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STAGE 2 MATHEMATICAL STUDIES

Matrices Folio ‘The Zombie Apocalypse’

By Jenna VeinbergSACE Number: 441628H

Word Count:

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Zombies are becoming an increasingly popular part of movie culture and are one of the most commonly spoken and written about fictional creatures. Zombies are often portrayed as mindless, violent creatures with a need for human flesh. In each text their distinct features can differ but there are many common ways their origins are depicted. Some of these are:

Reanimating corpses by magic Parasites that take over a human host Neurogenesis techniques involving stem cells to regrow dead brains Self-replicating nanobots that live after a human host has died A virus that can spread from person to person

This folio task mathematically determines what could happen in the case of a zombie apocalypse by considering different types of zombies, different methods of fighting back, and the effects that a zombie apocalypse would have on the human population.

PART 1: The Basic Zombie Humans are living, breathing creatures, that are classified as being alive. They can die, becoming a corpse or be turned in a zombie. Zombies are often killed by other zombies, or by fighting with humans and thus they also become corpses. A corpse has the ability to be turned into a zombie, but they cannot be turned back into humans, nor can zombies be turned back into humans.

It has been found that each week: 1% of humans will die from natural (non–zombie) causes, 15% of humans will be turned into zombies, 20% of all corpses will be turned into zombies; and 2% of the zombie population will die from fighting each other or humans.

A 3x3 transition matrix can be formed, to state the change between the alive (humans), the corpses (dead) and the zombies. This can help determine the population of each of these 3 ‘states’ after different periods of time. When forming this transition matrix, it is important to ensure each row is equal to 1 (or 100%), this is due to the fact there is no change in the population.

[0.84 0.01 0.150 0.8 0.20 0.02 0.98] This transition matrix is denoted as S0

The initial population for each of the three states is assumed: 35% of the population is alive, 64.7% of the population are corpses; and 0.3% of the worlds population are zombies.

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By multiplying the initial population with the transition matrix formed above, the population of each species can be found after 1 week. This is shown below. S1 = S0 x T1

[ A C Z0.35 0.647 0.003]×[0.84 0.01 0.15

0 0.8 0.20 0.02 0.98]

Therefore, the population of each species after 1 week is:((0.35 x 0.84) + 0 + 0)) x ((0.35 x 0.01) + (0.647 x 0.8) + (0.003 x 0.02)) x ((0.35 x 0.15) + (0.647 x 0.2) + (0.003 x 0.98)= [0.294 0.521 0.185 ]

This shows that after 1 week: 29.4% of the population will be alive and human, 52.1% of the population will be a corpse; and 18.5% of the population will have become zombies.

To find the population after 10 weeks the process is the same as above, except this time the transition matrix (which represents 1 week) will be raised to the power of 10 (the weeks we want to solve). S10 = S0 x T10

[ A C Z0.35 0.647 0.003]×[0.84 0.01 0.15

0 0.8 0.20 0.02 0.98]

10

= [0.062 0.132 0.807 ]

This shows that after 10 weeks: 6.2% of the population will be alive, 13.2% will be corpses; and 80.7% will be zombies.

To find the population of each species after 30 weeks is exactly the same as shown above, except instead of raising the transition matrix to the power of 10, it is raised to the power of 30. S30 = S0 x T30

[ A C Z0.35 0.647 0.003]×[0.84 0.01 0.15

0 0.8 0.20 0.02 0.98]

30

= [0.0018 0.0909 0.907 ]

This shows that after 10 weeks: 0.18% of the population will be human (alive) 9% of the population will be corpses 90.7% of the population will have become zombies.

Given the pattern found in the calculations above it can be predicted that the human population will eventually die out, thus only corpses and zombies will remain. This is shown

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by the vast drop of human population after just 10 and then 30 weeks. This means the human population will continue dropping until it ceases to exist, whilst the zombie, and corpse population will continue ton increase until it finds a steady ‘fixed’ state. This can be done by finding the steady state of the population matrix.

[0.84 0 00.01 0.8 0.020.15 0.2 0.98

¿alive¿corpse¿ zombie ]

Above is the original transition matrix after it has been rotated 90 degrees, making the rows the columns and visa versa. This has been done to ensure that each column equals 1 due to:

a (alive) + c (corpse) + z (zombie) = 1

This new matrix will be used to find the steady state for the corpse and zombie population. The alive population does not have to be included when finding the steady states as it will be whipped out and therefore will be 0. This means, that to find the steady state all columns must equal to 0.

