V. SIMPULAN DAN SARAN
5.2 Saran
Karena kestabilan sistem model SEIR terjadi setelah waktu yang relatif cukup lama untuk beberapa nilai parameter, khususnya pada subpopulasi manusia rentan, maka perlu dilakukan pengembangan model SEIR ini.
33
DAFTAR PUSTAKA
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Braun M. 1983. Differential Equations and Their Applications. New York. Springer-Verlag.
Derouich M, Boutayeb A, Twizell EH. 2003. A Model of Dengue Fever.
BioMedical Engineering OnLine 2 (4).
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Erickson RA, Presley SM, Allen LJS, Long KR, Cox SB. 2010. A Dengue Model with A Dynamic Aedes albopictus Vector Population. Ecological
Modelling 221, 2899–2908.
Estrada-Franco JGGB, Craig J. 1995. Biology, Disease Relationships, and Control of Aedes albopictus. Technical report. Pan American Health Organization.
Gratz NG. 2004. Critical Review of The Vector Status of Aedes albopictus.
Medical and Veterinary Entomology 18 (3), 215–227.
Gubler DJ. 1998. Dengue and Dengue Hemorrhagic Fever. Clinical Microbiology
Reviews 11 (3), 480–496.
Hawley WA. 1988. The Biology of Aedes albopictus. Journal of the American
Mosquito Control Association 4, 1–39.
Heymann DL. 2008. Control of Communicable Diseases Manual, 18th edition. American Public Health Association, Washington, DC.
Richards SL, Ponnusamy L, Unnasch TR, Hassan HK, Apperson CS. 2006. Host-Feeding Patterns of Aedes albopictus (Diptera: Culicidae) in Relation to Availability of Human and Domestic Animals in Suburban Landscapes of Central North Carolina. Journal of Medical
34 Tu PNV. 1994. Dinamical System, An Introduction with Applications in
Economics and Biology. New York: Springer-Verlag
van den Driessche P, Watmough J. 2008. Chapter 6: Further Notes on the Basic Reproduction Number. In: Brauer, F., van den Driessche, P., Wu, J. (Eds.), Mathematical Epidemiology, Vol. 1945. Lecture Notes in Mathematics, Springer, pp. 159–178.
Vazeille M, Mousson L, Failloux AB, 2003. Low Oral Receptivity for Dengue Type 2 Viruses of Aedes ablbopictus from Southeast Asia Compared with That of Aedes aegypti. American Journal of Tropical Medicine
and Hygiene 68 (2), 203–208.
Watmough J. 2008. Computation of The Basic Reproduction Number.
MITACS-PIMS Summer School on Mathematical Modelling of Infectious
35
Lampiran 1. Penentuan Titik Tetap
Clear@λ, α, µh, µv, τexh, τih, τexv, c, bi, pvh, n, nh, nv, sh, eh, ih, ev, ivD
H∗ Definisikan sistem persamaan diferensial 33 ∗L
f1@sh_, eh_, ih_, ev_, iv_D := λ − Hn ∗ c ∗ iv + µhL ∗ sh;
f2@sh_, eh_, ih_, ev_, iv_D := n ∗ c ∗ iv ∗ sh − Hτexh + µhL ∗ eh; f3@sh_, eh_, ih_, ev_, iv_D := τexh ∗ eh − Hτih + α + µhL ∗ ih; f4@sh_, eh_, ih_, ev_, iv_D :=
c ∗ ih ∗ H1 − ev − ivL − Hτexv + µvL ∗ ev;
f5@sh_, eh_, ih_, ev_, iv_D := τexv ∗ ev − µv ∗ iv;
sol = FullSimplify@Solve@
