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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "                          \+
                         " }{TEXT 256 19 "SIR Epidemic Model " }}
{PARA 0 "" 0 "" {TEXT -1 166 "\n   The SIR model of infectious disease
s was derived by Kermack\nand McKendrick (1927).  They studied the\npl
ague outbreak in the island of Bombay, India in 1905-1906. " }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 50 "restart;\nwith(DEtools):\nwith(linalg):\n
with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Define SIR equati
ons, parameters and initial conditions" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 176 "r:= 1.0: a := 1.0:\neqns := 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);\ninits:= SU(0)=2.8, IN(0)=0.2, R(0)=0.0;" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 29 "Generate a numerical solution" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 84 "soln := dsolve(\{eqns,inits\},\{SU(t),IN(t),R(t)\},
\ntype=numeric, output=listprocedure):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 25 "Generate string for title" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 106 "rr := convert(r,string):  aa := convert(a,string):\ncode := c
at(`SIR Models with rates r =`,rr,` a = `,aa);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 90 "Define functions for the susceptibles - s(t), the in
fectives - i(t) and\nthe removed - r(t)" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 66 "s := subs(soln,SU(t)): i:=subs(soln,IN(t)):\nre := subs(soln,R
(t)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "p1 := plot(\{s,i,r
e\},0..5,title=code,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "pt := textplot(\{[1,2.5,\"S(t)\"],[3, 1.75,\"R(t)\"],
[3,1,\"S(t)\"]\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "displ
ay(\{p1,pt\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 132 "Kermack and Mc
Kendrick obtained data on the weekly death total.\nThis is close to dr
/dt which is a*i(t).  We define a death function." }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "deaths := t -> a*i(t);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "plot(deaths,0..10,title=code);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 85 "eqns2 := diff(SU(t),t)=-r*SU(t)*IN(t),\n    \+
     diff(IN(t),t)= r*SU(t)*IN(t)-a*IN(t);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 66 "DEplot(\{eqns2\},[SU(t),IN(t)],t=0..10,SU=0..3,IN=0
..3,arrows=slim);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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