Two years ago, Assistant Professor Vikas Srivastava joined the Brown faculty and had the opportunity to develop a new graduate course, which he named Analytical Modeling for Biomechanical and Biomedical Systems. In it, he introduced an educational portion on the modeling of infectious disease spread and dynamics. A unique subset of problems solved in class now feels forebodingly familiar to those engineering graduate students who studied the concept of “flattening the curve” before it became a household catchphrase.
As a corporate engineer-turned-professor, Vikas Srivastava has specific goals in mind when it comes to what aspiring engineers and future researchers need to know. When he first arrived on campus in 2018, one of his first opportunities came in the form of creating a new biomedical engineering graduate level course, and formulating the topics that course would cover. It was important to him that the material be complementary to, and not already captured some other way under the existing engineering course curriculum. One topic that caught his eye was the idea of infectious disease modeling.
“In the infectious disease community, there is a widely accepted notion that it is just a matter of time before a new pandemic occurs,” he said. “I was thinking of what key and important problems this class could look at from an engineering point of view. And I was quite fascinated with the spread of infectious disease, and thought this would be something interesting for me to learn about, and teach my students.” At the time, he had no idea how ahead of the curve he was.
“I’m not an infectious disease researcher,” Srivastava said. “In fact, my research explores mechanics of materials for energy and biomedical applications, with no direct emphasis on infectious disease. Until now. But I know and enjoy mathematical and physical modeling. And I look for opportunities where these approaches can be applied.”
Srivastava estimates that about thirty percent of his course ENGN 2911R specifically involves discussion and modeling of infectious disease dynamics, although a handful of students indicated that was indeed one of the reasons for choosing the class. Second-year biomedical engineering Ph.D. student Kiara Lee said, “I had heard great things about the course, including about the infectious disease modeling, which I was very excited about because of my dual interests in global public health and engineering. At the same time as taking this course, I was also taking Introduction to Epidemiology in the School of Public Health, and the two courses did end up discussing similar topics, but from different lenses.”
Srivastava said, “When I began teaching, I pointed out the potential of a serious threat from a new infectious disease. At that time, I didn’t know how much students were able to appreciate this point. Now that part will be very clear. What was a short discussion about possible new diseases will now be extremely relevant given SARS-CoV-2.
“The mathematical modeling covered in the course examines highly infectious disease, and looks at what happens to an individual that gets infected. The person gets exposed to a virus, they are infectious, they fall sick, and then go through the recovery process or have mortality. While an individual is infectious, he can transfer (the disease) to several other people. You can look at the virus and infected individual’s response over time, selected population behavior and action in time and space, build a mathematical understanding of key phenomena, and then develop mathematical models to make predictions of infection spread and dynamics at the population level.”
Katherine Barry ’19, a first-year student in the Warren Alpert Medical School, said, “I remember a particular assignment where we were asked to use MATLAB to model the spread of a disease with certain parameters to determine how many people will be infected by a certain day. We were able to change those parameters in order to understand how they impact the numbers of susceptible, infected, and recovered individuals over time.”
Kiera Dwyer, a biomedical engineering graduate student, added, “I remember starting out the lecture by talking about the curves describing susceptible vs. infected vs. recovered populations and how looking at different diseases at different time scales influence the shape of them. When we learned specifically about infectious diseases, we talked about the different control of such a disease - treatment, vaccination, quarantine (which is especially relevant now), and how these interventions can also change the shape of the curves.”
“Everyone is talking about the curves,” said Srivastava. “Those curves come from the mathematical modeling.
“In the course, we model the critical factors that affect infection spread,” he said. “Some critical factors to understand are the length of the infectious period, and the rate of transmission per infected person. If the infectious period is high, social contact between individuals is high, and infection transmission potential is high, the disease is very infectious – infection can occur from one person to a lot more people at a rapid pace. Through mathematical modeling, we can view the value and significance of different infection characteristics on the infection spread and dynamics, and actually calculate these numbers. And then you start to realize how powerful this information is.
