When Humans Strike Back! Adaptive Strategies for Zombie Attacks Bard & Kyle Ermentrout November 4, 2010 Abstract We describe a novel model for interactions between zombies and humans such that (i) humans can have an advantage over zombies when the population density of the zombies is low; (ii) however, with greater numbers of zombies, the advantage is lost. This endows the system with bistability and hysteresis. We then extend the model to allow for the humans to become complacent when zombie attacks are infrequent. This provides a mechanism for periodic zombie outbreaks.

1

Introduction

In a recent article (Munz et al, 2009) suggested a number of models for zombie attacks on human populations, likening these to disease outbreaks. In addition to the trends of a standard disease outbreak, the zombie apocalypse may contain some more interesting dynamics such as multistability and oscillations. To see this, we allow for some reaction of humans to their zombie enemies. Rather than playing a defensive role against disease, humans are capable of becoming the aggressor against the undead. The zombie makes for a relatively vulnerable victim: it seems to lose any agility and speed that it may have had while living and it is mindless, meaning that it is incapable of any strategic action. The average person should easily be able to dispatch a lone walking corpse, provided they are equipped with the proper knowledge and minimal armament. The true danger of the zombie becomes apparent when he is in the company of his peers. Through involuntary vocalizations (that despair-inducing moan), the zombie unwittingly attracts other hungry 1

fellows when he becomes aware of his next potential meal (Brooks, 2003, p.16). Acting without fear, zombies will unhesitatingly swarm any source of warm flesh; if a human is not careful, he can easily become surrounded and overwhelmed. In this chapter we consider a number of mechanisms through which humans strike back at the zombie invaders over the long run. We introduce a new model for zombie-human interactions which allows for multiple stable end states where either zombies or humans are dominant. We then introduce a model in which humans can slowly adapt to their zombie attackers and find oscillations and other dynamics.

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Multiple zombies.

To motivate this first simple model, we consider the scene at the house early in Night of the Living Dead (Romero, 1968). Ben and others are able to fend off attacks by single zombies and successfully kill them (using the basic principle “Kill the brain and you kill the ghoul.”) This is because the zombies of this particular strain are slow and rather uncoordinated. Thus on one-onone encounters with alert and prepared susceptibles, the single zombie will generally be destroyed. What allows success in zombies is the sheer number of living dead. Groups of two or more zombies can successfully attack and defeat all but the most well-armed human. For example, a human with a chain gun is likely to survive a concerted attack by many zombies. However, a single person with a screwdriver (such as in the Dawn of the Dead, Romero, 1978), would probably be killed in an attack by multiple zombies. With these empirical observations, we consider our first simple model. Let Z(t) denote the population of zombies and S(t), the susceptible population. We describe interactions via the laws of mass action: ⋆ S S+Z S + 2Z Z

a

1 −→ a2 −→ a3 −→ a4 −→ a5 −→

S ⋆ S 3Z ⋆

(1) (2) (3) (4) (5)

Equation (1) represents the migration of new humans into the area of zombie infestation and equation (2) is the natural death rate of humans leaving the infested area. Equation (3) depicts the ability of the susceptible population 2

to pick off zombies when they occur individually. Equation (4) represents the idea that packs of zombies (here represented by two for simplicity) are able to kill susceptibles. To keep this initial model simple, we have assumed that the killed humans are instantly transformed into zombies. Equation (5) represents the death of zombies by natural or other means (such as the accidental beheading by the airplane in an early scene in Dawn of the Dead, Romero, 1978). Let (s, z) denote the population of zombies susceptible humans and zombies respectively. Then ds = a1 s0 − a2 s − a4 sz 2 dt dz = −a3 sz + a4 sz 2 − a5 z. dt Here s0 is the number of people outside of infested areas who are able to enter the region where the zombie outbreak has occurred. One of the first steps in modeling is to reduce the number of parameters and to make the equations dimensionless (see Edelstein-Keshet, 2005, Chapter 4). If we divide the first ˆ and equation by a2 , and let tˆ = a2 t be a dimensionless time, let s = SS ˆ ˆ z = SZ where S = a1 s0 /a2 , then we can eliminate several parameters from the model and obtain: dS = 1−S −a ˆ4 SZ 2 dtˆ dZ = −ˆ a3 SZ − a ˆ5 Z + a ˆ4 SZ 2 . ˆ dt

