Stochastic hybrid systems arise in numerous applications of systems with multiple models; e.g., air traffc management, flexible manufacturing systems, fault tolerant control systems etc. In a typical hybrid system, the state space is hybrid in the sense that some components take values in a Euclidean space, while some other components are discrete. In this paper we propose two stochastic hybrid...

The purpose of theses lectures is to introduce students to some of the interesting phenomena observed with stochastic processes which are evolving on two widely separated time scales. The study, and even definition of such processes involve a wide range of necessary concepts and tools from the theory of probability and stochastic processes. With the short time at our disposal, these notes are a...

Diverse modeling and simulation methods are being applied in the area of Systems Biology. Most models in Systems Biology can easily be located within the space that is spanned by three dimensions of modeling: continuous and discrete; quantitative and qualitative; stochastic and deterministic. These dimensions are not entirely independent nor are they exclusive. Many modeling approaches are hybr...

We study the competition between several advantageous mutants in an asexual population (clonal interference) as a function of the time between the appearance of the mutants ∆t, their selective advantages, and the rate of deleterious mutations. We find that the overall probability of fixation (the probability that at least one of the mutants becomes the ancestor of the entire population) does no...

We analyze the dynamics of growth of the number of congressmen supporting the resolution HR1207 to audit the Federal Reserve. The plot of the total number of co-sponsors as a function of time is of “Devil’s staircase” type. The distribution of the numbers of new co-sponsors joining during a particular day (step height) follows a power law. The distribution of the length of intervals between add...

High speed trains are currently meant to run faster and to carry heavier loads, while being less energy consuming and still respecting the security and comfort certification criteria. To face these challenges, a better understanding of the interaction between the dynamic train behavior and the track geometry is needed. As during its lifecycle, the train faces a great variability of track condit...

We review the topic of stochastic epidemic modeling with emphasis on compartmental stochastic models. A main theme is the usefulness of the correspondence between these and their large population deterministic limits, which describe dynamical systems. The dynamics of an ODE system informs us of the deterministic skeleton upon which the behavior of corresponding stochastic systems are built. In ...

Carcinogenesis is the transformation of normal cells into cancer cells. This process has been shown to be of a multistage nature, with stem cells that go through a series of genetic and epigenetic changes that eventually lead to a malignancy. Since the origins of the multistage theory in the 1950s, mathematical modeling has played a prominent role in the investigation of the mechanisms of carci...

We consider a firm moving towards a stochastic final destination, to be chosen from a discrete set after a decision period. The decision period itself may be deterministic or stochastic. We assume the firm can move at variable innovation (R&D) speed associated with a monotone nondecreasing variable cost, and it can also stop and move anywhere. There is a fixed cost per time unit "carried" by th...

The present work describes the structure of a generic modeling tool for stochastic dynamic programming problems (also called Markov Decision Processes – MDPs). Deterministic problems are handled as particular cases of MDP’s. The program is intended for general purpose problems and its structure permits independent modeling and solving. It is designed to capture the essential modeling elements o...