# American Institute of Mathematical Sciences

January  2012, 11(1): 97-113. doi: 10.3934/cpaa.2012.11.97

## Qualitative analysis and travelling wave solutions for the SI model with vertical transmission

 1 UMR CNRS 5251, I.M.B. and INRIA Bordeaux Sud-ouest Anubis, case 36, UFR Sciences et Modélisation, Université Victor Segalen Bordeaux 2, 3 ter, place de la Victoire - 33076 Bordeaux cedex 2 Institut de Mathématiques de Bordeaux, UMR CNRS 5251, INRIA Bordeaux sud-ouest, EPI Anubis, UFR Sciences de la Vie, Université Victor Segalen Bordeaux 2, 3 ter Place de la Victoire, 33076 Bordeaux, France

Received  January 2010 Revised  August 2010 Published  September 2011

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. For a bounded spatial domain we show global stability of the endemic state when it is feasible. Then we look at the existence of travelling wave solutions connecting the endemic and the disease free states.
Citation: Arnaud Ducrot, Michel Langlais, Pierre Magal. Qualitative analysis and travelling wave solutions for the SI model with vertical transmission. Communications on Pure and Applied Analysis, 2012, 11 (1) : 97-113. doi: 10.3934/cpaa.2012.11.97
##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford Univ. Press, Oxford, U.K., 1991. [2] A. Arapostathis, M. K. Ghosh and S. I. Marcus, Harnack's inequality for cooperative weakly coupled elliptic systems, Comm. Part. Diff. Eq., 24 (1999), 1555-1571. [3] F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000. [4] S. Busenberg and K. C. Cooke, "Vertically Transmitted Diseases," Biomathematics volume 23, Springer-Verlag, New York, 1993. [5] V. Capasso, "Mathematical Structures of Epidemic Systems," Lecture Notes in Biomathematics volume 97, Springer-Verlag, Berlin, 1993. [6] Z. Q. Chen and Z. Zhao, Harnack principle for weakly coupled elliptic system, J. Diff. Eq., 139 (1997), 261-282. [7] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," Wiley, Chichester, U.K., 2000. [8] R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. [9] W. E. Fitzgibbon and M. Langlais, A diffusive S.I.S. model describing the propagation of F.I.V., Communications of Applied Analysis, 7 (2003), 387-4038. [10] W. E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on non coincident spatial domains, p. 115-164, Lecture Notes in Mathematics (Mathematical Biosciences Subseries) volume 1936, P. Magal and S. Ruan eds, Springer-Verlag, New York, 2008. [11] W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model of the spread of Feline Leukemia Virus (FeLV) through a highly heterogeneous spatial domain, SIAM J. Math. Analysis, 33 (2001), p. 570-588. [12] B. S. Goh, Global stability in a class of predator-prey models, Bull. Math. Biol., 40 (1978), p. 525-533. [13] J. Hale, "Asymptotic Behavior of Dissipation Systems," American Mathematical Society, Providence, Rhode Island, 1988. [14] A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin Université d'Etat Moscou, Bjul. Moskowskogo Gos. Univ., 1937, 1-26. [15] J. J. Morgan, Boundedness and decay results for reaction diffusion systems, SIAM J. Math. Anal., 20 (1990), p. 1128-1149. [16] J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," Springer-Verlag, Berlin 2003. [17] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in "Mathematics for Life Science and Medicine," Eds. by Y. Takeuchi, K. Sato and Y. Iwasa, Springer-Verlag, Berlin, (2007) pp. 99-122. [18] S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in "Spatial Ecology,'' Chapman & Hall/CRC, Boca Raton, FL, (2009) pp. 293-316. [19] J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Springer-Verlag, New York, 1994. [20] H. R. Thieme, "Mathematics in Population Biology," Princeton Univ. Press, Princeton, NJ, 2003. [21] A. Volpert, Vit. Volpert and Vl. Volpert, "Travelling Wave Solutions of Parabolic Systems," Monographs, vol. 140, AMS Providence, RI, 1994.

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##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control," Oxford Univ. Press, Oxford, U.K., 1991. [2] A. Arapostathis, M. K. Ghosh and S. I. Marcus, Harnack's inequality for cooperative weakly coupled elliptic systems, Comm. Part. Diff. Eq., 24 (1999), 1555-1571. [3] F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000. [4] S. Busenberg and K. C. Cooke, "Vertically Transmitted Diseases," Biomathematics volume 23, Springer-Verlag, New York, 1993. [5] V. Capasso, "Mathematical Structures of Epidemic Systems," Lecture Notes in Biomathematics volume 97, Springer-Verlag, Berlin, 1993. [6] Z. Q. Chen and Z. Zhao, Harnack principle for weakly coupled elliptic system, J. Diff. Eq., 139 (1997), 261-282. [7] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," Wiley, Chichester, U.K., 2000. [8] R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. [9] W. E. Fitzgibbon and M. Langlais, A diffusive S.I.S. model describing the propagation of F.I.V., Communications of Applied Analysis, 7 (2003), 387-4038. [10] W. E. Fitzgibbon and M. Langlais, Simple models for the transmission of microparasites between host populations living on non coincident spatial domains, p. 115-164, Lecture Notes in Mathematics (Mathematical Biosciences Subseries) volume 1936, P. Magal and S. Ruan eds, Springer-Verlag, New York, 2008. [11] W. E. Fitzgibbon, M. Langlais and J. J. Morgan, A mathematical model of the spread of Feline Leukemia Virus (FeLV) through a highly heterogeneous spatial domain, SIAM J. Math. Analysis, 33 (2001), p. 570-588. [12] B. S. Goh, Global stability in a class of predator-prey models, Bull. Math. Biol., 40 (1978), p. 525-533. [13] J. Hale, "Asymptotic Behavior of Dissipation Systems," American Mathematical Society, Providence, Rhode Island, 1988. [14] A. N. Kolmogorov, I. G. Petrovsky and N. S. Piskunov, Étude de l'équation de la diffusion avec croissance de la quantité de matière et son application à un problème biologique, Bulletin Université d'Etat Moscou, Bjul. Moskowskogo Gos. Univ., 1937, 1-26. [15] J. J. Morgan, Boundedness and decay results for reaction diffusion systems, SIAM J. Math. Anal., 20 (1990), p. 1128-1149. [16] J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," Springer-Verlag, Berlin 2003. [17] S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in "Mathematics for Life Science and Medicine," Eds. by Y. Takeuchi, K. Sato and Y. Iwasa, Springer-Verlag, Berlin, (2007) pp. 99-122. [18] S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in "Spatial Ecology,'' Chapman & Hall/CRC, Boca Raton, FL, (2009) pp. 293-316. [19] J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Springer-Verlag, New York, 1994. [20] H. R. Thieme, "Mathematics in Population Biology," Princeton Univ. Press, Princeton, NJ, 2003. [21] A. Volpert, Vit. Volpert and Vl. Volpert, "Travelling Wave Solutions of Parabolic Systems," Monographs, vol. 140, AMS Providence, RI, 1994.
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