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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 25 "HIV and Antiviral Therapy
" }}{PARA 19 "" 0 "" {TEXT -1 85 "Robert M. Corless\nOntario Research \+
Centre for Computer Algebra\nhttp://www.orcca.on.ca" }}{PARA 0 "" 0 "
" {TEXT -1 209 "An exploration of \"Mathematical Models and the design
 of public health policy: HIV and Antiviral therapy\", by Sunetra Gupt
a, Roy M. Anderson, and Robert M. May, SIAM Reviw, Vol 35, No. 3, pp. \+
1-16, March 1993." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 22 "Notation of
 the paper:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 24 "p = proportion of cohort" }}{PARA 0 "" 0 "" {TEXT -1 36 "mu*N[0
] = sexually active population" }}{PARA 0 "" 0 "" {TEXT -1 30 "V[x] = \+
vaccinated susceptibles" }}{PARA 0 "" 0 "" {TEXT -1 29 "X = unvaccinat
ed susceptibles" }}{PARA 0 "" 0 "" {TEXT -1 34 "s = per capita rate of
 vaccination" }}{PARA 0 "" 0 "" {TEXT -1 44 "lambda = per partnership \+
probability of loss" }}{PARA 0 "" 0 "" {TEXT -1 31 "c = mean rate of p
artner change" }}{PARA 0 "" 0 "" {TEXT -1 57 "N[0] = stable population
 size in the absence of infection" }}{PARA 0 "" 0 "" {TEXT -1 51 "mu =
 inverse of average duration of sexual activity" }}{PARA 0 "" 0 "" 
{TEXT -1 33 "Y = untreated infected population" }}{PARA 0 "" 0 "" 
{TEXT -1 42 "v = per capita rate of development of AIDS" }}{PARA 0 "" 
0 "" {TEXT -1 55 "V[y] = number of infected individuals receiving ther
apy" }}{PARA 0 "" 0 "" {TEXT -1 70 "d = per capita rate of treated inf
ected population development of AIDS" }}{PARA 0 "" 0 "" {TEXT -1 67 "B
[1] = probability of transmission from an untreated carrier of HIV" }}
{PARA 0 "" 0 "" {TEXT -1 58 "B[2] = probability of transmission from a
 treated carrier." }}{PARA 0 "" 0 "" {TEXT -1 50 "N = current size of \+
the sexually active population" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 8 "restart;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "The assumptio
ns of the exposition in the paper (we can come back and change this la
ter and see how the conclusions change):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "s := 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG\"\"!
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "r := 0;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"rG\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
290 "The basic equations in differential form (with the dependence on \+
time suppressed for ease of working with equilibria):\nthe rate of cha
nge of unvaccinated susceptibles depends on how many come of age, and \+
how many are infected, and how many stop sexual activity, and how many
 are vaccinated:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eq1 := \+
