A singularly perturbed SIS model with age structure

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2013, 10(3): 499-521. doi: 10.3934/mbe.2013.10.499

A singularly perturbed SIS model with age structure

Jacek Banasiak ^1, , Eddy Kimba Phongi ^2, and MirosŁaw Lachowicz ^3,

1.	School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban
2.	School of Mathematics, Statistics and Computer Science, University of KwaZulu-Natal, Durban 4041, South Africa
3.	Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland

Received May 2012 Revised August 2012 Published April 2013

Abstract
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We present a preliminary study of an SIS model with a basic age structure and we focus on a disease with quick turnover, such as influenza or common cold. In such a case the difference between the characteristic demographic and epidemiological times naturally introduces two time scales in the model which makes it singularly perturbed. Using the Tikhonov theorem we prove that for certain classes of initial conditions the nonlinear structured SIS model can be approximated with very good accuracy by lower dimensional linear models.

Keywords: singularly perturbed dynamical systems, population dynamics, SIS system, Tikhonov theorem., age structure, multiple time scales.

Mathematics Subject Classification: Primary: 34E15, 92D30; Secondary: 34E1.

Citation: 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

References:

[1]	J. Banasiak and M. Lachowicz, Multiscale approach in mathematical biology. Comment on "Toward a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives" by Bellomo and Carbonaro, Physics of Life Reviews, 8 (2011), 19-20.
[2]	J. Banasiak and M. Lachowicz, Methods of small parameter in mathematical biology and other applications,, preprint., ().
[3]	J. Banasiak and M. Lachowicz, Singularly perturbed epidemiological models - behaviour close to non-isolated quasi steady states,, in preparation., ().
[4]	N. Bellomo and B. Carbonaro, Toward a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives, Physics of Life Reviews, 8 (2011), 1-18.
[5]	M. Braun, "Differential Equations and Their Applications," Springer-Verlag, New York, 1993.
[6]	J. Cronin, Electrically active cells and singular perturbation problems, Math. Intelligencer, 12 (1990), 57-64. doi: 10.1007/BF03024034.
[7]	D. J. D. Earn, A light introduction to modelling recurrent epidemics, in "Mathematical Epidemiology" (eds. F. Brauer, P. van den Driessche and J. Wu), LNM 1945, Springer, Berlin, (2008), 3-18. doi: 10.1007/978-3-540-78911-6_1.
[8]	N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.
[9]	G. Hek, Geometrical singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386. doi: 10.1007/s00285-009-0266-7.
[10]	F. C. Hoppensteadt, Stability with parameter, J. Math. Anal. Appl., 18 (1967), 129-134.
[11]	C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems" (ed. R. Johnson), LNM 1609, Springer, Berlin, (1995), 44-118. doi: 10.1007/BFb0095239.
[12]	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), 286-314. doi: 10.1137/S0036141099360919.
[13]	M. Lachowicz, Links between microscopic and macroscopic descriptions, in "Multiscale Problems in the Life Sciences. From Microscopic to Macroscopic" (eds. V. Capasso and M. Lachowicz), LNM 1940, Springer, (2008), 201-68. doi: 10.1007/978-3-540-78362-6_4.
[14]	M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems, Prob. Engin. Mech., 26 (2011), 54-60.
[15]	S. Muratori and S. Rinaldi, Low- and high-frequency oscillations in three-dimensional food chain systems, SIAM J. Appl. Math., 52 (1992), 1688-1706. doi: 10.1137/0152097.
[16]	J. D. Murray, "Mathematical Biology," Springer, New York, 2003. doi: 10.1007/b98869.
[17]	, "Common Cold Fact Sheet,", , ().
[18]	S. Rinaldi and S. Muratori, Slow-fast limit cycles in predator-prey models, Ecol. Model., 6 (1992), 287-308.
[19]	D. Schanzer, J. Vachon and L. Pelletier, Age-specific differences in influenza a epidemic curves: Do children drive the spread of influenza epidemics?, 174 (2011), 109-117. doi: 10.1093/aje/kwr037.
[20]	L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation, SIAM Reviews, 31 (1989), 446-477. doi: 10.1137/1031091.
[21]	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/2011-12.
[22]	Y. Sun, Z. Wang, Y. Zhang and J. Sundell, In China, students in crowded dormitories with a low ventilation rate have more common colds: Evidence for airborne transmission,, PLoS ONE, 6 (). doi: 10.1371/journal.pone.0027140.
[23]	H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 2003.
[24]	A. N. Tikhonov, A. B. Vasileva and A. G. Sveshnikov, "Differential Equations," Springer, Berlin, 1985. doi: 10.1007/978-3-642-82175-2.
[25]	A. B. Vasileva and V. F. Butuzov, "Asymptotic Expansions of Solutions of Singularly Perturbed Equations," Nauka, Moscow, 1973, in Russian.
[26]	A. B. Vasilieva and V. F. Butuzov, "Singularly Perturbed Equations in the Critical Cases," Moscow State University, 1978 (in Russian) (translation: Mathematical Research Center Technical Summary Report 2039, Madison, 1980).
[27]	A. B. Vasilieva, On the development of singular perturbation theory at Moscow State University and elsewhere, SIAM Review, 36 (1994), 440-452. doi: 10.1137/1036100.
[28]	F. Verhulst, "Methods and Applications of Singular Perturbations," Springer, New York, 2005. doi: 10.1007/0-387-28313-7.

