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Problem III Technical Paper – BMED 1300 – December 6, 2013 – Langley/Tucker D1 Cobras

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Abstract — For a zombie outbreak, as is true for most epidemics as well as generalized natural or man-made disasters, preventative measure, efficiency, and speed of response time are key factors in minimizing cost and loss of property or life. After running a slightly expanded SIR model accounting for ability of susceptible individuals to either escape or die instead of being only removed, decreased success rate for infected individuals to propagate the disease quickly enough to overcome their innate death rate from susceptible-infected interaction remains vital in preventing a disease outbreak from swelling to epidemic proportions. Costs of implementation can range anywhere from $7 million for erecting a barrier for a 50% estimated survival rate to nearly 100% if $1.5 billion is available for providing ammunition, training, and protection during an outbreak.

INTRODUCTION

According to the National Strategic Plan for Public Health Preparedness and Response, a recent publication by the CDC in 2011, preparation for any kind of natural disaster such as flooding stands to offer vast improvements in many other non-related disasters including wildfire, terrorism, or pandemic diseases. In preparing for a potential epidemic, we chose to model an outbreak of zombie-ism for its ease of capturing public attention and to identify general trends that can be related to other natural disasters. Mathematical modeling has been employed to predict the temporal transmission properties of many diseases, including influenza, Ebola, and malaria, often with highly accurate results (Praditsitthikorn 2013). Once correctly fitted with available data, they also have the potential to be powerful prognostic tools in predicting the effectiveness of future intervention strategies (Simonsen 2013). A mathematical model also offers additional advantages in terms of reproducibility and scale, as many potential diseases yet un-encountered in the real world would be prohibitively expensive and time-consuming, not to mention potentially unethical or even illegal, to create a

controlled test environment with real subjects. The potential epidemiological impacts of many such interventions in the face of disease outbreak were studied, and the results as well as a recommendation of the most effective approaches in response to a potential epidemic are discussed below.

I. METHODS

A. Modeling an Outbreak

Because no documented zombie outbreaks currently exist, a different source of data was needed to provide information on initial parameters for modeling. Since the base virus cited to cause zombie-ism was Ebola based, studies involving Ebola infection and transmissions were deemed to be useful as a baseline (Zaman 2009). Strategies for intervention involving a zombie outbreak centered at Emory University in Atlanta, Georgia were studied using a susceptible-infected-removed (SIR) model, with assumptions of a homogenously mixed population due to the short time span of the disease. Further assumptions were made involving the characteristics of the disease from the film 28 Days Later, specifically grouping the entire population of Atlanta into a single group of equal risk of infection regardless of age, that the virus spread only through contact of body fluids and thus was limited to ground-based contact with infected individuals, that infected individuals moved slowly enough to be contained within Georgia, and that zombies could not swim in deep running water nor scale barbed wire fences at least 5 feet high (Retrospect 2002). Other assumptions for infected individuals included an instant incubation period that indicated only infected individuals could spread the disease, complete erosion of higher thought in favor of blind aggression, the existence of a natural carrier with immunity to all symptoms of the disease save for a minor infection rate and easily identifiable red mark in the left eye, and would still be in full control of themselves and work actively with susceptible individuals to escape. Finally, infected individuals were assumed to both ignore and be completely ignored by all non-primates, and were not superhuman in strength and stamina and were still vulnerable to death from blunt force trauma, with no possibility of reanimation or returning to life.

Mathematical Modeling of a Zombie Virus Epidemic

K. Bai, R. Bonagura, A. Cavallaro, K. Fierro, P. Grob, M. Lewis, E. Lobben, A. Nicaretta, E. Peek


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For the basic SIR model (Figure I), the removed class was changed to two separate groups, dead (D) and escaped (E) in order to differentiate desired state of removal for susceptible versus infected individuals; ideally, all susceptible individuals during the initial outbreak would ultimately be able to escape, while all infected that formed due to the virus would die, either from intervention or natural causes. In addition to the susceptible class, a carrier class (C) was also added that was a different result from being infected by the disease; this group was still able to keep their mental faculties intact, and would be indistinguishable from a susceptible person aside from a small infection rate and a characteristic red dash in the left eye for easy identification.

