Analysis Of Dynamical Models With Impulsive Drug Effects

According to the latest research related to virus and bacterial infection of the basic process and drugs for the treatment, this paper establishes mathematical models with im-pulsive dosing in the treatment of both viral infection and bacterial infection, furthermore analyze the system dynamic behavior, finally their biological significance are discussed. The first chapter briefly introduces some basic knowledge on impulsive dynamics.Many studies focus on time-dependent pulse models in order to improve virus and bacteria models with the treatment. A study of a medical experiment, published in the famous international journals Cell, shows that the method of broadly neutralizing antibodies combined with inducer can prolong HIV virus rebound time in mice. The second chapter, based on the experiment mentioned above and the latest medical facts that a small part of healthy cells will first enter the latent period of HIV infection instead of being infected and inducers play a key role in stimulating latent cells, establishes a HIV drug-treatment dynamic model considering latent infection, which describes the process of receiving drug treatment after HIV virus infection in mice. Meanwhile, it defines the basic reproduction of the impulsive system and prove that infection-free state is globally asymptotically stable when Rq<1 and the infection of HIV is uniformly persistent when R0> 1. Through theoretical analysis we find the influence of inducer for the basic reproduction number is small while broadly neutralizing antibody play a key part in the basic reproduction number R0. However, through the numerical simulation we find that dose and injection interval of inducing agent has a significant effect to prolong viral rebound. Therefore, there remains to be further research on the basic reproductive number control optimization and numerical simulation for the optimal dosage regimen.In the third chapter, due to a study of Nature recently published a drug-resistant strain in the production of indole substances contribute to the protection of the wild type strain experiments, we build a model on it. Besides the antibiotics injected into the body of the drug concentration will gradually decay, efficacy also decreases as the concentration decreased, which is consistent with Michaelis equation in the pharmacoki-netics, thereby considering the time-dependent impulsive injection of antibiotics based on a four-dimensional double-strain chemostat model to close to the real as possible. Then we defined strain 1,2 of the basic reproduction number R01, R02 in the periodic system, proving the existence and uniqueness of solutions for periodic boundary strain 1 to obtain the reproduction number R021, that strain 2 invade strain 1. Mathematically we prove when R01<1, R02<1, bacteria-free equilibrium is globally asymptotically stable; when R01>1、R02<1, strain 1 is uniformly persistent; when R02>1、R021>1, strain 2 is uniformly persistent.In the last chapter, we make a brief review of the above conclusions, and present the impulsive dynamic point of view and practical meaning for these models.And we analyze some shortcomings of this paper and point out some questions and future work.