Mathematical Model for Lung Cancer the Effects of Second-Hand Smoke and Education, A

8/10/2019 Mathematical Model for Lung Cancer the Effects of Second-Hand Smoke and Education, A

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A Mathematical M del f r Lu g Ca cer: The E ect Sec d Ha d Sm ke a d du ati

BU-1525 M

Carlos A. Acevedo-EstefaniaUn ve s ty of Pue to co-Cayey

Chr st na Gonz lez

Texas A&M Un ve s tyKa en R Rios SotoUn ve s ty of Pue to co-Mayag ez

Er c . Summerv lleSt. Ma y's Un ve s ty

Baojun SongCo nell Un ve s ty

Carlos Cast llo ChavezCo nell Un ve s ty

ugus

b cIn the United States, lung cancer is the leading cause of cancer deaths. As o

today, cigarette smoking causes 85 percent of lung cancer deaths n this study, a nonlinear system of di erential equations is used to model t e dynamics of a populationwhich includes smokers. The parameters of the model are obtained om data publishedby cancer institu es, health and government organizations. The average number oindividuals who become smokers and the reduction of this average by an educationprogram are determined. The long-term impact of educating a susceptible class beforethey enter the population model and the e ect it has on the epidemic is also studied.Simulations using realistic parameters are carried out to illustrate our theoretical results.

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A Mathematical Model for Lung Cancer: The Effects ofSecond-Hand Smoke and Education

BU-1525-M

Carlos A. Acevedo-Estefania

University of Puerto Rico-CayeyChristina Gonzalez c

Texas A&M UniversityKaren R. Rios-Soto

University of Puerto Rico-Mayagiiez

Eric D. Summerville

St. Ma y sUniversity

Baoj un SongCornell University

Carlos Castillo-Chavez,Cornell University

August 2000

Abstract

In the United States, lung cancer is the leading cause of cancer deaths. As oftoday, cigarette smoking causes 85 percent of lung cancer deaths. In this study, a non-linear system of differential equations is used to model the dynamics of a populationWhich includes smokers. The parameters of the model are obtained from data publishedby cancer institutes, health and government organizations. The average number ofindividuals who become smokers and the reduction of this average by an educationprogram are determined. The long-term impact of educating a susceptible class before

theyenter the

populationmodel and the effect it has on the

epidemicis also

studied.Simulations using realistic parameters are carried out to illustrate our theoretical results.

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1 Introduction

Lung cance also efe ed to as b onchogen c ca c nomas s a majo cont buto ocance deaths n the Un ted States account ng fo2 pe cent of such deaths ].The de velopment of lung cance occu s on the l n ng glands wh ch conta ns damage cells that a elocated n ou lungs and b oncheal a ways known as the t acheob onch al system3 .Th spa tof a human be ng s mpo tant because th s system s suscept ble to be ng contam natedby nhaled a wh ch s a majo facto n the development of lung cance Sc ent sts bel evthat the majo cause of lung cance s due to cance -caus ng agents known as ca c nogenssuch as asbestos and adon Howeve esea ch and stat st cs show that the majo agent oflung cance s tobacco smoke wh ch conta ns ove60ca c nogens

Today c ga ette smoke s espons ble fo a g eat p opo t on of deaths w th n tobaccosmoke ach yea n the Un ted States app ox mately400,000people d e f om c ga ettesmoke wh ch accounts fo one n eve y ve deaths n the Un ted States14].The l kel hoodthat a smoke w ll develop lung cance f om c ga ette smoke depends on many aspects;such as the age at wh ch smok ng began how long the pe son has smoked the numbe ofc ga ettes smoked pe day and how deeply the smoke nhales10]. In 19 ,the Su geonGene al establ shed the add ct ve potent al of c ga ette smok ng by st ess ng that n cot neand othe agents n c ga ettes we e just as add ct ve as coca ne].The ab l ty of a smoketo qu t s ve y d cult because of the add ct on to n cot ne In fact90pe cent of smoke s

would l ke to qu t but can not12].Based on data of cu ent smoke s only34pe cent ofsmoke sattempt to qu t but only2.5pe cent succeed eve y yea ].The use of c ga ettesand the tox c a t c eates has been labeled as the s ngle most p eventable cause of p ematu e death n the Un ted States

The elat onsh p between c ga ette smoke w th espect to lung cance has been establ shed n 5 90pe cent of all lung cance cases(146,000case/yea ) the mo e anest mated3,000non-smoke s pe yea d e f om lung cance due to second-hand smoke (alsoknown as env onmental tobacco smoke TS)14].The numbe of deaths of non-smoke smay be lowe than act ve smoke s but acco d ng to the U S nv onmental P otect onAgency t s qu te la e when compa ed to those assoc ated w th othe ndoo and outdooenv onmentalpollutants Th s data has had a g eat mpact on publ c pol c es that p otectpeople f om second-hand smoke 9].

Based on the elat onsh p between lung cance and c ga ette smoke we want to showthe educt on of contact between non-smoke s and smoke s and how to dec ease the ate n wh ch non-smoke s and smoke sp og ess towa ds lung cance The a angement of sevend e ent classes w ll ass st us to de ne the total populat on we want to analyze Howeve the best way to deta l the t ans t on of each class s to use a m xtu e of pa amete s

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1 Introduction

Lung cancer, also referred to as bronchogenic carcinomas, is a major contributor ofcancer deaths in the United States, accounting for 28 percent of such deaths The de-velopment of lung cancer occurs on the lining glands, which contains damage cells that arelocated in our lungs and broncheal airways known as the tracheobronchial system Thispart of a human being is important because this system is susceptible to being contaminated

by inhaled air, which is a major factor in the development of lung cancer. Scientists believethat the major cause of lung cancer is due to cancer-causing agents known as carcinogens,such as asbestos and -radon. However, research and statistics show that the major agent oflung cancer is tobacco smoke, which contains over 60 carcinogens.

Today, cigarette smoke is responsible for a great proportion of deaths within tobaccosmoke. Each year in the United States, approximately 400,000 people die from cigarettesmoke, which accounts for one in every five deaths in the United States [14]. The likelihoodthat a smoker will develop lung cancer from cigarette smoke depends on many aspects;such as the age at which smoking began, how long the person has smoked, the number ofcigarettes smoked per day, and how deeply the smoker inhales [10]. In 1988, the SurgeonGeneral established the addictive potential of cigarette smoking by stressing that nicotineand other agents in cigarettes were just as addictive as cocaine The ability of a smokerto quit is very difficult because of the addiction to nicotine. In fact, 90 percent of smokerswould like to quit but can not [12]. Based on data of current smokers, only 34 percent ofsmokers attempt to quit, but only 2.5 percent succeed every year The use of cigarettesand the toxic air it creates has been labeled as the single most preventable cause of prema-ture death in the United States.

