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AdvancesinAppliedMathematics
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,2021,10(7),2359-2368
PublishedOnlineJuly2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.107247
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AsymptoticBehavioroftheSolutionof
aRandom
SIQS
EpidemicModelwith
NonlinearInfectionRate
XiangrongLiu
ZhongyuanInstituteofTechnology,ZhengzhouHenan
Received:Jun.12
th
,2021;accepted:Jul.1
st
,2021;published:Jul.14
th
,2021
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2359-2368.DOI:10.12677/aam.2021.107247
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Abstract
Thispap erstudiesakindofrandom
SIQS
infectiousdiseasemodelwithnonlinear
infectionrate.First,itisprovedthattherandom
SIQS
infectiousdiseasemodelhas
auniqueglobalpositivesolutiontotheinitialconditionsofthepositive.Then,by
constructinganappropriateLyapunovFunctionandcombinedwiththeapplication
ofIto’sformula,thegradualbehaviorofthesolutionoftherandom
SIQS
infectious
diseasemodelnearthedisease-freebalancepointandtheendemicdiseasebalance
pointisanalyzedanddiscussed.
Keywords
Non-LinearInfectionRate,Random
SIQS
EpidemicModel,
It
ˆ
o
Formula,Asymptotic
Behavior
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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(0))+
KT
≥
E
[1
Ω
k
V
((
S
(
τ
m
,w
)
,I
(
τ
m
,w
)
,Q
(
τ
m
,w
))]
≥
ε
(
m
−
1
−
ln
m
)
∧
1
m
−
1
−
ln
1
m
,
Ù
¥
,1
Ω
k
•
Ω
k
«
5
¼
ê
,
-
m
→∞
,
k
∞
>V
(
S
(0)
,E
(0)
,I
(0))+
KT
=
∞
.
g
ñ
.
u
´
k
τ
e
=
∞
.
(
Ø
y
.
3.
÷
Ã
¾
²
ï
:
ì
C
1
•
½
n
2
b
R
0
=
β
Λ
µ
(
γ
+
δ
+
µ
)
≤
1,
…
÷
v
^
‡
σ
2
1
<µ,σ
2
2
<
2
µ,σ
2
3
<µ
,
K
é
?
¿
‰
½
Ð
©
Š
(
S
(0)
,I
(0)
,Q
(0))
∈
R
3
+
,
.
(2)
)
(
S
(
t
)
,I
(
t
)
,Q
(
t
))
∈
R
3
+
k
X
e
5
Ÿ
:
limsup
t
→∞
1
t
E
Z
t
0
{
(
S
−
Λ
µ
)
2
+
I
2
+
Q
2
}
d
r
≤
K
1
M
1
,
Ù
¥
c
1
=
2
µε
+2
µ
(
δ
+2
µ
)
βε
,c
2
=
2
µ
ε
A
1
= (1+
c
3
)(
µ
−
σ
2
1
)
,A
2
= (1+
c
2
)(
µ
−
σ
2
2
2
)+
δc
2
,A
3
=
µ
−
σ
2
3
,
M
1
= min
{
B
1
,B
2
,B
3
}
,K
1
= (1+
c
2
)
σ
2
1
Λ
2
µ
2
.
y
²
Ä
k
Š
C
þ
O
†
u
=
S
−
Λ
µ
,v
=
I,w
=
Q
,
K
X
Ú
(2)
Œ
C
†
•
:
du
dt
=
−
µu
−
β
(
u
+
Λ
µ
)
v
1+
αv
+
rv
+
εw
+
σ
1
(
u
+
Λ
µ
)d
B
1
(
t
)
,
dv
dt
=
β
(
u
+
Λ
µ
)
v
1+
αv
−
(
r
+
δ
+
ε
)
v
+
σ
2
v
d
B
2
(
t
)
,
dw
dt
=
δv
−
(
µ
+
ε
)
w
+
σ
3
w
d
B
1
(
t
)
,
(3)
½
Â
¼
ê
:
V
1
=
1
2
(
u
+
v
+
w
)
2
,V
2
=
c
1
v,V
3
=
c
2
2
(
u
+
v
)
2
,
DOI:10.12677/aam.2021.1072472363
A^
ê
Æ
?