diff(X(t),t) = mu*N[0]*(1-p) - (c*lambda + mu + s)*X;" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$eq1G/-%%diffG6$-%\"XG6#%\"tGF,,&*(%#muG\"\"\"&%
\"NG6#\"\"!F0,&F0F0%\"pG!\"\"F0F0*&,&*&%\"cGF0%'lambdaGF0F0F/F0F0F*F0F
7" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 222 "Similarly, the rate of chan
ge of vaccinated susceptibles  depends on how many come of age and are
 vaccinated, and how many are infected, and how many stop sexual activ
ity, and how many are vaccinated from the X population." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "eq2 := diff(V[x](t),t) = mu*N[0]*p-
(c*lambda+mu)*V[x] + s*X;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq2G/-
%%diffG6$-&%\"VG6#%\"xG6#%\"tGF/,&*(%#muG\"\"\"&%\"NG6#\"\"!F3%\"pGF3F
3*&,&*&%\"cGF3%'lambdaGF3F3F2F3F3F*F3!\"\"" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 44 "eq3 := diff(Y(t),t) = c*lambda*X-(v+r+mu)*Y;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$eq3G/-%%diffG6$-%\"YG6#%\"tGF,,&*(%
\"cG\"\"\"%'lambdaGF0%\"XGF0F0*&,&%\"vGF0%#muGF0F0F*F0!\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "eq4 := diff(V[y](t),t) = c*l
ambda*V[x] + r*Y - (d+mu)*V[y];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$
eq4G/-%%diffG6$-&%\"VG6#%\"yG6#%\"tGF/,&*(%\"cG\"\"\"%'lambdaGF3&F+6#%
\"xGF3F3*&,&%\"dGF3%#muGF3F3F*F3!\"\"" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 126 "The paper doesn't say so, but the calculation only works
 out if we say that the total population is just the sum of the parts.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "NN := X + V[x] + Y + V[
y];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NNG,*%\"XG\"\"\"&%\"VG6#%\"x
GF'%\"YGF'&F)6#%\"yGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 123 "The pe
r partnership probability of infection depends on the infectiousness a
nd the likelihood that the partner is infected:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 34 "lambda := (B[1]*Y + B[2]*V[y])/NN;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%'lambdaG*&,&*&&%\"BG6#\"\"\"F+%\"YGF+F+*&&
F)6#\"\"#F+&%\"VG6#%\"yGF+F+F+,*%\"XGF+&F26#%\"xGF+F,F+F1F+!\"\"" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "We only care about the ultimate s
tate, when the populations don't change: therefore we set all derivati
ves to zero." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "equilibria \+
:= map(rhs,\{eq1,eq2,eq3,eq4\});" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
+equilibriaG<&,&*&*(%\"cG\"\"\",&*&&%\"BG6#F*F*%\"YGF*F**&&F.6#\"\"#F*
&%\"VG6#%\"yGF*F*F*%\"XGF*F*,*F9F*&F66#%\"xGF*F0F*F5F*!\"\"F**&,&%\"vG
F*%#muGF*F*F0F*F>,&*(FBF*&%\"NG6#\"\"!F*,&F*F*%\"pGF>F*F**&,&*&*&F)F*F
+F*F*F:F>F*FBF*F*F9F*F>,&*(FBF*FEF*FJF*F**&FLF*F;F*F>,&*&*(F)F*F+F*F;F
*F*F:F>F**&,&%\"dGF*FBF*F*F5F*F>" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
150 "Maple can solve those population equations, to give us solutions \+
