µh µv r
r r
r r r
r r r r
r r r r
r r r r
r r r r
r r
r r
r r r r
r r r r
r r r r
r r r r
r r r
r r
r µh
µh r
µh r
µh r
µh r
µh r
µh r
µh µh
µh r
µh r
µh r
µh r
µh r
µh r
µh µh
µh µh
µh r
r r
r r
r r
µh sh iv r r
r r r r
r r r r
r r r r
r r r r
r µh r µh r µh r µh r µh µh r µv r µv r µv r µv r µv
µhµv r r
r µh µv
r r
r r
r r
r r
r r
r µh
µh µh sh
, , w
= iv r r
r r r r
r r r r
r µh r µh r µh r µh µh r
r r
µh r r
r r
r µh µv
r r r r
r µh r
µh r µh r
µh r µh r
µh r µh r
µh r µh µh
µh µh
r r r
r r
r r
r r
µh µh
µh r r
r r
r r
r r
r µh
µh µh sh
r µh r r
r µh
r r
µh r r
µh r r
r r
µh µv r
µh r
r r
r µh
r r
r r
r r
µh sh
Lampiran 9 Pembuktian w
3
w
2
w
1
w
1 2
+ w
3 2
w dan w
3
0, w
1
0, w 0 untuk Endemik
μh=0.0000421;α=0.4;r1=120;r2=114;r3=1365;r4=12365;r5=13365;r6=130;r7=125;β1=0.0 25;β2=0.024;β3=0.03;β4=0.02;μv=120;A=110;β1=β1Aμv;Nh=1;β2=β2Aμv;β3=β3
Nh;β4=β4 Nh;δ=r1+r2+r3+r4+r6+r7;ρ=αr2+r3+r4+r5+r6+r7; sh
= r1 r3+r4+μh+r3+r4+μh r6+r7+μh+r2 r3+r4-r3 α+μh β3+β4 r3+r4+μh r5+μh+r2 r5 α+r5+r6+r7+r2 α μh+μh
2
+r3 r5+r6+r7+μh+r4 r5+r6+r7+μh μvβ3+β4 r3+r4+μh -r2 - r3-r4+r3 α-α β1-α β2-μh r5+μh+r1 r3+r4+μh r5+μh+r3+r4+μh r6+r7+μh β1+β2+μh+r5
r6+r7+β1+β2+μh; ih
=-r5+μh -β1+β2 β3+β4 r3+r4+μh+r1 r3+r4+μh+r3+r4+μh r6+r7+μh+r2 r3+r4-r3 α+μh μvβ3+β4 -r2 -r3-r4+r3 α-α β1-α β2-μh r5+μh+r1 r3+r4+μh r5+μh+r3+r4+μh
r6+r7+μh β1+β2+μh+r5 r6+r7+β1+β2+μh; dh
=-r2 α r5+μh -β1+β2 β3+β4 r3+r4+μh+r1 r3+r4+μh+r3+r4+μh r6+r7+μh+r2 r3+r4-r3 α+μh μvβ3+β4 r3+r4+μh -r2 -r3-r4+r3 α-α β1-α β2-μh r5+μh+r1 r3+r4+μh
r5+μh+r3+r4+μh r6+r7+μh β1+β2+μh+r5 r6+r7+β1+β2+μh; iv
=-r5+μh -β1+β2 β3+β4 r3+r4+μh+r1 r3+r4+μh+r3+r4+μh r6+r7+μh+r2 r3+r4-r3 α+μh μvβ1+β2 β3+β4 r3+r4+μh r5+μh+r2 r5 α+r5+r6+r7+r2 α μh+μh
2
+r3 r5+r6+r7+μh+r4 r5+r6+r7+μh μv;
sv iv ;
w30 True
w10 True
w00 True
w3w2w1w1
2
+w3
2
w0 True
Lampiran 10 Nilai Eigen Titik Tetap Bebas Endemik
, ,
, , ,
, , , ,
, , , ,
, , ,
, ,
, ,
, ,
µh r t
t t
t t t
t √ √ t
t t t t
t t t t
t t t
t √ √ t
t t t t
t t t t
, t
√ t
t t
t t t
√ √ t t
t t t t t
t t √
t t t
t √ √ t
t t t t
t t t t
√ √
√ √
Lampiran 11 Gambar Dinamika Populasi untuk R
= 0.00378437 .
; . ;
; ;
; ;
; ;
; .
; .
