Developmental Biology Center Friday Seminar Series

9 February 2024

matthew michalska-smith

This talk was given as part of the Developmental Biology Center's weekly seminar series on the UMN Minneapolis campus in the spring of 2024. Text here will supplement slide material.

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Some of the slides will have animations which require advancing the slide partway through these explanations. These will be indicated by "next slide" in these notes.

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Finally, this presentation will look best in full-screen, but should be more or less intelligible on smaller screens (no guarantees about phone screens). So, whenever you're ready...

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Because this talk was given to a non-network-focused audience, I begin with a brief explanation of what a network is: namely, a set of nodes

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connected by edges.

Because networks are intrinsically abstractions of real systems, there is flexibility in what the nodes and edges represent, here I show two islands, perhaps with species colonizing one from the other. But I could just as easily show animals moving between fragmented forest patches

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Or people travelling between cities

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Or livestock being moved between farms

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Critically, networks are usually not made up of just two things, but rather represent many things connected through a sometimes complex web of relationships.

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The relationship between the structure of these networks and the dynamics of the processes these neworks represent has been the central focus of my research for the past decade.

How does the structure of a network relate to the dynamical processes being depicted?

How does the structure of endophytic resource competition relate to the growth and health of host plants?

How does the positioning of parasites within food-webs relate to groupings of species based on interaction similarity?

How does the structure of bipartite ecological interaction networks differ between antagonism and mutualism?

How does the structure of competition networks relate to communities' dynamical stability?

How does the structure of metapopulations of host movement relate to local disease dynamics?

How does the structure of contact networks relate to the spread of disease?

Kimberly VanderWaal

Meggan Craft

$$ \begin{aligned} \frac{\textrm{d} S_i}{\textrm{d}t} &= {\color{bc4b51}r S \left(1 - \frac{N}{K}\right)} - {\color{50b99a}\beta S I}\\ \frac{\textrm{d} E_i}{\textrm{d}t} &= {\color{50b99a}\beta S I} - {\color{0b6884}\sigma E} - {\color{bc4b51}r\frac{N}{K} E} - {\color{bc4b51}\mu E}\\ \frac{\textrm{d} I_i}{\textrm{d}t} &= {\color{0b6884}\sigma E} - {\color{ff5e5b}\nu I} - {\color{bc4b51}r\frac{N}{K} I} - {\color{bc4b51}\mu I} \end{aligned} $$

${\color{bc4b51}r}$: growth (birth - death)
${\color{50b99a}\beta}$: infection
${\color{0b6884}\sigma}$: recovery
${\color{bc4b51}\mu}$: death
${\color{ff5e5b}\nu}$: disease-induced mortality

Anderson et al. 1981

$$ \mathbf{\Delta} = \begin{bmatrix} -\delta & 0 & 0 & \delta & 0 & 0 & 0\\ 0 & -\delta & 0 & 0 & 0 & \delta & 0\\ \delta & 0 & -3 \delta & \delta & \delta & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \delta & \delta & 0 & 0 & -2 \delta & 0 & 0\\ \delta & 0 & \delta & 0 & \delta & -4 \delta & \delta\\ 0 & 0 & \delta & \delta & 0 & 0 & -2 \delta \end{bmatrix} $$

$$ \begin{aligned} \frac{\textrm{d} S_i}{\textrm{d}t} &= {\color{bc4b51}r S \left(1 - \frac{N}{K}\right)} - {\color{50b99a}\beta S I} + {\color{7b678e}\sum_j \Delta_{ij} S_j}\\ \frac{\textrm{d} E_i}{\textrm{d}t} &= {\color{50b99a}\beta S I} - {\color{0b6884}\sigma E} - {\color{bc4b51}r\frac{N}{K} E} - {\color{bc4b51}\mu E} + {\color{7b678e}\sum_j \Delta_{ij} E_j}\\ \frac{\textrm{d} I_i}{\textrm{d}t} &= {\color{0b6884}\sigma E} - {\color{ff5e5b}\nu I} - {\color{bc4b51}r\frac{N}{K} I} - {\color{bc4b51}\mu I + {\color{7b678e}\sum_j \Delta_{ij} I_j}} \end{aligned} $$

${\color{bc4b51}r}$: growth (birth - death)
${\color{50b99a}\beta}$: infection
${\color{0b6884}\sigma}$: recovery
${\color{bc4b51}\mu}$: death
${\color{ff5e5b}\nu}$: disease-induced mortality
${\color{7b678e}\Delta_{ij}}$: host movement between patches

Anderson et al. 1981
Xiao, Zhou, & Tang 2011

Anderson et al. 1981

In the following slides, I will show timelines of disease prevalence to represent the disease dynamics. In general, the specifics of these timelines do not matter as much as the overall regime (oscillatory, steady-state, etc.) being shown. In these plots, I will use color to distinguish between farms, rather than to show dynamics as in earlier slides. First, we consider a toy system in which we have a set of farms parameterized to exhibit identical dynamics in isolation

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each showing the same oscillatory behavior.