that look a little different than those in the paper but that we can e
xperiment with:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "sols := \+
solve(equilibria, \{X,Y,V[x],V[y]\});" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#>%%solsG6$<&/&%\"VG6#%\"yG*&**,6**%\"cG\"\"\"&%\"BG6#\"\"#F1%\"pGF1
%\"vGF1!\"\"**F0F1&F36#F1F1%\"dGF1F6F1F1**F0F1F2F1%#muGF1F6F1F8**F0F1F
:F1F>F1F6F1F1*$)F>F5F1F1*(F0F1F:F1F<F1F8*&F7F1F>F1F1*&F<F1F>F1F1*&F<F1
F7F1F1*(F0F1F:F1F>F1F8F1F6F1&%\"NG6#\"\"!F1F>F1F1*&,&F<F1F>F1F1,6*(F>F
1F6F1F7F1F8FFF8F?F1F=F8FCF1*(F<F1F>F1F6F1F1FEF1F9F1F/F8FBF8F1F8/%\"YG,
$*&*(F>F1FGF1,FF@F8*&FAF1F6F1F1**F0F1F2F1)F6F5F1F>F1F8**F0F1F:F1FXF1F>
F1F1*,F5F1F0F1F:F1F>F1F6F1F8FNF1FOF1F=F1FDF8FCF8FFF1**F0F1F2F1FXF1F7F1
F8**F0F1F:F1FXF1F<F1F1F/F1*(F7F1F<F1F6F1F1*,F5F1F0F1F:F1F<F1F6F1F8FEF8
FBF1F1F1,H**F0F1F:F1F7F1F<F1F8*,F0F1F:F1F7F1F<F1F6F1F1*&)F7F5F1F<F1F1*
*F0F1F2F1F6F1F]oF1F8*&FAF1F7F1F1*(FAF1F<F1F6F1F1*(FAF1F6F1F7F1F8*(FAF1
F0F1F:F1F8*(F>F1F6F1F]oF1F8*(F>F1F<F1F7F1F1**FAF1F0F1F:F1F6F1F1**FAF1F
0F1F2F1F6F1F8*,F>F1F0F1F:F1F<F1F6F1F1*.F5F1F>F1F0F1F2F1F6F1F7F1F8*,F>F
1F0F1F:F1F6F1F7F1F1**F>F1F7F1F<F1F6F1F1**F>F1F0F1F:F1F<F1F8**F>F1F0F1F
:F1F7F1F8*&F>F1F]oF1F1F8F8/%\"XG*&*(,0*&F>F1F6F1F1F>F8*&FXF1F7F1F1F<F8
*(F5F1F<F1F6F1F1*&F<F1FXF1F8*&F6F1F7F1F8F1F>F1FGF1F1FMF8/&F)6#%\"xG,$*
&**,*F>F1FgpF1*&F<F1F6F1F8F<F1F1F6F1F>F1FGF1F1FMF8F8<&/FQFJ/F_p,&FGF1*
&FGF1F6F1F8/FipFeq/F(FJ" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 384 "One s
olution occurs if no-one is infected to start with, so Y = V[y] = 0.  \+
It is the other one we are interested in.  It is curious that there is
 no solution if X=V[x]=0; this is a result of an assumption in the pap
er that there is always a population of uninfected children \"coming o
f age\" to sexual maturity to supply uninfected susceptibles. (This is
 not pointed out in the paper)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 78 "Now let us check that the solutions we ge
t are the same as those in the paper." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "assign(sols[1]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "Y;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&*(%#muG\"\"
\"&%\"NG6#\"\"!F',F*$)F&\"\"#F'!\"\"*&F.F'%\"pGF'F'**%\"cGF'&%\"BG6#F/
F')F2F/F'F&F'F0**F4F'&F66#F'F'F8F'F&F'F'*,F/F'F4F'F:F'F&F'F2F'F0*(F&F'
F2F'%\"vGF'F'*(%\"dGF'F&F'F2F'F'**F4F'F5F'F&F'F2F'F'*&F@F'F&F'F0*&F>F'
F&F'F0*(F4F'F:F'F&F'F'**F4F'F5F'F8F'F>F'F0**F4F'F:F'F8F'F@F'F'**F4F'F5
F'F2F'F>F'F'*(F>F'F@F'F2F'F'*,F/F'F4F'F:F'F@F'F2F'F0*&F@F'F>F'F0*(F4F'
F:F'F@F'F'F'F',H**F4F'F:F'F>F'F@F'F0*,F4F'F:F'F>F'F@F'F2F'F'*&)F>F/F'F
@F'F'**F4F'F5F'F2F'FPF'F0*&F.F'F>F'F'*(F.F'F@F'F2F'F'*(F.F'F2F'F>F'F0*
(F.F'F4F'F:F'F0*(F&F'F2F'FPF'F0*(F&F'F@F'F>F'F'**F.F'F4F'F:F'F2F'F'**F
.F'F4F'F5F'F2F'F0*,F&F'F4F'F:F'F@F'F2F'F'*.F/F'F&F'F4F'F5F'F2F'F>F'F0*
,F&F'F4F'F:F'F2F'F>F'F'**F&F'F>F'F@F'F2F'F'**F&F'F4F'F:F'F@F'F0**F&F'F
