Stability And Hopf Bifurcation Analysis Of High-dimensional Infectious Disease System Subjected To Random Excitation Disturbance

Infectious diseases are prevalent all over the world.Although the development of the modern medical science and technology can effectively prevent and control many infectious diseases,but there are still some infectious disease outbreak,so the infectious disease model as a classic mathematical model,has been praised by the majority of scholars.There are many factors affecting the spread of infectious diseases.Indirect factors include differences in geography,climate,age,etc.,all of which may cause different rates of transmission.In addition,a variety of random factors,which are often overlooked for analysis,will inevitably affect their speed of transmission.Such as the temperature,humidity,p H,radiation and so on.Especially for nonlinear dynamic systems with higher dimensions,if the random factors are ignored,the analysis process is not rigorous and the influence is greater.Resulting in the results of the analysis is not conducive to dealing with practical problems.In fact,the system will inevitably be affected by the white noise of the internal or external environment,while the random excitation dissipation system is more realistic in describing the natural regular.In this paper,the stochastic dynamic behavior of the high-dimensional infectious disease system subjected to random excitation disturbance will be analyzed by the stochastically excited and dissipated Hamiltonian system theory.And the numerical simulation results show that the condition of the epidemic model to be stable and disturbed by Hopf bifurcation when disturbed by random excitation disturbance.The main contents of this paper are as follows:1.First of all,the research status and significance of infectious disease model are briefly described.And the research status and progress of stochastic dynamical systems are described.Then,the central manifold theorem and the related knowledge of Hamilton system theory of dissipative excitation are elaborated in detail.2.The dynamical behavior of a three-dimensional random SIR epidemic model was studied.The influence of the temperature,humidity,p H,radiation and other environmental noise effects are replaced by Gaussian white noise.Then a stochastic term is introduced into the stable SIR epidemic model.And the nonlinear differential equation of the random SIR epidemic model with stage structure are established.Then by applying stochastic center manifold and stochastic average method,the stochastic differential equation was reduced order and we got the corresponding Ito differential equation.The stochastic dynamic behavior of the system are analyzed by the stochastically excited and dissipated Hamiltonian system theory.Finally,the functional image of stationary probability density and jointly stationary probability density were simulated.3.In order to study the stochastic dynamics of a four-dimensional measles epidemic model,taking the random factors into account,the randomness are imported into the measles epidemic model with partial immunity and latency.The nonlinear differential equation of the random measles epidemic model with partial immunity and latency is established.Then,the original system is dimensionally reduced by using the same stochastic central manifold theorem.It is more complicated than 3D dimensionality reduction.It is reduced to a two-dimensional stochastic differential equation by polar transformation.The corresponding Ito stochastic differential equations are processed by the stochastic averaging method and related theorem.The stochastic dynamic behavior of the system are analyzed by the stochastically excited and dissipated Hamiltonian system theory.Finally,we also conduct numerical simulations to verify the probability and location of bifurcations.