Abstract
In this paper a class of SIRS epidemic dynamical models with nonlinear incidence rate \(\beta f(S)g(I)\), vaccination in susceptible and varying population size is studied. The positivity and boundedness of solutions and the existence of equilibria are obtained. By using the linearization method and the theory of persistence in dynamical systems, the local stability of equilibria and the permanence of the models are further obtained. By constructing new Lyapunov functions, the global stability of the equilibria for the models also is established. That is, under some additional assumptions for functions f(S) and g(I), the disease-free equilibrium is globally asymptotically stable if basic reproduction number \(\mathcal {R}_0\le 1\), and the endemic equilibrium is globally asymptotically stable if \(\mathcal {R}_0>1\). The numerical simulations show that, even if these additional assumptions do not hold, the global stability of the disease-free equilibrium and endemic equilibrium for the model may be completely determined only by basic reproduction number \(\mathcal {R}_0\).
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Acknowledgments
This work is supported by the National Natural Science Foundation of China (Grant Nos. 11271312, 11261056), the China Postdoctoral Science Foundation (Grant Nos. 20110491750, 2012T50836) and the Natural Science Foundation of Xinjiang (Grant Nos. 2011211B08).
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Communicated by Ahmad Izani Md Ismail.
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Tang, Q., Teng, Z. & Abdurahman, X. A New Lyapunov Function for SIRS Epidemic Models. Bull. Malays. Math. Sci. Soc. 40, 237–258 (2017). https://doi.org/10.1007/s40840-016-0315-5
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DOI: https://doi.org/10.1007/s40840-016-0315-5