Matching Items (5)
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- All Subjects: Stochastic Processes
- Creators: Lanchier, Nicolas
- Creators: Kang, Yun
Description
Mathematical modeling of infectious diseases can help public health officials to make decisions related to the mitigation of epidemic outbreaks. However, over or under estimations of the morbidity of any infectious disease can be problematic. Therefore, public health officials can always make use of better models to study the potential implication of their decisions and strategies prior to their implementation. Previous work focuses on the mechanisms underlying the different epidemic waves observed in Mexico during the novel swine origin influenza H1N1 pandemic of 2009 and showed extensions of classical models in epidemiology by adding temporal variations in different parameters that are likely to change during the time course of an epidemic, such as, the influence of media, social distancing, school closures, and how vaccination policies may affect different aspects of the dynamics of an epidemic. This current work further examines the influence of different factors considering the randomness of events by adding stochastic processes to meta-population models. I present three different approaches to compare different stochastic methods by considering discrete and continuous time. For the continuous time stochastic modeling approach I consider the continuous-time Markov chain process using forward Kolmogorov equations, for the discrete time stochastic modeling I consider stochastic differential equations using Wiener's increment and Poisson point increments, and also I consider the discrete-time Markov chain process. These first two stochastic modeling approaches will be presented in a one city and two city epidemic models using, as a base, our deterministic model. The last one will be discussed briefly on a one city SIS and SIR-type model.
ContributorsCruz-Aponte, Maytee (Author) / Wirkus, Stephen A. (Thesis advisor) / Castillo-Chavez, Carlos (Thesis advisor) / Camacho, Erika T. (Committee member) / Kang, Yun (Committee member) / Arizona State University (Publisher)
Created2014
Description
This dissertation examines six different models in the field of econophysics using interacting particle systems as the basis of exploration. In each model examined, the underlying structure is a graph G = (V , E ), where each x ∈ V represents an individual who is characterized by the number of coins in her possession at time t. At each time step t, an edge (x, y) ∈ E is chosen at random, resulting in an exchange of coins between individuals x and y according to the rules of the model. Random variables ξt, and ξt(x) keep track of the current configuration and number of coins individual x has at time t respectively. Of particular interest is the distribution of coins in the long run. Considered first are the uniform reshuffling model, immediate exchange model and model with saving propensity. For each of these models, the number of coins an individual can have is nonnegative and the total number of coins in the system is conserved for all time. It is shown here that the distribution of coins converges to the exponential distribution, gamma distribution and a pseudo gamma distribution respectively. The next two models introduce debt, however, the total number of coins again remains fixed. It is shown here that when there is an individual debt limit, the number of coins per individual converges to a shifted exponential distribution. Alternatively, when a collective debt limit is imposed on the whole population, a heuristic argument is given supporting the conjecture that the distribution of coins converges to an asymmetric Laplace distribution. The final model considered focuses on the effect of cooperation on a population. Unlike the previous models discussed here, the total number of coins in the system at any given time is not bounded and the process evolves in continuous time rather than in discrete time. For this model, death of an individual will occur if they run out of coins. It is shown here that the survival probability for the population is impacted by the level of cooperation along with how productive the population is as whole.
ContributorsReed, Stephanie Jo (Author) / Lanchier, Nicolas (Thesis advisor) / Smith, Hal (Committee member) / Gumel, Abba (Committee member) / Motsch, Sebastien (Committee member) / Camacho, Erika (Committee member) / Arizona State University (Publisher)
Created2019
Description
We model communication among social insects as an interacting particle system in which individuals perform one of two tasks and neighboring sites anti-mimic one another. Parameters of our model are a probability of defection 2 (0; 1) and relative cost ci > 0 to the individual performing task i. We examine this process on complete graphs, bipartite graphs, and the integers, answering questions about the relationship between communication, defection rates and the division of labor. Assuming the division of labor is ideal when exactly half of the colony is performing each task, we nd that on some bipartite graphs and the integers it can eventually be made arbitrarily close to optimal if defection rates are sufficiently small. On complete graphs the fraction of individuals performing each task is also closest to one half when there is no defection, but is bounded by a constant dependent on the relative costs of each task.
ContributorsArcuri, Alesandro Antonio (Author) / Lanchier, Nicolas (Thesis director) / Kang, Yun (Committee member) / Fewell, Jennifer (Committee member) / Barrett, The Honors College (Contributor) / School of International Letters and Cultures (Contributor) / Economics Program in CLAS (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2015-05
Description
This thesis explores and explains a stochastic model in Evolutionary Game Theory introduced by Dr. Nicolas Lanchier. The model is a continuous-time Markov chain that maps the two-dimensional lattice into the strategy space {1,2}. At every vertex in the grid there is exactly one player whose payoff is determined by its strategy and the strategies of its neighbors. Update times are exponential random variables with parameters equal to the absolute value of the respective cells' payoffs. The model is connected to an ordinary differential equation known as the replicator equation. This differential equation is analyzed to find its fixed points and stability. Then, by simulating the model using Java code and observing the change in dynamics which result from varying the parameters of the payoff matrix, the stochastic model's phase diagram is compared to the replicator equation's phase diagram to see what effect local interactions and stochastic update times have on the evolutionary stability of strategies. It is revealed that in the stochastic model altruistic strategies can be evolutionarily stable, and selfish strategies are only evolutionarily stable if they are more selfish than their opposing strategy. This contrasts with the replicator equation where selfishness is always evolutionarily stable and altruism never is.
ContributorsWehn, Austin Brent (Author) / Lanchier, Nicolas (Thesis director) / Kang, Yun (Committee member) / Motsch, Sebastien (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / School of International Letters and Cultures (Contributor)
Created2013-12
Description
We study two models of a competitive game in which players continuously receive points and wager them in one-on-one battles. In each model the loser of a battle has their points reset, while the points the winner receives is what sets the two models apart. In the knockout model the winner receives no new points, while in the winner-takes-all model the points that the loser had are added to the winner's total. Recurrence properties are assessed for both models: the knockout model is recurrent except for the all-zero state, and the winner-takes-all model is transient, but retains some aspect of recurrence. In addition, we study the population-level allocation of points; for the winner-takes-all model we show explicitly that the proportion of individuals having any number j of points, j=0,1,... approaches a stationary distribution that can be computed recursively. Graphs of numerical simulations are included to exemplify the results proved.
ContributorsVanKirk, Maxwell Joshua (Author) / Lanchier, Nicolas (Thesis director) / Foxall, Eric (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2016-12