8f1@sh, eh, ih, ev, ivD 0, f2@sh, eh, ih, ev, ivD 0,
f3@sh, eh, ih, ev, ivD 0, f4@sh, eh, ih, ev, ivD 0, f5@sh, eh, ih, ev, ivD 0<, 8sh, eh, ih, ev, iv<DD
:8sh → λ ê µh, eh → 0, ih → 0, ev → 0, iv → 0<,
:sh → HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL ê Hc τexh Hc n τexv + µh Hµv + τexvLLL,
eh → −I−c2n λ τexh τexv + µh3µvHµv + τexvL +
α µh µvHµh + τexhL Hµv + τexvL + µh µv τexh Hµv + τexvL τih + µh2µvHµv + τexvL Hτexh + τihLM ë
Hc τexh Hµh + τexhL Hc n τexv + µh Hµv + τexvLLL, ih →I−µh µv Hµv + τexvL + Ic2n λ τexh τexvM ë
HHµh + τexhL Hα + µh + τihLLM ë Hc Hc n τexv + µh Hµv + τexvLLL, ev → Iµv I−µh Hα + µhL µv2Hµh + τexhL −
I−c2n λ τexh + µhHα + µhL µv Hµh + τexhLM τexv − µh µvHµh + τexhL Hµv + τexvL τihMM ë
Hc n τexv Hµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL, iv → 1
n H−µh ê c + Hλ τexh Hc n τexv + µh Hµv + τexvLLL ê
Lampiran 2. Penentuan Bilangan Reproduksi Dasar H√0)
Dengan mengambil sistem persamaan (22) dan menyusunnya kembali dalam urutan sub-sub popuasi yang menyebabkan infeksi saja, yaitu
Eh, Ev, Ih, dan Iv
diperoleh sistem persamaan sebagai berikut: dEhë dt = ncIvSh−Hτexh+ µhL Eh dEvê dt = cIh H1 − Ev − Iv L − Hτexv+ µvL Ev dIhë dt = τexhEh −Hτih+ α + µhL Ih dIvê dt = τexvEv − µvIv
Dari sistem di atas, diperoleh matriks-matriks
= ncIvSh cIh H1 − Ev− Iv L 0 0 dan = Hτexh+ µhL Eh Hτexv+ µvL Ev − τexhEh +Hτih+ α + µhL Ih − τexvEv+ µvIv sehingga, F = 0 0 0 ncλ µh 0 0 c 0 0 0 0 0 0 0 0 0 dan V = τexh+ µh 0 0 0 0 τexv+ µv 0 0 − τexh 0 τih+ α + µh 0 0 − τexv 0 µv
Selanjutnya, dihitung matriks K = FV-1 sebagai berikut:
K =
0 c nλ τexv
µhµvHµv+τexvL 0 c nλ µhµv
c Hµv2τexh+µvτexhτexvL
µvHµh+τexhL Hµv+τexvL Hα+µh+τihL 0 α+µc
h+τih 0
0 0 0 0
0 0 0 0
sehingga bilangan reproduksi dasar adalah
√0 = c n λ τexh τexv µhµvHµh+τexhL Hµv+τexvL Hα+µh+τihL dan x = √02 = c2nλ τexhτexv µhµvHµh+τexhL Hµv+τexvL Hα+µh+τihL
Lampiran 3. Analisis Kestabilan Titik Tetap Tanpa Penyakit
Clear@jD
H∗ Definisikan Matriks Jacobi ∗L
j = D@8f1@sh, eh, ih, ev, ivD, f2@sh, eh, ih, ev, ivD,
f3@sh, eh, ih, ev, ivD, f4@sh, eh, ih, ev, ivD,
f5@sh, eh, ih, ev, ivD<, 88sh, eh, ih, ev, iv<<D;
Clear@jt1, p1, perskar1, λ, α, µh,
µv, τexh, τih, τexv, c, bi, pvh, n, nh, nvD
H∗ Matriks Jacobi untuk Titik Tetap Tanpa Penyakit ∗L
jt1 = FullSimplify@j ê. sol@@1DDD;
H∗ Nilai Eigen Pertama dan
Ruas Kiri Persamaan Karakteristik ∗L
p1 = FullSimplify@Det@jt1 − ψ IdentityMatrix@5DDD;
H∗ Ruas Kiri Persamaan Karakteristik ∗L
perskar1 = Collect@p1@@2DD, ψD;
H∗ Koefisien Persamaan Karakteristik ∗L
Clear@a0, a1, a2, a3, a4D
a0 = FullSimplify@CoefficientList@perskar1, ψD@@5DDD
1
a1 = FullSimplify@CoefficientList@perskar1, ψD@@4DDD
α +2 µh + 2 µv + τexh + τexv + τih
a2 = FullSimplify@CoefficientList@perskar1, ψD@@3DDD
µh2+ µv2+2 µv τexh + µv τexv + τexh τexv + α Hµh + 2 µv + τexh + τexvL + H2 µv + τexh + τexvL τih + µh H4 µv + τexh + 2 τexv + τihL
