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d
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[
1
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.
T
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ex
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tr
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[
2
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,
[
3
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.
Ho
wev
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with
o
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p
p
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p
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tr
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laten
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ay
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an
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[
4
]
-
[
6
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.
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I
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Hea
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Sci
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Vo
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14
,
No
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3
,
Sep
tem
b
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20
25
:
1
1
3
7
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Desp
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m
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in
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1
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[
7
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,
[
8
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.
W
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T
B
is
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r
a
b
le
with
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ly
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is
k
s
s
u
ch
as
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u
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to
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[
9
]
-
[
1
4
]
.
T
h
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s
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in
clu
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n
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u
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p
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ap
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tr
ateg
ies [
1
5
]
,
[
1
6
]
.
T
h
e
d
y
n
am
ics
o
f
T
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tr
an
s
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is
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o
d
u
ce
s
a
d
eter
m
in
is
tic
s
u
s
ce
p
tib
le,
ex
p
o
s
ed
,
in
f
ec
ted
,
m
ed
icate
d
(
SEI
M
)
m
o
d
el
to
an
aly
ze
T
B
tr
an
s
m
is
s
io
n
d
y
n
a
m
ics.
Un
lik
e
p
r
ev
io
u
s
s
tu
d
ies,
th
is
m
o
d
el
in
co
r
p
o
r
ates two
p
h
ases
o
f
in
f
ec
tiv
ity
(
laten
t a
n
d
ac
tiv
e
T
B
)
,
ac
co
u
n
ts
f
o
r
b
o
t
h
s
lo
w
an
d
r
ap
id
T
B
p
r
o
g
r
ess
io
n
,
an
d
e
v
alu
ates th
e
im
p
ac
t o
f
ef
f
ec
tiv
e
tr
ea
tm
en
t
o
n
d
is
ea
s
e
d
y
n
am
ics.
T
h
e
m
o
d
el
al
s
o
co
n
s
id
er
s
p
o
ten
tial
o
u
tco
m
es
o
f
m
ed
icatio
n
,
in
clu
d
in
g
MT
B
er
ad
icatio
n
,
d
o
r
m
an
cy
in
d
u
ctio
n
,
a
n
d
ad
v
e
r
s
e
d
r
u
g
r
ea
ctio
n
s
.
Usi
n
g
L
y
a
p
u
n
o
v
f
u
n
ctio
n
s
,
t
h
e
s
tu
d
y
in
v
esti
g
ates
th
e
g
lo
b
al
s
tab
ilit
y
o
f
en
d
em
ic
an
d
n
o
n
-
e
n
d
em
ic
eq
u
ilib
r
i
u
m
p
o
in
ts
,
p
r
o
v
id
i
n
g
n
ew
in
s
ig
h
ts
in
to
T
B
co
n
tr
o
l
s
tr
ateg
ies.
B
y
ad
d
r
ess
in
g
th
ese
d
y
n
am
ics,
th
e
s
tu
d
y
o
f
f
er
s
a
n
o
v
el
f
r
am
ewo
r
k
f
o
r
u
n
d
er
s
tan
d
in
g
an
d
m
itig
atin
g
T
B
tr
an
s
m
is
s
io
n
[
1
7
]
-
[
1
9
]
.
2.
M
E
T
H
O
D
2
.
1
.
M
o
del
f
o
r
m
ula
t
io
n
T
h
i
s
s
t
u
d
y
d
e
v
e
l
o
p
s
a
d
e
t
e
r
m
i
n
i
s
ti
c
m
a
t
h
e
m
a
ti
c
a
l
m
o
d
e
l
t
o
a
n
a
l
y
z
e
t
u
b
e
r
c
u
l
o
s
i
s
(
T
B
)
d
y
n
a
m
i
c
s
b
y
c
o
n
s
i
d
e
r
i
n
g
f
o
u
r
p
o
p
u
l
a
t
i
o
n
c
o
m
p
a
r
t
m
e
n
t
s
:
s
u
s
c
e
p
t
i
b
l
e
(
S
)
,
ex
p
o
s
e
d
(
E
)
,
i
n
f
e
c
t
e
d
(
I
)
,
a
n
d
m
e
d
i
c
a
t
e
d
(
M
)
.
T
h
e
p
r
i
m
a
r
y
o
b
j
e
c
t
i
v
e
i
s
t
o
a
s
s
es
s
th
e
s
t
a
b
i
li
t
y
o
f
e
q
u
i
l
i
b
r
i
u
m
p
o
i
n
t
s
a
n
d
e
v
a
l
u
a
t
e
t
h
e
i
n
f
l
u
e
n
c
e
o
f
k
e
y
p
a
r
a
m
e
t
e
r
s
o
n
T
B
t
r
a
n
s
m
is
s
i
o
n
.
T
h
e
m
at
h
e
m
a
t
i
c
a
l
m
o
d
e
l
i
s
r
e
p
r
e
s
e
n
te
d
b
y
a
s
y
s
t
e
m
o
f
d
i
f
f
e
r
e
n
t
i
al
e
q
u
a
ti
o
n
s
t
h
a
t
d
e
s
c
r
i
b
e
t
h
e
c
h
a
n
g
e
s
i
n
e
a
c
h
p
o
p
u
l
a
t
i
o
n
c
o
m
p
a
r
t
m
e
n
t
a
s
:
-
D
i
f
f
e
r
e
n
t
i
al
e
q
u
a
t
i
o
n
s
:
=
−
−
+
(
1
−
)
=
+
−
(
+
)
=
(
1
−
)
+
−
(
+
+
)
=
−
(
+
+
)
Her
e,
=
+
+
+
r
ep
r
esen
ts
th
e
to
tal
p
o
p
u
latio
n
s
ize.
-
V
a
r
i
a
b
l
es
a
n
d
p
a
r
a
m
e
t
e
r
s
:
=
Nu
m
b
er
o
f
s
u
s
ce
p
tib
le
in
d
i
v
id
u
als.
=
Nu
m
b
er
o
f
ex
p
o
s
ed
in
d
iv
id
u
a
ls
.
=
Nu
m
b
er
o
f
in
f
ec
te
d
in
d
iv
i
d
u
a
ls
.
=
Nu
m
b
er
o
f
in
d
i
v
id
u
als r
ec
eiv
i
n
g
m
ed
icatio
n
.
=
C
o
n
tact
r
ate
o
f
in
f
ec
tio
n
.
=
R
ec
r
u
itm
en
t r
ate
o
f
n
ew
in
d
iv
id
u
als in
to
th
e
s
u
s
ce
p
tib
le
co
m
p
ar
tm
en
t.
=
Natu
r
al
d
ea
th
r
ate.
=
R
ate
o
f
p
r
o
g
r
ess
io
n
to
ac
tiv
e
T
B
f
r
o
m
th
e
e
x
p
o
s
ed
s
tate.
=
Me
d
icatio
n
r
ate.
=
Dea
th
r
ate
d
u
e
t
o
T
B
.
=
T
r
an
s
itio
n
r
ate
f
r
o
m
m
e
d
icatio
n
to
s
u
s
ce
p
tib
le
o
r
ex
p
o
s
ed
s
tates.
=
Dea
th
r
ate
d
u
e
t
o
co
m
p
licatio
n
s
f
r
o
m
m
e
d
icatio
n
.
=
Pro
p
o
r
tio
n
o
f
p
atien
ts
wh
o
en
ter
a
d
o
r
m
an
t state.
2
.
2
.
A
n
a
l
y
s
is
t
e
ch
ni
qu
e
T
o
a
n
a
l
y
z
e
t
h
e
s
t
a
b
il
i
t
y
o
f
e
q
u
i
l
i
b
r
i
u
m
p
o
i
n
ts
,
w
e
u
s
e
L
y
a
p
u
n
o
v
f
u
n
c
t
i
o
n
t
e
c
h
n
i
q
u
e
s
.
T
h
e
a
n
a
l
y
s
is
is
c
o
n
d
u
c
t
e
d
f
o
r
t
w
o
p
r
i
m
a
r
y
e
q
u
i
l
i
b
r
i
u
m
p
o
i
n
ts
:
a.
N
o
n
-
e
n
d
e
m
i
c
e
q
u
i
l
i
b
r
i
u
m
p
o
i
n
t
:
A
c
h
i
e
v
e
d
w
h
e
n
ℜ
0
<
1
,
i
n
d
i
c
a
ti
n
g
th
e
d
i
s
e
as
e
-
f
r
e
e
s
t
a
t
e
.
b.
E
n
d
e
m
i
c
e
q
u
i
l
i
b
r
i
u
m
p
o
i
n
t:
A
ch
i
e
v
e
d
w
h
e
n
ℜ
0
>
1
,
s
i
g
n
i
f
y
i
n
g
t
h
e
p
er
s
i
s
t
e
n
c
e
o
f
T
B
w
it
h
i
n
t
h
e
p
o
p
u
l
a
t
i
o
n
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Pu
b
lic
Hea
lth
Sci
I
SS
N:
2252
-
8
8
0
6
Glo
b
a
l sta
b
ilit
y
o
f S
E
I
M tu
b
ercu
lo
s
is
mo
d
el
w
ith
tw
o
in
fecti
o
n
p
h
a
s
es
…
(
Jo
via
n
Dia
n
P
r
a
ta
ma
)
1139
L
y
ap
u
n
o
v
f
u
n
ctio
n
tec
h
n
iq
u
es a
r
e
em
p
lo
y
e
d
to
p
r
o
v
e
t
h
e
g
lo
b
al
s
tab
ilit
y
o
f
th
es
e
eq
u
ilib
r
iu
m
p
o
in
ts
,
d
eter
m
in
in
g
w
h
eth
er
th
e
s
o
lu
t
io
n
s
co
n
v
er
g
e
to
th
ese
p
o
in
ts
o
v
er
tim
e.
T
h
e
b
asic
r
ep
r
o
d
u
c
tio
n
n
u
m
b
er
ℜ
0
is
co
m
p
u
ted
to
e
v
alu
ate
th
e
p
o
ten
tial
f
o
r
T
B
s
p
r
ea
d
with
in
t
h
e
p
o
p
u
latio
n
.
ℜ
0
is
ca
lcu
lated
u
s
in
g
th
e
n
ex
t
-
g
en
er
atio
n
m
atr
ix
ap
p
r
o
a
c
h
:
ℜ
0
=
(
+
+
)
(
(
1
−
)
(
+
)
+
)
[
(
+
)
(
+
+
)
(
+
+
)
−
]
An
ℜ
0
v
alu
e
less
th
an
o
n
e
i
n
d
ica
tes
th
at
th
e
d
is
ea
s
e
will
d
ec
lin
e,
wh
ile
an
ℜ
0
v
alu
e
g
r
ea
ter
th
a
n
o
n
e
s
u
g
g
est
s
th
e
p
o
ten
tial f
o
r
en
d
em
icity
.
2
.
3
.
Num
er
ic
a
l
s
im
ula
t
io
ns
Nu
m
er
ical
s
im
u
latio
n
s
ar
e
p
er
f
o
r
m
ed
to
e
v
alu
ate
t
h
e
im
p
ac
t
o
f
k
ey
p
ar
am
ete
r
s
o
n
T
B
tr
an
s
m
is
s
io
n
.
T
h
e
m
o
d
el
is
im
p
lem
en
ted
u
s
in
g
m
ath
em
atica
l
s
o
f
twar
e
to
s
im
u
late
r
e
al
-
wo
r
ld
s
ce
n
ar
io
s
.
Nu
m
er
ical
s
im
u
latio
n
s
ar
e
p
er
f
o
r
m
e
d
to
e
v
alu
ate
th
e
in
f
lu
en
ce
o
f
m
o
d
el
p
ar
am
eter
s
o
n
T
B
d
y
n
a
m
ics.
T
h
e
s
im
u
latio
n
s
ar
e
im
p
lem
en
ted
u
s
in
g
m
ath
em
ati
ca
l
s
o
f
twar
e
b
ased
o
n
p
a
r
am
e
ter
v
alu
es
d
er
iv
ed
f
r
o
m
r
elev
a
n
t
ep
id
em
io
lo
g
ical
s
tu
d
ies.
Key
p
ar
am
eter
s
,
s
u
ch
as
in
f
ec
tio
n
r
ates,
co
n
tact
r
ates
,
an
d
m
e
d
icatio
n
r
ates,
ar
e
ta
k
e
n
f
r
o
m
th
e
liter
atu
r
e
to
en
s
u
r
e
r
ea
lis
m
an
d
r
eliab
il
ity
in
th
e
s
im
u
latio
n
s
.
A
s
en
s
itiv
ity
an
aly
s
is
is
co
n
d
u
cted
to
d
eter
m
in
e
h
o
w
v
ar
iatio
n
s
in
m
o
d
el
p
ar
a
m
eter
s
,
s
u
ch
as
(
c
o
n
tact
r
ate)
an
d
(
m
ed
icatio
n
r
ate)
,
af
f
ec
t
ℜ
0
an
d
T
B
tr
an
s
m
is
s
io
n
.
T
h
is
an
aly
s
is
h
elp
s
id
en
tify
th
e
m
o
s
t c
r
itical
f
ac
to
r
s
in
f
lu
en
c
in
g
d
is
ea
s
e
co
n
tr
o
l.
2
.
4
.
M
o
del
v
a
lid
a
t
io
n,
i
m
pli
ca
t
io
ns
,
a
nd
re
co
mm
enda
t
io
ns
Mo
d
el
v
alid
atio
n
is
ca
r
r
ied
o
u
t b
y
co
m
p
ar
in
g
s
im
u
latio
n
r
esu
lts
with
ac
tu
al
e
p
id
em
io
lo
g
ical
d
ata
a
n
d
p
r
ev
io
u
s
s
tu
d
ies.
T
h
is
en
s
u
r
es
th
at
th
e
d
ev
elo
p
ed
m
o
d
el
ac
cu
r
ately
r
ef
lects
T
B
d
y
n
am
ics.
T
h
e
r
esu
lts
f
r
o
m
th
is
m
o
d
el
p
r
o
v
id
e
in
s
ig
h
ts
in
to
e
f
f
ec
tiv
e
T
B
co
n
tr
o
l
s
tr
ateg
ies.
B
ased
o
n
th
e
an
aly
s
is
,
it
i
s
r
ec
o
m
m
en
d
ed
t
h
at
p
u
b
lic
h
ea
lth
p
o
licies
f
o
cu
s
o
n
r
ed
u
c
in
g
th
e
f
r
e
q
u
en
cy
o
f
co
n
tact
b
etwe
en
v
u
ln
e
r
ab
le
an
d
in
f
ec
t
ed
in
d
iv
id
u
als
an
d
im
p
r
o
v
in
g
m
ed
icatio
n
ad
h
er
en
ce
.
T
h
e
m
o
d
el
r
esu
lts
o
f
f
er
v
alu
ab
le
in
s
ig
h
ts
f
o
r
T
B
co
n
tr
o
l stra
teg
ies.
E
f
f
ec
tiv
e
in
ter
v
en
tio
n
s
i
n
clu
d
e:
a.
R
ed
u
cin
g
co
n
tact
r
ates b
etwe
en
v
u
ln
e
r
ab
le
an
d
in
f
ec
ted
in
d
i
v
id
u
als.
b.
E
n
h
an
cin
g
m
ed
icatio
n
ad
h
e
r
e
n
ce
to
lim
it T
B
p
r
o
g
r
ess
io
n
an
d
tr
an
s
m
is
s
io
n
.
Pu
b
lic
h
ea
lth
p
o
licies s
h
o
u
ld
in
co
r
p
o
r
ate
th
ese
f
in
d
in
g
s
to
s
t
r
en
g
th
en
T
B
er
ad
icatio
n
e
f
f
o
r
t
s
.
3.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
T
h
e
p
r
o
p
o
s
ed
SEI
M
m
o
d
el
o
f
f
er
s
u
n
iq
u
e
in
s
ig
h
ts
in
to
tu
b
er
c
u
lo
s
is
(
T
B
)
d
y
n
am
ics
b
y
f
o
cu
s
in
g
o
n
th
e
r
o
le
o
f
m
ed
icatio
n
an
d
its
s
id
e
ef
f
ec
ts
.
T
o
co
n
tex
tu
alize
th
es
e
f
in
d
in
g
s
,
th
e
r
esu
lts
ar
e
co
m
p
ar
ed
with
s
im
ilar
s
tu
d
ies:
a.
SEI
R
m
o
d
els with
v
ital d
y
n
a
m
ics
Stu
d
ies
u
s
in
g
SEI
R
f
r
am
ewo
r
k
s
h
ig
h
lig
h
t
t
h
e
r
o
le
o
f
late
n
cy
in
d
ela
y
in
g
th
e
s
p
r
ea
d
o
f
T
B
.
Fo
r
in
s
tan
ce
,
r
esear
ch
b
y
Sid
e
et
a
l
.
[
2
0
]
em
p
h
asizes
th
e
eq
u
ilib
r
i
u
m
s
tates
f
o
r
ℜ
0
<
1
an
d
ℜ
0
>
1
,
b
u
t
lack
s
f
o
cu
s
o
n
tr
ea
tm
en
t
ef
f
ec
ts
,
a
k
ey
f
ea
tu
r
e
in
o
u
r
m
o
d
el
[
2
1
]
.
Ou
r
SEI
M
m
o
d
el
co
m
p
lem
e
n
t
s
th
ese
f
in
d
in
g
s
b
y
in
teg
r
atin
g
d
r
u
g
-
in
d
u
ce
d
d
o
r
m
an
cy
a
n
d
tr
an
s
itio
n
s
b
etwe
e
n
co
m
p
ar
tm
en
ts
,
o
f
f
e
r
in
g
a
m
o
r
e
c
o
m
p
r
e
h
en
s
iv
e
an
aly
s
is
o
f
tr
ea
tm
en
t im
p
ac
t.
b.
Ag
e
-
s
tr
u
ctu
r
ed
m
o
d
els
R
esear
ch
o
n
ag
e
-
s
tr
u
ctu
r
ed
T
B
m
o
d
els
ex
p
lo
r
es
d
if
f
er
en
t
tr
an
s
m
is
s
io
n
r
ates
f
o
r
ch
ild
r
en
an
d
ad
u
lts
,
o
f
f
er
in
g
d
etailed
s
tr
atif
icatio
n
s
o
f
th
e
p
o
p
u
latio
n
.
Ho
wev
er
,
th
ese
m
o
d
els g
en
er
ally
o
v
er
lo
o
k
th
e
co
m
p
lex
ities
o
f
tr
ea
tm
en
t
p
ath
way
s
,
s
u
ch
as
to
x
icity
o
r
in
co
m
p
lete
r
ec
o
v
er
y
.
B
y
co
n
tr
as
t,
o
u
r
r
esu
lts
d
em
o
n
s
tr
ate
h
o
w
m
ed
icatio
n
in
f
lu
en
ce
s
T
B
d
y
n
am
ics,
p
r
o
v
id
i
n
g
ac
tio
n
a
b
le
in
s
ig
h
ts
in
to
th
er
ap
y
o
p
tim
izatio
n
[
2
2
]
,
[
2
3
]
.
c.
Vac
cin
atio
n
d
y
n
am
ics
Vac
cin
atio
n
m
o
d
els,
s
u
ch
as
t
h
o
s
e
p
r
esen
ted
b
y
Ald
ila
et
a
l.
[
2
4
]
,
p
r
im
a
r
ily
f
o
cu
s
o
n
r
ed
u
cin
g
th
e
b
asic
r
ep
r
o
d
u
ctio
n
n
u
m
b
er
(
ℜ
0
)
th
r
o
u
g
h
im
m
u
n
izatio
n
ca
m
p
aig
n
s
.
W
h
ile
ef
f
ec
tiv
e
f
o
r
p
r
ev
en
tio
n
,
th
ese
m
o
d
els
d
o
n
o
t
ev
alu
ate
tr
ea
t
m
en
t
o
u
tco
m
es.
Ou
r
SEI
M
m
o
d
el
ad
d
r
ess
es
th
is
g
ap
b
y
q
u
an
tify
in
g
m
e
d
icatio
n
ef
f
ec
ts
an
d
th
eir
p
o
ten
tial to
r
e
d
u
ce
ℜ
0
th
r
o
u
g
h
im
p
r
o
v
e
d
ad
h
e
r
en
ce
an
d
m
in
im
izin
g
to
x
icity
[
2
5
]
.
Fro
m
a
p
u
b
lic
h
ea
lth
p
er
s
p
ec
t
iv
e,
th
e
r
esu
lts
em
p
h
asize
th
r
e
e
k
ey
im
p
licatio
n
s
f
o
r
T
B
co
n
tr
o
l.
First,
r
ed
u
cin
g
m
e
d
icatio
n
to
x
icity
r
ates
(
)
an
d
im
p
r
o
v
in
g
tr
ea
tm
e
n
t
ad
h
er
en
ce
(
)
ca
n
s
ig
n
if
ican
tly
lo
wer
ℜ
0
,
h
ig
h
lig
h
tin
g
th
e
n
ec
ess
ity
f
o
r
s
af
er
an
d
m
o
r
e
ef
f
ec
tiv
e
d
r
u
g
r
e
g
im
en
s
.
Seco
n
d
,
t
h
e
in
cl
u
s
io
n
o
f
d
ir
ec
t
tr
an
s
itio
n
s
f
r
o
m
s
u
s
ce
p
tib
le
to
in
f
ec
ted
co
m
p
ar
tm
en
ts
(
1
−
)
s
u
g
g
ests
th
e
n
ee
d
f
o
r
tar
g
eted
in
ter
v
en
tio
n
s
to
p
r
ev
en
t
r
ap
i
d
d
is
e
ase
p
r
o
g
r
ess
io
n
am
o
n
g
h
i
g
h
-
r
is
k
p
o
p
u
latio
n
s
.
Fin
ally
,
th
e
s
tu
d
y
u
n
d
e
r
s
co
r
es th
e
im
p
o
r
tan
ce
o
f
co
m
b
i
n
in
g
p
r
ev
en
tiv
e
m
ea
s
u
r
es,
s
u
ch
as
v
ac
cin
atio
n
,
with
r
o
b
u
s
t
tr
ea
tm
en
t
s
tr
ateg
ies
to
en
s
u
r
e
co
m
p
r
eh
en
s
iv
e
T
B
co
n
tr
o
l.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
8
0
6
I
n
t J Pu
b
lic
Hea
lth
Sci
,
Vo
l.
14
,
No
.
3
,
Sep
tem
b
er
20
25
:
1
1
3
7
-
1
1
5
0
1140
T
o
tr
a
n
s
late
th
ese
f
in
d
in
g
s
in
t
o
ac
tio
n
a
b
le
p
o
licies,
in
v
estm
en
ts
in
s
af
er
m
ed
icatio
n
s
ar
e
cr
u
cial
to
m
in
im
ize
to
x
icity
-
r
elate
d
m
o
r
tality
.
Ad
d
itio
n
ally
,
s
tr
en
g
t
h
en
in
g
h
ea
lth
ca
r
e
s
y
s
tem
s
to
e
n
h
an
ce
tr
ea
t
m
en
t
ad
h
er
en
ce
an
d
im
p
lem
en
tin
g
m
u
ltifa
ce
ted
p
r
ev
en
tio
n
a
n
d
t
r
ea
tm
en
t
s
tr
ateg
ies
ar
e
v
ital
f
o
r
ac
h
ie
v
in
g
s
u
s
t
ain
ed
r
e
d
u
ctio
n
s
in
T
B
p
r
ev
alen
ce
.
T
h
e
SEI
M
m
o
d
el
th
u
s
b
r
id
g
es si
g
n
if
ican
t g
a
p
s
in
T
B
m
o
d
elin
g
r
esear
ch
,
p
r
o
v
id
in
g
ac
tio
n
ab
le
in
s
ig
h
ts
th
at
alig
n
with
p
u
b
lic
h
ea
lth
p
r
io
r
ities
.
3
.
1
.
M
o
del f
o
r
m
ula
t
io
n
T
h
r
ee
c
o
m
p
ar
tm
e
n
ts
co
m
p
r
i
s
e
th
e
tr
an
s
m
is
s
io
n
m
o
d
el:
s
u
s
ce
p
tib
le
in
d
iv
i
d
u
als
(
)
,
ex
p
o
s
ed
in
d
iv
id
u
als
(
th
o
s
e
in
f
ec
ted
with
.
tu
b
er
cu
lo
s
is
b
u
t
n
o
t
clin
i
ca
lly
ill
an
d
h
en
ce
n
o
n
in
f
ec
ti
o
u
s
)
(
)
,
in
f
ec
te
d
in
d
iv
id
u
als
(
th
o
s
e
with
ac
tiv
e
T
B
)
(
)
,
an
d
m
ed
icate
d
in
d
iv
id
u
als
(
th
o
s
e
r
ec
eiv
in
g
m
ed
icatio
n
)
(
)
.
(
)
=
(
)
+
(
)
+
(
)
+
(
)
is
th
e
to
tal
n
u
m
b
e
r
o
f
p
eo
p
le
at
tim
e
.
I
n
d
iv
id
u
als
in
th
e
s
u
s
ce
p
tib
le
class
(
)
h
av
e
a
c
o
n
tact
r
ate
w
ith
in
f
ec
ted
in
d
iv
i
d
u
als
o
f
/
,
w
h
er
e
0
<
≤
1
is
th
e
p
r
o
b
a
b
ilit
y
o
f
a
s
u
s
ce
p
tib
le
in
d
iv
i
d
u
al
b
ein
g
in
f
e
cted
b
y
an
in
f
ec
ted
in
d
i
v
id
u
al.
T
h
e
r
ec
r
u
itm
en
t
r
ate
in
t
h
e
s
u
s
ce
p
tib
le
s
u
b
p
o
p
u
latio
n
is
d
en
o
ted
b
y
.
Su
s
ce
p
tib
le
in
d
iv
id
u
als
ca
n
b
e
ex
p
o
s
ed
to
MTB
an
d
m
o
v
e
to
s
u
b
p
o
p
u
lati
o
n
with
a
p
r
o
b
ab
ilit
y
o
f
,
wh
er
e
0
<
≤
1
.
Su
s
ce
p
tib
le
in
d
i
v
id
u
als
in
f
e
cted
with
th
e
MT
B
v
ir
u
s
ca
n
b
ec
o
m
e
in
f
ec
ted
with
o
u
t
f
ir
s
t
b
ec
o
m
in
g
an
ex
p
o
s
ed
in
d
iv
id
u
al
w
ith
a
p
r
o
b
ab
ilit
y
o
f
(
1
−
)
.
E
x
p
o
s
ed
in
d
iv
i
d
u
als
ca
n
d
ev
e
lo
p
ac
tiv
e
T
B
at
th
e
r
ate
>
0
.
I
n
f
ec
ted
in
d
iv
id
u
als
u
n
d
e
r
g
o
m
ed
ic
atio
n
at
th
e
le
v
el
>
0
an
d
m
o
v
e
to
t
h
e
s
u
b
p
o
p
u
latio
n
M(
)
.
I
n
f
ec
ted
in
d
iv
i
d
u
als
h
a
v
e
a
r
is
k
o
f
d
e
ath
f
r
o
m
T
B
d
is
ea
s
e
at
th
e
r
ate
>
0
,
wh
ile
th
e
n
atu
r
al
d
ea
th
r
ate
in
ea
ch
s
u
b
p
o
p
u
latio
n
is
>
0
.
I
t
is
a
s
s
u
m
ed
th
at
ad
m
in
is
ter
in
g
d
r
u
g
s
to
i
n
d
iv
id
u
als
in
f
ec
ted
with
T
B
g
iv
es
th
r
ee
r
ea
ctio
n
s
.
T
h
e
f
i
r
s
t
r
ea
ctio
n
is
th
at
th
e
MT
B
in
th
e
b
o
d
y
d
is
ap
p
ea
r
s
s
o
th
at
th
e
in
d
iv
id
u
al
will
r
etu
r
n
to
b
ei
n
g
a
s
u
s
ce
p
tib
le
in
d
iv
id
u
al
with
a
r
ate
(
1
−
)
.
T
h
e
s
ec
o
n
d
r
ea
ctio
n
is
to
m
ak
e
th
e
ac
tiv
e
T
B
v
ir
u
s
in
ac
tiv
e
(
d
o
r
m
an
c
y
)
s
o
th
at
th
e
in
d
iv
id
u
al
will
r
etu
r
n
to
b
ein
g
an
e
x
p
o
s
ed
in
d
iv
id
u
al
with
a
r
ate
.
T
h
e
last
,
ad
m
in
i
s
ter
in
g
d
r
u
g
s
ca
n
ca
u
s
e
s
o
m
eo
n
e
to
d
ie
d
u
e
to
p
o
is
o
n
in
g
,
liv
er
d
a
m
ag
e,
o
r
o
v
er
d
o
s
e
with
a
r
ate
>
0
.
T
h
e
tr
an
s
m
is
s
io
n
m
o
d
el
ca
n
b
e
wr
itten
as th
e
s
y
s
tem
o
f
n
o
n
l
in
ea
r
d
if
f
e
r
en
tial e
q
u
atio
n
s
,
as in
(
1
)
.
=
−
−
+
(
1
−
)
=
+
−
(
+
)
=
(
1
−
)
+
−
(
+
+
)
=
−
(
+
+
)
(
1
)
Fig
u
r
e
1
d
is
p
lay
s
th
e
(
1
)
tr
a
n
s
f
er
d
iag
r
am
.
T
h
e
f
r
ac
tio
n
s
o
f
t
h
e
class
s
u
s
ce
p
tib
le,
ex
p
o
s
ed
,
in
f
ec
ted
,
an
d
m
ed
icate
d
(
d
r
u
g
g
ed
)
in
th
e
p
o
p
u
latio
n
ar
e
r
ep
r
esen
ted
,
r
esp
e
ctiv
ely
,
b
y
th
e
v
a
r
iab
les
=
,
=
,
=
,
an
d
=
.
Ver
if
y
in
g
t
h
at
,
,
,
an
d
f
u
lf
ill th
e
d
if
f
er
en
tial e
q
u
atio
n
s
y
s
tem
is
s
im
p
le,
=
−
−
+
(
1
−
)
=
+
−
(
+
)
=
(
1
−
)
+
−
(
+
+
)
=
−
(
+
+
)
(
2
)
s
u
b
ject
to
th
e
r
estrictio
n
+
+
+
=
1
.
Giv
en
th
at
(
2
)
d
e
p
icts
th
e
h
u
m
an
p
o
p
u
latio
n
,
it is
im
p
er
ativ
e
to
r
ev
ea
l th
at
e
v
er
y
s
tate
v
ar
ia
b
le
is
n
o
n
n
e
g
ativ
e.
T
h
e
f
o
llo
win
g
s
u
b
s
ec
tio
n
p
r
o
v
id
es a
n
ex
p
lan
atio
n
o
f
th
ese
f
u
n
d
am
en
tal
ch
a
r
ac
ter
is
tics
.
Fig
u
r
e
1
.
T
r
an
s
f
er
d
ia
g
r
am
f
o
r
d
y
n
am
ic
(
1
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Pu
b
lic
Hea
lth
Sci
I
SS
N:
2252
-
8
8
0
6
Glo
b
a
l sta
b
ilit
y
o
f S
E
I
M tu
b
ercu
lo
s
is
mo
d
el
w
ith
tw
o
in
fecti
o
n
p
h
a
s
es
…
(
Jo
via
n
Dia
n
P
r
a
ta
ma
)
1141
3
.
2
.
P
o
s
it
iv
it
y
o
f
s
o
lutio
ns
a
nd
inv
a
ria
nt
re
g
io
n
3
.
2
.
1
.
P
o
s
it
iv
it
y
o
f
s
o
lutio
ns
T
o
d
em
o
n
s
tr
ate
th
e
e
p
id
em
io
l
o
g
ical
s
ig
n
if
ican
ce
o
f
t
h
e
T
B
m
o
d
el,
we
s
h
all
d
em
o
n
s
tr
ate
t
h
at
ev
er
y
v
ar
iab
le
in
(
2
)
is
n
o
n
-
n
e
g
ativ
e
f
o
r
all
tim
es
>
0
.
T
h
is
lead
s
u
s
to
th
e
f
o
llo
win
g
lem
m
a.
L
em
m
a
1
.
T
h
e
s
o
lu
tio
n
s
(
)
,
(
)
,
(
)
,
an
d
(
)
o
f
th
e
(
2
)
a
r
e
p
o
s
itiv
e
f
o
r
al
l
>
0
if
(
0
)
≥
0
,
(
0
)
≥
0
,
(
0
)
≥
0
,
an
d
(
0
)
≥
0
.
Pro
o
f
:
t
h
e
f
ir
s
t
eq
u
atio
n
o
f
(
2
)
s
tates
th
at
if
(
0
)
,
(
0
)
,
(
0
)
,
an
d
(
0
)
ar
e
all
ze
r
o
,
th
en
(
)
=
−
(
)
(
)
−
(
)
+
(
1
−
)
(
)
.
R
ewr
itin
g
it wo
u
ld
lo
o
k
lik
e
t
h
is
(
3
)
.
(
)
+
(
)
(
)
+
(
)
=
+
(
1
−
)
(
)
(
)
+
(
)
(
(
)
+
)
(
+
∫
(
)
0
)
=
(
+
(
1
−
)
(
)
)
(
+
∫
(
)
0
)
(
(
)
(
+
∫
(
)
0
)
)
=
(
+
(
1
−
)
(
)
)
e
x
p
(
+
∫
(
)
0
)
(
3
)
I
n
teg
r
atin
g
b
o
th
s
id
es,
we
g
et
(
4
)
.
(
)
e
xp
(
+
∫
(
)
0
)
−
(
0
)
=
∫
(
+
(
1
−
)
(
)
)
0
e
xp
(
+
∫
(
)
0
)
(
4
)
T
h
er
ef
o
r
e,
(
)
an
s
wer
is
(
5
)
.
(
)
=
(
0
)
(
−
−
∫
(
)
0
)
+
e
xp
(
−
−
∫
(
)
0
)
∫
(
+
(
1
−
)
(
)
)
e
xp
(
+
∫
(
)
0
)
0
>
0
,
∀
>
0
.
(
5)
L
ik
ewise,
we
m
ay
d
em
o
n
s
tr
ate
th
at
(
)
,
(
)
,
an
d
(
)
>
0
.
T
h
u
s
,
f
o
r
a
n
y
>
0
,
th
e
(
2
)
s
o
lu
tio
n
s
(
)
,
(
)
,
(
)
,
an
d
(
)
ar
e
p
o
s
itiv
e.
So
,
we
h
av
e
(
6
)
.
(
)
=
−
(
)
(
)
−
(
)
+
(
1
−
)
(
)
.
(
6)
R
ewr
itin
g
it
(
6
)
wo
u
ld
lo
o
k
lik
e
th
is
:
(
)
+
(
)
(
)
+
(
)
=
+
(
1
−
)
(
)
.
3
.
2
.
2
.
I
nv
a
ria
nt
re
g
io
n
T
h
e
b
io
lo
g
ical
im
p
o
r
tan
ce
o
f
o
u
r
(
2
)
ca
n
b
e
d
eter
m
in
e
d
b
y
a
n
aly
zin
g
it
with
in
a
s
u
i
tab
le
an
d
p
r
ac
tical
r
eg
io
n
an
d
d
em
o
n
s
tr
atin
g
th
at
th
e
p
ar
am
eter
s
an
d
v
a
r
iab
les ar
e
n
o
n
-
n
eg
ativ
e
f
o
r
all
tim
es
≥
0
.
L
em
m
a
2
.
I
n
(
2
)
p
o
s
itiv
ely
in
v
ar
ian
tly
d
ef
in
es th
e
f
ea
s
ib
le
r
e
g
io
n
as
(
7
).
Ω
=
{
(
(
)
,
(
)
,
(
)
,
(
)
)
∈
ℝ
+
4
0
:
≤
(
)
+
(
)
+
(
)
+
(
)
≤
}
(
7)
W
ith
s
tar
tin
g
co
n
d
itio
n
(
0
)
≥
0
,
(
0
)
≥
0
,
(
0
)
≥
0
,
(
0
)
≥
0
.
Pro
o
f
.
In
(
2
)
ar
e
a
d
d
ed
t
o
y
iel
d
(
8
).
(
)
=
(
)
+
(
)
+
(
)
+
(
)
=
−
(
)
−
(
)
−
(
)
<
−
(
)
(
8)
I
n
(
2
)
is
ex
am
in
ed
to
d
eter
m
i
n
e
th
e
f
u
n
d
am
en
tal
r
e
p
r
o
d
u
ctio
n
r
atio
an
d
in
v
esti
g
ate
th
e
s
tab
i
lity
o
f
b
o
t
h
en
d
e
m
ic
an
d
n
o
n
-
en
d
em
ic
eq
u
ilib
r
iu
m
.
0
≤
(
)
≤
+
(
0
)
−
,
wh
er
e
(
0
)
r
e
p
r
esen
ts
th
e
s
tar
tin
g
v
alu
es
o
f
th
e
en
tire
p
o
p
u
latio
n
,
f
o
llo
ws.
T
h
u
s
,
as
→
∞
,
0
≤
(
)
≤
.
T
h
u
s
,
f
o
r
(
2
)
,
th
e
r
eg
i
o
n
(
7
)
is
a
p
o
s
itiv
ely
in
v
ar
ian
t
s
et.
T
h
e
d
y
n
a
m
ics o
f
eq
u
atio
n
s
(
2
)
o
n
th
e
r
eg
i
o
n
Ω
will b
e
e
x
am
in
ed
.
3
.
3
.
B
a
s
ic
re
pro
du
ct
io
n r
a
t
i
o
W
e
u
s
e
th
e
n
e
x
t
-
g
e
n
er
atio
n
m
atr
ix
to
d
er
iv
e
th
e
b
asic
r
e
p
r
o
d
u
ctio
n
r
atio
,
ℜ
0
,
to
an
al
y
ze
th
e
s
tab
ilit
y
o
f
n
o
n
-
en
d
em
ic
eq
u
ilib
r
i
um
[
2
6
]
,
[
2
7
]
.
0
=
(
,
0
,
0
,
0
)
is
th
e
n
o
n
-
en
d
e
m
ic
eq
u
ilib
r
iu
m
p
o
in
t
o
f
th
e
(
2
)
.
Acc
o
r
d
in
g
t
o
[
2
8
]
,
[
2
9
]
,
th
e
b
a
s
ic
r
ep
r
o
d
u
ctio
n
r
atio
ℜ
0
f
o
r
(
1
)
th
r
o
u
g
h
(
8
)
is
as
(
9
).
ℜ
0
=
(
+
+
)
(
(
1
−
)
(
+
)
+
)
[
(
+
)
(
+
+
)
(
+
+
)
−
]
(
9)
T
h
e
f
o
llo
win
g
s
u
b
s
ec
tio
n
p
r
esen
ts
th
e
lo
ca
l stab
ilit
y
o
f
eq
u
ilib
r
iu
m
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
8
0
6
I
n
t J Pu
b
lic
Hea
lth
Sci
,
Vo
l.
14
,
No
.
3
,
Sep
tem
b
er
20
25
:
1
1
3
7
-
1
1
5
0
1142
3
.
4
.
G
l
o
ba
l st
a
bil
it
y
o
f
no
n
-
endem
ic
equil
i
brium
W
e
em
p
lo
y
L
y
a
p
u
n
o
v
s
tab
ilit
y
to
d
em
o
n
s
tr
ate
th
e
g
lo
b
al
s
tr
en
g
th
o
f
t
h
e
n
o
n
-
en
d
em
ic
eq
u
ilib
r
iu
m
p
o
in
t
0
.
T
h
eo
r
em
1
ca
n
b
e
u
tili
ze
d
to
illu
s
tr
ate
th
e
g
lo
b
al
s
ta
b
ilit
y
o
f
0
.
T
h
eo
r
e
m
1
[
3
0
]
,
[
3
1
]
s
tates
th
at
wh
en
ℜ
0
<
1
th
en
eq
u
ilib
r
iu
m
p
o
in
t
0
is
g
lo
b
ally
asy
m
p
to
ticall
y
s
tab
le
.
Pro
o
f
.
G
iv
en
th
e
ca
n
d
id
ate
L
y
ap
u
n
o
v
f
u
n
ctio
n
,
as in
(
10
).
=
1
+
2
+
3
(
10)
W
h
ich
was
m
o
tiv
ated
b
y
Sav
a
do
go
et
a
l.
[
3
2
]
an
d
Aziz
-
Al
ao
u
i
et
a
l.
[
3
3
]
,
wh
er
e
1
,
2
,
an
d
3
ar
e
p
o
s
itiv
e
co
n
s
tan
ts
th
at
will b
e
f
o
u
n
d
la
ter
.
̇
=
1
̇
+
2
̇
+
3
̇
is
its
d
er
iv
ativ
e
alo
n
g
th
e
s
o
l
u
tio
n
s
to
(
2
)
,
we
h
av
e
(
1
1
)
.
̇
=
1
(
+
−
(
+
)
)
+
2
(
(
1
−
)
+
−
(
+
+
)
)
+
3
(
−
(
+
+
)
)
.
̇
=
(
−
1
(
+
)
+
2
)
+
(
−
2
(
+
+
)
+
3
+
1
+
2
(
1
−
)
)
+
(
1
−
3
(
+
+
)
)
(
11)
T
h
e
co
ef
f
icien
ts
o
f
ar
e
s
et
to
eq
u
al
ze
r
o
b
y
th
e
co
n
s
tan
ts
1
an
d
2
.
C
o
n
s
eq
u
en
tly
,
we
h
av
e
(
12
).
1
=
,
2
=
(
+
)
(
12)
I
n
a
s
im
ilar
v
ein
,
t
h
e
co
n
s
tan
t
3
is
s
elec
ted
s
o
th
at
y
'
s
co
ef
f
ici
en
ts
eq
u
al
ze
r
o
.
W
e
o
b
tain
(
1
3
)
.
3
=
1
(
+
+
)
(
13)
Sin
ce
1
=
,
we
h
av
e
(
14
).
3
=
(
+
+
)
(
14
)
Fin
ally
,
we
o
b
tain
(
1
5
)
a
f
ter
in
s
er
tin
g
in
th
e
p
o
s
itiv
e
co
n
s
tan
t
s
1
,
2
,
an
d
3
f
r
o
m
(
13
)
in
(
14
).
̇
=
[
−
(
+
)
(
+
+
)
+
(
+
+
)
+
+
(
+
)
(
1
−
)
]
̇
=
1
(
+
+
)
[
(
(
1
−
)
+
)
(
+
+
)
]
[
−
1
ℜ
0
]
(
15)
As
≤
0
,
we
p
o
s
s
ess
(
16
).
̇
≤
(
(
1
−
)
+
)
(
+
+
)
(
+
+
)
[
1
−
1
ℜ
0
]
(
16
)
Mo
r
eo
v
er
,
̇
=
0
o
n
ly
h
o
ld
s
f
o
r
ℜ
0
=
1
o
r
=
0
.
T
h
e
0
is
th
e
g
r
e
atest
in
v
ar
ian
t
s
et
in
{
(
,
,
,
)
:
̇
=
0
}
.
T
h
e
L
aSalle's
in
v
ar
ian
ce
p
r
in
cip
le
[
3
4
]
-
[
3
6
]
s
tates
th
at
0
is
g
lo
b
ally
s
tab
le
wh
en
ℜ
0
<
1
.
3
.
5
.
G
l
o
ba
l st
a
bil
it
y
o
f
no
n
-
endem
ic
equil
i
brium
T
h
e
(
2
)
h
as e
n
d
e
m
ic
eq
u
ilib
r
i
u
m
p
o
i
n
t
∗
=
(
∗
,
∗
,
∗
,
∗
)
,
wh
er
e
(
1
7
)
.
∗
=
ℜ
0
,
∗
=
(
(
1
−
)
+
(
+
+
)
(
+
+
)
)
(
(
+
)
(
+
+
)
(
+
+
)
−
)
(
+
+
)
(
+
(
1
−
)
(
+
)
)
(
(
+
)
(
+
+
)
(
+
+
)
−
−
(
1
−
)
(
+
(
1
−
)
(
+
)
)
)
(
ℜ
0
−
1
)
,
∗
=
(
(
+
)
(
+
+
)
(
+
+
)
−
)
(
(
+
)
(
+
+
)
(
+
+
)
−
−
(
1
−
)
(
+
(
1
−
)
(
+
)
)
)
(
ℜ
0
−
1
)
,
∗
=
(
(
+
)
(
+
+
)
(
+
+
)
−
)
(
+
+
)
(
(
+
)
(
+
+
)
(
+
+
)
−
−
(
1
−
)
(
+
(
1
−
)
(
+
)
)
)
(
ℜ
0
−
1
)
.
(
17)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Pu
b
lic
Hea
lth
Sci
I
SS
N:
2252
-
8
8
0
6
Glo
b
a
l sta
b
ilit
y
o
f S
E
I
M tu
b
ercu
lo
s
is
mo
d
el
w
ith
tw
o
in
fecti
o
n
p
h
a
s
es
…
(
Jo
via
n
Dia
n
P
r
a
ta
ma
)
1143
T
h
u
s
,
th
e
en
d
em
ic
eq
u
ilib
r
iu
m
is
p
r
esen
t
wh
en
ℜ
0
>
1
.
L
ater
,
th
e
g
lo
b
al
s
tab
ilit
y
o
f
∗
is
d
em
o
n
s
tr
ated
u
s
in
g
T
h
eo
r
em
2
.
T
h
eo
r
em
2
.
W
h
en
ℜ
0
>
1
,
th
e
en
d
em
ic
eq
u
ilib
r
iu
m
p
o
in
t
EEP
∗
is
g
lo
b
ally
asy
m
p
to
tically
s
tab
le.
Pro
o
f
.
Fo
r
th
e
en
d
em
ic
eq
u
ilib
r
iu
m
∗
,
∗
,
∗
,
∗
,
an
d
∗
s
atis
f
ies
(
1
8
)
.
−
∗
∗
−
∗
+
(
1
−
)
∗
=
0
∗
∗
+
∗
−
(
+
)
∗
=
0
(
1
−
)
∗
∗
+
∗
−
(
+
+
)
∗
=
0
∗
−
(
+
+
)
∗
=
0
(
18)
Mo
tiv
ated
b
y
[
3
0
]
,
[
3
1
]
,
we
c
o
n
s
id
er
th
e
L
y
ap
u
n
o
v
f
u
n
ctio
n
as (
1
9
).
=
(
−
∗
ln
)
+
1
(
−
∗
ln
)
+
2
(
−
∗
ln
)
+
3
(
−
∗
ln
)
(
19)
W
h
er
e
th
e
p
o
s
itiv
e
co
n
s
tan
ts
1
,
2
,
an
d
3
ar
e
to
b
e
f
o
u
n
d
s
u
b
s
eq
u
en
tly
.
As
we
d
if
f
er
en
tiate
co
n
ce
r
n
in
g
alo
n
g
th
e
(
2
)
s
o
lu
tio
n
s
,
we
o
b
t
ain
(
2
0
)
.
̇
=
(
1
−
∗
)
̇
+
1
(
1
−
∗
)
̇
+
2
(
1
−
∗
)
̇
+
3
(
1
−
∗
)
̇
̇
=
(
1
−
∗
)
(
−
−
+
(
1
−
)
)
+
1
(
1
−
∗
)
(
+
−
(
+
)
)
+
2
(
1
−
∗
)
(
(
1
−
)
+
−
(
+
+
)
)
+
3
(
1
−
∗
)
(
−
(
+
+
)
)
̇
=
(
+
1
(
+
)
∗
+
2
(
+
+
)
∗
+
3
(
+
+
)
∗
)
−
∗
+
(
−
1
(
+
)
+
2
)
+
(
∗
+
3
−
2
(
+
+
)
)
+
(
(
1
−
)
+
1
−
3
(
+
+
)
)
+
(
−
+
1
+
2
(
1
−
)
)
+
∗
−
−
(
(
1
−
)
∗
∗
∗
)
−
(
1
∗
∗
∗
∗
∗
+
1
∗
∗
∗
)
−
(
2
(
1
−
)
∗
∗
∗
+
2
∗
∗
∗
)
−
(
3
∗
∗
∗
)
(
20)
B
y
co
n
s
id
er
in
g
in
(
18
)
,
o
n
e
h
a
s
(
21
).
=
∗
∗
+
∗
−
(
1
−
)
∗
(
+
)
∗
=
∗
∗
+
∗
(
+
+
)
∗
=
(
1
−
)
∗
∗
+
∗
(
+
+
)
∗
=
∗
(
21)
B
y
u
s
in
g
f
ir
s
t e
q
u
atio
n
u
n
til f
o
r
th
eq
u
atio
n
o
f
s
y
s
tem
(
1
8
)
,
in
(
19
)
b
ec
o
m
es
(
20
).
̇
=
∗
∗
(
1
−
∗
)
−
(
1
−
)
∗
(
1
−
∗
)
+
1
∗
∗
(
1
−
∗
∗
∗
)
+
1
∗
(
1
−
∗
∗
)
+
2
(
1
−
)
∗
∗
(
1
−
∗
)
+
2
∗
(
1
−
∗
∗
)
+
3
∗
(
1
−
∗
∗
)
+
(
−
1
(
+
)
+
2
)
+
(
∗
+
3
−
2
(
+
+
)
)
+
(
(
1
−
)
+
1
−
3
(
+
+
)
)
+
(
−
+
1
+
2
(
1
−
)
)
+
∗
(
2
−
∗
−
∗
)
−
(
(
1
−
)
∗
∗
∗
)
(
22)
L
et
=
∗
,
=
∗
,
=
∗
,
=
∗
,
th
en
in
(
22
)
b
ec
o
m
es
(
23
).
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
8
0
6
I
n
t J Pu
b
lic
Hea
lth
Sci
,
Vo
l.
14
,
No
.
3
,
Sep
tem
b
er
20
25
:
1
1
3
7
-
1
1
5
0
1144
̇
=
∗
(
2
−
−
1
)
+
∗
∗
(
1
−
1
)
+
1
∗
∗
(
1
−
)
+
2
(
1
−
)
∗
∗
(
1
−
)
+
1
∗
(
1
−
)
+
2
∗
(
1
−
)
+
3
∗
(
1
−
)
−
(
1
−
)
∗
(
1
−
1
)
−
(
(
1
−
)
∗
)
+
(
−
1
(
+
)
+
2
)
+
(
∗
+
3
−
2
(
+
+
)
)
+
(
(
1
−
)
+
1
−
3
(
+
+
)
)
+
(
−
+
1
+
2
(
1
−
)
)
(
23
)
Ma
k
in
g
th
e
c
o
ef
f
icien
ts
,
,
,
ar
e
eq
u
al
to
0
,
s
o
we
h
av
e
th
e
r
elatio
n
s
h
ip
(
24
).
1
(
+
)
=
2
2
(
+
+
)
=
∗
+
3
3
(
+
+
)
=
(
1
−
)
+
1
1
+
2
(
1
−
)
=
1
(
24
)
W
e
o
b
tain
(
2
5
)
.
1
=
(
+
(
1
−
)
(
+
)
)
,
2
=
(
+
)
(
+
(
1
−
)
(
+
)
)
,
3
=
(
1
−
)
+
1
(
+
+
)
(
25
)
R
ep
lacin
g
th
e
ex
p
r
ess
io
n
s
o
f
1
,
2
an
d
3
in
(
23
)
th
en
we
g
et
(
26
).
̇
=
∗
(
2
−
−
1
)
+
1
∗
∗
(
1
−
1
)
+
2
(
1
−
)
∗
∗
(
1
−
1
)
+
1
∗
∗
(
1
−
)
+
2
(
1
−
)
∗
∗
(
1
−
)
+
1
∗
(
1
−
)
+
2
∗
(
1
−
)
+
3
∗
(
1
−
)
−
(
1
−
)
∗
(
1
−
1
)
−
(
(
1
−
)
∗
)
+
(
−
1
(
+
)
+
2
)
+
(
∗
+
3
−
2
(
+
+
)
)
+
(
(
1
−
)
+
1
−
3
(
+
+
)
)
+
(
−
+
1
+
2
(
1
−
)
)
(
26)
Usi
n
g
th
e
f
ac
t th
at
1
+
2
(
1
−
)
=
1
,
in
(
26
)
b
e
co
m
es
(
2
7
)
.
̇
=
∗
(
2
−
−
1
)
+
1
∗
∗
(
2
−
1
−
)
+
2
(
1
−
)
∗
∗
(
2
−
−
1
)
+
1
∗
(
1
−
)
+
2
∗
(
1
−
)
+
3
∗
(
1
−
)
−
(
1
−
)
∗
(
1
−
1
)
−
(
(
1
−
)
∗
)
+
(
−
1
(
+
)
+
2
)
+
(
∗
+
3
−
2
(
+
+
)
)
+
(
(
1
−
)
+
1
−
3
(
+
+
)
)
+
(
−
+
1
+
2
(
1
−
)
)
(
27)
Mu
ltip
ly
in
g
th
e
s
ec
o
n
d
eq
u
atio
n
o
f
(
1
3
)
b
y
1
an
d
th
e
f
ir
s
t e
q
u
atio
n
o
f
(
1
6
)
b
y
∗
g
iv
es
(
2
8
)
.
1
(
+
)
∗
=
1
∗
∗
+
1
∗
1
(
+
)
∗
=
2
∗
(
2
8
)
W
e
ca
n
d
ed
u
ce
th
at
,
1
∗
∗
+
1
∗
−
2
∗
=
0
(
2
9
)
No
w,
m
u
ltip
ly
in
g
th
e
ab
o
v
e
e
q
u
atio
n
b
y
1
(
)
wh
er
e
=
(
,
,
,
)
an
d
1
(
)
will
b
e
d
eter
m
i
n
ed
later
y
ield
s
(
30
).
1
∗
∗
1
(
)
+
1
∗
1
(
)
−
2
∗
1
(
)
=
0
(
30)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J Pu
b
lic
Hea
lth
Sci
I
SS
N:
2252
-
8
8
0
6
Glo
b
a
l sta
b
ilit
y
o
f S
E
I
M tu
b
ercu
lo
s
is
mo
d
el
w
ith
tw
o
in
fecti
o
n
p
h
a
s
es
…
(
Jo
via
n
Dia
n
P
r
a
ta
ma
)
1145
Nex
t,
m
u
ltip
ly
in
g
th
e
th
ir
d
eq
u
atio
n
o
f
(
21
)
b
y
2
an
d
th
e
s
ec
o
n
d
eq
u
atio
n
o
f
(
24
)
b
y
∗
g
iv
es
(
3
1
)
.
2
(
+
+
)
∗
=
2
(
1
−
)
∗
∗
+
2
∗
2
(
+
+
)
∗
=
∗
∗
+
3
∗
(
31)
W
e
ca
n
wr
ite
(
3
1
)
as (
3
2
)
.
2
(
1
−
)
∗
∗
+
2
∗
−
∗
∗
−
3
∗
=
0
(
32)
Usi
n
g
th
e
f
ac
t th
at
1
+
2
(
1
−
)
=
1
,
th
e
(
3
2
)
b
ec
o
m
es
(
3
3
)
.
2
∗
−
1
∗
∗
−
3
∗
=
0
(
33)
No
w,
m
u
ltip
ly
in
g
th
e
(
3
3
)
by
2
(
)
wh
er
e
=
(
,
,
,
)
an
d
2
(
)
will b
e
d
eter
m
in
e
d
later
y
ield
s
(
34
).
2
∗
2
(
)
−
1
∗
∗
2
(
)
−
3
∗
2
(
)
=
0
(
34)
Nex
t,
m
u
ltip
ly
in
g
th
e
last
eq
u
atio
n
o
f
(
21
)
b
y
3
an
d
t
h
e
th
ir
d
eq
u
atio
n
o
f
(
24
)
b
y
∗
g
iv
es
(
3
5
)
3
(
+
+
)
∗
=
3
∗
3
(
+
+
)
∗
=
(
1
−
)
∗
+
1
∗
(
35)
W
e
ca
n
s
tate
(
3
5
)
as (
3
6
)
.
3
∗
−
(
1
−
)
∗
−
1
∗
=
0
(
36)
M
u
l
t
i
p
l
y
i
n
g
t
h
e
(
3
6
)
by
3
(
)
w
h
e
r
e
=
(
,
,
,
)
a
n
d
3
(
)
w
i
l
l
b
e
d
e
te
r
m
i
n
e
d
l
a
te
r
y
i
e
l
d
s
(
3
7
)
.
3
∗
3
(
)
−
(
1
−
)
∗
3
(
)
−
1
∗
3
(
)
=
0
(
3
7
)
Ad
d
in
g
th
e
(
30
)
,
(
34
)
,
a
n
d
(
37
)
in
to
(
27
)
,
we
o
b
tain
(
3
8
)
.
̇
=
∗
(
2
−
−
1
)
+
1
∗
∗
(
2
−
1
−
−
1
(
)
+
2
(
)
)
+
2
(
1
−
)
∗
∗
(
2
−
−
1
)
+
1
∗
(
1
−
−
1
(
)
+
3
(
)
)
+
2
∗
(
1
−
+
1
(
)
−
2
(
)
)
+
3
∗
(
1
−
+
2
(
)
−
3
(
)
)
−
(
1
−
)
∗
(
1
−
1
−
3
(
)
)
−
(
(
1
−
)
∗
)
(
38)
T
h
e
f
u
n
ctio
n
s
1
(
)
,
2
(
)
,
an
d
3
(
)
ar
e
ch
o
s
en
s
u
ch
th
at
th
e
c
o
ef
f
icien
ts
o
f
∗
an
d
∗
ar
e
e
q
u
al
t
o
ze
r
o
.
I
n
th
is
ca
s
e,
we
h
av
e
(
39
).
1
(
)
=
2
−
1
−
,
2
(
)
=
3
−
−
1
−
,
an
d
3
(
)
=
1
−
1
(
39)
Fin
ally
,
we
g
et
(
40
).
̇
=
∗
(
2
−
−
1
)
+
1
∗
∗
(
3
−
1
−
−
)
+
2
(
1
−
)
∗
∗
(
2
−
−
1
)
+
3
∗
(
3
−
−
−
)
−
(
(
1
−
)
∗
)
(
40)
Giv
en
th
at
th
e
g
eo
m
etr
ical
m
e
an
eq
u
als
o
r
ex
ce
e
d
s
th
e
ar
ith
m
etica
l
m
ea
n
,
2
−
−
1
≤
0
f
o
r
>
0
an
d
2
−
−
1
=
0
if
a
n
d
o
n
ly
if
=
1
.
T
h
en
,
3
−
1
−
−
≤
0
f
o
r
,
,
>
0
an
d
3
−
1
−
−
=
0
if
a
n
d
o
n
ly
if
=
=
=
1
.
Als
o
,
3
−
−
−
≤
0
f
o
r
,
,
>
0
a
n
d
3
−
−
−
=
0
if
a
n
d
o
n
ly
if
=
=
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8
8
0
6
I
n
t J Pu
b
lic
Hea
lth
Sci
,
Vo
l.
14
,
No
.
3
,
Sep
tem
b
er
20
25
:
1
1
3
7
-
1
1
5
0
1146
=
1
.
T
h
er
e
f
o
r
e,
th
e
m
a
x
im
u
m
i
n
v
ar
ian
t
s
et
o
f
th
e
eq
u
atio
n
s
(
2
)
o
n
th
e
s
et
(
,
,
,
)
:
̇
=
0
is
(
1
,
1
,
1
,
1
)
,
an
d
̇
≤
0
f
o
r
,
,
,
>
0
an
d
̇
=
0
if
an
d
o
n
ly
if
=
=
=
=
1
.
Acc
o
r
d
in
g
to
th
e
L
aSalle
I
n
v
ar
ian
ce
Prin
cip
le
[
3
4
]
-
[
3
6
]
,
in
(
2
)
en
d
em
ic
eq
u
ilib
r
iu
m
p
o
in
t,
∗
,
is
th
u
s
g
lo
b
ally
asy
m
p
t
o
tically
s
tab
le
if
ℜ
0
>
1
.
3
.
6
.
Sens
it
iv
it
y
a
na
l
y
s
is
Sen
s
itiv
ity
an
aly
s
is
is
a
cr
u
cial
to
o
l
i
n
u
n
d
er
s
tan
d
in
g
th
e
r
o
b
u
s
tn
ess
o
f
e
p
id
em
io
lo
g
ic
al
m
o
d
els,
esp
ec
ially
in
d
eter
m
in
i
n
g
h
o
w
v
ar
io
u
s
p
ar
am
eter
s
af
f
ec
t
th
e
b
asic
r
ep
r
o
d
u
ctio
n
n
u
m
b
er
(
ℜ
0
)
,
wh
ich
is
a
k
ey
in
d
icato
r
o
f
d
is
ea
s
e
tr
an
s
m
is
s
io
n
p
o
ten
tial
[
3
7
]
.
Nu
m
er
o
u
s
s
tu
d
ies
h
av
e
d
em
o
n
s
tr
ated
th
e
im
p
o
r
ta
n
ce
o
f
an
aly
zin
g
th
e
s
en
s
itiv
ity
o
f
m
o
d
el
p
a
r
am
eter
s
in
co
n
t
r
o
llin
g
in
f
ec
tio
u
s
d
is
ea
s
es,
in
clu
d
in
g
tu
b
er
c
u
lo
s
is
(
T
B
)
[
3
8
]
,
[
3
9
]
.
B
y
id
en
tify
in
g
th
e
m
o
s
t
in
f
lu
en
tial
p
ar
am
eter
s
,
s
u
ch
as
co
n
tact
r
ates
o
r
tr
ea
tm
e
n
t
r
ates,
r
esear
ch
er
s
ca
n
b
etter
allo
ca
te
r
eso
u
r
ce
s
a
n
d
d
esig
n
ta
r
g
eted
in
ter
v
en
tio
n
s
.
W
e
p
er
f
o
r
m
s
en
s
itiv
ity
an
aly
s
is
o
f
r
ep
r
o
d
u
ctio
n
n
u
m
b
er
ℜ
0
in
th
is
s
ec
tio
n
to
s
ee
th
e
ef
f
ec
t
o
f
an
d
on
ℜ
0
.
T
h
e
r
e
p
r
o
d
u
ctio
n
n
u
m
b
e
r
ℜ
0
is
g
iv
en
b
y
th
e
f
o
llo
win
g
e
q
u
atio
n
in
(
5
)
,
s
o
th
at
(
4
1
)
.
