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Computation of $\mathcal R $ in age-structured epidemiological models with maternal and temporary immunity
Stefan problem, traveling fronts, and epidemic spread
1. | Mathematics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen |
References:
[1] |
W. Alt, Biased random walk model for chemotaxis and related diffusion approximation, J. Mathematical Biology, 9 (1980), 147-177.
doi: 10.1007/BF00275919. |
[2] |
D. G. Aronson, The asymptotic speed of propagation of a simple epidemic, Nonlinear diffusion (NSF-CBMS Regional Conf. Nonlinear Diffusion Equations, Univ. Houston, Houston, Tex., 1976), pp. 1-23. Res. Notes Math., No. 14, Pitman, London, 1977. |
[3] |
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve propagation, In Partial differential equations and related topics, (Program, Tulane Univ., New Orleans, La., 1974), pp. 5-49. Lecture Notes in Math. 446, Springer Verlag, 1975. |
[4] |
C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves, Math. Proc. Cambridge Philosophical Soc., 80 (1976), 315-330.
doi: 10.1017/S0305004100052944. |
[5] |
K. J. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Cambridge Philosophical Soc., 81 (1977), 431-433.
doi: 10.1017/S0305004100053494. |
[6] |
G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.
doi: 10.3934/nhm.2012.7.583. |
[7] |
O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Mathematical Biology, 6 (1978), 109-130.
doi: 10.1007/BF02450783. |
[8] |
Y. Du and Z. Lin, Spreading vanishing dichotomy in a diffusive logistic model with free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.
doi: 10.1137/090771089. |
[9] |
Y. Du and Z. Lin, Erratum: Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996.
doi: 10.1137/110822608. |
[10] |
Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary. II, J. Differential Equations, 250 (2011), 4336-4366.
doi: 10.1016/j.jde.2011.02.011. |
[11] |
Y. Du and Z. Guo, The Stefan problem for the Fisher-KPP equation, J. Differential Equations, 253 (2012), 996-1035.
doi: 10.1016/j.jde.2012.04.014. |
[12] |
Y. Du, H. Matano and K. Wang, Regularity and asymptotic behavior of nonlinear Stefan problems, Arch. Ration. Mech. Anal., 212 (2014), 957-1010.
doi: 10.1007/s00205-013-0710-0. |
[13] |
S. Dunbar, A branching random evolution and a nonlinear hyperbolic equation, SIAM J. Appl. Math., 48 (1988), 1510-1526.
doi: 10.1137/0148094. |
[14] |
P. C. Fife and L. A. Peletier, Nonlinear diffusion in population genetics, Arch. Ration. Mech. Anal., 64 (1977), 93-109.
doi: 10.1007/BF00280092. |
[15] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[16] |
A. Friedman and B. Hu, The Stefan problem for a hyperbolic heat equation, J. Math. Anal. Appl., 138 (1989), 249-279.
doi: 10.1016/0022-247X(89)90334-X. |
[17] |
Th. Gallay and G. Raugel, Stability of travelling waves for a damped hyperbolic equation, Z. Angew. Math. Phys., 48 (1997), 451-479.
doi: 10.1007/s000330050043. |
[18] |
J.-S. Guo and C.-H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Diff. Equat., 24 (2012), 873-895.
doi: 10.1007/s10884-012-9267-0. |
[19] |
K. P. Hadeler, Hyperbolic travelling fronts, Proc. Edinburgh Math. Soc., 31 (1988), 89-97.
doi: 10.1017/S001309150000660X. |
[20] |
K. P. Hadeler, Travelling fronts for correlated random walks, Canadian Applied Mathematics Quarterly, 2 (1994), 27-43. |
[21] |
K. P. Hadeler, Reaction telegraph equations and random walk systems, In Stochastic and Spatial Structures of Dynamical Systems, Royal Netherlands Academy of Arts and Sciences, 45 (1996), 133-161. |
[22] |
K. P. Hadeler, Travelling epidemic waves and correlated random walks, In M. Martelli et al., editor, Differential Equations and Applications to Biology and to Industry, World Scientific Publ., (1996), 145-156. |
[23] |
K. P. Hadeler, Reaction transport equations in biological modelling, In V. Capasso et al., editor, Mathematics inspired by biology, pp. 95-150. CIME Lectures 1997, Lecture Notes in Mathematics 1714, Springer, 1999.