a3 = FullSimplify@CoefficientList@perskar1, ψD@@2DDD
α µv H2 µh + µv + 2 τexhL + α Hµh + µv + τexhL τexv + µv τexh Hµv + τexvL + µh2H2 µv + τexvL + Hµv Hµv + 2 τexhL + Hµv + τexhL τexvL τih +
µh I2 µv2+ τexv Hτexh + τihL + 2 µv Hτexh + τexv + τihLM
a4 = FullSimplify@CoefficientList@perskar1, ψD@@1DDD
1 ê µh I−c2n λ τexh τexv +
µh3µv Hµv + τexvL + α µh µv Hµh + τexhL Hµv + τexvL +
Selanjutnya dilakukan penyederhanaan terhadap a4 sehingga diperoleh a4=1 ê µh
I−c2n λ τexh τexv + µh3µv Hµv + τexvL + α µh µv Hµh + τexhL Hµv + τexvL + µh µv τexh Hµv + τexvL τih + µh2µv Hµv + τexvL Hτexh + τihLM;
=1 ê µh I−c2n λ τexh τexv + µh µv Hµh + τexhL Hµv + τexvL Hα + µh + τihLM;
= µv Hµh + τexhL Hµv + τexvL Hα + µh + τihL H1 − ξL;
ξ =Ic2n λ τexh τexvM ë Hµh µv Hµh + τexhL Hµv + τexvL Hα + µh + τihLL
H∗ Masukan Nilai−nilai Parameter ∗L λ = 2.244 ∗ 10−5; α = 0.003; µh = 1ê 28 000; µv = 1ê 20; τexh = 1 ê 10; τih = 1 ê 4; τexv = 1ê 9; bi = 0.3; pvh = 0.4; c = bi pvh; nh = 1; nv = 1; n = nhê nv; H∗ Definisikan ξ ∗L Clear@ξD ξ =Ic2n λ τexh τexvM ë
Hµh µv Hµv + τexvL Hµh + τexhL Hµh + α + τihLL; H∗ Hitung a4 yang Belum Disederhanakan ∗L
a4
H∗ Hitung a4 yang Sudah Disederhanakan ∗L
Clear@a4D;
a4 = µvHµh + τexhL Hµv + τexvL Hα + µh + τihL H1 − ξL
0.000103376 0.000103376
Clear@ξ, a4D
H∗ Ambil sebarang nilai ξ < 1 ∗L ξ = 0.5;
H∗ Definisikan Koefisien a4 yang Sudah Disederhanakan ∗L
a4 = µvHµh + τexhL Hµv + τexvL Hα + µh + τihL H1 − ξL;
H∗ Pembuktian Kriteria Routh−Hurwitz ∗L
a1 > 0 a2 > 0 a3 > 0 a4 > 0 a1 a2 a3 > a32+ a12a4 True True True True True
Lampiran 4. Analisis Ketabilan Titik Tetap Endemik
Clear@jt2, brs, perskar2, λ, α, µh,
µv, τexh, τih, τexv, c, bi, pvh, n, nh, nvD
H∗ Matriks Jacobi untuk Titik Tetap Tanpa Penyakit ∗L
jt2 = FullSimplify@j ê. sol@@2DDD;
H∗ Unsur−unsur Baris Ke−p Matriks JT2 ∗L
brs@p_D := jt2@@pDD
H∗ Unsur−unsur Baris Ke−1 Matriks JT2 ∗L
brs@1D
8−Hc λ τexh Hc n τexv + µh Hµv + τexvLLL ê
HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL, 0, 0, 0,−Hn Hµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL ê
Hτexh Hc n τexv + µh Hµv + τexvLLL<
H∗ Unsur−unsur Baris Ke−2 Matriks JT2 ∗L
brs@2D
8c H−µh ê c + Hλ τexh Hc n τexv + µh Hµv + τexvLLL ê
HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLLL, −µh − τexh, 0, 0,Hn Hµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL ê Hτexh Hc n τexv + µh Hµv + τexvLLL<
H∗ Unsur−unsur Baris Ke−3 Matriks JT2 ∗L
brs@3D
80, τexh, −α − µh − τih, 0, 0<
H∗ Unsur−unsur Baris Ke−4 Matriks JT2 ∗L
brs@4D
90, 0, Hµv Hµh + τexhL Hc n τexv + µh Hµv + τexvLL Hα + µh + τihLL ê Hn τexv Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL,
τexvH−1 − Hc n Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL ê HHµh + τexhL Hc n τexv + µh Hµv + τexvLL Hα + µh + τihLLL, −I−µh µv Hµv + τexvL + Ic2nλ τexh τexvM ë
