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Archimedean copula and contagion modeling in epidemiology
Multiple travelling waves for an $SI$-epidemic model
1. | Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France, France, France |
References:
[1] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford Univ. Press, Oxford, U.K., 1991. |
[2] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
H. Berestycki, O. Diekmann, K. Nagelkerke and P. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2008), 399-429.
doi: 10.1007/s11538-008-9367-5. |
[4] |
H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed, I - The case of the whole space, Discrete Continuous Dynam. Systems - B, 21 (2008), 41-67.
doi: 10.3934/dcds.2008.21.41. |
[5] |
H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed, II - Cylindrical type domains, Discrete Continuous Dynam. Systems - A, 25 (2009), 19-61.
doi: 10.3934/dcds.2009.25.19. |
[6] |
M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 285 (1983). |
[7] |
F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000. |
[8] |
S. Busenberg and K. C. Cooke, "Vertically Transmitted Diseases," Biomathematics, 23, Springer-Verlag, New York, 1993.
doi: 10.1007/978-3-642-75301-5. |
[9] |
V. Capasso, "Mathematical Structures of Epidemic Systems," Lecture Notes in Biomathematics 97, Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-540-70514-7. |
[10] |
S.-N. Chow, C. Li and D. Wang, "Normal Forms and Bifurcation of Planar Vector Fields," Cambridge University Press, Cambridge, 1994.
doi: 10.1017/CBO9780511665639. |
[11] |
O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," Wiley, Chichester, U.K., 2000. |
[12] |
A. Ducrot and M. Langlais, Travelling waves in invasion processes with pathogens, Math. Models Methods Appl. Sci., 18 (2008), 325-349.
doi: 10.1142/S021820250800270X. |
[13] |
A. Ducrot, M. Langlais and P. Magal, Qualitative analysis and traveling wave solutions for the SI model with vertical transmission, Communications in Pure and Applied Analysis, 11 (2012), 97-113.
doi: 10.3934/cpaa.2012.11.97. |
[14] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. |
[15] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs 25, Amer. Math. Soc., Providence, RI, 1988. |
[16] |
F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions, J. Diff. Equ., 249 (2010), 1726-1745.
doi: 10.1016/j.jde.2010.06.025. |
[17] |
Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.
doi: 10.1142/S0218202595000504. |
[18] |
A. N. Kolmogorov, I. G. Petrovski and N. S. Piskunov, Etude de l'équation de la chaleur avec croissance de la quantité de matière et son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh., 1 (1937). |
[19] |
D. A. Larson, Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type, SIAM J. Appl. Math., 34 (1978), 93-103.
doi: 10.1137/0134008. |
[20] |
K.-S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky and Piscounov, J. Diff. Equ., 59 (1985), 44-70.
doi: 10.1016/0022-0396(85)90137-8. |
[21] |
F. Rothe, Convergence to travelling fronts in semilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 80 (1978), 213-234.
doi: 10.1017/S0308210500010258. |
[22] |
S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in "Mathematics for Life Science and Medicine" (eds. Y. Takeuchi, K. Sato and Y. Iwasa), Springer-Verlag, Berlin, (2007), 99-122. |
[23] |
S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in "Spatial Ecology", Chapman & Hall/CRC, Boca Raton, FL, (2009), 293-316. |
[24] |
H. L. Smith, "Monotone Dynamical Systems, an Introduction to the Theory of Competitive and Cooperative Systems," Math. Surveys and Monographs, 41, American Mathematical Society, Providence, Rhode Island 1995. |
[25] |
H. R. Thieme, "Mathematics in Population Biology," Princeton Univ. Press, Princeton, NJ, 2003. |
[26] |
K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508. |
[27] |
A. Volpert, V. Volpert and V. Volpert, "Travelling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, AMS Providence, RI, 1994. |
[28] |
V. A. Volpert and Y. M. Suhov, Stationary solutions of non-autonomous Kolmogorov-Petrovsky-Piskunov equations, Ergodic Theory and Dynamical Systems, 19 (1999), 809-835.
doi: 10.1017/S0143385799138823. |
show all references
References:
[1] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford Univ. Press, Oxford, U.K., 1991. |
[2] |
D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusions arising in population genetics, Adv. Math., 30 (1978), 33-76.
