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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 $R_0\ge 1$, 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.

MSC:
92D30Epidemiology
34D23Global stability of ODE
34D20Stability of ODE
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References:
[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) · 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)
[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) · Zbl 0154.09301
[20] Jr., R. H. Martin: 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