Setting the moment the first infected patient escapes from Emory, it was assumed that susceptible individuals would try to get away from infected zombies (parameter e), while zombies would try to infect as many people as they could, generating more infected (parameter a), or carriers (parameter g) if the susceptible individual was immune. During this time, the susceptible population would still experience a natural, if drastically altered from normal, birth (b) and death (c) rate. Similarly, zombies would also incur a small natural death rate from accidental injuries (d). Carriers would work with survivors to escape (e), but also have a small chance of infecting susceptible individuals (h). Survivors would also resist infection by killing off zombies as they escaped (l), and active intervention through armed individuals or army interactions would also kill off many infected (f), moving them into the dead group. The simulation would be run for as long as necessary for the initial population of 10 million susceptible people (Census 2013) and 1 infected to have all been moved into either the dead or escaped categories. Finally, a density-dependent parameter (k) was added to limit the number of interactions any two groups could have with each other in 1 day to avoid exponential crashes from assuming that 1 zombie could attack 10 million susceptible people on the first day of the outbreak. Full equations for each population are included in Appendix I.

Data for many parameters was not available due to lack of zombies existing in published literature. The infection variable (a) was estimated using the virulence of malaria, which has a 20-50% chance per bite of infected mosquito of passing the malaria causing protist on (Ahmed 2013). Malaria was chosen over Ebola because an infected patient with Ebola is typically bed-bound during the course of the infection, continually coughing up blood and drained of strength (Qiu 2013); the disease also has a staggering mortality rate of 90% (Nakayama 2013) and is slow to spread because it incapacitates infected individuals (Shi 2013), making it a poor model for zombie-ism. Malaria, in contrast, is a rapidly spreading blood-based

disease that relies on a mosquito vector, which is much less crippling on its victims and has a single-digit mortality rate in developed countries (Lusti-Narasimhan 2013). Most importantly, a carrier state for malaria also exists, having a much lower infection rate and displaying no adverse effects of the disease (Krause 2013). The estimate for parameter a was further refined using predator hunting success rates in Yellowstone national park, whose native wolf packs' successful kill rate averaged 30% over the course of 4 years (Vucitech 2011). We believed that an infected individual, working in small packs of highly aggressive individuals and relying primarily on their teeth and speed, would have a similar efficiency to packs of hunting dogs if left alone with unarmed civilians. Thus, the infection rate based on zombie-human interactions was set at 0.33, or 33 new infected individuals formed for every 100 encounters between a susceptible and a zombie.

A zombie outbreak would be a calamitous event, which would impact birth and death rates to values drastically different from their baseline values in the United States. We decided that increased levels of stress as well as general chaos would push conditions closer to a second or third world country during the short span of time as the infection spread. Thus, the birth rate (b) was set to the birth rate of rural Afghanistan, 20 births per thousand individuals per day (CIA 2013). Afghanistan was chosen because it was a country with very heavy U.S. military involvement and limited GDP and educational opportunities available for its citizens. Similarly, the death rate (c) was also taken from this location, 21.5 deaths per 1000 individuals per day (CIA 2013), which is one of the highest in the world due to lack of some modern medical facilities and basic utilities like water or electricity, which would also likely not be in full supply during an epidemic. Innate death rates for infected individuals was estimated based on the assumptions that animals ignored them, but they would still be vulnerable to the same accidental deaths that occurred for normal people, namely lightning, machinery-related injuries, and starvation. These rates (d) coincided with the annual death rate in the United States from accidental deaths from injury, approximately 39.1 per 100,000 people per year (CDC 2013).

Parameters for successful escape from an area rely on being able to successfully set up a quarantine boundary that separates the infected from the susceptible populations, and thus (e) was initially set to 0 in the baseline model to indicate no escape, which was varied in later simulations. Similarly, the (f) parameter that represented active intervention in killing zombies through force was also set to 0 and later varied in the base model to set up an estimate of time-span and damage a no-intervention policy would result in.


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Next, the rate at which interaction with an infected individual would become a carrier instead of a full infected due to natural immunity was modeled with (g); because there was no published estimate of the carrier rate, an estimate was derived using the movie setting of 28 Days Later, in which only three carriers were found in the entire population of Britain, or 3 carriers per 10 million people. While carriers were easily identified with a red dot in the left eye and actively helped other susceptible people escape, blood or saliva based contact from sharing food still posed a small risk of infection due to viruses present in bodily fluids. Hepatitis B, a similar blood-based viral disease, also has carriers with minor symptoms who are capable of infecting others throughout their lives, and thus (h), the infection parameter due to carrier interaction, was matched with published data concerning HBV and set to 12 infections per 100 interactions (WHO 2013).