HHµh + τexhL Hα + µh + τihLLM ë Hc n τexv + µh Hµv + τexvLL= H∗ Unsur−unsur Baris Ke−5 Matriks JT2 ∗L
brs@5D
H∗ Ruas Kiri Persamaan Karakteristik ∗L
perskar2 =
Collect@HDet@jt2 − ψ IdentityMatrix@5DDL ∗ H−1L, ψD;
H∗ Koefisien Persamaan Karakteristik ∗L
Clear@a0, a1, a2, a3, a4, a5D
a0 = CoefficientList@perskar2, ψD@@6DD; a1 = CoefficientList@perskar2, ψD@@5DD; a2 = CoefficientList@perskar2, ψD@@4DD; a3 = CoefficientList@perskar2, ψD@@3DD; a4 = CoefficientList@perskar2, ψD@@2DD; a5 = CoefficientList@perskar2, ψD@@1DD;
Selanjutnya dilakukan penyederhanaan terhadap a2, a3, a4, dan a5 sehingga diperoleh
a2= α µh + µh2+ α µv + 2 µh µv +
α τexh + µh τexh + µv τexh + µh τih + µv τih + τexh τih +
Hc λ τexh Hc n τexv + µh Hµv + τexvLL Hα + 2 µh + µv + τexh + τihLL ê HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL +
τexvHα + 2 µh + µv + τexh + τih + Hc λ τexh Hc n τexv + µh Hµv + τexvLLL ê HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLLL
H1 + Hc n Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL ê
HHµh + τexhL Hc n τexv + µh Hµv + τexvLL Hα + µh + τihLLL + Hµh µv τexv Hµv + τexvLL ê Hc n τexv + µh Hµv + τexvLL H−1 + ξL; a3= α µh µv + µh2µv + α µv τexh + µh µv τexh + µh µv τih +
µv τexh τih +Ic λ τexh Hc n τexv + µh Hµv + τexvLL Iµh2+ µv τexh + αHµh + µv + τexhL + Hµv + τexhL τih + µh H2 µv + τexh + τihLMM ë HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL +
τexvIα µh + µh2+ α µv + 2 µh µv + α τexh + µh τexh + µv τexh + µh τih + µv τih + τexh τih +Hc λ τexh Hc n τexv + µh Hµv + τexvLL τihL ê
HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLLM H1 + Hc n Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL ê
HHµh + τexhL Hc n τexv + µh Hµv + τexvLL Hα + µh + τihLLL + Hµh µv τexv Hµv + τexvL HHα + 2 µh + τexh + τihL ê
Hc n τexv + µh Hµv + τexvLL + Hc λ τexhL ê
HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLLLL H−1 + ξL; a4=
Ic λ τexh Iµv Hµh + τexhL2Hc n τexv + µh Hµv + τexvLL Hα + µh + τihL2+ τexv Ic2nλ τexh + c nHµh + τexhL Hµv + τexvL Hα + µh + τihL +
µhHµh + τexhL Hµv + τexvL Hα + µh + τihLM Iµh2+ µv τexh + αHµh + µv + τexhL + Hµv + τexhL τih + µh H2 µv + τexh + τihLMMM ë HHµh + τexhL Hµv + τexvL Hα + µh + τihL
Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL + IIµh µv Hµh + τexhL Hµv + τexvL2Hα + µh + τihLM ë
Hc n τexv + µh Hµv + τexvLL +
Hc λ µh µv τexh τexv Hα + 2 µh + τexh + τihLL ê
Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLM H−1 + ξL; a5= µh µvHµh + τexhL Hµv + τexvL Hα + µh + τihL H−1 + ξL;
H∗ Masukan Nilai−nilai Parameter ∗L λ = 2.244 ∗ 10−5; α = 0.003; µh = 1ê 28 000; µv = 1ê 20; τexh = 1 ê 10; τih = 1 ê 4; τexv = 1ê 9; bi = 0.3; pvh = 0.4; c = bi pvh; nh = 1; nv = 1; n = nhê nv; H∗ Definisikan ξ ∗L Clear@ξD ξ =
Ic2n λ τexh τexvM ë Hµh µv Hµv + τexvL Hµh + τexhL Hµh + α + τihLL;