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However, when we introduce movement between the orange and blue farms,

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we see a reduction in the cycle amplitude at the destination (blue) farm.

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A trend that is repeated as we add more farms to the chain.

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Remember, each of these farms would look like the orange one if there were no host movement between farms.

In the previous slide, we considered when identical farms were connected, but what if the connected farms differed in their dynamical regimes

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for instance, one having cyclical dynamics as before, but the other exhibiting steady state dynamics. What happens when we introduce a flow of individuals between these two populations?

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When the flow of individuals is from the cycles to the steady, we see a somewhat expected result: we introduce cycles to the blue farm. I say expected, because when prevalence is high in the orange farm, a higher proportion of hosts moving to the blue farm will be infectious, while it will be more susceptible and removed individuals when prevalence is low. Such variability in movement might likewise introduce variability into the destination. The more interesting result is that when the flow is reversed

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we see not only a reduction in cycles, but a qualitative change in dynamics, making the orange farm exhibit steady-state prevalence as well. In the supplementary slide below, we show that, while the rate of movement required to elicit each of these changes differs, very small rates of migration are needed to get this qualitative change.

Result (from small networks)

Prediction (in large networks)

Migration dampens cycles:

Reduced cycle amplitude and disease prevalence in networks with longer chains

Destinations inherit dynamics:

Interconnected parts of the network should have similar dynamics

Dynamics inherit asymmetrically:

Preponderance of cyclical and chaotic dynamics

To test these predictions, in the absence of good-quality, large empirical datasets, we generated a suite of random networks according to five distinct random-graph generating algorithms, an representative network of each is shown here. For each 100 random graphs in each of these general types, we randomized local parameters in each patch and then ran simulated epidemics and evaluated the qualitative dynamical regime of each patch after a sufficiently long time for transient dynamics to have passed.

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These are filled bar charts, where each patch in the network is a horizontal line, and vertical lines represent unique simulation runs. if the bar is completely one color (e.g. blue) from top to bottom, then every patch in the metapopulation had the same dynamics at the end of the simulation (e.g. cycles). If, instead, the color varies from top to bottom, then the dynamics vary across the network.

In accordance with our third prediction, we see mostly cyclical dynamics, but there was also an interesting difference between the left three network types and the right two. While the left networks tended to have either all cyclical or all steady state dynamics. On the right, we see much more heterogeneity across the metapopulation. Further analysis suggests that the key driver of this effect is variance in in-degree (i.e. some patches receive migrants from few other patches, while others receive from many).

Simple patterns emerge in small metapopulation networks

When linked with migration, cyclical dynamics are dampened further down chains
Migration tends to lead to inheritance of dynamical regimes between populations
When multiple source populations are considered, more complicated dynamics (cycles and chaos) dominate the inheritance

While there are patterns in larger networks, they are not simple

Patches with no immigration tend to have higher pathogen variance than patches that have at least one source of incoming host movement
Most patches exhibit oscillatory dynamics

Further research is needed to:

Identify relationships between more complex network structures and disease dynamics
Fit to empirical data, including empirical network structures

In summary, we found relatively simple, clear patterns when we consider the case of just a few patches connected in simple ways, as noted previously.

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In contrast, the results for larger networks were more complicated. While they were in agreement with some of our predictions from small networks, it is much more difficult to predict the dynamical regime of patches because of the complex, transitive relationships in larger, more realistc networks.

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Further research is really needed to dig into these complexities and identify relationships between measures of network structure and the dynamical regimes we observe. Of course, the collection of high-quality empirical data is also essential to assess the applicability of the theory outlined here.

Eva Enns

Lauren White

Marie Gilbertson

Meggan Craft

@City_Janesville on twitter.com

Art by Scusi

What is contact?

What makes a contact network realistic?

Home/Family
School
Work
Religious institutions
Hospitals
Nursing homes

image adapted from Monroe et al. 2021

To do this we had to answer a couple of basic questions, such as "What is a contact?" The answer to this question is obviously context specific, but with COVID-19, with its aerosolized transmission, we think about the risk of transmission increasing with the duration of people sharing the same air space.

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But of course the ways in which we are close to other people is also context dependent, leading to the key insight for this project: the structure of our interactions with other people is dependent on the different ways in which we interact with different facets of our social network.