(6) (7)

There are now just three dimensionless parameters, a ˆ3 = a3 a1 s0 /a22 , a ˆ4 = 2 2 3 a4 s0 a1 /a2 , and aˆ5 = a5 /a2 . For notational simplicity, we will drop the hats for the remainder of the paper. The reader should keep in mind that the three rate constants, a3,4,5 can all be expressed in terms of the rate of migration of fresh human meat and the steady state levels of humans in absence of attacks. For example, S = 0.25, Z = 0.75 means that the humans are reduced to 25% of their population before the attack and zombies represent 75% of the preattack population of humans. There can be up to three equilibrium points to this model and we will see that two of these can be stable. Recall that an equilibrium point satisfies, dS/dt = dZ/dt = 0 (Strogatz, 1994). The point (S, Z) = (1, 0) is always an equilbrium and, it is always asymptotically stable. This means that for low 3

density zombie attacks, humans always survive. Add equations (6) and (7) together to get 1 − S − a3 SZ − a5 Z = 0 and solve this to get S=

1 − a5 Z . 1 + a3 Z

Substituting this expression into equation (7), we get Z(a3 + a5 − a4 Z + a4 a5 Z 2 ) = 0. As expected, we find Z = 0, but, additionally, two other roots: Z=

a4 ±

q

a24 − 4a4 a5 (a3 + a5 ) . 2a4 a5

(8)

If the term inside the radical is positive, we have two positive roots and thus there are three equilibria. If the natural death rate of the zombies is small and the rate at which humans successfully kill zombies is small, then the zombies will be able to mount an attack that overwhelms the humans and wipe out a substantial portion of them. We still must show that the new equilibrium is stable. To determine stability, we linearize the right-hand sides of equations (6,7) about the equilibrium point. The eigenvalues of the resulting matrix (called the Jacobian) tell us whether or not the equilibrium is stable. If the eigenvalues have negative real parts then we have stability and if there is at least one eigenvalue with positive real parts, then the equilibrium is unstable (Strogatz, 1994). For a two-dimensional system, as we have here, stability is assured if the trace of the Jacobian matrix is negative and the determinant is positive (Edelstein-Keshet, 2005, Chapter 5). The Jacobian matrix for our model is J :=

−1 − a4 z 2 −2a4 zs −a3 z + a4 z 2 −a5 − a3 s + 2a4 zs

!

.

For Z = 0, it is clear that the trace is negative and the determinant positive. The nontrivial roots are a bit tougher to study. We note that for a5 small, that the middle root is approximately a3 /a4 and the large Z root is roughly 1/a5 . Plugging these approximations in, we find that the trace is negative for both roots and that the determinant is positive for the large root and 4

