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June  2016, 21(4): 1101-1117. doi: 10.3934/dcdsb.2016.21.1101

Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions

Attila Dénes 1, and  Gergely Röst 2,

1. 

Bolyai Institute, University of Szeged, Aradi vértanúk tere 1., H-6720 Szeged,, Hungary
2. 

Analysis and Stochastics Research Group, Hungarian Academy of Sciences, Bolyai Institute, University of Szeged, H-6720 Szeged, Aradi vértanúk tere 1.

Received  November 2014 Revised  September 2015 Published  March 2016

We prove the global asymptotic stability of the disease-free and the endemic equilibrium for general SIR and SIRS models with nonlinear incidence. Instead of the popular Volterra-type Lyapunov functions, we use the method of Dulac functions, which allows us to extend the previous global stability results to a wider class of SIR and SIRS systems, including nonlinear (density-dependent) removal terms as well. We show that this method is useful in cases that cannot be covered by Lyapunov functions, such as bistable situations. We completely describe the global attractor even in the scenario of a backward bifurcation, when multiple endemic equilibria coexist.
Keywords: Dulac functions., global stability, SIR and SIRS models.
Mathematics Subject Classification: Primary: 37C70, 92D30; Secondary: 34C23, 34D2.
Citation: Attila Dénes, Gergely Röst. Global stability for SIR and SIRS models with nonlinear incidence and removal terms via Dulac functions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1101-1117. doi: 10.3934/dcdsb.2016.21.1101
References:
[1]

M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96. doi: 10.1016/j.mbs.2004.01.003.  Google Scholar

[2]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Lecture Notes in Mathematics, No. 35, Springer-Verlag, Berlin-New York, 1967.  Google Scholar

[3]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61. doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[4]

H. Dulac, Recherche des cycles limites, C. R. Acad. Sci. Paris, 204 (1937), 1703-1706. Google Scholar

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L. Edelstein-Keshet, Mathematical Models in Biology, The Random House/Birkhäuser Mathematics Series, Random House, Inc., New York, 1988.  Google Scholar

[6]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907.  Google Scholar

[7]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9.  Google Scholar

[8]

D. H. Knipl and G. Röst, Backward bifurcation in SIVS model with immigration of non-infectives, Biomath, 2 (2013), 1312051, 14pp. doi: 10.11145/j.biomath.2013.12.051.  Google Scholar

[9]

M. A. Krasnoselskii, Positive Solutions of Operator Equations, P. Noordhoff Ltd. Groningen, 1964. doi: 10.11145/j.biomath.2013.12.051.  Google Scholar

[10]

W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: 10.1007/BF00276956.  Google Scholar

[11]

T. C. Reluga and J. Medlock, Resistance mechanisms matter in SIR models, Math. Biosci. Eng., 4 (2007), 553-563. doi: 10.3934/mbe.2007.4.553.  Google Scholar

[12]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[13]

W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71. doi: 10.1016/j.mbs.2005.12.022.  Google Scholar

[14]

D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429. doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

show all references

References:
[1]

M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96. doi: 10.1016/j.mbs.2004.01.003.  Google Scholar

[2]

N. P. Bhatia and G. P. Szegö, Dynamical Systems: Stability Theory and Applications, Lecture Notes in Mathematics, No. 35, Springer-Verlag, Berlin-New York, 1967.  Google Scholar

[3]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61. doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[4]

H. Dulac, Recherche des cycles limites, C. R. Acad. Sci. Paris, 204 (1937), 1703-1706. Google Scholar

[5]

L. Edelstein-Keshet, Mathematical Models in Biology, The Random House/Birkhäuser Mathematics Series, Random House, Inc., New York, 1988.  Google Scholar

[6]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907.  Google Scholar

[7]

A. Korobeinikov, Lyapunov functions and global stability for SIR and SIRS epidemiological models with non-linear transmission, Bull. Math. Biol., 68 (2006), 615-626. doi: 10.1007/s11538-005-9037-9.  Google Scholar

[8]

D. H. Knipl and G. Röst, Backward bifurcation in SIVS model with immigration of non-infectives, Biomath, 2 (2013), 1312051, 14pp. doi: 10.11145/j.biomath.2013.12.051.  Google Scholar

[9]

M. A. Krasnoselskii, Positive Solutions of Operator Equations, P. Noordhoff Ltd. Groningen, 1964. doi: 10.11145/j.biomath.2013.12.051.  Google Scholar

[10]

W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204. doi: 10.1007/BF00276956.  Google Scholar

[11]

T. C. Reluga and J. Medlock, Resistance mechanisms matter in SIR models, Math. Biosci. Eng., 4 (2007), 553-563. doi: 10.3934/mbe.2007.4.553.  Google Scholar

[12]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.  Google Scholar

[13]

W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71. doi: 10.1016/j.mbs.2005.12.022.  Google Scholar

[14]

D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429. doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

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