BAB IV PENYELESAIAN NUMERIS MODEL MATEMATIS…
C. Perbandingan Hasil Aproksimasi
Pada subbab ini akan dijabarkan hasil aproksimasi untuk beberapa nilai 𝑡 pada interval 𝑡 = 0 sampai 𝑡 = 50 dengan menggunakan fungsi 𝑜𝑑𝑒𝑖𝑛𝑡 pada Py-thon dan metode Runge-Kutta orde empat.
1. Populasi 𝑆𝑢𝑠𝑐𝑒𝑝𝑡𝑖𝑏𝑙𝑒 (𝑆)
Pada bagian ini akan ditampilkan hasil aproksimasi pada kelas 𝑆𝑢𝑠𝑐𝑒𝑝𝑡𝑖𝑏𝑙𝑒 (𝑆) yang disajikan pada Tabel 4.3.
Tabel 4.3. Hasil aproksimasi menggunakan fungsi 𝑜𝑑𝑒𝑖𝑛𝑡 dan metode Runge-Kutta orde empat.
t 𝑂𝑑𝑒𝑖𝑛𝑡 Runge-Kutta
orde empat Error mutlak Error relatif
0 1000 1000 0 0
5 974.588046 974.588046 2.61 × 10−8 2.68 × 10−11 10 890.1878908 890.1878848 6.01 × 10−6 6.75 × 10−9 15 585.950212 585.9501965 1.55 × 10−5 2.64 × 10−8 20 182.3766919 182.3766784 1.34 × 10−5 7.37 × 10−8 25 58.18510244 58.18509979 2.66 × 10−6 4.57 × 10−8 30 36.66311218 36.66311116 1.02 × 10−6 2.77 × 10−8 35 31.94440643 31.94440623 1.99 × 10−7 6.22 × 10−9 40 30.54455462 30.54455455 7.70 × 10−8 2.52 × 10−9 45 29.92693695 29.92693691 3.86 × 10−8 1.29 × 10−9 50 29.51241715 29.51241712 3.03 × 10−8 1.03 × 10−7 Rata-rata 3.54× 10−6 1. 74× 10−8
2. Populasi 𝐸𝑥𝑝𝑜𝑠𝑒𝑑 (𝐸)
Pada bagian ini akan ditampilkan hasil aproksimasi pada kelas 𝐸𝑥𝑝𝑜𝑠𝑒𝑑 (𝐸) yang disajikan pada Tabel 4.4.
Tabel 4.4. Hasil aproksimasi menggunakan fungsi 𝑜𝑑𝑒𝑖𝑛𝑡 dan metode Runge-Kutta orde empat.
t 𝑂𝑑𝑒𝑖𝑛𝑡 Runge-Kutta
orde empat Error mutlak Error relatif
0 0 0 0 0
5 8.063737787 8.063737771 1.57 × 10−8 1.95 × 10−9 10 46.45365385 46.4536578 3.95 × 10−6 8.51 × 10−8
15 178.6155349 178.6155396 4.69 × 10−6 2.63 × 10−8 20 224.9177437 224.917741 2.75 × 10−6 1.22 × 10−8 25 96.75844394 96.75843967 4.27 × 10−6 4.42 × 10−8 30 28.83899845 28.83899721 1.23 × 10−6 4.28 × 10−8 35 7.84847785 7.848477222 6.28 × 10−7 8.00 × 10−8 40 2.080008236 2.080008059 1.77 × 10−7 8.52 × 10−8 45 0.545971828 0.54597181 1.82 × 10−8 3.34 × 10−8 50 0.144579681 0.144579692 1.04 × 10−8 7.17 × 10−8 Rata-rata 1.61 × 10−6 4.39 × 10−8
3. Populasi 𝐼𝑛𝑓𝑒𝑐𝑡𝑒𝑑 (𝐼)
Pada bagian ini akan ditampilkan hasil aproksimasi pada kelas 𝐼𝑛𝑓𝑒𝑐𝑡𝑒𝑑 (𝐼) yang disajikan pada Tabel 4.5.
Tabel 4.5. Hasil aproksimasi menggunakan fungsi 𝑜𝑑𝑒𝑖𝑛𝑡 dan metode Runge-Kutta orde empat.
t 𝑂𝑑𝑒𝑖𝑛𝑡 Runge-Kutta
orde empat Error mutlak Error relatif
0 1 1 0 0
5 1.329259343 1.329259344 1.29 × 10−9 9.74 × 10−10 10 7.861675386 7.861676064 6.79 × 10−7 8.63 × 10−8 15 33.9479304 33.94793151 1.11 × 10−6 3.27 × 10−8 20 57.46064646 57.46064698 5.22 × 10−7 9.08 × 10−9 25 31.95579199 31.95579089 1.10 × 10−6 3.44 × 10−8 30 10.50355315 10.50355275 3.93 × 10−7 3.74 × 10−8 35 2.943104513 2.943104276 2.37 × 10−7 8.04 × 10−8 40 0.787167931 0.787167865 6.63 × 10−8 8.42 × 10−8 45 0.207330293 0.207330286 6.47 × 10−9 3.12 × 10−8 50 0.054996356 0.054996364 8.06 × 10−9 1.47 × 10−7
Rata-rata 3.74 × 10−7 4.94 × 10−8
4. Populasi 𝑄𝑢𝑎𝑟𝑎𝑛𝑡𝑖𝑛𝑒/𝐼𝑠𝑜𝑙𝑎𝑡𝑖𝑜𝑛 (𝑄)
Pada bagian ini akan ditampilkan hasil aproksimasi pada kelas 𝑄𝑢𝑎𝑟𝑎𝑛𝑡𝑖𝑛𝑒/𝐼𝑠𝑜𝑙𝑎𝑡𝑖𝑜𝑛 (𝑄) yang disajikan pada Tabel 4.6.
Tabel 4.6. Hasil aproksimasi menggunakan fungsi 𝑜𝑑𝑒𝑖𝑛𝑡 dan metode Runge-Kutta orde empat.
t 𝑂𝑑𝑒𝑖𝑛𝑡 Runge-Kutta
orde empat Error mutlak Error relatif
0 0 0 0 0
5 2.151389305 2.151389299 6.03 × 10−9 2.80 × 10−9 10 12.55868928 12.55868998 6.93 × 10−7 5.52 × 10−8 15 61.73455307 61.73455646 3.39 × 10−6 5.50 × 10−8 20 157.2240524 157.224057 4.58 × 10−6 2.91 × 10−8 25 160.7186396 160.7186378 1.75 × 10−6 1.09 × 10−8 30 91.8041533 91.80415058 2.72 × 10−6 2.96 × 10−8 35 39.76409539 39.7640937 1.70 × 10−6 4.27 × 10−8 40 15.00467953 15.00467872 8.15 × 10−7 5.43 × 10−8 45 5.239574416 5.239574057 3.59 × 10−7 6.85 × 10−8 50 1.76276329 1.762763185 1.05 × 10−7 5.98 × 10−8 Rata-rata 1.47 × 10−6 3.71 × 10−8
5. Populasi 𝐴𝑠𝑦𝑚𝑝𝑡𝑜𝑚𝑎𝑡𝑖𝑐 (𝐴)
Pada bagian ini akan ditampilkan hasil aproksimasi pada kelas 𝐴𝑠𝑦𝑚𝑝𝑡𝑜𝑚𝑎𝑡𝑖𝑐 (𝐴)yang disajikan pada Tabel 4.7.