To put it more concretely, interactions are structured differently among co-workers than among people at home, or at a religious institution. The differences in these interactions, including the strength of interaction and the interacting group size, can have consequences for the spread of disease.

Interaction properties

Node properties

So, we begin to assemble our networks: starting with a collection of individuals, whose ages are distributed according to some locality's census data and grouping them into households, based on the empirical distribution of household sizes. We then successively layer additional interaction types, including schools

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(based on empirical classroom sizes), work-places

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(according to empirical distributions of work-place sizes), and possibly others.

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Importantly, because each node has an age associated with it, we can also classify some individuals as being more vulnerable to adverse outcomes from COVID, using age-normalized hospitalization rates. Here, we dichotomize individuals into "Vulnerable" or "Not" for simplicity.

Disease Model

I want to take a minute to run through the underlying disease model, but will keep this brief as it is not the most important aspect of this project.

At each timestep in a simulation, every individual in the network has one of six states. They start out as Susceptible to disease and can become infected when they share an edge, representing a social contact, with an infectious individual.

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Upon infection, they enter a latency period and are classified as Exposed, but not yet infectious. Over time, they transition to being I/Infectious, and some fraction will develop symptoms.

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As more time passes, some of these will Recover,

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while others will sadly Die.

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The final piece of the puzzle is that those individuals that we classified as vulnerable have both longer disease courses (slower recovery),

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and are more likely to die, modifying these two parameters as shown.

What does this look like in a single simulation?

We can summarize the outcome of a single epidemic simulation in different ways, such as the epidemic peak (the maximum proportion of the population ever concurrently infectious), the percentage of vulnerable individuals in the population that get sick at some point during the simulation, or the total percent of the population that dies from the disease.

For these first results, we compare the two age/household-size distributions, a "Florida-like" and a "Texas-like" population. The latter sees a higher epidemic peak and percentage of vulnerable individuals infected. This is likely driven by the larger household sizes, driving faster and wider pathogen spread.

Counter-intuitively, however, significantly more people die in the Florida-like population. This is because, even though the epidemics are smaller, there is a much higher proportion of vulnerable individuals in the Florida-like population, leading to higher mortality rates.

We can also consider the relative contribution of each of our interaction network layers, here focusing on the contrast of classmate and co-worker interactions. We focus on this comparison because, at the time of writing, there was a large discussion about the relative merits of opening schools or workplaces as a phasing out of widespread lockdowns.

Our results showed that workplaces were more consequential (higher absolute value of effect size) for all measures of epidemic severity, including epidemic peak, total number of individuals infected, and rate of epidemic spread.

This is certainly influenced by our assumptions of network construction: we have more co-worker links in our networks than classmate. However, this decision of network construction is informed by the real-world: there are more working-age individuals than school-age children in the populations we consider and workplaces can be many times larger than schools and classrooms, so we do not view this as a limiation of the analysis per se.

Finally, we considered the effects of differential risk-tolerance through the population. For all previous results, we considered a case where all individuals participated in all of their respective interactions. Here we show the effect of different rates of transmission between classmates or coworkers on the epidemic peak and the total number of vulnerable individuals infected. As expected, higher rates of transmission result in more disease (top right of each heatmap).

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We next consider a situation where vulnerable individuals are less risk-tolerant, that is, they take steps to reduce their infection risk through avoiding school or workplace interactions. In this case, we see a moderate reduction in disease, but mostly at low transmission rates.

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We then asked, what if not only the vulnerable individuals themselves, but all members of their households likewise isolated. This would be like you deciding to work remotely because you live with your elderly parent/grandparent. In this case, we see a dramatic reduction in disease and a near-complete elimination of disease among vulnerable individuals.

Astute viewers might at this point be thinking "of course there is less disease, you've just removed a ton of links from the network!"

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Yet, it is not the removal of links per-se, but rather the selective fragmentation of the contact network that yields such dramatic effects. When we remove the same number of links randomly (i.e. irrespective of relation to vulnerable individuals), we re-capture the no-difference levels of disease.

Conclusions

Populations are highly interconnected, allowing disease to spread even to specific individuals that are taking precautions
Community action is needed to protect the most vulnerable
Demographic structure can have nuanced, contradictory effects on disease spread

Framework as an educational tool

Layered Interaction Network
COVID-19 Simulator

Developmental Biology Center Friday Seminar Series

How does the structure of a network relate to the dynamical processes being depicted?

How does the structure of endophytic resource competition relate to the growth and health of host plants?

How does the positioning of parasites within food-webs relate to groupings of species based on interaction similarity?

How does the structure of bipartite ecological interaction networks differ between antagonism and mutualism?

How does the structure of competition networks relate to communities' dynamical stability?

How does the structure of metapopulations of host movement relate to local disease dynamics?