Modelling Zombie Outbreak and Human Survival in The Walking Dead

Transcription

1 Modelling Zombie Outbreak and Human Survival in The Walking Dead A mathematical application on the zombie outbreak and human survival shown in The Walking Dead Academic year 2016/2017, semester II 20 April 2017 MH3110 Group 401 BASSEL ABDEL-QADER CHEE BING KUEN DANG THI MAI VY GINA TEO SHUN YING KIAM JIN YONG LOUISA ONG QING MEI Nanyang Technological University Singapore 1

2 Abstract The objective of the project is to investigate the likelihood of human survival in the event of a zombie outbreak based on the conditions in The Walking Dead series directed by Frank Darabont in 2010 [1]. Our group would like to propose a model with two non-linear ordinary differential equations showing the relationship between zombie and human populations. The analysis of the model shows that humans and zombies could exist together, but their population would fluctuate over time. 1 Introduction Zombie apocalypse has long been a popular topic of discussions. It has attracted millions of enthusiasts and inspired many theories and predictions about humanity s fate if such a catastrophe would ever occur. Intrigued by the topic and inspired by the research by Munz et al., 2009 [2], our group propose a system of two nonlinear differential equations to model human and zombie populations based on the famous TV series, The Walking Dead. It aims to provide a logical analysis investigating the fate of humanity in the event of a zombie apocalypse. Results from research have shown that humans would survive a zombie apocalypse. 2 Model construction The research by Munz, Hudea, Imad, and Smith, 2009 [2] use the basic model: S = Π βsz δs Z = βsz + ζr αsz R = δs + αsz ζr In the above model, S, Z and R represent the Susceptible humans, Zombies and the Removed, which are dead humans and zombies. Π is the birth rate, δ is the death parameter for humans (not due to zombies). β is the parameter for infection from zombies to humans and α is the kill parameter at which humans can kill zombies. ζ is the rate at which dead humans can be resurrected into zombies. We decide to adopt and modify this basic model: Firstly, their assumptions are in line with ours that susceptible humans become zombies when they encounter zombies and get bitten. As such, the infection rate depends on the density of humans and zombies. This is represented by isz in our model. 2

3 The research by Munz et al. assumes that zombies can be killed by humans, which is of similar context as The Walking Dead. Hence, we also include the kill factor, represented by ksz, in our model. However, the research by Munz et al. assumes that dead humans can be resurrected into zombies, which is not the case of The Walking Dead. Hence, we do not consider this factor in our model. The birth rate is assumed to be constant while the death rate not related to zombies is proportional to the current human population in the research. However, since we would like to explore the situation in the long time, we assume that both human natural death rate and birth rate are proportional to the current human population. We use the net growth rate to consider both of the birth and death rates. Our other assumptions based on The Walking Dead series include that there is no cure for zombification and that zombies cannot survive without humans to feed on, which is represented as αz in our proposed model. Based on the above analysis, we propose a system of non-linear differential equations to model human and zombie populations: Variables: dz dt ds dt = isz + gs = (i k)sz αz S = number of susceptible humans Z = number of zombies t = time (in years) Parameters: g = natural growth rate of world s human population i = infection rate of humans k = kill rate of zombies α = natural death rate of zombies 3

4 3 Data collection The data regarding the number of human characters killed or zombified, and the number of zombie characters killed were collected from web-based research and official figures from the series. Several assumptions were then used to process our data. The assumptions are as follows: Zombie outbreak happened by the end of 2012, based on the context of the movie series. Hence the initial human population at the time of outbreak is estimated to be 7 billion, according to the World Bank [3]. We assume at the time of zombie outbreak, the number of zombie is At that point in time, the world population growth rate, g, was , according to the World Bank [3]. Robert Kirkman, the author of The Walking Dead comic series, established that zombies outnumbered humans by a ratio of 5000 : 1 in The Walking Dead #10. Thus, we assume that in the first season, the ratio of zombies to humans is 5000 : 1 or about for human population and billions for zombie population at the begin of season 1. Based on seasons 1-6 of The Walking Dead television series, only 27.4% of human characters survived. Hence, from the initial human population of in season 1, only 27.4% of the initial human population remained by the end of season 6. The change in number of human and zombie characters in the television series is proportional to the change in the real world during a zombie apocalypse. Both the human and zombie populations are calculated at the end of each television season. Time is taken at the end of each season. This is in terms of days, but converted into years for easier handling of our data. Based on the above assumptions, we are able to get the following data: 4