Finally, to model interactions between the three groups of susceptible, infected, and carriers, a parameter (l) indicating the chance of successfully fighting off a zombie by killing it when it attacks was chosen to match the average survival rate in the United States from encountering a bear in the wilderness, as there had only been two wolf attacks in the past ten years (Wolf 2013). This value was set as 14 successes out of 100 encounters (Geetha 2012). Interactions between members of population were limited by time, and the final parameter (k) was established using information from the CDC's zombie outbreak model that limited members of all populations to interacting with a maximum of 21 other individuals per day, slightly less than 1 attack or encounter per hour in order to prevent a massive exponential crash in populations from assuming that 1 zombie would be able to attack all 10 million susceptible people on the first day that he or she was freed. All of the above variables are summarized in table I.

After the baseline model with no military intervention or escape was established, three more scenarios were explored: a quarantine model set around the Georgia area that would allow for escape over a wall followed by military action, an aggressive preparation model that would arm civilians and drill them in how to better fight off infected before escaping, and an escape model involving decreasing the successful infection rate of infected via an early system of warnings and evacuation drills.

The quarantine model assumed that zombies were unable to swim in running water deeper than five feet, ran at a maximum of 12 miles per hour similar to an average person (Rompottie 1972), and would be hindered by barbed wire fence at least five feet high. A proposed containment area was planned using a map of waterways in Georgia, with a

perimeter of 233 x 300 miles across (Georgia Ecoregion Descriptions 2007). Estimates for cost included barbed wire costing $1.23 per foot (Gerrish 2012) for a total of $6.9 million; this wall would take half a day to set up (t = 0.5) (Schulte 2011), and once established, the escape rate outlined in the Georgia Hurricane Evacuation Plan, a similar disaster evacuation scenario, indicated that the escape rate out of the wall would be (e = 0.2). After being alerted to the presence of an outbreak, it was estimated that the army would take 5 days to fully mobilize (Keegan 1999) and come in to actively eliminate infected; their intervention was predicted to have a 65% success rate when avoiding collateral damage (t = 5, f = 0.65, (Bradley 2003)).

The second model that was run was an armed civilian scenario. In this model, the escape routes are still present meaning the value of ‘e’ is non-zero. Evacuation drills are scheduled year-round meaning (e) is increased from the quarantine model, from 0.20 to 0.35 following data from Japan’s earthquake and typhoon evacuation plans and past data (Bouville 2013). More importantly, the focus of this plan is providing arms and ammunition for the adult population of Georgia. Any healthy individual at or above the legal age to bear arms would be provided one by the government, averaging around 200 dollars for a 12-gauge shotgun and ammunition per individual (Roger 1973). With an adult population in Georgia of about 7.5 million, this would cost the government approximately $1.498 billion dollars to implement. The ‘l’ value in this scenario was set at 0.5 assuming that an armed individual would be able to fight off a zombie about half the time, modeled after average kill rate for armed individuals during annual deer hunts (U.S. Fish and Wildlife Service, 2012).

The final model, equipping civilians to better escape, relied on preparation drills and preventative action once the initial outbreak started in order to prevent enough infected from forming to overcome their natural die-off rate. In a study done on wild wolves in Yellowstone National Park, it was suggested that changes in annual prey migration patterns combined with unusually thin snow during the year greatly decreased the wolf pack's successful kill rate (a) from 33% to 15% (Metz 2011). We would seek to emulate this effect through multiple publicized warnings on many radio and television stations, averaging $500 for a 30-second commercial, once an hour every day for 2 months after testing on the virus begins gives a cost of $720 thousand in addition to construction of the wall. The commercials would emphasize preparedness to evacuate, with emphasis on getting inside an automobile which is much more difficult for an infected to break into.

For all sets of parameters, simulations were run using PLAS


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with a time interval of 30 days, updating once every 0.1 days. The initial population of susceptible people was set to 10 million, the population of Georgia (CIA 2013), initial infected was set to 1 individual 'patient zero,' and initial escaped, dead, and carriers were all set to 0.

II. RESULTS

The default no intervention and no escape model (graph I) predicted a very quick population crash of susceptible people after 5 days. The initial infected outbreak grew exponentially starting from day 1, while the susceptible population also experienced a mild growth rate from background birth rate. However, by day 4, when the infected population became 30% of the susceptible population, it took only 1 more day for the infected to completely eliminate remaining susceptible individuals, driving the S population to 0 while the infected population peaked at 550,000. Remaining infected then slowly died off over the course of 3 weeks as they had no more new susceptible people to infect, and eventually all 10 million initial individuals ended up in the dead population. The carrier group remained extremely small (< 1000 individuals at all times) and did not make any major contribution to infections or infected death rates. The escaped population does not exist. This model was used as our baseline.

