The first tutorial was a bit of a blockbuster length-wise because there was a lot of ground to cover to get the basic model up and running. Moving forward, we’ll be able to go a bit more incrementally, adding elements to the basic model as we go. If you don’t feel comfortable with the original, go back and take a look at it and make sure that it makes sense to you before moving on.

We’re first going to deal with some of the efficiency issues in the first model. After this, we’ll make some basic changes to the architecture of the SIR program that makes it more amenable to contact patterns on a social network.

Finally, we’ll show you how to how to take the output of your epidemic model and generate animations like this one:

Blue nodes are exposed but uninfected, red nodes are infectious, and yellow ones have recovered.

The movie is a bit of a carrot to get you through the less flashy, but, I promise, important and actually interesting nuts and bolts of putting these kinds of models together.

This tutorial is going to cover the last two big things that we need to tackle before we get to the model of an outbreak on a network. So, here we go!

New Concepts

1. Arbitrarily Distributed Infectious Periods

First, we’re going to deal with the duration of the infectious period. The assumption of an exponentially distributed infectious period is unnecessarily restrictive for a general model of diffusion, and the way the original code goes about recovering individuals – drawing a random number on every step for every infectious individual – should strike you as both inelegant and computationally inefficient, particularly when the rate of recovery is slow and there are many infectious individuals.

In order to deal with this, we’re going to introduce two new tools. The first is the scipy.stats toolkit and the second is a neat (and very easy to use) data structure called a heap.

A heap is in very many ways what it sounds like: imagine a pile of trash in a landfill; the tires and rusting washing machines are on the bottom, while the pop cans and grocery store receipts are closer to the top.

As a programming tool, a heap is useful because it always keeps the smallest (or largest, depending on your preference) item at the top of the list.  It also allows for linear-time insertion and removal of objects. This means that the time it takes to execute an action grows proportionally to the size of the list, so if it has N items, it takes N*C steps (where C is a constant) to process the list, and if it has 2*N items, it takes 2*N*C steps. Other ways of sorting could take N^2 or worse steps to do the same.

In our outbreak model, the top item on the heap is always going to be the time at which the next individual recovers. By doing this, we can avoid the loop in the first tutorial (and replicated in one implementation here) that checks whether each infectious individual is going to recover on each step.

Looping over everyone is the most intuitive way to check if they’re going to recover, but it’s very inefficient, especially when infectious periods are long and the population is large. It’s also problematic from a theoretical perspective, because it chains us to exponentially distributed recovery periods.

Exponentially distributed infectious periods make analytic sense for a deterministic model, but your disease or *insert diffusible here* may have a constant or normally distributed ‘infectious’ period.

By using a heap-based implementation, as you will see, we can use arbitrary recovery periods, and Python’s implementation of the heap is very straightforward – just a small twist on the usual list using the heapq module.

2. Object Oriented Programming

One of Python’s strengths is that it supports a style of programming that mixes the best of object-oriented programming (OOP) and procedural or imperative programming.

We won’t go too deep into the details of OOP here, but the real strength of OOP implementations are that they allow code to be easily re-used in other programs (Python’s all-powerful ‘import‘ statement really makes this true) and also forces some structure on what functions have access to what variables, etc.

Click Below to Review the Implementation and Commented Code!

Implementation

I’ll walk you through the steps of implementing and running the model below. If you aren’t familiar with how Python does OOP, go take a look here to get the basics down. Python OOP is very friendly, although its relative simplicity of implementation can be a double-edged sword when compared to more strongly-typed languages.

For example, you could fill a list with a combination of the ‘simpleSIRModel’ objects below, some integers, other Python lists, etc., but then you need to know what to do with every object as it comes off the list.

This isn’t a very steep hill to climb, but it can be tricky if this is your first rodeo. By contrast, in Java, you would need to implement something known as a polymorphic array and then cast objects back to their type as you take them off of the list. That kind of type-safe implementation forces you to be aware of what objects are in what data structures, but you pay a penalty in terms of speed of coding and flexibility.