; . ;
. ; ;
; ;
; ;
; ;
; 0.00378437
, ,
, ,
, . ,
. , . ,
, ,
, , , ,
{{sh[t]→InterpolatingFunction[{{0.,100000.}},][t],ih[t]→Interpolat ingFunction[{{0.,100000.}},][t],dh[t]→InterpolatingFunction[{{0.,1
00000.}},][t],iv[t]→InterpolatingFunction[{{0.,100000.}},][t]}}
Gandi1=Plot[sh[t].bidsol,{t,0,1500},PlotRange →{0,1},FrameLabel→{Wa
ktu}, Frame
→{{True,False},{True,False}},PlotStyle→{Dashed,Red,Thick}]; Gandi2=Plot[ih[t].bidsol,{t,0,1500},PlotRange
→All,FrameLabel→{Wakt u},
Frame
→{{True,False},{True,False}},PlotStyle→{Dashed,Blue,Thick}]; Gandi3=Plot[dh[t].bidsol,{t,0,1500},PlotRange
→All,FrameLabel→{Wakt u},
Frame
→{{True,False},{True,False}},PlotStyle→{Dashed,Black,Thick}]; Gandi4=Plot[iv[t].bidsol,{t,0,1500},PlotRange
→All,FrameLabel→{Wakt u},
Frame
→{{True,False},{True,False}},PlotStyle→{Dashed,Green,Thick}];
, ,
, 25
Lampiran 12 Gambar Simulasi Populasi s
h
. ;
. ; ;
; ;
; ;
; ;
. ;
. ;
. ; . ;
; ;
; ;
; ;
; ;
bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1- sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2
ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv iv[t] iv[t],sh[0] 0,ih[0] 0.8,dh[0] 0.8,iv[0] 0.8},{sh[t],ih[t],dh[t],iv[t]},{t,0,100}]
{{sh[t]→InterpolatingFunction[{{0.,100.}},][t],ih[t]→InterpolatingFunction[{{0.,100.}},][t],dh[t] →InterpolatingFunction[{{0.,100.}},][t],iv[t]→InterpolatingFunction[{{0.,100.}},][t]}}
. , , ,
, ,
, ,
, ,
, ,
, ;
. ;
. ; ;
; ;
; ;
; ;
. ;
. ;
. ; . ;
; ;
; ;
; ;
; ;
bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1- sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2
ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv iv[t] iv[t],sh[0] 0,ih[0] 0.5,dh[0] 0.8,iv[0] 0.8},{sh[t],ih[t],dh[t],iv[t]},{t,0,100}]
{{sh[t]→InterpolatingFunction[{{0.,100.}},][t],ih[t]→InterpolatingFunction[{{0.,100.}},][t],dh[t] →InterpolatingFunction[{{0.,100.}},][t],iv[t]→InterpolatingFunction[{{0.,100.}},][t]}}
. , , ,
, ,
, ,
, ,
, ,
, ;
. ;
. ; ;
; ;
; ;
; ;
. ; .
; . ;
. ; ;
; ;
; ;
; ;
; bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1-
sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2 ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv
iv[t] iv[t],sh[0] 0,ih[0] 0.2,dh[0] 0.8,iv[0] 0.8},{sh[t],ih[t],dh[t],iv[t]},{t,0,100}] {{sh[t]→InterpolatingFunction[{{0.,100.}},][t],ih[t]→InterpolatingFunction[{{0.,100.}},][t],dh[t]
→InterpolatingFunction[{{0.,100.}},][t],iv[t]→InterpolatingFunction[{{0.,100.}},][t]}}
. , , ,
, ,
, ,
, ,
, ,
, ;
, ,
Lampiran 13 Gambar Dinamika Populasi untuk R
= 0.000105122 .
; . ;
; ;
; ;
; ;
; .
; .
; . ;
. ; ;
; ;
; ;
; ;
; R0=β1+β2β3+β4μh+r3+r4μv μh μh+δ+r3+r4+r1+r6+r6+r2 r3 1-α+r4
0.000105122 bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1-
sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2 ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv
iv[t] iv[t],sh[0] 0,ih[0] 0.8,dh[0] 0.8,iv[0] 0.8},{sh[t],ih[t],dh[t],iv[t]},{t,0,100000}] {{sh[t]→InterpolatingFunction[{{0.,100000.}},][t],ih[t]→InterpolatingFunction[{{0.,100000.}},][
t],dh[t]→InterpolatingFunction[{{0.,100000.}},][t],iv[t]→InterpolatingFunction[{{0.,100000.}},] [t]}}
Gandi1=Plot[sh[t].bidsol,{t,0,500},PlotRange→{0,1},FrameLabel→{Waktu}, Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Red,Thick}];
Gandi2=Plot[ih[t].bidsol,{t,0,500},PlotRange→All,FrameLabel→{Waktu}, Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Blue,Thick}];
Gandi3=Plot[dh[t].bidsol,{t,0,500},PlotRange→All,FrameLabel→{Waktu}, Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Black,Thick}];
Gandi4=Plot[iv[t].bidsol,{t,0,500},PlotRange→All,FrameLabel→{Waktu}, Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Green,Thick}];
, ,
,
Lampiran 14 Gambar Simulasi Populasi s
h
. ;
. ; ;
; ;
; ;
; ;
. ;
. ;
. ; . ;
; ;
; ;
; ;
; ;
bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1- sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2
ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv iv[t] iv[t],sh[0] 0,ih[0] 0.8,dh[0] 0.8,iv[0] 0.8},{sh[t],ih[t],dh[t],iv[t]},{t,0,100}]
{{sh[t]→InterpolatingFunction[{{0.,100.}},][t],ih[t]→InterpolatingFunction[{{0.,100.}},][t],dh[t] →InterpolatingFunction[{{0.,100.}},][t],iv[t]→InterpolatingFunction[{{0.,100.}},][t]}}
Gandi
Plot sh . bidsol, , , , PlotRange
All, FrameLabel Waktu , Frame
True, False , True, False , PlotStyle Dashed, Red, Thick ;
. ;
. ; ;
; ;
; ;
; ;
. ;
. ;
. ; . ;
; ;
; ;
; ;
; bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1-
sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2 ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv
iv[t] iv[t],sh[0] 0,ih[0] 0.5,dh[0] 0.8,iv[0] 0.8},{sh[t],ih[t],dh[t],iv[t]},{t,0,100}] 27
{{sh[t]→InterpolatingFunction[{{0.,100.}},][t],ih[t]→InterpolatingFunction[{{0.,100.}},][t],dh[t] →InterpolatingFunction[{{0.,100.}},][t],iv[t]→InterpolatingFunction[{{0.,100.}},][t]}}
. , , ,
, ,
, ,
, ,
, ,
, ;
. ;
. ; ;
; ;
; ;
; ;
. ; .