The quarantine model (graph II) with an estimated cost of $7 million, set the (e) parameter to 0.20 after half a day or 0.5 time units, and had a shallower susceptible crash over the course of 4 days. With the ability to escape safely, approximately 500,000 initial susceptible people and carriers, or half the population, was able to make it out of the area, while the maximum number of infected peak at only 250,000. Our modeled military intervention did not make a huge difference; after the population crash at 4 days, military intervention by day 5 only made the remaining number of infected individuals die off faster.

The aggressive intervention (graph III) with an estimated cost of $1.5 billion completely avoided the population crash of the previous 2 scenarios. Although the carrier population once again did not end up contributing very much, the higher escape and death of zombie from encounter with armed civilians rate was projected to completely wipe out the initial infection faster than it could reproduce itself, and thus the maximum number of infected, 76 individuals, were unable to reach a critical mass to overcome their die-out rate of combined parameters (c), (l), and (f). Over 90% of the population was able to evacuate in approximately 9 days, with every initial susceptible escaping within 2 weeks due to lack of risk from the infected group.

The lowered susceptibility scenario (graph IV) with an estimated cost of $7.7 million appeared to be a combination of the previous two scenarios. Once again, the lowered ability of the infected group to propagate itself, this time due to a lowered (a) value, prevented it from reaching the critical mass of 30% required to overcome the susceptible and carrier population’s ability to ward them off. However, because this time susceptible people were not better equipped to fight them off, the zombie population was able to avoid extinction, hovering around 100 individuals. Although it was insufficient to overcome the background birth rate, the deceased population continued to slowly increase as a few susceptible people were turned into infected and then promptly killed as the others escaped. Full evacuation took 18 days, by the end of which the dead population hovered around 2,300 individuals.

III. DISCUSSION

We investigated the results of an epidemic outbreak based on a highly infectious blood-based pathogen and the costs, time span, and overall effectiveness of different forms of intervention based on their potential to reduce the death toll due to infection. To address this question, we developed a modified SIR model with carrier and escaped groups to predict the various outcomes of different strategies.

For all scenarios, it appeared that the initial week after the infected was first released among the susceptible population was of critical importance; during this period, the infected population must successfully attack and infect enough susceptible people in order to overcome their own natural die-off rate, as well as later military action that would increase their death rate. If the infected population was ever able to reach at least 30% of the susceptible population size, a population crash would occur over the next day as the infected would have enough numbers to kill or infect all remaining members of the susceptible population before they could escape. However, if the infected were never able to reach this amount of individuals, the background death rate as well as deaths from attempted attacks on escaping susceptible people and carriers would cause their own population to quickly die off. Thus, it appeared that no equilibrium population dynamic between zombies and humans exists; either the susceptible people will fight off the initial tide of infected so that they cannot seriously hinder their escape and later die off from hunger or military action, or the infected will quickly explode out of control and convert all susceptible individuals into their own group before slowly dying off due to lack of new individuals to infect.

The background birth and death rates for susceptible people, as well as background death rates due to accidents from infected individuals had little to no impact on the projected


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changes in population. At the best during the no intervention scenarios or total kill scenarios of infected, the birth and death rates were only visible after one population had been completely wiped out, and served to increase or decrease the population by around 5% over the course of 2 weeks until the simulation ended. Modeled military intervention, while efficient at quickly exterminating infected, was not able to respond quickly enough to make a different and halt the initial population crash if it existed, or to kill enough infected individuals to prevent a crash in the first place. The carrier population and its own latent infection rate had such a small chance of forming that the carrier group was never a factor in any of the 4 described scenarios.

The most important parameters that dictated the outcome of an infection were (a) the infection rate, (e) the ability to escape to a safe area, and (l), the chance of a group of survivors being able to successfully fight off a group of zombies. Small decreases in (a) from the default 33% kill rate of zombies had the effect of ‘lengthening’ the time window that it took for zombies to approach critical mass, and during that time they would still be slowly dying off. This was most evident in comparing the default model to the lowered susceptibility scenario, in which the period before the susceptible population hit 0 was extended from 5 days to around 18 days. Decreased infection rate also greatly increased the number of individuals able to escape as there were fewer infected interfering. Higher escape (e) values had the effect of shortening the amount of time it took for the population to hit 0; with a higher escape rate, the susceptible population quickly emptied out to 0 as carriers and susceptible people either escaped outside of Georgia or succumbed to infection. However, because the pool of potential infected individuals shrank so much more quickly due to 2 separate ways of leaving the susceptible group, the total number of infected was much lower compared to the default model, and a significantly higher proportion of susceptible people were able to escape, around 80%. The (l) parameter from the last scenario indicated that doubling the chances of survivors being able to kill an infected that attacked them instead of becoming infected themselves was enough by itself to avoid a zombie epidemic by making it too ‘risky’ for a zombie to attack a person. Since zombies were only 33% successful to being with, an increase in kill rate from 14% to 30% meant that the initial zombie was only able to infect 3 people before he himself was killed. This strategy attacked the infected group right at the start, and prevented them from ever ballooning out of control. In fact, decreasing (a) or increasing (l) was efficient enough that evacuation does not even seem to be necessary as the infected population crashes to 0, eliminating the threat.