doi: 10.1007/BFb0092376. |
[24] |
K. P. Hadeler, The role of migration and contact distributions in epidemic spread, Bioterrorism, Frontiers Appl. Math., SIAM, Philadelphia, PA, 28 (2003), 199-210. |
[25] |
K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Mathematical Biology, 2 (1975), 251-263.
doi: 10.1007/BF00277154. |
[26] |
T. Hillen, Existence theory for correlated random walks on bounded domains, Canadian Applied Mathematics Quarterly, 18 (2010), 1-40. |
[27] |
T. Hillen and K. P. Hadeler, Nonlinear hyperbolic systems and transport equations in mathematical biology, In Analysis and numerics for conservation laws, pp. 257-279. Springer, 2005.
doi: 10.1007/3-540-27907-5_11. |
[28] |
E. E. Holmes, Are diffusion models too simple? a comparison with telegraph models of invasion, American Naturalist, 142 (1993), 779-795.
doi: 10.1086/285572. |
[29] |
A. Källén, Thresholds and travelling waves in an epidemic model for rabies, Nonlinear Analysis TMA, 8 (1984), 851-856.
doi: 10.1016/0362-546X(84)90107-X. |
[30] |
D. G. Kendall, Mathematical models of the spread of infection, In Mathematics and Computer Science, pp. 213-224. Medical Research Council, 1965. |
[31] |
A. Kolmogoroff, I. Petrovskij and N. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application a une problème biologique, Bull. Univ. Moscou, Ser. Internat., Sec. A, 1 (1937), 1-25. |
[32] |
C. Kuttler, Free boundary problem for a one-dimensional transport equation, Z. Anal. Anwendungen, 20 (2001), 859-881.
doi: 10.4171/ZAA/1049. |
[33] |
D. Mollison, Possible velocities for a simple epidemic, Adv. Appl. Prob., 4 (1972), 233-257.
doi: 10.2307/1425997. |
[34] |
D. Mollison, Spatial contact models for ecological and epidemic spread, J. Royal Statist. Soc., 39 (1977), 283-326. |
[35] |
H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Mathematical Biology, 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[36] |
G. Raugel and K. Kirchgässner, Stability of fronts for a kpp-system. II. the critical case, J. Differential Equations, 146 (1998), 399-456.
doi: 10.1006/jdeq.1997.3391. |
[37] |
F. Rothe, Convergence to travelling fronts in semilinear parabolic equations, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 80 (1978), 213-234.
doi: 10.1017/S0308210500010258. |
[38] |
K. Schumacher, Travelling-front solutions for integro-differential equations. I, Journal für die reine und angewandte Mathematik, 316 (1980), 54-70.
doi: 10.1515/crll.1980.316.54. |
[39] |
H. R. Schwetlick, Travelling fronts for multidimensional nonlinear transport equations, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 17 (2000), 523-550.
doi: 10.1016/S0294-1449(00)00127-X. |
[40] |
H. R. Thieme, Mathematics in Population Biology, Princeton University Press, 2003. |
[41] |
H. F. Weinberger, On sufficient conditions for a linearly determinate spreading speed, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2267-2280.
doi: 10.3934/dcdsb.2012.17.2267. |
show all references
References:
[1] |
W. Alt, Biased random walk model for chemotaxis and related diffusion approximation, J. Mathematical Biology, 9 (1980), 147-177.
