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

Stefan problem, traveling fronts, and epidemic spread

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.
Citation: Karl P. Hadeler. Stefan problem, traveling fronts, and epidemic spread. Discrete & 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. doi: 10.1007/BF00275919. Google Scholar

[2]

D. G. Aronson, The asymptotic speed of propagation of a simple epidemic,, Nonlinear diffusion (NSF-CBMS Regional Conf. Nonlinear Diffusion Equations, (1976), 1. Google Scholar

[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). Google Scholar

[4]

C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves,, Math. Proc. Cambridge Philosophical Soc., 80 (1976), 315. doi: 10.1017/S0305004100052944. Google Scholar

[5]

K. J. Brown and J. Carr, Deterministic epidemic waves of critical velocity,, Math. Proc. Cambridge Philosophical Soc., 81 (1977), 431. doi: 10.1017/S0305004100053494. Google Scholar

[6]

G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model,, Netw. Heterog. Media, 7 (2012), 583. doi: 10.3934/nhm.2012.7.583. Google Scholar

[7]

O. Diekmann, Thresholds and travelling waves for the geographical spread of infection,, J. Mathematical Biology, 6 (1978), 109. doi: 10.1007/BF02450783. Google Scholar

[8]

Y. Du and Z. Lin, Spreading vanishing dichotomy in a diffusive logistic model with free boundary,, SIAM J. Math. Anal., 42 (2010), 377. doi: 10.1137/090771089. Google Scholar

[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. doi: 10.1137/110822608. Google Scholar

[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. doi: 10.1016/j.jde.2011.02.011. Google Scholar

[11]

Y. Du and Z. Guo, The Stefan problem for the Fisher-KPP equation,, J. Differential Equations, 253 (2012), 996. doi: 10.1016/j.jde.2012.04.014. Google Scholar

[12]

Y. Du, H. Matano and K. Wang, Regularity and asymptotic behavior of nonlinear Stefan problems,, Arch. Ration. Mech. Anal., 212 (2014), 957. doi: 10.1007/s00205-013-0710-0. Google Scholar

[13]

S. Dunbar, A branching random evolution and a nonlinear hyperbolic equation,, SIAM J. Appl. Math., 48 (1988), 1510. doi: 10.1137/0148094. Google Scholar

[14]

P. C. Fife and L. A. Peletier, Nonlinear diffusion in population genetics,, Arch. Ration. Mech. Anal., 64 (1977), 93. doi: 10.1007/BF00280092. Google Scholar

[15]

R. A. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

[16]

A. Friedman and B. Hu, The Stefan problem for a hyperbolic heat equation,, J. Math. Anal. Appl., 138 (1989), 249. doi: 10.1016/0022-247X(89)90334-X. Google Scholar

[17]

Th. Gallay and G. Raugel, Stability of travelling waves for a damped hyperbolic equation,, Z. Angew. Math. Phys., 48 (1997), 451. doi: 10.1007/s000330050043. Google Scholar

[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. doi: 10.1007/s10884-012-9267-0. Google Scholar

[19]

K. P. Hadeler, Hyperbolic travelling fronts,, Proc. Edinburgh Math. Soc., 31 (1988), 89. doi: 10.1017/S001309150000660X. Google Scholar

[20]

K. P. Hadeler, Travelling fronts for correlated random walks,, Canadian Applied Mathematics Quarterly, 2 (1994), 27. Google Scholar

[21]

K. P. Hadeler, Reaction telegraph equations and random walk systems,, In Stochastic and Spatial Structures of Dynamical Systems, 45 (1996), 133. Google Scholar

[22]

K. P. Hadeler, Travelling epidemic waves and correlated random walks,, In M. Martelli et al., (1996), 145. Google Scholar

[23]

K. P. Hadeler, Reaction transport equations in biological modelling,, In V. Capasso et al., (1997), 95. doi: 10.1007/BFb0092376. Google Scholar

[24]

K. P. Hadeler, The role of migration and contact distributions in epidemic spread,, Bioterrorism, 28 (2003), 199. Google Scholar

[25]

K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations,, J. Mathematical Biology, 2 (1975), 251. doi: 10.1007/BF00277154. Google Scholar

[26]

T. Hillen, Existence theory for correlated random walks on bounded domains,, Canadian Applied Mathematics Quarterly, 18 (2010), 1. Google Scholar

[27]

T. Hillen and K. P. Hadeler, Nonlinear hyperbolic systems and transport equations in mathematical biology,, In Analysis and numerics for conservation laws, (2005), 257. doi: 10.1007/3-540-27907-5_11. Google Scholar

[28]

E. E. Holmes, Are diffusion models too simple? a comparison with telegraph models of invasion,, American Naturalist, 142 (1993), 779. doi: 10.1086/285572. Google Scholar