negative for the middle root. Thus, as suspected, the large root is stable and the middle root is unstable. It is convenient to sketch the phaseplane for this system in order to understand the qualitative dynamics. Figure 1A shows the (S, Z) phaseplane along with the nullclines dS/dt = 0 and dZ/dt = 0. Intersections show the three equilibrium points: two stable (circles) and one unstable (square). The unstable equilibrium is a saddle point and thus has a so-called stable manifold (labeled SM) (Strogatz, 1994). This means there is a pair of trajectories which go into the saddle point as t → ∞. They form a separatrix between the two stable equilibria: all initial conditions above the curve go to a persistent high zombie state while all those below go to a zero zombie state. To see this, we also show two trajectories starting at S = 1 and Z above (ii) and below (i) the stable manifold. Thus, if Z(0) is less than about 1.75, then the zombie population will collapse (i) while if Z(0) is larger than 1.75, the zombie population explodes and humans are nearly wiped out (ii). The key parameter that determines the threshold is a3 . This is the ability of a human to beat a zombie in a one-on-one interaction. If the humans are totally unprepared, then this term could be, in fact, negative and we would have to adjust the equations to reflect the fact that the susceptible human involved in the battle was lost to the population. Thus, to incorporate humans completely unready to do battle with zombies, we add the term +min(a3 SZ, 0) to the dS/dt equation. If a3 < 0, then we subtract from the population. We remark that when a4 = 0 and a3 < 0, we recover a simplified version of the original Munz et al model. Let us add one more small alteration to our present model. We allow for the possibility of zombie migration. That is, zombies from somewhere else are allowed to enter our local population. To this end, we add a small source term to the zombie equation and obtain the general two-dimensional model that includes group attacks by zombies: dS = 1 − S − a4 SZ 2 + min(a3 SZ, 0) dt dZ = −a3 SZ − a5 (Z − Z0 ) + a4 SZ 2 . dt

(9) (10)

It is no longer so simple to find the equilibria, so instead, we compute them numerically as the parameter a3 varies. Figure 1B shows the equilibrium 5

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Figure 1: Analysis of bistable zombie model. A: Phaseplane of simple zombie model. Z−nullclines (dZ/dt = 0) and S−nullclines (dS/dt = 0) are sjown. Lines (SM) curves are the stable manifolds of the saddle point (black square). Stable equilibria are shown with filled circles.(a3 = a4 = 0.25, a5 = 0.1.) B: Bifurcation diagram as a3 varies showing the equilbrium value of Z. A small amount of migration (Z0 = 0.5) is allowed, a4 = 0.25, a5 = 0.1. (Solid lines are stable and dashed are unstable. This and all other diagrams are computed using XPPAUT (Ermentrout, 2002).

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1

zombie population as the parameter a3 changes. Because of the small migration term, there is a positive number such that if a3 is smaller than this value, the lower (nearly zombie-free) state does not exist and zombies rule. Similarly, for large enough values of a3 , the zombies can never take hold and the population is largely zombie-free. (There will always be a few of them wandering around due to migration; I suspect you may have encountered them occasionally in your life.) In conclusion, in this section we have shown that preparedness for zombie attacks can be sufficient to maintain a low or zero population of zombies. However, if groups of them form, and there are enough, the zombies can overcome the defenses and rise to reign supreme.

3

Adaptive strategies for humans.

It would seem to take a good deal of effort to maintain a high preparedness for zombie attacks. Indeed, carrying around an ice pick or club whenever you want to go shopping is an inconvenience that most of us would rather avoid. Thus, we can suppose that if the zombie attacks remain infrequent, we might allow the parameter a3 to begin to fall, perhaps, even to fall below zero, where even an isolated zombie could attack and successfully kill such a citizen. For example, in one of the early scenes in Night of the Living Dead, Johnny is easily killed by a lone zombie due to his naive attitude toward the threat. One can then imagine that eventually the attacks would be very common as the isolated attacks increase the population of zombies and this is then amplified by the group attacks. As more and more zombies are created, the populace might be compelled to step up their readiness and thus increase the parameter a3 . Hence, in this section, we study the effects of adapting the readiness parameter, a3 to the zombie population. We amend equations (9) and (10) by now allowing a3 , the “readiness” parameter to evolve according to the zombie population: da3 = F (Z, a3). dt What would be a good choice for F ? We want a3 to increase if the zombie population is large and to decrease if it is small. How small or how large depends on ones tolerance to the presence of zombies. Obviously, some people find them so utterly abhorrant, that they will choose to set a minimum of 7

