Dynamic Modeling Of Spatio-temporal Transmission Of Infectious Diseases And Its Nonlinear Pattern Characteristics

All the great plagues in human history have caused large-scale spatial outbreaks,such as black death,smallpox,measles,Ebola,swine fever,avian influenza,SARS,COVID-19,etc.Due to the limitation of medical resources,the imbalance of economic development and the delay of prevention and control measures,the infected individuals will move in a large range in space,that is,the so-called non-local movement.Nowadays,it has become a hot topic to study how non-local movement affects the spread of infectious diseases and the spatial structure of infectious diseases.The nonlocal movement of infected individuals during the transmission of infectious diseases can be accurately characterized by the nonlocal delay reaction-diffusion equation.At present,researches of infectious disease model based on nonlocal delay reaction-diffusion equation mainly focus on traveling wave solution,global solution,wave front solution,wave series solution,minimum wave velocity and so on.Yet studies on pattern dynamics are still in the early stage.The local instability of uniform steady-state in nonlinear system leads to the formation of pattern.Specifically,infectious disease pattern is the spatio-temporal evolution behavior of nonlinear system.Phase transition can lead to the breaking of spatial symmetry or temporal symmetry.Nonlinear analysis method is an important tool to study the nonlinear characteristics of pattern.Based on the fact that the dynamics of infectious disease pattern can characterize the specific characteristics of infectious diseases in different spatial and temporal distribution,and describe the outbreak and extinction of infectious diseases,it is of theoretical and practical value to study the pattern dynamics of epidemic model with nonlocal delay.In this paper,two kinds of reaction-diffusion equation epidemic models with nonlocal delay and a class of infectious disease models with both nonlocal delay and chemotaxis are constructed in theory.And,the effects of the nonlocal and chemotaxis on the infectious disease pattern were studied in detail.The early dynamic of COVID-19 in Wuhan was studied by using the reaction diffusion equation.The specific research work is organized as follows.(1)Based on the fact that the nonlocal movement of infected individuals will lead to the competition of medical resources among the infected,a class of SIS infectious disease model with non-local delay is established.The approximate system near the endemic equilibrium point is obtained by using the linearization analysis theory,and the necessary conditions for the generation of Turing pattern are obtained by using Turing instability theory.Numerical simulation verifies the conclusion of Turing pattern analysis,and shows that delay not only inhibits the spread of infectious diseases,but also has a great influence on the spatial steady-state pattern.The spatial average density of the infected patients will decrease with the increase of delay,and the width of strip pattern will widen with the increase of delay;but delay increases to a certain value,the strip pattern will become mixed pattern.(2)When number of the population is large,the infection rate follows the standard incidence,then a spatio-temporal infectious disease model with standard nonlocal incidence and logistic growth is established.By using linear analysis and Turing instability theory,Turing space is determined by a series of inequalities.Nonlinear analysis method(multi-scale analysis)is used to derive the amplitude equation,and different steady-state solutions of the amplitude equation correspond to the steady-state patterns of different structures.Moreover,rich pattern structures are obtained through the change of control parameters(delay).Numerical simulation shows that the isolation degree of pattern increases with the increase of delay,and the density of the infected decreases with the increase of delay.Above all,delay prevents infectious disease from spreading in space.(3)Taking into account the fact that susceptible individuals actively move away from locations of high concentration of infected individuals and the nonlocal movement of infected individuals,a class of SI epidemic model with nonlocal delay and chemotaxis effect is established.And the linearized system with cross diffusion is obtained by linearization analysis.Using the Turing instability theory and selecting the chemotaxis as the control parameter,the constraint conditions of Turing pattern are obtained and the exact Turing space is found.The numerical simulation shows that there is a threshold of chemotaxis?~*,when?>?~*,infectious diseases will eventually die out,and the time of extinction of infectious disease will shorten with the increase of chemotaxis coefficient;when?>?~*,infectious diseases have a regular,heterogeneous macroscopic structure in space,and spatial density of infected individuals increases with the decrease of chemotactic coefficient,meanwhile,the spatial distribution of infected individuals presents a trend from sparse to dense.(4)Based on the early transmission of COVID-19 in Wuhan,a SEIR spatial infectious disease model was established.Combined with the published data,the least square method is proposed to solve the parameters of the reaction-diffusion equation,and parameters of the infection rate are estimated,and the optimal values of model parameters are obtained.The spatio-temporal transmission of infected and exposed individuals was numerically simulated,which of results show that infectious diseases will develop from single point outbreak to multi-point infection,and the number of the infected in fixed spatial location will increase with diffusing.At the same time,the sensitivity analysis of diffusion rate and infection rate was carried out.The results showed that the number of cumulative infected individuals increased with the increase of infection rate and diffusion rate,and the sensitivity to infection rate was greater than that to diffusion rate.According to the research results,effective prevention and control measures are proposed to cope with the outbreak of infectious diseases.For example,early detection and early isolation,home isolation measures,increasing social distance and wearing masks can all effectively reduce the number of infected group.