Tabel 4.7. Hasil aproksimasi menggunakan fungsi 𝑜𝑑𝑒𝑖𝑛𝑡 dan metode Runge-Kutta orde empat.
t 𝑂𝑑𝑒𝑖𝑛𝑡 Runge-Kutta
orde empat Error mutlak Error relatif
0 0 0 0 0
5 0.776329681 0.776329679 2.35 × 10−9 3.03 × 10−9 10 4.581562468 4.581562878 4.09 × 10−7 8.93 × 10−8 15 19.366537 19.36653761 6.17 × 10−7 3.18 × 10−8 20 30.76061906 30.76061914 7.68 × 10−8 2.50 × 10−9 25 15.90850341 15.90850279 6.22 × 10−7 3.91 × 10−8 30 5.049394842 5.049394645 1.97 × 10−7 3.89 × 10−8 35 1.398843865 1.398843759 1.07 × 10−7 7.62 × 10−8 40 0.372806637 0.372806606 3.15 × 10−8 8.45 × 10−8 45 0.098067682 0.09806768 2.29 × 10−9 2.34 × 10−8 50 0.025998246 0.025998247 1.15 × 10−9 4.41 × 10−8 Rata-rata 1.88 × 10−7 3.94 × 10−8
6. Populasi 𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑒𝑑 (𝑅)
Pada bagian ini akan ditampilkan hasil aproksimasi pada kelas 𝑅𝑒𝑐𝑜𝑣𝑒𝑟𝑒𝑑 (𝑅) yang disajikan pada Tabel 4.8.
Tabel 4.8. Hasil aproksimasi menggunakan fungsi 𝑜𝑑𝑒𝑖𝑛𝑡 dan metode Runge-Kutta orde empat.
t 𝑂𝑑𝑒𝑖𝑛𝑡 Runge-Kutta
orde empat Error mutlak Error relatif
0 0 0 0 0
5 2.22324442 2.223244417 3.28 × 10−9 1.47 × 10−9 10 15.7612822 15.76128248 2.77 × 10−7 1.75 × 10−8
15 86.2018059 86.20181157 5.67 × 10−6 6.58 × 10−8 20 301.6260617 301.6260727 1.10 × 10−5 3.65 × 10−8 25 579.5243687 579.5243791 1.04 × 10−5 1.80 × 10−8 30 759.0108556 759.0108611 5.56 × 10−6 7.32 × 10−9 35 836.9229482 836.9229511 2.87 × 10−6 3.43 × 10−9 40 861.1154871 861.1154883 1.17 × 10−6 1.36 × 10−9 45 863.0991159 863.0991163 4.24 × 10−7 4.92 × 10−10 50 857.0624347 857.0624348 1.16 × 10−7 1.36 × 10−10 Rata-rata 3.41 × 10−6 1.38 × 10−8
Berdasarkan hasil aproksimasi dari Tabel 4.3 sampai Tabel 4.8 dengan fungsi 𝑜𝑑𝑒𝑖𝑛𝑡 dan metode Runge-Kutta orde empat pada tiap populasi dapat dilihat bahwa galat dari metode Runge-Kutta orde untuk periode 𝑡 = 0 sampai 𝑡 = 50 relatif kecil. Hal ini menunjukkan bahwa metode Runge-Kutta orde empat mem-berikan hasil yang mendekati penyelesaian 𝑂𝑑𝑒𝑖𝑛𝑡. Catatan: penyelesaian odeint dipakai sebagai referensi penyelesaian eksak.
82 BAB V
KESIMPULAN DAN SARAN
Pada bab ini akan dipaparkan mengenai kesimpulan dari pemodelan ma-tematis penyebaran SARS-CoV-2 yang melibatkan kelas asimtomatik dan karan-tina dan saran yang akan diberikan kepada pembaca tugas akhir ini.
A. Kesimpulan
Model matematis yang disajikan pada tugas akhir ini adalah pengembangan model 𝑆𝐸𝐼𝑅 dengan menambahkan kelas asimtomatik dan karantina sehingga men-jadi model 𝑆𝐸𝐼𝑄𝐴𝑅 yang terbentuk dalam model sistem persamaan diferensial non-linear orde satu yaitu
𝑑𝑆
𝑑𝑡 = Ʌ − 𝛽𝑆(𝐼 + 𝑞𝐴)– 𝜇𝑆, 𝑑𝐸
𝑑𝑡 = 𝛽𝑆(𝐼 + 𝑞𝐴) – ( 𝜂 + 𝜃 + 𝜇)𝐸, 𝑑𝐼
𝑑𝑡= 𝑝𝜂𝐸 − (𝛼 + 𝑣 + 𝜇) + 𝜌𝐴, 𝑑𝑄
𝑑𝑡 = 𝛼𝐼 + 𝜃𝐸 − (𝛿 + 𝜇)𝑄, 𝑑𝐴
𝑑𝑡 = (1 − 𝑝)𝜂𝐸 − (𝜌 + 𝛾 + 𝜇) + 𝐴, 𝑑𝑅
𝑑𝑡 = 𝛾𝐴 + 𝛿𝑄 + 𝜐𝐼 − 𝜇𝑅.
Bilangan reproduksi dihitung dengan pendekatan matriks generasi beri-kutnya (𝑇ℎ𝑒 𝑁𝑒𝑥𝑡 𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑀𝑎𝑡𝑟𝑖𝑥). Kelas 𝑄𝑢𝑎𝑟𝑎𝑛𝑡𝑖𝑛𝑒 (𝑄) disediakan un-tuk dua kelas yaitu kelas 𝐸𝑥𝑝𝑜𝑠𝑒𝑑 (𝐸) dan kelas 𝐼𝑛𝑓𝑒𝑐𝑡𝑒𝑑(𝐼). Telah dibuktikan bahwa analisis titik ekuilibrium bebas penyakit akan stabil asimtotik jika ℛ0 < 1 dan tidak stabil jika ℛ0 > 1. Analisis titik ekuilibrium endemik akan stabil asimtotik ketika ℛ0 > 1 dan tidak stabil jika ℛ0 < 1. Hasil analisis sensitivitas ter-hadap lima parameter menujukkan bahwa adanya beberapa parameter yang
memberikan pengaruh positif dan beberapa parameter memberikan pengaruh negatif. Hal ini mengatakan bahwa model lebih sensitif terhadap laju transmisi dari kelas 𝐸𝑥𝑝𝑜𝑠𝑒𝑑 (𝐸) ke kelas 𝐼𝑛𝑓𝑒𝑐𝑡𝑒𝑑 (𝐼) daripada laju transmisi dari kelas 𝑆𝑢𝑠𝑐𝑒𝑝𝑡𝑖𝑏𝑙𝑒 (𝑆) ke kelas 𝐸𝑥𝑝𝑜𝑠𝑒𝑑 (𝐸).
