With Nonlinear Incidence Of Infectious Disease Model Branch Of Study

Recently, the epidemic models with non-linear incidence rate are concerned vastly. Inthis paper, we mainly investigate the stability of the equilibria and the phenomenon ofbifurcation of the epidemic models with non-linear incidence rate .In chapter 2, the stability of the equilibria of the epidemic model with non-linear in-cidence rate is researched. We derive the conditions for the stability of the disease-freeequilibrium and the endemic equilibrium in detail. Furthermore, we simulate the stabilityof equilibria. Our results show that the disease-free equilibrium is global asymptotic stablewhen the basic reproduction number is less than one and the endemic equilibrium is globalasymptotic stable when the basic reproduction number is greater than one.In chapter 3, the stability of the equilibria and the limit cycle of the epidemic modelwith non-linear incidence rate is researched. By calculating, the first Lyapunov coe?cient isless than zero. Therefore , the limit cycle is stable. Furthermore, we simulate the stabilityof the limit cycle. Our results show that the limit cycle is stable.In chapter 4, the stability of the limit cycle of the epidemic model with non-linearincidence rate and the pattern structures of the spatially extended model are researched. Bycalculating, the first Lyapunov coefficient is greater than zero. Therefore , the limit cycle isunstable. We derive the conditions for Turing instability of the spatially extended model indetail. Furthermore, we simulate the stability of the limit cycle and the pattern structures.Our results show that the limit cycle is unstable and the stripe and spot patterns emergemixed in distribution of the infected population density.