[29]

A. Källén, Thresholds and travelling waves in an epidemic model for rabies,, Nonlinear Analysis TMA, 8 (1984), 851. doi: 10.1016/0362-546X(84)90107-X. Google Scholar

[30]

D. G. Kendall, Mathematical models of the spread of infection,, In Mathematics and Computer Science, (1965), 213. Google Scholar

[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, 1 (1937), 1. Google Scholar

[32]

C. Kuttler, Free boundary problem for a one-dimensional transport equation,, Z. Anal. Anwendungen, 20 (2001), 859. doi: 10.4171/ZAA/1049. Google Scholar

[33]

D. Mollison, Possible velocities for a simple epidemic,, Adv. Appl. Prob., 4 (1972), 233. doi: 10.2307/1425997. Google Scholar

[34]

D. Mollison, Spatial contact models for ecological and epidemic spread,, J. Royal Statist. Soc., 39 (1977), 283. Google Scholar

[35]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Mathematical Biology, 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar

[36]

G. Raugel and K. Kirchgässner, Stability of fronts for a kpp-system. II. the critical case,, J. Differential Equations, 146 (1998), 399. doi: 10.1006/jdeq.1997.3391. Google Scholar

[37]

F. Rothe, Convergence to travelling fronts in semilinear parabolic equations,, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 80 (1978), 213. doi: 10.1017/S0308210500010258. Google Scholar

[38]

K. Schumacher, Travelling-front solutions for integro-differential equations. I,, Journal für die reine und angewandte Mathematik, 316 (1980), 54. doi: 10.1515/crll.1980.316.54. Google Scholar

[39]

H. R. Schwetlick, Travelling fronts for multidimensional nonlinear transport equations,, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 17 (2000), 523. doi: 10.1016/S0294-1449(00)00127-X. Google Scholar

[40]

H. R. Thieme, Mathematics in Population Biology,, Princeton University Press, (2003). Google Scholar

[41]

H. F. Weinberger, On sufficient conditions for a linearly determinate spreading speed,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2267. doi: 10.3934/dcdsb.2012.17.2267. Google Scholar

show all references

References:
[1]

W. Alt, Biased random walk model for chemotaxis and related diffusion approximation,, J. Mathematical Biology, 9 (1980), 147. doi: 10.1007/BF00275919. Google Scholar

[2]

D. G. Aronson, The asymptotic speed of propagation of a simple epidemic,, Nonlinear diffusion (NSF-CBMS Regional Conf. Nonlinear Diffusion Equations, (1976), 1. Google Scholar

[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). Google Scholar

[4]

C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves,, Math. Proc. Cambridge Philosophical Soc., 80 (1976), 315. doi: 10.1017/S0305004100052944. Google Scholar

[5]

K. J. Brown and J. Carr, Deterministic epidemic waves of critical velocity,, Math. Proc. Cambridge Philosophical Soc., 81 (1977), 431. doi: 10.1017/S0305004100053494. Google Scholar

[6]

G. Bunting, Y. Du and K. Krakowski, Spreading speed revisited: Analysis of a free boundary model,, Netw. Heterog. Media, 7 (2012), 583. doi: 10.3934/nhm.2012.7.583. Google Scholar

[7]

O. Diekmann, Thresholds and travelling waves for the geographical spread of infection,, J. Mathematical Biology, 6 (1978), 109. doi: 10.1007/BF02450783. Google Scholar

[8]

Y. Du and Z. Lin, Spreading vanishing dichotomy in a diffusive logistic model with free boundary,, SIAM J. Math. Anal., 42 (2010), 377. doi: 10.1137/090771089. Google Scholar

[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. doi: 10.1137/110822608. Google Scholar

[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. doi: 10.1016/j.jde.2011.02.011. Google Scholar

[11]

Y. Du and Z. Guo, The Stefan problem for the Fisher-KPP equation,, J. Differential Equations, 253 (2012), 996. doi: 10.1016/j.jde.2012.04.014. Google Scholar

[12]

Y. Du, H. Matano and K. Wang, Regularity and asymptotic behavior of nonlinear Stefan problems,, Arch. Ration. Mech. Anal., 212 (2014), 957. doi: 10.1007/s00205-013-0710-0. Google Scholar

[13]

S. Dunbar, A branching random evolution and a nonlinear hyperbolic equation,, SIAM J. Appl. Math., 48 (1988), 1510. doi: 10.1137/0148094. Google Scholar

[14]

P. C. Fife and L. A. Peletier, Nonlinear diffusion in population genetics,, Arch. Ration. Mech. Anal., 64 (1977), 93. doi: 10.1007/BF00280092. Google Scholar