4F'F:F'F>F'F0*&F&F'FPF'F'F0F0" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 
"This doesn't look like the solution in the paper.  We should see if w
e can simplify the computed Y to see if there is an error." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "Y/c/N[0]/mu/(1-p)*(v+mu); # = lamst
ar/(c*lamstar+mu)" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&*&,F*$)%#muG
\"\"#\"\"\"!\"\"*&F(F+%\"pGF+F+**%\"cGF+&%\"BG6#F*F+)F.F*F+F)F+F,**F0F
+&F26#F+F+F4F+F)F+F+*,F*F+F0F+F6F+F)F+F.F+F,*(F)F+F.F+%\"vGF+F+*(%\"dG
F+F)F+F.F+F+**F0F+F1F+F)F+F.F+F+*&F<F+F)F+F,*&F:F+F)F+F,*(F0F+F6F+F)F+
F+**F0F+F1F+F4F+F:F+F,**F0F+F6F+F4F+F<F+F+**F0F+F1F+F.F+F:F+F+*(F:F+F<
F+F.F+F+*,F*F+F0F+F6F+F<F+F.F+F,*&F<F+F:F+F,*(F0F+F6F+F<F+F+F+,&F:F+F)
F+F+F+*(,H**F0F+F6F+F:F+F<F+F,*,F0F+F6F+F:F+F<F+F.F+F+*&)F:F*F+F<F+F+*
*F0F+F1F+F.F+FNF+F,*&F(F+F:F+F+*(F(F+F<F+F.F+F+*(F(F+F.F+F:F+F,*(F(F+F
0F+F6F+F,*(F)F+F.F+FNF+F,*(F)F+F<F+F:F+F+**F(F+F0F+F6F+F.F+F+**F(F+F0F
+F1F+F.F+F,*,F)F+F0F+F6F+F<F+F.F+F+*.F*F+F)F+F0F+F1F+F.F+F:F+F,*,F)F+F
0F+F6F+F.F+F:F+F+**F)F+F:F+F<F+F.F+F+**F)F+F0F+F6F+F<F+F,**F)F+F0F+F6F
+F:F+F,*&F)F+FNF+F+F+F0F+,&F+F+F.F,F+F,F," }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 141 "We should get something like lambda/(c*lambda+mu), for a
 particular value of lambda---let us see if we can get something like \+
the paper does." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "solve(ls
tar/(c*lstar+mu)=%,lstar);;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,$*&,6*
*%\"cG\"\"\"&%\"BG6#\"\"#F(%\"pGF(%\"vGF(!\"\"**F'F(&F*6#F(F(%\"dGF(F-
F(F(**F'F(F)F(%#muGF(F-F(F/**F'F(F1F(F5F(F-F(F(*$)F5F,F(F(*(F'F(F1F(F3
F(F/*&F.F(F5F(F(*&F3F(F5F(F(*&F3F(F.F(F(*(F'F(F1F(F5F(F/F(*&F'F(,*F5F(
*&F-F(F.F(F(*&F3F(F-F(F/F3F(F(F/F/" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "lstar := %;" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&lst
arG,$*&,6**%\"cG\"\"\"&%\"BG6#\"\"#F*%\"pGF*%\"vGF*!\"\"**F)F*&F,6#F*F
*%\"dGF*F/F*F***F)F*F+F*%#muGF*F/F*F1**F)F*F3F*F7F*F/F*F**$)F7F.F*F**(
F)F*F3F*F5F*F1*&F0F*F7F*F**&F5F*F7F*F**&F5F*F0F*F**(F)F*F3F*F7F*F1F**&
F)F*,*F7F**&F/F*F0F*F**&F5F*F/F*F1F5F*F*F1F1" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "lstarf := map(factor,%);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%'lstarfG,$*&,6**%\"cG\"\"\"&%\"BG6#\"\"#F*%\"pGF*%\"v
GF*!\"\"**F)F*&F,6#F*F*%\"dGF*F/F*F***F)F*F+F*%#muGF*F/F*F1**F)F*F3F*F
7F*F/F*F**$)F7F.F*F**(F)F*F3F*F5F*F1*&F0F*F7F*F**&F5F*F7F*F**&F5F*F0F*
F**(F)F*F3F*F7F*F1F**&F)F*,*F7F**&F/F*F0F*F**&F5F*F/F*F1F5F*F*F1F1" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "phieq := B[1]*(1-p)/(v+mu) \+
+ B[2]*p/(d+mu);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&phieqG,&*&*&&%
\"BG6#\"\"\"F+,&F+F+%\"pG!\"\"F+F+,&%\"vGF+%#muGF+F.F+*&*&&F)6#\"\"#F+
F-F+F+,&%\"dGF+F1F+F.F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