; . ;
. ; ;
; ;
; ;
; ;
; bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1-
sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2 ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv
iv[t] iv[t],sh[0] 0,ih[0] 0.3,dh[0] 0.8,iv[0] 0.8},{sh[t],ih[t],dh[t],iv[t]},{t,0,100}] {{sh[t]→InterpolatingFunction[{{0.,100.}},][t],ih[t]→InterpolatingFunction[{{0.,100.}},][t],dh[t]
→InterpolatingFunction[{{0.,100.}},][t],iv[t]→InterpolatingFunction[{{0.,100.}},][t]}}
. , , ,
, ,
, ,
, ,
, ,
, ;
, ,
Lampiran 15 Gambar Dinamika Populasi untuk R
= 0.00378044 .
; .
; ;
; ;
; ;
; ;
. ;
. ;
. ; . ;
; ;
; ;
; ;
; ;
R0=β1+β2β3+β4μh+r3+r4μv μh μh+δ+r3+r4+r1+r6+r6+r2 r3 1-α+r4 0.00378044
bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1- sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2
ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv iv[t] iv[t],sh[0] 0,ih[0] 0.8,dh[0] 0.8,iv[0] 0.8},{sh[t],ih[t],dh[t],iv[t]},{t,0,100000}]
{{sh[t]→InterpolatingFunction[{{0.,100000.}},][t],ih[t]→InterpolatingFunction[{{0.,100000.}},][ t],dh[t]→InterpolatingFunction[{{0.,100000.}},][t],iv[t]→InterpolatingFunction[{{0.,100000.}},]
[t]}} Gandi1=Plot[sh[t].bidsol,{t,0,500},PlotRange→{0,1},FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Red,Thick}]; Gandi2=Plot[ih[t].bidsol,{t,0,500},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Blue,Thick}]; Gandi3=Plot[dh[t].bidsol,{t,0,500},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Black,Thick}]; Gandi4=Plot[iv[t].bidsol,{t,0,500},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Green,Thick}];
, ,
, 28
Lampiran 16 Gambar Dinamika Populasi untuk R
= 1.48298 .
; . ;
; ;
; ;
; ;
; . ;
. ; . ;
. ; ;
; ;
; ;
; ;
; R0=β1+β2β3+β4μh+r3+r4μv μh μh+δ+r3+r4+r1+r6+r6+r2 r3 1-α+r4
1.48298 bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1-
sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2 ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv
iv[t] iv[t],sh[0] 1,ih[0] 0.00002,dh[0] 0.00012,iv[0] 0.0002},{sh[t],ih[t],dh[t],iv[t]},{t,0,100000 }]
{{sh[t]→InterpolatingFunction[{{0.,100000.}},][t],ih[t]→InterpolatingFunction[{{0.,100000.}},][ t],dh[t]→InterpolatingFunction[{{0.,100000.}},][t],iv[t]→InterpolatingFunction[{{0.,100000.}},]
[t]}} Gandi1=Plot[sh[t].bidsol,{t,0,160},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Red,Thick}]; Gandi2=Plot[ih[t].bidsol,{t,0,160},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Blue,Thick}]; Gandi3=Plot[dh[t].bidsol,{t,0,160},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Black,Thick}]; Gandi4=Plot[iv[t].bidsol,{t,0,160},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Green,Thick}];
, ,
,
Lampiran 17 Gambar Simulasi Populasi s
h
. ;
. ; ;
; ;
; ;
; ;
. ; . ;
. ; . ;
; . ;
; ;
; ;
; ;
bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1- sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2
ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv iv[t] iv[t],sh[0] 1,ih[0] 0.00000002,dh[0] 0.00012,iv[0] 0.0002},{sh[t],ih[t],dh[t],iv[t]},{t,0,100
}] {{sh[t]→InterpolatingFunction[{{0.,100.}},][t],ih[t]→InterpolatingFunction[{{0.,100.}},][t],dh[t]
→InterpolatingFunction[{{0.,100.}},][t],iv[t]→InterpolatingFunction[{{0.,100.}},][t]}}
. , , ,
, ,
, ,
, ,
, ,
, ;
. ;
. ; ;
; ;
; ;
; ;
. ; . ;
. ; . ;
; . ;
; ;
; ;
; ;
bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1- sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2
ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv iv[t] iv[t],sh[0] 1,ih[0] 0.00000002,dh[0] 0.00012,iv[0] 0.0002},{sh[t],ih[t],dh[t],iv[t]},{t,0,100
}] {{sh[t]→InterpolatingFunction[{{0.,100.}},][t],ih[t]→InterpolatingFunction[{{0.,100.}},][t],dh[t]
→InterpolatingFunction[{{0.,100.}},][t],iv[t]→InterpolatingFunction[{{0.,100.}},][t]}}
. , , ,
, ,
, ,
, ,
, ,
, ;
. ;
. ; ;
; ;
; ;
; ;
. ; . ;
. ; . ;
; . ;
; ;
; ;
; ;
bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1- sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2
ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv iv[t] iv[t],sh[0] 1,ih[0] 0.00000002,dh[0] 0.00012,iv[0] 0.0002},{sh[t],ih[t],dh[t],iv[t]},{t,0,100
}] {{sh[t]→InterpolatingFunction[{{0.,100.}},][t],ih[t]→InterpolatingFunction[{{0.,100.}},][t],dh[t]
→InterpolatingFunction[{{0.,100.}},][t],iv[t]→InterpolatingFunction[{{0.,100.}},][t]}}
. , , ,
, ,
, ,
, ,
, ,
, ;
, ,
Lampiran 18 Gambar Dinamika Populasi untuk R
= 5.93191 .
