Mathematical model on the transmission of worms in wireless sensor network

Wireless sensor networks (WSNs) have received extensive attention due to their great potential in civil and military applications. The sensor nodes have limited power and radio communication capabilities. As sensor nodes are resource constrained, they generally have weak defense capabilities and are attractive targets for software attacks. Cyber attack by worm presents one of the most dangerous threats to the security and integrity of the computer and WSN. In this paper, we study the attacking behavior of possible worms in WSN. Using compartmental epidemic model, we propose susceptible – exposed – infectious – recovered – susceptible with a vaccination compartment (SEIRS-V) to describe the dynamics of worm propagation with respect to time in WSN. The proposed model captures both the spatial and temporal dynamics of worms spread process. Reproduction number, equilibria, and their stability are also found. If reproduction number is less than one, the infected fraction of the sensor nodes disappears and if the reproduction number is greater than one, the infected fraction persists and the feasible region is asymptotically stable region for the endemic equilibrium state. Numerical methods are employed to solve and simulate the systems of equations developed and also to validate our model. A critical analysis of vaccination class with respect to susceptible class and infectious class has been made for a positive impact of increasing security measures on worm propagation in WSN.     

Global dynamics and control of malicious signal transmission in wireless sensor networks

This paper studies the global dynamics of a discontinuous delayed model of malicious signal transmission in wireless sensor networks under the framework of differential inclusion. The local stability of two types of steady states are investigated for the discontinuous system by studying the corresponding characteristic equation. The sufficient conditions for the existence of two types of globally asymptotically stable steady states are obtained for the discontinuous system by using the comparison arguments method. Furthermore, the optimal control of the discontinuous system is investigated by using Pontryagin’s maximum principle. Numerical simulations of two examples are carried out to illustrate the main theoretical results. The obtained results can help us to better control and predict the spread of malicious signal transmission in wireless sensor networks.     

SEIQRS model for the transmission of malicious objects in computer network

Susceptible (S) – exposed (E) – infectious (I) – quarantined (Q) – recovered (R) model for the transmission of malicious objects in computer network is formulated. Thresholds, equilibria, and their stability are also found with cyber mass action incidence. Threshold R_cq determines the outcome of the disease. If R_cq ? 1, the infected fraction of the nodes disappear so the disease die out, while if R_cq > 1, the infected fraction persists and the feasible region is an asymptotic stability region for the endemic equilibrium state. Numerical methods are employed to solve and simulate the system of equations developed. The effect of quarantine on recovered nodes is analyzed. We have also analyzed the behavior of the susceptible, exposed, infected, quarantine, and recovered nodes in the computer network.     

Mathematical model on malicious attacks in a mobile wireless network with clustering

A mathematical model has been formulated for the analysis of a wireless epidemic on a clustered heterogeneous network. The model introduces mobility into the epidemic framework assuming that the component nodes have a tendency to be attached with a frequently visited home cluster. This underlines the inherent regularity in the mobility pattern of mobile nodes in a wireless network. The analysis focuses primarily on features that arise because of the mobility considerations compared in the larger scenario formed by the epidemic aspects. A result on the invariance of the home cluster populations with respect to time provides an important view-point of the long-term behavior of the system. The analysis also focuses on obtaining a basic threshold condition that guides the epidemic behavior of the system. Analytical as well as numerical results have also been obtained to establish the asymptotic behavior of the connected components of the network, and that of the whole network when the underlying graph turns out to be irreducible. Applications to proximity based attacks and to scenarios with high cluster density have also been outlined.     

Lifetime maximization in wireless directional sensor network

This paper addresses two versions of a lifetime maximization problem for target coverage with wireless directional sensor networks. The sensors used in these networks have a maximum sensing range and a limited sensing angle. In the first problem version, predefined sensing directions are assumed to be given, whereas sensing directions can be freely devised in the second problem version. In that case, a polynomial-time algorithm is provided for building sensing directions that allow to maximize the network lifetime. A column generation algorithm is proposed for both problem versions, the subproblem being addressed with a hybrid approach based on a genetic algorithm, and an integer linear programming formulation. Numerical results show that addressing the second problem version allows for significant improvements in terms of network lifetime while the computational effort is comparable for both problem versions.     

A mathematical model for the effect of malicious object on computer network immune system

In this paper, a nonlinear mathematical model is proposed and analyzed to study the effect of malicious object on the immune response of the computer network. Criteria for local stability, instability and global stability are obtained. It is shown that the immune response of the system decreases as the concentration of malicious objects increases, and certain criteria’s are obtained under which it settles down at its equilibrium level. This paper shows that the malicious objects have a grave effect on cyber defense mechanism. The paper has two parts – (i) in the first part a mathematical model is proposed in which dynamics of pathogen, immune response and relative characteristic of the damaged node in the network is investigated, (ii) in second part the effect of malicious object on the immune response of the network has been examined. Finally how and where to use this modeling is discussed.     