[−0.16 0 00.01 −0.2 0.020.15 0.2 −0.02

¿alive¿corpse¿ zombie ]

Above the matrix has been changed so that each column is equal to 0. This shows by looking at row 1, that -0.16 x a (as it is the alive row) = 0. Therefore, a must = 0. With this new information another matrix can be formed to remove all a values and thus to find the steady state of the corpse and zombie population.

[0 0 00 −0.2 0.020 0.2 −0.02

¿alive¿corpse¿ zombie]

By eliminating all of the 0s the matrix showing the steady state of the corpse and zombie population can be found and thus solved.

[−0.2 0.02 |00.2 −0.02 |0 ]

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As both rows have the same values, just reversed, only one row will be needed to find the steady state of the corpse and zombie population. Therefore, row 1, the corpse row values will be changed to 1s.

[ 1 1 |10.2 −0.02 |0 ] R2 R2 – 20/100 R1

SOLVE FOR STEADY STATE

By analysing the above calculations, it has become apparent that the steady state for the corpse population is…. and the steady state for the zombie population is….. This means that when each of the two species reach this population it will remain here (at the steady state), not increasing or decreasing. After 4 weeks of letting the zombies rage rampant the humans decided it was time to fight back. The began organizing themselves and effectively fought back. Every week they managed to kill 10% of the zombie population, and prevented the zombies from turning more than 5% of humans into their species.

Therefore, each week: 10% of zombies were killed (turned into corpses) 5% of the human population turned into zombies 1% of humans died form natural causes 20% of corpses where turned into zombies

After 4 weeks with no action the species populations would be the following:

[ A C Z0.35 0.647 0.003]×[0.84 0.01 0.15

0 0.8 0.20 0.02 0.98]

4

=[0.174 0.289 0.536 ]

After four weeks of no fighting back the population rates for each species would be: 17.4% would be humans (alive) 28.9% would be corpses 53.6% would be zombies

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Because the humans did decide to fight back, after four weeks the original matrix seen above would be changed, and therefore the first week of there fighting back (week 4 from when the zombies arrived) the population would look like:

[ A C Z0.35 0.647 0.003]×[0.94 0.01 0.05

0 0.8 0.20 0.1 0.9 ]

¿ [0.329 0.521 0.149 ]

Already it is apparent that by the humans fighting back, the population rate for the zombies is far lower than it would be for the same week if the humans did not fight back. The human population rate is far higher, as well as the corpse rate.

If the humans decided to take charge and fight back after four weeks: The human would take up 32.9% of the population, The corpses would be 52.1% of the population; and The zombies would only be 14.9% of the population.

To find the new population of each species after 10 weeks if the humans did decide to fight back after four weeks, the same process would be completed as it was originally, although his time, the new transition matrix would be used and instead of raising it to the power of 10, it would be raised to the power of 6, as it started on week 4 (4+6 = 10).

[ A C Z0.35 0.647 0.003]×[0.94 0.01 0.05

0 0.8 0.20 0.1 0.9 ]

6

¿ [0.120 0.290 0.588 ]

Therefore, after 10 weeks of the zombies being on earth, if the humans decided to fight back after 4 weeks, the population of each species would be:

12% of the population would be human, 29% of the population would be corpses; and 58.8% of the population would be zombies.

To find the new population after 30 weeks, the same process will be completed, this time, the transition matrix will be raised to the power of 26 (26 + 4 = 30)

[ A C Z0.35 0.647 0.003]×[0.94 0.01 0.05

0 0.8 0.20 0.1 0.9 ]

26

¿ [0.035 0.320 0.644 ]

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Therefore, after 30 weeks of the zombies being on earth, if the humans decided to fight back after 4 weeks, the population of each species would be:

3.5% would be humans, 32% would be corpses; and 64.4% would be zombies.

This shows that although the human population has still drastically dropped, by the humans fighting back, the human population will stay alive for longer than they would have if they did not fight back at all.

Many assumptions have been made in regards to the information and calculations made above. Assumptions beyond just the simple one of zombies existing. All of these assumptions that have been made have had a great impact on the results found and could pose great limitations on the answers and calculations that have been made.

An assumption made very early on in the calculation was that the population of the earth was constant. It was assumed that there was a set population which would not increase nor decrease throughout this apocalypse. If this was not assumed, and there was in fact a constantly changing population of the world it would be impossible to find the steady state of each species as they have been found above.

