On the global stability of SIS, SIR and SIRS epidemic models with standard incidence. (English) Zbl 1355.92130
Summary: In this paper, we establish the global stability conditions of classic SIS, SIR and SIRS epidemic models with constant recruitment, disease-induced death and standard incidence rate. We will make an ingenious linear combination of known functions, common quadratic and Volterra-type, and of a new class of functions, we call composite-Volterra function, to obtain a suitable Lyapunov functions. In particular, for the SIRS model we prove the global stability of the endemic equilibrium under a condition of parameters.
MSC:
| 92D30 | Epidemiology |
References:
| [1] | Beretta, E.; Capasso, V., On the general structure of epidemic systems. Global asymptotic stability, Comput Math Appl Part A, 12, 677-694 (1986) · Zbl 0622.92016 |
| [2] | Hethcote, W. H.; Zhien, M.; Shengbing, L., Effects of quarantine in six endemic models for infectious diseases, Math Biosci, 180, 141-160 (2002) · Zbl 1019.92030 |
| [3] | Korobeinikov, A.; Wake, G. C., Lyapunov functions and global stability for SIR* SIRS* and SIS epidemiological models, Appl Math Lett, 15, 8, 955-960 (2002) · Zbl 1022.34044 |
| [4] | Goh, B. S., Global stability in two species interactions, J Math Biol, 3, 313-318 (1976) · Zbl 0362.92013 |
| [5] | Korobeinikov, A., Lyapunov functions and global properties for SEIR and SEIS epidemic models, Math Med Biol, 21, 75-83 (2004) · Zbl 1055.92051 |
| [6] | Korobeinikov, A., Global properties of SIR and SEIR epidemic models with multiple parallel infectious stages, Bull Math Biol, 71, 1, 75-83 (2009) · Zbl 1169.92041 |
| [7] | Guo, H.; Li, M. Y., Global dynamics of a staged progression model for infectious diseases, Math Biosci Eng, 3, 3, 513-525 (2006) · Zbl 1092.92040 |
| [8] | Guo, H.; Li, M. Y.; Shuai, S., Global stability of the endemic equilibrium of multigroup SIR epidemic models, Can Appl Math Quart, 14, 259-284 (2006) · Zbl 1148.34039 |
| [9] | McCluskey, C. C., Lyapunov functions for tuberculosis models with fast and slow progression, Math Biosci Eng, 3, 4, 603-614 (2006) · Zbl 1113.92057 |
| [10] | Iggidr, A.; Mbang, J.; Sallet, G.; Tewa, J., Multi-compartment models, DCDS-B, 506-519 (2007) · Zbl 1163.34366 |
| [11] | Bame, N.; Bowong, S.; Mbang, J.; Sallet, G.; Tewa, J., Global stability analysis for SEIS models with n latent classes, Math Biosci Eng, 5, 1, 20-33 (2008) · Zbl 1155.34325 |
| [12] | McCluskey, C. C., Global stability for a class of mass action systems allowing for latency in tuberculosis, J Math Anal Appl, 338, 1, 518-535 (2008) · Zbl 1131.92042 |
| [13] | Guo, H.; Li, M. Y.; Shuai, S., A graph-theoretic approach to the method of global Lyapunov functions, Proc Amer Math Soc, 136, 2793-2802 (2008) · Zbl 1155.34028 |
| [14] | Vargas De León, C., Constructions of Lyapunov functions for classic SIS, SIR and SIRS epidemic models with variable population size, Revista Electrónica Foro Red Mat, 26, 1-12 (2009), <http://www.red-mat.unam.mx/foro/volumenes/vol026> |
| [15] | La Salle, J.; Lefschetz, S., Stability by Liapunovs direct method with applications (1961), Academic Press: Academic Press New York · Zbl 0098.06102 |
| [16] | Mena-Lorca, J.; Hethcote, H. W., Dynamic models of infectious diseases as regulators of population sizes, J Math Biol, 30, 7, 693-716 (1992) · Zbl 0748.92012 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
