Research On Dynamic Behaviors Of Stochastic Biological Systems

This thesis investigates asymptotic and stability properties of general stochastic functional di?erential equations, dynamics behavior of stochastic epidemic models and stochastic cells models. Based on the Lyapunov function method, Rhasminskii techniques,It?o stochastic calculus, graph theory and stochastic analysis techniques, some important results are obtain, the main work in this thesis can be summarized as follows:1. Background and research signi?cance of stochastic biological systems are reviewed,and brief introduction to the related de?nition and theorems in this thesis’ s area are presented.2. Asymptotic and stable properties of general stochastic functional di?erential equations are investigated by the multiple Lyapunov function method which admits nonnegative upper bounds for the stochastic derivatives of the Lyapunov functions, a theorem for asymptotic properties of the La Salle-type described by limit sets of the solutions of the equations is obtained. Based on it, stability theorems for the asymptotic stability of the equations are also established, which enable us to construct the Lyapunov functions more easily in application. Particularly the well-known classical theorem on stochastic stability is a special case of our result, the operator L V is not required to be negative which is more general to ful?l and the e?ect of the stochastic perturbation made an important use in it. An example is given to illustrate the usage of the method.3. We establish a general stochastic SIRS epidemic model based on the deterministic SIRS epidemic model. Using the Rhasminskii methods, the uniqueness and existence of the positive solution is proved. We de?ne the basic reproduction number RS 0and show its a sharp threshold for the dynamics of the stochastic model by calculating the Lyapunov exponents near the disease-free solution. If RS 0<1, the disease will die out; If RS 0>1,the disease will not die out. About the persistence issue of the disease, we can prove that the solution of the model has stationary distribution if RS 0>1. We also give some numerical examples to illustrate our theory results.4. Two different types of stochastic multigroup SIR epidemic models and one stochastic multigroup SEIR model is investigated. One of the stochastic multigroup SIR models is established by perturbing the transmission coe?cientβ, it guarantees the upper bound of the model’s solution. We can obtain the su?cient and necessary condition of the disease-free equilibrium by using a Lyapunov function which is established by graph theory. About the other two types of stochastic multigroup models, we prove the existence and uniqueness of the positive solution ?rstly. Then we ?nd out the threshold RS0 which determine the dynamics of the disease. We also investigate the persistence of the disease by collecting the de?nition of stochastic recurrence. At last, we give some numerical results to illustrate our theory results.5. A continuous time Markov chain model and a stochastic di?erential equation(SDE) epidemic model of CD4+T Cells are formulated, we de?ne the basic reproduction number RS 0and show that it is a sharp threshold for the dynamics of SDE model.Specially, if RS 0<1, the infection-free equilibrium is asymptotically stable, if RS 0>1,the infection-free equilibrium is unstable. For some parameter values, the SDE model exists bistability. The probability distribution associated with the infected cells exhibits bimodality with one model at the infection-free equilibrium and the other at the chronicinfection equilibrium. The model at the infection-free equilibrium will be higher if the stochastic perturbation becomes bigger. This is the phenomenon of our stochastic cells model’s backward bifurcation. Based on our theory results, we give a new method to control the infected cells.Finally, the dissertation’s summary is given and some future research issues are proposed.