phi :='phi';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiGF$" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "solve(phieq=phi,B[1]);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#,$*&*&,(*&&%\"BG6#\"\"#\"\"\"%\"pGF,!\"\"*&%$ph
iGF,%\"dGF,F,*&F0F,%#muGF,F,F,,&%\"vGF,F3F,F,F,*&,&F1F,F3F,F,,&F-F,F,F
.F,F.F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "subs(B[1]=%,lsta
rf);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,$*&,6**%\"cG\"\"\"&%\"BG6#\"
\"#F(%\"pGF(%\"vGF(!\"\"*&*,F'F(,(*&F)F(F-F(F/*&%$phiGF(%\"dGF(F(*&F5F
(%#muGF(F(F(,&F.F(F8F(F(F6F(F-F(F(*&,&F6F(F8F(F(,&F-F(F(F/F(F/F/**F'F(
F)F(F8F(F-F(F/*&*,F'F(F2F(F9F(F8F(F-F(F(*&F;F(F<F(F/F/*$)F8F,F(F(*&**F
'F(F2F(F9F(F6F(F(*&F;F(F<F(F/F(*&F.F(F8F(F(*&F6F(F8F(F(*&F6F(F.F(F(*&*
*F'F(F2F(F9F(F8F(F(*&F;F(F<F(F/F(F(*&F'F(,*F8F(*&F-F(F.F(F(*&F6F(F-F(F
/F6F(F(F/F/" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 15 "collect(%,phi);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#,&*&*&,**&**%\"cG\"\"\",&%\"vGF*%#muGF*F*F-F*%\"pGF*F*,
&F.F*F*!\"\"F0F0*&**F)F*F+F*%\"dGF*F.F*F*F/F0F0*&*(F)F*F+F*F3F*F*F/F0F
**&*(F)F*F+F*F-F*F*F/F0F*F*%$phiGF*F**&F)F*,*F-F**&F.F*F,F*F**&F3F*F.F
*F0F3F*F*F0F0*&,6**F)F*&%\"BG6#\"\"#F*F.F*F,F*F0*&*,F)F*F@F*)F.FCF*F+F
*F3F*F**&,&F3F*F-F*F*F/F*F0F**$)F-FCF*F***F)F*F@F*F-F*F.F*F0*&*,F)F*F@
F*FFF*F+F*F-F*F**&FHF*F/F*F0F**&F,F*F-F*F**&F3F*F-F*F**&*,F)F*F@F*F.F*
F+F*F3F*F**&FHF*F/F*F0F0*&*,F)F*F@F*F.F*F+F*F-F*F**&FHF*F/F*F0F0*&F3F*
F,F*F*F**&F)F*F:F*F0F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "c
ollect(%,phi,factor);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&*(,&%\"dG
\"\"\"%#muGF(F(,&%\"vGF(F)F(F(%$phiGF(F(,*F)F(*&%\"pGF(F+F(F(*&F'F(F/F
(!\"\"F'F(F1F(*&*&F&F(F*F(F(*&F-F(%\"cGF(F1F1" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 147 "This agrees with equations (6) and (8) of the paper (
if we don't use N=X+V[x]+Y+V[y]) and even if we do it looks as though \+
it is in the right form." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"phi := phieq;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG,&*&*&&%\"BG6
#\"\"\"F+,&F+F+%\"pG!\"\"F+F+,&%\"vGF+%#muGF+F.F+*&*&&F)6#\"\"#F+F-F+F
+,&%\"dGF+F1F+F.F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "lamst
ar := %%;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(lamstarG,&*&*(,&%\"dG
\"\"\"%#muGF*F*,&%\"vGF*F+F*F*,&*&*&&%\"BG6#F*F*,&F*F*%\"pG!\"\"F*F*F,
F6F**&*&&F26#\"\"#F*F5F*F*F(F6F*F*F*,*F+F**&F5F*F-F*F**&F)F*F5F*F6F)F*
F6F**&*&F(F*F,F*F**&F<F*%\"cGF*F6F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 174 "Now we compare our computed answer with the formula from the p
aper by subtracting the one from the other; if the difference is zero,
 then we know the paper result is correct." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 50 "Y - c*lamstar*mu*N[0]*(1-p)/(c*lamstar+mu)/(v+mu):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 117 "
