Qualitative Research On Several Kinds Of Epidemic Model With Multiple Patches

Recently, the multiple patches model with migration has been widely concerned by experts and scholars at home and abroad. According to the basic idea of compartment modeling and taking into account of the factors, such as isolation of patients, vaccination et al., several kinds of SEIQR epidemic models with population migration are established. The dynamics of the models are investigated by qualitative and numerical analysis and the influence of various factors on the development of the disease is explored. The contents of this paper are mainly arranged as follows.First, assuming that only the susceptible individuals travel between the two patches, an SEIQR epidemic model is formulated. The basic reproduction number R₀ is computed by the method of the next generation matrix. Besides, the existence and global stability of the disease free equilibrium, the boundary equilibrium and the endemic equilibrium are mainly discussed based on the theory of the Lyapunov stability theory, and the existence of the Hopf bifurcation is researched by the bifurcation theory.Second, assuming that the susceptible and the latent individuals or the susceptible and the infected individuals can travel between the two patches and the parameters in the two patches model are the same, the basic reproduction number R₀ of the disease is defined and the existence and global stability of the disease free equilibrium and endemic equilibrium are proved by the theory of Lyapunov stability. Numerical simulations indicate that all the trajectories of the model converge to the endemic equilibrium if R₀>1.Then, assuming that the susceptible, the latent and the infected individuals can travel between the two patches, the basic reproduction number R₀ is calculated by the next generation matrix method, and the existence and global stability of the disease free equilibrium and endemic equilibrium are studied. Numerical results indicate that predicted values of the model coincide with the clinical SARS data of Beijing and Shanxi province released by the ministry of health in 2003.Finally, The model of SEIQR is extended to n patches one and the basic reproduction number can be studied, and the existence and stability of the disease-free equilibrium are investigated.