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March  2016, 21(2): 417-436. doi: 10.3934/dcdsb.2016.21.417

Stefan problem, traveling fronts, and epidemic spread

Karl P. Hadeler 1,

1. 

Mathematics, University of Tübingen, Auf der Morgenstelle 10, 72076 Tübingen

Received  November 2014 Revised  April 2015 Published  November 2015

The scalar reaction diffusion equation with a nonlinearity of logistic type has a minimal speed $c_0$ for standard traveling fronts. It is shown that also for speeds $0 < c < c_0$ there are traveling fronts but these are solutions to free boundary value (Stefan) problems. Furthermore, these speeds depend in a monotone way on the Stefan coefficient which links the loss of matter at the free boundary to the displacement per time. The results are extended to correlated random walks, Cattaneo systems and, in particular, to models for epidemic spread. In the epidemic problems a dichotomy phenomenon shows up: For small values of the Stefan coefficient there are no fronts indicating that for such values and certain data the free boundary stays bounded.
Keywords: epidemic model, traveling front, Free boundary value problem, Stefan problem, correlated random walk..
Mathematics Subject Classification: Primary: 35C07, 92D30; Secondary: 35K57, 35L6.
Citation: Karl P. Hadeler. Stefan problem, traveling fronts, and epidemic spread. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 417-436. doi: 10.3934/dcdsb.2016.21.417
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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