×

zbMATH — the first resource for mathematics

Dynamics of SEIS epidemic models with varying population size. (English) Zbl 1145.34028
SEIS models with varying population size are considered. After rescaling, the main system has the following form \[ \left\{ \begin{aligned} \dot{s} = & \gamma i-(\lambda-\alpha)si,\\ \dot{e} = & bi+\alpha ei+\lambda si-\varepsilon e,\\ \dot{i} = & \varepsilon e-(\alpha+\gamma+b)i+\alpha i^2, \end{aligned} \right. \] subject to \(s+e+i=1\). Here \(s,e,i\) denote the proportions of susceptible, exposed, and infected individuals in a population respectively, \(b\) is the natural birth rate of the population, \(\alpha\) is the disease-related death rate, \(\gamma\) is the recovery rate of the infected, \(\lambda\) is the contact rate between the infected and other individuals, and \(\varepsilon\) is the rate at which the exposed become infected. Letting \(\sigma=\lambda/(\alpha+\gamma)\), the authors show that if \(\sigma<1\), the disease-free equilibrium \((1,0,0)\) of the system is globally asymptotically stable, while if \(\sigma>1\), \((1,0,0)\) is unstable and the system is uniformly persistent, meaning that the disease persists. If \(\sigma=1\), bifurcation occurs and leads to “the change of stability”. The authors also consider the system equipped with birth pulse and explore the dynamic complexity of SEIS epidemic models with varying population size through numerical simulation.

MSC:
34C60 Qualitative investigation and simulation of ordinary differential equation models
34D23 Global stability of solutions to ordinary differential equations
37N25 Dynamical systems in biology
92D30 Epidemiology
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson R. M., Nature 180 pp 361–
[2] DOI: 10.1098/rstb.1981.0005
[3] DOI: 10.1038/332228a0
[4] DOI: 10.1097/00002030-198906000-00001
[5] Anderson R. M., Infectious Disease of Humans, Dynamics and Control (1992)
[6] DOI: 10.1016/S0022-5193(84)80150-2
[7] Bainov D., System with Impulsive Effect: Stability, Theory and Applications (1989)
[8] Bainov D., Impulsive Differential Equations: Periodic Solutions and Applications (1993) · Zbl 0815.34001
[9] Bellenir K., Health Science Series 8, in: Contagious and Non-Contagious Infectious Disease Sourcebook (1996)
[10] Brauer F., J. Math. Biol. 28 pp 451–
[11] Busenberg S. N., Math. Biosci. 101 pp 41–
[12] DOI: 10.1007/978-3-642-75301-5
[13] DOI: 10.1090/S0002-9939-1986-0822433-4
[14] DOI: 10.1016/0022-0396(86)90049-5 · Zbl 0603.58033
[15] DOI: 10.1007/BF00290636
[16] Coppel W. A., Stability and Asymptotic Behavior of Differential Equations (1965) · Zbl 0154.09301
[17] DOI: 10.1016/S0025-5564(00)00067-5 · Zbl 1005.92030
[18] DOI: 10.1007/BF02218848 · Zbl 0811.34033
[19] Gao L. Q., J. Math. Biol. 30 pp 717–
[20] DOI: 10.1016/0025-5564(94)00071-7 · Zbl 0834.92021
[21] DOI: 10.1006/tpbi.1995.1006 · Zbl 0833.92018
[22] Hale J. K., Ordinary Differential Equations (1969) · Zbl 0186.40901
[23] DOI: 10.1016/0025-5564(76)90132-2 · Zbl 0326.92017
[24] Hethcote H. W., Periodicity in Epidemiological Models, in Applied Mathematical Ecology (1989)
[25] DOI: 10.1016/0025-5564(88)90078-8 · Zbl 0727.92025
[26] DOI: 10.1016/0025-5564(88)90031-4 · Zbl 0686.92016
[27] DOI: 10.1137/1.9781611970432
[28] DOI: 10.1006/jdeq.1993.1097 · Zbl 0786.34033
[29] DOI: 10.1137/S0036141094266449 · Zbl 0873.34041
[30] DOI: 10.1016/S0025-5564(99)00030-9 · Zbl 0974.92029
[31] Li M. Y., SIAM J. Appl. Math. 62 pp 58–
[32] DOI: 10.1016/0928-4869(93)90015-I
[33] DOI: 10.1038/261459a0 · Zbl 1369.37088
[34] Mena-Lorca J., J. Math. Biol. 30 pp 693–
[35] DOI: 10.1016/0040-5809(88)90019-6 · Zbl 0639.92012
[36] DOI: 10.2307/2373413 · Zbl 0167.21803
[37] Pugh C. C., Erg. Th. Dyn. Syst. 3 pp 261–
[38] Pugliese A., J. Math. Biol. 28 pp 65–
[39] Rössler O. E., Z. Natürforsch 31 pp 259–
[40] DOI: 10.1007/s002850000070 · Zbl 0977.92032
[41] DOI: 10.1017/S030821050001920X · Zbl 0622.34040
[42] DOI: 10.1016/0025-5564(92)90081-7 · Zbl 0782.92018
[43] DOI: 10.1007/BFb0083477
[44] DOI: 10.1216/rmjm/1181072473 · Zbl 0799.92020
[45] DOI: 10.1007/BF00168799 · Zbl 0823.92027
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.