Epidemiology and zombies

Transcription

1 Epidemiology and zombies The end is coming! You have just been recruited to be a secret agent in the most important and secretive Government agency there is: the Department of Zombies (DZ). As an agent in the DZ, it is your job to protect Australia from zombie plagues. Some typical zombies are shown in Figure 1. Word comes in that it has finally happened: the zombie apocalypse is here! Zombies are walking the streets! You receive a call from the Prime Minister of Australia: she wants to know what will happen next and how the zombie can be stopped. You must turn to the only thing more powerful than a wave of the undead: mathematics. Figure 1: Zombies are known for being many things, but friendly isn t one of them. 1 A simple zombie model We need to know if the zombies will eventually take over, and we need to figure this out without just waiting to see what happens. To do this we are going to make a mathematical model of the zombie plague and use it to predict what will happen next. 1

2 Cellular automata and the zombie model The model we will use is based on cellular automata. A cellular automaton is a grid of cells that may take on a number of different states, where the state of each cell is decided by the previous configuration. You may have already seen an example of this when playing the Game of Life. In our case, each cell will represent a person. Each cell is surrounded by nine other cells, which correspond to the people the first person comes into contact with each day. Each of these cells can take on two states: human or zombie. Initially, all but one of the cells will be humans. The central cell will start as a zombie, representing the initial outbreak of the plague. To complete the model we need a rule to govern how the zombie infection spreads. We will assume that, in order to become a zombie, a human needs to come into contact with a zombie. So, a zombie can turn any of the nine surrounding cells into a zombie. However, if someone encounters a zombie there is a chance they could get away without being infected. To account for this in the model we introduce a transmission probability β. This is a number between zero and one that represents how likely a person is to be infected if they come into contact with a zombie. If β = 0 they will never be infected, while if β = 1 they are guaranteed to be infected. For values in between, a person will sometimes be infected and will sometimes get away safely. A diagram of this transmission rule is shown in Figure 2. Logging in and using Matlab To login to the computers, the details are: Username: mathsci Password: M@thsci99 We will use a program called Matlab to run the zombie model. To start Matlab, double click on the Matlab icon on the desktop. If there is no link, Matlab can be found under the Programs tab on the start menu. Once Matlab is running you are ready to model some zombies. Matlab accepts commands through the command window (shown in Figure 3). To run the model, you will need to type zombie(β) into the command window next to the >> symbol and then press the enter key. Here, β is the probability of the zombie infection spreading from one person to another and should be a number between zero and one. For example, to run the model with infection probability β = 0.2, type zombie(0.2) into the command window and then press enter. This will bring up a new window showing an animation of the model that should look something like Figure 4. You may wish to make the plot window bigger so that you can see the animation more clearly. The plot on the left shows the distribution of humans and zombies. Humans are shown as green cells while zombies are shown in red. The number of days since the outbreak started is shown at the top of this plot. The plot on the right displays the number of humans and zombies using the same colouring. To begin with there are 2499 humans and 1 zombie. 2

3 β β Figure 2: The transmission rule for the zombie model. A zombie (the red cell) can infect any of the nine surrounding humans (the green cells) with probability β. The rule holds as the two diagrams are rotated about the central square, meaning the infection can spread in any direction. Figure 3: The Matlab command window. 3

4 Figure 4: Running zombie(0.2). 2 Running the model You are now ready to try some different scenarios. Exercises 1. Run the model with β = 0.1 (type zombie(0.1) into the command window and press enter ). (a) How many zombies are there after 50 days? (b) How does the infection spread? Does this seem realistic? 4

5 2. Run the model a few more times, again with β = 0.1 (hint: press the up-arrow key to bring up the previous command). Do your answers to the first two questions change? If so, why? Would you expect to see this same behaviour in the real world? 3. Try running the model a few times with β = 0.3. How many zombies are there after 50 days? 4. Do you think this model is missing anything? How could it be made more realistic? 5

6 3 Improving the model From the previous section it seems that things aren t looking very good. The results show that, even for small values of β, the zombies will eventually take over. If we are to avoid the zombie apocalypse we need to introduce a way to stop the zombie plague from spreading. It s time to fight back and kill some zombies. We can add this into the model and see what effect this has on the zombie plague. We model zombie deaths by allowing zombies to die at each time step with probability γ. You can specify γ in our model by giving zombie.m a second input separated by a comma. For example, to run the model with β = 0.2 and γ = 0.5, enter zombie(0.2,0.5). Again, as γ is a probability, it should be between zero and one. On the plot, dead zombies are represented by blue dots. The number of dead zombies is also plotted in blue. Exercises 1. Try running the model with β = 0.5 and γ = 0.2. What happens? Does anyone survive the zombie apocalypse? Run the model a few times to confirm you answer. 2. Suppose that we find a better way to kill the zombies so that γ increases to 0.5 (meaning it is more likely each zombie will die). What happens now? Can you recommend a strategy based on your observations? 4 Changing γ While working on the previous sections you might have noticed that running the same simulation twice does not always produce the same results. This is because of the probabilistic elements of the model: when a human encounters a zombie there is some probability that they will be infected, but the result will not always be the same. This reflects how a the random chance that would be inherent in a real-world zombie plague. Given that our model can produce different results when supplied with the same parameters (β and γ), how can we draw conclusions about how many humans will be infected? One way is to run the model a number of times and then take the average of all the results. The average result gives us an idea of the most likely result. To do this we now make one final extension to our zombie model. Giving zombie a third parameter N will run the simulation N times and then display the average number of humans remaining at the end of the simulation. For example, zombie(0.3,0.2,5) will run the model 5 times with β = 0.3 and γ = 0.2, and then display the average number of humans remaining across the 5 simulations. We will use this to explore the effect of changing the death parameter γ. 6

7 Exercises 1. Run zombie with β = 0.3 and each of γ = 0.1, 0.2, 0.3, 0.4 and 0.5, each time running the simulation three times (N = 3). Record the average number of humans each time. 2. Using the graph paper on the next page make a plot of the average number of humans versus γ. 3. Looking at your plot, what do you notice as γ increases? 5 Map of the Dead If you re still worried about the zombie apocalypse have a look at the Map of the Dead, which can be viewed at This map highlights areas of high risk during a zombie plague and also indicates where to locate useful resources to help you survive. Figure 5 shows the map of Adelaide, while Figure 6 shows Sydney. Exercises 1. What do you think makes one area safer than another? 2. How could you make a high-risk area safer? 7

8 Average number of humans versus for 3 simulations Average number of humans

9 Figure 5: The Map of the Dead for Adelaide. Figure 6: The Map of the Dead for Sydney. The icons represent resources that would be useful in a zombie apocalypse. 9