The relationship between cigarette smoke with respect to lung cancer has been es-tablished in 85-90 percent of all lung cancer cases (146,000 case/year). Furthermore, anestimated 3,000 non-smokers per year die from lung cancer due to second-hand smoke (alsoknown as enviromnental tobacco smoke, ETS) [14]. The number of deaths of non-smokersmay be lower than active smokers, but according to the U.S. Environmental Protection

Agency, it is quite large when compared to those associated with other indoor and outdoorenviromnental pollutants. This data has had a great impact on public policies that protectpeople from second-hand smoke

Based on the relationship between lung cancer and cigarette smoke, we want to showthe reduction of contact between non-smokers and smokers, and how to decrease the rate inwhich non-smokers and smokers progress towards lung cancer. The arrangement of sevendifferent classes will assist us to define the total population we want to analyze. How-ever, the best way to detail the transition of each class is to use a mixture of parameters,

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p obab l t es and ates. Based on the behav o of each class seven non-l nea d e e t

equat ons a e c eated. One of the ma n pu poses of the nonl nea equat ons s to obta nthe equ l b um po nts. The Jacobean Mat x s use to nd the bas c ep oduct ve numbeRo wh ch ep esents the ate that people get nfected. The ole ofRo s to dete m ne fsmok ng w ll d e out o nc ease. Th ough s mulat ons the model s analyzed to obta nd e ent s tuat ons that p oduce nte est ng esults among the spec c classes. Us ng eall fe data the model s bel eved to show how the nc ease of the educated class can lowethe p obab l ty of be ng d agnosed w th lung cance .

Ou nonl nea d fe ent al equat on model that focuses on the mpact of pee p essu eon non-smoke s and the p og ess on to lung cance v a st and second-hand smoke. Thedynam cs of add ct on a e shown to be gove ned by a s ngle non-d mens onal pa amete

Ro Rodenotes the numbe of seconda y a d ct ons gene ated by the st

(small class of

smoke s n a populat on of(m stly non-smoke s. Obv ously > 1 shows how theas thep evalence of add cts to n cot ne s h gh. O analys s then focus on the ole of educat onat a ous levels o the p og ess on cha n(to lung cance n the long-te m educt on of lungcance . Ou esults show that the most mpo tant facto n p event ng nd v duals f ombecom ng smoke s os educat on; wh le the second most mpo tant measu e s to conv nc ngheavy smoke s to qu t. Ou esults pa t ally ag ee w th those ecently publ shed byhaca

o al Howeve we d sag ee on the ecommendat on of focus ng educat on on smoke s.The p event on of smok ng s most e ect ve n the long un f t s focused on non-smoke

Ou pape s o gan zed as follows: sect on2, expla ns the populat on model wh le sect on3, expla ns the analys s of the smoke-f ee equ l b um the bas c ep oduct ve numbeand endem c equ lb um. In sect on4, we have an est mat on of pa amete s and nume calsolut ons; sect on 5, the conclus on; and sect on6, the futu e wo k.

2 A Population Model for the Risk of Getting Lung Cancer

We d v de the total populat on nto two sub-populat ons wh ch cons sts of nd v duals who have neve smoked that espond to p event on educat on and those who d d not. Theeducated populat on s denoted by nd v duals who neve become smoke sE(t) The lesseducated populat on s made up of s x classes. The non-smoke class ncludes the

nd v duals who do not smoke but a e suscept ble to smok ng; the l ght-smoke class(t),ncludes those who smoke 15 o less c ga ettes pe day; and the heavy smoke sI2(t). The e

ex sts th ee add t onal classes n the less-educated populat on:Q ),the qu tte class cons sts of nd v duals who used to smoke but stop pe manentlyS(t), who used to smoke anda e l kely to smoke aga n; and the lung cance class(t), nd v duals who have developed

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probabilities, and rates. Based on the behavior of each class, seven non-linear differential

equations are created. One of the main purposes of the nonl inear equations is to obtainthe equilibrium points. The Jacobean Matrix is use to find the basic reproductive number,RQ, which represents the rate that people get infected. The role of R0 is to determine ifsmoking will die out or increase. Through simulations, the model is analyzed to obtaindifferent situations that produce interesting results among the specific classes. Using reallife data, the model is believed to show how the increase of the educated class can lowerthe probability of being diagnosed with lung cancer.

Our nonlinear diferential equation model that focuses on the impact of peer pressureon non-smokers and the progression to lung cancer via first and second-hand smoke. Thedynamics of addiction are shown to be governed by a single non-dimensional parameter,

RQ. R0denotes the number of

secondaryaddictions

generated bythe first

(small)class of

smokers in a population of (mostly) non-smokers. Obviously, R0 > 1 shows how the as theprevalence of addicts to nicotine is high. Our analysis then focus on the role of educationat various levels of the progression chain (to lung cancer) in the long-term reduction of lungcancer. Our results show that the most important factor in preventing individuals frombecoming smokers os education; while the second most important measure is to convincingheavy smokers to quit. Our results partially agree with those recently published by IthacaJournal. However, we disagree on the recommendation of focusing education on smokers.The prevention of smoking is most effective in the long run, if it is focused on non-smokers.

Our paper is organized as follows: section 2, explains the population model, while sec-tion 3,

explainsthe

analysisof the smoke-free

equilibrium,the basic

reproductive number,and endemic equilbrium. In section 4, we have an estimation of parameters and numericalsolutions; section 5, the conclusion; and section 6, the f11ture work.

2 A Population Model for the Risk of Getting Lung Cancer

We divide the total population into two sub-populations, which consists of individualswho have never smoked that respond to prevention education and those who did not. Theeducated population i s denoted by individuals who never become smokers, The less-educated population, is made up of six classes. The non-smoker class, N (t), includes the

individuals who donot

smoke butare

susceptibleto

smoking; the light-smoker class, I1 (15),includes those who smoke 15 or less cigarettes per day; and the heavy smokers I2(t). Thereexists three additional classes in the less-educated population: Q(t), the quitter class, con-sists of individuals who used to smoke but stop permanently; S(t), who used to smoke andare likely to smoke again; and the lung cancer class, L(t), individuals who have developed

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ung cance We t eat the peop e that smoke as an n ected g oup n o de to show the

t ansm ss on at h ch the n ect on o smok ng occu sAn nd v dua can ente the popu at on n two d e ent ways One way s p oceed n

nto the educated c assE, w th a p obab ty o q o nto the non-smoke c ass w tha p obab ty o (1-q Ind v dua s n a c asses may deve op ung cance because o

mpact o second-hand smokeInd v dua s n the non smoke c ass can become a ght smoke( ) due to ack o edu-

cat on and pee p essu e o smoke s Once they become a ght smoke they can not moveback to o E The e o e a ght smoke may becom a heavy smoke(J ), o they maystop smok ng tempo a y(S) o pe manent y(Q). We assume that n o de o them tobecome a heavy smoke they must sta t o as a ght smoke

Once an nd v dua becomes a heavy smoke s/he may qu t tempo a y8 pe manent y(Q), o deve op ung cance In t eS c ass the nd v dua s can e the go back tosmok ng n wh ch we assume they sta t o as ght smoke s; o they ca deve op ung cace ( ). The c ass ep esents the numbe o nd v dua s who stop smok ng pe manent yHoweve they have a h ghe p obab ty o deve op ng ung cance than a on-smoke

We et the natu a death ate (pe cap taJ be the same a the c asses except othe c ass wh ch s cons de ed to have h ghe death ate

Ou mathemat ca mode s g ven by the o ow ng non- nea system o o d na y dent a equat ons

ddtdJdtdJdtdQ

tdSdtddt

dEdt

(1 A N I

+ h)

((1 n) + (1 ) S)( + h) ( 8 )IT c + + +

( + 8 + )h

J +W (8 q+ J QS( +h)

(1 )c + (1 I T S,(P n fN + P s fS + f eE)( + h ) 8 I 8 I 8 Q T + +q

J+ d) ,A E( +h) ET J '

whe eT = E+ + + ! Q+ S + ,

(1)

(2)

(3)

(4)

(5)

(6

(7

the pa amete s and the expected va ues a e sted n Tab e 1 and Tab e 2 espect ve y

524

lung cancer. We treat the people that smoke as an infected group, in order to show the

transmission at which the infection of smoking occurs.An individual can enter the population in two different ways. One way is proceeding

into the educated class, E, with a probability of q, or into the non-smoker class, N, withpa probability of (1-q). Individuals in all classes may develop lung cancer because of theimpact of second-hand smoke.