Berdasarkan uraian tersebut disimpulkan bahwa mengurangi nilai 𝜂 dapat membantu mengendalikan penyakit agar tidak meningkat secara signifikan pada 𝑡 tertentu secara lebih efisien. Laju transmisi 𝛽 juga mampu mengontrol laju infeksi tetapi kurang sensitif dibandingkan dengan 𝜂. Untuk parameter 𝜃 dan 𝜌kurang ber-pengaruh besar untuk mengontrol infeksi virus dalam skala besar.
B. Saran
Penulis menyadari masih banyak kekurangan dalam penulisan tugas akhir ini. Oleh karena itu, penulis mengharapkan adanya penelitian lanjutan terhadap ka-sus yang dipaparkan pada tugas akhir ini. Model yang disajikan melibatkan enam populasi tanpa memperhatikan populasi yang melakukan vaksinasi dan menyajikan keadaan bebas penyakit tanpa menyajikan keadaan saat endemik. Oleh karena itu, saran pertama dari penulis agar penelitian ini dapat dikembangkan dan terus diek-splorasi dengan menambahkan setiap detail yang belum bisa disajikan dalam tugas akhir ini. Saran kedua adalah agar pembaca benar-benar memahami bahwa agar pandemi ini segera berakhir seluruh masyarakat wajib mematuhi protokol kesehatan yang ditetapkan oleh pemerintah.
84
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86 LAMPIRAN
Berikut akan dilampirkan program 𝑂𝑑𝑒𝑖𝑛𝑡 dan metode Runge-Kutta orde em-pat untuk menentukan solusi numeris dengan bantuan program Python.
A. Program dengan fungsi 𝑂𝑑𝑒𝑖𝑛𝑡 from scipy.integrate import odeint import numpy as np
import matplotlib.pyplot as plt
def odes(x,t):
S = x[0]
E = x[1]
I = x[2]
Q = x[3]
A = x[4]
R = x[5]
N=1000
lamda = (0.2/365) miu=(0.87/365) beta=(2.65) p=(0.56) q=1
v=(10/228) eta=(0.3) teta=(0.01) rho=(0.5) alfa=(0.9) delta=(0.25) gamma=((0.5))
dSdt = lamda-(beta*S*(I+q*A))/(N-Q)-miu*S
dEdt = (beta*S*I + beta*S*q*A)/(N-Q) - E*(eta+teta+miu) dIdt = p*eta*E-(alfa+v+miu)*I+rho*A
dQdt = alfa*I + teta*E - Q*(delta+miu) dAdt = (1-p)*eta*E-(rho+gamma+miu)*A dRdt = gamma*A + delta*Q+ v*I- miu*R return [dSdt,dEdt,dIdt,dQdt,dAdt,dRdt]
x0= [1000, 0, 1, 0, 0, 0]
t= np.linspace(0, 50,1000) x= odeint(odes, x0, t) S = x[:,0]
E = x[:,1]
I = x[:,2]
Q = x[:,3]
A = x[:,4]
R = x[:,5]
plt.plot(t,S,"b",label="Susceptible (S)") plt.plot(t,E,"g",label="Exposed (E)") plt.plot(t,I,"r",label="Infected (I)") plt.plot(t,Q,"c",label="Quarantine (Q)") plt.plot(t,A,"m",label="Asymptomatic (A)") plt.plot(t,R,"y",label="Recovered (R)") plt.xlabel ("t (hari)")
plt.ylabel ("Populasi") plt.title("Grafik sistem") plt.legend (loc = "best") plt.axis([0,50, 0,1200]) plt.grid()
plt.show()
B. Program dengan metode Runge-Kutta orde empat
import matplotlib.pyplot as plt import numpy as np
S0 = 1000 E0 = 0 I0 = 1 Q0 = 0 A0 = 0 R0 = 0
N=1000
lamda = (0.2/365) miu=(0.87/365) beta=(2.65) p=(0.56) q=1
v=(10/228) eta=(0.3) teta=(0.01) rho=(0.5) alfa=(0.9) delta=(0.25) gamma=((0.5))
t0 = 0 tn = 50 ndata = 1000
t = np.linspace(t0,tn,ndata) h = t[2]-t[1]
S = np.zeros(ndata)
E = np.zeros(ndata) I = np.zeros(ndata) Q = np.zeros(ndata) A = np.zeros(ndata) R = np.zeros(ndata)
KS1 = np.zeros(ndata) KE1 = np.zeros(ndata) KI1 = np.zeros(ndata) KQ1 = np.zeros(ndata) KA1 = np.zeros(ndata) KR1 = np.zeros(ndata)
KS2 = np.zeros(ndata) KE2 = np.zeros(ndata) KI2 = np.zeros(ndata) KQ2 = np.zeros(ndata) KA2 = np.zeros(ndata) KR2 = np.zeros(ndata)
