Analysis On Dynamics Of Age-Structured Epidemic Model And Optimal Control

Using mathematical models to study infectious diseases is one of the important directions in the theoretical study of infectious diseases.There are significant differences in the modes of transmission of infectious diseases,Therefore,the establishment of different mathematical models for different infectious diseases is justified in the study of infectious disease models.The rate of transmission of many slowly progressive diseases,such as AIDS,may change with infection-age,the possibility of recovery or death may also depend on the time elapsed since infection.For this reason,many researchers have established mathematical models of age structure to describe these processes.The rapid development of theoretical research on age structure model has benefited from the deepening of relevant theoretical research on infinite dimensional space in recent years.Based on these theories,this paper studies the bifurcation and the global asymptotic stability of equilibrium states of several special infectious diseases,and fits the parameters of the model with actual data,so as to predict the spread of infectious diseases and give feasible control measures.At the same time,we also study the optimal control problem of a class of infectious diseases with life-long immunity under different control strategies.The main works of this paper are as follows:（1）Long-term case studies and pathological studies have shown that HIV virus has very strong robustness in the human body,In chapter 2,the dynamic analysis of the mathematical model is used to verify it.We investigate the Hopf bifurcation of an age-structured HIV model in which susceptible cells undergo mitosis after being stimulated by free HIV.Since the model is established in infinite dimensional space,the Hopf branch theorem in ordinary differential equations is not applicable.We will use the Hopf bifurcation theorem of the Cauchy problem with non-dense definitions recently given by Liu et al.to analyze the dynamics of the model.We rewrite the model as an abstract non-densely defined Cauchy problem and obtain the condition which guarantees the existence of the unique positive steady state.By linearizing the model at steady state and analysing the associated characteristic equations,we study the local asymptotic stability of the steady state.Furthermore,we show that Hopf bifurcation occurs at the positive steady state when bifurcating parameter crosses some critical values.Then,we perform some numerical simulations to illustrate our results.Finally,we find that mitosis is necessary for Hopf bifurcation in the model,and we believe that the transition from stable equilibrium state to stable periodic solution is a manifestation of the high robustness of HIV in the human body.（2）Malaria causes hundreds of millions of infections and hundreds of thousands of deaths worldwide every year.It is one of the infectious diseases that WHO focuses on.In chapter 3,we study the stability of a age-structured malaria model,which incorporates the age of prevention period of susceptible people,the age of latent period of human and the age of latent period of female Anopheles mosquitoes.We prove global existence and uniqueness of solutions,existence of a compact global attractor,and obtain a sufficient condition for uniform persistence of the solution semiflow.We define the basic reproduc-tion number R₀of the model from a biologically relevant perspective and show that R₀completely determines the global dynamics of the model,that is,if R₀<1,the disease-free equilibrium is globally asymptotically stable,if R₀>1,there exists a unique endemic equilibrium that attracts all solutions for which malaria transmission occurs.Finally,we perform some numerical simulations to illustrate our theoretical results.Moreover,we find that the threshold of the malaria model with age structure is smaller by comparison with the ordinary differential model,which indicates that compared with the ordinary differential model,malaria control is less difficult from the perspective of age structure model.（3）Since the possibility of developing symptoms of tuberculosis after infection with tuberculosis changes over time,a large number of studies have shown that the possibility of developing symptoms of tuberculosis gradually diminishes over time.Thus,in chapter3,we establish a age-structured tuberculosis model,the model includes latent age（i.e.,the time elapsed since the individual became infected but not infectious）and relapse age（i.e.,the time between cure and the symptoms returned）.We give an analytic expression for the basic reproduction number R₀,and show that R₀is the dynamic threshold of the model.If R₀<1,the disease-free equilibrium is globally asymptotically stable,which means that tuberculosis will disappear,and if R₀>1,there exists a unique endemic equilibrium that attracts all solutions for which tuberculosis transmission occurs.Based on the tuberculosis data in China from 2007 to 2018,we use GWO algorithm to find the optimal parameter values and initial values of the model.Furthermore,we perform uncertainty and sensitivity analysis to identify the most influential parameters that have significant impact on the basic reproduction number.Finally,we give some effective measures to reach the goal of WHO of reducing the incidence of tuberculosis by 80%by 2030 compared to 2015.Including increase media coverage,public education and extension of mandatory isolation time.（4）Cost control is a very important consideration in the process of infectious disease control.In chapter 5,we investigate optimal control problems for age-structured SIR epi-demic model with vaccination and treatment.By using the Banach contraction mapping principle and the Gronwall^′s lemma,we prove the uniqueness of the nonnegative solution of the age-structured infectious disease model and the continuous dependence of solution on control variables.By the use of tangent-normal cone technique,we give the necessary conditions for the optimal vaccination and treatment strategies under unconditional con-straints.According to Ekeland^′s variational principle,we give sufficient conditions for the existence and uniqueness of optimal vaccination and treatment strategies.Using the Dubovitskii-Milyutin theorem,we study optimal vaccination and treatment with terminal constraints under finite horizon,and give the necessary conditions for optimal vaccination and treatment strategies.