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As most disease causing pathogens require transmission from an infectious individual to a susceptible individual, continued persistence of the pathogen within the population requires the replenishment of susceptibles through births, immigration, or waning immunity. Consider the introduction of an unknown infectious disease into a fully susceptible population where it is not known how long immunity is conferred once an individual recovers from infection. If, initially, the prevalence of disease increases (that is, the infection takes off), the number of infectives will usually decrease to a low level after the first major outbreak. During this post-outbreak period, the disease dynamics may be influenced by stochastic effects and there is a non-zero probability that the epidemic will die out. Die out in this period following the first major outbreak is known as an epidemic fade-out. If the disease does not die out, the susceptible population may be replenished by the waning of immunity, and a second wave may start. In this study, we investigate if the rate of waning immunity (and other epidemiological parameters) can be reliably estimated from multiple outbreak data, in which some outbreaks display epidemic fade-out and others do not. We generated synthetic outbreak data from independent simulations of stochastic SIRS models in multiple communities. Some outbreaks faded-out and some did not. We conducted Bayesian parameter estimation under two alternative approaches: independently on each outbreak and under a hierarchical framework. When conducting independent estimation, the waning immunity rate was poorly estimated and biased towards zero when an epidemic fade-out was observed. However, under a hierarchical approach, we obtained more accurate and precise posterior estimates for the rate of waning immunity and other epidemiological parameters. The greatest improvement in estimates was obtained for those communities in which epidemic fade-out was observed. Our findings demonstrate the feasibility and value of adopting a Bayesian hierarchical approach for parameter inference for stochastic epidemic models. [ABSTRACT FROM AUTHOR]

We prove the conjecture formulated in Litvak and Ejov (2009), namely, that the trace of the fundamental matrix of a singularly perturbed Markov chain that corresponds to a stochastic policy feasible for a given graph is minimised at policies corresponding to Hamiltonian cycles. [ABSTRACT FROM AUTHOR]

A stochastic model is formulated and analyzed to study the advancements of messages under greedy routing in a sensor network with a power-saving scheme. The aim of this model is give a better understanding of stochastic dependencies arising in the system and to offer a method of computing the advancement of message under greedy routing. We observe that the majority of the stochastic dependence from a routing path is captured by including only the previous forwarding node location. We examine a simple uncoordinated power-saving scheme with a new understanding of its effects on local node density. We propose a method for sensibly limiting the number of transmission reattempts before concluding there is no node in the forwarding region. All expressions involving multidimensional integrals are derived and evaluated readily with quasi- Monte Carlo integration methods. An importance sampling function is derived to speed up the quasi-Monte Carlo methods. The integral expressions are compared to simulations and give very agreeable results. [ABSTRACT FROM AUTHOR]

In this paper we introduce a structure called the Markovian tree (MT). We define the MT and explore its alternative representation as a continuous-time Markovian Multitype Branching Process. We then develop two algorithms, the Depth and Order algorithms to determine the probability of eventual extinction of the MT process. We show that both of these algorithms have very natural physically intuitive interpretations and are analogues of the Neuts and U algorithms in Matrix-analytic Methods. Furthermore, we show that a special case of the Depth algorithm sheds new light on the interpretation of the sample paths of the Neuts algorithm. [ABSTRACT FROM AUTHOR]

We consider several known algorithms and introduce some new algorithms that can be used to calculate the probability of return to the initial level in the Markov stochastic fluid flow model. We give the physical interpretations of these algorithim within the fluid flow environment. The rates of convergence are explained in terms of the physical properties of the fluid flow processes. We compare these algorithms with respect to the numbers of iterations required and their complexity. The performance of the algorithms depends on the nature of the process considered in the analysis. We illustrate this with examples and give appropriate recommendations. [ABSTRACT FROM AUTHOR]

The existence of product form solutions in networks of queues has been known for many years. Recently the authors have shown that there is a class of stochastic Petri nets (SPN's), distinct from queueing networks, which also have a product form equilibrium distribution. Most practical SPN's, however, do not fall within this class. In this paper the class of SPN's which can be analyzed using these methods is extended in several directions. Using embedded discrete time processes we study a class of SPN's which have a closed form equilibrium distribution. These SPN's have probabilistic output bags, colored tokens, and alternating periods of arbitrarily distributed enabling and firing times (periods of time between transitions becoming enabled and absorption of tokens and between transitions absorbing tokens and depositing them in output places, respectively.) In addition an aggregation procedure is proposed which, in certain nets, not only reduces a complex SPN to a much simpler skeleton SPN but obtains results for the skeleton SPN which are exact marginal distributions for the original SPN. [ABSTRACT FROM AUTHOR]