A Lagrangean-based heuristics for the target covering problem in wireless sensor network

We study the target coverage problem in wireless sensor networks. The problem consists in maximizing the network lifetime by grouping the sensors in disjoint set covers of the targets. A binary integer programing model is formulated to maximize the network lifetime. Since the problem is NP-complete, we provide an iterative approximation based on Lagrangean relaxation and subgradient optimization.     

Application of generalized diagonal dominance in wireless sensor network optimization problems

The recent application of the diagonal dominance in the development of the optimization algorithms in the wireless sensor networks design, has been done by Yuan and Yu (2006) [14], extended in Yu et al. (2006) [9], and surveyed in Machado and Tekinay [11]. In this paper, we will use the concept of generalized diagonal dominance, to improve the obtained results regarding the power control game, in three directions. We also discuss the applicability of such improvements.     

Cryptanalysis of a chaos block cipher for wireless sensor network

Based on the analysis of a chaos block cipher for wireless sensor network (WSN), it is found that there is a fatal flaw in its security because the number of rounds is too small and the calculation precision of round function is too short. The scheme could be cryptanalyzed by utilizing differential cryptanalysis theory. First, the third round key is recovered by chosen plaintext attack according to the characteristics of the round function. Then, the second round key can be deduced from the relationship of the sub-keys between the second and the third rounds. Based on the above successful attacks, the first round key could also be broken by brute-force attack. Finally, by employing the characteristics of Feistel structure, the fourth round key could also be obtained. Since all round keys have been cryptanalyzed, the plaintext can then be decrypted. The encryption scheme is proven to be insecure consequently.     

A key design to prolong lifetime of wireless sensor network

In order to solve the contradiction between the connectivity of the wireless sensor network and the key storage consumption, under the premise of reducing network storage consumption, the key pre-distribution management scheme with higher connectivity rate is proposed using the hexagonal network deployment information, which adopts the idea of the matrix space for the square deployment information strategy to reduce the burden of the network storage. Ability against the capture attack is improved obviously. The results show that contradiction between the network connectivity rate and the energy consumption has a better solution, and the proposed algorithm is suitable for the wireless sensor networks of energy limited.     

Optimal relay node placement in delay constrained wireless sensor network design

The Delay Constrained Relay Node Placement Problem (DCRNPP) frequently arises in the Wireless Sensor Network (WSN) design. In WSN, Sensor Nodes are placed across a target geographical region to detect relevant signals. These signals are communicated to a central location, known as the Base Station, for further processing. The DCRNPP aims to place the minimum number of additional Relay Nodes at a subset of Candidate Relay Node locations in such a manner that signals from various Sensor Nodes can be communicated to the Base Station within a pre-specified delay bound. In this paper, we study the structure of the projection polyhedron of the problem and develop valid inequalities in form of the node-cut inequalities. We also derive conditions under which these inequalities are facet defining for the projection polyhedron. We formulate a branch-and-cut algorithm, based upon the projection formulation, to solve DCRNPP optimally. A Lagrangian relaxation based heuristic is used to generate a good initial solution for the problem that is used as an initial incumbent solution in the branch-and-cut approach. Computational results are reported on several randomly generated instances to demonstrate the efficacy of the proposed algorithm.     

A cooperative differential game model based on transmission rate in wireless networks

Network throughput and energy efficiency are paramount for network performance in an energy-constrained wireless network. However, it is difficult to achieve optimal objectives simultaneously. Therefore, it is necessary to find a rate control solution based on tradeoff between network throughput and energy efficiency. In this paper, we propose a cooperative differential game model and find an optimal rate control of each player to get the total minimal cost with tradeoff between network throughput and energy efficiency of the networks.     

Fuzzy epidemic model for the transmission of worms in computer network

An e-epidemic SIRS (susceptible–infectious–recovered–susceptible) model for the fuzzy transmission of worms in computer network is formulated. We have analyzed the comparison between classical basic reproduction number and fuzzy basic reproduction number, that is, when both coincide and when both differ. The three cases of epidemic control strategies of worms in the computer network–low, medium, and, high–are analyzed, which may help us to understand the attacking behavior and also may lead to control of worms. Numerical methods are employed to solve and simulate the system of equations developed.     