“Relationship and importance of some of the critical model factors is becoming clear now when we talk about social distancing, and being careful about cleaning, and washing hands. All these things should reduce transmission or prolong the transmission to avoid the overburdening of the capacity of our hospitals.”
Shruti Trivedi, a chemical engineering graduate student who is both working in Srivastava’s research lab at Brown and took ENGN 2911R, said, “During the ongoing COVID-19 pandemic, Professor Srivastava and I have had numerous discussions on conducting further research on this topic. As a follow-up to what I learned in class, I have started to work with him to develop advanced models for the spread and dynamics of COVID-19.”
Modeling can also help public health officials make decisions, by quantifying disease spread and dynamics and help them manage the disease by deployment of critical resources at the right time. Vaccinations is one example, Srivastava said. “If we have the opportunity to produce a vaccine, a model can help predict how many people will fall sick, for example 10 days from now, and this information can be used to produce the right number of vaccines.”
Joshua Daniel, a fifth year master’s student in biomedical engineering who had recently completed a six-month stint at the World Health Organization headquarters in Geneva, Switzerland, was specifically looking to develop a skill set in mathematical modeling when he signed up for the course. The WHO experience had exposed Daniel to the public health impact and resource allocation decisions of combating infectious diseases. He said, “Professor Srivastava did an excellent job in communicating that none of these models is absolutely perfect. Early in the module, he emphasized that the dynamics of an infection can be different over various geographic, cultural and economic conditions.”
“The fact that what we learned in class became so relevant to all of our lives only a few months later exemplified to me the importance of education in becoming informed members of society,” said first-year biomedical engineering Ph.D. student Josephine Kalshoven. “Having been in Professor Srivastava’s class, I was able to contextualize comments from ‘experts’ frequently interviewed on the news about the spread of COVID-19. When they presented their best approximations of the R0 of the disease (the basic reproductive ratio, which gives an approximation of how many people a single diseased individual may infect in a susceptible population), I could compare these conjectures to problems we’d solved for our class. In particular, I often think back to a homework assignment where we used real data from an outbreak of influenza in an English boarding school in the 1970s. Over the course of two weeks, the disease had swept through most of the school - 97 percent of students, we calculated - and from the data, we computed an R0 of about 3.7. This value, and the knowledge that the 1918 flu had an R0 of about 1.8, have helped me to understand the implications of the variety of COVID-19 R0 estimations, which have fluctuated wildly over the past months. I’ve seen R0s estimated anywhere from 1.4 to greater than 6.0. Having worked through this math for our class, I also understand that the numbers scientists ultimately obtain are limited by the accuracy and accessibility of data. When the population is the globe - and not a small English boarding school - conclusive disease parameter calculations are elusive.”
Barry agreed, “There are new articles about this pandemic being published constantly, and I think that the modeling background I gained in this course makes it easier for me to understand, and explain to my friends and family, how researchers are predicting what will happen and what strengths and limitations those models have.”
Where will Srivastava take this course next fall? “Of course, this COVID-19 virus might be one of the plug-ins of the homework problem, or a class project” he mused. “Hopefully by that time, we’ll have a complete set of data to use.
“With modeling, you can predict different scenarios. In real life, you only get one scenario - what you see. But with mathematical modeling, you can predict: If we had not done social distancing, what would happen? What if we had not done this? What if we had done that? What if we had acted two months earlier?
“The value of this course is that when students learn something new, they use this knowledge to build even more complicated models and solve these kinds of important problems. Students often learn things in class, and ask ‘Why all this complex math and physics? I’ll never see a day when I need to know this.’ But this is one example where they can acutely see the connection to the real world, now more than ever.
“I didn’t expect this to happen. Most people didn’t expect it to happen. But now my students can look at their experience and close the loop on the mathematical and engineering methods that can be applied to something so important and are now a very real part of our day-to-day life.”