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The Firefighter Problem models the spread of a fire across a network. The usual version of the model assumes the fire spreads deterministically in discrete-time, possibly allowing the placement of firefighters at various times to contain the fire. The goal of our project is to study a stochastic version of this problem. Instead of fire spreading, we prefer to think of an infection spreading across a network of individuals. Our framework assumes that a single individual is initially infected, and at each time step an infected individual will independently transmit the disease, with a certain probability, to each of its uninfected neighbors. We find probability distributions to describe the uncertain number of vertices that are infected in a network at a given time. We investigate this phenomenon using different types of networks such as star graphs, paths, cycles, complete graphs and binary trees. Our results show that the number of infected vertices in star graphs, paths and cycles are essentially binomially distributed. On the other hand, more complicated networks such as complete graphs and binary trees require a new analysis technique. We use Markov chains to explore a viral outbreak on complete graphs and binary trees. For these graphs we provide partial results on the distribution of the number of infected vertices. Moreover, we present an algorithm to find the number of infected vertices in binary trees at a given time.
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Nasim, Anam, "Spread of a Virus on Networks: A Probabilistic Approach" (2013). Senior Projects Spring 2013. 206.