[15]

R. A. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355. doi: 10.1111/j.1469-1809.1937.tb02153.x. Google Scholar

[16]

A. Friedman and B. Hu, The Stefan problem for a hyperbolic heat equation,, J. Math. Anal. Appl., 138 (1989), 249. doi: 10.1016/0022-247X(89)90334-X. Google Scholar

[17]

Th. Gallay and G. Raugel, Stability of travelling waves for a damped hyperbolic equation,, Z. Angew. Math. Phys., 48 (1997), 451. doi: 10.1007/s000330050043. Google Scholar

[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. doi: 10.1007/s10884-012-9267-0. Google Scholar

[19]

K. P. Hadeler, Hyperbolic travelling fronts,, Proc. Edinburgh Math. Soc., 31 (1988), 89. doi: 10.1017/S001309150000660X. Google Scholar

[20]

K. P. Hadeler, Travelling fronts for correlated random walks,, Canadian Applied Mathematics Quarterly, 2 (1994), 27. Google Scholar

[21]

K. P. Hadeler, Reaction telegraph equations and random walk systems,, In Stochastic and Spatial Structures of Dynamical Systems, 45 (1996), 133. Google Scholar

[22]

K. P. Hadeler, Travelling epidemic waves and correlated random walks,, In M. Martelli et al., (1996), 145. Google Scholar

[23]

K. P. Hadeler, Reaction transport equations in biological modelling,, In V. Capasso et al., (1997), 95. doi: 10.1007/BFb0092376. Google Scholar

[24]

K. P. Hadeler, The role of migration and contact distributions in epidemic spread,, Bioterrorism, 28 (2003), 199. Google Scholar

[25]

K. P. Hadeler and F. Rothe, Travelling fronts in nonlinear diffusion equations,, J. Mathematical Biology, 2 (1975), 251. doi: 10.1007/BF00277154. Google Scholar

[26]

T. Hillen, Existence theory for correlated random walks on bounded domains,, Canadian Applied Mathematics Quarterly, 18 (2010), 1. Google Scholar

[27]

T. Hillen and K. P. Hadeler, Nonlinear hyperbolic systems and transport equations in mathematical biology,, In Analysis and numerics for conservation laws, (2005), 257. doi: 10.1007/3-540-27907-5_11. Google Scholar

[28]

E. E. Holmes, Are diffusion models too simple? a comparison with telegraph models of invasion,, American Naturalist, 142 (1993), 779. doi: 10.1086/285572. Google Scholar

[29]

A. Källén, Thresholds and travelling waves in an epidemic model for rabies,, Nonlinear Analysis TMA, 8 (1984), 851. doi: 10.1016/0362-546X(84)90107-X. Google Scholar

[30]

D. G. Kendall, Mathematical models of the spread of infection,, In Mathematics and Computer Science, (1965), 213. Google Scholar

[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, 1 (1937), 1. Google Scholar

[32]

C. Kuttler, Free boundary problem for a one-dimensional transport equation,, Z. Anal. Anwendungen, 20 (2001), 859. doi: 10.4171/ZAA/1049. Google Scholar

[33]

D. Mollison, Possible velocities for a simple epidemic,, Adv. Appl. Prob., 4 (1972), 233. doi: 10.2307/1425997. Google Scholar

[34]

D. Mollison, Spatial contact models for ecological and epidemic spread,, J. Royal Statist. Soc., 39 (1977), 283. Google Scholar

[35]

H. G. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Mathematical Biology, 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar

[36]

G. Raugel and K. Kirchgässner, Stability of fronts for a kpp-system. II. the critical case,, J. Differential Equations, 146 (1998), 399. doi: 10.1006/jdeq.1997.3391. Google Scholar

[37]

F. Rothe, Convergence to travelling fronts in semilinear parabolic equations,, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 80 (1978), 213. doi: 10.1017/S0308210500010258. Google Scholar

[38]

K. Schumacher, Travelling-front solutions for integro-differential equations. I,, Journal für die reine und angewandte Mathematik, 316 (1980), 54. doi: 10.1515/crll.1980.316.54. Google Scholar

[39]

H. R. Schwetlick, Travelling fronts for multidimensional nonlinear transport equations,, Annales de l'Institut Henri Poincaré (C) Non Linear Analysis, 17 (2000), 523. doi: 10.1016/S0294-1449(00)00127-X. Google Scholar

[40]

H. R. Thieme, Mathematics in Population Biology,, Princeton University Press, (2003). Google Scholar

[41]

H. F. Weinberger, On sufficient conditions for a linearly determinate spreading speed,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2267. doi: 10.3934/dcdsb.2012.17.2267. Google Scholar

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