show all references

References:

[1]	J. Banasiak and M. Lachowicz, Multiscale approach in mathematical biology. Comment on "Toward a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives" by Bellomo and Carbonaro, Physics of Life Reviews, 8 (2011), 19-20.
[2]	J. Banasiak and M. Lachowicz, Methods of small parameter in mathematical biology and other applications,, preprint., ().
[3]	J. Banasiak and M. Lachowicz, Singularly perturbed epidemiological models - behaviour close to non-isolated quasi steady states,, in preparation., ().
[4]	N. Bellomo and B. Carbonaro, Toward a mathematical theory of living systems focusing on developmental biology and evolution: A review and perspectives, Physics of Life Reviews, 8 (2011), 1-18.
[5]	M. Braun, "Differential Equations and Their Applications," Springer-Verlag, New York, 1993.
[6]	J. Cronin, Electrically active cells and singular perturbation problems, Math. Intelligencer, 12 (1990), 57-64. doi: 10.1007/BF03024034.
[7]	D. J. D. Earn, A light introduction to modelling recurrent epidemics, in "Mathematical Epidemiology" (eds. F. Brauer, P. van den Driessche and J. Wu), LNM 1945, Springer, Berlin, (2008), 3-18. doi: 10.1007/978-3-540-78911-6_1.
[8]	N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31 (1979), 53-98. doi: 10.1016/0022-0396(79)90152-9.
[9]	G. Hek, Geometrical singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347-386. doi: 10.1007/s00285-009-0266-7.
[10]	F. C. Hoppensteadt, Stability with parameter, J. Math. Anal. Appl., 18 (1967), 129-134.
[11]	C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems" (ed. R. Johnson), LNM 1609, Springer, Berlin, (1995), 44-118. doi: 10.1007/BFb0095239.
[12]	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), 286-314. doi: 10.1137/S0036141099360919.
[13]	M. Lachowicz, Links between microscopic and macroscopic descriptions, in "Multiscale Problems in the Life Sciences. From Microscopic to Macroscopic" (eds. V. Capasso and M. Lachowicz), LNM 1940, Springer, (2008), 201-68. doi: 10.1007/978-3-540-78362-6_4.
[14]	M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems, Prob. Engin. Mech., 26 (2011), 54-60.
[15]	S. Muratori and S. Rinaldi, Low- and high-frequency oscillations in three-dimensional food chain systems, SIAM J. Appl. Math., 52 (1992), 1688-1706. doi: 10.1137/0152097.
[16]	J. D. Murray, "Mathematical Biology," Springer, New York, 2003. doi: 10.1007/b98869.
[17]	, "Common Cold Fact Sheet,", , ().
[18]	S. Rinaldi and S. Muratori, Slow-fast limit cycles in predator-prey models, Ecol. Model., 6 (1992), 287-308.
[19]	D. Schanzer, J. Vachon and L. Pelletier, Age-specific differences in influenza a epidemic curves: Do children drive the spread of influenza epidemics?, 174 (2011), 109-117. doi: 10.1093/aje/kwr037.
[20]	L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation, SIAM Reviews, 31 (1989), 446-477. doi: 10.1137/1031091.
[21]	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/2011-12.
[22]	Y. Sun, Z. Wang, Y. Zhang and J. Sundell, In China, students in crowded dormitories with a low ventilation rate have more common colds: Evidence for airborne transmission,, PLoS ONE, 6 (). doi: 10.1371/journal.pone.0027140.
[23]	H. R. Thieme, "Mathematics in Population Biology," Princeton University Press, Princeton, 2003.
[24]	A. N. Tikhonov, A. B. Vasileva and A. G. Sveshnikov, "Differential Equations," Springer, Berlin, 1985. doi: 10.1007/978-3-642-82175-2.
[25]	A. B. Vasileva and V. F. Butuzov, "Asymptotic Expansions of Solutions of Singularly Perturbed Equations," Nauka, Moscow, 1973, in Russian.
[26]	A. B. Vasilieva and V. F. Butuzov, "Singularly Perturbed Equations in the Critical Cases," Moscow State University, 1978 (in Russian) (translation: Mathematical Research Center Technical Summary Report 2039, Madison, 1980).
[27]	A. B. Vasilieva, On the development of singular perturbation theory at Moscow State University and elsewhere, SIAM Review, 36 (1994), 440-452. doi: 10.1137/1036100.
[28]	F. Verhulst, "Methods and Applications of Singular Perturbations," Springer, New York, 2005. doi: 10.1007/0-387-28313-7.

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