This dissertation contributes to a methodology and a better understanding that can be used to study the effects of strategies during a pandemic, especially in multi-community networks. The dissertation is structured as the following: In the first chapter, we introduce the concept of networks and its properties, and node and link percolation, which is an important process embedded in networks. Then we discuss different epidemic models, among which the SIR model is representative of many infectious diseases, and can also be mapped into a link percolation problem. We bring up two quantities that are most important in evaluating the effectiveness of epidemic strategies, one is the total fraction of individuals ever been infected by the final steady state of the SIR model, the other is the peak fraction of infected throughout the process, the second of which has seldom been studied before. There have been many researches on epidemic models within isolated networks, but recently people start getting more interested in network of networks, due to its better representation of real world systems. So we study those two quantities and their dependence on the fraction of bridge nodes in multi-community networks, in the second and third chapters: In the second chapter, we look at the final steady state of the SIR (Susceptible-Infected-Recovered) model, which can be mapped as one cluster in a link percolation problem. Using the scaling relations for the cluster size distributions around the critical point within isolated networks, we find multiple regimes in a network with two communities so that the total fraction of individuals ever been infected asymptotically follows different power laws with the fraction of bridge nodes within each regime. We also find crossovers between neighbor regimes so that the power law exponent changes from one regime to the other. It is interesting to note that the power-law relations get steeper in regimes with smaller transmissibilities, so those epidemic strategies that reduce connections between communities are more effective in those regimes. In the third chapter, we look at the peak fraction of infected of the SIR model, which also shows power law relations with the fraction of bridge nodes in different regimes, as well as crossovers between regimes. We also find that the power-law relation for the peak fraction of infected with the fraction of bridge nodes is steeper than the one for the total fraction of individuals ever been infected in the same regime, which indicates that the peak fraction of infected is more sensitive to strategies that reduce connections between communities. This explains why strategies to flatten the curve are usually taken when there are limited medical resources.

This work describes a model of the piezoresistive behavior in nanocomposite sensors. These sensors are also called flexible sensors because the polymer matrix allows for large deformations without failure. The sensors have conductive nanoparticles dispersed through an insulative polymer matrix. The insulative polymer gaps between nanoparticles are assumed to be possible locations for electron tunneling. When the distance between two nanoparticles is small enough, electrons can tunnel from one nanoparticle to the next and ultimately through the entire sensor. The evolution of this gap distance with strain is important to understand the overall conductivity of the strain sensor. The gap evolution was modeled in two ways: (1) applying Poisson's contraction to the sensor as a homogenous material, referred to as Simple Poisson's Contraction (SPC) and (2) modeling the nanoparticle-polymer system with Finite Element Analysis (FEA). These two gap evolution models were tested in a random resistor network model where each polymer gap was treated as a single resistor in the network. The overall resistance was calculated by solving the resistor network system. The SPC approach, although much simpler, was sufficient for cases where various orientations of nanoparticles were used in the same sensor. The SPC model differed significantly from the FEA, however, in cases where nanoparticles had specific alignment, e.g. all nanoparticles parallel to the tensile axis. It was also found that the distribution used to determine initial gap sizes for the polymer gaps as well as the mean of that distribution significantly impacted the overall resistivity of the sensor.Another key part of this work was to determine if the piezoresistivity in the sensors follows a percolation type behavior under strain. The conductance versus strain curve showed the characteristic s-curve behavior of a percolative system. The conductance-strain curve was also compared to the effective medium and generalized effective medium equations and the latter (which includes percolation theory) fit the random resistor network much more closely. Percolation theory is, therefore, an accurate way to describe this polymer-nanoparticle piezoresistive system.Finally, the FEA and SPC models were compared against experimental data to verify their accuracy. There are also two design problems addressed: one to find the sensor with the largest gauge factor and another to determine how to remove the characteristic initial spike in resistivity seen in nanocomposite sensors.

An invasion percolation (IP) model was used to illustrate the effects of gravity on DNAPL migration into a horizontal water saturated fracture. While gravity is typically neglected in the conventional approach, this work demonstrated that gravity should often be included when modelling DNAPL invasion in water saturated fractures and provides an equation estimating the difference in invasion pattern between simulations including or neglecting gravity. The IP model was further utilized to examine the invasion of DNAPL saturated fractures by water. These simulated experiments focus on cases where covariance (COV), the ratio of the mean of the aperture field to the standard deviation of the aperture field) as well as when the fracture is inclined or declined from horizontal. Results show that when COV is greater than 0.1, then DNAPL will always remain in the fracture after waterflooding. Furthermore, fracture angles below -15 degrees permit the complete removal of DNAPL, while fractures oriented at higher angles do not. In order to study the transport of particles in water saturated fractures, physical experiments measuring the transport of 0.046 um and 0.55 um microspheres were undertaken on fractures where the geometry could be imported into a computer for comparative simulation analysis. Results demonstrated that during advection, particles generally travel at less than the velocity of the surrounding fluid. As well, hydrodynamic effects such as shear were shown to influence the effluent concentrations by increasing dispersion. Finally, the physical geometry of the fracture was shown to influence the particle pathway during transport and can limit the chances of particles adhering to a fracture wall, thus reducing dispersion and increasing peak concentration. The combined results of these studies show that fracture geometry has a significant effect on the mechanisms of transport in saturated fractures. / Thesis / Doctor of Philosophy (PhD) / This thesis describes the transport of contaminants in rock fractures in the environment. Specifically, the transport of denser than water liquids that are immiscible in water and particles are modelled and analysed. This work used experiments in order to calibrate these models for analysis. It was found that the local geometry of the fracture walls heavily influences the invasion pattern of immiscible dense fluids as well as the retention of the fluids after waterflooding (a first step in remediation). Particle transport was found to be heavily affected by the local geometry in the fracture, specifically lowering the likelihood of attachment to fracture walls limiting the filtration effects, and thus allowing greater contaminants to exit the fracture. Ultimately, these results lead to a greater understanding of the mechanisms of transport in fractured media.

