Progressive Temporal Models for a Zombie Apocalypse P R E S E N T A T I O N BY A A R O N M O S H E R

Transcription

1 Pogessive Tempoal Models fo a Zomie Apocalypse P R E S E T A T I O BY A A R O M O S H E R

2 Topic Oveview Imagine that you ae the CDC faced with the possiility of an impending epidemic What do you want to know? How fast will the disease spead? How many people will it affect? What is the est esponse? What you need: A mathematical desciption of the disease, the population, and the geneal envionment. Tools to analyze the esults

3 Astact Define asic assumptions and conditions What is a Zomie? What can e assumed aout the population? What can e assumed aout the location? Popose a seies of pogessive ODE models y gadually loosening conditions Basic model to cumulative MBER model Test and compae models with numeical and empiical esults Zomie tag tial umeical methods Lineaization Pesent and intepet esults If time, discuss a PDE model using chemotaxis/diffusion to model decision factos

4 Base assumptions and conditions D E F I I T I O O F A Z O M B I E P O P U L A T I O A S S U M P T I O S S P A T I A L C O D I T I O S

5 Concening Zomies The zomie myth oiginated as pat of the Haitian voodoo legend Evil socees wee thought to possess the powe to evive the ecently dead as mindless sevants Became a popula pat of Ameican film cultue ight of the Living Dead 1968) Dawn of the Dead 1978), emake in 4 8 Days Late ) Taditionally thought to e the esult of cuses/magic. Moden films often depict zomies as the esult of a iological agent teestial o othewise) Fo the puposes of modeling, these ae equivalent

6 Definition In the following models, we define a zomie as any infected o cused suject that exhiits the following qualities: A loss of highe ode easoning functions An amplified hunge/suvival esponse Is capale of speading the infection/cuse via diect contact i.e. a ite) Remains moile, and is capale of speading infection indefinitely unless emoved y an extenal souce ote: By the aove definition, zomies will actively wok to spead infection. This diffes geatly fom othe diseases

7 Othe Conditions Like the aove films, the model will focus on a single isolated population within a elatively shot time span Ignoe gloal mass tansit, and assume that thee is zeo flux in o out of the defined aea Assume that natual iths/deaths have a negligile impact on the oveall population i.e. total population is constant) Assume that the population is unifomly mixed Evey individual has an equal chance of encounteing any othe individual Some conditions may e loosened in late models

8 Odinay Diffeential Models G E E R A L D E R I V A T I O S B A S E C U M U L A T I V E S I M B E R

9 Model Foundations We popose that the ate of new infections is popotional to the ate of contact etween sujects who can catch the disease susceptiles) and sujects that can tansmit the disease infectives) Popose that the ate of contact, is popotional to the poaility of contact etween susceptiles and infectives Rate of infection = c * P contact ) fo some constant c Suppose we have a constant, isolated, unifomly mixed population P that has sujects. We patition P into distinct classes P 1,P,P 3,,P n each with 1,, 3,, n sujects.

10 continued) Because P is unifomly mixed, we know that the poaility that any individual will encounte a suject fom class P i is i 1 Thus, the poaility that a suject fom class P i will contact a suject fom class P j is i j

11 Assumptions of model We conside a population of sujects that is split into two classes S is the nume of people who can ecome zomies susceptiles) I is the nume of zomies infectives) We know that if I =, thee will e new infections. Similaly, if S = thee cannot e any new infections Assume that zomies ae immotal and indestuctile Assume thee is no delay, o incuation time, until zomies ecome infective Gaphically we have Susceptiles S Bitten y zomies Infectives I

12 Base Model is a positive, eal valued constant Model: ds di S I The system can e solved y a sepaation of vaiales Analytic Solution: I t) t 1 Ce S t) 1 Ce t C I I o

13 Discussion of model Population is conseved d/ = ) otice that the constant has units of sujects/second Can e though of as a conditional ate Has many nice popeties Has an analytic solution Repesents a wost case scenaio that always esults in epidemic

14 Intoduction of B Suppose it is found that afte a susceptile is itten, it takes some aveage time T efoe they ecome fully infective. We split the population into 3 classes: Susceptiles S Infectives I Bitten B Let v = 1/T, then an aveage of vb exposed sujects will ecome infective pe unit time Gaphically Susceptiles S Bitten y Zomies Bitten B Become infective afte time T Infectives I

15 B Model, v ae positive eal valued constants S ds I B db vb di vb

16 Intoduction of M and R ext, we allow zomies to e emoved/quaantined Pehaps due to a tained esponde class M Susceptiles ae ale to fight ack Like infection, we popose that the ate of emoval will e popotional to the poaility of contact M may e moe esistant and/o efficient than S Susceptiles S Bitten B Infectives I Medical M Removed R

17 S I M B R MBR Model s, m ae the infection constants fo the susceptiles and espondes espectively mi is the emoval constant of I y M. Similaly fo m, si, s *ote: With a cetain selection of constants, this model can e educed to the R model discussed in [3] ds dm db di dr s m s MI si vb s m mi m SB s MI MI si MB m SB mi MB vb MI

18 Intoduction of E Logic would dictate that, given the option, people would attempt to evacuate o find shelte Assume that such shelte can only e povided y tained espondes The ate of evacuation will e popotional to the ate of contact etween S and M Evacuated E Susceptiles S Bitten B Infectives I Medical M Removed R

19 MBER Model e is the conditional ate of evacuation y class M MI MB SB dr SM e de MI vb di vb MB SB MI db MI dm SM e ds mi si m s mi si m s m s m s

