The diseases in question include three types of worm (hookworm, roundworm and whipworm), a number of helminths (elephantitis, river blindness, Guinea worm disease and schistosomiasis), protozoans (leishmaniasis, Chagas’ Disease, sleeping sickness) and bacterial infections (the Buruli ulcer, leprosy and trachoma). Approximately 4.2 billion people — more than half the population of the Earth — are at risk for hookworm alone, with 807 million currently infected.

What characterizes these particular diseases isn’t that — unlike more sensational diseases like HIV/AIDS, malaria and TB — they kill huge numbers of people (about 530,000 people per year, although that’s still not nothing). Instead, they’re responsible for massive levels of disfigurement and disability, impairing childhood development and economic productivity. They’re found in every tropical country (including Australia) and yet are neglected at the community, national and international levels, largely because they affect the poor, the powerless and the stigmatised.

For example, Chagas’ disease kills 50,000 people a year (far more than West Nile virus, Bird Flu and swine flu combined), but you probably haven’t heard of it because it’s a disease of the poor. If your house is made of sticks, the bugs that carry the disease burrow through your walls and bite you under the eye. But if you can afford plaster, then you’re completely safe. So it’s a widespread disease in poor, rural South America (where the average life of a dog is about two years, thanks to the disease), but doesn’t kill anyone who might be in a position to lobby governments, advocate for medical interventions or mobilize advertising campaigns.

Rather than simply count deaths, the World Health Organization has developed a measure of the number of years of life lost from premature death or disability, or DALYs (Disability-Adjusted Life Years). The number of DALYs per year for HIV/AIDS is 84.5 million. That is, without HIV/AIDS we’d have about 84,500,000 years of healthy life back. But NTDs are collectively the next largest burden on the world, with DALYs of 56.6 (diarrhoeal diseases are third, followed by childhood and vaccine preventable diseases, then malaria and TB). So despite being neglected, the NTDs are one of the largest problems human beings face today.

Treatments exist for some NTDs, although often control occurs through less “sexy” methods, such as mass dewormings in schools, insecticides, safe water and, in some cases, arsenic and amputation. (Seriously. Arsenic is still used to treat sleeping sickness, while the only treatment for the Buruli ulcer is to amputate infected limbs. NTDs ain’t pretty.) Part of the problem is that there’s no money in them: why would a profit-driven pharmaceutical company waste time developing treatments for diseases whose sufferers can’t pay? Of the 1600 drugs developed between 1974 and 2004, only 18 were for tropical diseases (and 3 for TB).

So what’s to be done? Fortunately, there are a couple of success stories. Guinea worm disease has been all but eliminated, despite having no vaccine, no drug and no immunity. Instead, behavior changes (convincing people not to put infected limbs in the water, distributing cloth filters to villages and outfitting nomadic people with drinking pipes) have led to a massive reduction in cases and already eliminated the disease from Asia and the Middle East.

Who made this miraculous feat happen? It’s thanks to the efforts of one man: former president Jimmy Carter, who did the unglamorous but important work of mobilizing public-private partnerships, delivering education messages to remote populations and even negotiating a “Guinea worm ceasefire” in the Sudan civil war so that NGOs could go in and educate those most at risk. As a result, Guinea worm disease has been almost eradicated from the planet. It’s not only going to be the first parasitic disease to be eradicated, it’s also going to be the first to be eliminated using behavior changes alone. That’s an incredible achievement.

Another success story is river blindness, and this is where mathematical modelling comes into the picture. The West African river blindness program was developed as a co-production between the World Health Organization, the World Bank, the UN, and 20 donor countries and agencies in 1974. Mathematical modelling was used at the outset to predict long-term outcomes; by including modelling in the design of the program, sceptical donors were convinced that control was feasible. When the drug ivermectin was made available in the late eighties, mathematical models were able to adapt to its inclusion. After the program was completed, modelling retained a prominent role in subsequent policy discussions.

One of the great advantages of mathematical modelling is that it’s cheap. A lot can be done with a little, so many potential scenarios can be investigated even when data is limited. In a way, this makes NTDs an ideal subject for modelling to tackle. There are a great many problems that urgently need to be solved that mathematical models could help with.

Unfortunately, the NTDs are as neglected by modelling as they are by everyone else. Only sleeping sickness has received any substantial theoretical modelling. There are no models at all for the Buruli ulcer and only one for Guinea worm disease. When models do exist for NTDs, they’re usually confined to one lab and its collaborators per NTD. What we urgently need is a diversity of voices.

