Analysis of an SIR model with bilinear incidence rate

An SIR vector disease model with incubation time is studied under the assumption that the susceptible of host population satisfies the logistic equation and the incidence rate is the simple mass action incidence. Threshold quantity ℜ0ℜ0 is derived which determines whether the disease dies out or remains endemic. If ℜ0<1ℜ0<1, the disease-free equilibrium is globally asymptotically stable and the disease eventually disappears. If ℜ0>1ℜ0>1, there will be an endemic and the disease is permanent if it initially exists. Using the time delay (i.e., incubation time) as a bifurcation parameter, the local stability of the endemic equilibrium is investigated, and the conditions for Hopf bifurcation to occur are derived. Numerical simulations are presented to illustrate our main results.