Dynamic Analysis Of A Competition Model Of The Bacteria And Immune System With Switches

In this paper,an immune dynamics model with two switch functions is established,and the dynamic properties of the model are systematically described,including the existence of the model's sliding mode?sliding region?,the existence and stability of true?false?equilibrium and pseudo-equilibrium points.In addition,combined with the related knowledge and theory of Filippov system,the locally sliding boundary bifurcation of the model is discussed.The first chapter mainly introduces the background of system's shell and immune system,the research progress of the right discontinuous model and related theoretical basis.The second chapter assumes that the threshold concentration of pathogenic bacteria B₀₁,controlling the cell membrane gate switch is equal to the threshold value of pathogenic bacteria that elicited immune response B₀₂.A discontinuous model of pathogenic bacteria and immune system competition is established,and the effect of pathogen density threshold on the cell membrane boundary gate and immune response is described.Using the Filippov convex combination theory,the dynamic behavior of the sliding region of the model is studied and the bifurcation phenomenon of the boundary equilibrium point is analyzed.At the same time,the global dynamic behavior of the model is discussed.The results show that the system does not have standard limit cycle,crossing limit cycle,and limit cycle tangenting to or intersecting the sliding mode region.The solution of the system eventually tends to the positive equilibrium of the subsystem or the pseudo-equilibrium point of the system,which indicate that when the bacteria cannot be completely eliminated,the threshold and other parameters can be properly controlled,so that the ratio of bacteria and immune cells is stable to a certain level.The third chapter assumes that the concentration threshold of the bacteria B₀₁,that controls the cell membrane gate switch and the pathogenic threshold that elicited the immune response B₀₂ are different,i.e.B₀₂<B₀₂,B₀₁>B₀₂.Then,our model is developed to a discontinuous one with two switches.The existence and stability of equilibrium points of each subsystem are discussed respectively.In addition,the dynamic behaviors on the two switching surfaces are analyzed using the convex combination theory.Finally,using the software Xppaunt,the theoretical results are verified,and rich dynamic properties are obtained.