zbMATH — the first resource for mathematics

Qualitative analyses of SIS epidemic model with vaccination and varying total population size. (English) Zbl 1045.92039
Summary: An SIS epidemic model with vaccination, temporary immunity, and varying total population size is studied. Three threshold parameters \(R_0\), \(R_1\), and \(R_2\) are identified. The disease-free equilibrium is globally stable if \(R_0 \leq 1\) and unstable if \(R_0 > 1\), the endemic equilibrium is globally stable if \(R_0 > 1\). The disease cannot break out if \(R_1 < 1\), the disease may break out when the fractions of the susceptible and the infectious satisfy some condition if \(R_1 > 1\) and \(R_0 \leq 1\). The population becomes extinct ultimately and the disease always exists in the population if \(R_0 > 1\) and \(R_2 \leq 1\). There is a really endemic disease if \(R_0 > 1\) and \(R_2 > 1\).
The global stability of the disease-free equilibrium and the existence and global stability of the endemic equilibrium are proved by means of LaSalle’s invariance principle, the method of estimating values and Stokes’ theorem, respectively. The results with vaccination and without vaccination are compared, and the measures and effects of vaccination are discussed.

92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
Full Text: DOI
[1] Anderson, R.M.; May, R.M., Population biology of infectious disease: part I, Nature, 280, 361-376, (1979)
[2] May, R.M.; Anderson, R.M., Population biology of infectious disease: part II, Nature, 280, 455-461, (1979)
[3] Hethcote, H.W., Quantitative analyses of communicable disease models, Math. biosci, 28, 335-356, (1976) · Zbl 0326.92017
[4] Hethcote, H.W.; Tudor, D.W., Integral equations models for endemic infectious disease, J. math. biol, 9, 37-47, (1980) · Zbl 0433.92026
[5] Hethcote, H.W.; Stech, H.W.; Van Den Driessche, P., Periodicity and stability in epidemic models: A survey, (), 65-82
[6] Mena-Lorca, J.; Hethcote, H.W., Dynamic models of infectious disease as regulators of population sizes, J. math. biol, 30, 693-716, (1992) · Zbl 0748.92012
[7] Gao, L.Q.; Hethcote, H.W., Disease transmission models with density-dependent demographics, J. math. biol, 30, 717-731, (1992) · Zbl 0774.92018
[8] Zhou, J.; Hethcote, H.W., Population size dependent incidence in models for diseases without immunity, J. math. biol, 32, 809-834, (1994) · Zbl 0823.92027
[9] Greenhalgh, D., Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Mathl. comput. modelling, 25, 2, 85-107, (1997) · Zbl 0877.92023
[10] Greenhalgh, D.; Das, R., Modelling epidemics with variable contact rates, Theoret. popn. biol, 47, 129-179, (1995) · Zbl 0833.92018
[11] Greenhalgh, D., Some results for an SEIR epidemic model with density dependence in the death rate, IMA J. math. appl. med. biol, 9, 67-106, (1992) · Zbl 0805.92025
[12] Benenson, A.S., ()
[13] Lasalle, J.P., The stability of dynamical systems, Regional conference series in applied mathematics, (1976), SIAM Philadelphia, PA · Zbl 0364.93002
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.