Global Stability Of Multi-group Epidemic Dynamical Models

In recent years,research on the dynamics of infectious diseases based on differential equation models has played a positive role in predicting the spread of diseases and analyzing control measures.The research on the(global)stability of the dynamic model of infectious diseases not only has practical guiding significance for disease prevention and control,but also has the value of differential equation theory research.With the deepening of research on the dynamics of infectious diseases,the factors considered in dynamic modeling are more and more complicated.For example,when considering factors such as age,occupation and region,it is often necessary to establish a multi-group infectious disease dynamic model with network relationships.The research on the(global)stability of the high-dimensional coupled differential equation models is a key and difficult problem.When modeling multi-group infectious disease dynamics,the connection between different groups can be abstracted as a kind of network relationship,each same group can be abstracted as a node in the network,and the mutual connection and interaction between different groups can be abstracted as the connecting edges in the network.Li and Shuai(2010)discussed the stability of the high-dimensional coupled epidemic dynamics model under a strongly connected network.This paper further discussed the global stability of the SIR highdimensional coupled epidemic dynamics model on a weakly connected network structure.It mainly includes the following contents.Chapter 1 introduces the research background of multi-group infectious diseases,the research significance of the global stability of differential equations,and the current research status at home and abroad.Chapter 2 summarizes the theoretical knowledge related to the global stability of the multi-group infectious disease dynamic model.Chapter 3 studies the global dynamics of a multi-group SIR epidemic model on a weakly connected network structure.Firstly,the system on the weakly connected network is transformed into an equivalent system coupled between several strongly connected subsystems;according to Li and Shuai(2010),the respective solutions of the strongly connected subsystems can converge to a single equilibrium point(ie,boundary equilibrium point or positive balance point).Secondly,based on the theory of asymptotic autonomous systems,by constructing the Lyapunov function,it is proved that under the condition of a weakly connected network,the solution of the system on the weakly connected network structure may also converge to a mixed equilibrium point(the coordinates are in some subsystems,while other on the border).Finally,taking cholera in Haiti in 2010 as an example,a weakly connected multi-regional infectious disease dynamic model is established and numerical simulations are performed to verify the theoretical findings.Chapter 4 studies the global stability of the SIR epidemic dynamics model with time delays on weakly connected networks.Taking into account that it takes a period of time for susceptible persons to become infectious after being infected,a time lag is introduced to the multi-group SIR infectious disease model.Using a similar method in Chapter 3,the weakly connected network model is transformed into a coupled model between strongly connected subsystems.The global asymptotic stability of the model is proved by constructing the Lyapunov function and the theory of asymptotic autonomous systems.Finally,a numerical simulation is performed to verify the theoretical results.