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Land application of domestic wastewater effluent by rapid rate infiltration (i.e., percolation ponds) is a very successful and cost-effective method for wastewater management. Municipal percolation pond systems have been successfully operated in the United States for about 100 years. The disposal concept depends on a relatively high rate of secondary wastewater effluent infiltration into the soil by rapid percolation, either vertically or horizontally, away from the application surface area. This study was accomplished to determine infiltration rates at two working percolation pond systems and the variability of these rates, to compare the operating results with the initial design, and to create a stochastic computer based simulation program for design and operation. The initial study site was located west of Orlando, Florida, and consisted of a system of two percolation ponds. Daily readings were obtained on evaporation, rainfall, flow to the ponds, pond depth and groundwater table elevations. A mass balance inventory equation was formulated and the infiltration parameter was determined. A frequency distribution was created for the rainfall, evaporation and calculated infiltration from the initial site, and then a stochastic computer based simulation program was written with this data. The program calculated results which compared favorably with the design for this initial percolation pond site. A second site was chosen, also located in the Orlando area, to confirm the usefulness of the program and its operational capabilities.

Stochastic and deterministic approaches for modeling complex networks are presented. The methodology combines analysis of the structure formed by the interconnections among the elements of a network with an assessment of the vulnerability towards the propagation of cascading failures. The goal is to understand the mutual interplay between the structure of the network connections and the propagation of cascading failures. Two fundamental issues related to the optimal design and operation of complex networks are addressed. The first concerns the impact that cascading failures have on networks due to the connectivity pattern linking their components. If the state of load on the network components is high, the risk of cascade spreadings becomes significant. In this case, the needed reduction of the connectivity efficiency to prevent the propagation of failures affecting the entire system is quantified. The second issue concerns the realization of the most efficient connectivity in a network that minimizes the propagations of cascading failures. It is found that a system that routinely approaches the critical load for the onset of cascading failures during its operation should have a larger efficiency value. This allows for a smoother transition to the cascade region and for a reasonable reaction time to counteract the onset of significant cascading failures. The interplay between the structure of the network connections and the propagation of cascading failures is assessed also in interdependent networks. In these systems, the linking among several network infrastructures is necessary for their optimal and economical operation. Yet, the interdependencies introduce weaknesses due to the fact that failures may cascade from one system to other interdependent systems, possibly affecting their overall functioning. Inspired by the global efficiency, a measure of the communication capabilities among interdependent systems, i.e. the interdependency efficiency, is defined. The relations between the structural parameters, i.e. the system links and the interdependency links, and the interdependency efficiency, are also quantified, as well as the relations between the structural parameters and the vulnerability towards the propagation of cascading failures. Resorting to this knowledge, the optimal interdependency connectivity is identified. Similar to the spreading of failures, the formation of a giant component is a critical phenomenon emerging as a result of the connectivity pattern in a network. This structural transition is exploited to identify the formation of macrometastases in the developed model for metastatic colonization in tumor growth. The methods of network theory proves particularly suitable to reproduce the local interactions among tumor cells that lead to the emergent global behavior of the metastasis as a community. This model for intercellular sensing reproduces the stepwise behavior characteristic of metastatic colonization. Moreover, it prompts the consideration of a curative intervention that hinders intercellular communication, even in the presence of a significant tumor cell population. / Ph. D.

In this thesis, we focus on the steady state properties of two systems which are genuinely out of equilibrium. The first project is an application of dynamic field theory to a specific non equilibrium critical phenomenon, while the second project involves both simulations and analytical calculations. The methods of field theory are used on both these projects. In the first part of this thesis, we investigate a generalization of the well-known field theory for directed percolation (DP). The DP theory is known to describe an evolving population, near extinction. We have coupled this evolving population to an environment with its own nontrivial spatio-temporal dynamics. Here, we consider the special case where the environment follows a simple relaxational (model A) dynamics. We find two marginal couplings with upper critical dimension of four, which couple the two theories in a nontrivial way. While the Wilson-Fisher fixed point remains completely unaffected, a mismatch of time scales destabilizes the usual DP fixed point. Some open questions and future work remain. In the second project, we focus on a simple particle transport model far from equilibrium, namely, the totally asymmetric simple exclusion process (TASEP). While its stationary properties are well studied, many of its dynamic features remain unexplored. Here, we focus on the power spectrum of the total particle occupancy in the system. This quantity exhibits unexpected oscillations in the low density phase. Using standard Monte Carlo simulations and analytic calculations, we probe the dependence of these oscillations on boundary effects, the system size, and the overall particle density. Our simulations are fitted to the predictions of a linearized theory for the fluctuation of the particle density. Two of the fit parameters, namely the diffusion constant and the noise strength, deviate from their naive bare values [6]. In particular, the former increases significantly with the system size. Since this behavior can only be caused by nonlinear effects, we calculate the lowest order corrections in perturbation theory. Several open questions and future work are discussed. / Ph. D.