Epidemics on Networks


Building a Network Model of Infectious Disease

&

Some Thoughts on Media Coverage of Mathematical Models


Giselle Cohen

This illustration, created at the Centers for Disease Control and Prevention (CDC), reveals ultrastructural morphology exhibited by coronaviruses.


Living during this Covid-19 pandemic, I began to see a contradiction inherent in epidemic modeling. The vast majority of interventions we had access to must be adopted at an individual level to have any efficacy, but our predictive models focused on large-scale trends and were often unable to handle the nuance of individual behavior.

As part of the MSCS Capstone course Network Science taught by Dr. Andrew Beveridge, I worked with Jennifer DeJong to develop a potential predictive model that was specifically designed to attempt to show the impact of household-level precautionary behaviors on the spread of a novel disease across a state. In order to accomplish this, we build a dual-layer network to model community connectivity that we could run an SIR model across to predict the spread of a novel disease.

Our Network Model

The network we built actually works by using a base layer that is made up of nodes that represent households and, when connected, form a county.

The second layer turns each of those counties into a node and is connected to form a State. Because this model is fully theoretical, we connected the counties to be a highly-connected slightly random ring called a small-world graph.

The counties within the state are connected according to a power-law that allows us to have counties with a variety of levels of influence and connectivity. Building in these kinds of connectivity, influence, and randomness is essential because we need the connectivity of our network to resemble how a state might reasonably be connected. In our network, an edge means a contact across which transmission could occur.

Building this network provides a theoretical State on which to run an SIR model. However, because we wanted to focus on household-level interventions like mask-wearing, hand washing, and physical distancing we introduced a variable called a safety level. Assuming that everyone within a county falls into a relatively normal distribution of how many of these precautions they are actively taking, we can then skew that distribution towards safer behaviors or towards less safe behaviors. Within this distribution, safety levels are assigned randomly to nodes in our county networks. Safer behaviors correspond to a lower probability of transmission across a given edge in a network. This allowed us to run an SIR model that would take into account the level of adoption of household-level precautions.


After we built our model, we ran it on a set of toy networks. This means that we build small-scale networks that matched our understanding of the relationships between households and counties within a theoretical state. Then we ran our SIR model to get a sense of the impact that skewing the behavior distributions safer or less safe would have on the spread of a novel infectious disease. We ran an infection across our invented state and looked at the results on individual counties. The network analyzed on the left is pictured at the second level where each node is a county.

The parameter Safety represents the skew of the distribution of safety ratings. So a negative Safety parameter says that overall the counties are skewed towards unsafe behavior, a positive safety parameter skews towards safer behavior (more precautions in place), and a Safety of zero means that the safety ratings of the households fall into a normal distribution.


The top row is skewed towards unsafe behavior, the middle row is an unskewed distribution of safety behaviors, and the final row is skewed towards safe behaviors. Notice that the scale changes and the peak of the safer graph is actually much lower than that of the others though it may not look that way. For more information on the impact of skewed safety behaviors on the spread across a network, please visit the mathematical breakdown on our website.

How this math informs Covid-19 Models


A graphic published by the New York Times in April of 2020 showing the outcomes of 5 different predictive models looking at future Covid trends in New York.
https://www.nytimes.com/interactive/2020/04/22/upshot/coronavirus-models.html

Mathematical models are essential to our understanding of the spread of disease and the impact of other public health concerns on a community. During the media coverage of the COVID-19 pandemic, this became incredibly clear. However, it also became clear that an increased literacy in mathematical modeling was necessary for the public to make use of this data. There are two major types of models - predictive and analytic. Predictive models extrapolate from current data to try to predict future trends. These models are similar to weather forecasts: they are beneficial, but they yield likely trends rather than precise numbers. Predictive models are more accurate at closer time steps, and they should be used as a guide, not absolute truth. Analytic models look at the data of what has already occurred to find themes, trends, and data analytics. Often these results, in addition to standing on their own, are used to improve predictive models.


From this Blog Post discussing responsible use of data and modeling

Image from the National Museum of Health and Medicine showing four different flatten the curve graphics

Using my experience in building a potential mathematical model, I am extrapolating to consider the use of mathematical models in media coverage of the COVID-19 pandemic. For example, the idea of “flattening the curve” became incredibly popular during the early days of the COVID-19 pandemic. This was based on a set of mathematical models that analyzed current hospitalization trends, transmission rates, and predicted future levels of hospitalized cases of COVID-19.


The concept of “flattening the curve” also used a second set of predictive models which showed the potential impact of social distancing measures to slow the rate of transmission. But these models are quite different from the graphs which look at current hospitalizations or numbers of cases. One of the other complications with the COVID-19 pandemic is we have also used predictive models to look at potential cases that existed before testing was widespread, so there are some analytic models that use only positive test values to denote cases and others that use a larger number of likely cases. Additionally, in a public health context, these models are vital to study transmission rates and look for hotspots like super spreading events and behaviors. So understanding how to read and interpret these models is key to understanding how we look at the pandemic.

For more on the mathematical model, visit Giselle Cohen and Jennifer DeJong's Website:

About the Author

Giselle Cohen is a Mathematics Major with a concentration in Community and Global Health as part of the class of 2021 from Macalester College. She is from Portland, OR, and – in addition to her academic pursuits – she enjoys kayaking, backpacking, wilderness medicine, swing dancing, cooking, and calligraphy. Flamboyantly nerdy, she will happily talk about mathematics for hours to anyone who will listen and she is similarly passionate about her work in Emergency Medicine, Sports Medicine, and In-home medical care. She has an unusual fondness for whiteboards, ice cream, and rain.