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Probabilistic Modeling of Car Traffic Accidents
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| Pengarang | : | Simone Gottlich, Thomas Schillinger, Andrea Tosin |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 3) |
| Halaman | : | 1099-1120 |
| Abstrak | : | We introduce a counting process to model the random occurrence in time of car traffic accidents, taking into account some aspects of the self-excitation typical of this phenomenon. By combining methods from probability and differential equations, we study this stochastic process in terms of its statistical moments and large-time trend. Moreover, we derive analytically the probability density functions of the times of occurrence of traffic accidents and of the time elapsing between two consecutive accidents. Finally, we demonstrate the suitability of our modeling approach by means of numerical simulations, which also address a comparison with real data of weekly trends of traffic accidents. |
On a Spatial Mosquito-Borne Disease Model with Density-Dependent Dispersal
| Pengarang | : | Kai Wang, Peng Wu, Qian Ding, Jianshe Yu |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 3) |
| Halaman | : | 1067-1098 |
| Abstrak | : | Chemotaxis of mosquito movement has been widely confirmed, but it has rarely been incorporated into infectious disease modeling. This paper proposes a novel mosquito-borne disease model driven by density-suppressed motility and spatial heterogeneity. Via overcoming the difficulties caused by dispersal mechanism and heterogeneity, we establish the global existence and ultimate boundedness of solutions for the model following a priori estimates and the parabolic regularity theory. We define the basic reproduction ratio R0 and derive its upper bound ˜R0 and lower bound ˆR0, and obtain the threshold dynamics of model. Specifically, we prove that the system is uniformly persistent as ˆR0 >1 and the disease will disappear as ˜R0 <1 by invoking the comparison principle to the corresponding integral system. In the case of spatial homogeneity, the global asymptotic stability of endemic equilibrium is achieved with the aid of a suitable Lyapunov functional and the eigenvalue theory. Numerically, we apply the model to dengue transmission and demonstrate that the chemotaxis of infected mosquitoes may exacerbate the disease in some regions which highlights the impacts of mosquito taxis on the spread of dengue. |
Mass Action Systems: Two Criteria for Hopf Bifurcation Without Hurwitz
| Pengarang | : | Nicola Vassena |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 3) |
| Halaman | : | 1046-1066 |
| Abstrak | : | We state two sufficient criteria for periodic oscillations in mass action systems. Neither criterion requires a computation of the Hurwitz determinants. Instead, both criteria exploit the linear algebra concepts of ????-stability and ????-matrices. The criteria are complementary: the first is based on a stable matrix that is not a ????− matrix, while the second is based on a ????− matrix that is not stable. In analogy, a qualitatively different interpretation follows: the first criterion relates to positive feedback in the network, while the second concerns negative feedback. We present examples that showcase the applicability of both criteria. As a final independent remark, we prove that for the special case of fully open networks, the capacity for Hopf bifurcation is just equivalent to the capacity for a steady state with a complex pair of eigenvalues with positive real part. |
Tristability and Elite Control in an HIV Infection Model with Natural Killer Cells Effect and Immune Impairment
| Pengarang | : | Shaoli Wang, Tengfei Wang, Jianhong Wu, Tianhai Tian |
| Nama Majalah/Jurnal | : | Siam Journal On Applied Mathematics |
| Volume / Edisi | : | 85 (No. 3) |
| Halaman | : | 1023-1045 |
| Abstrak | : | Recent biomedical studies demonstrate that natural killer (NK) cells play an important role in inhibiting HIV infection. However, the detailed mechanisms of this inhibiting function are not clear. In this paper, we develop a mathematical model that incorporates the effects of NK cells and immune impairment into the dynamics of HIV infection. Multiple threshold values are introduced to classify the dynamic behaviors of the model, including the previously reported bistability and a novel tristability. Bifurcation analysis is conducted to illustrate how bistability and, in particular, tristability take place through saddle-node bifurcation, backward bifurcation, and forward bifurcation. We also introduce a definition of robustness for systems with multiple equilibria and analyze the robustness properties of the proposed system under various conditions. Compared to models without the effects of NK cells, our results suggest that the NK cells can effectively reduce the virus load, which implies that it is less possible for virus to rebound if the functions of NK cells are effective. |
Second Linearization of Galois Theory
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 7) |
| Halaman | : | 676-691 |
| Abstrak | : | This paper presents an alternative exposition of the fundamental theorem of Galois theory and the Abel–Ruffini theorem which is based on linear algebra and self-contained up to explicitly stated prerequisites. Notable features of the exposition include a linear-algebraic definition of Galois extensions based on a diagonalizability condition, a proof of the fundamental theorem of Galois theory based on extension of scalars and linear independence of field automorphisms, a proof of the Abel–Ruffini theorem based on eigenvalues of permutations, and several discussions on mathematical and pedagogical topics of independent interest. |
Modeling Still Matters: A Surprising Instance of Catastrophic Floating Point Errors in Mathematical Biology and Numerical Methods for ODEs
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 67 (No. 3) |
| Halaman | : | 624-641 |
| Abstrak | : | We guide the reader on a journey through mathematical modeling and numerical analysis, emphasizing the crucial interplay of both disciplines. Targeting undergraduate students with basic knowledge of dynamical systems and numerical methods for ordinary differential equations, we explore a model from mathematical biology where numerical methods fail badly due to catastrophic floating point errors. We analyze the reasons for this behavior by studying the steady states of the model and use the theory of invariants to develop an alternative model suited for numerical simulations. Our story is intended to motivate the combining of analytical knowledge and numerical knowledge, even in those cases where the world looks fine at first sight. We have set up an online repository containing an interactive notebook with all the numerical experiments in this article to make this study fully reproducible and useful for classroom teaching. |
