American Institute of Mathematical Sciences

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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

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.   Google Scholar

[2]

J. Banasiak and M. Lachowicz, Methods of small parameter in mathematical biology and other applications,, preprint., ().   Google Scholar

[3]

J. Banasiak and M. Lachowicz, Singularly perturbed epidemiological models - behaviour close to non-isolated quasi steady states,, in preparation., ().   Google Scholar

[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.   Google Scholar

[5]

M. Braun, "Differential Equations and Their Applications,", Springer-Verlag, (1993).   Google Scholar

[6]

J. Cronin, Electrically active cells and singular perturbation problems,, Math. Intelligencer, 12 (1990), 57.  doi: 10.1007/BF03024034.  Google Scholar

[7]

D. J. D. Earn, A light introduction to modelling recurrent epidemics,, in, (2008), 3.  doi: 10.1007/978-3-540-78911-6_1.  Google Scholar

[8]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, J. Differ. Equ., 31 (1979), 53.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[9]

G. Hek, Geometrical singular perturbation theory in biological practice,, J. Math. Biol., 60 (2010), 347.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

[10]

F. C. Hoppensteadt, Stability with parameter,, J. Math. Anal. Appl., 18 (1967), 129.   Google Scholar

[11]

C. K. R. T. Jones, Geometric singular perturbation theory,, in, (1995), 44.  doi: 10.1007/BFb0095239.  Google Scholar

[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.  doi: 10.1137/S0036141099360919.  Google Scholar

[13]

M. Lachowicz, Links between microscopic and macroscopic descriptions,, in, (2008), 201.  doi: 10.1007/978-3-540-78362-6_4.  Google Scholar

[14]

M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems,, Prob. Engin. Mech., 26 (2011), 54.   Google Scholar

[15]

S. Muratori and S. Rinaldi, Low- and high-frequency oscillations in three-dimensional food chain systems,, SIAM J. Appl. Math., 52 (1992), 1688.  doi: 10.1137/0152097.  Google Scholar

[16]

J. D. Murray, "Mathematical Biology,", Springer, (2003).  doi: 10.1007/b98869.  Google Scholar

[17]

, "Common Cold Fact Sheet,", , ().   Google Scholar

[18]

S. Rinaldi and S. Muratori, Slow-fast limit cycles in predator-prey models,, Ecol. Model., 6 (1992), 287.   Google Scholar

[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), 174 (2011), 109.  doi: 10.1093/aje/kwr037.  Google Scholar

[20]

L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation,, SIAM Reviews, 31 (1989), 446.  doi: 10.1137/1031091.  Google Scholar

[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, (2011), 2011.   Google Scholar

[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.  Google Scholar

[23]

H. R. Thieme, "Mathematics in Population Biology,", Princeton University Press, (2003).   Google Scholar

[24]

A. N. Tikhonov, A. B. Vasileva and A. G. Sveshnikov, "Differential Equations,", Springer, (1985).  doi: 10.1007/978-3-642-82175-2.  Google Scholar

[25]

A. B. Vasileva and V. F. Butuzov, "Asymptotic Expansions of Solutions of Singularly Perturbed Equations,", Nauka, (1973).   Google Scholar

[26]

A. B. Vasilieva and V. F. Butuzov, "Singularly Perturbed Equations in the Critical Cases,", Moscow State University, (1978).   Google Scholar

[27]

A. B. Vasilieva, On the development of singular perturbation theory at Moscow State University and elsewhere,, SIAM Review, 36 (1994), 440.  doi: 10.1137/1036100.  Google Scholar

[28]

F. Verhulst, "Methods and Applications of Singular Perturbations,", Springer, (2005).  doi: 10.1007/0-387-28313-7.  Google Scholar

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.   Google Scholar

[2]

J. Banasiak and M. Lachowicz, Methods of small parameter in mathematical biology and other applications,, preprint., ().   Google Scholar

[3]

J. Banasiak and M. Lachowicz, Singularly perturbed epidemiological models - behaviour close to non-isolated quasi steady states,, in preparation., ().   Google Scholar

[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.   Google Scholar

[5]

M. Braun, "Differential Equations and Their Applications,", Springer-Verlag, (1993).   Google Scholar

[6]

J. Cronin, Electrically active cells and singular perturbation problems,, Math. Intelligencer, 12 (1990), 57.  doi: 10.1007/BF03024034.  Google Scholar

[7]

D. J. D. Earn, A light introduction to modelling recurrent epidemics,, in, (2008), 3.  doi: 10.1007/978-3-540-78911-6_1.  Google Scholar

[8]

N. Fenichel, Geometric singular perturbation theory for ordinary differential equations,, J. Differ. Equ., 31 (1979), 53.  doi: 10.1016/0022-0396(79)90152-9.  Google Scholar

[9]

G. Hek, Geometrical singular perturbation theory in biological practice,, J. Math. Biol., 60 (2010), 347.  doi: 10.1007/s00285-009-0266-7.  Google Scholar

[10]

F. C. Hoppensteadt, Stability with parameter,, J. Math. Anal. Appl., 18 (1967), 129.   Google Scholar

[11]

C. K. R. T. Jones, Geometric singular perturbation theory,, in, (1995), 44.  doi: 10.1007/BFb0095239.  Google Scholar

[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.  doi: 10.1137/S0036141099360919.  Google Scholar

[13]

M. Lachowicz, Links between microscopic and macroscopic descriptions,, in, (2008), 201.  doi: 10.1007/978-3-540-78362-6_4.  Google Scholar

[14]

M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems,, Prob. Engin. Mech., 26 (2011), 54.   Google Scholar

[15]

S. Muratori and S. Rinaldi, Low- and high-frequency oscillations in three-dimensional food chain systems,, SIAM J. Appl. Math., 52 (1992), 1688.  doi: 10.1137/0152097.  Google Scholar

[16]

J. D. Murray, "Mathematical Biology,", Springer, (2003).  doi: 10.1007/b98869.  Google Scholar

[17]

, "Common Cold Fact Sheet,", , ().   Google Scholar

[18]

S. Rinaldi and S. Muratori, Slow-fast limit cycles in predator-prey models,, Ecol. Model., 6 (1992), 287.   Google Scholar

[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), 174 (2011), 109.  doi: 10.1093/aje/kwr037.  Google Scholar

[20]

L. A. Segel and M. Slemrod, The quasi-steady-state assumption: A case study in perturbation,, SIAM Reviews, 31 (1989), 446.  doi: 10.1137/1031091.  Google Scholar

[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, (2011), 2011.   Google Scholar

[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.  Google Scholar

[23]

H. R. Thieme, "Mathematics in Population Biology,", Princeton University Press, (2003).   Google Scholar

[24]

A. N. Tikhonov, A. B. Vasileva and A. G. Sveshnikov, "Differential Equations,", Springer, (1985).  doi: 10.1007/978-3-642-82175-2.  Google Scholar

[25]

A. B. Vasileva and V. F. Butuzov, "Asymptotic Expansions of Solutions of Singularly Perturbed Equations,", Nauka, (1973).   Google Scholar

[26]

A. B. Vasilieva and V. F. Butuzov, "Singularly Perturbed Equations in the Critical Cases,", Moscow State University, (1978).   Google Scholar

[27]

A. B. Vasilieva, On the development of singular perturbation theory at Moscow State University and elsewhere,, SIAM Review, 36 (1994), 440.  doi: 10.1137/1036100.  Google Scholar

[28]

F. Verhulst, "Methods and Applications of Singular Perturbations,", Springer, (2005).  doi: 10.1007/0-387-28313-7.  Google Scholar

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