w w 0 untuk Endemik

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, , ,PlotRange฀All,FrameLabel฀ Waktu , Frame฀ True,False , True,False ,PlotStyle฀ Dashed,Blue,Thick ; 32 Gandi Plot dh t .bidsol, t, , ,PlotRange฀All,FrameLabel฀ Waktu , Frame฀ True,False , True,False ,PlotStyle฀ Dashed,Black,Thick ; Gandi Plot iv t .bidsol, t, , ,PlotRange฀All,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, , ,PlotRange฀All,FrameLabel฀ Waktu , Frame฀ True,False , True,False ,PlotStyle฀ Dashed,Blue,Thick ; Gandi Plot dh t .bidsol, t, , ,PlotRange฀All,FrameLabel฀ Waktu , Frame฀ True,False , True,False ,PlotStyle฀ Dashed,Black,Thick ; Gandi Plot iv t .bidsol, t, , ,PlotRange฀All,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