Saran - KESIMPULAN DAN SARAN - Skripsi. Diajukan Untuk Memenuhi Salah Satu Syarat Memperoleh Ge

BAB V KESIMPULAN DAN SARAN

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.

DAFTAR PUSTAKA

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Boyce, W. E. and Diprima, R. C. (2012). Elementary Differential Equations and Boundary Value Problems (9^𝑡ℎ Edition). Hoboken: John Wiley

& Sons.

Budhi, W. S. (2001). Kalkulus Peubah Banyak dan Penggunaannya. Ban-dung: Penerbit ITB.

Burden, R.L., and Faires, J.D. (2011). Numerical Analysis (Ninth Edition).

Boston: Brooks/Cole, Cengange Learning.

Djafri, D. (2015). Pemodelan Epidemiologi Penyakit Menular. Jurnal Kesehatan Masyarakat Andalas, 10(1), 1-2.

Fowkes, N.D., and Mahony, J.J. (1994). An Introduction to Mathematical Modelling. Chichester: John Wiley & Sons.

Giordano, F. R., William, P. F., and Steven, B. H. (2014). A First Course in Mathematical Modeling (5th Edition). Boston: Brooks/Cole, Cengange Learning.

Hahn, W. (1967). Stability of Motion. New York: Springer-Verlag.

Mamo, D. K. (2020). Model the transmission of COVID-19 propagation with public health intervention. Chaos, Solitons & Fractals, 7, 1-14.

Marsudi dan Marjono. (2012). Aljabar Linear. Malang: Universitas Brawi-jaya Press (UB Press.

Martcheva, M. (2015). An Introduction to Mathematical Epidemiology.

New York: Springer.

Matthews, J. H. and Kurtis, D. F. (2004). Numerical Methods Using MATLAB (4^𝑡ℎ Edition). New Jersey: Prentice Hall.

Mishra, A.M., Purohit, S.D., Owolabi, K.M., and Sharma Y.D. (2020). A nonlinear epidemiological model considering asymptotic and quar-antine classes for SARS-CoV-2 virus. Chaos, Solitons & Fractals, 138, 1-10.

Nishiura, H., Kinoshita, R., Jung, S., and Yuan, B. (2020). Estimation of the asymptomatic ratio of novel coronavirus infections (COVID-19).

Chaos, Solitons & Fractals, 94, 154-155.

Ndii, M. Z. (2018). Pemodelan Matematika Dinamika Populasi dan Penyebaran Penyakit: Teori, Aplikasi, dan Numerik. Yogyakarta:

Deepublish.

Perko, L. (1991). Differential Equation and Dynamical System. New York:

Springer-Verlag Berlin Heidelberg.

Ricardo, H. J. (2021). A Modern Introduction to Differential Equations (3^𝑟𝑑 Edition). United States: Elsevier Inc.

Ross, S. L. (1989). Introduction to Ordinary Differential Equations (4^𝑟𝑑 Edition). New York: John Wiley & Sons.

Rost, G., and Jianhong, W. (2008). SEIR Epidemiological Model with Var-ying Infectivity and Infinite Delay. Mathematical Biosciences and Engineering. 8(2): 389-402. Tersedia di http://www.mberjurnal.org [diakses 01 November 2021]

Schoenbach, Victor J, and Wayne D. S. (2000). Understanding the Funda-mentals of Epidemiology. Chapel Hill: University of North Carolina.

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.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)

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()

Dalam dokumen Skripsi. Diajukan Untuk Memenuhi Salah Satu Syarat Memperoleh Gelar Sarjana Matematika Program Studi Matematika (Halaman 96-118)