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epidemics using barcode printing for visual studio .net control to generate, create code 3/9 image in visual studio .net applications. Web application framework when all contagion a visual .net Code 3/9 ttempts simultaneously fail for tI steps in a row, and at this point it will be over. Thus, a key question with an SIS epidemic on a given contact network is to understand how long the outbreak will last and how many individuals will be affected at different points in time.

For contact networks where the structure is mathematically tractable, researchers have in fact proved knife-edge results for the SIS model similar to our dichotomy for branching processes. These results, for particular classes of contact networks, show that at a particular critical value of the contagion probability p, an SIS epidemic on the network will undergo a rapid shift from one that dies out quickly to one that persists for a very long time [52, 278]. This type of analysis tends to be mathematically quite complex, and this critical value of the contagion probability p depends in subtle ways on the structure of the network.

A Connection Between SIR and SIS Epidemics. Despite the differences between the SIR and SIS models, in fact it is possible to represent some of the basic variants of the SIS model as special cases of the SIR model. This surprising relationship is further evidence of the exibility of the basic epidemic models, in which formalisms de ned in different ways turn out to have very close connections to each other.

We describe the relationship for the SIS model with tI = 1, when each node is infectious for a single step before recovering. The key insight is that if we think about a node v as in fact being a different individual at each time step, then we can represent things so that nodes are never reinfected. Speci cally, given an instance of the SIS model with tI = 1, we create a separate copy of each node for each time step t = 0, 1, 2, 3 and onward.

We will call this the time-expanded contact network. Now, for each edge in the original contact network, linking a node v to a node w, we create edges in the time-expanded contact network from the copy of v at time t to the copy of w at time t + 1; this simply encodes the idea that node w can potentially catch the disease at time t + 1 if node v is infected at time t. Figure 21.

6(a) shows this construction applied to the contact network from Figure 21.5. The point is that the same SIS disease dynamics that previously circulated around in the original contact network can now ow forward in time through the time-expanded contact network, with copies of nodes that are in the I state at time t producing new infections in copies of nodes at time t + 1.

But on this time-expanded graph we have an SIR process, since any copy of a node can be treated as removed (R) once its one time step of infection is over; with this view of the process, we have the same distribution of outcomes as the original SIS process. Figure 21.6(b) shows the course of the SIR epidemic that corresponds to the SIS epidemic in Figure 21.

5.. 21.5 Synchronization The models we ve dev .net framework bar code 39 eloped give us a framework for thinking about various broader issues in the spread of disease. We already encountered one of these issues in the dichotomy for branching processes, which provided a formal basis for the sensitivity of outbreaks to small variations in contagiousness and for the crucial role of the basic reproductive number.

We now look at a related issue in the global dynamics of a.
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