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Dynamical behavior of an epidemic model with a nonlinear incidence rate. (English) Zbl 1028.34046
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.

##### MSC:
 34C60 Qualitative investigation and simulation of ordinary differential equation models 92D30 Epidemiology 34D05 Asymptotic properties of solutions to ordinary differential equations 34C23 Bifurcation theory for ordinary differential equations
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