## SIAM Nuggets

## Mathematical models with complicated dynamics for disease study

Philadelphia, PA – “The impact of human mobility on disease dynamics has been the focus of mathematical epidemiology for many years, especially since the 2002-03 SARS outbreak, which showed that an infectious agent can spread across the globe very rapidly via transportation networks,” says mathematician Gergely Röst. Röst is co-author of a paper published today in the *SIAM Journal on Applied Dynamical Systems* that presents a mathematical model to study the effects of individual movement on infectious disease spread.

“More recently, the risk of polio in Europe has been elevated by human migration, and many countries were concerned about the possibility of Ebola getting out of West-Africa,” continues Röst, who co-authored the paper with Diana Knipl and Paweł Pilarczyk. “There are several mathematical tools that can help in assessing how mobility facilitates disease spread, including multipatch compartmental models which are suitable to describe local disease dynamics as well as travel patterns between distinct locations, such as major cities.” Read the rest of this entry »

## Should a political party form a coalition? Using math to decide

Philadelphia, PA–Mathematical ideas and tools are often used to describe aspects of large macroscopic systems. Examples abound in areas as varied as finance to psychology. In a paper published last month in the *SIAM Journal on Applied Mathematics*, author Fabio Bagarello proposes mathematical models to analyze political decision-making. Using a dynamical approach which accounts for interactions between political parties and their constituents, the model tries to deduce whether parties should form coalitions under various circumstances.

“Mathematics is important in many aspects of social behavior. Politics is just one of these aspects, since

some of the typical behavior in politics can be characterized by suitable quantities which, usually, evolve in time,” says Bagarello. “In other words, political parties are examples of dynamical systems.” Read the rest of this entry »

## Decision cascades in social networks

*Math can shed light on spread of behaviors in online interactions*

Philadelphia-PA—How do people in a social network behave? How are opinions, decisions and behaviors of individuals influenced by their online networks? Can the application of math help answer these questions?

“The way in which information, decisions, and behaviors spread through a network is a fundamental social phenomenon, and the past several decades have shown that it is a phenomenon that can be studied using rich mathematical models,” says Flavio Chierichetti who co-authored a paper that studies online behavior, published this month in the *SIAM Journal on Computing*. “At one level, these processes have elements in common with biological contagion, which is also inherently based on a mechanism of spread through a network. But at another level, the processes are different — the spread of behavior is based on individual decision-making, and as such, can exhibit richer and more complex behavior than the more direct mechanics of biological contagion.”

Along with co-authors Jon Kleinberg and Alessandro Panconesi, Chierichetti studies how people in social networks are often influenced by each other’s decisions, resulting in a run of behaviors in which their choices become highly correlated, thus causing a cascade of decisions. The authors focus on the problem of ordering in a cascade with the end goal of maximizing the expected number of “favorable” decisions. “Often, cascading behavior in a social network is guided by an entity that wants to achieve a certain outcome,” says Alessandro Panconesi. “For example, a company might be trying to guide the adoption of a product by word-of-mouth effects, or a political movement might be trying to guide the success of its message in a population.” Read the rest of this entry »

## When vaccines are imperfect: What math can tell us about their effects on disease propagation

**Philadelphia, PA—**The control of certain childhood diseases is difficult, despite high vaccination coverage in many countries. One of the possible reasons for this is “imperfect vaccines,” that is, vaccines that fail either due to “leakiness,” lack of effectiveness on certain individuals in a population, or shorter duration of potency.

In a paper publishing today in the *SIAM Journal on Applied Mathematics*, authors Felicia Magpantay, Maria Riolo, Matthieu Domenech de Celles, Aaron King, and Pejman Rohani use a mathematical model to determine the consequences of vaccine failure and resulting disease dynamics.

“We examined the effects of individual-level vaccine failure on the propagation of a disease through a population,” says author Felicia Magpantay. “Specifically, we took into account different ways in which vaccines may fail. We distinguished between vaccine-induced immunity that is ‘leaky’, whereby vaccination reduces the probability of infection upon exposure but does not eliminate it; ‘all-or-nothing’, which leads to perfect protection in some individuals, but none in others; and ‘waning’, which reflects transient protection—or some combination of all three.” Read the rest of this entry »

## 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 »