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Global dynamics of a SEIR model with varying total population size. (English) Zbl 0974.92029

Summary: A SEIR model for the transmission of an infectious disease that spreads in a population through direct contact of the hosts is studied. The force of infection is of proportionate mixing type. A threshold \(\sigma\) is identified which determines the outcome of the disease; if \(\sigma \leq 1\), the infected fraction of the population disappears so the disease dies out, while if \(\sigma>1\), the infected fraction persists and a unique endemic equilibrium state is shown, under a mild restriction on the parameters, to be globally asymptotically stable in the interior of the feasible region. Two other threshold parameters \(\sigma'\) and \(\overline\sigma\) are also identified; they determine the dynamics of the population sizes in the cases when the disease dies out and when it is endemic, respectively.

MSC:

92D30 Epidemiology
34D23 Global stability of solutions to ordinary differential equations
37N25 Dynamical systems in biology
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[1] Anderson, R. M.; May, R. M., Population biology of infectious diseases I, Nature, 180, 361 (1979)
[2] May, R. M.; Anderson, R. M., Population biology of infectious diseases II, Nature, 280, 455 (1979)
[3] Busenberg, S. N.; van den Driessche, P., Analysis of a disease transmission model in a population with varying size, J. Math. Biol., 28, 257 (1990) · Zbl 0725.92021
[4] Nold, A., Heterogeneity in disease-transmission modeling, Math. Biosci., 52, 227 (1980) · Zbl 0454.92020
[5] Mena-Lorca, J.; Hethcote, H. W., Dynamic models of infectious diseases as regulator of population sizes, J. Math. Biol., 30, 693 (1992) · Zbl 0748.92012
[6] M.C.M. de Jong, O. Diekmann, H. Heesterbeek, How does transmission of infection depend on population size? in: Denis Mollison (Ed.), Epidemic Models: Their Structure and Relation to Data, Publications of the Newton Institute, vol. 5, Cambridge University, Cambridge, 1995, p. 84; M.C.M. de Jong, O. Diekmann, H. Heesterbeek, How does transmission of infection depend on population size? in: Denis Mollison (Ed.), Epidemic Models: Their Structure and Relation to Data, Publications of the Newton Institute, vol. 5, Cambridge University, Cambridge, 1995, p. 84 · Zbl 0850.92042
[7] Liu, W.-M.; Hethcote, H. W.; Levin, S. A., Dynamical behavior of epidemiological models with non-linear incidence rate, J. Math. Biol., 25, 359 (1987) · Zbl 0621.92014
[8] Greenhalgh, D., Hopf bifurcation in epidemic models with a latent period and non-permanent immunity, Math. Comput. Modelling, 25, 85 (1997) · Zbl 0877.92023
[9] Li, M. Y.; Muldowney, J. S., Global stability for the SEIR model in epidemiology, Math. Biosci., 125, 155 (1995) · Zbl 0821.92022
[10] Greenhalgh, D.; Das, R., Modeling epidemics with variable contact rates, Theoret. Popu. Biol., 47, 129 (1995) · Zbl 0833.92018
[11] Thieme, H., Epidemic and demographic interaction in the spread of potentially fatal diseases in growing populations, Math. Biosci., 111, 99 (1992) · Zbl 0782.92018
[12] Muldowney, J. S., Compound matrices and ordinary differential equations, Rocky Mount. J. Math., 20, 857 (1990) · Zbl 0725.34049
[13] Cook, K. L.; van den Driessche, P., Analysis of an SEIRS epidemic model with two delays, J. Math. Biol., 35, 240 (1996) · Zbl 0865.92019
[14] H.L. Smith, Monotone dynamical systems, an introduction to the theory of competitive and cooperative systems, Am. Math. Soc., Providence (1995); H.L. Smith, Monotone dynamical systems, an introduction to the theory of competitive and cooperative systems, Am. Math. Soc., Providence (1995) · Zbl 0821.34003
[15] Hirsch, M. W., Systems of differential equations which are competitive or cooperative IV: Structural stability in three dimensional systems SIAM, J. Math. Anal., 21, 1225 (1990) · Zbl 0734.34042
[16] Smith, H. L., Periodic orbits of competitive and cooperative systems, J. Different. Eq., 65, 361 (1986) · Zbl 0615.34027
[17] J.P. LaSalle, The stability of dynamical systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976; J.P. LaSalle, The stability of dynamical systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976 · Zbl 0364.93002
[18] Butler, G. J.; Waltman, P., Persistence in dynamical systems, Proc. Am. Math. Soc., 96, 425 (1986) · Zbl 0603.34043
[19] 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; 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
[20] 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
[21] R.A. Usmani, Applied Linear Algebra, Marcel Dekker, New York, 1987; R.A. Usmani, Applied Linear Algebra, Marcel Dekker, New York, 1987 · Zbl 0602.15001
[22] J.K. Hale, Ordinary Differential Equations, Wiley, New York, 1969; J.K. Hale, Ordinary Differential Equations, Wiley, New York, 1969 · Zbl 0186.40901
[23] Martin, R. H., Logarithmic norms and projections applied to linear differential systems, J. Math. Anal. Appl., 45, 432 (1974) · Zbl 0293.34018
[24] L. Markus, Asymptotically autonomous differential systems, Contributions to the Theory of Non-linear Oscillations, vol. 3, Princeton University, Princeton, NJ, 1956, p. 17; L. Markus, Asymptotically autonomous differential systems, Contributions to the Theory of Non-linear Oscillations, vol. 3, Princeton University, Princeton, NJ, 1956, p. 17 · Zbl 0071.08501
[25] Thieme, H., Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30, 755 (1992) · Zbl 0761.34039
[26] M.Y. Li, Geometrical studies on the global asymptotic behaviour of dissipative dynamical systems, PhD thesis, University of Alberta, 1993; M.Y. Li, Geometrical studies on the global asymptotic behaviour of dissipative dynamical systems, PhD thesis, University of Alberta, 1993
[27] H.W. Hethcote, S.A. Levin, Periodicity in epidemiological models, in: L. Gross, S.A. Levin (Eds.), Applied Mathematical Ecology, Springer, New York, 1989, p. 193; H.W. Hethcote, S.A. Levin, Periodicity in epidemiological models, in: L. Gross, S.A. Levin (Eds.), Applied Mathematical Ecology, Springer, New York, 1989, p. 193
[28] H.W. Hethcote, H.W. Stech, P. van den Driessche, Periodicity and stability in epidemic models: A survey, in: K.L. Cook (Ed.), Differential Equations and Applications in Ecology, Epidemics and Population Problems, Academic Press, New York, 1981, p. 65; H.W. Hethcote, H.W. Stech, P. van den Driessche, Periodicity and stability in epidemic models: A survey, in: K.L. Cook (Ed.), Differential Equations and Applications in Ecology, Epidemics and Population Problems, Academic Press, New York, 1981, p. 65 · Zbl 0477.92014
[29] Hethcote, H. W.; van den Driessche, P., Some epidemiological models with non-linear incidence, J. Math. Biol., 29, 271 (1991) · Zbl 0722.92015
[30] Fiedler, M., Additive compound matrices and inequality for eigenvalues of stochastic matrices, Czech. Math. J., 99, 392 (1974) · Zbl 0345.15013
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