Optimal Control Models Of Computer Viruses And HIV Viruses

With the development of software/hardware technology and network communication technology, the spread of computer virus over a computer network has come to be a serious threat for the security of the computer network. In order to protect a computer network from the spread of computer viruses, a variety of control strategies, such as installing anti-virus software, reinstalling the operating system, and installing a firewall, is introduced. However, all of these control strategies would consume lots of computing resources. Therefore, it is of practical importance to keep a low number of infected computers at a low cost.Another issue is the optimal control of the HIV virus. The HIV virus, which in 2009 caused about 18 million people dead, has become a great hazard to human health. There are two kinds of drugs that can control the HIV effectively, with significant toxic side effects on the human body. Consequently, it is worth study how the HIV virus can be controlled effectively by using as little amount of drugs as possible.The main contributions made in this thesis are summarized as follows:1: Inspired by the previous computer viruses models (the SAIR model and the improved SAIR model) and by attempting to keep a low number of infected nodes at the lowest cost, two optimal control models of computer viruses are proposed. In these models, the dynamical behaviors of the virus-free equilibrium and the virus-endemic equilibrium are analyzed. Then, the objective functional for this system is presented. The existence of an optimal control strategy is proved. Finally, by means of the Pontryagin's Maximun Principle we derive the corresponding optimality system as well as the optimal control. Simulation experiments show that the spread of computer virus can be controlled effectively with proper control strategy.2: A delayed optimal control model of computer viruses is suggested. In reality, the spread of computer viruses is often accompanied with time delay. By incorporating the objective of keeping a low number of infected nodes at the lowest cost into a known computer virus model (the delayed SIRS model), a novel model is described. Then the objective functional of this system is given. An optimal control strategy is proved to be existent. Again by calling the Pontryagin's Maximun Principle, we derive the optimal control and the optimality system characterizing the optimal control. Numerical simulations show that the spread of computer virus can be controlled effectively with proper control strategy. To our knowledge, this is the first time the delay factor is considered in the optimal control model of computer viruses.3: Under a novel HIV model, an efficient therapy strategy is devised. It is known that (1) the HIV virus can infect two types of benefit cells (T cells and macrophages), (2) HIV infection can be divided into three stages, and (3) two kinds of drugs (PIs and RTIs) can treat and control the HIV effectively in the first and third stages of the HIV infection. In this context, we propose an optimal control model of HIV, which is intended to maximize the number of healthy cells at the least cost of chemotherapy. Next the objective functional of this system is described. The existence of an optimal control is shown. Finally, we derive the optimal control and the optimality system. Numerical results show that HIV can be controlled effectively by using our method. To our knowledge, there is no known work about the optimal control of HIV multi-cell infection.In summary, this thesis proposes and investigates three computer viruses models and one HIV model. This work opens a door to the effective control of computer virus and HIV.