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From the guest editors
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 |
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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