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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. 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).
|
[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. 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. |
[16] |
J. D. Murray, "Mathematical Biology,", Springer, (2003).
doi: 10.1007/b98869. |
[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. |
[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. 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. |
[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. 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).
|
[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. 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. |
[16] |
J. D. Murray, "Mathematical Biology,", Springer, (2003).
doi: 10.1007/b98869. |
[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. |
[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. 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. |
[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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