Another assumption that was made was that the initial victims begin to cause havoc within the human race quickly and that the transition to a zombie form was fast. All of the calculation were done on a weekly basis, so if it takes longer than a week for a human or a corpse to become a zombie the data found above would be irrelevant. It was also an assumption made that all people took the same amount of time to become zombies, if this was found to be incorrect with variants between human to human or corpse, the data and results would become irregular. It was also assumed that the entire worlds population would be able to be infected by zombies. This means that the zombies not only would have to be able to reach each piece of land inhabited by humans but they would also have to have the ability to dig up and find corpses after that had previously been buried. If this is not true, and is in fact a false assumption, humans would not go extinct as the zombies may not have a way to reach other countries. If they cannot reach other countries, then the calculations above are irrelevant as only the population for the zombie inhabited land should be used for the above calculations.

It is also assumed that all given data is accurate. The data values for how many zombies are killed, humans and corpses turned into zombies, and human deaths have all been assumed above. It was also assumed with these values that they remain the same each week and that there are no changes with the population of deaths each week.

If this was all incorrectly assumed the calculations and results found would deem irrelevant and new, accurate data should be used to re calculate the results to increase the reasonableness and to decrease the limitations that these assumptions would cause to the results.

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PART 2: The Modern Zombie In recent years’ media has shown that zombies are often created by viruses, or even parasites. Because of this they can be cured and thus returned to human form. These zombies, although can be cured are found to be faster, smarter and stronger than usual. Using the same three states for the population as part 1, being alive, corpse and zombie. It is found that with these new curable zombies every week:

4% of people die from non-zombie related causes, 50% of the alive population become zombies, 5% of the zombie population become corpses, 20% of corpses become zombies, 20% of zombies are cured and become human again, The human population increases by 2% due to births; and Every week the humans burn all the zombies they kill (assume 3% of zombie

population) and also burn an additional 5% of remaining corpses.

To model the changes in an alive/corpse/zombie state over a one-week period unlike part 1, a simple proportion transition matrix will not be sufficient. To calculate in this case all the information above can be put into a transition matrix except for the last two points. They have to form another matrix and will then using matrix addition will be combined with the first matrix. By putting the known values into a transition matrix the rest of the matrix can then be calculated around these. It is known that each row must equal 1, it is also known that corpses cannot turn back into humans so this must equal 0. From there, the following transition matrix can be calculated:

[0.46 0.04 0.050 0.8 0.20.2 0.05 0.75]

This above transition matrix represents the first 5 points recorded above. Below, another matrix will be formed with the last 2 points, these 3 numbers, will be put into a matrix and the rest will be filled out with 0’s.

[0.02 0 00 −0.05 00 −0.03 0]

Now by adding the two matrices together the final matrix representing the changes in an alive/zombie/corpse state over one week can be calculated.

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[0.46 0.04 0.050 0.8 0.20.2 0.05 0.75] + [0.02 0 0

0 −0.05 00 −0.03 0] = [0.48 0.04 0.5

0 0.75 0.20.2 0.02 0.75]

It is apparent, by looking at the final matrix the population of each state is changing. This can be seen as the rows no longer equal 1 like they did originally.

The initial population for each species is assumed: 0.3% of the population are zombies 64.7% of the population are corpses 35% of the population are humans (alive)

Like earlier in part 1 to find the population percentages of each species after week 1 the formula S1 = S0 x T1 will be used. By multiplying the initial population of each species with the matrix the population for each species will be found.

[ A C Z0.35 0.647 0.003]×[0.48 0.04 0.5

0 0.75 0.20.2 0.02 0.75]

Therefore, the population of each species after 1 week is:((0.35 x 0.48) + (0.647 x 0) + (0.003 x 0.2)) x ((0.35 x 0.04) + (0.647 x 0.75) + (0.003 x 0.02)) x ((0.35 x 0.5) + (0.647 x 0.2) + (0.003 x 0.75)¿ [0.169 0.499 0.307 ]

This shows that after 1 week: The human population would have dropped to 16.9%, The corpse population dropped to 49.9%; and The zombie population increased to 30.7%.

As in part 1, to calculate the population after 10 weeks S10 = S0 x T10. The initial population will be multiplied with the matrix raised to the power of 10.

[ A C Z0.35 0.647 0.003]×[0.48 0.04 0.5

0 0.75 0.20.2 0.02 0.75]

10

¿ [0.189 0.100 0.484 ]

This shows that after 10 weeks: The human population increased from week 1, to 18.9%, The corpse population dropped again to 10%; and The zombie population once again increased to 48.4%.

