American Institute of Mathematical Sciences

Advanced
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

Arnaud Ducrot 1, , Michel Langlais 1, and  Pierre Magal 2,

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.
Keywords: travelling wave solutions., Spatially structured SI model, global stability.
Mathematics Subject Classification: Primary: 35A18, 35B35, 35B40, 35K57; Secondary: 92D3.
Citation: Arnaud Ducrot, Michel Langlais, Pierre Magal. Qualitative analysis and travelling wave solutions for the SI model with vertical transmission. Communications on Pure & 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. Google Scholar

[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.  Google Scholar

[3]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000.  Google Scholar

[4]

S. Busenberg and K. C. Cooke, "Vertically Transmitted Diseases," Biomathematics volume 23, Springer-Verlag, New York, 1993.  Google Scholar

[5]

V. Capasso, "Mathematical Structures of Epidemic Systems," Lecture Notes in Biomathematics volume 97, Springer-Verlag, Berlin, 1993.  Google Scholar

[6]

Z. Q. Chen and Z. Zhao, Harnack principle for weakly coupled elliptic system, J. Diff. Eq., 139 (1997), 261-282.  Google Scholar

[7]

O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," Wiley, Chichester, U.K., 2000.  Google Scholar

[8]

R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

B. S. Goh, Global stability in a class of predator-prey models, Bull. Math. Biol., 40 (1978), p. 525-533.  Google Scholar

[13]

J. Hale, "Asymptotic Behavior of Dissipation Systems," American Mathematical Society, Providence, Rhode Island, 1988.  Google Scholar

[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. Google Scholar

[15]

J. J. Morgan, Boundedness and decay results for reaction diffusion systems, SIAM J. Math. Anal., 20 (1990), p. 1128-1149.  Google Scholar

[16]

J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," Springer-Verlag, Berlin 2003.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[19]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Springer-Verlag, New York, 1994.  Google Scholar

[20]

H. R. Thieme, "Mathematics in Population Biology," Princeton Univ. Press, Princeton, NJ, 2003.  Google Scholar

[21]

A. Volpert, Vit. Volpert and Vl. Volpert, "Travelling Wave Solutions of Parabolic Systems," Monographs, vol. 140, AMS Providence, RI, 1994.  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]

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.  Google Scholar

[3]

F. Brauer and C. Castillo-Chavez, "Mathematical Models in Population Biology and Epidemiology," Springer, New York, 2000.  Google Scholar

[4]

S. Busenberg and K. C. Cooke, "Vertically Transmitted Diseases," Biomathematics volume 23, Springer-Verlag, New York, 1993.  Google Scholar

[5]

V. Capasso, "Mathematical Structures of Epidemic Systems," Lecture Notes in Biomathematics volume 97, Springer-Verlag, Berlin, 1993.  Google Scholar

[6]

Z. Q. Chen and Z. Zhao, Harnack principle for weakly coupled elliptic system, J. Diff. Eq., 139 (1997), 261-282.  Google Scholar

[7]

O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis and Interpretation," Wiley, Chichester, U.K., 2000.  Google Scholar

[8]

R. A. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

B. S. Goh, Global stability in a class of predator-prey models, Bull. Math. Biol., 40 (1978), p. 525-533.  Google Scholar

[13]

J. Hale, "Asymptotic Behavior of Dissipation Systems," American Mathematical Society, Providence, Rhode Island, 1988.  Google Scholar

[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. Google Scholar

[15]

J. J. Morgan, Boundedness and decay results for reaction diffusion systems, SIAM J. Math. Anal., 20 (1990), p. 1128-1149.  Google Scholar

[16]

J. D. Murray, "Mathematical Biology II: Spatial Models and Biomedical Applications," Springer-Verlag, Berlin 2003.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[19]

J. Smoller, "Shock Waves and Reaction-Diffusion Equations," 2nd edition, Springer-Verlag, New York, 1994.  Google Scholar

[20]

H. R. Thieme, "Mathematics in Population Biology," Princeton Univ. Press, Princeton, NJ, 2003.  Google Scholar

[21]

A. Volpert, Vit. Volpert and Vl. Volpert, "Travelling Wave Solutions of Parabolic Systems," Monographs, vol. 140, AMS Providence, RI, 1994.  Google Scholar

[1]

Antoine Perasso. Global stability and uniform persistence for an infection load-structured SI model with exponential growth velocity. Communications on Pure & Applied Analysis, 2019, 18 (1) : 15-32. doi: 10.3934/cpaa.2019002
[2]

