Abstract
A detailed analysis of a general class of SIRS epidemic models is given. Sufficient conditions are derived which guarantee the global stability of the endemic equilibrium solution. Further conditions are found which ensure instability for the equilibrium. Finally, the dependence of the stability on the contact number and the ratio of the mean length of infection to the mean removed time is considered.
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Grossman, S. I., Miller, R. K.: Perturbation theory for Volterra integrodifferential systems. J. Diff. Equations 8, 457–474 (1971)
Grossman, S. I., Miller, R. K.: Nonlinear Volterra integrodifferential systems with L 1-kernels. J. Diff. Equations 13, 551–556 (1973)
Hethcote, H. W.: Qualitative analyses of communicable disease models. Math. Biosci. 28, 335 -356 (1976)
Hethcote, H. W., Stech, H. W., van den Driessche, P.: Nonlinear oscillations in epidemic models. SIAM J. Appl. Math., in press (1981)
Hille, E., Phillips, R. S.: Functional analysis and semi-groups. Amer. Math. Soc. Colloq. Publ. Vol. 31, Providence, R.I., 1957
Londen, S. O.: On the variation of the solutions of a nonlinear integral equation. J. Math. Anal. Appl. 52, 430–449 (1975)
Miller, R. K., Nohel, J. A.: A stable manifold theorem for a system of Volterra integrodifferential equations. SIAM J. Math. Anal. 6, 506–522 (1975)
Stech, H. W.: The effect of time lags on the stability of the equilibrium state of a population growth equation. J. Math. Biol. 5, 115–130 (1978)
Stein, E. M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton Univ. Press, 1971
Wang, F. J. S.: Perturbation and maximum number of infectives of some SIR epidemic models. SIAM J. Math. Anal. 10, 721–727 (1979)
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Stech, H., Williams, M. Stability in a class of cyclic epidemic models with delay. J. Math. Biology 11, 95–103 (1981). https://doi.org/10.1007/BF00275827
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DOI: https://doi.org/10.1007/BF00275827