The Applied Research Of Mixed Control Strategy On Epidemic Dynamics

Epidemic dynamics as an important part of biomathematics is attracting wide attention of many researchers.In this dissertation,we study several epidemic models and its local and global dynamic behaviors by means of reproduction number theory,RouthHurwitz criterion,limiting system theory,Lyapunov function,second compound matrix,Floquet theory,Kamke theorem and so on.Analyze the transmission process,forecast the trend of disease and the corresponding control strategy are given.With the development of medical conditions,the measures we can use to control diseases are becoming increasingly diverse.The combination of different control measures can provide a faster and more effective way to eradicate the infectious diseases.So the mixed control strategy is becoming the main direction of development for disease control.By applying Pontryagin maximum principle,Runge-Kutta method and CMSC method to numerical simulations,the mixed control strategy under different parameters is shown.Our study has wide applications,and it can offer the reference for infectious disease prevention and control in the future.The arrangement of the dissertation is as follows.In Chapter 1,a brief introduction is given for the background and basic method for research of mathematical modeling on infectious diseases.Some preliminaries are presented and a survey of the results in this thesis is also given.In Chapter 2,we study the TB trends mainly focused on the full coverage of DOTS strategy in China in the recent two decades.The mathematical model to explore the impact of control strategy on the transmission dynamics of tuberculosis is formulated.Combined qualitative theory,limit thought and comparison method,theoretical analysis for the epidemic model are given.Based on the data reported by National Bureau of Statistics of China,the control reproduction number of each stage is estimated and compared.We study the efforts of DOTS strategy through the basic reproduction number of the two stages.The full coverage of DOTS strategy has reduced the level of TB obviously,but the disease still exists(R0> 1).Then we discuss further control which is focused on development of new vaccine and improvement of treatment.Optimal control problem for minimizing the total number of infectious individuals with the lowest cost has been proposed and analyzed by Pontryagin's maximum principle.Numerical simulations are provided to illustrate the theoretical results and give an intuitionistic comparison between the optimal control strategy and the current control strategy.Our study helps to design further control strategy for eradicating the disease in practice.In Chapter 3,according to the End TB strategy proposed by WHO,it is well known that the disease should be controlled well in limited time.So we make some modifications for the model.The standard incident rate is used to describe the transmission and estimate the parameters.To reach the new End TB goal raised by WHO in 2015 and considering the health system in China,we design a mixed vaccination strategy.Based on the Floquet theorem and the normalized method,the threshold condition for eradicating the disease and the maximum impulsive period are given.Theoretical analysis indicates that the infectious population asymptotically tends to zero with the new vaccination strategy which is the combination of constant vaccination and pulse vaccination.We obtain that the control of TB is quicker to be achieved with the mixed vaccination.The new strategy can make the best of current constant vaccination,and the periodic routine health examination provides an operable environment for implementing pulse vaccination in China.Numerical simulations are provided to illustrate the theoretical results and help to design the final mixed vaccination strategy once the new vaccine comes out.In Chapter 4,we formulate an SEIRV S epidemic model with different vaccination strategies to investigate the elimination of the chronic disease.Theoretical analysis and threshold conditions for eradicating the disease are given.The local(R0(T)< 1)and global(R1(T)< 1)asymptotically stable conditions for the elimination of the disease are studied theoretically.Using the comparison method and Kamke theorem,we give a more concise and clearer proof of the stability of infection-free periodic solution.Then we propose an optimal control problem,due to the discontinuity caused by pulse,the optimal control problem is difficult to handle.We transform the optimal control problem into nonlinear program and solve the optimal scheduling of the mixed vaccination strategy through the combined multiple shooting and collocation(CMSC)method.Numerical simulations show that the exposed and infected populations decrease more rapidly inoptimal mixed vaccination strategy than in optimal constant vaccination strategy.It means that we can eradicate the disease in a shorter time under optimal mixed vaccination strategy.It helps greatly to avoid more complex problems like antimicrobial resistance caused by longer treatment period.If the vaccinated population is limited,our optimal scheduling can provide a balanced strategy which the constant vaccination maintains in a moderate level and the number of pulse reaches the maximal boundary while the quantity decreases progressively.Theoretical results and numerical simulations can help to design the final mixed vaccination strategy for the optimal control of the chronic disease once the new vaccine comes into use.Finally,we conclude the dissertation with a summary of the main results and establish the goal and direction of research in the future.