Individuals in the non-smoker class can become a light smoker (I1) due to lack of edu-

cation and peer pressure of smokers. Once they become a light smoker, they can not moveback to N or E. Therefore, a light smoker may become a heavy smoker (I2), or they maystop smoking temporarily (S) or permanently We assume that in order for them tobecome a heavy smoker, they must start off as a light smoker.

Once an individual becomes a heavy smoker, s/he may quit temporarily (S), perma-

nently (Q), or develop lung cancer. In the S class, the individuals can either go back tosmoking, in which we assume they start off as light smokers; or they can develop lung can-cer The Q class represents the number of individuals who stop smoking permanently.However, they have a higher probability of developing lung cancer than a non-smoker.

We let the natural death rate (percapita), p,, be the same for all the classes except for

the L class, which is considered to have higher death rate.Our mathematical model is given by the following non-linear system of ordinary differ-

ential equations.

-A

-- N( L+2)

- HN,

gg - --- P ") 3N+ 1 S S 1+ - (01+/1+51+/011, gif = 7111 _ (72 +52 +/A)-72, (3)gig* P27212 +P10'1I1 _ (5q+ MQ, (4)555 (1 -P1)f'1I1 + (1 -P2)'v2I2 - -giggigu-S, (5)

9% + 61,1 + 62,2 + ,QQ y- (H+ d)L,

`

h

t (6)y

fig? qA_5eE( 1+-72)_pE, (7)where T: E-|-N-l-I1 +I2+Q-I-S-I-L,

the parameters and their expected values are listed in Table 1 and Table 2, respectively.

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3 Analysis

3.1 Smoke-Fee Equilibrium and he Basic Reproduc ive Number

In th s sect on we ana yze the non nea d e ent a equat on mode We so ve the systeo non- nea d e ent a equat ons to nd the equ b um po nts w th the ass stance omathemat ca p og am Map e 6. F st we so ve o the smoke- ee equ b um, wh ch

A(l-q) ).J ' ' ' ' ' 'JTh oughout o th s pape we cons stant y use1an 2 wh ch a e

The Jacob an Mat x at the smoke ee equ b um s

- p000000

- (1- )(1- n) (1- ) -'lPWl

(1- PI In (1 - )+ + 8

(1 )(1 - n) (1 - )

- 2P2'2(1- P2h2

n (1 - ) + + 82-

Th s mat x has 5 negat ve e genva ues, wh ch a e

000

-( 8q+ p)0 8q

0

- p, -( 8q+ p ,- - + d) - .The est o the e genva ues a e om the sub mat x

0000

- p00

( (1 - n) (1- ) - (1 n) (1 ))l - 2

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0 00 00 00 00 0

- (p+ d) 00 - p

3 Analysis

3.1 Smoke-Free Equilibrium and the Basic Reproductive Number

In this section we analyze the non-linear differential equation model. We solve the systemof non-linear differential equations to find the equilibrium points, with the assistance of themathematical program Maple 6. First, we solve for the smoke-free equilibrium, which is:

-W 00000H pl

7 7 7 7 7 7pl

'

Throughout of this paper we consistantly use 21 an 22 which are:

E1 = Y1+U1+51+Ma 11d22 = 'Y2-I-52+#

The Jacobian Matrix at the smoke-free equilibrium is:

-u _B(1 - q) ~B(1 ~ q) 0 0 0 00 (1 _ P,,,)B(1 q) - 21 (1 - P,,,)B(1 q) 0 0 0 00 W1 -22 0 0 0 00 P101 P272 -(6.1 + H) 0 0 00 (1 ' P1)U1 (1 ' P2)')'2 0 _H 0 00 Pn5(1 _ f1)+ Bef]+ 51 Pn5(1 _ 11) Beq + 52 511 0 _(H + d) 00 "Beg _Beg 0 0 0 _/1'

This matrix has 5 negative eigenvalues, which are:

_u _(5q + H), _M _(M + fi) -M-

The rest of the eigenvalues are from the sub-matrix:

(1 _ PH-)5(1 fl) _ 21. (1 _ P 5 1fl)'Y1 -22

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Table 1: Table of Parame ers

a amet Dermi onA Rec i ent rate.

Mort ity rate (per-capita).Tr nsmission rate

i Rate in which e educated class developslung c ncer due to second h nd smoke

B I Rate for developing ung c cerh Rate for deve oping lung c ncer

C Q -Rate for deve oping ung c ncer' I Rate in which ight smokers become heavy

smokers' 2 Rate in which a heavy smoke quits

smokin Rate in which a light smoker stopssmokin

d Mort ity rate in which a p rson d es ofun c ncer

q Probabili that n incoming in vidu lenters into the educated c ass

(1-q) Probabili that a non-educated in viduente s the non smoker class

P Probabili that a non-smoker developsung c ncer(1-P Probabili that a non-smoker becomes a

ight smoker.P Probabi i of get ing ung c cer via

second ry smoke, if you go inS.(1 P ) Probabi i in which a person who stopped

smo ing tempor ri y becomes a ightsmoker

PI Probabili that a ight smoker quitssmoking perm ent y.

(1 p ) Probabi i that a ight smoker quitssmo ing tempor y

p2 Probabili that a heavy smoker quitssmo ing ermanent y(1-p 2 ) Probabili that a heavy smoker quits

smoking tempora y

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Table 1: Table of Parameters

Recruitment rate.

Mortality rate (per-capitaTransmission rate.

Rate in which the educated class developslun cancer due to second-hand smoke.I1 .Rate for developing lung cancer.I2- Rate for developinglung cancer.

Q-Rate for

developing lungcancer.

Rate in which light smokers become heavysmokers.

Rate in which aheavy smoker. quitssm oking.Rate in which alight smoker stops

i

5, a 51 _

_ _

_ sm okin _ .lun cancer.

"

enters i nt o t he ed uc at ed class.

enters the non-sm oker c lass .

lun cancer.P"

light sm oker.

seconclar smoke, if ou o in S. P

sm oking temporarily becomes a lightsm oker.

smokin ermanentll

_

(1-p1) Probability that a light sm oker quits_sm oking permanently.

sm oking temp orarily.


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L cal asympt tical stability is gua anteed p vided that the dete minant is g eate than

ze , that is, if( 1 Pn (1 q 1 2 ((1 Pn (1 q / 1> 0

which is equivalent t

Hence, we de ne:

1 > (1 (1 Pn ( + 22 1

R _ (1 (1 P n ( + 20 2 1 '

(

(9)

(10

and c nclude that if 1 then the sm ke- ee equilib ium p int is l cally asympt ticallystable. implies a sm ke- ee p pulati n. N te that can be ew itten as:

(11

Hence, the basic ep ductive numbe , , gives the numbe f the sec nda y sm ke sp duced by a typical sm ke du ing his life as a sm ke .

