# American Institute of Mathematical Sciences

March  2013, 8(1): 171-190. doi: 10.3934/nhm.2013.8.171

## Multiple travelling waves for an $SI$-epidemic model

 1 Univ. Bordeaux, IMB, UMR 5251, F-33400 Talence, France, France, France

Received  March 2012 Revised  September 2012 Published  April 2013

In this note we analyze a spatially structured SI epidemic model with vertical transmission, a logistic effect on vital dynamics and a density dependent incidence. The dynamics of the underlying system of ordinary differential equations are first shown to exhibit an infinite number of heteroclinic orbits connecting the trivial equilibrium with an interior equilibrium. Our mathematical study of the corresponding reaction-diffusion system is concerned with travelling wave solutions. Based on a detailed study of the center-unstable manifold around the interior equilibrium, we are able to prove the existence of an infinite number of travelling wave solutions connecting the trivial equilibrium and the interior equilibrium.
Citation: Arnaud Ducrot, Michel Langlais, Pierre Magal. Multiple travelling waves for an $SI$-epidemic model. Networks & Heterogeneous Media, 2013, 8 (1) : 171-190. doi: 10.3934/nhm.2013.8.171
##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford Univ. Press, Oxford, U.K., 1991. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 285 (1983).  Google Scholar [7] F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," Wiley, Chichester, U.K., 2000.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. Google Scholar [15] J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs 25, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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). Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [25] H. R. Thieme, "Mathematics in Population Biology," Princeton Univ. Press, Princeton, NJ, 2003.  Google Scholar [26] K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.  Google Scholar [27] A. Volpert, V. Volpert and V. Volpert, "Travelling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, AMS Providence, RI, 1994.  Google Scholar [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.  Google Scholar

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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [6] M. Bramson, Convergence of solutions of the Kolmogorov equation to travelling waves, Mem. Amer. Math. Soc., 285 (1983).  Google Scholar [7] F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," Wiley, Chichester, U.K., 2000.  Google Scholar [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.  Google Scholar [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.  Google Scholar [14] R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. Google Scholar [15] J. K. Hale, "Asymptotic Behavior of Dissipative Systems," Mathematical Surveys and Monographs 25, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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). Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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. Google Scholar [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.  Google Scholar [25] H. R. Thieme, "Mathematics in Population Biology," Princeton Univ. Press, Princeton, NJ, 2003.  Google Scholar [26] K. Uchiyama, The behavior of solutions of some nonlinear diffusion equations for large time, J. Math. Kyoto Univ., 18 (1978), 453-508.  Google Scholar [27] A. Volpert, V. Volpert and V. Volpert, "Travelling Wave Solutions of Parabolic Systems," Translations of Mathematical Monographs, 140, AMS Providence, RI, 1994.  Google Scholar [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.  Google Scholar
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