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Zbl 1028.34046
Ruan, Shigui; Wang, Wendi
Dynamical behavior of an epidemic model with a nonlinear incidence rate. (English)
[J] J. Differ. Equations 188, No.1, 135-163 (2003). ISSN 0022-0396

Here, an SIRS model of disease spread with nonlinear incidence rate is studied, assuming that the total population is constant ($S+I+R=N_0>0$). The rescaled system $${{dI}\over{dt}} = {{I^2}\over{1 + p I^2}} (A-I-R) - mI,\quad {{dR}\over{dt}} = q I - R,$$ is analyzed by means of qualitative analysis. The authors provide conditions under which the disease can persist (with one or two positive equilibria apart from the origin) or vanishes. Moreover, they give conditions for stability of the equilibria and existence and stability of periodic orbits. They show that for some parameters at least two periodic orbits appear, and examine cases under which the system undergoes Bogdanov-Takens bifurcation, supercritical and subcritical Hopf bifurcations, and homoclinic bifurcation.
[David S.Boukal (České Budějovice)]

MSC 2000:: *34C60 Applications of qualitative theory of ODE
92D30 Epidemiology
34D05 Asymptotic stability of ODE
34C23 Bifurcation (periodic solutions)

Keywords: epidemiology; nonlinear incidence; global analysis; bifurcation; limit cycle

Cited in: Zbl 1171.34033

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