Ð
4
•
J
Ù
¥
c
1
,c
2
•
?
¿
~
ê
,
A^
ž
B
ú
ª
[12],
·
‚
Œ
X
e
(
Ø
:
LV
1
=(
u
+
v
+
w
)[
−
µ
(
u
+
v
+
w
)]+
1
2
σ
2
1
(
u
+
Λ
µ
)
2
+
1
2
σ
2
2
v
2
+
1
2
σ
2
3
w
2
=
−
µu
2
−
µv
2
−
µw
2
−
2
µuv
−
2
µuw
−
2
µvw
+
1
2
σ
2
1
(
u
+
Λ
µ
)
2
+
1
2
σ
2
2
v
2
+
1
2
σ
2
3
w
2
,
(3.1)
LV
2
=
c
1
[
β
(
u
+
Λ
µ
)
v
1+
αv
−
(
r
+
δ
+
µ
)
v
]
≤
c
1
βuv
+
c
1
[
β
Λ
µ
−
(
r
+
δ
+
µ
)]
v,
(3.2)
LV
3
=
c
2
(
u
+
v
)[
−
µu
−
(
δ
+
µ
)
v
+
εw
]+
1
2
c
2
σ
2
1
(
u
+
Λ
µ
)
2
+
1
2
c
2
σ
2
2
v
2
=
−
c
2
µu
2
−
c
2
(
δ
+
µ
)
v
2
−
c
2
(
δ
+2
µ
)
uv
+
c
2
εuw
+
c
2
εvw
+
1
2
c
2
σ
2
1
(
u
+
Λ
µ
)
2
+
1
2
c
2
σ
2
2
v
2
.
(3.3)
¼
ê
V
=
V
1
+
V
2
+
V
3
,
=ò
(3
.
1)
,
(3
.
2)
,
(3
.
3)
ª
ƒ
\
,
LV
=
−
(
µ
+
c
2
µ
)
u
2
−
[
µ
+
c
2
(
δ
+
µ
)
v
2
]
−
µw
2
+[
c
1
β
−
2
µ
−
c
2
(2
µ
+
δ
)
uv
]
+(
c
2
ε
−
2
µ
)
uw
+(
c
2
ε
−
2
µ
)
vw
+
c
1
(
r
+
δ
+
µ
)(
R
0
−
1)
v
+
1
2
σ
2
1
(
u
+
Λ
µ
)
2
+
1
2
σ
2
2
v
2
+
1
2
σ
2
3
w
2
+
1
2
c
2
σ
2
1
(
u
+
Λ
µ
)
2
+
1
2
c
2
σ
2
2
v
2
≤−
(1+
c
2
)(
µ
−
σ
2
1
)
u
2
−
[(1+
c
2
)(
µ
−
σ
2
2
2
)+
δc
2
]
v
2
−
(
µ
−
σ
2
3
)
w
2
(1+
c
2
)
σ
2
1
Λ
2
µ
2
,
(3.4)
Ù
¥
c
1
=
2
µε
+2
µ
(
δ
+2
µ
)
βε
,c
2
=
2
µ
ε
,
K
k
c
2
ε
−
2
µ
= 0,
c
1
β
−
2
µ
−
c
2
(2
µ
+
δ
),
…
÷
v
R
0
≤
1.
-
A
1
= (1+
c
3
)(
µ
−
σ
2
1
)
,A
2
= (1+
c
2
)(
µ
−
σ
2
2
2
)+
δc
2
,A
3
=
µ
−
σ
2
1
,K
1
= (1+
c
2
)
σ
2
1
Λ
2
µ
2
,
qd
½
n
2
^
‡
Œ
:
dV
=
≤
(
−
A
1
u
2
−
A
2
v
2
−
A
3
w
2
+
K
1
)dt+
σ
1
(
u
+
Λ
µ
)[
u
+
v
+
w
+
c
2
(
u
+
v
)]dB
1
(
t
)
+
σ
2
v
[
c
1
+
u
+
v
+
w
+
c
2
(
u
+
v
)]dB
2
(
t
)+
σ
3
w
(
u
+
v
+
w
)dB
3
(
t
)
(3.5)
DOI:10.12677/aam.2021.1072472364
A^
ê
Æ
?
