# zbMATH — the first resource for mathematics

Dynamics of a delayed epidemic model with non-monotonic incidence rate. (English) Zbl 1221.34197
Summary: A delayed epidemic model with non-monotonic incidence rate which describes the psychological effect of certain serious on the community when the number of infectives is getting larger is studied. The disease-free equilibrium is globally asymptotically stable when $$R_{0}<1$$ and is globally attractive when $$R_{0}=1$$ are derived. On the other hand, The disease is permanent when $$R_{0}>1$$ is also obtained. Numerical simulation results are given to support the theoretical predictions.

##### MSC:
 34K20 Stability theory of functional-differential equations 92D30 Epidemiology
Full Text:
##### References:
 [1] Xiao, D.; Ruan, S., Global analysis of an epidemic model with nonmonotone incidence rate, Mathematical biosciences, 208, 419-429, (2007) · Zbl 1119.92042 [2] Capasso, V., () [3] Hale, J.K., Theory of functional differential equations, (1977), Springer New York [4] Hethcote, H.W.; Tudor, D.W., Integral equation models for endemic infectious diseases, J math biol, 9, 37-47, (1980) · Zbl 0433.92026 [5] Kuang, Y., Delay differential equations with applications in population dynamics, (1993), Academic Press New York · Zbl 0777.34002 [6] Hale, J.K.; Waltman, P., Persistence in infinite-dimensional systems, SIAM J math anal, 20, 388-395, (1989) · Zbl 0692.34053 [7] Zhang, T.; Teng, Z., Global behavior and permanence of SIRS epidemic model with time delay, Nonlin anal real world appl, (2007) [8] Faina, B.; Georgy, K.; Song, B.; Castillo-Chavez, C., A simple epidemic model with surprising dynamics, Math biosci eng, 2, 133-152, (2005) · Zbl 1061.92052 [9] Song, B.; Castillo-Chavez, C.; Aparicio, J.P., Tuberculosis models with fast and slow dynamics: the role of close and casual contacts, Math biosci, 180, 187-205, (2002) · Zbl 1015.92025 [10] Wang, H.; Li, J.; Kuang, Y., Mathematical modeling and qualitative analysis of insulin therapies, Math biosci, 210, 17-33, (2007) · Zbl 1138.92021 [11] Zhen, J.; Ma, Z.; Han, M., Global stability of an SIRS epidemic model with delays, Acta Mathematica scientia, 26B, 2, 291-306, (2006) · Zbl 1090.92044 [12] Zhen, J.; Ma, Z., The stability of an SIR epidemic model with time delays, Math biosci, 3, 101-109, (2006) · Zbl 1089.92045 [13] Capasso, V., () [14] Levin, S.A.; Hallam, T.G.; Gross, L.J., Applied mathematical ecology, (1989), Springer New York [15] Capasso, V.; Serio, G., A generalization of the kermack – mckendrick deterministic epidemic model, Math biosci, 42, 43, (1978) · Zbl 0398.92026 [16] Lin, J.; Andreasen, V.; Levin, S.A., Dynamics of influenza A drift: the linear three-strain model, Math biosci, 162, 33, (1999) · Zbl 0947.92017 [17] Liu, W.M.; Hethcote, H.W.; Levin, S.A., Dynamical behavior of epidemiological models with nonlinear incidence rates, J math biol, 25, 359, (1987) · Zbl 0621.92014 [18] Yorke, J.A.; London, W.P., Recurrent outbreaks of measles,chickenpox and mumps II, Am J epidemiol, 98, 469, (1973)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.