KS3 = np.zeros(ndata) KE3 = np.zeros(ndata) KI3 = np.zeros(ndata) KQ3 = np.zeros(ndata) KA3 = np.zeros(ndata) KA3 = np.zeros(ndata) KR3 = np.zeros(ndata)
KS4 = np.zeros(ndata) KE4 = np.zeros(ndata) KI4 = np.zeros(ndata)
KQ4 = np.zeros(ndata) KA4 = np.zeros(ndata) KR4 = np.zeros(ndata)
S[0] = S0 E[0] = E0 I[0] = I0 Q[0] = Q0 A[0] = A0 R[0] = R0
for ii in range (1,ndata):
KS1[ii] =lamda-(beta*S[ii-1]*(I[ii-1]+q*A[ii-1]))/(N-Q[ii-1])-miu*S[ii-1]
KE1[ii]=beta*S[ii-1]*I[ii-1]/(N-Q[ii-1])+beta*S[ii-1]*q*A[ii-1]/(N-Q[ii-1])-E[ii-1]*(eta+teta+miu)
KI1[ii] = p*eta*E[ii-1]-(alfa+v+miu)*I[ii-1]+rho*A[ii-1]
KQ1[ii] = alfa*I[ii-1] + teta*E[ii-1] - Q[ii-1]*(delta+miu) KA1[ii] = (1-p)*eta*E[ii-1]-(rho+gamma+miu)*A[ii-1]
KR1[ii] = gamma*A[ii-1] + delta*Q[ii-1] + v*I[ii-1]- miu*R[ii-1]
KS2[ii] =lamda-(beta*(S[ii-1]+0.5*KS1[ii]*h)*((I[ii-1]+0.5*KI1[ii]*h) +q*(A[ii-1]+0.5*KA1[ii]*h)))/(N-(Q[ii-1]+0.5*KQ1[ii]*h))-miu*(S[ii-1]
+0.5*KS1[ii]*h)
KE2[ii] = beta*(S[ii-1]+0.5*KS1[ii]*h)*(I[ii-1]+0.5*KI1[ii]*h)/(N-(Q[ii-1]+0.5*KQ1[ii]*h)) +
beta*(S[ii-1]+0.5*KS1[ii]*h)*q*(A[ii-1]+0.5*KA1[ii]*h)/(N-(Q[ii-1]+0.5*KQ1[ii]*h)) -(E[ii-1]+0.5*KE1[ii]
*h)*(eta+teta+miu)
KI2[ii] = p*eta*(E[ii-1]+0.5*KE1[ii]*h)-(alfa+v+miu)*(I[ii-1]+0.5*KI1[ii]*h)+rho*(A[ii-1]+0.5*KA1[ii]*h)
KQ2[ii] = alfa*(I[ii-1]+0.5*KI1[ii]*h) + teta*(E[ii-1]+0.5*KE1[ii]*h) - (Q[ii-1]+0.5*KQ1[ii]*h)*(delta+miu)
KA2[ii] = (1-p)*eta*(E[ii-1]+0.5*KE1[ii]*h)-(rho+gamma+miu)*(A[ii-1]+0.5*KA1[ii]*h)
KR2[ii] = gamma*(A[ii-1]+0.5*KA1[ii]*h) + delta*(Q[ii-1]+0.5*KQ1[ii]*h) + v*(I[ii-1]+0.5*KI1[ii]*h)- miu*(R[ii-1]+0.5*KR1[ii]*h)
KS3[ii] = lamda-(beta*(S[ii-1]+0.5*KS2[ii]*h)*((I[ii- 1]+0.5*KI2[ii]*h)+q*(A[ii-1]+0.5*KA2[ii]*h)))/(N-(Q[ii-1]+0.5*KQ2[ii]*h))-miu*(S[ii-1]+0.5*KS2[ii]*h)
KE3[ii] = beta*(S[ii-1]+0.5*KS2[ii]*h)*(I[ii-1]+0.5*KI2[ii]*h)/(N-(Q[ii-1]+0.5*KQ2[ii]*h)) +
beta*(S[ii-1]+0.5*KS2[ii]*h)*q*(A[ii-1]+0.5*KA2[ii]*h)/(N-(Q[ii-1]+0.5*KQ2[ii]*h)) -(E[ii-1]+0.5*KE2[ii]*h)*(eta+teta+miu)
KI3[ii] = p*eta*(E[ii-1]+0.5*KE2[ii]*h)-(alfa+v+miu)*(I[ii-1]+0.5*KI2[ii]*h)+rho*(A[ii-1]+0.5*KA2[ii]*h)
KQ3[ii] = alfa*(I[ii-1]+0.5*KI2[ii]*h) + teta*(E[ii-1]+0.5*KE2[ii]*h) - (Q[ii-1]+0.5*KQ2[ii]*h)*(delta+miu)
KA3[ii] = (1-p)*eta*(E[ii-1]+0.5*KE2[ii]*h)-(rho+gamma+miu)*(A[ii-1]+0.5*KA2[ii]*h)
KR3[ii] = gamma*(A[ii-1]+0.5*KA2[ii]*h) + delta*(Q[ii-1]+0.5*KQ2[ii]*h) + v*(I[ii-1]+0.5*KI2[ii]*h)- miu*(R[ii-1]+0.5*KR2[ii]*h)
KS4[ii] = lamda-(beta*(S[ii-1]+KS3[ii]*h)*((I[ii-1]+KI3[ii]*h)+q*(A[ii-1]+KA3[ii]*h)))/(N-(Q[ii-1]+KQ3[ii]*h))-miu*(S[ii-1]+KS3[ii]*h) KE4[ii] =
beta*(S[ii-1]+KS3[ii]*h)*(I[ii-1]+KI3[ii]*h)/(N-(Q[ii-1]+KQ3[ii]*h)) + beta*(S[ii-1]+KS3[ii]*h)*q*(A[ii-1]+KA3[ii]*h)/(N-(Q[ii-1]+KQ3[ii]*h)) -(E[ii-1]+KE3[ii]*h)*(eta+teta+miu)
KI4[ii] = p*eta*(E[ii-1]+KE3[ii]*h)-(alfa+v+miu)*(I[ii-1]+KI3[ii]*h) +rho*(A[ii-1]+KA3[ii]*h)
KQ4[ii] = alfa*(I[ii-1]+KI3[ii]*h) + teta*(E[ii-1]+KE3[ii]*h) - (Q[ii-1]+
KQ3[ii]*h)*(delta+miu)
KA4[ii] = (1-p)*eta*(E[ii-1]+KE3[ii]*h)-(rho+gamma+miu)*(A[ii1]
+KA3[ii]*h)
KR4[ii] = gamma*(A[ii-1]+KA3[ii]*h) + delta*(Q[ii-1]+KQ3[ii]*h) + v*
(I[ii-1]+KI3[ii]*h)- miu*(R[ii-1]+KR3[ii]*h)