It is, so at least the formula for Y is right.  But the formula for V[
y] also looks different from that of the paper:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 5 "V[y];" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#*&**,6
**%\"cG\"\"\"&%\"BG6#\"\"#F(%\"pGF(%\"vGF(!\"\"**F'F(&F*6#F(F(%\"dGF(F
-F(F(**F'F(F)F(%#muGF(F-F(F/**F'F(F1F(F5F(F-F(F(*$)F5F,F(F(*(F'F(F1F(F
3F(F/*&F.F(F5F(F(*&F3F(F5F(F(*&F3F(F.F(F(*(F'F(F1F(F5F(F/F(F-F(&%\"NG6
#\"\"!F(F5F(F(*&,&F3F(F5F(F(,6*(F5F(F-F(F.F(F/F=F/F6F(F4F/F:F(*(F3F(F5
F(F-F(F(F<F(F0F(F&F/F9F/F(F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "A
gain we subtract the formula from the paper and see if it is zero." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "normal( V[y] - c*lamstar*mu
*N[0]*p/(c*lamstar+mu)/(d+mu) );" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "It is. Now we see if we c
an recapitulate their argument to get the equlibrium population in the
 presence of infection:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "
normal( lamstar - c*lamstar*mu*N[0]*phi/(c*lamstar+mu)/Nstar):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "solve(%,Nstar):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#*&*,,*%#muG\"\"\"*&%\"pGF'%\"vGF'F'*&%\"dGF'F)F'!\"\"F,
F'F',.*(F&F'&%\"BG6#\"\"#F'F)F'F-*(F&F'&F16#F'F'F)F'F'*&F&F'F5F'F-*(F)
F'F0F'F*F'F-*(F5F'F,F'F)F'F'*&F5F'F,F'F-F'%\"cGF'F&F'&%\"NG6#\"\"!F'F'
*(,&F,F'F&F'F',&F*F'F&F'F',6*(F&F'F)F'F*F'F-*(F;F'F5F'F&F'F-**F;F'F5F'
F&F'F)F'F'**F;F'F0F'F&F'F)F'F-*&F*F'F&F'F'*(F,F'F&F'F)F'F'*&F,F'F*F'F'
**F;F'F5F'F,F'F)F'F'**F;F'F0F'F)F'F*F'F-*(F;F'F5F'F,F'F-F'F-" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Nstar := %;" }}{PARA 12 "" 
1 "" {XPPMATH 20 "6#>%&NstarG*&*,,*%#muG\"\"\"*&%\"pGF)%\"vGF)F)*&%\"d
GF)F+F)!\"\"F.F)F),.*(F(F)&%\"BG6#\"\"#F)F+F)F/*(F(F)&F36#F)F)F+F)F)*&
F(F)F7F)F/*(F+F)F2F)F,F)F/*(F7F)F.F)F+F)F)*&F7F)F.F)F/F)%\"cGF)F(F)&%
\"NG6#\"\"!F)F)*(,&F.F)F(F)F),&F,F)F(F)F),6*(F(F)F+F)F,F)F/*(F=F)F7F)F
(F)F/**F=F)F7F)F(F)F+F)F)**F=F)F2F)F(F)F+F)F/*&F,F)F(F)F)*(F.F)F(F)F+F
)F)*&F.F)F,F)F)**F=F)F7F)F.F)F+F)F)**F=F)F2F)F+F)F,F)F/*(F=F)F7F)F.F)F
/F)F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 126 "Again, that formula loo
ks different from the one in the paper, which defines things in terms \+
of an auxiliary parameter, Omega." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 43 "Nstar = N[0]*(1-Omega)/(1-(1/c)*Omega/phi);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#/*&*,,*%#muG\"\"\"*&%\"pGF(%\"vGF(F(*&%\"dGF
(F*F(!\"\"F-F(F(,.*(F'F(&%\"BG6#\"\"#F(F*F(F.*(F'F(&F26#F(F(F*F(F(*&F'
F(F6F(F.*(F*F(F1F(F+F(F.*(F6F(F-F(F*F(F(*&F6F(F-F(F.F(%\"cGF(F'F(&%\"N
G6#\"\"!F(F(*(,&F-F(F'F(F(,&F+F(F'F(F(,6*(F'F(F*F(F+F(F.*(F<F(F6F(F'F(
F.**F<F(F6F(F'F(F*F(F(**F<F(F1F(F'F(F*F(F.*&F+F(F'F(F(*(F-F(F'F(F*F(F(
