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

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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

Abstract
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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 and 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.
[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.
[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.
[4]	H. Dulac, Recherche des cycles limites, C. R. Acad. Sci. Paris, 204 (1937), 1703-1706.
[5]	L. Edelstein-Keshet, Mathematical Models in Biology, The Random House/Birkhäuser Mathematics Series, Random House, Inc., New York, 1988.
[6]	H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907.
[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.
[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.
[9]	M. A. Krasnoselskii, Positive Solutions of Operator Equations, P. Noordhoff Ltd. Groningen, 1964. doi: 10.11145/j.biomath.2013.12.051.
[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.
[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.
[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.
[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.
[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.

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.
[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.
[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.
[4]	H. Dulac, Recherche des cycles limites, C. R. Acad. Sci. Paris, 204 (1937), 1703-1706.
[5]	L. Edelstein-Keshet, Mathematical Models in Biology, The Random House/Birkhäuser Mathematics Series, Random House, Inc., New York, 1988.
[6]	H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907.
[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.
[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.
[9]	M. A. Krasnoselskii, Positive Solutions of Operator Equations, P. Noordhoff Ltd. Groningen, 1964. doi: 10.11145/j.biomath.2013.12.051.
[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.
[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.
[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.
[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.
[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.

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