S[ii] = S[ii-1] + ((KS1[ii]+2*KS2[ii]+2*KS3[ii]+KS4[ii]))*h/6 E[ii] = E[ii-1] + ((KE1[ii]+2*KE2[ii]+2*KE3[ii]+KE4[ii]))*h/6 I[ii] = I[ii-1] + ((KI1[ii]+2*KI2[ii]+2*KI3[ii]+KI4[ii]))*h/6 Q[ii] = Q[ii-1] + ((KQ1[ii]+2*KQ2[ii]+2*KQ3[ii]+KQ4[ii]))*h/6 A[ii] = A[ii-1] + ((KA1[ii]+2*KA2[ii]+2*KA3[ii]+KA4[ii]))*h/6 R[ii] = R[ii-1] + ((KR1[ii]+2*KR2[ii]+2*KR3[ii]+KR4[ii]))*h/6
plt.plot(t,S,"b",label="Susceptible (S)") plt.plot(t,E,"g",label="Exposed (E)") plt.plot(t,I,"r",label="Infected (I)") plt.plot(t,Q,"c",label="Quarantine (Q)") plt.plot(t,A,"m",label="Asymptomatic (A)") plt.plot(t,R,"y",label="Recovered (R)") plt.xlabel ("t (hari)")
plt.ylabel ("Populasi") plt.title("Grafik sistem") plt.legend (loc = "best") plt.axis([0,50, 0,1200]) plt.grid()
plt.show()
C. Analisis sensitivitas terhadap parameter alfa import matplotlib.pyplot as plt
import numpy as np
S0 = 1000 E0 = 0 I0 = 1 Q0 = 0 A0 = 0 R0 = 0 z0 = 0 N=1000
lamda = (0.2/365) miu=(0.87/365) beta=(2.65) p=(0.56) q=1
v=(10/228) eta=(0.3) teta=(0.01) rho=(0.5) alfa=(0.7) alfab=(0.8) alfac=(0.9) delta=(0.25)
gamma=((0.5)) t0 = 0
tn = 100 ndata = 1000
t = np.linspace(t0,tn,ndata) h = (t[2]-t[1])
S = np.zeros(ndata) E = np.zeros(ndata) I = np.zeros(ndata) Q = np.zeros(ndata) A = np.zeros(ndata) R = np.zeros(ndata) Sb = np.zeros(ndata) Eb = np.zeros(ndata) Ib = np.zeros(ndata) Qb = np.zeros(ndata) Ab = np.zeros(ndata) Rb = np.zeros(ndata) Sc = np.zeros(ndata) Ec = np.zeros(ndata) Ic = np.zeros(ndata) Qc = np.zeros(ndata)
Ac = np.zeros(ndata) Rc = np.zeros(ndata)
KS1 = np.zeros(ndata) KE1 = np.zeros(ndata) KI1 = np.zeros(ndata) KQ1 = np.zeros(ndata) KA1 = np.zeros(ndata) KR1 = np.zeros(ndata)
KS2 = np.zeros(ndata) KE2 = np.zeros(ndata) KI2 = np.zeros(ndata) KQ2 = np.zeros(ndata) KA2 = np.zeros(ndata) KR2 = np.zeros(ndata) KS3 = np.zeros(ndata) KE3 = np.zeros(ndata) KI3 = np.zeros(ndata) KQ3 = np.zeros(ndata) KA3 = np.zeros(ndata) KA3 = np.zeros(ndata)
KR3 = np.zeros(ndata) KS4 = np.zeros(ndata) KE4 = np.zeros(ndata) KI4 = np.zeros(ndata) KQ4 = np.zeros(ndata) KA4 = np.zeros(ndata) KR4 = np.zeros(ndata)
KS1b = np.zeros(ndata) KE1b = np.zeros(ndata) KI1b = np.zeros(ndata) KQ1b = np.zeros(ndata) KA1b = np.zeros(ndata) KR1b = np.zeros(ndata) KS2b = np.zeros(ndata) KE2b = np.zeros(ndata) KI2b = np.zeros(ndata) KQ2b = np.zeros(ndata) KA2b = np.zeros(ndata) KR2b = np.zeros(ndata) KS3b = np.zeros(ndata) KE3b = np.zeros(ndata)
KI3b = np.zeros(ndata) KQ3b = np.zeros(ndata) KA3b = np.zeros(ndata) KR3b = np.zeros(ndata) KS4b = np.zeros(ndata) KE4b = np.zeros(ndata) KI4b = np.zeros(ndata) KQ4b = np.zeros(ndata) KA4b = np.zeros(ndata) KR4b = np.zeros(ndata)
KS1c = np.zeros(ndata) KE1c = np.zeros(ndata) KI1c = np.zeros(ndata) KQ1c = np.zeros(ndata) KA1c = np.zeros(ndata) KR1c = np.zeros(ndata) KS2c = np.zeros(ndata) KE2c = np.zeros(ndata) KI2c = np.zeros(ndata) KQ2c = np.zeros(ndata) KA2c = np.zeros(ndata)
KR2c = np.zeros(ndata) KS3c = np.zeros(ndata) KE3c = np.zeros(ndata) KI3c = np.zeros(ndata) KQ3c = np.zeros(ndata) KA3c = np.zeros(ndata) KR3c = np.zeros(ndata) KS4c = np.zeros(ndata) KE4c = np.zeros(ndata) KI4c = np.zeros(ndata) KQ4c = np.zeros(ndata) KA4c = np.zeros(ndata) KR4c = np.zeros(ndata) S[0] = S0
E[0] = E0 I[0] = I0 Q[0] = Q0 A[0] = A0 R[0] = R0 Sb[0] = S0 Eb[0] = E0 Ib[0] = I0
Qb[0] = Q0 Ab[0] = A0 Rb[0] = R0 Sc[0] = S0 Ec[0] = E0 Ic[0] = I0 Qc[0] = Q0 Ac[0] = A0 Rc[0] = R0