doi: 10.1007/BF00275919. |
[2] |
D. G. Aronson, The asymptotic speed of propagation of a simple epidemic, Nonlinear diffusion (NSF-CBMS Regional Conf. Nonlinear Diffusion Equations, Univ. Houston, Houston, Tex., 1976), pp. 1-23. Res. Notes Math., No. 14, Pitman, London, 1977. |
[3] |
D. G. Aronson and H. F. Weinberger, Nonlinear diffusion in population genetics, combustion, and nerve propagation, In Partial differential equations and related topics, (Program, Tulane Univ., New Orleans, La., 1974), pp. 5-49. Lecture Notes in Math. 446, Springer Verlag, 1975. |
[4] |
C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves, Math. Proc. Cambridge Philosophical Soc., 80 (1976), 315-330.
doi: 10.1017/S0305004100052944. |
[5] |
K. J. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Cambridge Philosophical Soc., 81 (1977), 431-433.
doi: 10.1017/S0305004100053494. |
[6] |
G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model, Netw. Heterog. Media, 7 (2012), 583-603.
doi: 10.3934/nhm.2012.7.583. |
[7] |
O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Mathematical Biology, 6 (1978), 109-130.
doi: 10.1007/BF02450783. |
[8] |
Y. Du and Z. Lin, Spreading vanishing dichotomy in a diffusive logistic model with free boundary, SIAM J. Math. Anal., 42 (2010), 377-405.
doi: 10.1137/090771089. |
[9] |
Y. Du and Z. Lin, Erratum: Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 45 (2013), 1995-1996.
doi: 10.1137/110822608. |
[10] |
Y. Du and Z. Guo, Spreading-vanishing dichotomy in a diffusive logistic model with a free boundary. II, J. Differential Equations, 250 (2011), 4336-4366.
doi: 10.1016/j.jde.2011.02.011. |
[11] |
Y. Du and Z. Guo, The Stefan problem for the Fisher-KPP equation, J. Differential Equations, 253 (2012), 996-1035.
doi: 10.1016/j.jde.2012.04.014. |
[12] |
Y. Du, H. Matano and K. Wang, Regularity and asymptotic behavior of nonlinear Stefan problems, Arch. Ration. Mech. Anal., 212 (2014), 957-1010.
doi: 10.1007/s00205-013-0710-0. |
[13] |
S. Dunbar, A branching random evolution and a nonlinear hyperbolic equation, SIAM J. Appl. Math., 48 (1988), 1510-1526.
doi: 10.1137/0148094. |
[14] |
P. C. Fife and L. A. Peletier, Nonlinear diffusion in population genetics, Arch. Ration. Mech. Anal., 64 (1977), 93-109.
doi: 10.1007/BF00280092. |
[15] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[16] |
A. Friedman and B. Hu, The Stefan problem for a hyperbolic heat equation, J. Math. Anal. Appl., 138 (1989), 249-279.
doi: 10.1016/0022-247X(89)90334-X. |
[17] |
Th. Gallay and G. Raugel, Stability of travelling waves for a damped hyperbolic equation, Z. Angew. Math. Phys., 48 (1997), 451-479.
doi: 10.1007/s000330050043. |
[18] |
J.-S. Guo and C.-H. Wu, On a free boundary problem for a two-species weak competition system, J. Dyn. Diff. Equat., 24 (2012), 873-895.
doi: 10.1007/s10884-012-9267-0. |
[19] |
K. P. Hadeler, Hyperbolic travelling fronts, Proc. Edinburgh Math. Soc., 31 (1988), 89-97.
doi: 10.1017/S001309150000660X. |
[20] |
K. P. Hadeler, Travelling fronts for correlated random walks, Canadian Applied Mathematics Quarterly, 2 (1994), 27-43. |
[21] |
K. P. Hadeler, Reaction telegraph equations and random walk systems, In Stochastic and Spatial Structures of Dynamical Systems, Royal Netherlands Academy of Arts and Sciences, 45 (1996), 133-161. |
[22] |
K. P. Hadeler, Travelling epidemic waves and correlated random walks, In M. Martelli et al., editor, Differential Equations and Applications to Biology and to Industry, World Scientific Publ., (1996), 145-156. |
[23] |
K. P. Hadeler, Reaction transport equations in biological modelling, In V. Capasso et al., editor, Mathematics inspired by biology, pp. 95-150. CIME Lectures 1997, Lecture Notes in Mathematics 1714, Springer, 1999.