Obse ve that J1is the ave age am unt f time a pe s n stays in the light sm ke class( ) J2 is the ave age am unt f time that a pe s n stays in the heavy sm ke class J 2;(1 Pn is the ate in which a n nsm ke bec me a light sm ke pe unit f time and,(1 q ep esent the p babilit f a n n-educated pe s n ente ing the n n sm ke class.Hence, q Pn. ep esents the new sm ke s f m light sm ke sis the p p ti n fa light sm ke s wh bec me m heavy sm ke s while () q Pn. ep esents newsm ke s m heavy sm ke s 1 implies a n n sm ke s ciety.

L king at the , we can analyze the sensitivity f the system by bse ving th pa amete s that can d astically change the value f . The value fq, which is the p bability

f getting int he educated class, w uld have an imp tant e ect, pa ticula ly if it is cl set ne f we make it app ximately equal t ne cl se en ugh, we get the t be lessthan ne, that is, u p pulati n bec mes sm ke ee. fqis cl se t ze , then m st f thep pulati n will g t the class

Othe pa amete that g eatly a ect is , since this is the t ansmissi n ate betweencl ses. t is bvi us by l king f that if we inc ease the am unt f , will inc ease,

52

Local asymptotical stability is guaranteed provided that the determinant is greater than

zero, that is, if

((1 - Pn)5(1 - 11) 21)(-22) _ ((1 - Pn)5(1 - 1))v1 0, (8)

which is equivalent to

(1 - )(1 - PMB( +2)1 > Hence, we define:

_(1- )(1 -PMB( +2)

R0 -

,(10)

and conclude that if R0 < 1, then the smoke-free equilibrium point is locally asymptoticallystable. R0 implies a smoke-free population. Note that R0 can be rewritten as:

Ro: (1-q)g1-un)F ()(2-oo (11)Hence, the basic reproductive number, Rg, gives the number of the secondary smokers

produced by a typical smoker during his life as a smoker.

Observe that i is the average amount of time a person stays in the light smoker class(I1); 5 is the average amount of time that a person stays in the heavy smoker class (I2);(1 - P,,_),Bs the rate in which a nonsmoker become a light smoker per unit of time; and,(1 - q) represent the probability of a non-educated person entering the non-smoker class.Hence, Q-'llxzgmepresents the new smokers from light smokers; 35%s the proportion ofa light smokers who become from heavy smokers; while ( ) represents newsmokers from heavy smokers. R0 < 1 implies a non-smoker society.

Looking at the Rg, we can analyze the sensitivity of the system by observing the- para-meters that can drastically change the value of Rg. The value of q, which is the probabilityof getting into the educated class, would have an important effect, particularly if it is closerto one. If we make it approximately equal to one or close enough, we get the R0 to be lessthan one, that is, our population becomes smoke-free. If q is close to zero, then most of thepopulation will go to the N class.

Other parameter that greatly affect R0 is ,3, since this is the transmission rate betweenclasses. It is obvious by looking of R0 that if we increase the amount of ,B, R0 will increase,

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educing the c ntact with sm ke s

The the pa amete that seems imp tant isP h weve P is ve y small n the casefRo 1, inc easingP leads t the inc ease f pe ple devel ping lung cance As t gets

la ge , the numbe f lung cance cases g es t ze .

3 2 Endemic Equilibria

The p evi us subsecti n sh ws that ifRo< 1, then the sm ke -f ee equilib ium is l callyasympt tically stable, meaning eventually that the e will be n n-sm ke s T l k at thecase Ro> 1 we s lve the f ll ing algeb aic equati ns in de t nd ut whethe n ta p sitive equi ib ium is p ssible

0 = (1 A- {N +h) N,

000

0

0

0

((1- P ) N + (1 P ) S)(I + I ) ( 8 )T + / 1 , ( + 8 + )I ,

I+ W (8 q + )Q,{S( + I )1 ) + 1 h T j S,

(P N+ P S + E)( + I ) 8 I 8 8 QT + + +q + d L,

A E( + I ) E T whe eT = E+ N+ + I + Q+ S + L

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(12

(13(14(15)

(16

(1 7 )

(1 8 )

reducing the contact with smokers.

The other parameter that seems important is Pn; however Pn, is very small. In the caseof R0 < 1, increasing Pn leads to the increase of people developing lung cancer. As t getslarger, the number of lung cancer cases goes to zero.

3.2 Endemic Equilibria

The previous subsection shows that if R0 < 1, then the smoker-free equilibrium is locallyasymptotically stable, meaning eventually that there will be non-smokers. To look at thecase R0 > 1, we solve the following algebraic equations in order to find out whether or nota positive equilibrium is possible.

0 = (1-q)A- -MN, (12)0 = -------- PMN + QTP S +


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If we let:

Then using (14 and (15 , we can represent J2 and Q in terms of h , respectivly,

(19) (20)

Multiplying equation (12 by S and equation (16 by N, e nd a lin ar relationshipbetween S and N, namely

S- OfN

- (1- q)A

sing equations (13 and (21 , we can solve for !,

N ( 1 ) ((1- q)A)T = (1+ ) )(ul) ) < >

Using equations (12) and (22), we can solve for N,

(1 - q)A 21 I1(1 - q)AN = --i - - _-_ . 23,, (,,)(


11/28

And using eq ations (21) and (22 we solve fo ,

(1 + ) 1 then :8 + 8 + 8 B+ t d(1+ )+ (f) < yd , A(1 ) and

+ g1(1 ) > 0o

This shows that the e exists at least an endemic equilib ium solution

We have shown the existence of an endemic Which means that the smoking o ulations a e esent. t is ha d to dete mine its stability since we do not have the ex licitfo mula fo the endemic. Ou nume ical simulations su o t ou conjectu e that this endemic is locally asym totically stable.

531

And using equations (21) and (22), we solve for g,

S 21119-_ = -_-_- 27T nu + A>>

Adding equations (12) through (18), allows us to solve for L in terms of T,

L = 9%-11. (28)

Substituting equations (19) through (28) into (17), we have an equilibrium equation for I1:

6

= P,,211-q)AI1 PS21zgI1)2 . AI1FU1) (I1) + (I1) 1' E1


12/28

4 Estimation of Parameters and Numerical Solutions

4.1 Es ima ed Parame ers

We st estimated the pa amete s by available data, then used Matlab t nume ically s lvethe system f dina y di e ential equati ns (1) (7)

stimated Values By Data

- m tality ate, is estimated by the ave age life spanP & P 2- p bability given by data that 2 5% f sm ke s quit pe manently[8]/ - p bability given by the data that 60% f sm ke s a e in the2class [17]

given by the data that individuals in the class quit at a highe ate [18]/2 - given by data that individuals in the h class quit at a l we ate [18]d- m tality ate, given by data that pe ple wh devel p lung cance have a m tality ate

f 7 yea s less thanJ 11]

Assumed Values

8 assuming that 15 ut f 1000 individuals devel p lung cance82 - assuming that 30 ut f 1000 2individuals devel p lung cance8q assuming that 5 ut f 1000 Q individuals develp lung cance

P - assuming that the p bability f devel ping lung cance due t p evi us sm king sec nda y sm king is l wPn- assuming that the p bability f devel ping lung cance due t sec nda y sm ke is

ve y l w - assuming that the t ansmissi n ate between the p pulati n and and 2 is ve yl wA- assuming that the e is a c nstant p pulati n that ente s u m del; it has t be g eatethan 1

532

4 Estimation of Parameters and Numerical Solutions

4.1 Estimated Parameters

We first estimated the parameters by available data, then used Matlab to numerically solvethe system of ordinary differential equations (1)-(7).