for ii in range (1,ndata):
KS1[ii] = lamda-(beta*S[ii-1]*(I[ii-1]+q*A[ii-1]))/(N-Q[ii-1])-miu*S[ii-1]
KE1[ii] = beta*S[ii-1]*I[ii-1]/(N-Q[ii-1]) + beta*S[ii-1]*q*A[ii-1]/(N-Q[ii-1]) - E[ii-1]*(eta+teta+miu)
KI1[ii] = p*eta*E[ii-1]-(alfa+v+miu)*I[ii-1]+rho*A[ii-1]
KQ1[ii] = alfa*I[ii-1] + teta*E[ii-1] - Q[ii-1]*(delta+miu) KA1[ii] = (1-p)*eta*E[ii-1]-(rho+gamma+miu)*A[ii-1]
KR1[ii] = gamma*A[ii-1] + delta*Q[ii-1] + v*I[ii-1]- miu*R[ii-1]
KS2[ii] = lamda-(beta*(S[ii-1]+0.5*KS1[ii]*h)*((I[ii-1]+0.5*KI1[ii]*h) +q*(A[ii-1]+0.5*KA1[ii]*h)))/(N-(Q[ii-1]+0.5*KQ1[ii]*h))-miu*(S[ii-1]
+0.5*KS1[ii]*h)
KE2[ii] = beta*(S[ii-1]+0.5*KS1[ii]*h)*(I[ii-1]+0.5*KI1[ii]*h)/(N-(Q[ii-1]+0.5*KQ1[ii]*h)) + beta*(S[ii-1]+0.5*KS1[ii]*h)*q*(A[ii-1]+0.5*KA1[ii]
*h)/(N-(Q[ii-1]+0.5*KQ1[ii]*h)) -(E[ii-1]+0.5*KE1[ii]*h)*(eta+teta+miu) KI2[ii] = p*eta*(E[ii-1]+0.5*KE1[ii]*h)-(alfa+v+miu)*(I[ii1]+0.5*KI1[ii]
*h)+rho*(A[ii-1]+0.5*KA1[ii]*h)
KQ2[ii] = alfa*(I[ii-1]+0.5*KI1[ii]*h) + teta*(E[ii-1]+0.5*KE1[ii]*h) - (Q[ii-1]+0.5*KQ1[ii]*h)*(delta+miu)
KA2[ii] = (1-p)*eta*(E[ii-1]+0.5*KE1[ii]*h)-(rho+gamma+miu)*(A[ii-1]
+0.5*KA1[ii]*h)
KR2[ii] = gamma*(A[ii-1]+0.5*KA1[ii]*h) + delta*(Q[ii-1]+0.5*KQ1[ii]
*h) + v*(I[ii-1]+0.5*KI1[ii]*h)- miu*(R[ii-1]+0.5*KR1[ii]*h)
KS3[ii] = lamda-(beta*(S[ii-1]+0.5*KS2[ii]*h)*((I[ii-1]+0.5*KI2[ii]
*h)+q*(A[ii-1]+0.5*KA2[ii]*h)))/(N-(Q[ii-1]+0.5*KQ2[ii]*h))-miu*(S[ii-1]
+0.5*KS2[ii]*h)
KE3[ii] = beta*(S[ii-1]+0.5*KS2[ii]*h)*(I[ii-1]+0.5*KI2[ii]*h)/(N-(Q[ii-1]+0.5*KQ2[ii]*h)) + beta*(S[ii-1]+0.5*KS2[ii]*h)*q*(A[ii-1]+0.5*KA2[ii]
*h)/(N-(Q[ii-1]+0.5*KQ2[ii]*h)) -(E[ii-1]+0.5*KE2[ii]*h)*(eta+teta+miu) KI3[ii] = p*eta*(E[ii-1]+0.5*KE2[ii]*h)-(alfa+v+miu)*(I[ii1]+0.5*KI2[ii]
*h)+rho*(A[ii-1]+0.5*KA2[ii]*h)
KQ3[ii] = alfa*(I[ii-1]+0.5*KI2[ii]*h) + teta*(E[ii-1]+0.5*KE2[ii]*h) - (Q[ii-1]+0.5*KQ2[ii]*h)*(delta+miu)
KA3[ii] = (1-p)*eta*(E[ii-1]+0.5*KE2[ii]*h)-(rho+gamma+miu)*(A[ii-1]
+0.5*KA2[ii]*h)
KR3[ii] = gamma*(A[ii-1]+0.5*KA2[ii]*h)+delta*(Q[ii1]+0.5*KQ2[ii]*h) + v*(I[ii-1]+0.5*KI2[ii]*h)- miu*(R[ii-1]+0.5*KR2[ii]*h)
KS4[ii] = lamda-(beta*(S[ii-1]+KS3[ii]*h)*((I[ii-1]+KI3[ii]*h)+q*(A[ii-1]+KA3[ii]*h)))/(N-(Q[ii-1]+KQ3[ii]*h))-miu*(S[ii-1]+KS3[ii]*h)
KE4[ii] = beta*(S[ii-1]+KS3[ii]*h)*(I[ii-1]+KI3[ii]*h)/(N-(Q[ii-1]
+KQ3[ii]*h)) + beta*(S[ii-1]+KS3[ii]*h)*q*(A[ii-1]+KA3[ii]*h)/(N-(Q[ii-1]+KQ3[ii]*h)) -(E[ii-1]+KE3[ii]*h)*(eta+teta+miu)
KI4[ii] = p*eta*(E[ii-1]+KE3[ii]*h)-(alfa+v+miu)*(I[ii-1]+KI3[ii]*h) +rho*(A[ii-1]+KA3[ii]*h)
KQ4[ii] = alfa*(I[ii-1]+KI3[ii]*h) + teta*(E[ii-1]+KE3[ii]*h) - (Q[ii-1]+KQ3[ii]*h)*(delta+miu)
KA4[ii] = (1-p)*eta*(E[ii-1]+KE3[ii]*h)-(rho+gamma+miu)*(A[ii-1]
+KA3[ii]*h)
KR4[ii] = gamma*(A[ii-1]+KA3[ii]*h) + delta*(Q[ii-1]+KQ3[ii]*h) + v*(I[ii-1]+KI3[ii]*h)- miu*(R[ii-1]+KR3[ii]*h)
S[ii] = S[ii-1] + ((KS1[ii]+2*KS2[ii]+2*KS3[ii]+KS4[ii]))*h/6 E[ii] = E[ii-1] + ((KE1[ii]+2*KE2[ii]+2*KE3[ii]+KE4[ii]))*h/6 I[ii] = I[ii-1] + ((KI1[ii]+2*KI2[ii]+2*KI3[ii]+KI4[ii]))*h/6 Q[ii] = Q[ii-1] + ((KQ1[ii]+2*KQ2[ii]+2*KQ3[ii]+KQ4[ii]))*h/6 A[ii] = A[ii-1] + ((KA1[ii]+2*KA2[ii]+2*KA3[ii]+KA4[ii]))*h/6 R[ii] = R[ii-1] + ((KR1[ii]+2*KR2[ii]+2*KR3[ii]+KR4[ii]))*h/6 for i in range(1,ndata):
KS1b[i] = lamda-(beta*Sb[i-1]*(Ib[i-1]+q*Ab[i-1]))/(N-Qb[i-1])-miu*