Specific problems might include adapting malaria pesticide models for vector control in Chagas’ disease or leishmaniasis. Spatial modelling is critical: access to resources depends critically upon geographical constraints, so models that accounts for distance to hospitals, swamps, mountains and road networks are crucial. Co-infection models — between other NTDs and also major diseases like HIV — are also desperately needed.

Modelling could also help categorize the costs to developing economies of disabling NTDs: if treating NTDs is shown to save more money than it costs in productivity, this will help motivate action. Another, slightly meta, approach might be to model research funding itself: if granting agencies are requiring researchers to provide “at home” benefits, this could be standing in the way of significant work on diseases that might help a very large number of people.

In summary, NTDs require immediate attention. They extract an enormous price in suffering, lack of economic development and the promotion of poverty. Mathematical models can be used to inform policy at minimal cost, solving problems that may not be theoretically complex, but which have the potential to deliver enormous benefits.

NTDs are the low-hanging fruit of mathematical modelling. A great many problems could be solved, relatively easily, by harnessing the power of mathematical modelling. The price — political and otherwise — for such a huge improvement in the quality of life for one sixth of the world’s population is tiny.

Robert Smith?
The Department of Mathematics
The University of Ottawa
585 King Edward Ave
Ottawa, ON K1S 0S1
Canada

The report analyzes the current state of various fields under the umbrella of the mathematical sciences, presenting ideas to ensure that the discipline remains in a strong position, and is capable of maximizing its contributions to the nation in 2025. The report recommends the reassessment of training for future generations of mathematical scientists in light of the growing cross-disciplinary nature of the field. Download the full report at the link above.

Nonlinear, dispersive evolution equations and systems of such equations arise as models for wave motion in a very wide variety of physical, biological and engineering. Since the 1960′s, there has been a steady increase of interest in the theory and applications of such equations. On the mathematical side, the pioneering work of Ginibre and Velo and Kenig, Ponce and Vega was followed by the spectacular progress of Bourgain, Tao and their collaborators, as well as many others.

If one tries to use the pure initial-value formulations in practice, one is immediately beset by the difficulty of determining accurately a wave profile in the entire spatial domain of its definition at a single instant of time. Generally speaking, this is not possible to accomplish with any semblance of accuracy. Moreover, when these equations are used in engineering and science, the natural way to pose them is with specified, not necessarily homogeneous boundary conditions. And, problems of control of dispersive equations demand a firm grasp of boundary-value problems as a starting point for developing cogent theory.

By contrast with the initial-value problem, theory for boundary-value problems other than those featuring periodicity has generally lagged behind the developments for the pure initial-value problems. The overall goal of the workshop is to advance the study of boundary-value problems for nonlinear dispersive wave equations. Some specific topics that are being considered are:

1. Investigating the smoothing properties enjoyed by solutions of boundary-value problems and associated well-posedness theory.

2. Investigating the controllability and stabilizability of solutions of nonlinear, dispersive wave equations. Experience shows that results from the first topic above will be central to such an investigation.

3. Extending the theory to multi-space dimensional problems arising in geophysical applications such as coastal dynamics and elsewhere.

Estelle Basor
AIM

This year’s Snowbird meeting will host a Featured Minisymposium on “Dynamics of Planet Earth” as part of MPE2013. The Featured Minisymposium (MS38) is organized by Hans Kaper, chair of the SIAG/DS, and will take place on Monday, May 20, 2:30 p.m. – 4:45 p.m., in Ballroom I.

The minisymposium will feature an overview talk by the organizer and four talks on specific applications of dynamical systems and bifurcation theory to the Earth’s climate system. Chris Danforth (U Vermont) will demonstrate a novel method for improving forecasts during integration of a weather model. Mary Silber (Northwestern U) will discuss tipping points in the context of bifurcation theory, using case studies of possible tipping points in models of Arctic sea-ice retreat and desertification. Marty Anderies (Arizona State U), who is interested in land use and the carbon cycle, will explore the relationship between nonlinear dynamics and planetary boundaries. Mary Lou Zeeman (Bowdoin College and Cornell U) will focus on issues of sustainability and will explore how a decision-support viewpoint may inspire new questions for dynamical systems.

The biennial Snowbird meetings offer a unique opportunity to learn about the application of dynamical systems theory to areas outside of mathematics. These application areas are diverse and multidisciplinary, ranging over all areas of applied science and engineering, including biology, chemistry, climate, geophysics, physics, finance, and industrial applied mathematics. This conference strives to achieve a blend of application-oriented material and the mathematics that informs and supports it. The goals of the meeting are a cross-fertilization of ideas from different application areas, and increased communication between the mathematicians who develop dynamical systems techniques and applied scientists who use them.

