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A black swan and canard cascades in an SIR infectious disease model

1 Centre de Recerca Matemática, Campus de Bellaterra, 08193 Bellaterra, Barcelona, Spain
2 Samara National Research University, 34 Moskovskoye shosse, Samara, 443086, Russia

Special Issues: Advances in Mathematical Modelling and Analysis of Bioprocesses

Models of the spread of infectious diseases commonly have to deal with the problem of multiple timescales which naturally occur in the epidemic models. In the most cases, this problem is implicitly avoided with the use of the so-called “constant population size” assumption. However, applicability of this assumption can require a justification (which is typically omitted).
In this paper we consider some multiscale phenomena that arise in a reasonably simple SusceptibleInfected-Removed (SIR) model with variable population size. In particular, we discuss examples of the canard cascades and a black swan that arise in this model.
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1. E. Shchepakina and V. Sobolev, Black Swans and Canards in Laser and Combustion Models, in Singular Perturbation and Hysteresis (eds. M P Mortell et al.), SIAM, Philadelphia, (2005), 207-255.

2. E. Shchepakina, V. Sobolev and M. P. Mortell, Singular Perturbations: Introduction to System Order Reduction Methods with Applications, Springer Lecture Notes in Mathematics, Vol. 2114, Springer, Basel, 2014.

3. V. Sobolev, Geometry of Singular Perturbations: Critical Cases, Singular Perturbation and Hysteresis, (eds. M.P. Mortell et al.), SIAM, Philadelphia, (2005), 153-206.

4. N. N. Bogolyubov, On Some Statistical Methods in Mathematical Physics, Izd. Ukrain. Akad. Nauk, Kiev (1945).

5. N. N. Bogolyubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York, 1961.

6. N. N. Bogolyubov and Y. A. Mitropolsky, The method of integral manifolds in nonlinear mechanics, Contributions to Differential Equations, 2, Wiley, New York, (1963), 123-196.

7. J. Hale, Integral manifolds of perturbed differential systems annals of mathematics, Ann. Math., 73 (1961), 496-531.

8. J. Hale, Oscillations in Nonlinear Systems, MacGraw-Hill, New York, 1963.

9. J. Hale, Ordinary Differential Equations, Wiley, New York, 1969.

10. K. V. Zadiraka, On the integral manifold of a system of differential equations containing a small parameter, Dokl. Akad. Nauk SSSR, 115 (1957), 646-649.

11. K. V. Zadiraka, On a non-local integral manifold of a singularly perturbed differential system, Ukrain. Math. Z., 17 (1965), 47-63; also AMS Transl. Ser. 2, 89 (1970), 29-49.

12. V. A. Sobolev and V. V. Strygin, Permissibility of changing over to precession equations of gyroscopic systems, Mech. Solids, 13 (1978), 10-17.

13. V. V. Strygin and V. A. Sobolev, Effect of geometric and kinetic parameters and energy dissipation on orientation stability of satellites with double spin, Cosmic Res., 14 (1976), 331-335.

14. V. V. Strygin and V. A. Sobolev, Asymptotic methods in the problem of stabilization of rotating bodies by using passive dampers, Mech. Solids, 5 (1977), 19-25.

15. N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equations, 31 (1979), 53-98.

16. C. K. R. T. Jones, Geometric singular perturbation theory, Lecture Notes in Mathematics, 1609 (1994), 44-118.

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

18. E. Shchepakina and V. Sobolev, Integral manifolds, canards and black swans, Nonlinear Anal. Theor. Methods Appl., 44 (2001), 897-908.

19. Z. Feng, Y. Yi and H. Zhu, Fast and slow dynamics of malaria and the s-gene frequency, J. Dyn. Differ. Equ., 16 (2004), 869-896.

20. M. Li, W. Liu, C. Shan, et al., Turning points and relaxation oscillation cycles in simple oscillation cycles in simple epidemic models, SIAM J. Appl. Math., 76 (2018), 663-687.

21. W. Liu, D. Xiao and Y. Yi, Relaxation oscillations in a class of predator-prey systems, J. Differ. Equations, 188 (2003), 306-331.

22. E. Benoit, J. L. Calot, F. Diener, et al., Chasse au canard, Collect. Math., 31-32(1-3) (1980-1981), 37-119.

23. E. Shchepakina, Black swans and canards in self-ignition problem, Nonlinear Anal-Real, 4 (2003), 45-50.

24. E. Shchepakina and V. Sobolev, Invariant surfaces of variable stability, J. Phys. Conf. Ser., 727 (2016), 012016.

25. V. Sobolev, Canard Cascades, Discrete Cont. Dyn-S, 18 (2013), 513-521.

26. A. Korobeinikov, E. Shchepakina and V. Sobolev, Paradox of enrichment and system order reduction: bacteriophages dynamics as case study, Math. Med. Biol., 33 (2016), 359-369.

27. M. P. Mortell, R. E. O'Malley, A. Pokrovskii, et al. (eds): Singular Perturbation and Hysteresis, SIAM, Philadelphia, 2005.

28. G. Ledder, Scaling for Dynamical Systems in Biology, Bull. Math. Biol., 79 (2017), 2747-2772.

29. M. Rosenzweig, The paradox of enrichment, Science, 171 (1971), 385-387.

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