Lachowicz, "Methods of Small Parameter in Mathematical Biology'',, Birkhäuser Springer, (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 (2014), 2383.   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 (1981), 31.   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), 1973.   Google Scholar

[7]

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

[8]

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

[9]

W. Eckhaus, Relaxation oscillations including a standard chase on French ducks,, in Asymptotic analysis II (ed. F. Verhulst), (1983), 449.   Google Scholar

[10]

G. Hek, Geometrical singular perturbation theory in biological practice,, J. Math. Biol., 60 (2010), 347.   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 (2001), 286.   Google Scholar

[12]

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

[13]

H. L. Smith and P. Waltman, "The Theory of the Chemostat'',, Cambridge University Press, (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, (2011).   Google Scholar

[15]

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

[1]

Jacek Banasiak, Eddy Kimba Phongi, MirosŁaw Lachowicz. A singularly perturbed SIS model with age structure. Mathematical Biosciences & Engineering, 2013, 10 (3) : 499-521. doi: 10.3934/mbe.2013.10.499
[2]

Weichung Wang, Tsung-Fang Wu, Chien-Hsiang Liu. On the multiple spike solutions for singularly perturbed elliptic systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237
[3]

Jianhe Shen, Maoan Han. Bifurcations of canard limit cycles in several singularly perturbed generalized polynomial Liénard systems. Discrete & Continuous Dynamical Systems - A, 2013, 33 (7) : 3085-3108. doi: 10.3934/dcds.2013.33.3085
[4]

Bedr'Eddine Ainseba. Age-dependent population dynamics diffusive systems. Discrete & Continuous Dynamical Systems - B, 2004, 4 (4) : 1233-1247. doi: 10.3934/dcdsb.2004.4.1233
[5]

Jacek Banasiak, Amartya Goswami. Singularly perturbed population models with reducible migration matrix 1. Sova-Kurtz theorem and the convergence to the aggregated model. Discrete & Continuous Dynamical Systems - A, 2015, 35 (2) : 617-635. doi: 10.3934/dcds.2015.35.617
[6]

Mostafa Fazly, Mahmoud Hesaaraki. Periodic solutions for a semi-ratio-dependent predator-prey dynamical system with a class of functional responses on time scales. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 267-279. doi: 10.3934/dcdsb.2008.9.267
[7]

Tristan Roget. On the long-time behaviour of age and trait structured population dynamics. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2551-2576. doi: 10.3934/dcdsb.2018265
[8]

Flaviano Battelli, Ken Palmer. Heteroclinic connections in singularly perturbed systems. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 431-461. doi: 10.3934/dcdsb.2008.9.431
[9]

Yicang Zhou, Paolo Fergola. Dynamics of a discrete age-structured SIS models. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 841-850. doi: 10.3934/dcdsb.2004.4.841
[10]

Jacek Banasiak, Rodrigue Yves M'pika Massoukou. A singularly perturbed age structured SIRS model with fast recovery. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2383-2399. doi: 10.3934/dcdsb.2014.19.2383
[11]

Lei Liu, Shaoying Lu, Cunwu Han, Chao Li, Zejin Feng. Fault estimation and optimization for uncertain disturbed singularly perturbed systems with time-delay. Numerical Algebra, Control & Optimization, 2020, 10 (3) : 367-379. doi: 10.3934/naco.2020008
[12]

Alexandru Kristály, Ildikó-Ilona Mezei. Multiple solutions for a perturbed system on strip-like domains. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 789-796. doi: 10.3934/dcdss.2012.5.789
[13]

MirosŁaw Lachowicz, Tatiana Ryabukha. Equilibrium solutions for microscopic stochastic systems in population dynamics. Mathematical Biosciences & Engineering, 2013, 10 (3) : 777-786. doi: 10.3934/mbe.2013.10.777
[14]

Karl P. Hadeler. Quiescent phases and stability in discrete time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (1) : 129-152. doi: 10.3934/dcdsb.2015.20.129
[15]

Christian Pötzsche, Stefan Siegmund, Fabian Wirth. A spectral characterization of exponential stability for linear time-invariant systems on time scales. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1223-1241. doi: 10.3934/dcds.2003.9.1223
[16]

Yunfei Peng, X. Xiang, W. Wei. Backward problems of nonlinear dynamical equations on time scales. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1553-1564. doi: 10.3934/dcdss.2011.4.1553
[17]

Xiang-Ping Yan, Wan-Tong Li. Stability and Hopf bifurcations for a delayed diffusion system in population dynamics. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 367-399. doi: 10.3934/dcdsb.2012.17.367
[18]

Yicang Zhou, Zhien Ma. Global stability of a class of discrete age-structured SIS models with immigration. Mathematical Biosciences & Engineering, 2009, 6 (2) : 409-425. doi: 10.3934/mbe.2009.6.409
[19]

Sung Kyu Choi, Namjip Koo. Stability of linear dynamic equations on time scales. Conference Publications, 2009, 2009 (Special) : 161-170. doi: 10.3934/proc.2009.2009.161
[20]

Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 2095-2115. doi: 10.3934/cpaa.2015.14.2095

 Impact Factor: 

Tools

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences