Stochastic Process and Modelling Solution

7/24/2019 Stochastic Process and Modelling Solution

1/18

5th

k

(k+ 2)

th

t

t exp () t N(t) t

P[N(t) = n] =

t

nexp

t

n!

to (k+ 2) p

n=k+2

P[N(to) = n] p

1k+1n=0

P[N(to) = n] p

f(to) = 1k+1n=0

to

nexp

to

n!

p

to f(to) to to

1k+1n=0

to

nexp

to

n!

=p

to

f(to) to to= 0

to= 10 (k+ 2)

to = 10 (k+ 2) (k+ 2) k+ 2


2/18

k0 1 2 3 4 5

t(inhours)

3

3.5

4

4.5

5

5.5

66.5

7

7.5

8

8.5

9

9.5

10

10.5

11

k to

= 1 hour p= 0.9 k= 0, 1, 2, 3, 4, 5 to k to

to k

n=k+2P[N(to) = n] = 1

k+1

n=0to

nexp

to

n!

k= 0 = 1 hour to = 90 min= 1.5 hour

11

n=0

1.5n exp(1.5)

n! = 1 (exp(1.5) + 1.5exp(1.5)) = 1 2.5exp(1.5) 0.44

k= 0 2nd 1st

P[N(1.5) = 0] = exp (1.5) 0.22

to k = 0 3.89 hours

0.44 0.9


3/18

k

0 10 20 30 40 50 60 70 80 90 100

TheRatio

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

k to(k+2)

p

k

k+ 2 (k+ 2) k

to(k+2) k

to(k+2) to k

p k

k

1k+1n=0

to

nexp

to

n!

=

n=k+2

to

nexp

to

n!

= p

to

k+2exp

to

(k+ 2)!

+

to

k+3exp

to

(k+ 3)!

+

to

k+4exp

to

(k+ 4)!

+ = p

exp( (k+ 2) x)

(k+ 2)

k+2xk+2

(k+ 2)! +

(k+ 2)k+3

xk+3

(k+ 3)! +

(k+ 2)k+4

xk+4

(k+ 4)! +

p where x=

to

(k+ 2)

n

ln (n!) n ln (n) n = n! nn

en

exp( (k+ 2) x)

(k+ 2)

k+2xk+2

(k+ 2)(k+2)

e(k+2)+

(k+ 2)k+3

xk+3

(k+ 3)(k+3)

e(k+3)+

(k+ 2)k+4

xk+4

(k+ 4)(k+4)

e(k+4)+

= p

exp( (k+ 2) x)

xk+2

e(k+2)

1

0

k+ 2

k+2+

xk+3

e(k+3)

1

1

k+ 3

k+3+

xk+4

e(k+4)

1

2

k+ 4

k+4+

= p

limn

1 a

n

n=ea


4/18

exp( (k+ 2) x)

xk+2

e(k+2)+

xk+3

e(k+3)e1 +

xk+4

e(k+4)e2 +

= p

exp( (k+ 2) x)

exp( (k+ 2)) xk+2

1 + x + x2 +

= p

xe1x

1 + x + x2 + 1k+2 = p

1k+2

k 1k+2 0

p 1k+2

1

xe1x

1 + x + x2 + 1k+2 = 1

0< x 1 1 x= 1 k x 1

to

M ln

S = {0, 1, 2, . . . , M }

pAj

j=0

pAj j

j=0

pAj = 1

pLr (l)

r=0

pLr (l) r

l r=0

pLr (l) = 1; l pLr (l)

l

pDi (l)

li=0

pDi (l)

i

l

l pDi l

li=0

pDi (l) = 1 ; l pD0 (0) = 1

pJk(l) =j=k

pAj

j

k

1 pLjk(l)

k pLjk(l)

jk

x= 1

0 < x


5/18

pJk(l) k j pAj j k k

jk

1 pLjk(l)

k pLjk(l)

jk

j

ln ln+1 0 ln+1 M 1 k i k i

ln ln+1

k i= ln+1 ln; i {0, 1, . . . , ln} ; k 0

0 ln+1 M 1 ln ln+1

p (ln, ln+1) =

ki=ln+1ln; i{0,1,...,ln} ;k0

pDi (ln)pJk(ln)

ln+1 ln

p (ln, ln+1) =

ln+1k=0

pDk+lnln+1 (ln)pJk(ln)

ln+1 > ln

p (ln, ln+1) =

lni=0

pDi (ln)pJi+ln+1ln(ln)

ln ln+1 ln+1 = M 1

p (ln, M) = 1M1i=0

p (ln, i)