Z 12

Z 10

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*

4

c

2 0

# 0

b 0.2

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a3 a3

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Figure 2: Different levels of zombie tolerance. Parameters as in figure 1B. zero. But, as we noted earlier, such preparedness comes at a cost (at the least, inconvenience, and more seriously, in collateral damage due to accidents with anti-zombie devices). A simple linear equation would seem to suffice: τ

da3 = Z − Z¯ − ca3 . dt

(11)

The parameter Z¯ sets the level of zombies that you are willing to tolerate. The parameter c is just a decay of a3 and could optionally be set to zero. A nonzero value of c means that there is some natural decay of readiness to a neutral (a3 = 0) value. Finally, the parameter τ sets the time scale for the reaction to the zombies. If humans are slow to react to the increasing zombie attacks, then we should make τ large. On the other hand, if the humans are acutely aware of the zombies and react quickly, then τ should be small. Figure 2 repeats figure 1B and thus shows the steady state population of zombies as a function of a3 . Equilibria of equation (11) satisfy, Z = Z¯ + ca3 . This is just a straight line. If we plot this line along with the diagram in figure 1B, intersections tell us the equilibrium values. For example, if c = 0, ¯ then, the equilibrium values are found by drawing horizontal lines at Z = Z. Suppose you have a high tolerance of zombies such as line (a) in the figure. Then you can choose a3 quite low. Of course you and most of your neighbors will be nearly exterminated, but, hey, you didn’t spend any energy, so if you are one of the lucky ones to survive, you can leave your icepick at home. On 8

the other hand, if you choose to tolerate very few zombies, then you might want to set your tolerance to be very low, such as line (b) in the figure. In fact, if you want to avoid any zombie attacks at all, then set the line low enough so that a3 is larger than about 0.6 in this particular example. For then, there can never be a dominant zombie presence. Suppose that you hedge your bets and pick an intermediate tolerance of zombies, say, like line (c) in the figure. Then, the intersection is on the “unstable” part of the zombie equilibrium curve, and, in reasonable circumstances, you can expect to see periodic fluctuations in the zombie and human populations as well as in the readiness parameter, a3 . Figure 3 shows an example simulation of such an oscillation. The zombie population rises and wipes out a substantial fraction of the people. The remainder arm themselves to the teeth and cut down the zombies to a low level. They then become complacent allowing the zombie population to once more rise. We can understand this oscillation by looking again to figure 2. Suppose that τ is very large so that the people adapt really slowly. At a ¯ da3 /dt is positive and a3 starts to high zombie population, since Z > Z, increase (dashed arrow on the top of figure 2). This increase in weaponry causes the zombie population to slowly decrease until the point marked with (*) is reached. At this point, the zombie population crashes to the nearly ¯ the tolerance level, a3 begins to decrease zero level. Since Z is now below Z, (complaceny sets in, shown by the dashed line at the bottom). The zombie level rises slightly, but almost imperceptibly until the point (#) is reached where there is a sudden explosion in the zombie attacks and the zombie population rises to dominance once again. If we treat tolerance as a parameter, we can numerically determine how ¯ Figure 4A shows the behavior of the dynamics of the model change with Z. equations (9,10,11) as we change the tolerance from low to higher values (e.g., we move the dashed line (c) in figure 2). The equilbrium value of a3 is stable for low tolerance, but at a fairly low value, it loses stability (first arrow) as a Hopf bifurcation (Strogatz, 1994). That is, the equilibrium switches from having a damped oscillation to a growing oscillation. A periodic solution emerges as the only stable behavior (the curve labeled SPO) and a3 , Z, and S all oscillate. The oscillation grows in amplitude until it is abruptly lost ((*) in the figure) and there is a return to a stable equilibrium point. Thus, if an intermediate tolerance is chosen, then the zombie attacks wax and wane in a rhythmic manner. Interestingly, there is a region of Z¯ where there is both rhythmicity and stable equilbrium behavior. We will explore this shortly. 9

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Figure 3: Behavior of the adaptive strategy model. (A) Solution in phase space of a limit cycle behavior (B) Time series over several cycles. Parameters are a4 = 2, a5 = 0.25, Z0 = 1, Z¯ = 1.1, c = 0.2, τ = 2.