5 Population / Ten millions Season Time Human population Zombie population (Years) (Ten millions) (Ten millions) Human population Zombie population Time / Years Figure 1: Human and Zombie populations 4 Fitting parameters of the model into data By Particle Swarm Optimization from Khelil, Djerou, Rahmani, and Dakhia, 2010 [4] in MATLAB, the parameters are determined as follows: g = , i = 0.986, k = 0.197, α = In Figure 1., both graphs are based on the condition that the initial human population is 7 billion and the initial zombie population is The left graph also has data points derived from the data table of human and zombie populations versus time. They show that the human population decreases rapidly while the zombie population increases rapidly in a short period of time. 5

6 Population (in tens of millions) Population/Ten millions Humans Zombies Human Zombie Years Years Figure 2: Solution plot The right graph is plotted over a much longer period of time (100 years) than the graph on the left (0.05 years) of Figure 1. From the graph on the right, it may be predicted that both humans and zombies will die off over time. However, the range of the vertical axis is so large that even if the two populations approach very small numbers over time, it would look like they approach zero. Hence, further analysis would be required for a more concrete conclusion. 5 Rigorous analysis In Figure 2., both graphs show the same phase portrait of the system (but with different initial points). The critical points are coloured in green. The right graph is in its normal scale and the left graph is in a modified scale to clearly show that the non-zero critical point is, indeed, a centre. Proposed system of differential equations to model zombie outbreak: ds dt = isz + gs; dz dt = (i k)sz αz [ ] g - iz -is Jacobian matrix of the linearized sysem: A =. (i-k)z (i-k)s - α 6

7 Zombies Population / Ten Millions Zombies Population / Ten Millions Susceptible Humans Population / Ten Millions Susceptible Humans Population / Ten Millions Figure 3: Phase portraits To solve for critical points, let: ds dz = 0 and dt dt = 0 Two critical points are found: (0,0) and ( g i, when substituting the parameters found. α ), which is ( , ) i k For critical point (0,0), the corresponding Jacobian matrix is A = [ ] g 0 0 α and two distinct eigenvalues : λ 1 = g, λ 2 = α Given that all our parameters are positive, this Jacobian matrix has 2 distinct eigenvalues, one positive and one negative. Thus, (0,0) is a saddle point in the linearized system and the original system. [ ] 1 Moreover, the corresponding eigenvectors for λ 1 = g is and for λ 0 2 = α [ ] 0 is, corresponding to the horizontal and vertical axes on the SZ-plane 1 respectively. This implies that if the initial populations are S 0 = 0 and Z 0 > 0 for zombies and humans respectively; (S 0, Z 0 ) falls on the Z-axis, Z 0 will decrease to 0, the equilibrium population. If the initial populations are S 0 > 0, Z 0 = 0; the human population, S, will increase while the zombie population, Z, remain zero. 7

8 Population (in tens of millions) Humans Zombies Years Figure 4: Human and zombie populations with initial condition S 0 = 0.3, Z 0 = 0.04 For the critical point ( g i, α ), it is either a center, a spiral source or a spiral i k sink. To analyse the nature of this critical point, we will find the trajectories implicitly by the method of separation of variables: dz ds = Z((i k)s α) S(g iz) g ln Z iz = (i k)s α ln S + C g ln Z iz (i k)s + α ln S C = 0 Let f(s, Z) = g ln Z iz (i k)s + α ln S C. Different values of C yields different level curves for f(s, Z). We can see that f(s, Z) is continuous for S > 0, Z > 0 thus f(s, Z) is continuous at ( g i, α i k ). If ( g i, α ) is a spiral source or sink, this critical point would lie on many i k level curves. If we denote f i (S, Z) to a corresponding C = C i, a distinct value of C, then: f 1 ( g i, α i k ) = f 2( g i, α i k ) = f 3( g i, 8 α i k ) =...