Sb[i-1]
KE1b[i] = beta*Sb[i-1]*Ib[i-1]/(N-Qb[i-1]) + beta*Sb[i-1]*q*Ab[i-1]/(N-Qb[i-1]) -Eb[i-1]*(eta+teta+miu)
KI1b[i] = p*eta*Eb[i-1]-(alfab+v+miu)*Ib[i-1]+rho*Ab[i-1]
KQ1b[i] = alfab*Ib[i-1] + teta*Eb[i-1] - Qb[i-1]*(delta+miu) KA1b[i] = (1-p)*eta*Eb[i-1]-(rho+gamma+miu)*Ab[i-1]
KR1b[i] = gamma*Ab[i-1] + delta*Qb[i-1] + v*Ib[i-1]- miu*Rb[i-1]
KS2b[i] = lamda-(beta*(Sb[i-1]+0.5*KS1b[i]*h)*((Ib[i-1]+0.5*KI1b[i]
*h)+q*(Ab[i-1]+0.5*KA1b[i]*h)))/(N-(Qb[i-1]+0.5*KQ1b[i]*h))-miu*(Sb[i-1]+0.5*KS1b[i]*h)
KE2b[i] = beta*(Sb[i-1]+0.5*KS1b[i]*h)*(Ib[i-1]+0.5*KI1b[i]*h)/(N-(Qb[i-1]+0.5*KQ1b[i]*h)) + beta*(Sb[i-1]+0.5*KS1b[i]*h)*q*(Ab[i-1]
+0.5*KA1b[i]*h)/(N-(Qb[i-1]+0.5*KQ1b[i]*h))-(Eb[i-1]+0.5*KE1b[i]*h)
*(eta+teta+miu)
KI2b[i] = p*eta*(Eb[i-1]+0.5*KE1b[i]*h)-(alfab+v+miu)*(Ib[i-1]+0.5
*KI1b[i]*h)+rho*(Ab[i-1]+0.5*KA1b[i]*h)
KQ2b[i] = alfab*(Ib[i-1]+0.5*KI1b[i]*h) + teta*(Eb[i-1]+0.5*KE1b[i]*h) - (Qb[i-1]+0.5*KQ1b[i]*h)*(delta+miu)
KA2b[i] = (1-p)*eta*(Eb[i-1]+0.5*KE1b[i]*h)-(rho+gamma+miu)*(Ab[i-1]+0.5*KA1b[i]*h)
KR2b[i] = gamma*(Ab[i-1]+0.5*KA1b[i]*h) + delta*(Qb[i-1]+0.5 *KQ1b[i]*h) + v*(Ib[i-1]+0.5*KI1b[i]*h)- miu*(Rb[i-1]+0.5*KR1b[i]*h)
KS3b[i] = lamda-(beta*(Sb[i-1]+0.5*KS2b[i]*h)*((Ib[i-1]+0.5*KI2b[i]
*h)+q*(Ab[i-1]+0.5*KA2b[i]*h)))/(N-(Qb[i-1]+0.5*KQ2b[i]*h))-miu*(Sb[i-1]+0.5*KS2b[i]*h)
KE3b[i] = beta*(Sb[i-1]+0.5*KS2b[i]*h)*(Ib[i-1]+0.5*KI2b[i]*h)/(N-(Qb[i-1]+0.5*KQ2b[i]*h)) + beta*(Sb[i-1]+0.5*KS2b[i]*h)*q*(Ab[i-1]+0.5
*KA2b[i]*h)/(N-(Qb[i-1]+0.5*KQ2b[i]*h))-(Eb[i1]+0.5*KE2b[i]*h)
*(eta+teta+miu)
KI3b[i] = p*eta*(Eb[i-1]+0.5*KE2b[i]*h)-(alfab+v+miu)*(Ib[i-1]+0.5
*KI2b[i]*h)+rho*(Ab[i-1]+0.5*KA2b[i]*h)
KQ3b[i] = alfab*(Ib[i-1]+0.5*KI2b[i]*h) + teta*(Eb[i-1]+0.5*KE2b[i]*h) - (Qb[i-1]+0.5*KQ2b[i]*h)*(delta+miu)
KA3b[i] = (1-p)*eta*(Eb[i-1]+0.5*KE2b[i]*h)-(rho+gamma+miu)*(Ab[i-1]+0.5*KA2b[i]*h)
KR3b[i] = gamma*(Ab[i-1]+0.5*KA2b[i]*h)+delta*(Qb[i1]+0.5*KQ2b[i]
*h) + v*(Ib[i-1]+0.5*KI2b[i]*h)- miu*(Rb[i-1]+0.5*KR2b[i]*h) KS4b[i] = lamda-(beta*(Sb[i-1]+KS3b[i]*h)*((Ib[i-1]+KI3b[i]*h)+q
*(Ab[i-1]+KA3b[i]*h)))/(N-(Qb[i-1]+KQ3b[i]*h))-miu*(Sb[i1]+KS3b[i]*h) KE4b[i] = beta*(Sb[i-1]+KS3b[i]*h)*(Ib[i-1]+KI3b[i]*h)/(N-(Qb[i-1]
+KQ3b[i]*h)) + beta*(Sb[i-1]+KS3b[i]*h)*q*(Ab[i-1]+KA3b[i]*h)
/ (N(Qb[i-1]+KQ3b[i]*h)) -(Eb[i-1]+KE3b[i]*h)*(eta+teta+miu)
KI4b[i] = p*eta*(Eb[i-1]+KE3b[i]*h)-(alfab+v+miu)*(Ib[i-1]+KI3b[i]*h) +rho*(Ab[i-1]+KA3b[i]*h)
KQ4b[i] = alfab*(Ib[i-1]+KI3b[i]*h) + teta*(Eb[i-1]+KE3b[i]*h) - (Qb[i1]
+KQ3b[i]*h)*(delta+miu)
KA4b[i] = (1-p)*eta*(Eb[i-1]+KE3b[i]*h)-(rho+gamma+miu)*(Ab[i-1]
+KA3b[i]*h)
KR4b[i] = gamma*(Ab[i-1]+KA3b[i]*h) + delta*(Qb[i-1]+KQ3b[i]*h) + v *(Ib[i-1]+KI3b[i]*h)- miu*(Rb[i-1]+KR3b[i]*h)
Sb[i] = Sb[i-1] + ((KS1b[i]+2*KS2b[i]+2*KS3b[i]+KS4b[i]))*h/6 Eb[i] = Eb[i-1] + ((KE1b[i]+2*KE2b[i]+2*KE3b[i]+KE4b[i]))*h/6 Ib[i] = Ib[i-1] + ((KI1b[i]+2*KI2b[i]+2*KI3b[i]+KI4b[i]))*h/6 Qb[i] = Qb[i-1] + ((KQ1b[i]+2*KQ2b[i]+2*KQ3b[i]+KQ4b[i]))*h/6 Ab[i] = Ab[i-1] + ((KA1b[i]+2*KA2b[i]+2*KA3b[i]+KA4b[i]))*h/6 Rb[i] = Rb[i-1] + ((KR1b[i]+2*KR2b[i]+2*KR3b[i]+KR4b[i]))*h/6 KS1c[i]=lamda-(beta*Sc[i-1]*(Ic[i-1]+q*Ac[i-1]))/(N-Qc[i-1])-miu *Sc[i- 1]
KE1c[i] = beta*Sc[i-1]*Ic[i-1]/(N-Qc[i-1]) + beta*Sc[i-1]*q*Ac[i-1]/(N- Qc[i-1])-Ec[i-1]*(eta+teta+miu)