; . ;
; ;
; ;
; ;
; . ;
. ; . ;
. ; ;
; ;
; ;
; ;
; R0=β1+β2β3+β4μh+r3+r4μv μh μh+δ+r3+r4+r1+r6+r6+r2 r3 1-α+r4
5.93191 bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1-
sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2 ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv
iv[t] iv[t],sh[0] 1,ih[0] 0.000002,dh[0] 0.00012,iv[0] 0.0002},{sh[t],ih[t],dh[t],iv[t]},{t,0,1000 00}]
{{sh[t]→InterpolatingFunction[{{0.,100000.}},][t],ih[t]→InterpolatingFunction[{{0.,100000.}},][ t],dh[t]→InterpolatingFunction[{{0.,100000.}},][t],iv[t]→InterpolatingFunction[{{0.,100000.}},]
[t]}} Gandi1=Plot[sh[t].bidsol,{t,0,100},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Red,Thick}]; Gandi2=Plot[ih[t].bidsol,{t,0,100},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Blue,Thick}]; Gandi3=Plot[dh[t].bidsol,{t,0,100},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Black,Thick}]; Gandi4=Plot[iv[t].bidsol,{t,0,100},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Green,Thick}];
, ,
, 30
Lampiran 19 Gambar Simulasi Populasi s
h
. ;
. ; ;
; ;
; ;
; ;
. ; . ;
. ; . ;
; . ;
; ;
; ;
; ;
bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1- sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2
ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv iv[t] iv[t],sh[0] 1,ih[0] 0.000002,dh[0] 0.00012,iv[0] 0.0002},{sh[t],ih[t],dh[t],iv[t]},{t,0,100}]
{{sh[t]→InterpolatingFunction[{{0.,100.}},][t],ih[t]→InterpolatingFunction[{{0.,100.}},][t],dh[t] →InterpolatingFunction[{{0.,100.}},][t],iv[t]→InterpolatingFunction[{{0.,100.}},][t]}}
. , , ,
, ,
, ,
, ,
, ,
, ;
. ;
. ; ;
; ;
; ;
; ;
. ; . ;
. ; . ;
; . ;
; ;
; ;
; ;
bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1- sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2
ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv iv[t] iv[t],sh[0] 1,ih[0] 0.000002,dh[0] 0.00012,iv[0] 0.0002},{sh[t],ih[t],dh[t],iv[t]},{t,0,100}]
{{sh[t]→InterpolatingFunction[{{0.,100.}},][t],ih[t]→InterpolatingFunction[{{0.,100.}},][t],dh[t] →InterpolatingFunction[{{0.,100.}},][t],iv[t]→InterpolatingFunction[{{0.,100.}},][t]}}
. , , ,
, ,
, ,
, ,
, ,
, ;
. ;
. ; ;
; ;
; ;
; ;
. ; . ;
. ; . ;
; . ;
; ;
; ;
; ;
bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1- sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2
ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv iv[t] iv[t],sh[0] 1,ih[0] 0.000002,dh[0] 0.00012,iv[0] 0.0002},{sh[t],ih[t],dh[t],iv[t]},{t,0,100}]
{{sh[t]→InterpolatingFunction[{{0.,100.}},][t],ih[t]→InterpolatingFunction[{{0.,100.}},][t],dh[t] →InterpolatingFunction[{{0.,100.}},][t],iv[t]→InterpolatingFunction[{{0.,100.}},][t]}}
. , , ,
, ,
, ,
, ,
, ,
, ;
, ,
31
Lampiran 20 Gambar Dinamika Populasi untuk R
= 1.4837 .
; .
; ;
; ;
; ;
; ;
. ; . ;
. ; . ;
; ;
; ;
; ;
; ;
R0=β1+β2β3+β4μh+r3+r4μv μh μh+δ+r3+r4+r1+r6+r6+r2 r3 1-α+r4 1.4897
bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1- sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2
ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv iv[t] iv[t],sh[0] 1,ih[0] 0.000002,dh[0] 0.00012,iv[0] 0.0002},{sh[t],ih[t],dh[t],iv[t]},{t,0,10000
0}] {{sh[t]→InterpolatingFunction[{{0.,100000.}},][t],ih[t]→InterpolatingFunction[{{0.,100000.}},][
t],dh[t]→InterpolatingFunction[{{0.,100000.}},][t],iv[t]→InterpolatingFunction[{{0.,100000.}},] [t]}}
Gandi1=Plot[sh[t].bidsol,{t,0,100},PlotRange→All,FrameLabel→{Waktu}, Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Red,Thick}];
Gandi2=Plot[ih[t].bidsol,{t,0,100},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Blue,Thick}]; Gandi3=Plot[dh[t].bidsol,{t,0,100},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Black,Thick}]; Gandi4=Plot[iv[t].bidsol,{t,0,100},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Green,Thick}];
, ,
,
Lampiran 21 Gambar Dinamika populasi ketika vektor awal terinfeksi 5 untuk R
1 .
; . ;
; ;
; ;
; ;
; .
; .