9 where f 1, f 2, f 3... corresponds to C 1, C 2, C 3... and these values of C are pair-wise different. However, based on the general function f(s, Z) = g ln Z iz (i k)s + α ln S C, with the same values of S, Z but different values of C (C 1 C 2 ) then f 1 (S, Z) f 2 (S, Z). Thus, there is a contradiction if ( g i, spiral source or sink. Hence, we can conclude that ( g i, α i k ) is a α ) is a stable center critical point with i k many limit cycles around it. By definition, these limit cycles will be neutrally stable with no trajectories approaching or going away from them in the neighborhood of the center. Based on our model, if the initial populations of humans and zombies, (S 0, Z 0 ), is close to the critical point ( g i, α ), they will oscillate over time i k with constant magnitudes as shown in Figure 3. When the human population is high, zombies are able to find and attack humans easily resulting in an increase in the zombie population and a decrease in the human population. As zombies need to feed on humans to survive, the zombie population will decrease when the human population is low. When the zombie population decreases, humans can fight and/or escape with more ease, hence the survival rate increases and subsequently the human population will increase as well. 6 Conclusion In summary, this project fundamentally focuses on the competition between humans and zombies. However, we did not include some other factors that may affect human and zombie populations. These factors include humans gaining experience in escaping and fighting with zombies, internal conflicts in human community and so on. The lack of these factors may affect the accuracy of our analysis but due to time and resource constraints, we are not able to analyse them here. From the analysis of the model, it has been shown that humans and zombies have the chance to survive concurrently. The presence of limiting cycles means that when beginning with certain initial population sizes, humans and zombies can co-exist with their populations oscillating over time. As the series is still on-going, there could be an introduction of a new cure to the zombie virus or the zombie virus might mutate, leading to a different outcome.therefore, more in-depth research can be done with reference to future developments in the movie plot. 9

10 References [1] Frank Darabont. The Walking Dead [2] Philip Munz et al. When zombies attack!: mathematical modelling of an outbreak of zombie infection. In: Infectious Disease Modelling Research Progress 4 (2009), pp [3] World Bank Group. Total Population url: http : / / data. worldbank.org/indicator/sp.pop.totl?end=2013&start=2012 (visited on 2013). [4] N Khelil et al. On the application of Particle Swarm Optimization to Differential Equations. In: Particle Swarm Optimization (2010). 10

11 Appendix A: Matlab codes PSO Main 1 clc ; 2 clear ; 3 close all ; 4 5 %% Problem Definiton 6 7 S = readtable ('zombie. xlsx '); 8 CostFunction lse (S,x); 9 nvar = 3; 10 VarSize = [1 nvar ]; 11 VarMin = 0.1; 12 VarMax = 0.99; %% Parameters of PSO MaxIt = 1000; 17 PopSize = 50; 18 w = 1; 19 wdamp = 0.99; 20 c1 = 2; 21 c2 = 2; 22 ShowIterInfo = true ; 23 MaxVelocity = 0.2*( VarMax - VarMin ); 24 MinVelocity = - MaxVelocity ; %% Initialization % The Particle Template 29 p0. Position = []; 30 p0. Velocity = []; 31 p0. Cost = []; 32 p0. Best. Position = []; 33 p0. Best. Cost = []; % Create Population Array 36 particle = repmat ( p0, PopSize, 1); 37 11

12 38 % Initialize Global Best 39 GlobalBest. Cost = inf ; % Initialize Population Members 42 for i =1: PopSize % Generate Random Solution 45 particle (i). Position = unifrnd ( VarMin, VarMax, VarSize ); % Initialize Velocity 48 particle (i). Velocity = zeros ( VarSize ); % Evaluation 51 particle (i). Cost = CostFunction ( particle (i). Position ); % Update the Personal Best 54 particle (i). Best. Position = particle (i). Position ; 55 particle (i). Best. Cost = particle (i). Cost ; % Update Global Best 58 if particle (i). Best. Cost < GlobalBest. Cost 59 GlobalBest = particle (i). Best ; 60 end end % Array to Hold Best Cost Value on Each Iteration 65 BestCosts = zeros ( MaxIt, 1); %% Main Loop of PSO for it =1: MaxIt for i =1: PopSize % Update Velocity 74 particle (i). Velocity = w* particle (i). Velocity c1* rand ( VarSize ).*( particle (i). Best. Position - particle (i). Position ) 76 + c2* rand ( VarSize ).*( GlobalBest. Position - particle (i). Position ); % Apply Velocity Limits 12