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To find the population of each species after 30 weeks the initial population will once again be multiplied with the matrix but this time the matrix will be raised to the power of 30. S30 = S0 x T30.

[ A C Z0.35 0.647 0.003]×[0.48 0.04 0.5

0 0.75 0.20.2 0.02 0.75]

30

¿ [0.134 0.0524 0.336 ]

These above calculations show that after 30 weeks: The human population has dropped to 13.4%, The corpse population has dropped to just 5.24%; and And the zombie population has also dropped to 33.6%.

Some of the humans came up with a plan, they managed to escape to a remote island on a boat and thought they were safe from any zombie activity.

After a week of being on the island however there were: 10 dead bodies (corpses), 52 zombies; and 42 people left human.

After one week of the people moving to this island the population is represented as:

[ A C Z42 10 52] This can be denoted as S1.

S1 = S0T As the population after one week is known, but the initial population still needs to be found, this equation will need to be rearranged in order for the initial population to be found. To do this T must be canceled from each side and therefore T-1 must be multiplied to each side.

S1T-1 = S0TT-1

= S0

S0 = S1T-1

Now by using the calculations above and the matrices that have been developed prior to this the population after one week and the matrix can be multiplied to find the initial population of island inhibitors.

[ A C Z42 10 52]×[0.48 0.04 0.5

0 0.75 0.20.2 0.02 0.75]

−1

¿ [82.48 8.613 12.05 ]

Each number will be rounded up to the nearest whole number in order to know the initial number of each species that moved to the island.

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Therefore, the initial number of each species that moved to the island where: 82 humans, 9 corpses; and 12 zombies.

Like part 1 of this folio multiple assumptions have been made in order to complete the above calculations. Assumptions far beyond that of zombies existing.

The first thing that has been assumed true is all of the data given. The percentage of each human killed, turned into a zombie, the percentage of corpses being turned into zombies, the percentage of zombies being killed etc. All of this information has been assumed in order to complete the above calculations. If any of these where changed the entirety of the results would change so it is an incredibly important assumption that has been made.

The initial population of each species was also assumed. Along with the fact that their must have already been zombies on this island in order for those populations to grow. Therefor it has been assumed that either zombies snuck over with the humans, or they had already inhabited this island.

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PART 3: Survival Plan It is a common conception that the ‘zombie gene’ is passed on in a virus like manor. It has been studied and found that like the influenza, or a cold the zombie virus can be passed between humans, and even to corpses through contact. This virus is fatal to the human population, and if left un treated the human population could be diminished forever.

Humans can catch this virus as easily as the common cold, they must only come into contact with a zombie and by breathing, coughing, exchanging bodily fluids or biting can cause the human to catch this virus. Once the human has caught this virus they have one week before they have completed their transformation into a zombie. This week is incredibly painful and they become ill during their transition. The only way to currently prevent one from turning into a zombie after infection is by turning into a corpse. Unfortunately, though, not even corpses are safe from this virus, if a zombie bites a corpse, the corpse will become a zombie itself.

It has been found that each week:

3% of humans die form natural causes 30% of humans will be infected by the virus and begin their transition From the 30% infected only 20% will become zombies thus 10% will become corpses 25% of corpses will be turned into zombies each week 5% of zombies will be killed

Above a table has been formed in order to create the transition matrix below. There are four different states that are seen in this table, Alive, Corpse, Zombie, and Transitioning. The table has been completed based on the information given above.

[0.47 0.03 0.3 0.20 0.75 0 0.2500

0.10.05

0.1 0.80 0.95 ] This can be denoted by S0 and is the transition matrix.

The initial population for each of the states: The initial population for the humans is 0.4 The initial population for the corpses are 0.52 The zombie initial population is 0.05; and The initial population for those in transition is 0.03

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A C T Z

A 0.47 0.03 0.3 0.2

C 0 0.75 0 0.25

T 0 0.1 0.1 0.8

Z 0 0.05 0 0.95

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Like in part 1 and 2 the population for each species can be found at different times by multiplying the initial population with the transition matrix above raised to the power of how ever many weeks are being found.