; . ;
. ; ;
; ;
; ;
; ;
; bidsol NDSolve µh
sh r ih
iv sh r
r ih r dh
r sh
ih dh
sh ,
iv sh r
r ih µhih
r r ih
r dh ih
, r ih r
r µh dh
dh ,
ih iv
µviv iv
, sh , ih
. , dh . , iv
. , sh , ih , dh , iv , , ,
sh t InterpolatingFunction .,
. , t ,ih t InterpolatingFunction
., . ,
t ,dh t InterpolatingFunction .,
. , t ,iv t InterpolatingFunction
., . ,
t Gandi Plot sh t .bidsol, t, ,
,PlotRange , ,FrameLabel Waktu , Frame True,False , True,False ,PlotStyle Dashed,Red,Thick ;
Gandi Plot ih t .bidsol, t, , ,PlotRangeAll,FrameLabel Waktu ,
Frame True,False , True,False ,PlotStyle Dashed,Blue,Thick ; 32
Gandi Plot dh t .bidsol, t, , ,PlotRangeAll,FrameLabel Waktu ,
Frame True,False , True,False ,PlotStyle Dashed,Black,Thick ; Gandi Plot iv t .bidsol, t, ,
,PlotRangeAll,FrameLabel Waktu , Frame True,False , True,False ,PlotStyle Dashed,Green,Thick ;
, ,
,
Lampiran 22 Gambar Dinamika populasi ketika vektor awal terinfeksi 50 untuk R
1 .
; . ;
; ;
; ;
; ;
; .
; .
; . ;
. ; ;
; ;
; ;
; ;
; bidsol NDSolve µh
sh r ih
iv sh r
r ih r dh
r sh
ih dh
sh ,
iv sh r
r ih µhih
r r ih
r dh ih
, r ih r
r µh dh
dh ,
ih iv
µviv iv
, sh , ih
. , dh . , iv
. , sh , ih , dh , iv , , ,
sh t InterpolatingFunction .,
. , t ,ih t InterpolatingFunction
., . ,
t ,dh t InterpolatingFunction .,
. , t ,iv t InterpolatingFunction
., . ,
t Gandi Plot sh t .bidsol, t, ,
,PlotRange , ,FrameLabel Waktu , Frame True,False , True,False ,PlotStyle Dashed,Red,Thick ;
Gandi Plot ih t .bidsol, t, , ,PlotRangeAll,FrameLabel Waktu ,
Frame True,False , True,False ,PlotStyle Dashed,Blue,Thick ; Gandi Plot dh t .bidsol, t, ,
,PlotRangeAll,FrameLabel Waktu , Frame True,False , True,False ,PlotStyle Dashed,Black,Thick ;
Gandi Plot iv t .bidsol, t, , ,PlotRangeAll,FrameLabel Waktu ,
Frame True,False , True,False ,PlotStyle Dashed,Green,Thick ; ,
, ,
Lampiran 23 Gambar Dinamika populasi ketika vektor awal terinfeksi 5 untuk R
1 .
; . ;
; ;
; ;
; ;
; . ;
. ; . ;
. ; ;
; ;
; ;
; ;
; R0=β1+β2β3+β4μh+r3+r4μv μh μh+δ+r3+r4+r1+r6+r6+r2 r3 1-α+r4
bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1- sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2
ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv iv[t] iv[t],sh[0] 1,ih[0] 0.00002,dh[0] 0.00012,iv[0] 0.0002},{sh[t],ih[t],dh[t],iv[t]},{t,0,100000
}] 33
{{sh[t]→InterpolatingFunction[{{0.,100000.}},][t],ih[t]→InterpolatingFunction[{{0.,100000.}},][ t],dh[t]→InterpolatingFunction[{{0.,100000.}},][t],iv[t]→InterpolatingFunction[{{0.,100000.}},]
[t]}} Gandi1=Plot[sh[t].bidsol,{t,0,160},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Red,Thick}]; Gandi2=Plot[ih[t].bidsol,{t,0,160},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Blue,Thick}]; Gandi3=Plot[dh[t].bidsol,{t,0,160},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Black,Thick}]; Gandi4=Plot[iv[t].bidsol,{t,0,160},PlotRange→All,FrameLabel→{Waktu},
Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Green,Thick}];
, ,
,
Lampiran 24 Gambar Dinamika populasi ketika vektor awal terinfeksi 50 untuk R
1 .
; . ;
; ;
; ;
; ;
; . ;
. ; . ;
. ; ;
; ;
; ;
; ;
; R0=β1+β2β3+β4μh+r3+r4μv μh μh+δ+r3+r4+r1+r6+r6+r2 r3 1-α+r4
bidsol=NDSolve[{μh 1-sh[t]-α r2 ih[t]-β1+β2iv[t] sh[t]+r1+r2ih[t]+r4 dh[t]+r5 1- sh[t]+ih[t]+dh[t] sh[t],β1+β2iv[t] sh[t]-r1+r2ih[t]-μh ih[t]-r7+r6ih[t]+r3 dh[t] ih[t],α r2
ih[t]-r3+r4+μhdh[t] dh[t],β3+β4ih[t]1-iv[t]-μv iv[t] iv[t],sh[0] 1,ih[0] 0.00002,dh[0] 0.00012,iv[0] 0.0002},{sh[t],ih[t],dh[t],iv[t]},{t,0,100000
}] {{sh[t]→InterpolatingFunction[{{0.,100000.}},][t],ih[t]→InterpolatingFunction[{{0.,100000.}},][
t],dh[t]→InterpolatingFunction[{{0.,100000.}},][t],iv[t]→InterpolatingFunction[{{0.,100000.}},] [t]}}
Gandi1=Plot[sh[t].bidsol,{t,0,160},PlotRange→All,FrameLabel→{Waktu}, Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Red,Thick}];
Gandi2=Plot[ih[t].bidsol,{t,0,160},PlotRange→All,FrameLabel→{Waktu}, Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Blue,Thick}];
Gandi3=Plot[dh[t].bidsol,{t,0,160},PlotRange→All,FrameLabel→{Waktu}, Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Black,Thick}];
Gandi4=Plot[iv[t].bidsol,{t,0,160},PlotRange→All,FrameLabel→{Waktu}, Frame→{{True,False},{True,False}},PlotStyle→{Dashed,Green,Thick}];
, ,
, 34
L A M P I R A N
Lampiran 1 Pembuktian Teorema 1
Teorema 1. Misalkan A, B, C bilangan-bilangan real. Bagian real dari setiap nilai eigen persamaan karakteristik
p λ = λ
3
+A λ
2
+ B λ + C = 0
adalah negatif jika dan hanya jika A, C, positif dan AB C. Bukti :
Dari persamaan
p λ = λ
3
+A λ
2
+ B λ + C, maka
a = 1, a
1
= A, a
2
= B, a
3
= C dan a
i
= 0 jika i selainnya. Berdasarkan kriteria Routh-Hurwitz, maka bagian real dari setiap akar polynomial p
λ = λ
3
+A λ
2
+ B λ + C adalah negatif jika dan hanya jika
│M
1
│.│M
2
│.│M
3
│positif, dimana :
│M
1
│=│ a
1
│= │A│= A 0 1
│M
2
│= =
= AB – C 0 2
│ M
3
│ = =
= A B C – C
2
0 3 Dari 1 maka diperoleh A 0
Dari 2 maka diperoleh AB – C 0 Dari 3 maka diperoleh ABC – C
2
0 yang dapat diubah dalam bentuk C AB – C 0, sehingga dari 2 diperoleh nilai C 0.
