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To explore the impacts of pulse vaccination on the dynamical behaviors of virus, the current paper investigates the persistence and global attractivity of a diffusive viral infection model incorporated with impulsive CTL immune response while the stronger logistic growth mechanism. Based on the comparison principle and the proper upper and lower solutions, we first discuss the existence of the infection-free steady state, and define the crucial CTL-activated viral infection reproduction number $ R_0 $ R 0 . Then, the fundamental extinction and uniform persistence behaviors of virus are distinguished by various cases of the threshold parameter $ R_0 $ R 0 . Finally, the existence and global attractivity of the positive steady state are still established by employing Brouwer's fixed point theorem and auxiliary functional. Our results indicate that high vaccination rates will stimulate the CTL response more effectively, and thus eventually can force the virus to eradicate. [ABSTRACT FROM AUTHOR]

We study here, spatially homogeneous mechanistic mathematical models describing the interactions between a malignant tumor and the immune system and phase-space analysis of a certain mathematical model of tumor growth with an immune responses. Mathematical modeling of this process is viewed as a potentially powerful tool in the development of improved treatment regimens. We study some features of local and global behavior the multipoint problem for the three-dimensional tumour growth model system obtained by de Pillis and Radunskaya in 2003, with dynamics described in terms of densities of three cells populations: tumor cells, healthy host cells and effector immune cells. We found sufficient conditions, under which trajectories from the positive domain of feasible multipoint initial conditions tend to one of equilibrium points. The addition of a drug term to the system can move the solution trajectory into a desirable basin of attraction. We show that the solutions of the model with a time-varying drug term approach can be evaluated in a more fruitful way and down to earth style from the point of practical importance than the solutions of the system without drug treatment, in the condition of stimulated immune processes, only. [ABSTRACT FROM AUTHOR]