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Analysis of a delayed SIR model with nonlinear incidence rate. (English) Zbl 1159.92037
Summary: An SIR epidemic model with incubation time and saturated incidence rate is formulated, where the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. The threshold value $\Re _{0}$ determining whether the disease dies out is found. The results obtained show that the global dynamics are completely determined by the values of the threshold value $\Re _{0}$ and the time delay (i.e., incubation time length). If $\Re _{0}$ is less than one, the disease-free equilibrium is globally asymptotically stable and the disease always dies out, while if it exceeds one it will be endemic. By using the time delay as a bifurcation parameter, the local stability for the endemic equilibrium is investigated, and the conditions with respect to the system to be absolutely stable and conditionally stable are derived. Numerical results demonstrate that the system with time delay exhibits rich complex dynamics, such as quasiperiodic and chaotic patterns.

##### MSC:
 92D30 Epidemiology 34K60 Qualitative investigation and simulation of models 34K20 Stability theory of functional-differential equations 34D23 Global stability of ODE 34D05 Asymptotic stability of ODE
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