Dengan demikian diperoleh bahwa bagian real dari setiap akar polynomial p λ
= λ
3
+A λ
2
+ B λ + C adalah negatif jika dan hanya jika A 0, C 0 serta AB C.
Terbukti ■
Lampiran 2 Pembuktian Teorema 2
Teorema 2. Misalkan A, B, C dan D bilangan-bilangan real. Bagian real dari setiap nilai eigen persamaan karakteristik
p λ = λ
4
+A λ
3
+ B λ
2
+ C λ + D = 0
adalah negatif jika dan hanya jika A, C dan D positif dan ABC C
2
+ A
2
D. Bukti :
Dari persamaan
p λ = λ
4
+A λ
3
+ B λ
2
+ C λ + D, maka
a = 1, a
1
= A, a
2
= B, a
3
= C, a
4
= D dan a
i
= 0 jika i selainnya. Berdasarkan kriteria Routh-Hurwitz, maka bagian real dari setiap akar polinomial p
λ = λ
4
+A λ
3
+ B λ
2
+ C λ + D adalah negatif jika dan
hanya jika
│M
1
│,│M
2
│,│M
3
│,│M
4
│ positif, dimana :
│M
1
│=│ a
1
│= │A│= A 0 │M
2
│= =
= AB – C 0 │M
3
│= =
= ABC – A
2
D – C
2
│M
4
│= =
= D ABC – A
2
D – C
2
Dari 1 maka diperoleh A 0 Dari 3 dan 4 diperoleh D 0
Dari 2 dan 3, maka dapat ditulis CAB – C A
2
D, karena A
2
D 0 dan AB – C 0, sehingga diperoleh nilai C 0.
Persamaan 4 benar jika D 0 dan ABC C
2
+ A
2
D. Dengan demikian diperoleh bahwa bagian real dari setiap akar polinomial p
λ =
λ
3
+A λ
2
+ B λ + C adalah negatif jika dan hanya jika A 0, C 0, D 0 serta ABC
C
2
A
2
D.
Lampiran 3 Penurunan Persamaan 3.7 – 3.10
S
h
t ΛN
h
‐ r I
h
t ‐μ
h
S
h
t ‐ I
V
t S
h
t r r I
h
t r D
h
t r R
h
t s
h
t N
h
ΛN
h
‐ r i
h
t N
h
‐μ
h
s
h
t N
h
‐ i
v
t
A µ
V
s
h
t N
h
r r i
h
t N
h
r d
h
t N
h
r r
h
t N
h
s
h
t Λ‐ r i
h
t ‐μ
h
s
h
t ‐
A µ
V
A µ
V
i
v
t s
h
t r r i
h
t r d
h
t r r
h
t s
h
t Λ‐μ
h
s
h
t ‐ r i
h
t ‐
A µ
V
A µ
V
i
v
t s
h
t r r i
h
t r d
h
t r r
h
t s
h
t μ
h
‐s
h
t ‐ r i
h
t ‐ i
V
t s
h
t r r i
h
t r d
h
t r ‐ s
h t
i
h
t d
h
t
I
h
t I
V
t S
h
t ‐ r r I
h
t ‐μ
h
I
h
t ‐ r r I
h
t r D
h
t i
h
t N
h
i
v
t
A µ
V
s
h
t N
h
‐ r r i
h
t N
h
‐μ
h
i
h
t N
h
‐ r r i
h
t N
h
r d
h
t N
h
i
h
t
A µ
V
A µ
V
i
v
t
A µ
V
s
h
t ‐ r r i
h
t ‐μ
h
i
h
t ‐ r r i
h
t r d
h
t i
h
t i
v
t s
h
t ‐ r r i
h
t ‐μ
h
i
h
t ‐ r r i
h
t r d
h
t
D
h
t r I
h
t ‐ r r μ
h
D
h
t d
h
t N
h
r i
h
t N
h
‐ r r μ
h
d
h
t N
h
d
h
t r i
h
t ‐ r r μ
h
d
h
t I
V
t I
h
t S
V
t ‐μ
V
I
V
t i
v
t
A µ
V
i
h
t N
h
s
v
t
A µ
V
‐μ
V
i
v
t
A µ
V
i
v
t i
h
t s
V
t ‐μ
V
i
v
t i
v
t i
h
t ‐i
V
t ‐μ
V
i
v
t
Lampiran 4 Mencari Titik Tetap
Titik tetap akan diperoleh dengan menetapkan µ
h
1-s
h
t- αr i
h
t- +
i
v
t s
h
t+ r +r i
h
t r d
h
t+ r 1-s
ht
+i
h
t+d
h
t = 0 i
+ i
v
t s
h
t- r +r i
h
t-µ
h
i
h
t- r r i
h
t+ r d
h
t = 0 ii
αr i
h
t - r +r +µ
h
d
h
t = 0 iii
+ i
h
t1-i
v
t-µ
v
i
v
t = 0
iv
1.
