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Determining Key Parameters in Riots Using Lexicographic Directional Differentiation
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| Pengarang | : | Matthew Ackley |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1303-1331 |
| Abstrak | : | This article investigates parametric sensitivities of rioting activity, based on a model of the 2005 French riots. Previously validated against a comprehensive data set, the aforementioned modeling efforts were multiscale in nature and applied at the provincial scale for all of France and the municipal scale in the I?le-de-France region around Paris. The governing dynamics at the municipal scale include a bandwagon effect, observed in rioting activity wherein an outburst of activity occurs once a “tipping point” threshold is reached in the people engaging in the activity. This tipping point mechanism is modeled using a nonsmooth function, yielding a framework of nonsmooth ODEs, which makes conventional theory and methods unsuitable. In this article, we perform a sensitivity analysis of the municipal-scale model by adopting a recently developed tool in nonsmooth analysis called the lexicographic directional derivative. Numerical solutions are provided from which we can conclude the relative importance of different parameters in the model, in order to discover the underlying mechanisms that play a crucial role in driving the riot activities. It is shown that the exit/removal rate from rioting activity and geographic proximity have the greatest impact on the dynamics of the social contagion. On the other hand, the initial rioting activity and the tipping point seem less influential. |
On the Characterization of Nonradiating Sources for the Elastic Waves in Anisotropic Inhomogeneous Media
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 4) |
| Halaman | : | 1530-1551 |
| Abstrak | : | In this paper, we would like to characterize nonradiating volume and surface (faulting) sources for the elastic waves in anisotropic inhomogeneous media. Each type of source can be decomposed into a radiating part and a nonradiating part. The radiating part can be uniquely determined by an explicit formula containing the near-field measurements. On the other hand, the nonradiating part does not induce scattered waves at a certain frequency. In other words, such nonradiating source cannot be detected by measuring the field at a single frequency in a region outside of the domain where the source is located. |
Diffusion in a Disk with a Circular Inclusion
| Pengarang | : | Remus Stana |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1287-1302 |
| Abstrak | : | We consider diffusion in a disk, representing a cell with a circular interior compartment. Using bipolar coordinates, we perform exact calculations, not restricted by the size or location of the intracellular compartment. We find Green functions, hitting densities and mean times to move from the compartment to the cellular surface and vice versa. For molecules with diffusivity ????, mean times are proportional to ????2/????, where ???? is the radius of the cell. We find explicit expressions for the dependence on ????2 (the fraction of the cell occupied by the intracellular compartment) and on the displacement of the compartment from the center of the cell. We consider distributions of initial conditions that are (i) uniform on the nuclear surface, (ii) uniform on the cellular surface, or (iii) given by the hitting density of particles diffusing from the nuclear to the cellular surface. |
Mathematical Analysis of a Three-Tiered Model of Anaerobic Digestion
| Pengarang | : | Sarra Nouaoura |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1264-1286 |
| Abstrak | : | In this paper, we are interested in a mechanistic model describing the anaerobic mineralization of chlorophenol in a three-step food web. The model is a six-dimensional system of ordinary differential equations. In our study, we take into account the phenol and the hydrogen input concentrations as well as the maintenance terms. Moreover, we consider the case of a large class of growth rates, instead of specific kinetics. In this general case, a recent study shows that the system can have up to eight steady states, and their existence conditions were analytically determined. We focus here on the necessary and sufficient conditions of the local stability of the steady states, according to the four operating parameters of the process, which are the dilution rate and the chlorophenol, phenol, and hydrogen input concentrations. In previous studies, this stability analysis was performed only numerically. Using the Lie?nard--Chipart stability criterion, we show that the positive steady state can be unstable and we give numerical evidence for a supercritical Hopf bifurcation with the appearance of a stable periodic orbit. We give two bifurcation diagrams with respect to the dilution rate, first, and then to the chlorophenol input concentration as the bifurcating parameters, showing that the system can present rich behavior, including bistability, coexistence, and occurrence of limit cycle. |
Stable Steady-State Solutions of Some Biological Aggregation Models
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1248-1263 |
