| Melanoma is a skin cancer with high mortality rate,which seriously affects people’s health and safety.Medical research shows that the number of CD8 + T cells in melanoma patients directly determines the development of melanoma.Therefore,the number of CD8 + T cells in the body can be maintained at a higher level by increasing the activation rate of antigen presenting cells to CD8 + T cells and reducing the apoptosis rate of CD8 + T cells,thus inhibiting the further proliferation of melanoma.Therefore,a kind of melanoma model with immune response is studied.First,the nonnegativity and boundedness of the solution of the system are proved by the ordinary differential equations comparison theorem.It is found that the zero equilibrium point of the system is always unstable and there is no boundary equilibrium point.Then,it is proved that there may be one,two or three positive equilibrium points in the system by combining algebra method and geometric method.By studying the distribution of the root of the characteristic equation of the positive equilibrium point of the system,the stability condition of the positive equilibrium point is given.At the same time,it is proved that the system exists saddle-node bifurcation,and the sufficient condition for Hopf bifurcation is obtained at the positive equilibrium point.By using the center manifold theorem and the normal form theory,the reduced equation of the system limited to the center manifold is calculated,and the properties of Hopf bifurcation are obtained.Second,the effects of apoptosis rate of CD8 + T cells and the semi-saturation constant on immunotherapy of melanoma patients are studied.It is proved that Bogdanov-Takens bifurcation and Bautin bifurcation will occur in the system with the two parameters as bifurcation parameters.The normal forms of Bogdanov-Takens bifurcation and Bautin bifurcation are obtained by calculation,and the expression of homoclinic bifurcation is given.Finally,the practical parameters are selected for numerical simulation,so that the theoretical results are explained.By fixing parameters,the bifurcation diagram of system in two-parameters plane is obtained.Then,the stable periodic solution generated by Hopf bifurcation is simulated with fixed the semi-saturation constant on immunotherapy of melanoma and changing the apoptosis rate of CD8 + T cells.With the appearance of homoclinic orbit of Bogdanov-Takens bifurcation,the periodic solution will disappear.At the same time,it can be seen from the numerical simulation that when there is a positive equilibrium point,the positive equilibrium point is globally stable.When there are three positive equilibrium points,two are stable and one is unstable.The parameters are taken near Bautin bifurcation point to simulate the different periodic solutions and asymptotic stability phenomenons of the system.By simulating the immune response process of melanoma patients in different periods,we can get that the activation rate of CD8 + T cells can be increased by appropriate immunotherapy and the apoptosis rate of CD8 + T cells is reduced.Different dynamic phenomena will appear in the system.It is found that the development of the final condition of melanoma patients is mainly related to the number of melanoma cells and the activation rate of CD8 + T cells. |
