×

zbMATH — the first resource for mathematics

Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate. (English) Zbl 1219.92060
Summary: We study an SEIR epidemic model with saturated recovery rate. A backward bifurcation leading to bistability possibly occurs, and global dynamics are shown by compound matrices and geometric approaches. Numerical simulations are presented to illustrate the results.

MSC:
92D30 Epidemiology
34C23 Bifurcation theory for ordinary differential equations
34D23 Global stability of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
PDF BibTeX Cite
Full Text: DOI
References:
[1] Kermack, W.O.; McKendrick, A.G., A contribution to the mathematical theory of epidemics, Proc R soc lond B biol sci, 115, 700-721, (1927) · JFM 53.0517.01
[2] Kermack, W.O.; McKendrick, A.G., Contributions to the mathematical theory of epidemics (part II), Proc R soc lond B biol sci, 138, 55-83, (1932) · Zbl 0005.30501
[3] Kermack, W.O.; McKendrick, A.G., Contributions to the mathematical theory of epidemics (part III), Proc R soc lond B biol sci, 141, 94-112, (1932) · Zbl 0007.31502
[4] Jin, Z.; Haque, M.; Liu, Q.X., Pulse vaccination in the periodic infection rate SIR epidemic model, Int J biomath, 1, 4, 409-432, (2008) · Zbl 1156.92028
[5] Huo, H.F.; Ma, Z.P., Dynamics of a delayed epidemic model with non-monotonic incidence rate, Commun nonlinear sci numer simul, 15, 2, 459-468, (2010) · Zbl 1221.34197
[6] van den Driessche, P.; Watmough, J., A simple SIS epidemic model with a backward bifurcation, J math biol, 40, 525-540, (2000) · Zbl 0961.92029
[7] Yang, J.Y.; Zhang, F.Q.; Wang, X.Y., SIV epidemic models with age of infection, Int J biomath, 2, 1, 61-67, (2009) · Zbl 1342.92291
[8] Li, Y.F.; Cui, J.A., The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage, Commun nonlinear sci numer simul, 14, 5, 2353-2365, (2009) · Zbl 1221.34034
[9] Derrick, W.R.; van den Driessche, P., Homoclinic orbits in a disease transmission model with non-linear incidence and nonconstant population, Discret contin dyn syst ser B, 3, 299-309, (2003) · Zbl 1126.34337
[10] Cui, J.A.; Mu, X.X.; Wan, H., Saturation recovery leads to multiple endemic equilibria and backward bifurcation, J theor biol, 254, 273-285, (2008) · Zbl 1400.92472
[11] Van den Driessche, P.; Watmough, J., Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math biosci, 180, 29-48, (2002) · Zbl 1015.92036
[12] Castillo-Chavez, C.; Song, B.J., Dynamical models of tuberculosis and their applications, Math biosci eng, 1, 361-404, (2004) · Zbl 1060.92041
[13] Guckenheimer, J.; Holmes, P., Nonlinear oscillations, Dynamical systems and bifurcations of vector fields, (1983), Springer Berlin · Zbl 0515.34001
[14] Hale, J.K., Ordinary differential equations, (1969), Wiley New York · Zbl 0186.40901
[15] Arino, J.; McCluskey, C.C.; van den Driessche, P., Global results for an epidemic model with vaccination that exhibits backward bifurcations, SIAM J appl math, 64, 260-276, (2003) · Zbl 1034.92025
[16] Li, M.Y.; Muldowney, J.S., On R.A. smith’s autonomous convergence theorem, Rocky mountain J. math., 25, 365-379, (1995) · Zbl 0841.34052
[17] Li, M.Y.; Muldowney, J.S., A geometric approach to globle stability problems, SIAM J math anal, 27, 1070-1083, (1996) · Zbl 0873.34041
[18] Li, M.Y.; Muldowney, J.S., On bendixson’s criterion, J differ equat, 106, 27-39, (1993) · Zbl 0786.34033
[19] Muldowney, J.S., Compound matrices and ordinary differential equations, Rocky mount J math, 20, 857-872, (1990) · Zbl 0725.34049
[20] Li, M.Y.; Smith, H.L.; Wang, L., Global dynamics of an SEIR epidemic model with vertical transmission, SIAM J appl math, 62, 58-69, (2001) · Zbl 0991.92029
[21] Li, M.Y.; Muldowney, J.S., Global stability for the SEIR model in epidemiology, Math biosci, 125, 155-164, (1995) · Zbl 0821.92022
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.