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

A singularly perturbed SIS model with age structure

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.
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.

[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.

[5]

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

[6]

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

[7]

D. J. D. Earn, A light introduction to modelling recurrent epidemics,, in, (2008), 3. 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. doi: 10.1016/0022-0396(79)90152-9.

[9]

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

[10]

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

[11]

C. K. R. T. Jones, Geometric singular perturbation theory,, in, (1995), 44. 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. doi: 10.1137/S0036141099360919.

[13]

M. Lachowicz, Links between microscopic and macroscopic descriptions,, in, (2008), 201. 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.

[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.

[16]

J. D. Murray, "Mathematical Biology,", Springer, (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.

[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.

[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.

[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.

[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, (2003).

[24]

A. N. Tikhonov, A. B. Vasileva and A. G. Sveshnikov, "Differential Equations,", Springer, (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, (1973).

[26]

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

[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.

[28]

F. Verhulst, "Methods and Applications of Singular Perturbations,", Springer, (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.

[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.

[5]

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

[6]

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

[7]

D. J. D. Earn, A light introduction to modelling recurrent epidemics,, in, (2008), 3. 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. doi: 10.1016/0022-0396(79)90152-9.

[9]

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

[10]

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

[11]

C. K. R. T. Jones, Geometric singular perturbation theory,, in, (1995), 44. 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. doi: 10.1137/S0036141099360919.

[13]

M. Lachowicz, Links between microscopic and macroscopic descriptions,, in, (2008), 201. 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.

[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.

[16]

J. D. Murray, "Mathematical Biology,", Springer, (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.

[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.

[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.

[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.

[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, (2003).

[24]

A. N. Tikhonov, A. B. Vasileva and A. G. Sveshnikov, "Differential Equations,", Springer, (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, (1973).

[26]

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

[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.

[28]

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

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