H∗ Hitung a2 yang Belum Disederhanakan ∗L
a2
H∗ Hitung a2 yang Sudah Disederhanakan ∗L
Clear@a2D
a2 = α µh + µh2+ α µv + 2 µh µv + α τexh +
µh τexh + µv τexh + µh τih + µv τih + τexh τih +
Hc λ τexh Hc n τexv + µh Hµv + τexvLL Hα + 2 µh + µv + τexh + τihLL ê HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL + τexv Hα + 2 µh + µv + τexh + τih + Hc λ τexh Hc n τexv + µh Hµv + τexvLLL ê
HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLLL H1 + Hc n Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL ê
HHµh + τexhL Hc n τexv + µh Hµv + τexvLL Hα + µh + τihLLL + Hµh µv τexv Hµv + τexvLL ê Hc n τexv + µh Hµv + τexvLL H−1 + ξL 0.10791
0.10791
H∗ Hitung a3 yang Belum Disederhanakan ∗L
a3
H∗ Hitung a3 yang Sudah Disederhanakan ∗L
Clear@a3D
a3 = α µh µv + µh2µv + α µv τexh +
µh µv τexh + µh µv τih + µv τexh τih +Ic λ τexh
Hc n τexv + µh Hµv + τexvLL Iµh2+ µv τexh + αHµh + µv + τexhL + Hµv + τexhL τih + µh H2 µv + τexh + τihLMM ë
HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL +
τexvIα µh + µh2+ α µv + 2 µh µv + α τexh + µh τexh + µv τexh + µh τih + µv τih + τexh τih +Hc λ τexh Hc n τexv + µh Hµv + τexvLL τihL ê
HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLLM H1 + Hc n Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL ê
HHµh + τexhL Hc n τexv + µh Hµv + τexvLL Hα + µh + τihLLL + Hµh µv τexv Hµv + τexvL HHα + 2 µh + τexh + τihL ê
Hc n τexv + µh Hµv + τexvLL + Hc λ τexhL ê HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLLLL H−1 + ξL 0.00818896
H∗ Hitung a4 yang Belum Disederhanakan ∗L
a4
H∗ Hitung a4 yang Sudah Disederhanakan ∗L
Clear@a4D
a4 =Ic λ τexh
Iµv Hµh + τexhL2Hc n τexv + µh Hµv + τexvLL Hα + µh + τihL2+
τexvIc2n λ τexh + c nHµh + τexhL Hµv + τexvL Hα + µh + τihL +
µhHµh + τexhL Hµv + τexvL Hα + µh + τihLM
Iµh2+ µv τexh + αHµh + µv + τexhL + Hµv + τexhL τih + µhH2 µv + τexh + τihLMMM ë
HHµh + τexhL Hµv + τexvL Hα + µh + τihL
Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL + IIµh µv Hµh + τexhL Hµv + τexvL2Hα + µh + τihLM ë
Hc n τexv + µh Hµv + τexvLL +
Hc λ µh µv τexh τexv Hα + 2 µh + τexh + τihLL ê Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLM H−1 + ξL 9.9597× 10−8
9.9597× 10−8
H∗ Hitung a5 yang Belum Disederhanakan ∗L
a5
H∗ Hitung a5 yang Sudah Disederhanakan ∗L
Clear@a5D
a5 = µh µvHµh + τexhL Hµv + τexvL Hα + µh + τihL H−1 + ξL
− 3.692 × 10−9 − 3.692 × 10−9
Clear@ξ, a2, a3, a4, a5D
H∗ Ambil sebarang nilai ξ > 1 ∗L ξ = 1.5;
H∗ Definisikan Koefisien−koefisien a2,a3,
a4 dan a5 yang Sudah Disederhanakan ∗L
a2 = α µh + µh2+ α µv + 2 µh µv + α τexh +
µh τexh + µv τexh + µh τih + µv τih + τexh τih +
Hc λ τexh Hc n τexv + µh Hµv + τexvLL Hα + 2 µh + µv + τexh + τihLL ê HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL +
τexvHα + 2 µh + µv + τexh + τih +
Hc λ τexh Hc n τexv + µh Hµv + τexvLLL ê
HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLLL H1 + Hc n Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL ê
HHµh + τexhL Hc n τexv + µh Hµv + τexvLL Hα + µh + τihLLL + Hµh µv τexv Hµv + τexvLL ê Hc n τexv + µh Hµv + τexvLL H−1 + ξL;
a3 = α µh µv + µh2µv + α µv τexh +
µh µv τexh + µh µv τih + µv τexh τih +Ic λ τexh
Hc n τexv + µh Hµv + τexvLL Iµh2+ µv τexh + αHµh + µv + τexhL + Hµv + τexhL τih + µh H2 µv + τexh + τihLMM ë
HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL + τexv Iα µh + µh2+ α µv + 2 µh µv + α τexh + µh τexh + µv τexh + µh τih +
µv τih + τexh τih +Hc λ τexh Hc n τexv + µh Hµv + τexvLL τihL ê HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLLM H1 + Hc n Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL ê
HHµh + τexhL Hc n τexv + µh Hµv + τexvLL Hα + µh + τihLLL + Hµh µv τexv Hµv + τexvL HHα + 2 µh + τexh + τihL ê
Hc n τexv + µh Hµv + τexvLL + Hc λ τexhL ê HHµv + τexvL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLLLL H−1 + ξL;
a4 =Ic λ τexh Iµv Hµh + τexhL2Hc n τexv + µh Hµv + τexvLL Hα + µh + τihL2+ τexv
Ic2n λ τexh + c nHµh + τexhL Hµv + τexvL Hα + µh + τihL +
µhHµh + τexhL Hµv + τexvL Hα + µh + τihLM
Iµh2+ µv τexh + αHµh + µv + τexhL + Hµv + τexhL τih + µhH2 µv + τexh + τihLMMM ë
HHµh + τexhL Hµv + τexvL Hα + µh + τihL Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLL + IIµh µv Hµh + τexhL Hµv + τexvL2Hα + µh + τihLM ë
Hc n τexv + µh Hµv + τexvLL +
Hc λ µh µv τexh τexv Hα + 2 µh + τexh + τihLL ê
Hc λ τexh + µv Hµh + τexhL Hα + µh + τihLLM H−1 + ξL;
a5 = µh µvHµh + τexhL Hµv + τexvL Hα + µh + τihL H−1 + ξL;
H∗ Pembuktian Kriteria Routh−Hurwitz ∗L
a1 > 0 a2 > 0 a3 > 0 a4 > 0 a5 > 0 a1 a2 a3 > a32+ a12a4
Ha1 a4 − a5L Ia1 a2 a3 − a32− a12a4M > a5 Ha1 a2 − a3L2+ a1 a52
True True True True True True True
Lampiran 5. Program Simulasi untuk Kondisi ¬0< 1
H∗ Scipt Gambar 3 ∗L
Clear@µv, c, bi, T, sh0, eh0, ih0, ev0, iv0D;
Needs@"PlotLegends`"D;
ManipulateAModuleA8sol1, sol2, sh, eh, ih, rh, sv, ev, iv, t<,
rh@t_D := nh − Hsh@tD + eh@tD + ih@tDL;
sv@t_D := nv − Hev@tD + iv@tDL; sol1 = With@8c = bi ∗ pvh<,
NDSolve@8sh '@tD λ − Hn ∗ c ∗ iv@tD + µhL ∗ sh@tD,
eh '@tD n ∗ c ∗ iv@tD ∗ sh@tD − Hτexh + µhL ∗ eh@tD,
ih '@tD τexh ∗ eh@tD − Hτih + α + µhL ∗ ih@tD, ev'@tD
c ∗ ih@tD ∗ H1 − ev@tD − iv@tDL − Hτexv + µvL ∗ ev@tD,
iv '@tD τexv ∗ ev@tD − µv ∗ iv@tD, sh@0D sh0,
eh@0D eh0, ih@0D ih0, ev@0D ev0, iv@0D iv0<, 8sh, eh, ih, ev, iv<, 8t, 0, T<DD;
sol2@x_D := First@x ê. sol1D;
IfAvector, Plot@8sol2@sv@tDD, sol2@ev@tDD, sol2@iv@tDD<, 8t, 0, T<, PlotRange → 880, T<, 80, 1<<,
ImageSize → 400, AxesLabel →8"Hari t"<, PlotStyle →
88Black, Thick<, 8Blue, Thick<, 8Red, Thick<<,
PlotLegend →8"Sv HtL", "Ev HtL", "Iv HtL"<, LegendSize → 0.45, LegendPosition →81, − 0.25<, ShadowBackground → WhiteD,
PlotA8sol2@sh@tDD, sol2@eh@tDD, sol2@ih@tDD,
sol2@rh@tDD<, 8t, 0, T<, PlotRange → 880, T<, 80, 1<<,