S = {0, 1, 2, . . . , M 1} ln M ln= M ln+1 = M+ 1 M

ln ln+1 0 ln+1 M1 ln 1 1 k k

k 1 = ln+1 ln; k 0

k ln+1 ln 1 0 ln+1 M 1 ln 1 ln ln+1

p (ln, ln+1) =

0 ; ln+1 < ln 1

pJln+1ln+1(ln) ; ln+1 ln 1


6/18

ln= 0 k k

k= ln+1; k 0

0 ln+1 M 1 ln= 0 p (0, ln+1) = p

Jln+1

(ln)

ln+1 = M

0 1 2 3 M 1

0 pJ0 (0) pJ1 (0) p

J2 (0) p

J3 (0) 1

M2i=0

pJi (0)

1 pJ0 (1) pJ1 (1) p

J2 (1) p

J3 (1) 1

M2i=0

pJi (1)

2 0 pJ0 (2) pJ1 (2) p

J2 (2) 1

M3i=0

pJi (2)

3 0 0 pJ0 (3) pJ1 (3) 1

M4i=0

pJi (3)

M 1 0 0 0 pJ0(M 1) 1 pJ0 (M 1)

x

x

kth

kth

kth

kth

r

X=

x1n x2n x3n

xin 1 ith nth 0

Rin 1 ith

nth 0 Rin= 1 q

Mijn 1 ith

jth nth 0 Mijn = 1 r Mijn =M

jin

Iijn 1 ith

jth nth 0 Iijn = 1 p


7/18

x1n+1 =

M12n

I12n

x2n

M13n

I13n

x3n ; x

1n= 0

R1n ; x1n= 1

1st nth x1n = 0 M12n

I12n

x2n 1

st 2nd

M13n

I13n

x3n x

1n+1 = 1 M

12n

I12n

x2n = 1 M

13n

I13n

x3n= 1

1st nth x1n= 1 x1n+1 = 0 1

st

R1n = 1 x1n+1 = 1 1st R1n = 0

x1n+1 =

M12n

I12n

x2n

M13n

I13n

x3n

x1n

R1n

x1n

x2n+1 =

M21n

I21n

x1n

M23n

I23n

x3n

x2n

R2n

x2n

x3n+1 =

M31n

I31n

x1n

M32n

I32n

x2n

x3n

R3n

x3n

N

xin+1 =

j{1,2,...,N} ; j=i

Mijn

Iijn

xjn

xinRinxin

Ai= A1

A2

AN

xn S = {0, 1, 2, 3} P xn xn+1

n= 1Pn1

n 1

nth 1st

1st 1 =

0 1 0 0

n En nth

En = n

0123

n

th

En=

0 1 0 0

Pn1

0123

P N

ith y (N y)

k (N y) y


8/18

pI(y, k) =

N y

k

(1 (1 rp)

y)k

((1 rp)y

)Nyk

k {0, 1, 2, . . . , N y} (1 rp)y y (1 (1 rp)y) y

y m

pR(y, m) =

y

m

qm (1 q)

m

k {0, 1, 2, . . . , y}

xn xn+1 k

m k m

xn xn+1

k m= xn+1 xn; k {0, 1, . . . , N y} ; m {0, 1, . . . , y}

xn xn+1

p (xn, xn+1) =

km=xn+1xn; k{0,1,...,Ny} ;m{0,1,...,y}pI(y, k)pR(y, m)

s = rp

P=

1 0 0 0

(1 s)2

q (1 s)

2(1 q)

+2s (1 s) q2s (1 s) (1 q)

+s2q s2 (1 q)

(1 s)2

q2

2 (1 s)2

q(1 q)

+

1 (1 s)2

q2

(1 s)2

(1 q)2

+2

1 (1 s)2

q(1 q)

1 (1 s)2

(1 q)2

q3 3q2 (1 q) 3q(1 q)2 (1 q)3

nth En(r,p,q) r p s = rp En(r,p,q) En(s, q) En(s, q) s q En(s, q) n


9/18

En(s, q) n

n En(s, q) s q

r

n= 1

Mijn n Mijn =M

ij Mij

n= 1 Mij =Mji

x1n+1 =

M12

I12n

x2n

M13

I13n

x3n

x1n+ R1n

x1n

x2n+1 =

M21

I21n

x1n

M23

I23n

x3n

x2n+ R

2n

x2n

x3n+1 =

M31

I31n

x1n

M32

I32n

x2n

x3n+ R

3n

x3n

xin+1 =

j{1,2,...,N} ; j=i

Mij

Iijn

xjn

xinRinxin

Xn+1 = f(Xn, Zn) Xn Zn Zn M

ij

Mij

x1n+1 =

M12n

I12n

x2n

M13n

I13n

x3n

x1n+ R

1n

x1n

x2n+1 =

M21n

I21n

x1n

M23n

I23n

x3n

x2n+ R

2n

x2n

x3n+1 = M31n

I31n x1n

M32n I32n

x2nx3n+ R

3n

x3n


10/18

Mij1 =M

ij2 =M

ij3 = M

ijn =M

ij i, j

Mijn

Xn+1 = f(Xn, Zn, Y) Y Mij

x1n+1 =

M12n

I12n

x2n

M13n

I13n

x3n

x1n+ R

1n

x1n

x2n+1 =

M12n

I21n

x1n

M23n

I23n

x3n

x2n+ R

2n

x2n

x3n+1 =

M13n

I31n

x1n

M23n

I32n

x2n

x3n+ R

3n

x3n

M12n+1 = M12n

M13n+1 = M13n

M23n+1 = M23n

p= 0.8 q= 0.5 r= 0.6

Day1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6 37 3 8 39 4 0

0

1Individual 1

Day

1 2 3 4 5 6 7 8 9 1 0 11 1 2 1 3 1 4 15 1 6 1 7 18 1 9 2 0 21 2 2 2 3 24 25 2 6 2 7 28 2 9 3 0 3 1 32 3 3 3 4 35 3 6 3 7 38 3 9 4 00

1Individual 2

Day

1 2 3 4 5 6 7 8 9 1 0 11 1 2 1 3 14 1 5 1 6 17 1 8 1 9 20 2 1 22 2 3 2 4 25 2 6 2 7 28 2 9 3 0 31 3 2 3 3 34 3 5 3 6 37 3 8 3 9 400

1Individual 3

p = 0.8

q= 0.5

r= 0.6

p= 0.5 q= 0.5 r= 0.5


11/18

Day

1 2 3 4 5 6 7 8 9 1 0 11 1 2 1 3 14 1 5 16 1 7 1 8 19 2 0 21 22 2 3 24 2 5 2 6 27 2 8 29 30 3 1 32 3 3 34 3 5 36 3 7 38 39 4 00

1Individual 1

Day

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 26 2 7 28 2 9 30 3 1 32 3 3 34 3 5 36 3 7 38 3 9 400

1Individual 2

Day

1 2 3 4 5 6 7 8 9 1 0 11 1 2 1 3 14 1 5 16 1 7 18 1 9 20 2 1 22 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6 37 3 8 39 4 00

1Individual 3

p = 0.5q= 0.5 r= 0.5

p r

N = 15 p= 0.8 q= 0.5 r= 0.6

Day0 5 10 15 20 25 30 35 40

0

1Individual 1

Day0 5 10 15 20 25 30 35 40

0

1Individual 2

Day0 5 10 15 20 25 30 35 40

0

1Individual 3

Day0 5 10 15 20 25 30 35 40

0

1Individual 4

Day0 5 10 15 20 25 30 35 40

0

1Individual 5

Day0 5 10 15 20 25 30 35 40

0

1Individual 6

Day0 5 10 15 20 25 30 35 40

0

1Individual 7

Day0 5 10 15 20 25 30 35 40

0

1Individual 8

Day0 5 10 15 20 25 30 35 40

0

1Individual 9

Day

0 5 10 15 20 25 30 35 400

1Individual 10

Day

0 5 10 15 20 25 30 35 400

1Individual 11

Day0 5 10 15 20 25 30 35 40

0

1Individual 12

Day0 5 10 15 20 25 30 35 40

0

1Individual 13

Day0 5 10 15 20 25 30 35 40

0

1Individual 14

Day0 5 10 15 20 25 30 35 40

0

1Individual 15

N= 15 p= 0.8 q= 0.5 r= 0.6

p= 0.8 q= 0.5 r= 0.6


12/18

Day

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 400

1Individual 1

Day

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 400

1Individual 2

Day

1 2 3 4 5 6 7 8 9 1 0 11 1 2 1 3 14 1 5 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 2 7 28 2 9 30 3 1 3 2 33 3 4 3 5 36 3 7 3 8 39 4 0

Individual 3

p = 0.8q= 0.5 r= 0.6

p= 0.5 q= 0.5 r= 0.5

Day1 2 3 4 5 6 7 8 9 10 1 1 12 1 3 14 1 5 16 1 7 18 1 9 20 2 1 22 2 3 24 2 5 26 2 7 28 2 9 30 3 1 32 3 3 34 3 5 36 3 7 38 3 9 40

0

1Individual 1

Day

1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 1 6 17 1 8 19 2 0 21 2 2 23 2 4 25 2 6 27 2 8 29 3 0 31 3 2 33 3 4 35 3 6 37 3 8 39 4 00

1Individual 2

Day1 2 3 4 5 6 7 8 9 1 0 11 1 2 13 1 4 15 16 1 7 18 1 9 20 2 1 22 23 2 4 25 2 6 27 2 8 29 30 3 1 32 3 3 34 3 5 36 37 3 8 39 4 0

0

1Individual 3

p = 0.5q= 0.5 r= 0.5

p r

N = 15 p= 0.8 q= 0.5 r= 0.6


13/18

Day0 5 10 15 20 25 30 35 40

0

1Individual 1

Day0 5 10 15 20 25 30 35 40

0

1Individual 2

Day0 5 10 15 20 25 30 35 40

0

1Individual 3

Day0 5 10 15 20 25 30 35 40

0

1Individual 4

Day

0 5 10 15 20 25 30 35 400

1Individual 5

Day

0 5 10 15 20 25 30 35 400

1Individual 6

Day0 5 10 15 20 25 30 35 40

0

1Individual 7

Day0 5 10 15 20 25 30 35 40

0

1Individual 8

Day0 5 10 15 20 25 30 35 40

0

1Individual 9

Day0 5 10 15 20 25 30 35 40

0

1Individual 10

Day

0 5 10 15 20 25 30 35 400

1Individual 11

Day0 5 10 15 20 25 30 35 40

0

1Individual 12

Day

0 5 10 15 20 25 30 35 400

1Individual 13

Day0 5 10 15 20 25 30 35 40

0

1Individual 14

Day

0 5 10 15 20 25 30 35 400

1Individual 15

N= 15 p= 0.8 q= 0.5 r= 0.6

En(r,p,q) EIn(r,p,q)

En(r,p,q) EIn(r,p,q) 0 n

En(r,p,q) EIn(r,p,q) n

En(r,p,q) EIn(r,p,q) n q q

En(r,p,q) EIn(r,p,q)

En(r,p,q) EIn(r,p,q) n p r

p r

EIn(r,p,q)

x1n+1 =

M12n

I12n

x2n

M13n

I13n

x3n

x1n+ R1n

x1n

x2n+1 =

M21n

I21n

x1n

M23n

I23n

x3n

x2n+ R

2n

x2n

x3n+1 =

M31n

I31n

x1n

M32n

I32n

x2n

x3n+ R

3n

x3n

I1n 1st

I1n = 1 1st

nth I1n= 0 I1n

I1n+1 = I1n

x1nR1n


14/18

x1n

x1n+1 =

M12n

I12n

x2n

M13n

I13n

x3n

x1n+ R

1n

x1n

I1n

xin+1 =

j{1,2,...,N} ; j=i

Mijn

Iijn

xjn

xinRinxinIin ; iIin+1 = I

in

xin

Rin

; i


15/18

10/6/15 4:16 PM C:\Us...\Inverse_Poisson_UpperCDF_Solver.m 1 of 1

fun!ion"!o# $ Inverse_Poisson_UpperCDF_Solver%!&u'('p)

*_l+$0, - Ini!i&l oer oun

*_u+$102%(3)2!&u, - Ini!i&l Upper oun

!resol$0.001, - *resol for error in p

fl&$1,

ilefl&$$1

mi$%*_l+3*_u+)/,

v&l$PoissonUpperCDF%!&u'('mi),

if%&+s%v&l7p)8$!resol)

fl&$0,

elseif%v&l8p)

*_l+$mi,

else

*_u+$mi,

en

en

!o$mi,

en

fun!ion"v&l# $ PoissonUpperCDF%!&u'('!)

v&l$1,

forn$0:1:%(31)

v&l$v&l7%!/!&u)9%n)2ep%7!/!&u)/f&!ori&l%n),

en

en

Appendix A (Inverse Poisson Upper CDF Solver: MATLAB Code)


16/18

10/7/15 1:10 PM C:\Users\sahah_000...\disease_expectancy.m 1 of 1

function! " disease_expectancy#n$

fort"1:1:n

i"1%

fors"0:0.05:1

&"1%

for'"0:0.05:1

P"Pro()ransMatrix#s*'$%

+#&*i$"0 1 0 0!,#P-#t1$$,0%1%%!%

&"&1%

end

i"i1%

end

2"0:0.05:1%

3"0:0.05:1%

surf#2*3*+$%

4rid on

xa(e#6s6$%

ya(e#6'6$%

a(e#6+6$%

str"sprintf#68ay 9d6*t$%

tite#str$%

pause#$

end

end

functionP!"Pro()ransMatrix#s*'$

1"1 0 0 0!%

"##1s$-$,' 0 0 #s-$,#1'$!%

"##1s$-$,'- 0 0 ##1'$-$,#1#1s$-$!%

;"'- ,#'-$,#1'$ ,',#1'$- #1'$-!%

P"1%%%;!%

P#*$"##1s$-$,#1'$,',s,#1s$%

P#*$",#1'$,s,#1s$#s-$,'%

P#*$",',#1'$,#1s$-#1#1s$-$,'-%

P#*$"##1s$-$,##1'$-$,',#1'$,#1#1s$-$%

end

Appendix B (Disease Expectancy: MATLAB Code)


17/18

10/7/15 5:10 AM C:\Users\sahah_000\Deskto...\disease_sim.m 1 of 2

clear

clc

% Model arameters: !"A#"

$&' % $(m)er of eo*le

*0.+' % ro)a)ilit, of -etti- ifected

0.5' % ro)a)ilit, of -etti- c(red

r0.' % ro)a)ilit, of Meeti-/ro)a)ilit, of )ei- a terati- air

sceario2' % sceario1 !tatic Meeti- Model 3teracti- air !ceario4

% sceario2 D,amic Meeti- Model

% Model arameters: $D

% Miscellaeo(s arameters: !"A#"

"sim60' % !im(latio "ime 3i Da,s4

o(t*(t1' %

% Miscellaeo(s arameters: $D

% itial !tate: !"A#"

8eros3$914'

tem*rad'

idema3ceil3tem*;$4914'

3ide41'

% itial !tate: $D

% !im(latio: !"A#"

arr8eros3$9"sim4'

arr3:914'

fort2:1:"sim

% #eco?er, @ector: $D

% *44914'

fori1:1:$

3i9i40'

ed

%


18/18

10/7/15 5:10 AM C:\Users\sahah_000\Deskto...\disease_sim.m 2 of 2

% r44914'

fori1:1:$

for1:1:i

if3i4

M3i940'

else

M3i94M39i4'

ed

ed

ed

ed

%

Stochastic Process and Modelling Solution

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