Mathematical Analysis Of A Dynamical Model Of The Innate Immune Response To Initial Pulmonary Bacterial Infections

Pneumonia is a common disease of respiratory system and detrimental to health,as well as the second most common nosocomial infections of critically ill patients,so diagnose and control pneumonia as early as possible is very important.Based on current knowledge of the biology of pulmonary innate immunity and the known fea?tures of the initial infection exhibited,a dynamical model of the initial pulmonary innate system response to bacteria was proposed by Young and other people(Young et al.Mathematical Bioscienses,235,2012),they mainly considered the situation under the constraint that ?/?>?/?.That is,when the amount of the main effector PMNs and other reactive factors z is not too much,large number of bacteria would break the innate immune system.Under this constraint,they analyzed the type of equilibrium and got a complete picture of the dynamics of the system,the boundary equilibrium was always a stable node,the unique positive equilibrium was a saddle,and there was no limit cycle.But when the amount of 2:is not too much,large number of bacteria might not,break the innate immune system.So in this paper,we will continue analyze this model under the opposite condition of?/?>?/? and study the complex dynamics generated by this model.We have obtained the conclusion that the system undergoes repelling and attracting Bogdanov-Takens bifurcations at different parameter values.That is,the system undergoes saddle-node bifurcation,Hopf bifurcation,and homoclinic bifurcation.Numerical simulations including bifur?cation diagram and phase portraits:as well as the generation process of subcritical and supercritical Hopf bifurcations are shown,and numerical results also show that the system could have stable and unstable limit cycles or homoclinic orbits.These demonstrate that the interaction between the bacteria and the innate immune system is not only very sensitive to the disturbance of parameters,but also depending on the initial concentration of bacteria.Eventually the bacteria will die out,or maybe exist persistently in the form of periodic oscillations or positive states of equilibrium,which is very important for the control of lung infections.