KI1c[i] = p*eta*Ec[i-1]-(alfac+v+miu)*Ic[i-1]+rho*Ac[i-1]
KQ1c[i] = alfac*Ic[i-1] + teta*Ec[i-1] - Qc[i-1]*(delta+miu) KA1c[i] = (1-p)*eta*Ec[i-1]-(rho+gamma+miu)*Ac[i-1]
KR1c[i] = gamma*Ac[i-1] + delta*Qc[i-1] + v*Ic[i-1]- miu*Rc[i-1]
KS2c[i] = lamda-(beta*(Sc[i-1]+0.5*KS1c[i]*h)*((Ic[i-1]+0.5*KI1c[i]*h) +q*(Ac[i-1]+0.5*KA1c[i]*h)))/(N-(Qc[i-1]+0.5*KQ1c[i]*h))-miu *(Sc[i-1]+0.5*KS1c[i]*h)
KE2c[i] = beta*(Sc[i-1]+0.5*KS1c[i]*h)*(Ic[i-1]+0.5*KI1c[i]*h)/(N-
(Qc[i-1]+0.5*KQ1c[i]*h)) + beta*(Sc[i-1]+0.5*KS1c[i]*h)*q*(Ac[i-1]
+0.5*KA1c[i]*h)/(N-(Qc[i-1]+0.5*KQ1c[i]*h))-(Ec[i-1]+0.5*KE1c[i]*h) *(eta+teta+miu)
KI2c[i] = p*eta*(Ec[i-1]+0.5*KE1c[i]*h)-(alfac+v+miu)*(Ic[i-1]+0.5 *KI1c[i]*h)+rho*(Ac[i-1]+0.5*KA1c[i]*h)
KQ2c[i] = alfac*(Ic[i-1]+0.5*KI1c[i]*h) + teta*(Ec[i-1]+0.5*KE1c[i]*h)- (Qc[i-1]+0.5*KQ1c[i]*h)*(delta+miu)
KA2c[i] = (1-p)*eta*(Ec[i-1]+0.5*KE1c[i]*h)-(rho+gamma+miu) *(Ac[i-1]+0.5*KA1c[i]*h)
KR2c[i] = gamma*(Ac[i-1]+0.5*KA1c[i]*h) + delta*(Qc[i-1]+0.5
*KQ1c[i]*h) + v*(Ic[i-1]+0.5*KI1c[i]*h)- miu*(Rc[i-1]+0.5*KR1c[i]*h) KS3c[i] = lamda-(beta*(Sc[i-1]+0.5*KS2c[i]*h)*((Ic[i-1]+0.5*KI2c[i]*h) +q*(Ac[i-1]+0.5*KA2c[i]*h)))/(N-(Qc[i-1]+0.5*KQ2c[i]*h))-miu*(Sc[i-1]
+0.5*KS2c[i]*h)
KE3c[i] = beta*(Sc[i-1]+0.5*KS2c[i]*h)*(Ic[i-1]+0.5*KI2c[i]*h)/(N- (Qc[i-1]+0.5*KQ2c[i]*h)) + beta*(Sc[i-1]+0.5*KS2c[i]*h)*q*(Ac[i-1]+0.5 *KA2c[i]*h)/(N-(Qc[i-1]+0.5*KQ2c[i]*h))-(Ec[i-1]+0.5*KE2c[i]*h) *(eta+teta+miu)
KI3c[i] = p*eta*(Ec[i-1]+0.5*KE2c[i]*h)-(alfac+v+miu)*(Ic[i-1]+0.5 *KI2c[i]*h)+rho*(Ac[i-1]+0.5*KA2c[i]*h)
KQ3c[i] = alfac*(Ic[i-1]+0.5*KI2c[i]*h) + teta*(Ec[i-1]+0.5*KE2c[i]*h)- (Qc[i-1]+0.5*KQ2c[i]*h)*(delta+miu)
KA3c[i] = (1-p)*eta*(Ec[i-1]+0.5*KE2c[i]*h)-(rho+gamma+miu)*(Ac[i- 1]+0.5*KA2c[i]*h)
KR3c[i] = gamma*(Ac[i-1]+0.5*KA2c[i]*h) + delta*(Qc[i-1]+0.5
*KQ2c[i]*h) + v*(Ic[i-1]+0.5*KI2c[i]*h)- miu*(Rc[i-1]+0.5*KR2c[i]*h) KS4c[i] = lamda-(beta*(Sc[i-1]+KS3c[i]*h)*((Ic[i-1]+KI3c[i]*h)
+q*(Ac[i-1]+KA3c[i]*h)))/(N-(Qc[i-1]+KQ3c[i]*h))-miu*(Sc[i-1]
+KS3c[i]*h)
KE4c[i] = beta*(Sc[i-1]+KS3c[i]*h)*(Ic[i-1]+KI3c[i]*h)/(N-(Qc[i-1]
+KQ3c[i]*h)) + beta*(Sc[i-1]+KS3c[i]*h)*q*(Ac[i-1]+KA3c[i]*h)
/(N-(Qc[i-1]+KQ3c[i]*h)) -(Ec[i-1]+KE3c[i]*h)*(eta+teta+miu) KI4c[i] = p*eta*(Ec[i-1]+KE3c[i]*h)-(alfac+v+miu)*(Ic[i-1]+KI3c[i]
*h)+rho*(Ac[i-1]+KA3c[i]*h)
KQ4c[i] = alfac*(Ic[i-1]+KI3c[i]*h) + teta*(Ec[i-1]+KE3c[i]*h)-(Qc[i- 1]+KQ3c[i]*h)*(delta+miu)
KA4c[i] = (1-p)*eta*(Ec[i-1]+KE3c[i]*h)-(rho+gamma+miu)*(Ac[i-1]
+KA3c[i]*h)
KR4c[i] = gamma*(Ac[i-1]+KA3c[i]*h) + delta*(Qc[i-1]+KQ3c[i]*h) + v*(Ic[i-1]+KI3c[i]*h)- miu*(Rc[i-1]+KR3c[i]*h)
Sc[i] = Sc[i-1] + ((KS1c[i]+2*KS2c[i]+2*KS3c[i]+KS4c[i]))*h/6 Ec[i] = Ec[i-1] + ((KE1c[i]+2*KE2c[i]+2*KE3c[i]+KE4c[i]))*h/6 Ic[i] = Ic[i-1] + ((KI1c[i]+2*KI2c[i]+2*KI3c[i]+KI4c[i]))*h/6 Qc[i] = Qc[i-1] + ((KQ1c[i]+2*KQ2c[i]+2*KQ3c[i]+KQ4c[i]))*h/6 Ac[i] = Ac[i-1] + ((KA1c[i]+2*KA2c[i]+2*KA3c[i]+KA4c[i]))*h/6 Rc[i] = Rc[i-1] + ((KR1c[i]+2*KR2c[i]+2*KR3c[i]+KR4c[i]))*h/6 plt.plot(t,I,"r",label="alfa = 0.7")
plt.plot(t,Ib,"g",label="alfa = 0.8") plt.plot(t,Ic,"b",label="alfa = 0.9") plt.legend (loc = "best")
plt.axis([0,60, 0,100]) plt.xlabel ("t (hari)")
plt.ylabel ("Populasi Infected")
plt.title("Analisis sensitivitas terhadap alfa") plt.grid()
plt.show()