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Global stability of an SEIS epidemic model with recruitment and a varying total population size. (English) Zbl 1005.92030
Summary: This paper considers an SEIS epidemic model that incorporates constant recruitment, disease-caused death and disease latency. The incidence term is of the bilinear mass-action form. It is shown that the global dynamics is completely determined by the basic reproduction number \(R_0\). If \(R_0\geq 1\), the disease-free equilibrium is globally stable and the disease dies out. If \(R_0>1\), a unique endemic equilibrium is globally stable in the interior of the feasible region and the disease persists at the endemic equilibrium.

92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
34D20 Stability of solutions to ordinary differential equations
Full Text: DOI
[1] Anderson, R.M.; May, R.M., Population biology of infectious diseases: part 1, Nature, 280, 361, (1979)
[2] Hethcote, H.W., Qualitative analysis of communicable disease models, Math. biosci., 28, 335, (1976) · Zbl 0326.92017
[3] Li, M.Y.; Muldowney, J.S., A geometric approach to global-stability problems, SIAM J. math. anal., 27, 1070, (1996) · Zbl 0873.34041
[4] Greenhalgh, D., Hopf bifurcation in epidemic models with a latent period and nonpermanent immunity, Math. comput. modeling, 25, 85, (1997) · Zbl 0877.92023
[5] H.W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000) 5999 · Zbl 0993.92033
[6] L. Genik, P. van den Driessche, A model for diseases without immunity in a variable size population, in: Geoffrey J. Butler Memorial Conference in Differential Equations and Mathematical Biology, Edmonton, AB, 1996; Canadian. Appl. Math. Quart. 6 (1998) 5 · Zbl 0940.92024
[7] Gao, L.Q.; Mena-Lorca, J.; Hethcote, H.W., Four SEI endemic models with periodicity and separatrices, Math. biosci., 128, 157, (1995) · Zbl 0834.92021
[8] LaSalle, J.P., The stability of dynamical systems, (1976), SIAM Philadelphia, PA · Zbl 0364.93002
[9] Butler, G.J.; Waltman, P., Persistence in dynamical systems, J. differential equations, 63, 255, (1986) · Zbl 0603.58033
[10] P. Waltman, A brief survey of persistence, in: S. Busenberg, M. Martelli (Eds.), Delay Differential Equations and Dynamical Systems, Springer, New York, 1991, p. 31 · Zbl 0756.34054
[11] Freedman, H.I.; Tang, M.X.; Ruan, S.G., Uniform persistence and flows near a closed positively invariant set, J. dynam. diff. equat., 6, 583, (1994) · Zbl 0811.34033
[12] Li, M.Y.; Graef, J.R.; Wang, L.; Karsai, J., Global dynamics of a SEIR model with a varying total population size, Math. biosci., 160, 191, (1999) · Zbl 0974.92029
[13] Smith, R.A., Some applications of Hausdorff dimension inequalities for orinary differential equations, Proc. R. soc. Edinburgh A, 104, 235, (1986)
[14] Li, M.Y.; Muldowney, J.S., On R.A. Smith’s autonomous convergence theorem, Rocky mount. J. math., 25, 365, (1995) · Zbl 0841.34052
[15] Li, M.Y.; Muldowney, J.S., On Bendixson’s criterion, J. different. eq., 106, 27, (1994) · Zbl 0786.34033
[16] Hirsch, M.W., Systems of differential equations that are competitive or cooperative. VI: A local Cr closing lemma for 3-dimensional systems, Ergod. theor. dynam. sys., 11, 443, (1991) · Zbl 0747.34027
[17] Pugh, C.C., An improved closing lemma and a general density theorem, Am. J. math., 89, 1010, (1967) · Zbl 0167.21804
[18] Pugh, C.C.; Robinson, C., The C1 closing lemma including Hamiltonians, Ergod. theor. dynam. sys., 3, 261, (1983) · Zbl 0548.58012
[19] Coppel, W.A., Stability and asymptotic behavior of differential equations, (1965), Heath Boston · Zbl 0154.09301
[20] Martin, R.H., Logarithmic norms and projections applied to linear differential systems, J. math. anal. appl., 45, 432, (1974) · Zbl 0293.34018
[21] Fiedler, M., Additive compound matrices and inequality for eigenvalues of stochastic matrices, Czech math. J., 99, 392, (1974) · Zbl 0345.15013
[22] Muldowney, J.S., Compound matrices and ordinary differential equations, Rocky mount. J. math., 20, 857, (1990) · Zbl 0725.34049
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