SIS Stochastic Model: Distributions and Proportional Probability

In this post, I'll describe how to go from the SIS model to a zombie model. The results are a stochastic simulation, shown below. There are few items on the model TODO list: add spatial variables and zombie deaths/kills.

SIS Stochastic Model: On the Road to Zombies

Epidemiology has many interesting models for describing disease outbreaks. A large class of these models, the only ones with which I'm familiar in fact, are known as compartment models. One of the most amusing extensions of these models is incorporating a zombie compartment, or population. A researcher named Robert J. Smith? has developed some deterministic models for describing these zombie models. Not totally sure why he publishes with a question mark; I assume to differentiate his citation counts from all the other Robert Smiths. Anyway, his publications can be found here. It's interesting stuff, but I think I can do it with a stochastic version. In Part 1 of an unknown number, I'll introduce the SIS model and some matrix techniques for working with it. In Part 2, to be written, I'll move to Markov probability distributions instead of matrices and introduce zombies.

Latex to PNG to Blogger

I just wanted to put at the top what you need to follow along with this post. Everything below is written assuming you're using a Unix terminal and you have pdflatex and imagemagick installed. For Ubunu, the de facto linux distribution, type this into your terminal window to get these programs:

sudo apt-get install imagemagick
sudo apt-get install pdflatex

Due to my failure in making nice looking equations in my last posts, I worked on discovering a solution. Naturally, I tried to google Latex and Blogger. Latex is what I usually use for formatting equations, so that is what I was hoping to find. Unfortunately, most of the ideas I found for inserting equations into Blogger relied on hosted scripts and random people providing image hosting. I tried two solutions described in blogs and both didn't work, due to the site hosting the rendering engine being down. If you would like to convert LaTeX code directly into PNGs on the web, you can try codecogs, but I find the website to appear pretty Spammy and you cannot have very much control over your results. But, hey, it works.

Next idea: use Latex to write equations, use pdflatex to turn them into PDFs, and then use ImageMagick to convert the PDFs into PNGs. Easy, right? Well, here are the results:

Predator-Prey Models and Global Warming Followup

I thought I'd follow up my last post with some information on some of the tools I used. I'm not going to talk about how I made the equations, because they came out terrible and I'm trying to come up with a new way. I guess if anyone is curious: I made them in LaTeX and used gimp to convert them to PNGs.

Ok, so let's see how I made the graphics. Here's the first graphic, in case anyone forgot.

Predator-Prey Models and Global Warming

For my first look at mathematical models, I thought I would see what happens when we put Metabolic Theory of Ecology, which predicts changes of organism metabolic related trait rates with temperature, together with classic predator-prey models! Sounds exciting, right?! I think so, mostly because predator prey models are interesting differential equation models by themselves and combing them with temperature dependence sounds cool. So, the question I'll be trying to answer is: what happens as temperature increases in predator-prey systems and can the systems go from stable to unstable populations? Another question I've stumbled across while working on this blogpost: why does blogger not support equations and why do my equations look terrible?

Short answer: With some temperature increases about the same as global warming, we see increasing instabilities in the predator-prey models. See the cool figure below and read the math for more information.