GLOBAL DYNAMICS AND BIFURCATIONS IN A SIRS EPIDEMIC MODEL WITH A NONMONOTONE INCIDENCE RATE AND A PIECEWISE-SMOOTH TREATMENT RATE

摘要

In this paper, we analyze a SIRS epidemic model with a nonmonotone incidence rate and a piecewise-smooth treatment rate. The nonmonotone incidence rate describes the "psychological effect": when the number of the infected individuals (denoted by I) exceeds a certain level, the incidence rate is a decreasing function with respect to I. The piecewise-smooth treatment rate describes the situation where the community has limited medical resources, treatment rises linearly with I until the treatment capacity is reached, after which constant treatment (i.e., the maximum treatment) is taken. Our analysis indicates that there exists a critical value (I) over tilde (0) (= b/d) for the infective level I-0 at which the health care system reaches its capacity such that: (i) When I-0 >= (I) over tilde (0), the transmission dynamics of the model is determined by the basic reproduction number R-0: R-0 = 1 separates disease persistence from disease eradication. (ii) When I-0 < <(I)over tilde>(0) bifurcations, such as multiple endemic equilibria, periodic orbits, homoclinic orbits, Bogdanov-Takens bifurcations, and subcritical Hopf bifurcation.

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