Events
Bard Ermentrout
COVID - 19: Modeling through Mathematics
With the advent of the novel corona virus, SARS-COV2, the world is gripped by a new pandemic viral disease that is relatively easy to transmit through respiratory droplets and causes a wide variety of symptoms. The novelty of the illness, called COVID19, and its high fatality rate (higher than seasonal influenza) has led to an explosion of scientific research on how to stop the spread and how to mitigate the symptoms. Mathematicians and theoreticians from around the world including Pitt have been involved from the earliest stages in modeling this new disease from the population and epidemiological level down to the immune response and the molecular level.
The weekly Math Biology Seminar series has been devoted to the presentation of new papers on the epidemiology, critiques of these papers, and models of the physiology that underlies the pathology. In addition, several faculty and alumni have worked on new models of this disease. In the mathbio seminar, the first several presentations focused on the spread of this disease. Most epidemiological models start with a very simple concept: A susceptible population, $S$ interacts with an infected population, $I$ which produces a new infection. Infected individuals remain in the population until they enter the removed or recovered population, $R$. A simple differential equation for this interaction is:
\begin{eqnarray*}
\frac{dS}{dt} &=& -\beta I S \\
\frac{dI}{dt} &=& \beta I S - \gamma I
\frac{dR}{dt} &=& \gamma I.
\end{eqnarray*}
The way to think about these parameters is that $\gamma$ is the rate that one leaves the infected pool, so that $1/\gamma$ is roughly the number of days that the disease is transmissible. If the total susceptible population is $N$, then $\beta N$ is the rate of infection. If the rate of infection exceeds the rate of recovery, then the disease takes off, that is $\beta N>\gamma$ or written another way, $R_0=\beta N/\gamma > 1$. The disease parameter, $R_0$ is roughly the average number of infections that one infected person will cause. If $R_0<1$ the disease will die out. For example, for measles, $R_0$ is about 15, so each person with measles can be expected to give it to about 15 others! $R_0=1.4--2.8$ for seasonal flu and $R_0=2--6$ for COVID-19. These numbers assume a completely naive population with no vaccine. In the mathbio seminar, we looked at a number of studies that built on this simple epidemiological model and how different interventions could help prevent the spread. A simple extension of the model is to introduce an exposed population, $E$ which is not infectious but transitions into the infection stage at some rate. Figuring out $R_0$ is not the end of the modeling of disease, particularly COVID-19 where many people are not symptomatic and do not even spread the virus while others are so-called ``superspreaders''. A superspreader event happened in early October infecting many people at the White House. For this reason, there are many more complicated models that have been developed. For example the Imperial College model developed in London operates at the level of individual households and tracks the demographic data such as the ages of children, how far people commute, whether there are school kids, etc and thus has thousands of parameters and variables. In a recent mathbio seminar, Tom Hales examined some of the thousands of lines of computer code and demonstrated a number of inconsistencies and errors. Whiler it is not known how these have affected the results of this massive model, it is important to note that a number of political decisions in the UK and elsewhere were based on thr simulations.
Another tack in the applications of mathematics to disease is to consider how the actual viral infection interacts with the innate and adaptive immune systems. The innate immune system is the first responder to infection and is responsible for the symptoms such as fever, headache, coughing, and stuffy nose; all a consequence of the inflammatory response. The adaptive immune system kicks in later and provides the body with the memory of past infections so that the invading virus is attacked much sooner. This is how vaccines work. David Swigon, Jonathan Rubin, and Bard Ermentrout, along with their students have written numerous papers on the interaction of pathogens with the innate immune system. The intriguing aspects of the modeling and what makes them interesting to a mathematician are the multiple positive and negative feedback loops that are involved. We want our immune response to be strong enough to remove the virus but then we want to calm it down before it does too much damage to our own cells. Previous Pitt Math PhD, Judy Day (now at University of Tennessee) gave a presentation on her recent work on the adaptive immune system and COVID-19 as a guest in the mathbio seminar. With collaborators at University of Indiana, Ermentrout and former student Ericka Mochen (now on the faculty at Carlow) and two undergraduate students, have reconfigured her model for response to influenza to a model for infection of the lower and upper respiratory tracts. We have used this model to study the effects of both anti-viral drugs (should be given early) and anti-inflammatory drugs (there is a narrow window of effectiveness). COVID-19 seems to evoke a very strong response of the immune system compared to many other diseases. In a recent talk at out mathbio seminar, Pitt Computational Biology professor Madhavi Ganapathiraju, presented her modeling work on protein-protein interactions where she showed that COVID-19 proteins interact with far more human proteins than do other viruses such as SARS1.