Estimated Values By Data

p.- mortality rate, is estimated by the average life span

P1 & P2 - probability given by data that 2.5% of smokers quit permanently[8].71 - probability given by the data that 60% of smokers are in the I2 class [17].01 - given by the data that individuals in the I1 class quit at a higher rate [18].72 - given by data that individuals in the I2 class quit at a lower rate [18].ci - mortality rate, given by data that people who develop lung cancer have a mortality rateof 7 years less than p, [11].

Assumed Values

61 - assuming that 15 out of 1000 I1 individuals develop lung cancer.62 - assuming that 30 out of 1000 I2 individuals develop lung cancer.

6,1 assuming that 5 out of 1000 Q individuals develp lung cancer.

Ps-

assuming that the probability of developing lung cancer due to previous smoking orsecondary smoking is low.Pn - assuming that the probability of developing lung cancer due to secondary smoke isvery low.

,Be assuming that the transmission rate between the E population and I1 and I2 is verylow.

A - assuming that there is a constant population that enters our model; it has to be greaterthan 1.

532


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Table 2: Es ima ion of Parame ers

P amet s Values1 4 +

1 0.014 *p 2

"'

Pn 0 00001+1-Pn 0.99999+

o10 015*

1 0 60*p1 0 025*-p 0.975*p2 0 025*

l-p2) 0 975*c1 0 50*

2 0 25*Ps 0 0001+

1-Ps) 0.9999+

0 005*o 0.03*d 0.016*q 0 25

( 1-q 0 .7y

Pe 0 00001

The data was btained f m di e ent ganizati ns such at the CDC (Cente f DiseaseC nt l) Ame ican Lung Cance S ciety Nati nal Cance nstitute (NC ) and t e n np t and g ve nment agencies.

* Estimated by the use of data + Fee Parameters= values ass med in order to try to makerea istic "-" Values that be random y changed to see the behavior of o

533

Table 2: Estimation of Parameters

I Parameters I Values II A I 14+ II

I

I 0.014 *

I I I I 2 I Pu I 0_00001+

I

I g1-1=Q I 0_99999+ I

I 51 I 0_015* II I I 0150* II pl I 0.025* II 1- 1 I 0925* II E2 I 0025* II 1- 2 I 0925* II U1

I

I 0.50*I 5 I 025*I Ps I 00001+I Q1-Psg I 0_9999+ II ag I 0005* II 52 I 003* II a I 0016* 1 II q I U 2 5 II (1-q) I U 3 5 II @, I 000001+

~

I

- The data was obtained from different organizations such at the CDC (Center for Disease- Control), American Lung Cancer Society, National Cancer Institute (NCI), and other non-

profit and government agencies.

* Estimated by the use of data, + Hee Parameters = values assumed in order to try to make the model

realistic, Values that will be randomly changed to see the behavior of our model.

533


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4 2 Numerical Solutions

We simulated our model in order to obtain the cases when < 1 and > 1 esultsshow that when < 1, the population of susceptibles to the infection would eventually dieout. This agrees with our anaylsis in section3. The results of our simulations show that when > 1 an endemic stable state is established The two simulations shown describegraphically what we have explained on the behavior of .

Figure 2: Graph shows the simulation for 1. Resultsshow that when R0 < 1, the population of susceptibles to the infection would eventually dieout. This agrees with our anaylsis in section 3. The results of our simulations show thatwhen R0 > 1, an endemic stable state is established. The two simulations shown describe

graphically what we have explained on the behavior of RO.

W W % % 7%& 7 ?W 'ZCWWZ W , V~ Z

/

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'

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w~=~g >v% > g yWy /W w ~ QW# 5 l /,%X,~ -sm , V///// %%j%$%/@ > ig, /ffwf ,, / xzfp//Q ~y%%//%,% f rf W'/5 Ms

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Cv \4% . _ m Q~4,/,-f..;_ ,_ 4 , / , -f..;_ ,_

3 2% , , < we =

= _.-; f 52

gpm < % > . W/ / /#J /f / / / W // / ////f/;/d/ f / / / / f , /1 /` A/ / f // ff/W M

X

wif/ swx m? 5 @ w> _Q g _ # n % 4 4%m? f /5 @ w> Q;/gi//_#fn/% // /44%

Figure2:

Graphshows the simulation for

R0< 1. The smoke-free

equilibriumis stab

534

le


15/28

Figu e3: G aph shows the simulation fo > 1 The endemic equilib ium is stable

535

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f

fI \\ \

, fQ? _ f~_ ~ Jaw ~ $11 ;_

v

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f_ , __= >_ _ _ _ _ 1 N _ _ /1 ,, N \

~ _ _ N r Q 9/ ,W '_ _ A af _ y\ 52 , . , . . , , _ . , , ., . _ . ______ _ __ _ __ _ _._ ,- __ i =:

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_ ~ - K _ _ _ __ _ _ _ ; _ " _ V ; -_ " 51 -_ _

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//~/ y, /,%/ ,/4%/4///N ~ ~ '~ __~ < _ _ 2 ,/M2 > 3 , ~ ._ M5//,M ' = A A i A ~

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_________ ?~ ~ _#A > N 2 >% > >C MQ ? W~ ~ _#A { ' > N > % > > C MQ W'z' "2 > % > , / ff f . ->Cf'. .` MQ ? W

f v/ M %/,// ,f /, Mwgwr . _f2~w%r/>- =>W% ~ -_ M ff/X/%~%f%}w~._~ f >fwe W 5% a a ? W x Q M V ___ 9 _ _ d W @ 2Q MW_M Wm > 7 ~ M Q _ X _ affair? , W fx QM /V ___ 9__ fd W@ /, /2 Q f MW _M Wm/>Wm/> ;/ -7 ~ f M Q/_ ' X/ _ f f- 2_ f % _ , M , Mf v -f _ / 'fi %< Q/ /f~M@ ' af, %@ =2 v ~ g w aea /_ /39,2 f "

Figure 3: Graph shows the simulation for R0 > 1. The endemic equilibrium is stable.


16/28

5 Results of our Deterministic Model

To obtain a realistic representati n on the e ects of smoking on a given population weformulated a deterministic model Simulations of our model were conducted using parameters and estimations from real life studies We mainly varied two parameters to study thesensitivity of our results due to smoking within a population; these were the probabilitythat incoming individuals would enter our educated class(E); and { the transmission rate

We produce several simulations showing the e ects of smoking Based on the analysisof in the previous section and our endemic-equilibrium point we established that theepidemic of smoking will establish itself provided that > 1 Otherwise smoking willdie out in our population When > 1 we got = 0 25 and = 2 which allowed us toobtain = 4 04(Figure 3) By looking at the seven classes we can see that when > 1then all seven classes will establish themselves If we make = 0 85 and = 2 then =0 808(Figure 2) In this simulation the population of smokers and ex-smokers eventuallydie out The sum of the non-smoker class and the educated class E reaches the approximate -dependent equilibrium values This simulation shows that for the given valuesthere will be no smoke-induced population

