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

Canard-type solutions in epidemiological models

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.
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]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[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]

Xiangjin Xu. Multiple solutions of super-quadratic second order dynamical systems. Conference Publications, 2003, 2003 (Special) : 926-934. doi: 10.3934/proc.2003.2003.926

 Impact Factor: 

Metrics

  • PDF downloads (29)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]