Diffusion Models for Generative Artificial Intelligence: An Introduction for Applied Mathematicians
| Pengarang | : | Catherine Higham |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 67 (No. 3) |
| Halaman | : | 607-623 |
| Abstrak | : | Generative artificial intelligence (GAI) refers to algorithms that create synthetic but realistic output. Diffusion models currently offer state-of-the-art performance in GAI for images. They also form a key component in more general tools, including text-to-image generators and large language models. Diffusion models work by adding noise to the available training data and then learning how to reverse the process. The reverse operation may then be applied to new random data in order to produce new outputs. We provide a brief introduction to diffusion models for applied mathematicians and statisticians. Our key aims are to (a) present illustrative computational examples, (b) give a careful derivation of the underlying mathematical formulas involved, and (c) draw a connection with partial differential equation (PDE) diffusion models. We provide code for the computational experiments. We hope that this topic will be of interest to advanced undergraduate and postgraduate students. Portions of the material may also provide useful motivational examples for those who teach courses in stochastic processes, inference, machine learning, PDEs, or scientific computing. |
Alexandrov’s Soap Bubble Theorem for Polygons
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 7) |
| Halaman | : | 666-675 |
| Abstrak | : | Regular polygons are characterized as area-constrained critical points of the perimeter functional with respect to particular families of perturbations in the class of polygons with a fixed number of sides. We also review recent results in the literature involving other shape functionals as well as further open problems. |
Optimal Survival Strategies for Diffusive Flows: A Schrödinger Bridge Approach to Unbalanced Transport
| Pengarang | : | Yongxin Chen |
| Nama Majalah/Jurnal | : | Siam Review |
| Volume / Edisi | : | 67 (No. 3) |
| Halaman | : | 579-604 |
| Abstrak | : | Diffusive flows, and their discrete counterparts, are ubiquitous in the physical and engineering sciences. In many important examples, the total mass is not preserved and therefore standard probabilistic models are not suitable. Examples include electrons which may be absorbed by the medium in which they travel. In population genetics, some individuals may “disappear” due to their genotype. In traffic flows over a network, some vehicles might simply exit the circulation and park. In this more general situation, where some of the mass may be lost, it is of particular interest to reconcile the observed initial and final marginal distributions with a given prior. In the case when the two marginals are probability distributions, and thus of equal mass, this problem was posed and, to a considerable extent, solved by E. Schrödinger in 1931/32. It is now known as the Schrödinger Bridge Problem (SBP). It turns out that Schrödinger’s problem can be viewed as both a modeling and a control problem. Due to the fundamental significance of this problem, interest in the SBP and in its deterministic (zero-noise limit) counterpart of optimal mass transport (OMT) has in recent years enticed scientists from a broad spectrum of disciplines, including physics, stochastic control, computer science, probability theory, and geometry. Yet, while the mathematics and applications of SBP/OMT have been developing at a considerable pace, accounting for marginals of unequal mass has received scant attention. The problem of interpolating between “unbalanced” marginals has been approached by introducing source/sink terms into the transport equations in an ad hoc manner, chiefly driven by applications in image registration. Nevertheless, as hinted at above, losses are inherent in many physical processes and, thereby, models that account for lossy transport may also need to be reconciled with observed marginals following Schrödinger’s quest, that is, to adjust the probability of trajectories of particles, including those that do not make it to the terminal observation point, so that the updated evolution represents the most likely way that particles may have been transported, or vanished, at some intermediate point. Thus, the purpose of this work is to develop such a natural generalization of the SBP for diffusive evolution with losses, whereupon particles are “killed” (jump into a coffin/extinction state) according to a probabilistic law, and thereby mass is gradually lost along their stochastically driven flow. Through a suitable embedding, which appears to be novel, we turn the problem into an SBP for stochastic processes that combine diffusive and jump characteristics. Then, following a large-deviations formalism in the style of E. Schrödinger, given a prior law that allows for losses, we ask for the most probable evolution of particles along with the most likely killing rate as the particles transition between the specified marginals. Our approach differs sharply from previous work involving a Feynman–Kac multiplicative reweighing of the reference measure. The latter, as we argue, is far from Schrödinger’s quest. An iterative scheme, generalizing the celebrated Fortet–IPF–Sinkhorn algorithm, permits the computation of the new drift and the new killing rate of the path-space solution measure. We also formulate and solve a related fluid-dynamic control problem for the flow of one-time marginals where both the drift and the new killing rate play the role of control variable. A numerical example illustrating the new theoretical results is also presented. |
Can the Quadratrix Truly Square the Circle?
| Pengarang | : | - |
| Nama Majalah/Jurnal | : | The American Mathematical Monthly |
| Volume / Edisi | : | 132 (No. 7) |
| Halaman | : | 647-665 |
| Abstrak | : | The quadratrix received its name from the circle quadrature, squaring the circle, but it only solves it if completed by taking a limit, as pointed out already in antiquity. We ask if it can square the circle without limits and restrict its use accordingly, to converting ratios of angles and segments into each other. The problem is then translated into algebra by analogy to straightedge and compass constructions, and leads to an open question in transcendental number theory. In particular, Lindemann’s impossibility result no longer suffices, and the answer depends on whether ???? belongs to the analog of Ritt’s exponential-logarithmic field with an algebraic base. We then derive that it does not from the well-known Schanuel conjecture. Thus the quadratrix so restricted cannot square the circle after all. |