ℜ
0
=
(
+
+
)
(
(
1
−
)
(
+
)
+
)
[
(
+
)
(
+
+
)
(
+
+
)
−
]
(
4
1
)
we
lo
o
k
f
o
r
th
e
d
er
iv
ativ
e
wit
h
r
esp
ec
t to
, w
e
o
b
tain
(
4
2
)
.
ℜ
0
=
(
+
+
)
(
(
1
−
)
(
+
)
+
)
[
(
+
)
(
+
+
)
(
+
+
)
−
]
(
42)
W
h
e
n
(
+
)
(
+
+
)
(
+
+
)
>
,
w
e
h
a
v
e
ℜ
0
>
0
.
I
t
m
e
a
n
s
t
h
a
t
ℜ
0
in
c
r
e
a
s
es
w
h
e
n
i
n
c
r
e
a
s
es
.
Af
ter
th
at,
we
lo
o
k
f
o
r
th
e
d
e
r
iv
ativ
e
with
r
esp
ec
t to
,
we
o
b
t
ain
(
4
3
)
.
ℜ
0
=
−
(
(
1
−
)
(
+
)
+
)
(
(
+
)
(
+
+
)
−
)
(
(
+
)
(
+
+
)
(
+
+
)
−
)
2
(
43)
W
h
en
(
(
1
−
)
(
+
)
+
)
(
(
+
)
(
+
+
)
−
)
>
0
,
we
h
a
v
e
ℜ
0
<
0
.
I
t
m
ea
n
s
th
at
ℜ
0
d
ec
r
ea
s
es wh
en
in
cr
ea
s
es.
B
io
lo
g
ically
,
th
is
m
ea
n
s
th
at
i
n
f
ec
ted
in
d
iv
id
u
als
wh
ich
ar
e
m
ed
icate
d
h
av
e
g
r
ea
t
n
eg
ativ
e
in
f
lu
en
ce
o
n
th
e
s
p
r
ea
d
o
f
t
h
e
tu
b
er
c
u
lo
s
is
.
Fig
u
r
e
2
s
h
o
ws
th
e
r
elatio
n
s
h
ip
am
o
n
g
ℜ
0
,
,
an
d
wh
en
we
s
et
=
0
.
05
,
=
1
,
=
0
.
03
,
=
0
.
2
,
=
0
.
9
,
=
0
.
12
,
=
0
.
5
,
=
0
.
5
,
=
0
.
09
,
an
d
=
0
.
02
.
3
.
7
.
Num
er
ic
a
l sim
ula
t
io
n
Nu
m
er
ical
s
im
u
latio
n
s
p
la
y
a
c
r
itical
r
o
le
in
v
alid
atin
g
th
e
o
r
et
ical
r
esu
lts
an
d
p
r
o
v
id
i
n
g
v
is
u
al
in
s
ig
h
ts
in
to
th
e
d
y
n
a
m
ic
b
eh
av
io
r
o
f
i
n
f
ec
tio
u
s
d
is
ea
s
e
m
o
d
els.
T
h
ey
h
elp
d
em
o
n
s
tr
ate
h
o
w
d
if
f
e
r
en
t
p
ar
am
eter
v
alu
es
af
f
ec
t
th
e
tr
ajec
to
r
y
o
f
th
e
d
is
ea
s
e
o
v
er
tim
e,
esp
ec
ially
in
m
o
d
els
o
f
c
o
m
p
lex
d
is
ea
s
es
s
u
ch
as
tu
b
er
c
u
lo
s
is
(
T
B
)
[
4
0
]
.
Pre
v
io
u
s
s
tu
d
ies
h
a
v
e
s
h
o
wn
th
e
im
p
o
r
tan
ce
o
f
s
im
u
latin
g
T
B
m
o
d
els
to
u
n
d
er
s
tan
d
th
e
lo
n
g
-
ter
m
ef
f
ec
ts
o
f
i
n
ter
v
en
tio
n
s
an
d
th
e
b
eh
av
i
o
r
o
f
d
is
ea
s
e
tr
an
s
m
is
s
io
n
in
d
i
f
f
er
en
t
p
o
p
u
latio
n
g
r
o
u
p
s
[
4
1
]
,
[
4
2
]
.
B
y
ad
ju
s
tin
g
k
ey
p
ar
a
m
eter
s
s
u
ch
as
th
e
tr
an
s
m
is
s
io
n
r
ate
(
)
an
d
th
e
m
e
d
icatio
n
r
ate
(
)
,
we
c
an
o
b
s
er
v
e
h
o
w
th
ese
f
ac
to
r
s
in
f
lu
en
ce
t
h
e
s
tab
ilit
y
o
f
th
e
d
is
ea
s
e
-
f
r
ee
an
d
e
n
d
em
ic
eq
u
ilib
r
ia.
W
e
will
g
iv
e
s
o
m
e
s
im
u
la
tio
n
s
to
illu
s
tr
ate
th
e
th
eo
r
etica
l
an
aly
s
is
o
f
(
2
)
.
First,
we
s
et
=
0
.
05
,
=
1
,
=
0
.
03
,
=
0
.
2
,
=
0
.
9
,
=
0
.
12
,
=
0
.
5
,
=
0
.
5
,
=
0
.
09
,
=
0
.
02
an
d
we
h
a
v
e
ℜ
0
=
0
.
7733
<
1
.
Fig
u
r
e
3
p
r
esen
ts
th
e
tr
ajec
to
r
y
p
lo
t
o
f
(
2
)
wh
en
ℜ
0
<
1
.
Fig
u
r
e
3
illu
s
tr
ates
h
o
w
(
2
)
tr
ajec
to
r
y
m
ap
c
o
n
v
er
g
es
to
th
e
n
o
n
-
e
n
d
em
ic
eq
u
ilib
r
iu
m
.
I
n
o
th
er
wo
r
d
s
,
as
T
h
eo
r
em
1
ill
u
s
tr
ates,
th
e
d
is
ea
s
e
ev
en
tu
ally
d
is
ap
p
ea
r
s
f
r
o
m
th
e
p
o
p
u
latio
n
.
Seco
n
d
,
we
ch
an
g
e
th
e
v
al
u
e
o
f
an
d
.
W
e
ch
o
o
s
e
=
0
.
43
,
=
0
.
01
an
d
we
h
av
e
ℜ
0
=
12
.
6023
>
1
.
B
ased
o
n
T
h
eo
r
e
m
2
,
wh
e
n
ℜ
0
>
1
in
d
icate
s
th
at
th
e
d
is
ea
s
e
will
p
er
s
is
t
s
in
th
e
p
o
p
u
latio
n
.
Fig
u
r
e
4
co
n
f
ir
m
s
th
e
tr
ajec
to
r
y
p
lo
t
o
f
(
2
)
wh
en
ℜ
0
>
1
.
All
th
e
p
o
p
u
latio
n
co
m
p
ar
tm
en
ts
co
n
v
er
g
e
to
th
eir
en
d
em
ic
e
q
u
ilib
r
iu
m
.
Fro
m
Fig
u
r
e
s
3
an
d
4
,
we
ca
n
ea
s
ily
s
ee
th
at
th
e
r
ate
o
f
in
f
ec
ted
in
d
iv
id
u
als
p
er
f
o
r
m
m
ed
ica
tio
n
an
d
th
e
co
n
tact
r
ate
h
av
e
s
ig
n
if
ican
t
ef
f
ec
t
o
n
th
e
d
y
n
am
ics
o
f
T
B
p
o
p
u
latio
n
.
E
x
p
o
s
ed
,
in
f
ec
t
ed
,
an
d
m
ed
icate
d
s
u
b
p
o
p
u
latio
n
s
in
T
B
d
y
n
am
i
cs
ex
p
er
ien
ce
a
d
ec
r
ea
s
e
in
p
o
p
u
latio
n
n
u
m
b
e
r
s
f
r
o
m
th
e
in
i
tial
v
alu
e
u
n
til
th
ey
r
ea
ch
a
s
tab
le
co
n
d
itio
n
at
t
h
e
eq
u
ilib
r
iu
m
p
o
in
t.
W
h
en
we
d
ec
r
ea
s
e
th
e
v
alu
e
o
f
,
th
e
n
u
m
b
er
o
f
ex
p
o
s
ed
a
n
d
in
f
ec
ted
in
d
iv
id
u
als
in
cr
ea
s
e.
T
h
is
s
h
o
w
th
at
th
e
ad
m
in
is
tr
a
tio
n
o
f
m
e
d
icatio
n
h
as
n
eg
ati
v
e
in
f
l
u
en
ce
o
n
th
e
s
p
r
ea
d
o
f
th
e
T
B
d
is
ea
s
e.
W
h
ile
we
in
cr
ea
s
e
th
e
v
alu
e
o
f
,
th
e
n
u
m
b
er
o
f
ex
p
o
s
ed
an
d
in
f
ec
ted
in
d
iv
id
u
als
in
cr
ea
s
e.
T
h
is
s
h
o
w
th
at
th
e
c
o
n
tact
r
ate
h
as p
o
s
itiv
e
in
f
lu
en
ce
o
n
th
e
s
p
r
ea
d
o
f
th
e
T
B
d
is
ea
s
e.
Evaluation Warning : The document was created with Spire.PDF for Python.