A singularly perturbed SIS model with age structure
Pages: 499  521,
Issue 3,
June
2013
doi:10.3934/mbe.2013.10.499 Abstract
References
Full text (447.7K)
Related Articles
Jacek Banasiak  School of Mathematics, Statistics and Computer Science, University of KwaZuluNatal, Durban, South Africa (email)
Eddy Kimba Phongi  School of Mathematics, Statistics and Computer Science, University of KwaZuluNatal, Durban 4041, South Africa (email)
MirosÅaw Lachowicz  Institute of Applied Mathematics and Mechanics, University of Warsaw, Warsaw, Poland (email)
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), 1920. 

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 nonisolated 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), 118. 

5 
M. Braun, "Differential Equations and Their Applications," SpringerVerlag, New York, 1993. 

6 
J. Cronin, Electrically active cells and singular perturbation problems, Math. Intelligencer, 12 (1990), 5764. 

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), 318. 

8 
N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equ., 31 (1979), 5398. 

9 
G. Hek, Geometrical singular perturbation theory in biological practice, J. Math. Biol., 60 (2010), 347386. 

10 
F. C. Hoppensteadt, Stability with parameter, J. Math. Anal. Appl., 18 (1967), 129134. 

11 
C. K. R. T. Jones, Geometric singular perturbation theory, in "Dynamical Systems" (ed. R. Johnson), LNM 1609, Springer, Berlin, (1995), 44118. 

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), 286314. 

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), 20168. 

14 
M. Lachowicz, Microscopic, mesoscopic and macroscopic descriptions of complex systems, Prob. Engin. Mech., 26 (2011), 5460. 

15 
S. Muratori and S. Rinaldi, Low and highfrequency oscillations in threedimensional food chain systems, SIAM J. Appl. Math., 52 (1992), 16881706. 

16 
J. D. Murray, "Mathematical Biology," Springer, New York, 2003. 

17 
"Common Cold Fact Sheet," http://www.tdi.texas.gov/pubs/videoresource/fscommoncold.pdf. 

18 
S. Rinaldi and S. Muratori, Slowfast limit cycles in predatorprey models, Ecol. Model., 6 (1992), 287308. 

19 
D. Schanzer, J. Vachon and L. Pelletier, Agespecific differences in influenza a epidemic curves: Do children drive the spread of influenza epidemics?, 174 (2011), 109117. 

20 
L. A. Segel and M. Slemrod, The quasisteadystate assumption: A case study in perturbation, SIAM Reviews, 31 (1989), 446477. 

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/201112. 

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, e27140. 

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. 

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), 440452. 

28 
F. Verhulst, "Methods and Applications of Singular Perturbations," Springer, New York, 2005. 

Go to top