By contrast, Python does some behind-the-scenes work to deal with lists of arbitrarily typed objects, but it makes your life a lot easier. I’ll discuss why you might want to do this after we walk through the code.

For this one, you’re definitely going to need Python 2.4+ and a working install of relatively recent versions of Scipy and Numpy.

(Mac users, just do yourself a favor and install the Enthought Python Distribution.)

I think the justification for using a heap of recovery times is straightforward: it’s flexible, may be faster, and lets us use arbitrary recovery periods.  But why go all OOP?

If you take a look back at the model in the first tutorial, it should become clear that the main way you’re going to re-use that code is by cutting and pasting.

By contrast, this program can either be run using the Python interpreter (the code below “if __name__ == ‘__main__’:” only runs when it is used as a standalone program), or imported into another python script. We can then create multiple outbreak objects, subclass their methods, etc.

For example, I have used the model in the code example below to simulate outbreaks in a collection of households that were all exposed to a gastrointestinal pathogen called norovirus. In order to do this, I import this code and create a new ‘communityOutbreak’ object type that is filled with simpleSIRModel objects. When I call this object’s ‘run’ method, it calls the run method of each simpleSIRModel object contained within and records the results.

When we get to the networked outbreak model, I’ll show you another scenario where it can be helpful to import components from one model into another.

Ok, some food for thought for those of you playing along at home:

1. How could we better use objects here to use arbitrary recovery time distributions?
2. How could you make this even more flexible as a command-line program? Try to add command line arguments using either Python’s argv or getopt tools.
3. Try to run this a multiple times and output averages of the runs over time to get a sense of the model’s asymptotic behavior.

In epidemiology, it is common to model the transmission of a pathogen from one person to another. In the social sciences and law, we may be interested in thinking about the way in which individuals influence each other’s opinions, ideology and actions.

These two examples are different, but in many ways analogous: it is not difficult to imagine the influence that one individual has on another as being similar to the infectivity of a virus in the sense that both have the ability to change the state of an individual. One may go from being susceptible to being infected, or from unconvinced to convinced.

Additionally, social networks have become an important area of study for epidemiological modelers. We can imagine that the nature of the network is different than the ones we think about in the social sciences: when studying outbreaks of a sexually transmitted disease, one doesn’t care all that much about the friendship networks of the people involved, while this would be very important for understanding the impact of social influence on depression and anxiety.

As someone who spends a lot of time working in the space where epidemiology and sociology overlap, I end up thinking a lot about these models – and their potential application to new and different problems and am really excited to share them with a broader audience here. In this first tutorial, I’m going to introduce a simple Susceptible-Infected-Recovered (SIR) model from infectious disease epidemiology and show a simple, Pythonic implementation of it. We’ll work through the process of writing and optimizing this kind of model in Python and, in the final tutorials, will cover how to include a social network in the simulation model.

In order to use the example below, all you need to have installed is a current version of Python (2.4+ is probably best) and the excellent Python plotting package Matplotlib in order to view output. If you don’t have Matplotlib and don’t want to go and install it (although I guarantee you won’t regret it), just comment out import for Pylab and any lines related to plotting.

Model Assumptions

1. State Space / Markov Model

Before getting into the mechanics of the model, let’s talk about the theory and assumptions behind the model as it is implemented here:

The SIR model is an example of a ‘state space‘ model, and the version we’ll be talking about here is a discrete time, stochastic implementation that has the Markov property, which is to say that its state at time t+1 is only conditional on the parameters of the model and its state at time t.

A simple state-space diagram (courtesy of Wikipedia)

For the uninitiated, in a state-space model, we imagine that each individual in the system can only be in one state at a time and transitions from state to state as a function of the model parameters, i.e., the infectivity of the pathogen or idea and the rate of recovery from infection…and the other states of the system. In other words, the system has endogenous dynamics. This is what makes it both interesting and in some ways difficult to work with.

In the SIR model, we assume that each infectious individual infects each susceptible individual at rate beta. So, if beta = .5, there is a 50% chance that each susceptible individual will be infected by an exposure to an infectious individual. For this reason, as the number of infected individuals in the system grows, the rate at which the remaining susceptible individuals is infected also grows until the pool of susceptible individuals is depleted and the epidemic dies out.