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Figure 4: Bifurcation diagrams for the adaptive zombie model. (A) Full three-dimensional model. Arrows denote Hopf bifurcations and (*) denotes the collision of unstable (UPO) with stable (SPO) periodic solutions. (B) Same as (A) with the reduced model. Parameters as in figure 3

10

1.8

2

Three-dimensional dynamics is more difficult to understand than twodimensions, so we might ask if there is a way to reduce our three variable system to a simpler one. For a moment, returning to figure 1A, the (Z, S)−model where a3 is fixed. Two trajectories are drawn (black arrows) and both of them appear to move horizontally until they hit the S−nullcline where they essentially follow it nearly perfectly to the equilibrium. This suggests that the dynamics of S are much faster than Z; a reasonable assumption, given that classic zombies are slow compared to humans. Thus, we could let S reach its equilibrium value found by setting equation (9) to zero: 1 S = Seq (Z) := . 2 1 + a4 Z − min(a3 Z, 0) If we make this substitution, then the three-dimensional model becomes a two-dimensional model: dZ = −a3 Seq (Z)Z + a4 Z 2 Seq (Z) − a5 Z dt da3 τ = Z¯ − Z − ca3 . dt

(12) (13)

Very little is lost in making this simplification. Indeed, figure 4B is almost identical both quantitatively and qualitatively to figure 4A. Figure 5A shows the phaseplane and nullclines for the same parameters as figure 3. Z, a3 −nullclines are shown as well as the limit cycle. Comparing the time series for figure 5B to that of figure 3B, shows little difference. As Z¯ changes, we can get the a3 −nullcline to intersect in different parts of the Z−nullcline and thus vary the qualitative dynamics. Intersections away from the middle part of the Z−nullcline will lead to stable equilibria and thus a stable adaptation to zombie attacks. We close our discussion of the adaptive model by returning to the region in figure 4 where there was both a stable equilibrium solution and a stable limit cycle solution (the values of Z¯ between the right arrow and the asterisk. Figure 6 shows the phaseplane behavior for the reduced (Z, a3 ) dynamics when Z¯ = 1.4. The closed circle (UPO) is an unstable periodic solution and divides the plane into two regions. Any initial values of a3 , Z starting inside the UPO are attracted to the stable equilibrium point at the intersection of the nullclines. On the other hand, initial values outside the UPO will go to the limit cycle where the zombie population oscillates. This suggests an interesting phenomenon. Suppose that we are at the stable equilibrium. A 11

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Figure 5: Reduced adaptive model where S = 1/(1 + a4 Z 2 − min(a3 Z, 0). (A) Phaseplane showing the Z− nullclines and the a3 −nullclines as well as the limit cycle. The single equilibrium point is unstable. (B) Time series for the cycle shown in (A). Parameters as in figure 3. brief influx on new zombies could push the initial conditions to the right, past the UPO and result in a massive decrease in zombies, subsequent complacency, and a rebound to an oscillation. Only a carefully timed culling of the zombies could take you back to the stable equilibrium. In conclusion, we see that having a zombie-dependent behavior on readiness to confront the undead hordes can lead to instabilities that result in waxing and waning of the total zombie population. This is reminiscent of many other disease and population cycles seen in nature (Hethcote and Levin, 1989).