ImageSize → 400, AxesLabel →8"Hari t"<,
PlotStyle →88Black, Thick<, 8Blue, Thick<,
8Red, Thick<, 8Green, Thick<<,
PlotLegend →9"Sh HtL", "Eh HtL", "Ih HtL", "Rh HtL"=, LegendSize → 0.6, LegendPosition →81, − 0.3<,
ShadowBackground → WhiteEEE,
88vector, False, "Lihat Populasi Nyamuk"<, 8False, True<<,
88µv, 0.07, "Laju Kematian Nyamuk µv"<, 0.01,
0.09, 0.01, Appearance → "Labeled"<, 88bi, 0.3, "Rata−rata Gigitan Nyamuk bi"<,
0.3, 0.45, 0.01, Appearance → "Labeled"<,
H∗ Scipt Gambar 4 dan 5 ∗L
Clear@sol3, plot1, plot2, x, y, z,
µv, c, bi, R0, sh, eh, ih, rh, sv, ev, ivD;
rh@t_D := nh − Hsh@tD + eh@tD + ih@tDL;
sv@t_D := nv − Hev@tD + iv@tDL;
sol3@x_, µv_, bi_D := First@x ê. With@8c = bi ∗ pvh<,
NDSolve@8sh '@tD λ − Hn ∗ c ∗ iv@tD + µhL ∗ sh@tD,
eh '@tD n ∗ c ∗ iv@tD ∗ sh@tD − Hτexh + µhL ∗ eh@tD,
ih '@tD τexh ∗ eh@tD − Hτih + α + µhL ∗ ih@tD, ev'@tD
c ∗ ih@tD ∗ H1 − ev@tD − iv@tDL − Hτexv + µvL ∗ ev@tD,
iv '@tD τexv ∗ ev@tD − µv ∗ iv@tD, sh@0D sh0,
eh@0D eh0, ih@0D ih0, ev@0D ev0, iv@0D iv0<, 8sh, eh, ih, ev, iv<, 8t, 0, T<DDD;
plot1@x_, y_, z_, xx_StringD := Plot@
Evaluate@Table@sol3@x, i, 0.30D, 8i, 0.03, 0.09, 0.02<DD,
8t, 0, T<, PlotRange → 880, 40<, 8y, z<<,
AxesLabel →8"Hari t", xx<,
PlotStyle →88Black, Thick<, 8Blue, Thick<,
8Red, Thick<, 8Green, Thick<<, PlotLegend →
8"µv = 0.03", "µv = 0.05", "µv = 0.07", "µv = 0.09"<,
LegendPosition →81, − 0.3<, LegendTextSpace → 3,
LegendSize → 0.5, ShadowBackground → WhiteD;
plot2@x_, y_, z_, xx_StringD := Plot@
Evaluate@Table@sol3@x, 0.05, jD, 8j, 0.25, 0.40, 0.05<DD,
8t, 0, T<, PlotRange → 880, 40<, 8y, z<<,
AxesLabel →8"Hari t", xx<,
PlotStyle →88Black, Thick<, 8Blue, Thick<,
8Red, Thick<, 8Green, Thick<<, PlotLegend →
8"bi = 0.25", "bi = 0.30", "bi = 0.35", "bi = 0.40"<,
LegendPosition →81, − 0.3<, LegendTextSpace → 3,
LegendSize → 0.5, ShadowBackground → WhiteD;
plot1Ash@tD, 0.4, 1, "Sh
HtL"E;
plot1Aeh@tD, 0, 0.15, "Eh
HtL"E;
plot1Aih@tD, 0, 0.05, "Ih
HtL"E;
plot1Arh@tD, 0, 0.5, "Rh
HtL"E;
plot1@sv@tD, 0.75, 1, "Sv
HtL"D;
plot1@ev@tD, 0, 0.04, "Ev
HtL"D;
plot1@iv@tD, 0, 0.2, "Iv
HtL"D;
plot2Ash@tD, 0.4, 1, "Sh
HtL"E;
plot2Aeh@tD, 0, 0.15, "Eh
HtL"E;
plot2Aih@tD, 0, 0.06, "Ih
HtL"E;
plot2Arh@tD, 0, 0.4, "Rh
HtL"E;
plot2@sv@tD, 0.75, 1, "Sv
HtL"D;
plot2@ev@tD, 0, 0.05, "Ev
HtL"D;
plot2@iv@tD, 0.03, 0.2, "Iv
Lampiran 6. Program Simulasi untuk Kondisi ¬0> 1
H∗ Scipt Gambar 6 ∗L
Clear@µv, c, biD
ManipulateA
ModuleA8sol4, sol5, sh, eh, ih, rh, sv, ev, iv, t<,
rh@t_D := nh − Hsh@tD + eh@tD + ih@tDL;
sv@t_D := nv − Hev@tD + iv@tDL; sol4 = With@8c = bi ∗ pvh<,
NDSolve@8sh'@tD λ − Hn ∗ c ∗ iv@tD + µhL ∗ sh@tD,
eh '@tD n ∗ c ∗ iv@tD ∗ sh@tD − Hτexh + µhL ∗ eh@tD,
ih '@tD τexh ∗ eh@tD − Hτih + α + µhL ∗ ih@tD, ev'@tD
c ∗ ih@tD ∗ H1 − ev@tD − iv@tDL − Hτexv + µvL ∗ ev@tD,
iv '@tD τexv ∗ ev@tD − µv ∗ iv@tD, sh@0D sh0,
