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Abstract We consider an SIRS epidemic model with a more generalized non-monotone incidence: χ(I)=κIp1+Iq $\chi(I)=\frac{\kappa I^{p}}{1+I^{q}}$ with 0<p<q $0< p< q$, describing the psychological effect of some serious diseases when the number of infective individuals is getting larger. By analyzing the existence and stability of disease-free and endemic equilibrium, we show that the dynamical behaviors of p<1 $p<1$, p=1 $p=1$ and p>1 $p>1$ distinctly vary. On one hand, the number and stability of disease-free and endemic equilibrium are different. On the other hand, when p≤1 $p\leq1$, there do not exist any closed orbits and when p>1 $p>1$, by qualitative and bifurcation analyses, we show that the model undergoes a saddle-node bifurcation, a Hopf bifurcation and a Bogdanov–Takens bifurcation of codimension 2. Besides, for p=2 $p=2$, q=3 $q=3$, we prove that the maximal multiplicity of weak focus is at least 2, which means at least 2 limit cycles can arise from this weak focus. And numerical examples of 1 limit cycle, 2 limit cycles and homoclinic loops are also given.