In this paper a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. We investigate how these quantities vary with the clustering in the graph and find that, as the clustering increases, the epidemic threshold decreases. The network is modeled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if there is at least one group that they are both members of.
In this paper, a branching process approximation for the spread of a Reed-Frost epidemic on a network with tunable clustering is derived. The approximation gives rise to expressions for the epidemic threshold and the probability of a large outbreak in the epidemic. It is investigated how these quantities varies with the clustering in the graph and it turns out for instance that, as the clustering increases, the epidemic threshold decreases. The network is modelled by a random intersection graph, in which individuals are independently members of a number of groups and two individuals are linked to each other if and only if they share at least one group.
We present an approach to market-consistent multi-period valuation of insurance liability cash flows based on a two-stage valuation procedure. First, a portfolio of traded financial instrument aimed at replicating the liability cash flow is fixed. Then the residual cash flow is managed by repeated one-period replication using only cash funds. The latter part takes capital requirements and costs into account, as well as limited liability and risk averseness of capital providers. The cost-ofcapital margin is the value of the residual cash flow. We set up a general framework for the cost-of-capital margin and relate it to dynamic risk measurement. Moreover, we present explicit formulas and properties of the cost-of-capital margin under further assumptions on the model for the liability cash flow and on the conditional risk measures and utility functions. Finally, we highlight computational aspects of the cost-ofcapital margin, and related quantities, in terms of an example from life insurance.
We consider a random network evolving in continuous time in which new nodes are born and old may die, and where undirected edges between nodes are created randomly and may also disappear. The node population is Markovian and so is the creation and deletion of edges, given the node population. Each node is equipped with a random social index and the intensity at which a node creates new edges is proportional to the social index, and the neighbour is either chosen uniformly or proportional to the its social index in a modification of the model. We derive properties of the network as time and the node population tends to infinity. The degree distribution is shown to have a mixed Poisson distribution which may exhibit a heavy tail if the social index distribution has a heavy tail.Key words: Random networks, dynamic networks, birth and death process, mixed Poisson distribution. * Postal address: Mathematical Statistics, Stockholm University, SE-106 91, Sweden. E-mail: tomb@math.su.se, lindholm@math.su.se. Dynamic random networks in dynamic populationsTom Britton, Stockholm University * Mathias Lindholm, Stockholm University † May 13, 2009Abstract We consider a random network evolving in continuous time in which new nodes are born and old may die, and where undirected edges between nodes are created randomly and may also disappear. The node population is Markovian and so is the creation and deletion of edges, given the node population. Each node is equipped with a random social index and the intensity at which a node creates new edges is proportional to the social index, and the neighbour is either chosen uniformly or proportional to the its social index in a modification of the model. We derive properties of the network as time and the node population tends to infinity. The degree distribution is shown to have a mixed Poisson distribution which may exhibit a heavy tail if the social index distribution has a heavy tail.
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