| Abstrak | : | Aggregation phenomena occur across the biological sciences, from cell adhesion to insect swarms, animal home ranges to human cities. Understanding the mechanisms by which they may spontaneously emerge has therefore generated much interest from applied mathematicians. Partial differential equations (PDEs) with nonlocal advection offer a popular formalism for studying aggregations. However, the inherent nonlocality, often necessary for ensuring continuum models are well-posed, makes their study technically challenging. Here, we take a different approach by studying a discrete-space system that can be formally related to classical nonlocal PDE approaches via a limiting procedure. We show how to find expressions for the asymptotically stable steady-states of this discrete-space system via an energy functional approach. This allows us to predict the size of aggregations as a function of the underlying movement mechanisms of individual organisms. We apply this to a recent model of cell adhesion, revealing a hysteresis property whereby the existing aggregations may persist even as the adhesion tendency decreases past the bifurcation point. We compare this to numerical solutions of the associated nonlocal PDE system, showing that the hysteresis property predicted by the discrete-space expressions is also present in the continuum system. |
From Life-Saving to Life-Threatening: A Mathematical Model to Simulate Bacterial Infections in Surgical Procedures
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1226-1247 |
| Abstrak | : | Following the implantation of indwelling medical devices, bacteria inoculated during the surgery or coming from a preexistent focus of infection race for the medical surface where they attach. Adaptation to survive is a common feature of life, and microorganisms are not an exception. Bacteria form, in short periods of time, a habitat---the biofilm---where they develop multiresistance and tolerance to antibiotics and to the host immune system. To avoid its formation, researchers in the biomedical sciences showed evidence that coating medical devices with antibacterial agents---antibiotics---is a promising strategy. We present a mathematical model to simulate the action of an antibiotic, released from a medical surface, to fight bacterial infection. The model is composed by a system of partial differential equations that describe the distribution of drug and the evolution of a bacterial population. The preexistence of infection focus, the inoculation of bacteria during the surgery, the race for the medical surface, the resistance and tolerance of the population are taken into account. Analytical estimates of the bacterial density show the crucial importance of aseptic surgical procedures and of timely detection of preexisting infection focus. Numerical simulations illustrate several scenario. |
Synchronization in a Power Balance System With Inertia and Nonlinear Derivatives
| Pengarang | : | Sun-Ho Choi |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1202-1225 |
| Abstrak | : | We consider synchronization of the all-to-all power grid model with inertia and nonlinear terms of the first- and second-order derivatives. This model was derived from the energy conservation law in [G. Filatrella, A. H. Nielsen, and N. F. Pedersen, Eur. Phys. J. B, 61 (2008), pp. 485--491]. Since the model contains a nonlinear differential operator, we cannot use a Lyapunov functional and an a priori estimate method directly. In this paper, we provide a new framework to obtain complete phase and frequency synchronization of the power grid model. We assume that the coupling strength and the moment of inertia are positive and the initial data are near the synchronization state. After applying an appropriate transform, we derive a system of differential equations whose coefficient matrix is negative definite. The initial data assumption and the negative definite coefficient matrix lead to the global existence and synchronization of the power grid model with a nonlinear differential operator. For both identical and nonidentical cases, the exponential decay for the frequency difference is determined and the square of frequencies converges to the average of the normalized energy feeding rates. |
Boundary Value Problem on a Weighted Graph Relevant to the Static Analysis of Truss Structures
| Pengarang | : | Costy Kodsi |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1190-1201 |
| Abstrak | : | The boundary value problem on a weighted graph relevant to the static analysis of truss structures with linear elastic bars is presented along with strategies for assembling the corresponding algebraic system of equations. |
Impact of Varying Community Networks on Disease Invasion
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1166-1189 |
| Abstrak | : | We consider the spread of an infectious disease in a heterogeneous environment modeled as a network of patches. We focus on the invasibility of the disease, as quantified by the corresponding value of an approximation to the network basic reproduction number, R0, and study how changes in the network structure affect the value of R0. We provide a detailed analysis for two model networks, a star and a path, and discuss the changes to the corresponding network structure that yield the largest decrease in R0. We develop both combinatorial and matrix analytic techniques, and we illustrate our theoretical results by simulations with the exact R0. |
Shape Optimization of Fixed-Bed Reactors in Process Engineering
| Pengarang | : | Alexis Courtais |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 81 (No. 3) |
| Halaman | : | 1141-1165 |
| Abstrak | : | This paper deals with geometric shape optimization of a parallelepiped shaped fixed-bed reactor with single phase liquid flow where a chemical reaction takes place. The packing is a sort of static mixer made up of solid cylindrical obstacles uniformly distributed in the reactor. The reactive flow is described by means of momentum and mass transport equations in laminar flow regime. The objective is to determine the shape of the packing that maximizes the reaction conversion rate subjected to some specified manufacturing constraints. The optimization approach developed is based on the adjoint system method and the results show that the optimal shape obtained allows one to significantly improve the reaction conversion rate. Furthermore, the optimal shape is printed by means of an additive manufacturing technique and several manufacturing constraints mainly related to the thickness of packing are considered. |