When and 2 consists of high values that is when light and heavy smokers quitpermanently and at a faster rate then likely smokers becoming smokers then shouldeventually be less than 1 If individuals from the light-smoking class quit at a higher ratethan individuals that become heavy smokers then we will be left with a smaller populationof smokers in general If we make > 1 but close to 1 in the simulations then the smokers (J and h) will have a small population but still have a very large portion of the totalpopulation in the temporary quitters (S) meaning that individuals are still susceptible tosmoking again If we make high enough so that < 1 then the total population willbe concentrated in the likely-smoker class ( and the educated population (E)

Simulations were 2 is varied can a ect the values of signi cantly if we let2 then the equation could be less than 1 This e ects the equation by eliminate the contribution of heavy smokers but since we still have the contribution of light-smokers wecan not necessarily say that the equation for will be lower than 1(Figure 6 and7

When we ran simulations varying we found out that if we made high enough then

the smoking populations would establish themselves and the prevalence of smokers (hgrows Also when is a high value the smokers will convert faster the likely smoker ; thenour and the risk of lung cancer increases(Figure 4) If we decrease to a point whereit is close enough to zero then less individuals will start smoking due to peer pressure

ventually could be less than 1(Figure 5)

536

5 Results of our Deterministic Model

To obtain a realistic representation on the effects of smoking on a given population, weformulated a deterministic model. Simulations of our model were conducted using parame-ters and est imat ions from real life studies. We mainly varied two parameters to study thesensitivity of our results due to smoking within a population; these were q, the probabilitythat incoming individuals would enter our educated class(E); and H, the transmission rate.

We produce several simulations showing the effects of smoking. Based on the analysisof R0 in the previous section and our endemic-equilibrium point, we established that theepidemic of smoking wil l es tabli sh itself, provided that R0 > 1. Otherwise, smoking willdie out in our population. VVhen R0 > 1, we got q = 0.25 and ,3 = 2, wl1ich allowed us toobtain R5 = 4.04(Figure3). By looking at the seven classes, we can see that when R0 > 1,then all seven classes will establish themselves. If we make q = 0.85 and ,3 = 2, then R0 =

0.808(Figure2). In this simulation, the population of smokers and ex-smokers eventuallydie out . The sum of the non-smoker class, N, and the educated class, E, reaches the ap-proximate q-dependent equilibrirun values. This simulation shows that for the given values,there will be no smoke-induced population. V

When 01 and 72 consists of l1igh values, that is when light and heavy smokers quitpermanently and at a faster rate then likely smokers becoming smokers, then R0 shouldeventually be less than 1. If individuals from the light-smoking class quit at a higher ratethan individuals that become heavy smokers, then we will be left with a smaller populationof smokers in general. If we make R0 > 1, but close to 1 in the simulations, then the smok-ers (I1 and I2) will have a small population, but still have a very large portion of the totalpopulation in the temporary quitters (S), meaning t hat individual s are still susceptible tosmoking again. If we make 01 high enough so that R0 < 1, then the total population willbe concentrated in the likely-smoker class (N) and the educated population

Simulations were 72 i s va ri ed can affect the values of R0 significantly, if we let 72 -> oo,then the R0 equation could be less than 1. This effects the equation by eliminate the con-tribution of heavy smokers, but, since we still have the contribution of light-smokers, wecan not necessarily say that the equation for R0 will be lower than 1(Figure 6 and 7).

VVhen we ran simulations varying ,3, we found out that if we made ,3 high enough, then

the smoking populations would establish themselves and the prevalence of smokers grows. Also, when ,3 is a l1ighvalue, the smokers will convert faster the likely smokers; then,our R0 and the risk of lung cancer increases(Figure4). If we decrease ,3 to a point whereit is close enough to zero, then less individuals will start smoking due to peer p ressur e.Eventually, R0 could be less than 1(Figure5).

536


17/28

he pa amete s that we need to change in o de to educe the p eva ence o smokingand ung cance a eq OI,and 2 Idea one must concent ate on the most sensitivepa amete s which a eq and . We sa that because shown b the data o the pa amete sOl and 2,it is much ha de to a ect individua s since the e is a high pe centage o smoke stht wou d ike to quit howeve on 2 5 pe cent o those do it.

om ou simu ations we obse ved thatPn( . 1) does not have a big e ect at thepopu ation eve o ung cance FoPnto have a signi cant change inRothe va ue wou dhave to change d amatica ; howeve the data indicates the opposite.

6 ConclusionIn ou mode the use o non- inea di e entia equation was c ucia to stud the d nam

ics o ung cance at the popu ation eve caused b smoking and second and smoke Bbui ding this popu ation we ound an impo tant aspect o mathematica bio ogRo whichcont o s the d namics o ou mode .

On ugust 5 2 an a tic e ased on ung cance was pu ished in theIthaca ou al, which came om aBritish ou a of Medicine. his a tic e stated that i we dec ease theeducation on non-smoke s and concent ate on smo e s than the p eva ence o a smokedeve oping ung cance is ow Howeve using ou mode a ong with ou simu ations

a gue that when the e is an inc ease o the numbe individua s that a e educated thanthei p obabi it o becoming smoke s dec eases and eventua we wi have a smoke- epopu ation Ro< 1) Howeve iRo> 1 then ou popu ation o ight and heav smoke s wi estab ish themse ves B changing we ound that it had a signi cant e ect on thenumbe o individua s that we e in ected Howeve the g eatest di e ence ocu ed whe ethe va ue oq changed and when we educated a high numbe o individua s in ou popu ation.

In conc usion the best wa to owe the numbe o smoke s and individua s who deve opung cance is b inc easing the numbe o individua s that a e we -educated on the e ect

o smoking

7 Future Workven though we conside ed the tota popu ation o smoke s in ou mode we can add

to ou conditions a numbe o va iations. An age st uctu e and ethnicit dive si cationcan be added that wi stud and ana ze the p eva ence o ung cance his is due to

537

The parameters that we need to change in order to reduce the prevalence of smokingand lung cancer are q, ,U, 01, and 72. Ideally, one must concentrate on the most sensitiveparameters, wl1ich are q and ,U. We say that because shown by the data of the parameters01, and 72, it is much harder to affect individuals since there is a high percentage of smokerstht would like to quit, however, only 2.5 percent of those do it.

From our simulations, we observed that Pn (0.00001) does not have a big effect at thepopulation level of lung cancer. For Pn to have a significant change in RO, the value wouldhave to change dramatically; however, the data indicates the opposite.

6 Conclusion

In our model the use of non-linear differential equation was crucial to study the dynam-ics of lung cancer at the population level caused by smoking and second~hand smoke. Bybuilding tl1is population, we found an important aspect of mathematical biology, RQ, wl1ichcontrols the dynamics of our model.

On August 5, 2000, an article based on lung cancer was pulished in the Ithaca Journal,wl1ich came from a British Journal of Medicine. Thisarticle stated that if we decrease theeducation on non-smokers and concentrate on smokers, than the prevalence of a smoker

developing lung cancer is low. However,using our model along with our simulations, we

argue that when there isan

increase of the number individuals thatare

educated, thantheir probability of becoming smokers decreases and eventually we will have a smoke-freepopulation (RO< 1). However, if R0 > 1, then our population of light and heavy smokerswill establish themselves. By changing ,3, we found that it had a significant effect on thenumber of individuals that were infected. However, the greatest difference ocurred wherethe value of q changed and when we educated a l1igh number of individuals in our population.

In conclusion, the best way to lower the number of smokers and individuals who developlung cancer is by increasing the number of individuals that are well-educated on the effectof smoking.