Dari persamaan iv dapat disederhanakan agar diperoleh nilai i
v
+ i
h
t1-i
v
t-µ
v
i
v
t = 0
i
h
t = 0 dan i
V
t = 0 atau o
Dari Persamaan iii αr i
h
t - r +r +µ
h
d
h
t = 0 d
h
t = -
.
o Dari Persamaan i
µ
h
1-s
h
t- αr i
h
t- +
i
v
t s
h
t+ r +r i
h
t r d
h
t+ r 1-s
ht
+i
h
t+d
h
t = 0 µ
h
1-s
h
t- α.0- + .0 s
h
t+ r +r .0 r .0 + r 1-s
ht
+0+0 = 0 µ
h
1-s
h
t-0-0+0 0 + r 1-s
ht
= 0 µ
h
1-s
h
t+ r 1-s
ht
= 0 1-s
h
t µ
h
+ r = 0
1-s
h
t = 0 s
h
t = 1 17
2. Dari Persamaan iii dapat disederhanakan agar diperoleh d
h
αr i
h
t - r +r +µ
h
d
h
t = 0 r +r +µ
h
d
h
t = αr i
h
t d =
o Dari Persamaan ii
+ i
v
t s
h
t- r +r i
h
t-µ
h
i
h
t- r r i
h
t = 0 r +r i
h
t-µ
h
i
h
t- r r i
h
t+ r d
h
t = +
i
v
t s
h
t + r d
h
t i =
i = o
Dari Persamaan i µ
h
1-s
h
t- αr i
h
t- +
i
v
t s
h
t+ r +r i
h
t r d
h
t+ r 1-s
ht
+i
h
t+d
h
t = 0 µ
h
- µ
h
s
h
t- αr i
h
t- +
i
v
t s
h
t+ r +r i
h
t r d
h
t+ r -r s
h
t- r i
h
t- r d
h
t = 0 µ
h
s
h
t+ +
i
v
t s
h
t+ r s
h
t = µ
h
- αr i
h
t+ r +r i
h
t r d
h
t+ r - r i
h
t- r d
h
t s
h
t = s
h
t = s
h
t = s
o Dari Persamaan iv
+ i
h
t1-i
v
t-µ
v
i
v
t = 0
+ i
h
t +
i
h
ti
v
t - µ
v
i
v
t = 0
+ i
h
ti
v
t - µ
v
i
v
t =
+ i
h
t i
Untuk membuktikan titik tetap pertama dan titik tetap kedua digunakan Mathematica 7 seperti berikut :
Clear[μh,α,r1,r2,r3,r4,r5,r6,r7,β1,β2,β3,β4,μv,R0,Nh,A] δ=r1+r2+r3+r4+r6+r7;
R0=β1+β2β3+β4μh+r3+r4μv μh μh+δ+r3+r4+r1+r6+r6+r2 r3 1-α+r4; titik tetap
titet=Solve[{μh 1-sh-α r2 ih-β1+β2iv sh+r1+r2ih+r4 dh+r5 1-sh+ih+dh 0,β1+β2iv sh- r1+r2ih-μh ih-r7+r6ih+r3 dh 0,α r2 ih-r3+r4+μhdh 0,β3+β4ih 1-iv-μv
iv 0},{sh,ih,dh,iv}]FullSimplify; keadaan bebas endemik Subscript[E, 0]
titet[[1]] {dh→0,sh→1,iv→0,ih→0}
keadaan endemik Subscript[E, 1] titet[[2]]
{dh→-r2 α r5+μh -β1+β2 β3+β4 r3+r4+μh+r1 r3+r4+μh+r3+r4+μh r6+r7+μh+r2 r3+r4-r3 α+μh μvβ3+β4 r3+r4+μh -r2 -r3-r4+r3 α-α β1-α β2-μh r5+μh+r1 r3+r4+μh
r5+μh+r3+r4+μh r6+r7+μh β1+β2+μh+r5 r6+r7+β1+β2+μh, sh→r1 r3+r4+μh+r3+r4+μh r6+r7+μh+r2 r3+r4-r3 α+μh β3+β4 r3+r4+μh r5+μh+r2
r5 α+r5+r6+r7+r2 α μh+μh
2
+r3 r5+r6+r7+μh+r4 r5+r6+r7+μh μvβ3+β4 r3+r4+μh -r2 - r3-r4+r3 α-α β1-α β2-μh r5+μh+r1 r3+r4+μh r5+μh+r3+r4+μh r6+r7+μh β1+β2+μh+r5
r6+r7+β1+β2+μh, iv→-r5+μh -β1+β2 β3+β4 r3+r4+μh+r1 r3+r4+μh+r3+r4+μh r6+r7+μh+r2 r3+r4-r3
α+μh μvβ1+β2 β3+β4 r3+r4+μh r5+μh+r2 r5 α+r5+r6+r7+r2 α μh+μh
2
+r3 r5+r6+r7+μh+r4 r5+r6+r7+μh μv,
ih→-r5+μh -β1+β2 β3+β4 r3+r4+μh+r1 r3+r4+μh+r3+r4+μh r6+r7+μh+r2 r3+r4-r3 α+μh μvβ3+β4 -r2 -r3-r4+r3 α-α β1-α β2-μh r5+μh+r1 r3+r4+μh r5+μh+r3+r4+μh
r6+r7+μh β1+β2+μh+r5 r6+r7+β1+β2+μh} Sehingga disederhanakan :
s µ
r d i r
r α
r r
d µ β
β i r
i d r
β β i s r
r r
r µ
d αr i t r
r µ
i β
β i β β i
µ
Lampiran 5 Mencari Matriks Jacobi
Mencari Matriks Jacobi dengan menggunakan software Mathematica 7 sebagai berikut : Jacobi
Simplify µh
sh r ih
iv sh r
r ih r dh
r sh
ih dh
, iv sh
r r ih
µhih r
r ih r dh , r ih
r r
µh dh , ih
iv µviv
, sh , ih , dh , iv MatrixForm
r µh
iv iv
r r
r r
r r
sh iv
r r
r r
µh r
sh r
r r
µh iv
µv ih
Untuk Jacobi Bebas Endemik :
, ,
, ,
, ,
, .
