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If denotes the infection rate and denotes the removal rate of infected individuals, then assuming a homogeneous mixing model (i.e., each of the susceptibles can get in contact with any of the infectives), it is simple to observe that in time t, there are xy t new infections and y t removals. Therefore, the basic differential equations that describe the rate of change of susceptibles, infectives, and recovered individuals are given by dX(t) = XY, dt dY (t) = XY Y, dt dZ(t) = Y dt The above equations can be solved either approximately or precisely based on some boundary conditions, such as, at the start of the epidemic, when t = 0, (X, Y, Z) can take the values (x0 , y0 , 0). Note that, in particular, if y0 is very small, x0 is approximately equal to N. It also follows that if the relative removal rate, = / , is greater than x0 , only then can an epidemic start to build up as this condition will result in [dY (t)/dt]t=0 > 0, i.e. Y (t) will have a positive slope. Therefore, the relative removal rate = x0 gives a threshold density of susceptibles. On the other hand, the S-I-S model does not have the recovered subset Z(t), and those who are infected fall back into the susceptible subset S(t) after their infectivity duration. An important aspect that is of particular interest in epidemiological studies is the phenomenon of phase transition of the spreading process that is dependent on a threshold value of the epidemic parameter; that is, if the epidemic parameter is above the threshold, the infection will spread out and become persistent; on the contrary, if the parameter is below the threshold, the infection will die out. Identi cation of this threshold value is critical in the study of how an epidemic spreads and how it can be controlled. Apart from modeling technique based on the continuous differential rate equation, the study of epidemics has often been performed by treating the population as a network graph, with the nodes representing each individual and the edges their interaction. This form of analysis [3] has mainly been used in scenarios where the end result of the epidemic spread is more important than the temporal dynamics of the propagation. Several works have spawned from this formulation [3 8], where the spread of diseases have been studied by modeling the social network as a scalefree topology. Several other works also exist that model the spread of computer viruses [9, 10]. Epidemic Theory has found special attention in the design and modeling of several phenomena and protocols in sensor networks wherever there is a scope of information distribution on a large scale, preferably from a small number of sources to a large number of recipients. Among the popular phenomena in sensor and ad hoc networks
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where this theory has been adopted are data dissemination, broadcast protocols, and routing. We will delve into some of these areas where Epidemic Theory has been used to study and model several processes and functions of sensor networks.
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3.3 DATA DISSEMINATION IN SENSOR NETWORKS: MODEL AND PROTOCOLS The problem of reliable data dissemination in the context of wireless sensor networks is very critical. Reliable data dissemination to all nodes is absolutely necessary for the propagation of queries, code updates, and other sensitive information in a wireless sensor network. This is not a trivial task since the number of nodes in a sensor network can be quite huge and the environment is dynamic (i.e., nodes can die or move), thus making the topology change constantly. Since data dissemination primarily deals with the transfer of messages from one node to all nodes of a network, algorithms based on epidemiological formulations are a perfect t. Accordingly, these algorithms have been successfully used in disseminating information in sensor networks and, depending on the application, the dissemination can start at a single node, such as a base station, or at multiple sensor nodes. The decentralized and distributed nature of wireless sensor networks ts the context of epidemic algorithms aptly. One of the prominent works of data dissemination in sensor networks is SPIN [11]. An obvious problem with normal epidemic broadcast-based dissemination is the inef cient use of bandwidth and other resources. Therefore, the basic epidemic strategy needs to be optimized for sensor networks. In reference 11, the authors proposed the concept of meta data or data descriptors to eliminate the chance of redundant transmissions in sensor networks. Their work focuses on the ef cient dissemination of individual sensor observations to all the sensors in a network. Their main contribution was based on the basic de ciencies of classic ooding, namely, Implosion, Overlap, and Resource Blindness. Implosion is sending data redundantly to one s neighbors regardless of whether they already received it. Coverage overlap of nodes can make them gather the same data and ood it to common neighbors. Classic ooding can be blind to the availability of resources when it is ooding data across the network. The use of metadata allows nodes to negotiate between themselves and prevent redundantly transmitting the same information. Also, in SPIN, each node has a local resource manager that keeps track of its resources and helps a node decide whether to transmit or process data. SPIN rst broadcasts metadata to its neighbors. Then, if it receives a request for the data from any neighbor it sends the data to that node. There are four protocols in the SPIN family. The rst two, SPIN-PP and SPIN-BC, tackle the basic problem of data dissemination under ideal conditions. The other two, SPIN-EC and SPIN-RL, are modi ed versions of the rst two. SPIN-PP is optimized for communicating in a point-to-point mode, where for each data transmission between neighbors, a three-stage handshaking (ADV-REQ-DATA) is
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