The other parameter we care about is gamma, or the rate of recovery. If gamma is also equal to .5, we assume that the average individual has a 50% chance of recovering on a given day, and the average duration of infectiousness will be 1/gamma, or 2 days.

We refer to the ratio beta/gamma as the basic reproductive ratio, or Ro (‘R naught’). When this number is less than one, we typically expect outbreaks to die out quickly. When this quantity is greater than one, we expect that the epidemic will grow and potentially saturate the whole population.

2. Homogeneous Mixing:

We’re assuming a world in which everyone has simultaneous contact with everyone else. In other words, we’re thinking of a totally connected social network. If you’re a regular reader of this blog, a social network enthusiast, or in some other way a thinking person, this assumption probably seems unreasonable. It turns out, however, that for many diseases, this assumption of homogeneous or ‘mass-action’ mixing, which was actually originally borrowed from chemistry,  turns out to be a reasonable approximation.

For instance, if we are trying to approximate the transmission dynamics of a very infectious pathogen like measles in a city or town, we can usually overlook social network effects at this scale and obtain a very good fit to the data. This is because even very weak contacts can transmit measles, so that friendships and other types of close contacts are not good predictors of risk. Instead, we we are better off looking at a higher level of organization – the pattern of connection between towns and cities to understand outbreaks. In a social context, something like panic may be thought of as being super-infectious (for a really interesting study about the potential relationship between social panic and flu dynamics, see this paper by Josh Epstein).

This is, however, a generally problematic assumption for most problems of social influence, but an understanding of this most basic version of the model is necessary to move on to more complicated contact patterns.

3. Exponentially distributed infectious periods:

In the most basic implementation of the SIR model, we assume that each infectious individual has some probability of recovering on every step. If our model steps forwards in days and individuals have a .5 probability of recovery on each day, we should expect that the time to recovery follows an exponential distribution. This means that most people will be pretty close to the mean, but some  will take a relatively long time to recover. This is accurate for a lot of cases, but definitely not for all. In some diseases, recovery times may be lognormal, power-law or bimodally disributed.  For social models, the notion of an ‘infectious period’ may not make a tremendous amount of sense at all. But it allows for a very simple and transparent implementation, so we’ll use it here.

CLICK THROUGH TO SEE THE IMPLEMENTATION and RELEVANT PYTHON CODE!

Implementation

Ok, so now it’s time to move on to actually writing and running the model. If you have matplotlib installed, you should be able to just copy or save the code into a new file called ‘simpleSIR.py’ and then run it from the command line using the command ‘python simpleSIR.py’.

I’ll walk you through the substance of the model in the code:

And that’s really it for a basic SIR model. This implementation leaves a lot to be desired, but that’s not really the point of this tutorial.

In 36 lines of actual code (the vast majority of the program is comments), we implemented a stochastic, dynamic model and were able to take a look at the results.

This is really a strength of using a Python based approach: it’s simple, easy to read (for those of you who are Matlab users, it should look pretty familiar), and quick to put together. The code in this file took me about 20 minutes to write.

Your mileage may vary, but the point is that Python is great for getting up and going, prototyping your ideas and seeing what they look like. You can go very far with an all-Python model. If speed becomes a serious issue, you may think about blending in some C++ for the really computationally intensive methods.

For those of you playing along at home, before moving on to the next tutorials, spend some time with the following questions:

1. Would this model scale well? Where are there potential bottlenecks in the code?
2. What would an object-oriented implementation look like? Why would you want to try one?
3. How would you change the assumptions behind the model to look more like a more social scientific problem you care about?
4. What would have to (fundamentally) change in the code to be useful for thinking about outbreaks on a network?

Subsequent posts will cover the following topics (in roughly the following order):

#2) Tips and Tricks for writing more efficient model code.  (And why you should care!)

#3) Writing Individual/Agent-Based models in Python

#4) Implementing an SIR Model on a Network in Python

#5) Animating networked outbreaks using rSonia