doi: 10.1016/0001-8708(78)90130-5. |
[3] |
H. Berestycki, O. Diekmann, K. Nagelkerke and P. Zegeling, Can a species keep pace with a shifting climate?, Bull. Math. Biol., 71 (2008), 399-429.
doi: 10.1007/s11538-008-9367-5. |
[4] |
H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed, I - The case of the whole space, Discrete Continuous Dynam. Systems - B, 21 (2008), 41-67.
doi: 10.3934/dcds.2008.21.41. |
[5] |
H. Berestycki and L. Rossi, Reaction-diffusion equations for population dynamics with forced speed, II - Cylindrical type domains, Discrete Continuous Dynam. Systems - A, 25 (2009), 19-61.
doi: 10.3934/dcds.2009.25.19. |
[6] |
M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 285 (1983). |
[7] |
F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000. |
[8] |
S. Busenberg and K. C. Cooke, "Vertically Transmitted Diseases," Biomathematics, 23, Springer-Verlag, New York, 1993.
doi: 10.1007/978-3-642-75301-5. |
[9] |
V. Capasso, "Mathematical Structures of Epidemic Systems," Lecture Notes in Biomathematics 97, Springer-Verlag, Berlin, 1993.
doi: 10.1007/978-3-540-70514-7. |
[10] |
S.-N. Chow, C. Li and D. Wang, "Normal Forms and Bifurcation of Planar Vector Fields," Cambridge University Press, Cambridge, 1994.
doi: 10.1017/CBO9780511665639. |
[11] |
O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," Wiley, Chichester, U.K., 2000. |
[12] |
A. Ducrot and M. Langlais, Travelling waves in invasion processes with pathogens, Math. Models Methods Appl. Sci., 18 (2008), 325-349.
doi: 10.1142/S021820250800270X. |
[13] |
A. Ducrot, M. Langlais and P. Magal, Qualitative analysis and traveling wave solutions for the SI model with vertical transmission, Communications in Pure and Applied Analysis, 11 (2012), 97-113.
doi: 10.3934/cpaa.2012.11.97. |
[14] |
R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. |
[15] |
J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs 25, Amer. Math. Soc., Providence, RI, 1988. |
[16] |
F. Hamel and L. Roques, Fast propagation for KPP equations with slowly decaying initial conditions, J. Diff. Equ., 249 (2010), 1726-1745.
doi: 10.1016/j.jde.2010.06.025. |
[17] |
Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.
doi: 10.1142/S0218202595000504. |
[18] |
A. N. Kolmogorov, I. G. Petrovski and N. S. Piskunov, Etude de l'équation de la chaleur avec croissance de la quantité de matière et son application à un problème biologique, Bull. Moskov. Gos. Univ. Mat. Mekh., 1 (1937). |
[19] |
D. A. Larson, Transient bounds and time-asymptotic behavior of solutions to nonlinear equations of Fisher type, SIAM J. Appl. Math., 34 (1978), 93-103.
doi: 10.1137/0134008. |
[20] |
K.-S. Lau, On the nonlinear diffusion equation of Kolmogorov, Petrovsky and Piscounov, J. Diff. Equ., 59 (1985), 44-70.
doi: 10.1016/0022-0396(85)90137-8. |
[21] |
F. Rothe, Convergence to travelling fronts in semilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 80 (1978), 213-234.
doi: 10.1017/S0308210500010258. |
[22] |
S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in "Mathematics for Life Science and Medicine" (eds. Y. Takeuchi, K. Sato and Y. Iwasa), Springer-Verlag, Berlin, (2007), 99-122. |
[23] |
S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in "Spatial Ecology", Chapman & Hall/CRC, Boca Raton, FL, (2009), 293-316. |
[24] |
H. L. Smith, "Monotone Dynamical Systems, an Introduction to the Theory of Competitive and Cooperative Systems," Math. Surveys and Monographs, 41, American Mathematical Society, Providence, Rhode Island 1995. |
[25] |
H. R. Thieme, "Mathematics in Population Biology," Princeton Univ. Press, Princeton, NJ, 2003. |
[26] |
K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508. |
[27] |
A. Volpert, V. Volpert and V. Volpert, "Travelling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, AMS Providence, RI, 1994. |
[28] |
V. A. Volpert and Y. M. Suhov, Stationary solutions of non-autonomous Kolmogorov-Petrovsky-Piskunov equations, Ergodic Theory and Dynamical Systems, 19 (1999), 809-835.
doi: 10.1017/S0143385799138823. |
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