*&F-F(F+F(F(**F<F(F6F(F-F(F*F(F(**F<F(F1F(F*F(F+F(F.*(F<F(F6F(F-F(F.F(
F.*&*&F=F(,&F(F(%&OmegaGF.F(F(,&F(F(*&FRF(*&F<F(,&*&*&F6F(,&F(F(F*F.F(
F(FCF.F(*&*&F1F(F*F(F(FBF.F(F(F.F.F." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "tst := solve(%,Omega);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$tstG,$*&,**(%#muG\"\"\"%\"pGF*%\"vGF*F**&F,F*F)F*!\"\"*(%\"dG
F*F)F*F+F*F.*&F0F*F,F*F.F*,*F1F**&F0F*F)F*F*F-F**$)F)\"\"#F*F*F.F." }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "That again looks different from t
he formula in the paper:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 
"tst - v*(1-p)/(v+mu) - d*p/(d+mu);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#,(*&,**(%#muG\"\"\"%\"pGF(%\"vGF(F(*&F*F(F'F(!\"\"*(%\"dGF(F'F(F)F(F,
*&F.F(F*F(F,F(,*F/F(*&F.F(F'F(F(F+F(*$)F'\"\"#F(F(F,F,*&*&F*F(,&F(F(F)
F,F(F(,&F*F(F'F(F,F,*&*&F.F(F)F(F(,&F.F(F'F(F,F," }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 10 "normal(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "But it is not.  Therefo
re, all the formulas in the paper are correct." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Now, we can extend the pr
inted results and work out what the uninfected population NA would be:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "eval(Nstar,p=0):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "NA := factor(%);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#NAG*&**&%\"NG6#\"\"!\"\"\"%\"cGF+&%\"BG6#
F+F+%#muGF+F+*&,&*&F,F+F-F+F+%\"vG!\"\"F+,&F4F+F0F+F+F5" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "effectiveness_ratio := Nstar/NA;" }
}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%4effectiveness_ratioG*&*(,*%#muG\"
\"\"*&%\"pGF)%\"vGF)F)*&%\"dGF)F+F)!\"\"F.F)F),.*(F(F)&%\"BG6#\"\"#F)F
+F)F/*(F(F)&F36#F)F)F+F)F)*&F(F)F7F)F/*(F+F)F2F)F,F)F/*(F7F)F.F)F+F)F)
*&F7F)F.F)F/F),&*&%\"cGF)F7F)F)F,F/F)F)*(,&F.F)F(F)F),6*(F(F)F+F)F,F)F
/*(F?F)F7F)F(F)F/**F?F)F7F)F(F)F+F)F)**F?F)F2F)F(F)F+F)F/*&F,F)F(F)F)*
(F.F)F(F)F+F)F)*&F.F)F,F)F)**F?F)F7F)F.F)F+F)F)**F?F)F2F)F+F)F,F)F/*(F
?F)F7F)F.F)F/F)F7F)F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "ev
al(%,p=1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*(,&%\"vG\"\"\"%#muGF'
F',&*&F(F'&%\"BG6#\"\"#F'!\"\"*&F+F'F&F'F/F',&*&%\"cGF'&F,6#F'F'F'F&F/
F'F'*(,&%\"dGF'F(F'F',**(F3F'F+F'F(F'F/*&F8F'F(F'F'*&F8F'F&F'F'*(F3F'F
+F'F&F'F/F'F4F'F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "normal
(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#*&*(,&*&%\"cG\"\"\"&%\"BG6#F(F
(F(%\"vG!\"\"F(&F*6#\"\"#F(,&F,F(%#muGF(F(F(*(F)F(,&*&F'F(F.F(F(%\"dGF
-F(,&F6F(F2F(F(F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 138 "This short \+
exploration showed an example of using Maple to verify the results in \+
an area of mathematical biology and public health policy." }}}}{MARK "
0 0 0" 12 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 
1 }