## SIAM Nuggets

## Insightful mathematics for an optimal run

*Mathematical equations can help improve athletic performance*

Sure, we can become better runners by hydrating well, eating right, cross training, and practice. But getting to an optimal running strategy with equations? That’s exactly what a pair of mathematicians from France propose in a paper published this month in the *SIAM Journal on Applied Mathematics*.

“By modeling running in the form of equations and then solving them, we can predict the optimal strategy to run a given distance in the shortest amount of time,” says Amandine Aftalion, who co-authored the paper with Frederic Bonnans.

The model uses a system of ordinary differential equations. Aftalion explains: “Our model relies on two basic principles: energy is preserved, and acceleration (or variations of velocity) is equal to the sum of all forces. This leads to a system of differential equations coupling the unknown variables of the runner (velocity, propulsive force and anaerobic energy), and dependent on physiological parameters such as maximal oxygen uptake and total available anaerobic energy.” Read the rest of this entry »

## Where is that spacecraft?

*Statistically measuring uncertainty for space surveillance *

**Philadelphia, PA—**Space surveillance is inherently challenging when compared to other tracking environments due to various reasons, not least of which is the long time gap between surveillance updates. “Unlike the air and missile defense environments where objects are frequently observed, the space surveillance environment data is starved, with many objects going several orbital periods between observations,” according to researcher Joshua Horwood. “Thus, it is more challenging to predict the future location of these sparsely-seen objects and they have a tendency to get lost using traditional methods. A new way of tracking them, the Gauss von Mises (GVM) distribution, has improved predictive capabilities that permit one to more effectively maintain custody of infrequently-observed space objects.”

In a paper published in July in the* SIAM/ASA Journal on Uncertainty Quantification*, authors Horwood and Aubrey Poore, both of Numerica Corporation, propose a more statistically rigorous treatment of uncertainty in the near-Earth space environment than currently available. The method proposed uses a new class of multivariate probability density functions, called the Gauss von Mises (GVM) family of distributions. Read the rest of this entry »

## Using math to analyze movement of cells, organisms, and disease

Philadelphia, PA—A few recent SIAM journal papers you should know about:

*Traveling waves model tumor invasion*

Cell migration, which is involved in wound healing, cancer and tumor growth, and embryonic growth and development, has been a topic of interest to mathematicians and biologists for decades.

In a paper published recently in the *SIAM Journal on Applied Dynamical Systems*, authors Kristen Harley, Peter van Heijster, Robert Marangell, Graeme Pettet, and Martin Wechselberger study a model describing cell invasion through directional outgrowth or movement in the context of malignant tumors, in particular, melanoma or skin cancer. Tumor cells move up a gradient, based on the presence of a chemical or chemoattractant – this process is called haptotaxis. Receptors on the exterior of cell walls detect and allow passing of the chemoattractant. Based on the locations of these receptors, cells determine the most favorable migration direction. Read the rest of this entry »

## Putting a number on opinion dynamics in a population

*How math helps us analyze information distribution and assimilation *

Philadelphia, PA—Opinion formation in a large population is influenced by both endogenous factors, such as interaction with one’s peers—in-person and via social media—as well as exogenous factors, such as the media, of which mainstream media is one of the most influential factors. For example, according to a study conducted by the National Bureau of Economic Research in 2006, after the introduction and expansion of Fox News in the United States between 1996 and 2000, an estimated 3-28% of the audience was persuaded to vote Republican.

In a recent paper published in the *SIAM Journal of Applied Dynamical Systems*, authors Anahita Mirtabatabaei, Peng Jia, and Francesco Bullo use a mathematical model to study the process of information assimilation in a population resulting from such exogenous inputs. Read the rest of this entry »

## Overcoming structural uncertainty in computer models

*What is good enough to aid health economics decision making?*

Philadelphia, PA—A computer model is a representation of the functional relationship between one set of parameters, which forms the model input, and a corresponding set of target parameters, which forms the model output. A true model for a particular problem can rarely be defined with certainty. The most we can do to mitigate error is to quantify the uncertainty in the model.

In a recent paper published in the *SIAM/ASA Journal on Uncertainty Quantification*, authors Mark Strong and Jeremy Oakley offer a method to incorporate judgments into a model about structural uncertainty that results from building an “incorrect” model.

“Given that ‘all models are wrong,’ it is important that we develop methods for quantifying our uncertainty in model structure such that we can know when our model is ‘good enough’,” author Mark Strong says. “Better models mean better decisions.” Read the rest of this entry »

## Applying math to cancer, climate, crime and cameras

*Some SIAM journal papers you should know about: *

*Improving radiation therapies for cancer mathematically*

In a paper published in December in the *SIAM Journal on Scientific Computing*, authors Li-Tien Cheng, Bin Dong, Chunhua Men, Xun Jia, and Steve Jiang propose a method to optimize radiation therapy treatments in cancer patients.