20 Loosening Conditions What if the population isn t constant o isolated? otice that evacuated o emoved sujects no longe influence the population What if thee is a goup of K espondes outside the nomal population that can e deployed in the case of an emegency We define a vaiale as the cuent nume of active sujects S I M B We popose that, up to the maximum capacity K, espondes will e deployed at a ate popotional to the seveity of infection e.g. the nume of infected cases I dm d 1 M K I

21 Cumulative MBER Model Adjusts fo inactive potions of the population, and allows fo a deployment of tained pesonnel MI MB SB dr SM e de MI vb di vb MB SB MI db MI I K M d dm SM e ds mi si m s mi si m s m s m s 1

22 Analysis of Odinay Models M O D E L V A L I D A T I O L I E A R I Z A T I O A D B A S I C R E P R O D U C T I O U M B E R D E V E L O P M E T O F C O T R O L P R O C E D U R E S

23 Model Veification To test the validity of the model, we would like to compae esults to eal wold data Due to the natue of the disease, thee is no eadily availale statistical data fo compaison Simple /B model can e appoximated y a modified vesion of tag Tial of 9 sujects, with one initial it Evey peson tagged ecomes an it and emains infected until the end of tial Sujects exhiited an incuation time on the ode of 1 second

24 Infected pesons It) Expeimental Tial Fo Tial 1, was found to e ~.48, and v was assumed to e ~ Tial I Analytic I umeic B time s)

25 Expeimental Tial Infected Pesons It) Fo Tail, was found to e ~., v~1 1 Tial Analytic B umeic time s)

26 Infected Pesons It) Expeimental Tail Tial 3 Analytic 1 Fo Tial 3, e was found to e ~.785, v~1 umeic B

27 Infected Pesons It) Expeimental Tial Fo Tial 4, was found to e ~.41, v~ Tial 4 Analytic umeic B

28 Discussion of Expeimental Tials Possile Souces of Eo: Small population, so infection ove time is a discete function Measuement eo Wide vaiation of movement speed and fitness levels The effect of incuation time is moe ponounced within smalle time intevals Significant fo P-value of.1 ased on an F test esults geate than 8.

29 Computation of R The asic epoduction nume R is defined as the nume of seconday cases geneated y a pimay infectious case in an entiely susceptile population [5] Clealy, if R >1 then an infection will spead, and if R <1 an infection will die out R can e computed fom a linea appoximation of a system aout some equiliium point

30 Lineaization of MBR The fist step is to find an equiliium point such that all deivatives ae zeo Fo MBR: Let X=[ S, I, M, B, R ] T, then X * =[ S,, M,, ] T ow, conside only the compatments whee the infection is in active pogession B and I) Define F j as the ate of appeaance of new infection is compatment j Define V j as the ate of tansfe in/out of compatment j y all othe means V j+ is the ate at which individuals ente V j- is the ate at which individuals leave Then, fo any compatment j dx j F j V j V j ) F j V j

31 continued) Clealy then, fo the MBR model we have ow we use the Jacoian of F and V aout X* vb MI X F m s ) ) MI vb MB SB X V mi si m s ) * v M S X J m s F * ) M S v M S X J mi si m s V

32 Theoem Fom [5] we have that R =ρj F J V -1 ), whee ρa) denotes the maximum eigenvalue of a matix A Doing the computations we find. R si S v M mi ) s ss S M m m M ) v ) It can then e shown the system is locally asymptotically stale if R <1, and locally asymptotically unstale if R >1.

33 Model Staility R =.75 < 1 R = 1.5 > 1

34 Reminde of Impotant Figues Model: MBER Model: R : t t Ce t S Ce t I 1 ) 1 ) ) ) ) v M S M S M S v R m s mi si m s MI MB SB dr SM e de MI vb di vb MB SB MI db MI I K M d dm SM e ds mi si m s mi si m s m s m s 1 di ds

35 Development of Contol Pocedues Rememe that the model was said to have nice popeties Existence of an analytic solution otice that fo a given set of constants and initial conditions, di/ ) di/ MBER). Thus a solution to the MBER model will e ounded aove y the analytic solution to the model Fom hee, one can pedict the magnitude of esponse necessay to contain infection Statistically detemine model constants ) Use the Analytic solution to pedict the seveity of an infection at a specified esponse time 3) Use the expession fo R to solve fo the nume of espondes, M, necessay to contain a zomie infection

36 Pause fo Questions

37 Concluding Remaks Results Developed seveal models to fit a wide ange of assumptions Analyzed models though lineaization to compute the asic epoduction nume R. Developed a seies of Patial diffeential models fo futhe study Othe applications of Zomie Models With a caeful selection of constants, can e used to model the spead of gossip/misinfomation Topics of futhe study: Solution and/o analysis of Patial Diffeential models Geate exploation of umeical methods Staility Efficiency on egula ounday conditions Questions?

38 Acknoledgements D. Jessica Skla: Capstone Pofesso D. Byan Done: Refeences to the heat equation D. Mei Zhu: Capstone go-to adviso

39 Biliogaphy [1] Buden, Richad L, umeical Analysis, 8 th edition, pgs , Thomson Books/Cole 5) [] Keyszig, Ewin, Advanced Engineeing Mathematics, 7 th edition, pgs , John Wiley & Sons, Inc, ew Yok, 1993) [3] Muay, J.D, Mathematical Biology nd edition, pgs 36-49, , , Spinge 1989) [4] Weinege, Hans F, A Fist Couse in Patial Diffeential Equations, 1 st edition, pgs 58-6, Blaisdell Pulishing, ew Yok 1965) [5] Y. Dumont, F. Chioleu, C. Domeg, "On a tempoal model fo the Chikun-gunya disease: Modeling theoy, and numeics", Mathematical Bio-sciences, 13 8): 8-91