Dynamic Analysis And Optimal Control Of Several Infectious Disease Models Based On Complex Networks

In the 21 st century,infectious diseases are still an important problem threatening the safety of the whole human being.It is significant to study the transmission mechanism of infectious diseases for formulating its control strategy.The dynamics of infectious disease is a subject that studies the mechanism of infectious disease transmission theoretically by establishing mathematical models.The infectious disease model based on complex network takes the heterogeneity of contact into account,which is closer to reality than the traditional infectious disease model.In this paper,we use the mean field theory of complex network to establish model of complex network based on degree distribution for the spread of infectious diseases,and study the global dynamics,including finding the feasible region of model solution,calculating the basic reproduction number,discussing the local and global asymptotic stability of disease-free equilibrium and endemic equilibrium,and simulating the model numerically.The main contents of this paper are as follows:(1)Asymptomatic infectors can transmit diseases,but it is difficult to find them.A model is established on complex networks to describe the transmission of infectious diseases with asymptomatic infectors,and the global behavior of the dynamics is studied.We discuss the feasible region of the solution of the model,use the next generation matrix method to solve the basic reproduction number,and give the conditions for the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium.The expression of basic reproduction number indicates that asymptomatic patients will reduce the threshold in the process of disease transmission.Finally,the numerical simulation also shows that the role of asymptomatic infection in the spread of network diseases cannot be ignored.(2)In view of the fact that seasonality and the regularity of contact patterns will produce periodic dynamics of infectious diseases,we establish a model with periodic infection rate on complex network to study the synergistic effects of periodic transmission and contact heterogeneity on the infection threshold and dynamics of seasonal diseases.We discuss the feasible region of the solution of the model,give the formula of the basic reproduction number,and obtain the conditions of the global asymptotic stability of the disease-free equilibrium and the existence and global asymptotic stability of the endemic periodic solution.Finally,the numerical simulation shows that the contact heterogeneity plays an important role in accelerating the spread of disease and increasing the amplitude of periodic steady-state solution.These results suggest that we need to pay attention to the factors that lead to the cyclical patterns and contact patterns of seasonal diseases when formulating policies to control the epidemic.(3)For some diseases,such as herpes and tuberculosis,there are incomplete recovery and relapse in the evolution process of infected people.We establish an infectious disease model with incomplete recovery and relapse on the complex network,study the global dynamics and the impact of incomplete recovery and relapse.We discuss the feasible region of the solution of the model,and use the next generation matrix method to solve the expression of the basic reproduction number,and analyze the stability of the equilibria of the model.In addition,considering the vaccination control strategy,the vaccination optimal control problem is established and solved.Finally,the theoretical results are simulated.The results showed that relapse would reduce the threshold of disease transmission and affect the effect of vaccination.(4)Considering vaccination,isolation and treatment,we establish a set of time-varying optimal control problems on complex network.We discuss the existence of optimal control solve the optimal control problems.Through a series of comparisons,we analyze the advantages and disadvantages of different strategies.The results show that the effect of short-term vaccination was weak,but constant coefficient vaccination is very helpful.(5)Aiming at the spread of covid-19,we establish an infectious disease model and discuss the global dynamics.We discuss the feasible region of the solution of the model,use the next generation matrix method to solve the expression of the basic reproduction number,and obtain the conditions for the global asymptotic stability of the disease-free equilibrium and the endemic equilibrium.Finally,the model is extended to complex networks and the global stability of disease-free equilibrium is studied.