The Study Of Numerical Method Based On POD For Diffusion Epidemic Models With Age-structured Populations

In this paper, we include Brownian motion and Poisson jumps into the diffusion epidemic model with age-structured populations which is the basic model of the paper, respectively. Therefore, two kinds of stochastic diffusion epidemic models are established. Meanwhile, A proper orthogonal decomposi-tion(POD) method is applied to a usual finite element(FE) formulation for the three models such that the models are reduced into the reduced FE formations with lower dimensions and highly enough accuracy, respectively. The error estimates of the models between the reduced POD FE solutions and the usual FE solutions are provided. The main contents of the paper are shown as following.1. A proper orthogonal decomposition(POD) method is taken into account in a usual finite elemen-t(FE) formulation for the diffusion epidemic models with age-structured populations. The error estimates between the reduced POD FE solutions and the usual FE solutions are provided according to the Dis-crete Gronwall Lemma and some basic inequalities. The numerical results are consistent with theoretical conclusions and validate the feasibility and efficiency of POD method.2. Brownian motion symbolized as the white noise is considered into the basic model so that a stochastic diffusion epidemic model is established. Moreover we applied the POD method to a usual finite element(FE) formulation and establish the reduced POD FE formations for the numerical solution of the model. Then we solve the error estimates between the reduced POD FE solutions and the usual FE solutions by using Burkholder-Davis-Gundy inequality, Discrete Gronwall Lemma and so on. What’s more, the numerical examples are given to show whether the theoretical conclusions are efficient.3. We take advantage of Poisson jumps to simulate the accidental disasters such as the tsunami, earthquake, debris flow and so forth in mathematics. Then we take it into account in the basic model of the paper to establish another stochastic diffusion epidemic model with Poisson jumps, we introduce a reduced POD FE formations base on the usual FE formulation for solving the numerical results of the newly established model which has lower dimensions and highly enough accuracy. The error estimates between the reduced POD FE solutions of the model are analysed by making use of Burkholder-Davis-Gundy inequality, Discrete Gronwall Lemma and other basic inequalities. At last, the numerical examples validate the feasibility and efficiency of numerical method which is shown in the paper.