13 79 particle (i). Velocity = max ( particle (i). Velocity, MinVelocity ); 80 particle (i). Velocity = min ( particle (i). Velocity, MaxVelocity ); % Update Position 83 particle (i). Position = particle (i). Position + particle (i). Velocity ; % Apply Lower and Upper Bound Limits 86 particle (i). Position = max ( particle (i). Position, VarMin ); 87 particle (i). Position = min ( particle (i). Position, VarMax ); % Evaluation 90 particle (i). Cost = CostFunction ( particle (i). Position ); % Update Personal Best 93 if particle (i). Cost < particle (i). Best. Cost particle (i). Best. Position = particle (i). Position ; 96 particle (i). Best. Cost = particle (i). Cost ; % Update Global Best 99 if particle (i). Best. Cost < GlobalBest. Cost 100 GlobalBest = particle ( i). Best ; 101 end end end % Store the Best Cost Value 108 BestCosts ( it) = GlobalBest. Cost ; % Display Iteration Information 111 if ShowIterInfo 112 disp ([ ' Iteration ' num2str ( it) ': Best Cost = ' 113 num2str ( BestCosts (it))]); 114 end % Damping Inertia Coefficient 117 w = w * wdamp ; end 13

14 pop = particle ; 122 BestSol = GlobalBest ; %% Results fprintf ('a = %f, b = %f, c = %f\n',bestsol. Position ); 127 a = BestSol. Position (1) ; 128 b = BestSol. Position (2) ; 129 c = BestSol. Position (3) ; Least Square Error 1 function z = lse2 (S,x) 2 3 H=S {:,2}; 4 Z=S {:,3}; 5 rh=s {:,5}; 6 rz=s {:,6}; 7 8 z= sum ((-x (1) *H.*Z * H- rh).^2+((( x (1) -x (2) )*H.*Z- x (3) *Z-rz).^2) ) 9 10 end lvrhs 1 function ret = lvrhs (t,y) 2 % set up ret in correct form ( i. e. a column vector ) 3 ret = zeros (2,1) ; 4 % extract current values of V and P 5 A = y (1) ; %a= human 6 N = y (2) ; %n= zombies 7 % return updates 8 ret (1) = * A * A* N; % change to human rate 9 ret (2) = ( ) * N*A -0.1* N; % change to zombies 14

15 Graph of Zombies and Humans Population over Time 1 trange = [0:0.001:0.5]; 2 % set up range for t 3 yzero = [700,0.001]; %(x0,y0) 4 % and an initial condition 5 [myt, myy ]= ode45 (@ lvrhs, trange, yzero ); 6 % solve 7 plot (myt, myy (:,1),'r'); 8 hold on 9 plot (myt, myy (:,2) ); yo = [ ]; 12 ho = [ ]; 13 zo = [ ]; 14 plot (yo,ho,'o'); 15 plot (yo,zo,'*'); % change legend and labels 19 legend ('Human ','Zombie '); 20 xlabel ('Years '); 21 ylabel ('Population / Ten millions '); Phase portrait 1 function phaseplot 2 clf 3 hold on 4 grid off 5 6 a = ; b =0.4; 7 c = ; d =0.2; 8 m =20; n =20; 9 10 axis ([a b c d]) [x,y]= meshgrid (a:(b-a)/m:b,c:(d-c)/n:d); 13 dx = F_matlab (x,y); 14 dy = G_matlab (x,y); 15

16 15 r = sqrt (dx.^2+ dy.^2) ; 16 dx = dx./r; 17 dy = dy./r; 18 quiver (x,y,dx,dy,0.4, 'color ','black '); tspan =0:0.01:1000; 21 [t, X]= ode45 (@ rhs_matlab, tspan,[0.07;0.0001]) ; 22 plot (X(:,1),X (:,2),'color ','blue ') 23 plot (0,0, 'o', ' markerfacecolor ', 'green ') 24 plot ( , , 'o',' markerfacecolor ','green ') 25 plot (0.07,0.0001, 'o',' markerfacecolor ','yellow ') 26 xlabel (' Susceptible Humans Population / Ten Millions ') 27 ylabel (' Zombies Population / Ten Millions ') function RHS = rhs_matlab (t, input_vector ) 31 F = F_matlab ( input_vector (1), input_vector (2) ); 32 G = G_matlab ( input_vector (1), input_vector (2) ); 33 RHS = [F; G]; 34 end function F = F_matlab (x, y) 37 F = x.*( * y); 38 end function G = G_matlab (x, y) 41 G = y.*(( ).*x -0.1) ; 42 end 43 end 16

17 Appendix B: Data time humans zombies time_change rate_human rate_zombie E-05 17