eh@0D eh0, ih@0D ih0, ev@0D ev0, iv@0D iv0<, 8sh, eh, ih, ev, iv<, 8t, 0, T<DD;
sol5@x_D := First@x ê. sol4D;
IfAvector,
Plot@8sol5@sv@tDD, sol5@ev@tDD, sol5@iv@tDD<, 8t, 0, T<, PlotRange → 880, T<, 80, 1<<,
ImageSize → 400, AxesLabel →8"Hari t"<, PlotStyle →
88Black, Thick<, 8Blue, Thick<, 8Red, Thick<<,
PlotLegend →8"Sv HtL", "Ev HtL", "Iv HtL"<, LegendSize → 0.45, LegendPosition →81, −0.25<, ShadowBackground → WhiteD,
PlotA8sol5@sh@tDD, sol5@eh@tDD, sol5@ih@tDD,
sol5@rh@tDD<, 8t, 0, T<, PlotRange → 880, T<, 80, 1<<,
ImageSize → 400, AxesLabel →8"Hari t"<,
PlotStyle →88Black, Thick<, 8Blue, Thick<,
8Red, Thick<, 8Green, Thick<<,
PlotLegend →9"Sh HtL", "Eh HtL", "Ih HtL", "Rh HtL"=, LegendSize → 0.6, LegendPosition →81, −0.3<,
ShadowBackground → WhiteEEE,
88vector, False, "Lihat Populasi Nyamuk"<, 8False, True<<, 88µv, 0.01, "Laju Kematian Nyamuk HµvL"<,
0.01, 0.09, 0.01, Appearance → "Labeled"<, 88bi, 0.41, "Rata−rata Gigitan Nyamuk HbiL"<,
0.3, 0.45, 0.01, Appearance → "Labeled"<,
H∗ Scipt Gambar 7 dan 8 ∗L
Clear@sol6, plot3, plot4, x, y, z,
µv, c, bi, R0, sh, eh, ih, rh, sv, ev, ivD
rh@t_D := nh − Hsh@tD + eh@tD + ih@tDL;
sv@t_D := nv − Hev@tD + iv@tDL;
sol6@x_, µv_, bi_D := First@x ê. With@8c = bi ∗ pvh<,
NDSolve@8sh'@tD λ − Hn ∗ c ∗ iv@tD + µhL ∗ sh@tD,
eh '@tD n ∗ c ∗ iv@tD ∗ sh@tD − Hτexh + µhL ∗ eh@tD,
ih '@tD τexh ∗ eh@tD − Hτih + α + µhL ∗ ih@tD, ev'@tD
c ∗ ih@tD ∗ H1 − ev@tD − iv@tDL − Hτexv + µvL ∗ ev@tD,
iv '@tD τexv ∗ ev@tD − µv ∗ iv@tD, sh@0D sh0,
eh@0D eh0, ih@0D ih0, ev@0D ev0, iv@0D iv0<, 8sh, eh, ih, ev, iv<, 8t, 0, T<DDD;
plot3@x_, y_, z_, xx_StringD := Plot@
Evaluate@Table@sol6@x, i, 0.40D, 8i, 0.01, 0.07, 0.02<DD,
8t, 0, T<, PlotRange → 880, 45<, 8y, z<<,
AxesLabel →8"Hari t", xx<,
PlotStyle →88Black, Thick<, 8Blue, Thick<,
8Red, Thick<, 8Green, Thick<<, PlotLegend →
8"µv = 0.01", "µv = 0.03", "µv = 0.05", "µv = 0.07"<,
LegendPosition →81, − 0.3<, LegendTextSpace → 3,
LegendSize → 0.5, ShadowBackground → WhiteD;
plot4@x_, y_, z_, xx_StringD := Plot@
Evaluate@Table@sol6@x, 0.03, jD, 8j, 0.40, 0.55, 0.05<DD,
8t, 0, T<, PlotRange → 880, 45<, 8y, z<<,
AxesLabel →8"Hari t", xx<,
PlotStyle →88Black, Thick<, 8Blue, Thick<,
8Red, Thick<, 8Green, Thick<<, PlotLegend →
8"bi = 0.40", "bi = 0.45", "bi = 0.50", "bi = 0.55"<,
LegendPosition →81, − 0.3<, LegendTextSpace → 3,
LegendSize → 0.5, ShadowBackground → WhiteD;
plot3Ash@tD, 0, 1, "Sh
HtL"E;
plot3Aeh@tD, 0, 0.2, "Eh
HtL"E;
plot3Aih@tD, 0, 0.08, "Ih
HtL"E;
plot3Arh@tD, 0, 0.7, "Rh
HtL"E;
plot3@sv@tD, 0.6, 1, "Sv
HtL"D;
plot3@ev@tD, 0, 0.07, "Ev
HtL"D;
plot3@iv@tD, 0, 0.35, "Iv
HtL"D;
plot4Ash@tD, 0, 1, "Sh
HtL"E;
plot4Aeh@tD, 0, 0.25, "Eh
HtL"E;
plot4Aih@tD, 0, 0.08, "Ih
HtL"E;
plot4Arh@tD, 0, 0.7, "Rh
HtL"E;
plot4@sv@tD, 0.6, 0.85, "Sv
HtL"D;
plot4@ev@tD, 0, 0.09, "Ev
HtL"D;
plot4@iv@tD, 0.1, 0.3, "Iv