Analysis On Within-host Dynamical Models Of Tuberculosis And Prostate Cancer

Using mathematical models to study the effects of immunity and treatment on the occurrence and development of diseases can provide intuitive understanding of the prevention and control of diseases.In this thesis,several mathematical models are established to study the different infection processes in tuberculosis and the strategy selection problem in the therapy of prostate cancer.And treatment strategies of diseases are explored through the study of their dynamical behaviors.In the second chapter,by considering the acceleration effect of T cells on their own recruitment,a four dimensional within-host model of Mycobacterium tuberculosis is established to study the causes of different processes in tuberculosis infection.Firstly,the threshold for Mycobacterium tuberculosis and immune response is obtained.Then the existence and stability of equilibria are studied.We obtain that reducing the extracellular proliferation of Mycobacterium tuberculosis and improving the bactericidal ability of macrophages are key factors to eliminate Mycobacterium tuberculosis.And analytical conditions for the existence of focus and elliptic type of nilpotent singularities with codimension 3 are obtained by bifurcation analysis.Finally,complex dynamical behaviors such as homoclinic loop,saddle-node bifurcation of limit cycle,slow-fast periodic solution with large-amplitude or small-amplitude and co-existence of two limit cycles are revealed in numerical simulations.Especially,the slow-fast periodic solution provides a perfect explanation for the rapid or slow reactivation of latent infection.In the third and fourth chapters,for the selection of continuous and intermittent hormone deprivation treatment in prostate cancer,mathematical models with single or multiple drug inhibition effects are considered respectively,and the corresponding personalised treatment regimens are explored.In the third chapter,by considering the incomplete inhibition effects of drugs on serum androgen production and the pharcodynamics of drugs,two models with generalized proliferation and apoptosis functions are built for the early continuous and intermittent treatment.By carrying out qualitative analysis,we obtain that crossing bifurcation occurs in the intermittent treatment system,and a global stable periodic solution appears.Then we compare the efficacy of continuous and intermittent treatment and obtain the generalized results on the selection of treatment strategy.We further generalize the results for the models with specific proliferation and apoptosis functions.It shows that the optimal treatment strategy may be different due to different proliferation and apoptosis functions.Finally,numerical results further show the optimal intermittent therapy.In the last chapter,based on the comprehensive blocking regimen of hormone production and intracellular androgens receptors proposed in clinical androgen deprivation therapy,we built models of the early continuous and intermittent therapy.Firstly,we obtain that a global stable periodic solution appears in the intermittent treatment system.Then by comparing the efficacy of continuous and intermittent treatment,we obtain some results of strategy selection between intermittent and continuous therapy,and find that the comprehensive inhibition effect of drugs is more beneficial for the control of continuous treatment effect.Numerical results are conducted to illustrate our conclusions and show that enhancing the inhibition effect on the binding of androgen and intracellular receptors can avoid the "flare" phenomenon in the treatment of prostate cancer.