4

Discussion

In this paper, we have introduced a new type of interaction between zombies and humans that is based on some empirical observations from classic films. Specifically, we hypothesize that low populations of zombies are easily overcome by humans that are suitably armed. These defenses can be overcome when zombies attack in groups. It is well known (Brooks, 2003) that zombies like to congregate, so this latter part of the hypothesis also seems 12

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Figure 6: Reduced adaptive model with Z¯ = 1.4, in the bistable region. Unstable limit cycle is labeled UPO. Parameters as in figure 3. well-founded. By introducing this novel interaction, we are able to endow zombie/human interactions with various interesting nonlinear behaviors. In particular, the existence of two types of interactions (zombies lose at low density and win at high density), we can find model systems where there can be two qualitatively distinct outcomes for exactly the same parameters: humans dominate or zombies dominate. This winner-take-all behavior is common in population models where there is competition. Here, there is not direct competition, rather, the prey can become the predators if they are well-armed. We also introduced adaptive strategy for humans. That is, we suggested that when zombie levels are low, then the alertness and readiness of humans became low. This results in a massive amplification of zombie attacks and a near decimation of humans until they slowly return to their vigilant and armed condition. Then, the inevitable return to complacency and the cycle begins anew. We can imagine several ways to extend and generalize the present model. In (Munz, 2009), humans killed by zombies did not immediately turn into zombies. Rather there was a waiting period to become living dead. This addition could be added to the present model. Another generalization could be to break the human population into two groups, those who are well armed and those who are not. As the zombie population drifts downward, the well13

armed drop their weapons and become members of the passive crowd until the zombies attack again. We could also include a “penalty” for increasing the vigilance as this could be viewed as taking resources away from other useful tasks such as working, building roads, etc. Indeed, too many “complacent” humans (free riders) depending on a few well-armed individuals could lead to decimation of the humans, an example of the so-called tragedy of the commons (Hardin, 1968). As humans actually are able to cooperate and communicate, we could also introduce a situation where multiple humans could defeat multiple zombies. Recent evidence has accumulated indicating that zombies, themselves, may cooperate. In Zombie Strippers (Lee, 2008), the female zombies cooperated for the most part in attacking male humans and on occasion fought among themselves. The male zombies in this film, however, behaved like classic zombies. Thus, one might be tempted to add some degree of zombie cooperation into the model. Finally, it is known that zombies in temperate latitudes slow down as winter arrives (Brooks, 2003). They are technically dead so that their blood must get colder and thus they might approach a torpid state during the winter. Hence, one could envision adding a seasonal fluctuation to the parameter a4 such that is is quite small during the winter but rises to a large value in the spring as thousands of hungry zombies awaken and search for fresh brains. Periodic forcing of oscillatory and nearly oscillatory systems can often lead to complex dynamics including chaos. And, what could be more chaotic than a horde of flesh-eating undead descending unpredictably on their human hosts!

References [1] Brooks, M. The Zombie Survival Guide: Complete Protection from the Living Dead. New York: Three Rivers, 2003. [2] Edelstein-Keshet, L. Mathematical Models in Biology, SIAM, Philadelphia, PA, 2005 [3] Ermentrout, B. Simulating, analyzing, and animating dynamical systems, SIAM, Philadelphia, PA 2002. [4] Hardin, G. The Tragedy of the Commons, Science, 162:1243-1248, 1968.

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[5] Hethcote, H. and Levin, S.A., Periodicity in Epidemiological Models, in Applied Mathematical Ecology Biomathematics 18 (Levin, S.A. and Hallam, T, eds), Springer-Verlag, Berlin, 1989. [6] Lee, J. (writer, director) Zombie Strippers, 2008. [7] Munz, P., Hudea, I., Imad, J, Smith?, R.J., When Zombies Attack!:Mathemetical modelling of an outbreak of zombie infection, in Infections Disease Modeling Research Progress, (J.M. Tchuenche and C. Chiyaka, eds) pp 133-156, Nova Science Pub, 2009. [8] Romero, G. A. (writer, director), Night of the Living Dead, 1968. [9] Romero, G. A. (writer, director), Dawn of the Dead, 1978. [10] Strogatz, S, Nonlinear Dynamics and Chaos, Perseus Books, Cambridge, MA, 1994.

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