| The epidemic pattern is an important part of the dynamics of epidemics and one of the main methods to study the spread of epidemics.The pattern structure directly reflects the infection distribution of epidemics in space,and plays an important role in predicting the spread trend and speed of epidemics.In the real world,the spread of epidemics is influenced by the distribution of populations and how closely they are connected.The population is not continuously distributed in the space,but discrete distribution in each area,so the network epidemic pattern has come into being.Reaction-diffusion system is an important tool to study epidemic pattern,but its research is mainly based on continuous space,so the establishment of the reaction-diffusion system on the network to obtain epidemic pattern has more practical significance.At present,the theoretical research on the reaction-diffusion system of epidemics in continuous space has been very mature.However,the average degree of the network,the network structure,and the ultra-high dimensionality caused by the number of network nodes pose many challenges for the study of complex network reaction-diffusion systems,resulting in less research on network epidemics.In this paper,relevant research is carried out on the problems of network average degree,network structure and pattern control faced by complex network reaction-diffusion systems,and the network pattern and pattern control of a class of SI epidemic models on complex networks are analyzed and simulated.The main research content of this paper is as follows:(1)Aiming at the problem of how the mortality due to disease,the average degree of network and the network structure affect the network pattern,a pattern simulation algorithm for SI network epidemic models has been proposed,the theoretical analysis and simulation experiment of the pattern dynamics of the network reaction-diffusion system are provided.First,a continuous space SI epidemic model is extended to the network,and the self-diffusion coefficient and Laplace matrix are introduced to establish a network reaction-diffusion system;Secondly,the theory of pattern dynamics is used to analyze the Turing stability and deduce the Turing instability conditions.Finally,the network reaction-diffusion system is combined with the pattern simulation algorithm to conduct experiments on different networks,discussing the generated Turing pattern,revealing the influence of mortality due to disease,network average degree and network structure on the network pattern.(2)Aiming at the optimal control problem of how to achieve the best control effect while reducing the control cost,a variable step gradient algorithm for pattern control of epidemic models in SI networks has been proposed,the optimal control theory analysis and optimal control experiment of the network reaction-diffusion system are provided.First,the transmission rate and mortality due to disease have been determined as the control parameters,so as to set the objective function.Secondly,the Lagrange function is constructed,and the first-order optimality condition is derived by combining the optimal control theory,and the adjoint equation and the objective gradient function are obtained.Finally,the network reaction-diffusion system is combined with the variable step-size gradient algorithm to conduct experiments,and the control effect and performance of the variable step-size gradient algorithm are evaluated by comparing and analyzing the target pattern and the controlled pattern.(3)Based on the above-mentioned pattern simulation algorithm and variable step-size gradient algorithm,a space-time patterns evolution and analysis system for complex network epidemics was designed and implemented.The research results obtained in this paper not only enrich the research content of Turing pattern and optimal control of network reaction-diffusion systems,but also provide theoretical support and application value for epidemic prevention and control. |
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