There is widespread interest in finding and designing spacecraft trajectories to the Moon, Mars, other planets, or other celestial bodies (comets, asteroids), which require as little fuel as possible. This is justified mostly by the cost of the missions: each extra pound of load for a spacecraft roughly costs 1 million dollars. Hence, for robotic space missions, which typically conduct numerous observations and measurements over long periods of time, in order to maximize the equipment load, it is imperative to minimize the fuel consumption of the propulsion system. One way to achieve this is to cleverly exploit, in a mathematically explicit way, the gravitational forces of the Earth, Moon, Sun, etc.

To illustrate the concept, suppose that one would like to design a low-energy transfer from the Earth to the Moon. Of course, first one has to place the spacecraft on some orbit around the Earth, which is unavoidably energy expensive. Surprisingly though, the second leg of the trajectory, to take the spacecraft from near the Earth to some prescribed orbit about the Moon, can be done at a low energy cost (in theory, even for free). Say that we would like to insert the spacecraft at the periapsis of an elliptic orbit about the Moon, of prescribed eccentricity and at some prescribed angle with respect to the Earth-Moon axis. We would like to do that without having to slow down the spacecraft at the arrival (or maybe just a little), thus saving the fuel necessary for such an operation. Imagine that we run the “movie” of the trajectory backwards, from the moment when the spacecraft is on the elliptic orbit. Since the eccentricity is fixed, the semi-major axis determines the velocity of the spacecraft at the periapsis. If the semi-major axis is too short (or, equivalently, the velocity is too low) the trajectory will turn around the Moon without leaving the Moon region; such a trajectory is redeemed as ‘stable’. By gradually increasing the semi-major axis (hence, the velocity), one will find a trajectory that leaves the Moon region and makes a transfer to the Earth region; such a trajectory is redeemed as ‘unstable’. See Fig. 1.

Fig. 1. Stable and unstable trajectories; P1 denotes the Earth and P2 the Moon.

The critical values of the semi-major axis which delineates the stable motions from the unstable ones are called “weak stability boundary” points. Exploring all angles of insertion and all values of the eccentricity of the elliptical orbit yields a “weak stability boundary” set; this appears to be some sort of a fractal. See Fig. 2.

Fig. 2. Weak Stability Boundary.

All points in the weak stability boundary correspond to arrival points of low energy transfers from the Earth to the Moon. The spacecraft trajectories designed by this method yield fuel savings of 10-15%.

The notion of a weak stability boundary was introduced by Edward Belbruno (Princeton University) in 1987; a documentary trailer on the discovery of this concept can be found on YouTube. The method was successfully applied, for the first time, for the rescue of the Japanese lunar mission Hiten in 1991. The recent mission GRAIL (Gravity Recovery and Interior Laboratory) of NASA, which took place in 2012, used the same transfer as Hiten. The purpose of this mission was to obtain a high-resolution mapping of the gravitational field of the Moon. For this purpose, two spacecrafts were placed on the same orbit about the Moon, and their instruments measured the changes in the relative velocity very precisely; such changes were translated into changes of the gravitational field. (This technique had been tested previously for the mapping of Earth’s gravity, as a part of the mission GRACE – Gravity Recovery and Climate Experiment – a joint mission of NASA and the German Aerospace Center, since 2002.) A key point for the GRAIL mission was to place the two spacecrafts on precisely the same lunar orbit; the weak stability boundary concept was quite suitable for this purpose.

A deeper understanding of the weak stability boundary can be achieved from studying hyperbolic invariant manifolds. The motion of a spacecraft relative to the Earth-Moon system can be modeled through the three-body problem. In this model, the intertwining gravitational fields of the Earth and the Moon determine some “invisible pathways,” called stable and unstable manifolds, on which optimal transport is possible. These manifolds have been found in the works of Henri Poincaré. It turns out, surprisingly, that these manifolds are deeply related to the weak stability boundary. More precisely, under some energy restriction, it can be geometrically proved that the weak stability boundary points lie on certain stable manifolds.

Here are some recent references:
Belbruno, E.; Gidea, M; Topputo, F. Weak stability boundary and invariant manifolds. SIAM J. Appl. Dyn. Syst. 9 (2010), no. 3, 1061–1089.
Belbruno, E.; Gidea, M.; Topputo, F. Geometry of Weak Stability Boundaries. Qual. Theory Dyn. Syst. 12 (2013), no. 1, 53–66.