7 Future WorkEven though we considered the total population of smokers in our model, we can add

to our conditions a number of variations. An age structure and ethnicity diversificationcan be added that will study and analyze the prevalence of lung cancer. This is due to

537


18/28

the fa t that smoking and its onse uen es a e di e ent if we take into a ount age, sex,

and ethni ity. A so, stu ying a mo e ea isti mode that dea s with the impa t of smokingand the behavio it has on the p eva en e of ung an e . One examp e is studying e tainb ands of iga ettes. A so, we ou d bui d a mode that wou d in o po ate the e ove y atefo ung an e , meaning to add anothe ass, a e ove y ass(R), we e the popu ationof the ung an e ass( L ou d go. Looking into the deve opment of ung an e s, we

ou d take into onside ation eating a mode that ooks at the e e ts of two types of ungan e s,sin e on e an individua e ove s f om un an e the st time ,Type 1, then they

have a han e of getting a new type of ung an e , ype 2. Fina y, we ou d fo wa dou esea h by ooking at the e e ts of edu ing the impa t of pee p essu e on ike y newsmoke s, su h as u ent smoke s and the mass media.

8 AcknowledgementThis study was suppo ted by the fo owing institutions and g ants: Nationa S ien e

Foundation (NSF G ant DMS-997 919); Nationa Se u ity Agen y (NSA G ants MDA-904-00-1-0006); P esidentia Fa u ty Fe owship Awa d (NSF G ant D B) and the P esidentiaMento ing Awa d (NSF G ant H D) to Ca os Casti o-Ch vez and the o e of the p ovostof Co ne Unive sity; Inte Te hno ogy fo du ation 2000 uipment G ant.

Thanks to ou adviso s Baojun Song and Ca os Casti o-Ch vez, fo you he p andideas. A so we want to thanks Ca os He n ndez, Steve i kus, B isa Ney San hez andthe MTBI p og am fo a owing us to have the oppo tunity of doing this esea h. Fina ythanks to MTBI students fo you mo a suppo .

eferences

[1] B aue , ed.A Mo e fo an SI Disease in Age St uctu e opu ation

2 B aue , ed and Ca os Casti o-Chavez.Metho s an incip es in Mathematica Bioogy with App ications to Epi emio ogy opu ation Bio ogy an Resou ce Management(2000) pp. 271- 94.

[3 Cook, A an .The New Cance Sou cebookOmnig aphi s (1996) pp. 6-10, 41-48,251-282.

4 Gaude man, James W and John L. Mo ison.Evi ence fo Age Genetic Re ative Risksin Lung Cance Ame i an Jou na of pidemio ogy. Vo ume 151, Numbe 1, Janua y1, 2000 pp.41-49.

5 Heeste beek, Hans.R pp.3-115

538

the fact that smoking and its consequences are different if we take into account age, sex,

and ethnicity. Also, studying a more realistic model that deals with the impact of smokingand the behavior it has on the prevalence of lung cancer. One example is studying certainbrands of cigarettes. Also, we could build a model that would incorporate the recovery ratesfor lung cancer, meaning to add another class, a recovery class (R), were the populationof the lung cancer class (L) could go. Looking into the development of lung cancers, wecould take into consideration creating a model that looks at the effects of two types of lungcancers,since once an individual recovers from lung cancer the first time ,Type 1, then theyhave a chance of getting a new type of lung cancer, Type 2. Finally, we could forwardour research by looking at the effects of reducing the impact of p ee r pr ess ur e on likely newsmokers, such as current smokers and the mass media.

8 AcknowledgementThis study was supported by the following institutions and grants: National Science

Foundation (NSF Grant DMS-9977919); ational Security Agency (NSA Grants MDA-904-00-1-0006); residential Faculty Fellowship Award (NSF Grant DEB) and the Pres ident ialMentoring Award (NSF Grant HRD) to Carlos Castillo-Chavez and the office of the provostof Cornell University; Intel Technology for Education 2000 Equipment Grant.

Thanks to our advisors Baojun Song and Carlos Castillo-Chavez, for your help andideas. Also we want to thanks Carlos Hernandez, Steve Wirkus, Brisa Ney Sanchez andthe MTBI program for allowing us to have the opportunity of doing t l1 is research. Finallythanks to MTBI students for your moral support.

References

[1] Brauer, Fred. A Model for an SI Disease in Age-Structured Population.

[2] Brauer, Fred and Carlos Castillo-Chavez. Methods and Principles in Mathematical Biol-ogy with Applications to Epidemiology Population Biology, and Resource Management.(2000) pp. 271-294.

[3] Cook, Allan R. The New [Cancerourcebook. Onmigraphics (1996) pp. 6-10, 41-48,251-282.

[4] Gauderman,James W. and John L. Morrison. Evidence

for AgeGenetic Relative Risks

in Lung Cancer. American Journal of Epidemiology. Volume 151, Number 1, January1, 2000 pp.41-49.

[5] Heesterbeek, Hans. Rg. pp.3-115.

538


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5 3 9

United;States Department of Health & Human Services. SEER Cancer Statistics Re-

view, 1973-1994. Ed. Lynn A. Gloeckler Ries, M.S. pp 15-105.

Zhang, Tongxiao. Role of Peer Pressure of Smoking.' Smoking as an Epidemic. Thesis(1999).

http://www.aacr.org/5000/5001/5110g.htm1.Lung Cancer. March 1998. pp. 1-5

http://www.cancer.org/statistics/cff2000/tobacco.htm1.000 Facts Figures: TobaccoUse pp. 3-4.

http://cancernet_nci.nih.gov/wyntk_pubs/lung.htm#5 Lung Cancer: Who s at Risk?pp. 1-2.

http://www.1ungcheck.com/info/stats.htm.Lung Check-Statistical Facts. pp. 1-3.

http://rex.nci.nig.gov/NCI_Pub_Interface/raterisk/risks70.html. Risk Factors.' Ciga-rette Smoking as a Cause of Cancer pp.1-2.

http: / /www.cdc.gov/tobaccotab_3.htm. CDC S TIPS- Current Smoking StatusAmong Adults. (1965-95) pp.1-3.

http://www.cdc.gov/tobacco/morta1i.htm. CDC S TIPS - Cigarette Smoking-RelatedMortality.

http://wwW.cdc.goV/tobacco/initfact.htm. CDC S TIPS - Incidence of Initiation ofCigarette Smoking Among US Teens-Fact Sheet. pp.1-2.

http://www.cdc.goV/tobacco/adstat1.htm. CDC S TIPS - Adult Prevalence Data.

http://www.cdc.gov/tobacco/adstat3.htm. CDC S TIPS - Adult Prevalence Data.

http://www.cdc.goV/tobacco/adstat4.htm. CDC S TIPS - Adult Former SmokerPrevalence Data.

