Dynamical Analysis Of Epidemic Models In A Patchy Environment

In recent years,with the development of transportation and urbanization,population migration across regions becomes more frequent.The increasing mobility among regions might lead to the spread of the infectious diseases regionally much faster than ever before.The epidemic model with migration has become one of the most popular research fields.The main aim is to disuss the dynamical behaviors of two patchy models.By analysing and summarizing the research status of population models,and according to ordinary differential equations theory and applying stability,and graph theory,we aim to determine the basic reproduction number and the stability of equilibrium points by employing eigenvalue analysis method,Lyapunov function method and Matlab simulations.The dynamic analysis and simulations proved some interesting phenomena and some meaningful results.The paper is organized as follows:1.We discuss two-patch model with vaccination and treatment,and investigate global stability.By using the next general matrix,the basic reproduction number R₀ is obtained.Secondly,we showed that the disease-free equilibrium point of asymmetrical patch existence and it is locally asymptotically stable.At the same time,we proved the model is uniformly persistent.By applying Lyapunov function method and LaSalle invariant principle,the disease-free equilibrium point and the endemic equilibrium point are proved to be globally asymptotically stable.In comparing the size of the four basic reproduction numbers R₀,R_NV,R_NT,R_NVT,it illustrates that R₀ is the least,where as R_NVT is the greatest.It is showed that the simultaneous application of vaccination as well as treatment controls is the most appropriate technique to control the disease.Finally,the simulations are carried out to conform to the migration,vaccination and treatment affect the peak of disease,studied the relationship between the contact rate and basic reproduction number.2.A n-patch epidemic model with vaccination is considered.Firstly,by the next generation matrix,the basic reproduction number R₀ is obtained.Then,by the non-negative matrix and its eigenvalues we showed that the disease-free equilibrium point is locally asymptotically stable whenR₀＜1.When R₀＜1,by applying Lyapunov function method,the endemic equilibrium point is globally asymptotically stable.Finally,numerical simulations verify the conclusions.