# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Stability and bifurcation of an SIR epidemic model with nonlinear incidence and treatment. (English) Zbl 1159.92036
Summary: The dynamical behavior of an SIR epidemic model with nonlinear incidence and treatment is investigated. It is assumed that the treatment rate is proportional to the number of infectives below the capacity and is a constant when the number of infectives is greater than the capacity. It is found that a backward bifurcation occurs if the capacity is small. It is also found that there exist bistable endemic equilibria if the capacity is low. Theoretical and numerical results suggest that decreasing the basic reproduction number below one is insufficient for disease eradication.
##### MSC:
 92D30 Epidemiology 34D23 Global stability of ODE 34C60 Qualitative investigation and simulation of models (ODE)
##### Keywords:
nonlinear incidence rate; backward bifurcation; stability
##### References:
 [1] Hethcote, H. W.; Den Driessche, P. Van: Some epidemiological models with nonlinear incidence, J. math. Biol. 29, 271-287 (1991) · Zbl 0722.92015 · doi:10.1007/BF00160539 [2] Liu, W. M.; Levin, S. A.; Iwasa, Y.: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. math. Biol. 23, 187-204 (1986) · Zbl 0582.92023 · doi:10.1007/BF00276956 [3] Capasso, V.; Serio, G.: A generalization of the kermack C. Mckendrick deterministic epidemic model, Math. biosci. 42, 43-75 (1978) · Zbl 0398.92026 · doi:10.1016/0025-5564(78)90006-8 [4] Feng, Z.; Thieme, H. R.: Recurrent outbreaks of childhood diseases revisited: the impact of isolation, Math. biosci. 128, 93-114 (1995) · Zbl 0833.92017 · doi:10.1016/0025-5564(94)00069-C [5] Wu, L.; Feng, Z.: Homoclinic bifurcation in an SIQR model for childhood diseases, J. diff. Equat. 168, 150-167 (2000) · Zbl 0969.34042 · doi:10.1006/jdeq.2000.3882 [6] Hyman, J. M.; Li, J.: Modeling the effectiveness of isolation strategies in preventing STD epidemics, SIAM J. Appl. math. 58, 912-925 (1998) · Zbl 0905.92027 · doi:10.1137/S003613999630561X [7] Wang, W.; Ruan, S.: Bifurcation in an epidemic model with constant removal rate of the infectives, J. math. Anal. appl. 291, 75-793 (2004) · Zbl 1054.34071 · doi:10.1016/j.jmaa.2003.11.043 [8] Wang, W.: Backward bifurcation of an epidemic model with treatment, Math. biosci. 201, 58-71 (2006) · Zbl 1093.92054 · doi:10.1016/j.mbs.2005.12.022