Numerical investigation of bifurcations of equilibria and Hopf bifurcations in disease transmission models

One of the general SIRS disease transmission model is considered under the assumptions that the size of the population varies, the incidence rate is nonlinear, and the recovered (removed) class may also be directly reinfected. A combination of analytical and numerical techniques is used to show that (for some parameters) the bifurcations of equilibria can occur and also asymptotically orbitally stable periodic solutions with asymptotic phase can arise through Hopf bifurcations. The investigation is based on computer simulation of bifurcation manifolds in the parameter space. Hopf bifurcations are investigated on the base of center manifold theory by the computation of bifurcation parameters and the approximation of Hopf-bifurcating cycles by bifurcation formulas. This method finds the limit cycle to a good approximation and also its stability. For computer simulations the necessary computer oriented algorithms were developed and encoded by C++. Some results of computer simulations are presented and numerical evidence of existence of bifurcations of equilibria and Hopf bifurcations for the considered model is provided.