Study On The Dynamics Of HIV Epidemic Models With Switching Parameters And Pulse Control

The research on dynamics of HIV(Human Immunodeficiency Virus) epidemic has been a hot topic, and mathematical models for the dynamics of HIV infection have been developed to study the spread and treatment of infectious disease, and specific issues(such as the stability of the disease-free equailibrium and persistence of the disease) have been addressed. In most existing HIV models, the model parameters are assumed to be constant. However, during the HIV infection period, the infectious diseases are inevitably influenced by internal or external environmental factors such as temperature, the immunological state of the host, which causes that the parameters are no longer than constant. Thus, it is more reasonable to assume that these parameters are time-varying and switching in time. Based on Razumikhintype method, Lyapunov function, and Ito Lemma, this paper establishes HIV models with switching parameters and pulse control, stochastic impulsive switched HIV models, stochastic switched HIV/AIDS(Human Immunodeficiency Virus/Acquired Immune Deficiency Syndrome) models with two types of control schemes, and studies their dynamics. The effectiveness of the proposed results is illustrated by some simulation examples. The main contents and results are as follows:In first part, HIV models with switching parameters and pulse control are investigated. First, the models parameters are assumed to be time-varying and switching in time and the HIV model with switching parameters is presented. Some stability criteria are established to determine persistence or extinction of the disease by defining the basic production number on the basis of Razumikhin-type method. The results show that the basic production number of the whole system is less than one, then the disease dies out, regardless of whether the basic production number of the subsystems is less than one or lager than one. In other words, regardless of whether the subsystems are unstable or stable, the basic production number of the whole system is less than one, which guarantees that the whole system is stable. Then, both control strategies are applied to infected cells and uninfected cells of the above model, and some stability conditions for disease-free period solution are obtained to ensure that pulse treatment can eradicate the HIV infection. Numerical simulations are performed to demonstrate the analytical results.In second part, the dynamics of new HIV models with switching nonlinear incidence functions and pulse control are studied. Nonlinear incidence functions are first assumed to be time-varying functions and switching functional forms in time, which have more realistic significance to model infectious disease models. By using the novel type of common Lyapunov function and Razumikhin-type approach, new threshold conditions are derived to guarantee eradication of the disease. The results show that the proper switching conditions chosen can increase the counts of CD4 T-cells whereas reduce viral load. Then, pulse control scheme is applied to the above model, and new sufficient conditions for the eradication of the disease are presented in the terms of the pulse effect and the switching effect. Furthermore, when the switching rule is periodic, the effects of switching parameters and pulse parameters to behavior of disease are analyzed in detail. Finally, several numerical examples are given to show the effectiveness of the proposed results.In third part, the stochastic dynamics of epidemic HIV models with switching parameters and pulse control are studied. First, the stochasticity is incorporated into a switched HIV model. Stochastic switched HIV models are presented and investigated. When the switching rule is periodic or not, some new criteria ensuring stochastic stability for the above models are obtained by utilizing stochastic Ito lemma. It is found that the stochastic solutions always fluctuate around the deterministic solutions and the stochastic perturbations finally converge to the deterministic solutions. Furthermore, pulse control strategies are applied to infected cells, uninfected cells and infected cells, respectively. Each control strategy is analyzed to guarantee its success in eradicating the disease. A critical vaccination rate is defined to ensure that pulse vaccination can successfully clear the disease. Finally, the effectiveness of the proposed results is illustrated by some simulation examples.In fourth part, the dynamics behavior of HIV infectious disease model with switching parameters and combined bounded noise and Gaussian white noise are investigated. Based on stochastic Ito lemma and Razumikhin-type approach, some threshold conditions are established which ensure the disease eradication or persistence. Results show that the basic production number of the whole system is less than one, then the disease-free equilibrium is stochastically asmpotically stable, which implies that the disease could die out; The disease becomes persistent if the basic production number is larger than one. Numerical examples are given to verify the obtained results. It is shown form the simulations that the stochastic solutions of the stochastic model always fluctuate around the deterministic solutions its corresponding deterministic model. The fluctuations reduce as the amplitude of bounded noise decreases if stochastic noise intensity is fixed. When the amplitude of bounded noise is fixed, the increase of stochastic noise intensity may cause that the stochastic model quickly converges to the disease-free equilibrium.In fifth part, stochastic stability for stochastic switched HIV/AIDS models with constant and impulsive control schemes is investigated. The stochasticity is introduced via the technique of parameter perturbation and the switching is assumed that the models parameters are time-varying functions and switch their forms in time. First, a stochastic switched AIDS model with constant control schemes is proposed and studied. By using the Lyapunov-Razumikhin method, new sufficient conditions are established. The results show that the system is stable if the basic production number of the whole system is less than one, regardless of whether the subsystems are unstable or stable, which implies that the disease could be eradicated theoretically. Furthermore, impulsive control schemes are applied into a stochastic switched AIDS model. Threshold conditions on the basic reproduction number are developed which guarantee the system is stochastically stable. In addition, complex dynamic behavior for the positive periodic solution is analyzed, and the results imply that pulse control strategies could lead theoretically the disease to die out. Numerical examples are employed to verify the main results.