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
[3]

A. Ducrot. Travelling wave solutions for a scalar age-structured equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (2) : 251-273. doi: 10.3934/dcdsb.2007.7.251
[4]

Fred Brauer. A model for an SI disease in an age - structured population. Discrete & Continuous Dynamical Systems - B, 2002, 2 (2) : 257-264. doi: 10.3934/dcdsb.2002.2.257
[5]

Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences & Engineering, 2014, 11 (3) : 641-665. doi: 10.3934/mbe.2014.11.641
[6]

Claude-Michael Brauner, Josephus Hulshof, J.-F. Ripoll. Existence of travelling wave solutions in a combustion-radiation model. Discrete & Continuous Dynamical Systems - B, 2001, 1 (2) : 193-208. doi: 10.3934/dcdsb.2001.1.193
[7]

Sebastian Aniţa, Vincenzo Capasso, Ana-Maria Moşneagu. Global eradication for spatially structured populations by regional control. Discrete & Continuous Dynamical Systems - B, 2019, 24 (6) : 2511-2533. doi: 10.3934/dcdsb.2018263
[8]

Janet Dyson, Eva Sánchez, Rosanna Villella-Bressan, Glenn F. Webb. An age and spatially structured model of tumor invasion with haptotaxis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 45-60. doi: 10.3934/dcdsb.2007.8.45
[9]

Geni Gupur, Xue-Zhi Li. Global stability of an age-structured SIRS epidemic model with vaccination. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 643-652. doi: 10.3934/dcdsb.2004.4.643
[10]

Wei Wang, Wanbiao Ma. Global dynamics and travelling wave solutions for a class of non-cooperative reaction-diffusion systems with nonlocal infections. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3213-3235. doi: 10.3934/dcdsb.2018242
[11]

Chueh-Hsin Chang, Chiun-Chuan Chen. Travelling wave solutions of a free boundary problem for a two-species competitive model. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1065-1074. doi: 10.3934/cpaa.2013.12.1065
[12]

Tiberiu Harko, Man Kwong Mak. Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach. Mathematical Biosciences & Engineering, 2015, 12 (1) : 41-69. doi: 10.3934/mbe.2015.12.41
[13]

Zhihua Liu, Hui Tang, Pierre Magal. Hopf bifurcation for a spatially and age structured population dynamics model. Discrete & Continuous Dynamical Systems - B, 2015, 20 (6) : 1735-1757. doi: 10.3934/dcdsb.2015.20.1735
[14]

Claude-Michel Brauner, Josephus Hulshof, Luca Lorenzi. Stability of the travelling wave in a 2D weakly nonlinear Stefan problem. Kinetic & Related Models, 2009, 2 (1) : 109-134. doi: 10.3934/krm.2009.2.109
[15]

Yu Yang, Shigui Ruan, Dongmei Xiao. Global stability of an age-structured virus dynamics model with Beddington-DeAngelis infection function. Mathematical Biosciences & Engineering, 2015, 12 (4) : 859-877. doi: 10.3934/mbe.2015.12.859
[16]

Ting Liu, Guo-Bao Zhang. Global stability of traveling waves for a spatially discrete diffusion system with time delay. Electronic Research Archive, , () : -. doi: 10.3934/era.2021003
[17]

Christopher K. R. T. Jones, Robert Marangell. The spectrum of travelling wave solutions to the Sine-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (5) : 925-937. doi: 10.3934/dcdss.2012.5.925
[18]

Mostafa Adimy, Fabien Crauste, Laurent Pujo-Menjouet. On the stability of a nonlinear maturity structured model of cellular proliferation. Discrete & Continuous Dynamical Systems, 2005, 12 (3) : 501-522. doi: 10.3934/dcds.2005.12.501
[19]

Mohammed Nor Frioui, Tarik Mohammed Touaoula, Bedreddine Ainseba. Global dynamics of an age-structured model with relapse. Discrete & Continuous Dynamical Systems - B, 2020, 25 (6) : 2245-2270. doi: 10.3934/dcdsb.2019226
[20]

Seung-Yeal Ha, Mitsuru Yamazaki. $L^p$-stability estimates for the spatially inhomogeneous discrete velocity Boltzmann model. Discrete & Continuous Dynamical Systems - B, 2009, 11 (2) : 353-364. doi: 10.3934/dcdsb.2009.11.353

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (85)
  • HTML views (0)
  • Cited by (35)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences