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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1
45 "Transmission of a Disease Within a Population" }}{PARA 257 "" 0 "
" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 9 "Jim Herod" }}{PARA
259 "" 0 "" {TEXT -1 21 "School of Mathematics" }}{PARA 260 "" 0 ""
{TEXT -1 12 "Georgia Tech" }}{PARA 261 "" 0 "" {TEXT -1 21 "herod@math
.gatech.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 367 " The model in this work
sheet is commonly used in introductory discussions of mathematical bio
logy. There are two suppositions. The first is that an infectious dise
ase has infected a group of people in a community. The second supposit
ion is that the people in the community can be divided into a suscept
ible class, an infected class, and a recovered class. The" }{TEXT 258
18 " susceptible class" }{TEXT -1 111 " consists of those who are not \+
infected, but who are capable of catching the disease and becoming inf
ected. The" }{TEXT 259 15 " infected class" }{TEXT -1 88 " consists of
the individuals who are capable of transmitting the disease to others
. The " }{TEXT 260 15 "recovered class" }{TEXT -1 226 " consists of th
ose who have had the disease, but are no longer infectious. The dyna
mics begins with the small group of individuals who are infected with \+
the contagious disease coming into contact with the larger population.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 88 " \+
A mathematical representation of this problem is often construct
ed as follows." }}{PARA 15 "" 0 "" {TEXT -1 170 "The susceptible class
is decreased in proportion to the rate of encounter of members of thi
s class with the infected class. The constant of proportionality is ca
lled the " }{TEXT 257 15 "infectious rate" }{TEXT -1 1 "." }}{PARA 15
"" 0 "" {TEXT -1 203 "The infected class increases at the same rate as
the susceptible class decreases. The infected class also decreases as
members move into the recovered class. The constant of proportionalit
y is called the" }{TEXT 261 13 " recovery rat" }{TEXT -1 2 "e." }}
{PARA 15 "" 0 "" {TEXT -1 109 "The recovered class grows as members mo
ve from the infected class. (In some models, this class is called the \+
" }{TEXT 256 7 "removed" }{TEXT -1 108 " class.) It is assumed in this
first model that the individuals in this class are now immune to the \+
disease." }}{PARA 0 "" 0 "" {TEXT -1 8 " " }}{PARA 0 "" 0 ""
{TEXT -1 147 " We cannot get analytic solutions for the resultin
g system of equations. Rather, we use Maple's numerical procedures for
obtaining solutions." }}{PARA 0 "" 0 "" {TEXT -1 501 " An obser
vation to make with this model is that an illness can cease to move th
rough a community, not only because all the community has become infec
ted, but also because of the dissaparance of the infected community. O
bservations such as this can be made by performing \"what if ...\" exp
eriments. We perform one analysis, and recommend a different collectio
n of assumptions to obtain a significantly different scenario. A littl
e imagination will suggest other situations that a reader might try."
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1
0 11 "r:=1; a:=1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "sol:=
dsolve(\{diff(SU(t),t)=-r*SU(t)*IN(t),\n diff(IN(t),t)=r*S
U(t)*IN(t)-a*IN(t),\n diff(R(t),t)=a*IN(t),\n SU(0)=2.