Radiation therapy is one of the primary methods used for cancer treatment, along with chemotherapy and surgery. While doses of radiation are delivered to eliminate cancerous tissue, care is taken to keep radiation within acceptable levels so as not to affect neighboring tissues and organs. The most common type of therapy delivers high-energy radiation via a medical linear accelerator mounted on a rotating apparatus to adjust the direction, and a collimator to shape the beam of radiation. In the recently developed volumetric modulated arc therapy (VMAT), beams continuously deliver doses as the delivery device rotates around the patient. Enhancement of radiotherapy treatment is challenged by complexities of shape optimization, due to the mechanics of the equipment involved as well as the apertures of devices delivering the beams of radiation. Read the rest of this entry »

## Statistics research could build consensus around climate predictions

Philadelphia, PA—Vast amounts of data related to climate change are being compiled by research groups all over the world. Data from these many and varied sources results in diﬀerent climate projections; hence, the need arises to combine information across data sets to arrive at a consensus regarding future climate estimates.

In a paper published last December in the *SIAM Journal on Uncertainty Quantification*, authors Matthew Heaton, Tamara Greasby, and Stephan Sain propose a statistical hierarchical Bayesian model that consolidates climate change information from observation-based data sets and climate models.

“The vast array of climate data—from reconstructions of historic temperatures and modern observational temperature measurements to climate model projections of future climate—seems to agree that global temperatures are changing,” says author Matthew Heaton. “Where these data sources disagree, however, is by how much temperatures have changed and are expected to change in the future. Our research seeks to combine many different sources of climate data, in a statistically rigorous way, to determine a consensus on how much temperatures are changing.” Read the rest of this entry »

## Algorithms enhance imaging techniques for breast cancer diagnosis

**Philadelphia, PA—**Over 200,000 women are diagnosed with breast cancer every year. The five-year survival rate for afflicted women is 97% if the cancer is localized and discovered before it spreads to other parts of the body.

“Currently, mammography is the technique used most often for breast cancer screening, but since it gives only two-dimensional (2D) projection information of a three-dimensional (3D) anatomical structure, inaccuracies in screening often occur,” says James Nagy, co-author of a paper on breast tomosynthesis image reconstruction published last fall in the *SIAM Journal on Scientific Computing (*along with Veronica Mejia Bustamante, Steve Feng, and Ioannis Sechopoulos)*. *

While conventional x-ray mammography produces 2D projection images of 3D objects, digital technologies such as tomosynthesis can produce 3D image information of an object by using slightly modified conventional digital x-ray systems. Conventional mammography is limited by superposition of breast tissue, which can sometimes mimic or obscure malignant pathology. Read the rest of this entry »

## Math models analyze the evolution of epidemics during air travel

Philadelphia, PA—Air travel connects distant worldwide territories like never before. The benefits of high-connectedness are numerous, but epidemic outbreaks in recent decades highlight one pitfall of the global air travel network. Commercial flights remain a leading factor in the spread of commonly-known infectious diseases like tuberculosis, measles, and seasonal influenza.

In a paper published this fall in the* SIAM Journal on Applied Dynamical Systems, *Diána H. Knipl, Gergely Röst, and Jianhong Wu formulate a model to describe the evolution of an epidemic in regions connected by international flights. Using the 2009 influenza A(H1N1) epidemic in Mexico and Canada as an example, the model describes the spread of disease within and between two regions. Read the rest of this entry »

## A mathematical perspective of seasonal variations in Lyme disease transmission

Philadelphia, PA—Lyme disease is a common tick-borne illness caused by a bacterium, which is transmitted to humans through the bite of infected ticks. The transmission dynamics of Lyme disease is dependent on a variety of factors, including the length of the tick’s life cycle, availability of hosts, climatic conditions and seasonal influences, which are important to understand for control strategies.

In a paper published last month in the *SIAM Journal on Applied Mathematics*, authors Yuxiang Zhang and Xiao-Qiang Zhao propose a reaction-diffusion model to study transmission dynamics of Lyme disease while taking into account seasonality.

Ticks live for roughly 2 years, and their life cycle includes three stages: larva, nymph and adult. Ticks climb on to host animals who brush against vegetation from the tips of grasses and shrubs. Once they attach themselves, they feed on blood by inserting their mouthparts into the skin of a host, thus transmitting the disease. After obtaining a blood meal—which can take anywhere between 3 and 5 days—ticks drop off their hosts and prepare for the next stage of the life cycle. Read the rest of this entry »