Stability analysis of dynamic collaboration model with control signals on two lanes

In this paper, the influence of control signals on the stability of two-lane traffic flow is mainly studied by applying control theory with lane changing behaviors. We present the two-lane dynamic collaboration model with lateral friction and the expressions of feedback control signals. What is more, utilizing the delayed feedback control theory to the two-lane dynamic collaboration model with control signals, we investigate the stability of traffic flow theoretically and the stability conditions for both lanes are derived with finding that the forward and lateral feedback signals can improve the stability of traffic flow while the backward feedback signals cannot achieve it. Besides, direct simulations are conducted to verify the results of theoretical analysis, which shows that the feedback signals have a significant effect on the running state of two vehicle groups, and the results are same with the theoretical analysis.     

Dynamics of a congestion control model in a wireless access network

In this paper, a congestion control algorithm with heterogeneous delays in a wireless access network is considered. We regard the communication time delay as a bifurcating parameter to study the dynamical behaviors, i.e., local asymptotical stability, Hopf bifurcation and resonant codimension-two bifurcation. By analyzing the associated characteristic equation, the Hopf bifurcation occurs when the delay passes through a sequence of critical value. Furthermore, the direction and stability of the bifurcating periodic solutions are derived by applying the normal form theory and the center manifold theorem. In the meantime, the resonant codimension-two bifurcation is also found in this model. Some numerical examples are finally performed to verify the theoretical results.     

Dynamic model of worms with vertical transmission in computer network

An e-epidemic SEIRS model for the transmission of worms in computer network through vertical transmission is formulated. It has been observed that if the basic reproduction number is less than or equal to one, the infected part of the nodes disappear and the worm dies out, but if the basic reproduction number is greater than one, the infected nodes exists and the worms persist at an endemic equilibrium state. Numerical methods are employed to solve and simulate the system of equations developed. We have analyzed the behavior of the susceptible, exposed, infected and recovered nodes in the computer network with real parametric values.     

Effect of time-delay on a food chain model

This paper aims to study the effect of discrete time-delay on a tritrophic food chain model with Holling type-II functional responses. Dynamical behaviours such as boundedness, stability, persistence and bifurcation of the model are studied. Our analytical findings are illustrated through computer simulation. Biological implications of our analytical findings are addressed critically.     

Maximizing lifetime of a wireless sensor network via joint optimizing sink placement and sensor-to-sink routing

Wireless sensor networks typically contain hundreds of sensors. The sensors collect data and relay it to sinks through single hop or multiple hop paths. Sink deployment significantly influences the performance of a network. Since the energy capacity of each sensor is limited, optimizing sink deployment and sensor-to-sink routing is crucial. In this paper, this problem is modeled as a mixed integer optimization problem. Then, a novel layer-based diffusion particle swarm optimization method is proposed to solve this large-scaled optimization problem. In particular, two sensor-to-sink binding algorithms are combined as inner layer optimization to evaluate the fitness values of the solutions. Compared to existing methods that the sinks are selected from candidate positions, our method can achieve better performance since they can be placed freely within a geometrical plane. Several numerical examples are used to validate and demonstrate the performance of our method. The reported numerical results show that our method is superior to those existing. Furthermore, our method has good scalability which can be used to deploy a large-scaled sensor network.     

动态传播率模型及其在疫情分析中的应用

应用数据驱动的动态传播率来代替基本传染数$R_0$,在全国和省市两个层面上研究COVID-19疫情发展的特点和趋势。首先,基于动态增长率建立传染病常微分方程,推导得出动态传播率模型。其次,选择幂函数作为动态传播率的拟合函数,以3天作为最优滑窗期,对各地拐点进行了估计。最后,通过动态模型对各地不同程度尾声开始的起点进行了预测,并在13个省市间进行9个疫情相关指标的对比分析。结果显示,各地动态传播率在经过短暂的波动后均稳步下降,疫情得到有效控制;估计的拐点主要集中在2月中旬,而预测的尾声都将在3月底之前到来;同时,各地疫情发展特点和趋势、防控措施力度和效果存在一定差异。     

Cooperative differential game model based on trade-off between energy and delay for wireless sensor networks

A wireless sensor network (WSN) consists of a large number of unattended sensors with limited storage, battery power, computation, and communication capabilities, where battery power (or energy) is the most crucial resource for sensor nodes. The information sensed by sensors needs to be transmitted to sink quickly especially for the applications with delay restriction. However, it is difficult to achieve optimal energy efficiency and source-to-sink delay simultaneously. So it is very necessary to find a power control solution based tradeoff between energy and delay. In this paper, a cooperative differential game model is proposed, and a power solution is obtained which determines a fair distribution of the total cooperative cost among sources.