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Epidemiology 202

Bruno Gonçalves

Welcome to the latest post of the “Epidemic Modeling” series where I aim to guide you through the mathematical and conceptual details underlying epidemic modeling with a focus on what’s most relevant to the current CoVID-19 pandemic.

In this post, we explore some simple network models in order to better understand the role that degree correlations play in disease spreading. I aim to make each post self contained, but you might want to check out our previous posts to brush up on the basics of epidemic modeling:

Each post comes with a corresponding Jupyter notebook that implements the models and generates the figures. You can find them over at the GitHub repository dedicated to this series:

These are simply my personal views and opinions, as an individual with some background real world experience in this fascinating field and shouldn’t reflect on any group or institution I might be affiliated with.

We saw in a previous post that we can explore network structure to better understand the spread of epidemics and some of the empirical effects we observe in real world networks.

Here we will look in more detail at how the structure of the network impacts the spread of the disease, even when the neighborhood of each node looks, on average, the same. Let us consider the Simple Susceptible-Infectious model:

Susceptible-Infectious model

In this simplified view of epidemic spreading, we consider only two possible compartments, Susceptible (healthy) and Infectious (sick). Under these assumptions, every node in the network will eventually become infected since an infected node remains infectious forever and never recovers.

Erdős-Rényi Network

For the network structure, let us use a simple Erdős-Rényi Network. Under this model we simply connect every pair of nodes with a fixed probability p, resulting in an average degree of:

With 300 nodes and p=0.0199 we expect an average degree of 5.95. Numerically, find:

The degree of each node in an ER network realization

Due to the complexity of the network model, we can’t just integrate the underlying equations so we must simulate the spreading process using random numbers. As a consequence, each run will look slightly different. We simulate 100 runs and take the median of the values across all runs at each point in time. The end result is:

Number of infected nodes as s function of time for each of the individual runs and the median value

We can get a simple check that this is the expected shape of the curve, by going back to the Differential Equation formulation:

where we have used the N=S+I identity. We can solve this equation to find:

We can verify this expression using a numerical fit, with excellent results:

Comparison between the empirical results and the analytical fit in the ER network

where we find the numerical value of β to be 0.217, which is the equivalent value of β in the continuous approximation where we intrinsically assume that every node is connected to every other node with no internal structure.

Barabási-Albert Network

For comparison with the previous example, we consider the case of the Barabási-Albert model where nodes are connected to other nodes proportionally to their degree. In this case, we find a similar behavior:

Naturally, we can perform a numerical fit just as we did above with reasonable results:

Comparison between the empirical results and the analytical fit in the BA network.

In this case, we find that β to be 0.297, a higher value than before and comparatively worse results. This is a direct result of the fact that we now have a broader degree distribution with some hubs being connected with over 40 others while most nodes having degrees of just 3 or 4:

The degree of each node in an BA network realization

The effect of Topology

We make the difference between these two scenarios plain to see by plotting two median curves next to each other:

Comparing the median results for ER and BA network with similar average degree

The difference is entirely due to the way in which the nodes are connected and the resulting correlations.

Network structure, the way in which nodes connect to one another, is one of the most important details that determine how an epidemic will spread. As we will see in future posts, we can take advantage of this fact to better understand and even control epidemics.

Resources:

Thank you for making it through another post in this series. Next time we’ll get a lot deeper into network structure and how they impact epidemic spreading. In the meantime, you can brush up on our previous posts.

To help you along the process of digesting all this new information, I’ve published all all the code necessary to implement the models described above and to generate the figures in all posts. You can find it in the series GitHub repository:

You might also enjoy my weekly newsletter where I share the latest news and developments in Machine Learning and Data Science as well as any future blog posts I write.



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