So after 1 week the population of each of the 4 specie states can be found by:

[ A C Z T0.4 0.52 0.05 0.03] x [0.47 0.03 0.3 0.2

0 0.75 0 0.2500

0.10.05

0.1 0.80 0.95 ]

S1 = S0 x T1 and therefore after 1 week the population of each species is:¿ [0.188 0.4085 0.125 0.278 ]

Therefore, after 1 week the new population of each species are: Human population has dropped to 18.8% Corpse population has dropped to 40.9% Zombie population has increased to 12.5%; and, The transitioning population has increased to 27.8%

The population was found after 10 and 50 weeks by using the same method as above:S10 = S0 x T10 ¿ [2.1 x10−4 0.178 1.7 x10−4 0.822 ]S50 = S0 x T50 ¿ [1.6 x10−17 0.166 1.3 x 10−17 0.833 ]

After this initial week the human population decided it was time to take charge and came up with 3 ways to fight back against the zombies.

The first plan was that scientists created an anti-dote. This was injected into 30% of the zombie population per week. This antidote caused the zombies to become human again, increasing the human population and thus decreasing the population of the zombies.

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A C T Z

A 0.47 0.03 0.3 0.2

C 0 0.75 0 0.25

T 0 0.1 0.1 0.8

Z 0.30 0.05 0 0.65

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[0.47 0.03 0.3 0.20 0.75 0 0.2500.30

0.10.05

0.1 0.80 0.65 ]

Above, a new transition matrix has been formed, this time the data of the zombies turning back into humans has been entered.

After 1 week of this new antidote the population of each species looks like:

[ A C Z T0.4 0.52 0.05 0.03] x [0.47 0.03 0.3 0.2

0 0.75 0 0.2500.30

0.10.05

0.1 0.80 0.65 ]

¿ [0.197 0.4085 0.125 0.26 ]

The population after one week of each species: 19.7% of the population where humans 40.8% where corpses 12.5% where zombies; and 26% where transitioning

The population was also found for each species after 10 and 50 weeks:S10 = S0 x T10 ¿ [0.265 0.173 0.088 0.478 ]S50 = S0 x T50 ¿ [0.269 0.163 0.089 0.476 ]

The calculations above show that this antidote although originally increases the population of humans and zombies, it decreases the population of corpses and those transitioning. After the increase at 10 weeks the results show that the population of each specie remains steady and therefore this may not be the best way to fight against the zombies. It would however, be a good way to keep the human population constant once its at a comfortable level.

The second plan to fight back against the zombies was by a flu shot for the zombie virus, preventing anyone who has had this flu shot to catch the virus. This flu shot was given out to humans around the world and caused the the transitioning population to drop from 30% to 10% and only 5% of this 10% would actually turn into a zombie.

After 1 week of the rotation of this new flu shot the population of each species looked like:

[ A C Z T0.4 0.52 0.05 0.03] x [0.82 0.03 0.1 0.05

0 0.75 0 0.2500

0.050.05

0.1 0.850 0.95 ]

¿ [0.328 0.406 0.045 0.221 ]

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Therefore, after 1 week of this flu shot being in rotation the species population looks like: 32.8% of the population are humans 40.6% are corpses 4.5% are zombies; and, 22.1% are transitioning

The same method was used to find the population of each species 10 and 50 weeks after this flu shot was introduced. The results are below:S10 = S0 x T10 ¿ [0.054 0.169 0.076 0.768 ]S50 = S0 x T50 ¿ [1.96 x10−5 0.166 2.725 x10−6 0.833 ]

These calculations show that this defence may also not be as affective as the scientists originally would have hoped. It has not done the predicted job of keeping the zombie population to minimum, possibly even getting rid of it altogether.

The third and final solution to fixing the zombie problem was an increase in the human army fighting the zombie population. Each week the army killed 85% of the zombie population. A new transition matrix is made and the population after one week of the humans killing 85 % of the zombies is found:

[ A C Z T0.4 0.52 0.05 0.03]x[0.82 0.03 0.1 0.05

0 0.75 0 0.2500

0.050.85

0.1 0.850 0.15 ]

¿ [0.328 0.430 0.049 0.221 ]

Therefore, after 1 week of the army increase the population of each species is as follows: The human population is 32.8% The corpse population is 43% The zombie population is 4.9%; and The population of those transitioning is 22.1%

The same method was used to find the population of each species 10 and 50 weeks after the increase in the army. The results are below:S10 = S0 x T10 ¿ [0.054 7.832 0.688 5.83 ]S50 = S0 x T50 ¿ [1.96 x10−5 3650000 316600 2710000 ]

Like the first two solutions the third and finally solution did not prove to be useful. Therefore, none of the suggested improvements would help against the zombie population with the data given. Each method may be of use if other methods where used at the same time, but if just one of these methods where used on their own the zombie population would continue increasing and thus the human population would slowly be diminished.

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