, ,
, r
µh r
r r
r r
r r
r r
r µh
r r
r r
µh µv
Untuk Jacobi Endemik : 19
, ,
, ,
, ,
, .
, ,
, r
µh i
i r
r r
r r
r s
i r
r r
r µh
r s
r r
r µh
i µv
i
Lampiran 6 Persamaan Karakteristik tanpa Penyakit
Menentukan Persamaan Karakteristik Bebas endemik
Dengan μ
μ
μ
μ Akan diambil kolom untuk menentukan determinannya :
• μ
μ •
μ μ
μ μ
μ μ
μ •
μ μ
μ μ
μ μ
μ μ
μ μ μ
μ μ
μ μ μ
μ μ
μ μ
μ μ
μ μ
μ μ
μ μ
μ μ
μ •
μ μ
μ μ
μ μ
μ μ
μ μ
μ μ
μ μ
μ μ
μ μ
μ μ
μ μ
μ μ
μ μ
μ μ
μ μ
.
μ μ
μ μ
μ Maka :
μ μ
μ μ
μ μ
μ μ
μ μ
+µ + t
t t = 0
Dengan t
2
= μ
μ t
1
= μ
μ μ
μ r r
r r r r
r r r
r r
r t
= μ
μ μ
r r r r
r r r
r r
r r
r r
r r
R =
µ µ µ µ
Lampiran 7 Pembuktian t
2
. t
1
– t 0 untuk Tanpa Penyakit
Membuktikan t
2
. t
1
– t t
2
. t
1
– t = [
μ μ
][ μ
μ μ
μ r r
r r r r
r r r
r r
r μ
μ μ
r r r r
r r r
r r
= μ
μ μ
μ μ
r r r r
r r r
r r r
r r
μ μ μ
μ μ μ
μ r r μ r r
r μ r r
r μ r r
r r
μ μ
μ μ
r r r r
r r r
r r
μ μ μ μ
μ r r μ r r
μ r r r
r r
= μ
μ μ
μ μ
r r r r
r r r
r r r
r r
μ μ μ
μ μ r r
μ μ
μ μ
r r r r
r r r
r r 0
Lampiran 8 Mencari nilai w
3
, w
2
, w
1
dan w untuk
Persamaan Karakteristik Endemik
, ,
, ,
, , ,
, , ,
, , , , ,
r µh
i i
r r
r r
r r
s i
r r
r r
µh r
s r
r r
µh i
µv i
, ;
, , w
3
= r
r r
r r
r r
µh µv iv
sh , ,
w
2
= r r
r r r r
r r r r
r r r r
r r r r
r r r r
r r r r
r r r r
r µh r µh
r µh r µh
r µh r µh
r µh µh
r µv r µv r µv r µv r µv r µv r µv µhµv
r r
r r
r r
r µh sh
iv r r
r r
r r
µh µv sh
, , w
1
= r r r
r r r r r r
r r r r r r
r r r r r r
r r r r r r
r r µh r r µh r r µh r r µh r r µh r r µh r r µh r r µh r r µh r r µh r r µh r r µh r r µh r r µh r r µh r µh
r µh r µh
r µh r µh
r µh r µh
µh r r µv r r µv r r µv r r µv r r µv
r r µv r r µv r r µv r r µv r r µv r r µv r r µv r r µv r r µv r r µv
r µhµv r µhµv
r µhµv r µhµv
r µhµv r µhµv
r µhµv 22
µh µv r
r r
r r r
r r r r
r r r r
r r r r
r r r r
r r
r r
r r r r
r r r r
r r r r
r r r r
r r r
r r
r µh
µh r
µh r
µh r
µh r
µh r
µh r
µh µh
µh r
µh r
µh r
µh r
µh r
µh r
µh µh
µh µh
µh r
r r
r r
r r
µh sh iv r r
r r r r
r r r r
r r r r
r r r r
r µh r µh r µh r µh r µh µh r µv r µv r µv r µv r µv
µhµv r r
r µh µv
r r
r r
r r
r r
r r
r µh
µh µh sh
, , w