Applications Of Optimal Control Governed By Dierential Equations In Epidemic And Population Models

Optimal control problem governed by differential equation is one of the most important areas in control theory,applied mathematics and computational science.It is widely used in materials and engineering design,shape design,petroleum exploration,aerospace and so on.With the in-depth research and rapid development of the infectious diseases and population ecological dynamics in the field of biomathematics,the optimal control theory governed by differential equations has gradually penetrated into the study of such models,and our research will also be carried out in this area.In this thesis,we study two types of problems,i.e.,vaccination strategies of an SIR pair approximation model on complex networks and optimal control of pattern formation in a predator-prey system with Allee effect.In the first part,a constant vaccination rate is introduced firstly for an SIR pair approx-imation model with demographic on network.The basic reproduction number and endemic which relate to vaccination rate are studied,and a critical value of vaccination rate is ob-tained.When vaccination level exceeds the critical value,the basic reproduction number is less than 1 and the disease would be die out.Next,we study optimal vaccination strategy with the help of optimal control governed by differential equation,in which economic costs are taken into consideration.We aim to control disease transmission simultaneously to re-duce vaccination costs through optimal strategies.Based on this,we investigate an optimal control problem of the SIR approximation model with demographic.The existence of a so-lution and optimality condition are given.Finally,for the constant vaccination model,we provide stochastic simulation on different initial networks as well as numerical simulation for model equations.For the optimal vaccination model numerical simulations are performed.The second part of this thesis is devoted to study a reaction diffusion predator-prey model with Allee effect.We first discuss the well-posedness of model solution,the stability of constant steady state solution and the limit cycle.On the basis,rich pattern formations of the model are researched.The effects of spatial dimension and initial conditions on population sustainability and pattern formations are investigated by means of numerical simulation.From stability analysis and numerical simulation results,it can be seen that,for a given region and population initial distribution,the population either become extinct or persistence.While in the case of persistence,the population density distribution either take on chaotic character or present regular structure in space and time.Next,we apply optimal control governed by differential equation to study the predator-prey system.This is mainly motivated by the following two considerations.First,if the population becomes extinct eventually with spontaneous development,how to effectively exert control to cause it to persist?Second,if the population can persist with spontaneous development,how to effectively exert control to cause it to distribute regularly?For the control problem governed by predator-prey system,the existence of a solution and optimality condition are obtained.Finally,a numerical algorithm is given.Using this algorithm numerical simulations are provided for the optimal control problems and satisfying results are achieved.