Research On A Collective Population Model With A Reaction Diffusion Process

Metapopulation model is not only used to describe population reproduction,competition and death,but also describe the spread of disease within and among populations.Compared with the classical compartment model,the metapopulation epidemic dynamics model not only takes account of the evolution process of the subpopulation,but also considers the interaction in the subpopulations.It is obvious that the topology of the network and the rules of individual migration have a significant influence on the spread of epidemic in the population.Therefore,the epidemic dynamics model in metapopulation networks with reaction diffusion process has become a hotspot of research.In order to describe the influence of individual migration in the large scale space on the spread of the disease,the propagation of epidemic on the metapopulation network is studied in this thesis.On the basis of previous studies,the diffusion problem is considered mainly.By considering the two factors of heterogeneous diffusion rate and heterogeneous traffic flow,the differential equation dynamics models were established and analyzed.In the second chapter,considering the heterogeneity of scale and transportation degree of cities,the diffusion rate also has some heterogeneity.The epidemic dynamics model with heterogeneous diffusion rate is established by taking the city as the node,which more accurately described the network structure and the influence of the migration on the spread of disease.Firstly,using the qualitative and stability theory of differential equation,the local stability of the system at the disease-free equilibrium is given,and the threshold condition is calculated.Secondly,by using the matrix theory,it is proved that the system has an unique endemic equilibrium when the threshold condition is satisfied.Moreover,the method of eigenvalues estimation is applied to prove the endemic equilibrium is always locally asymptotically stable.Finally,numerical simulations are given,and the results are consistent with the theoretical.It is found that the population density,connectivity and the heterogeneity of traffic flow are all factors affecting the diffusion rate.And the relationship is obtained among diffusion rate,connectivity of cities and the heterogeneity parameter of traffic flow.In the third chapter,considering the different possibility of individuals choosing each road to migrate,there will be some heterogeneity in traffic flow.According to the degree of city,the epidemic model with heterogeneous traffic flow is established.Firstly,qualitative and quantitative of the system is analyzed in the same way,and the threshold condition for the instability of the disease-free equilibrium is obtained by using the next-generation matrix.Then it is proved that the system has an unique endemic equilibrium,and the endemic equilibrium is always locally asymptotically stable,when the threshold condition is satisfied.Finally,the numerical simulation results are in agreement with the theoretical.The results show that the population density,the degree of traffic development and the heterogeneity of traffic flow will affect the spread of the epidemic.Obviously,cities with better transportation conditions are more likely to be the target cities for population migration.In addition,developed cities have relatively sound medical equipment.Therefore,whether susceptible or infected individuals,the preference for migration to those cities will be higher.In contrast,when an epidemic outbreaks,the risk of disease in big cities will be greater.Therefore,strengthening the control about the disease can be carried out in two aspects: First of all,people should have some awareness about disease prevention.Both infected and susceptible individuals should minimize the frequency of trips to major cities.Secondly,the big cities themselves must focus on preventing diseases,and make a set of relatively reasonable and complete preventive measures and response plans.