计算机病毒和HIV病毒最优控制模型
摘要
随着计算机软硬件技术及网络通信技术的发展,计算机病毒的传播能力越来越强,目前计算机病毒的传播已经对计算机网络安全构成了很大的威胁。为了减少计算机病毒传播对网络安全的威胁,人们采取了一些相应的控制策略,包括安装杀毒软件,重新安装操作系统,安装防火墙等等,而这些控制策略都会消耗不少系统的计算资源。因此如何使用较小的系统资源,使得网络中感染计算机的数目达到最少,成为摆在我们面前一个现实而严峻的问题。
本文研究的另一个课题是HIV病毒的最优控制模型。在2009年,HIV病毒造成了大约1800万人死亡,成为危害人类健康的主要杀手之一。世界上主要有两种药物对HIV病毒能起到良好的治疗和控制效果,然而这些药物对人体都有很多毒副作用。因此,如何使用尽量少的药物用量,有效地控制HIV病毒,也是一个值得研究的课题。
本文主要工作如下:
1,提出了两个计算机病毒最优控制模型。本文在已经存在的计算机病毒传播模型(SAIR模型,改进的SAIR模型)的基础上,考虑到人为控制操作对网络计算机感染数量的影响,引入目标泛函,建立了计算机病毒的最优控制模型。并且运用最优控制理论的相关原理和方法,证明了最优控制函数的存在性,然后使用Pontryagin极大值原理推导出最优控制以及描述最优控制的最优系统。数值仿真结果表明:使用适当的控制策略后,计算机病毒的传播得到了有效的控制。
2,提出了时滞计算机病毒最优控制模型。时滞效应能够很好的刻画计算机病毒传播的规律。因此,我们利用已经存在的计算机病毒传播模型(时滞SIRS模型),考虑到人为控制策略对网络中计算机感染数量的影响,引入目标泛函,建立了时滞计算机病毒最优控制模型。并且运用最优控制理论的相关原理和方法,证明了最优控制函数的存在性,然后使用Pontryagin极大值原理推导出最优控制以及描述最优控制的最优系统。数值仿真结果表明:使用适当的控制策略后,计算机病毒的传播得到了有效的控制。据我们所知,目前还没有相关文献把最优控制理论引入时滞计算机病毒模型。
3,提出了一个HIV病毒最优控制模型。我们首先注意到HIV病毒能感染T细胞和巨噬细胞这两类有益细胞。其次HIV病毒的感染细胞可以分为3个阶段。然后两类药物分别是PIs和RTIs能对HIV病毒的感染两个阶段起到良好的治疗和控制效果。在这种背景之下,提出了自己的HIV最优控制模型,目的是让这两类有益细胞的数量的数量到达最大,HIV病毒细胞的数量达到最小,并且用药量达到最小。然后,我们提出了自己的目标泛函,其次,运用最优控制理论的相关原理和方法,证明了最优控制函数的存在性。最后,使用Pontryagin极大值原理推导出最优控制以及描述最优控制的最优系统。数值仿真结果表明:HIV病毒能够在我们的方法下,得到有效的控制。据我们所知,这项工作也是最早的将最优控制理论引入HIV多细胞感染模型。
简而言之,本文运用最优控制理论对计算机病毒传播模型和HIV病毒模型进行了研究,提出了一些控制这两类病毒的方法,为有效地控制这两类病毒开辟了新的途径。
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.
本文研究的另一个课题是HIV病毒的最优控制模型。在2009年,HIV病毒造成了大约1800万人死亡,成为危害人类健康的主要杀手之一。世界上主要有两种药物对HIV病毒能起到良好的治疗和控制效果,然而这些药物对人体都有很多毒副作用。因此,如何使用尽量少的药物用量,有效地控制HIV病毒,也是一个值得研究的课题。
本文主要工作如下:
1,提出了两个计算机病毒最优控制模型。本文在已经存在的计算机病毒传播模型(SAIR模型,改进的SAIR模型)的基础上,考虑到人为控制操作对网络计算机感染数量的影响,引入目标泛函,建立了计算机病毒的最优控制模型。并且运用最优控制理论的相关原理和方法,证明了最优控制函数的存在性,然后使用Pontryagin极大值原理推导出最优控制以及描述最优控制的最优系统。数值仿真结果表明:使用适当的控制策略后,计算机病毒的传播得到了有效的控制。
2,提出了时滞计算机病毒最优控制模型。时滞效应能够很好的刻画计算机病毒传播的规律。因此,我们利用已经存在的计算机病毒传播模型(时滞SIRS模型),考虑到人为控制策略对网络中计算机感染数量的影响,引入目标泛函,建立了时滞计算机病毒最优控制模型。并且运用最优控制理论的相关原理和方法,证明了最优控制函数的存在性,然后使用Pontryagin极大值原理推导出最优控制以及描述最优控制的最优系统。数值仿真结果表明:使用适当的控制策略后,计算机病毒的传播得到了有效的控制。据我们所知,目前还没有相关文献把最优控制理论引入时滞计算机病毒模型。
3,提出了一个HIV病毒最优控制模型。我们首先注意到HIV病毒能感染T细胞和巨噬细胞这两类有益细胞。其次HIV病毒的感染细胞可以分为3个阶段。然后两类药物分别是PIs和RTIs能对HIV病毒的感染两个阶段起到良好的治疗和控制效果。在这种背景之下,提出了自己的HIV最优控制模型,目的是让这两类有益细胞的数量的数量到达最大,HIV病毒细胞的数量达到最小,并且用药量达到最小。然后,我们提出了自己的目标泛函,其次,运用最优控制理论的相关原理和方法,证明了最优控制函数的存在性。最后,使用Pontryagin极大值原理推导出最优控制以及描述最优控制的最优系统。数值仿真结果表明:HIV病毒能够在我们的方法下,得到有效的控制。据我们所知,这项工作也是最早的将最优控制理论引入HIV多细胞感染模型。
简而言之,本文运用最优控制理论对计算机病毒传播模型和HIV病毒模型进行了研究,提出了一些控制这两类病毒的方法,为有效地控制这两类病毒开辟了新的途径。
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.
引文
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