8, IN(0)=0.2, R(0)=0\}, \{SU(t),IN(t),R(t)\},\n type=numeric,outpu
t=listprocedure);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f:=sub
s(sol,SU(t)); g:=subs(sol,IN(t)); h:=subs(sol,R(t));" }}}{EXCHG {PARA
0 "> " 0 "" {MPLTEXT 1 0 32 "plot(\{f,g,h\},0..20,color=BLACK);" }}}
{PARA 0 "" 0 "" {TEXT -1 32 "We can predict the steady state." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "solve(\{-r*SU(t)*IN(t)=0,r*S
U(t)*IN(t)-a*IN(t)=0,a*IN(t)=0\},\n \{SU(t),IN(t),R(t)\});" }}}
{PARA 0 "" 0 "" {TEXT -1 74 " Different situations arise by cha
nging the parameters. Try having " }{XPPEDIT 18 0 "a" "6#%\"aG" }
{TEXT -1 19 " twice as large as " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT
-1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "r:=1; a:=2;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 224 "sol:=dsolve(\{diff(SU(t),t)
=-r*SU(t)*IN(t),\n diff(IN(t),t)=r*SU(t)*IN(t)-a*IN(t),\n \+
diff(R(t),t)=a*IN(t),\n SU(0)=2.8, IN(0)=0.2, R(0)=0\}
, \{SU(t),IN(t),R(t)\},\n type=numeric,output=listprocedure);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "f:=subs(sol,SU(t)); g:=subs(
sol,IN(t)); h:=subs(sol,R(t));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
1 0 32 "plot(\{f,g,h\},0..50,color=BLACK);" }}}{PARA 0 "" 0 "" {TEXT
-1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }}{SECT 1 {PARA 3 "" 0 ""
{TEXT -1 26 "Exercises for the student." }}{PARA 0 "" 0 "" {TEXT -1 0
"" }}{PARA 0 "" 0 "" {TEXT -1 219 "1. Re-do the model twice: firs
t, double the rate of infection to see what is the effect if the disea
se is transmitted easily. Then, halve the rate of infection. Compare a
nd contrast the effects for these two cases." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 424 "2. An alternate model is to su
ppose that at some time after an individual has recovered from the dis
ease, immunity is lost. This allows people to move back into the susc
eptible class after recovery. What follows is the syntax for creating \+
such a model. Predict what you think the results will be. Then, run th
e program below to see if the model worked as you predicted. Explain y
our prediction and the results you computed." }}{PARA 0 "" 0 "" {TEXT
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "This procedure may take a littl
e while to run." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "restart:\nreadlib(spline);" }}}{EXCHG {PARA 0
"> " 0 "" {MPLTEXT 1 0 12 "r:=1; a:=1;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 76 "f[0]:=t->2.8; g[0]:=t->0.2; h[0]:=t->0;\n J||0:=plot(
\{f[0],g[0],h[0]\},-1..0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0
31 "x0seq:=[seq(j/5,j=0..5)]: N:=5;" }}}{EXCHG {PARA 0 "> " 0 ""
{MPLTEXT 1 0 658 "for n from 1 to N do\n\011sol:=dsolve(\{diff(SU(t),t
)=-r*SU(t)*IN(t)+'h[n-1](t-1)',\n\011\011diff(IN(t),t)=r*SU(t)*IN(t)-a
*IN(t),\n\011\011diff(R(t),t)=a*IN(t)-'h[n-1](t-1)',\n\011\011\011SU(n
-1)=f[n-1](n-1),\n\011\011\011IN(n-1)=g[n-1](n-1),\n\011\011\011R(n-1)
=h[n-1](n-1)\}, \n\011\011\{SU(t),IN(t),R(t)\}, numeric, output=listpr
ocedure):\n\011f:=subs(sol,SU(t)): g:=subs(sol,IN(t)): h:=subs(sol,R(t
)):\n\011xseq:=map(t->t+n-1,x0seq):\n fxseq:=map(f,xseq): gxseq:=map(g
,xseq): hxseq:=map(h,xseq):\n\011\011F:=spline(xseq,fxseq,t,cubic): \+
\n\011\011G:=spline(xseq,gxseq,t,cubic):\n\011\011H:=spline(xseq,hxseq
,t,cubic): \n\011f[n]:=unapply(F,t): \n\011g[n]:=unapply(G,t): \n\011h
[n]:=unapply(H,t):\n\011J[n]:= plot(\{f[n],g[n],h[n]\}, (n-1)..n,color
=BLACK):\n\011od:\011" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "pl
ots[display](\{J[1],J[2],J[3],J[4],J[5]\});" }}}{EXCHG {PARA 0 "> " 0
"" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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