doi: 10.1007/BFb0092376. |
[24] |
K. P. Hadeler, The role of migration and contact distributions in epidemic spread, Bioterrorism, Frontiers Appl. Math., SIAM, Philadelphia, PA, 28 (2003), 199-210. |
[25] |
K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations, J. Mathematical Biology, 2 (1975), 251-263.
doi: 10.1007/BF00277154. |
[26] |
T. Hillen, Existence theory for correlated random walks on bounded domains, Canadian Applied Mathematics Quarterly, 18 (2010), 1-40. |
[27] |
T. Hillen and K. P. Hadeler, Nonlinear hyperbolic systems and transport equations in mathematical biology, In Analysis and numerics for conservation laws, pp. 257-279. Springer, 2005.
doi: 10.1007/3-540-27907-5_11. |
[28] |
E. E. Holmes, Are diffusion models too simple? a comparison with telegraph models of invasion, American Naturalist, 142 (1993), 779-795.
doi: 10.1086/285572. |
[29] |
A. Källén, Thresholds and travelling waves in an epidemic model for rabies, Nonlinear Analysis TMA, 8 (1984), 851-856.
doi: 10.1016/0362-546X(84)90107-X. |
[30] |
D. G. Kendall, Mathematical models of the spread of infection, In Mathematics and Computer Science, pp. 213-224. Medical Research Council, 1965. |
[31] |
A. Kolmogoroff, I. Petrovskij and N. Piskunov, Etude de l'équation de la diffusion avec croissance de la quantité de matière et son application a une problème biologique, Bull. Univ. Moscou, Ser. Internat., Sec. A, 1 (1937), 1-25. |
[32] |
C. Kuttler, Free boundary problem for a one-dimensional transport equation, Z. Anal. Anwendungen, 20 (2001), 859-881.
doi: 10.4171/ZAA/1049. |
[33] |
D. Mollison, Possible velocities for a simple epidemic, Adv. Appl. Prob., 4 (1972), 233-257.
doi: 10.2307/1425997. |
[34] |
D. Mollison, Spatial contact models for ecological and epidemic spread, J. Royal Statist. Soc., 39 (1977), 283-326. |
[35] |
H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Mathematical Biology, 26 (1988), 263-298.
doi: 10.1007/BF00277392. |
[36] |
G. Raugel and K. Kirchgässner, Stability of fronts for a kpp-system. II. the critical case, J. Differential Equations, 146 (1998), 399-456.
doi: 10.1006/jdeq.1997.3391. |
[37] |
F. Rothe, Convergence to travelling fronts in semilinear parabolic equations, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 80 (1978), 213-234.
doi: 10.1017/S0308210500010258. |
[38] |
K. Schumacher, Travelling-front solutions for integro-differential equations. I, Journal für die reine und angewandte Mathematik, 316 (1980), 54-70.
doi: 10.1515/crll.1980.316.54. |
[39] |
H. R. Schwetlick, Travelling fronts for multidimensional nonlinear transport equations, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 17 (2000), 523-550.
doi: 10.1016/S0294-1449(00)00127-X. |
[40] |
H. R. Thieme, Mathematics in Population Biology, Princeton University Press, 2003. |
[41] |
H. F. Weinberger, On sufficient conditions for a linearly determinate spreading speed, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2267-2280.
doi: 10.3934/dcdsb.2012.17.2267. |
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