With these data in mind, our recommended approach to a localized epidemic infection such as zombie-ism would be to have some kind of containment border around Georgia that is at least half a day’s running distance outside of a potential disaster ground zero area. With a base cost of around $7 million in construction costs, such a border could be quickly established in a manner similar to the Berlin Wall once an alarm was raised, and combined with natural barriers such as rivers and mountains, would offer an easily enforceable barrier that could be strengthened as needed with addition supplies.

Assuming that it was possible to tell susceptible people from infected apart, this action alone would provide a 50% survival rate from the disease as long as people began fleeing when the outbreak began. Since it is much more expensive to arm the entire population, and proactive preventions are already established with the CDC and Emory’s biohazard level 4 protocols, lowering infection rate seems difficult to predict depending on strain of virus, as well as infeasible to cover all existing experiments. Therefore, regular preparation for evacuation such as informing citizens of optimized escape routes (increasing e), as well as a small emergency source of either firearms or protective equipment and supplies during a time of crisis (increasing l), would be the most cost-efficient approach to maximizing survival rate and minimizing costs. We recommend enough public dispensation of information through radio, T.V., or social media through a $700,000 budget, enough to show once on every public TV station for a year hourly, to increase the escape rate enough through preparation to 25% that the initial infection does not gain enough momentum to cause a population crash. This approach of prevention over reaction would also be likely to be very useful in other natural disaster scenarios such as evacuating in the face of a flood or hurricane. These results are summarized in table II and graph V below. It should be noted that if finances are of no concern, a $1.5 billion investment in arming and educating the general populace is enough to ensure a 100% survival rate for this type of disaster, but real expenses are likely to be a key deciding factor in how prepared one chooses to be.

Overall, this study has many limitations in assuming homogenous population density, extrapolations of disease infection and escape rates, and somewhat unrealistic assumptions that available food, power, medicine, or clean water would remain unchanged during this specific type of crisis, and that all populations of individuals are equally vulnerable. Further studies could be targeted in an area larger than Atlanta or Georgia, and incorporate many other risk factors like disease, malnutrition, and unevenly distributed population.


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IV. CONCLUSION

For a zombie outbreak, as is true for most epidemics as well as generalized natural or man-made disasters, preventative measure, efficiency, and speed of response time are key factors in minimizing cost and loss of property or life. After running a slightly expanded SIR model accounting for ability of susceptible individuals to either escape or die instead of being only removed, decreased success rate for infected individuals to propagate the disease quickly enough to overcome their innate death rate from susceptible-infected interaction remains vital in preventing a disease outbreak from swelling to epidemic proportions. Costs of implementation can range anywhere from $7 million for erecting a barrier for a 50% estimated survival rate to nearly 100% if $1.5 billion is available for providing ammunition, training, and protection during an outbreak.

APPENDIX I. Differential Equations Used for Modeling

Figure I

Fig.1 Original SIR Model

Figure II

Fig. II Augmented SCIDE Model

Table I

*Modified in later scenarios Table I Parameter Values for Default Model Graph I

Graph I Default Model Survival Rate Graph II

Graph II Quarantine Scenario Survival Rate

Parameter Meaning Value

a Zombie infection rate 0.33*

b Susceptible birth rate 0.081

c Susceptible/carrier death rate 0.00215

d Zombie death rate (natural) 0.0000391

e Escape rate 0*

f Zombie death rate (military) 0*

g Carrier from infection rate 0.0000000316

h Carrier infection rate 0.12

l Survivor fighting off zombie rate 0.144*


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Graph III

Graph III Aggressive Intervention Scenario Survival Rate

Graph IV

Graph IV Lowered Susceptibility Scenario Survival Rate Graph V

Graph V Recommended Intervention Survival Rate

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