Marian Gidea

School of Mathematics
Institute for Advanced Study
Princeton

and

Department of Mathematics
Northeastern Illinois University
Chicago

The mathematical model is a domain-based coupled system of Stokes equations with Darcy’s law describing the rate at which a fluid flows through a permeable medium. An important part of the modeling problem is the choice of the conditions at the interface between the free flow region and the porous medium.

A first condition states that the normal component of the velocity must be continuous across the interface. A second condition, referred to as the Beavers-Joseph-Saffman law, relates the tangential component of the free flow velocity with its shear stress. The latter law was derived from simple experiments of laminar tangential flows over a porous bed and was confirmed by homogenization technique for periodic porous media with circular pores [1,2,3]. A third interface condition involving the pressure of the fluid remained controversial until quite recently. Some scientists claimed that the pressure in the free flow must be continuous and equal to the Darcy pressure at the interface, while others claimed that there must be a jump in the pressure. Two recent works by Mikelic and co-authors [4,5] show theoretically and numerically that the pressure is discontinuous across the interface and that the pressure jump is proportional to the free fluid shear. It is interesting to note that if the pores of the media are isotropic (circular), the discontinuity in the pressure vanishes.

Does this contradict the physical law of continuous pressure? No. The fluid pressure in the pores remains continuous. The pressure that appears in Darcy’s law is an average of the physical pressure of the fluid in the pores over a representative elementary volume. At the micro-scale, there is no real interface between the free flow and porous media. At the macro-scale, for general porous media, the continuum pressure is discontinuous across the interface, and the interface itself is part of the model.

There are still many open questions related to the coupled free flows and porous media problem. An interdisciplinary effort combining analytical, experimental and numerical techniques is a key to gain a significant understanding of these coupled flows. For those interested in this problem, there will be a whole session dedicated to the modeling of these interface conditions at the 5th International Conference on Porous Media & Annual Meeting organized this month in Prague.

[1] G.S. Beavers and D.D. Joseph, Boundary conditions at a naturally permeable wall, J. Fluid Mech., 30, p.197-207, 1967.
[2] P.G. Saffman, On the boundary condition at the interface of a porous medium, Studies in Applied Mathematics, 1, p. 93-101, 1971.
[3] W. Jager, A. Mikelic, On the interface boundary conditions by Beavers, Joseph and Saffman, SIAM J. Appl. Math., 60, p. 1111-1127, 2000.
[4] A. Marciniak-Czochra, A. Mikelic, Effective pressure interface law for transport phenomena between an unconfined fluid and a porous medium using homogenization, SIAM Multiscale modeling and simulation, 10, p. 285-305, 2012.
[5] T. Carraro, C. Goll, A. Marciniak-Czochra, A. Mikelic, Pressure jump interface law for the Stokes-Darcy coupling: confirmation by direct numerical simulations, preprint arXiv:1301:6580 [math.NA], 2013.

Beatrice Riviere
Associate Professor
Department of Computational and Applied Mathematics
Rice University
riviere@rice.edu

So begins the SIAM News article by Erica Klarreich, Examining the Dynamics of Ocean Mixing (SIAM News, May 1, 2013). The text of the full article may be found here.

Africa is a continent that has been and still is being plagued with infectious diseases. Most notably are the current epidemics caused by HIV, Tuberculosis and Malaria. But there are many other diseases, both treatable and preventable, that also affect African populations. The workshop focused on HIV, Tuberculosis, Malaria, Polio, Neglected Tropical Diseases and surveillance. One day was devoted to each of the “big three” (HIV, Tuberculosis and Malaria) and two days to Neglected Tropical Diseases, Polio, surveillance, and a discussion of the effects of disease on children. Each day featured four plenary talks and two discussion sessions. In the discussion sessions, participants identified gaps in our knowledge and discussed the role of mathematical modelling in the particular theme areas. A group of researchers will follow through on these discussions and initiate a new collaborative network.

The objectives of the workshop were:
(a) To combine the expertise of public health officials and researchers in biology and the mathematical sciences in the areas of infectious diseases relevant to Africa;
(b) To encourage and seek participation of African colleagues, to foster collaborations between Canadian and African researchers;

(c) To compare public health policies and experiences, helping all participants develop a better understanding of this difficult yet crucial aspect; and

(d) To train junior researchers, postdoctoral fellows and graduate students.