In sum, the quantification of SARS-COV1 and the underlying disease, COVID-19, requires modeling at all levels from susceptible populations (epidemiology) to the physiological effects of the disease (immunology) to the molecular level (protein-protein interactions). The language of modeling is mathematics, so with luck, perseverance, and numbers, we hope to be able to contribute to a better understanding of this novel disease.
Alberto Bressan
A Posteriori Error Estimates for Numberical Solutions to Hyperbolic Conservation Laws
One of the consequences of the COVID-19 pandemic is the lack of in-person seminars and colloquia. Despite of the necessary restrictions, the department colloquia successfully continued with 11 online talks during the Fall 2020 semester. Leading mathematicians shared their research from a diverse range of topics in pure and applied mathematics, including nonlocal PDEs, tropical geometry, Verlet time-stepping methods, cluster algebras, toric degenerations, and self-organization dynamics.
One highlight of the colloquium series was the lecture, “A posteriori error estimates for numerical solutions to hyperbolic conservation laws” given by Alberto Bressan on October 9, 2020. In this talk, Professor Bressan discussed the inherent difficulty of obtaining a priori estimates for numerical schemes for n x n hyperbolic systems of conservation laws. While is it is well known that the Cauchy problem has a unique entropy-weak solution, depending continuously on the initial data, a priori error estimates of fully discrete schemes (e.g., Lax-Friedrichs or Godunov schemes) are still open problems.
In his talk, Professor Bressan explained the obstruction toward a priori error bounds for such discrete schemes. Taking a different point of view, he presented some recent results on a posteriori error estimates, achieved by a "post-processing algorithm" that checks the total variation of the numerically computed solution, and computes its oscillation on suitable subdomains.
Professor Alberto Bressan is an Italian Mathematician at Penn State University, where he is an Eberly Family Chair of Mathematics and Director of the Center for Interdisciplinary Mathematics. His research expertise is in hyperbolic conservation laws and nonlinear wave equations, optimal control, and dynamic blocking problems. Professor Bressan is an AMS fellow and a recipient of the Bocher Memorial Prize and the SIAM Analysis of PDEs prize.
SIAM Job Fair
In October, the Pitt SIAM chapter held a jobs panel consisting of alumni in various stages of their careers, both in industry and academia. On the academic side, we hosted recent alumni Dr. Youngmin Park, completing his postdoc at Brandeis University, and Dr. Wanying Fu, assistant professor of actuarial science at Lebanon Valley College. On the more industrial side, we hosted Dr. Ross Ingram of Bettis Atomic Power Laboratory and Dr. Yong Li, formerly a quant and Goldman Sachs and now a software engineer at Google. In the middle, we had Dr. Abigail Snyder, who had just completed a postdoc at the Pacific Northwest National Laboratory and is now a full researcher there.
The event led to good discussion of the different lifestyles of each path, and things for graduate students to keep in mind when preparing to graduate. Perhaps the biggest takeaway was that, when positions hire mathematicians, they are hiring for our skill as mathematicians, without necessarily requiring that we are experts in the particular research topic at hand; and in an industrial setting you will be working with people from a diverse set of backgrounds, bringing viewpoints and expertise that others may not have.
All in all, it was a helpful and enjoyable experience. Our panelists have graciously agreed to share their email addresses for any advice or questions students may have.
Dr. Park: ympark88@gmail.com
Dr. Fu: wfu@lvc.edu
Dr. Ingram: ross.ingram@unnpp.gov
Dr. Li: yakaqi@gmail.com
Dr. Snyder: abigail.c.snyder@gmail.com
Roger Penrose
On October 6th 2020, the Nobel Prize Committee awarded the Physics Nobel Prize for this year to Roger Penrose, Reinhard Genzel and Andrea Ghez:
- To Penrose for proving that black holes were a generic prediction of the theory of gravity due to Albert Einstein.
- To Genzel and Ghez for proving by extensive observations over a thirty-year period that the center of our galaxy harbors a supermassive black hole.
It is now understood that black holes are ubiquitous; indeed, they sometimes collide giving rise to spectacular releases of gravitational waves, whose recent observation on Earth has revolutionized astronomy.
Penrose has visited the University of Pittsburgh many times, giving many inspiring talks, variously in the Mathematics, Physics, and Philosophy Departments. In 2011 he gave our annual Edmund R. Michalik Distinguised Lecture in the Mathematical Sciences: “Can We See Through the Big Bang, into Another World?” Although his affirmative answer to the question raised in the title is controversial, he put forward a cogent, coherent case (as he always does on any matter), which is very hard to directly refute!
We extend our heartiest congratulations to Roger Penrose as he accepts the award! His acceptance speech is on Tuesday 8th December at about 3.00 am Pittsburgh time. See: https://www.kva.se/sv/kalendarium/the-nobel-lectures-2020
His title is: ”Black Holes, Cosmology, and Space-Time Singularities"