539


20/28

9 APPENDIX

n o de to wo k on the simu ations we needed to c eate a p og am in MATLA that was composed basica of the di e entia equations the data we found and t e p otting othe g aphs

n MATLA we needed to bui d two p og ams in o de to un the simu ationsP og am1tspan=[ 1000]xO= [500 200 200 200 200 200 200]y0=[250 100 100 100 100 100100]z0=[750 350 350 350 350 350 350q= 0 25f = 0 014( - 2 - '81= 0 01582= 0 038q= 001A= 14P =0 0001Pn= 0 00001

P = 0 025 P2= 0 025

= 0 2 = 025

d = 001fe = 0001

[t x] = ode45(' u g',tsp ,x, [] p 8 82 AFs Fn PbP2 1 a" 2 8q d e);[s y] = ode45(' u g',tsp , y [] p 8 82 A F Fn PbP2 at 2 8 q d qfe);[r z] = ode45(' u g'tsp ,z []p 81 82 A FFn PbP2 a1 28 q d qfe);R = (1 q* ((((1 Pn * )/(! + 8 + a1+ p))+

+ ((! * (1 Pn * )/(( 1+ 81+ a + p) *( '2+ 82+ p))))gu e

subp ot(231)ho d on

p ot(t x(: 1) 'c')p ot(s (: 1) 'b')p ot( z(: 1) 'm')tit e([' o= ' num2st ( ) )

x abe ('time')

540

9 APPENDIX

In order to work on the simulatlons, we needed to create a program m MATLAB thatwas composed basically of the differentlal equat1ons, the data we found, and the plott1ng ofthe graphs.

In MATLAB, we needed to bu1ld two programs m order to run the s1mulat1onsProgram 1

tspan=[0 1000];[500 200 200 200 200 200 200][250 100 100 100 100 100 100][750 350 350 350 350 350 350]

:1:0=

2/0=z0=

q = 0.25;/.L = 0.014;H = 2;61 = 0.015;62 = 0.03;6q = 0.01;A = 14;Ps = 0.0001;P" = 0.00001;P1 = 0.025;p2 = 0025;71 = 0-6;'y2 = 0.25;d = 0.016;,3e =

[t,a:] = o d e 4 5 ( u n g s pa n r 1 r 0l# 5 51 62 A P Pu P1 P2 'Y1 U1 'Y2 5q d Q 5[s,y] = ode45('lung',tspa,n,y0, ],y, ,H 61 62 A P, P 1 P2 'Y1 U1 Y 25q d fl

I I P 6 d[fr-,z] ode45(lung,tspa,n,z0,[],y, H 61 62 A Ps ,,, P1 P2 'Y1 01 Y2 q 11R0= (1-


21/28

yla el('# Individuals ( '

hold osu plot(232)hold onplot(t,x(:,2), 'r')plot(r,z(:,2) 'g')plot(s,y(:,2), ' ')title( 'q = ',num2str( ) );

xla el('time') yla el('# Individuals (I ) )hold osu plot(233)

hold onplot(t,x(:,3), r')plot(s,y(:,3),' ')plot(r,z(:,3), 'g') xla el('time') yla el('# Individuals (I ))title([' = ',num2str( )]);hold osu plot(236)hold onplot(t,x(:,4), 'g')plot(s,y(:,4),'m')plot(r,z(:,4),'y')

xla el('time') yla el('# Individuals (Q)')hol osu plot(235)hold onplot( t,x(:,5),'g')plot(s,y(:,5), 'm')plot(r,z( :,5),' ') xla el('time') yla el('# Individuals ( ) )hold osu plot(234)hold onplot( t,x(:,6),'g ')plot(s,y(:,6), 'y. ')

541

y a b e ( #ndividuals (Ny)hold Offsubplot(232)hold on

P 1 0 G ( v X ( = 2 ) ) P 1 0 G ( 1 2 ( = 2 ) g ) p 1 0 v ( S y ( = 2 ) b ) e ( q n u m 2 s r ( q ) ) x a b e ( m e ) y a b e ( #ndividuals ( 1 ) )hold off

subplot(233)

hold on p 1 0 v ( v X ( = 3 ) ) p 1 0 v ( S y ( = 3 ) b ) P 1 0 G ( 1 2 ( = 3 ) g ) '

x a b e ( m e ) I

y a b e ( #ndividuals ( 2 ) ) 1 e ( H n u m 2 s < @ > 1 >hold off

subplot(236)hold on

p 1 0 v ( v X ( = 4 ) g )

p 1 0 ( S y ( = 4 ) m ) P 1 0 G ( 1 2 ( = 4 ) > " ) x a b e ( m e ) y a b e ( #ndividuals ( Q ) )hold off

subplot(235)hold on

p 1 0 v ( v X ( = 5 ) g ) p 1 0 ( S y ( = 5 ) m ) P 1 0 G ( 1 2 ( = 5 ) b ) x a b e ( m e )

y a b e ( #ndividuals

( S ) )hold offsubplot(234)hold on

p 1 0 v ( v X ( = 6 ) g - ) p 1 0 ( S y ( = 6 ) y - )


22/28

p ot(r z(:,6),'k. ')

p ot(t x(:7),'r')p ot(s y(:,7),'m')p ot(r z(:,7),'b') x abe ('time') y abe ('# Individuals ( &.L ) ' )ho d o

Program 2

functiondx= u g(t, x, f g,j, /, 8 1, 8 2,A,F ,Fn, , , , , , 8 q, d, ,

N = x(1);I = x(2); h = x(3);Q = x(4);S = x(5); L= x(6);E = x(7);T = N+I l + 2+ Q+S+ L+ E;e 1 = (1- ) *A- * * ( + )/ - * ;e 2= (1- Pn)*/ * N * ( + )/ + (1- Ps* * S * ( +)/ - (a + + 8 + * ;e 3 = * - ( +8 + * h;e 4 = * 2* h+ * a * - ( 8 q + *Q;e 5 = (1- ) * a * +(1 )* * - * S * ( + h)/ - * S;e 6 P * * * ( + h)/ + Ps* * S * ( + )/ + *E * ( + h)/ + 8 * +8 * h+ 8qe 7 = *A- *E * (I l + 2)/ - *E;

dx = [e 1;e 2;e 3;e 4;e 5;e 6;e 7];

542

p1o z 6 k

P10 G G X = 7 1"1H0tG%y(a7)Hf)P10 G 1 2 = 7 b x1abe1 me y1abe1 #ndividuals (E & L hold off

Program 2

function

d E= Lg $lag;/'51 /616]-1621A1PS1PM-1p11p21'Y110'11'Y216q1d1(11166)

N = a:(1);I1 = a:(2); I2 = :1:(3);Q a:(4); S = a:(5);L = a:(6);E = a:(7);T=N+I1+I2+Q+S+L-I-E;eq1= (1-q)=|-,u,=| -,u,=|eq2eq3eq4eq5eq6eq7

da:

=(1-Pn)*,6*N*(I1+I2)/T-l-(1-Ps)=|='Y1 *I1-('Y2+52+U)*-72;=P2*'Y2*I2+P1*U1*I1-(5q+U)*Q 9=(1-P1)*U1*I1+(1"'P2)*'Y2*I2-5*S*(I1+-72)/7'U*S=P,,=|=q=|

[@q1;eq2;eq3;eq4;eq5;eq6;eq7k

542


23/28

In this section we will show some othe simulations that will explain the behavio o

ou model if we va y some othe pa amete s that we e supposed to changed the value oRsigni cantly

Figu e This g aph is with = .

5 3

In this section, we will show some other simulations that will explain the behavior of

our model if we Vary some other parameters that were supposed to changed the Value of R0significantly.

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24/28

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544


25/28

Figu e6 This g aph is with"2 = 0 25

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26/28

Figu e This g aph is with/2 = 3

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27/28

Figure 8: This graph is th 1= 0 5

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Figure 8: This graph is with U1 = 0.5.

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' _Figure 9: This graph is with U1 = 4.

Mathematical Model for Lung Cancer the Effects of Second-Hand Smoke and Education, A

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