The organizers of this workshop were Jane Heffernan (York University) and Julien Arino (University of Manitoba), both affiliated with the Centre for Disease Modelling at York University.

More information regarding the workshop can be found here.

Note added by the editor:
A post on Neglected Tropical Diseases by Robert Smith? is scheduled for publication on the MPE2013 Daily Blog on May 18, 2013.

Cats were the subject of a recent, surprising news item. A group of computer scientists at Google and Stanford University fed YouTube videos to a computer that was running a “machine learning” program. This program “trains” on the input to find clusters of similar images and once it’s trained, the computer can classify new images as belonging to one of the clusters. After training on images from ten million YouTube videos, the computer learned to reliably identify images of cats. Like a newborn baby, the computer started with no knowledge but learned to identify objects – in this case cats – based on what it had already seen. This exercise illustrates the ability of machine learning to enable recognition tasks such as speech recognition, as well as classification tasks such as identifying cat faces as a distinct category of images.

Batteries deserve attention on this website because of their essential role in any strategy for sustainable energy. Batteries are a primary means for storing, transporting and accessing electrical energy. For example, they provide storage of excess energy from wind and solar sources and enable electrical power for cars and satellites. Today’s hybrid and electric vehicles depend on lithium-ion batteries, but the performance of these vehicles is limited by the energy density and lifetime of these batteries. To match the performance of internal combustion vehicles, researchers estimate that the energy density of current batteries would need to increase by a factor of 2 to 5.

Strategies for achieving these gains depend on identifying new materials with higher energy densities. The traditional method for finding new materials is to propose a material based on previous experience, fabricating the new material and measuring its properties, all of which can be expensive and time consuming. More recently, computational methods, such as density functional theory, have been used to accurately predict the properties of hypothetical materials. This removes the fabrication step but can involve large-scale computing. Although both of these methods have produced many successful new materials, the time and expense of the methods limit their applicability.

Cats – more precisely, the machine learning program that recognized cats – could come to the rescue. Instead of watching YouTube videos, a machine learning method could train on existing databases (from both experiment and computation) of properties for known materials and learn to predict the properties for new materials. Once the machine learning method is trained (which can be a lengthy process), its prediction of material properties should be very fast. This would enable a thorough search through chemical space for candidate materials. Machine learning methods have not yet been used for finding materials for batteries, but they have been used for prediction of structural properties, atomization energies, and chemical reaction pathways. Their use in materials science is growing rapidly, and we expect that they will soon be applied to materials for batteries and other energy applications.

Russ Caflisch, Director
Institute for Pure & Applied Mathematics (IPAM)

Guinea worms are a parasite that humans acquire by drinking unclean water. The worms can grow to several feet in length then painfully emerge from the body basically any way they can.

Carter’s foundation invented a strainer made of a fine material — something like parachute silk. They manufactured enough of these sieves to distribute to every afflicted village. That was the easy part. Then they had to persuade people to drink the water from their ponds only after straining it through the sieve. One major issue they encountered was that sometimes the local people regarded the pond water as holy and didn’t want to cause offense to the powers that be by introducing a foreign device. It was necessary to convince people almost literally one-by-one that the worms were basically aliens that had invaded the holy water and it was ok to strain them out.

You are probably asking, “Where is the mathematics in here?” I see several analogies. One is that the solution was counterintuitive and was arrived at only after many trials and many errors. Also, the solution, or variations of it, had to be applied on a case-by-case basis, not unlike the case-by-case analysis that is present in many mathematical proofs, such as the proof of the 4-color theorem or of Kepler’s conjecture. The solution also required the sustained collaborative effort of many individuals to go to each village and make the necessary arguments that would persuade the villagers to behave in an unfamiliar and even abhorrent way that was not part of their culture. I also see elements of the logistical analysis of operations research in the solution here. The exact steps have to be carried out in the right order.

The Carter Center’s persistence in developing a solution and carrying it out over a 25-year period also reminds me of the dogged determination that many mathematicians exhibit in the relentless pursuit of a solution that many would have earlier given up on. Finally, I see the success of a large-scale collaborative effort, which reminds me of some ongoing large-scale collaborative mathematical efforts requiring the cooperation of thousands of individuals, such as finding new Mersenne primes, or verifying that trillions zeros of the Riemann zeta-function are all on the critical line, or Tim Gowers’ polymath projects.

To see the clip of Jimmy Carter on the Daily Show, click here.

I also recommend this article about guinea worms.