Ð
4
•
J
é
(3
.
5)
ª
ü
à
l
0
t
È
©
2
¦
Ï
"
,
k
0
≤
E
[
V
(
S
(
t
)
,I
(
t
)
,Q
(
t
))]
≤
E
[
V
(
S
(0)
,I
(0)
,Q
(0))]+
E
Z
t
0
(
−
A
1
u
2
−
A
2
v
2
−
A
3
w
2
+
K
1
)d
r,
l
í
Ñ
E
Z
t
0
(
−
A
1
u
2
−
A
2
v
2
−
A
3
w
2
)d
r
≤
E
[
V
(
S
(0)
,I
(0)
,Q
(0))]+
K
1
)+
K
1
t,
þ
ª
ü
à
Ó
ž
Ø
±
t
,
2
-
t
→∞
,
Œ
limsup
t
→∞
1
t
E
Z
t
0
(
−
A
1
u
2
−
A
2
v
2
−
A
3
w
2
)d
r
≤
K
1
,
M
1
= min
{
A
1
,A
2
,A
3
}
,
´
:
limsup
t
→∞
1
t
E
Z
t
0
{
(
S
−
Λ
µ
)
2
+
I
2
+
Q
2
}
d
r
≤
K
1
M
1
.
(
Ø
y
.
4.
÷
/
•
¾
²
ï
:
ì
C
1
•
½
n
3
b
R
0
=
β
Λ
µ
(
γ
+
δ
+
µ
)
>
1,
…
÷
v
^
‡
σ
2
1
<µ,σ
2
2
<µ,σ
2
3
<µ
,
K
é
?
¿
‰
½
Ð
©
Š
(
S
(0)
,I
(0)
,Q
(0))
∈
R
3
+
,
.
(2)
)
(
S
(
t
)
,I
(
t
)
,Q
(
t
))
∈
R
3
+
k
X
e
5
Ÿ
:
limsup
t
→∞
1
t
E
Z
t
0
{
(
S
−
S
∗
)
2
+(
I
−
I
∗
)
2
+(
Q
−
Q
∗
)
2
}
d
r
≤
K
2
M
2
,
Ù
¥
c
1
=
2
µε
+2
µ
(
δ
+2
µ
)
βε
,c
2
=
2
µ
ε
B
1
= (1+
c
3
)(
µ
−
σ
2
1
)
,B
2
= (1+
c
2
)(
µ
−
σ
2
1
)+
δc
2
,B
3
=
µ
−
σ
2
3
,
M
2
= min
{
A
1
,A
2
,A
3
}
,K
2
= 2
σ
2
1
(
S
∗
)
2
+2
σ
2
2
(
I
∗
)
2
+
σ
2
3
(
Q
∗
)
2
+
c
2
2
I
∗
σ
2
2
.
y
²
½
Â
¼
ê
:
V
1
=
1
2
(
S
−
S
∗
+
I
−
I
∗
+
Q
−
Q
∗
)
2
,V
2
=
c
1
(
I
−
I
∗
−
I
∗
ln
I
I
∗
)
,
DOI:10.12677/aam.2021.1072472365
A^
ê
Æ
?
Ð
4
•
J
V
3
=
c
2
2
(
S
−
S
∗
+
I
−
I
∗
)
2
,V
=
V
1
+
V
2
+
V
3
,
Ù
¥
c
1
,c
2
•
?
¿
~
ê
,
A^
ž
B
ú
ª
[12],
·
‚
Œ
X
e
(
Ø
:
LV
1
=(
S
−
S
∗
+
I
−
I
∗
+
Q
−
Q
∗
)[
−
µ
(
S
−
S
∗
)
−
µ
(
I
−
I
∗
)
−
µ
(
Q
−
Q
∗
)]
+
1
2
σ
2
1
S
2
+
1
2
σ
2
2
I
2
+
1
2
σ
2
3
Q
2
≤−
(
µ
−
σ
2
1
)(
S
−
S
∗
)
2
−
(
µ
−
σ
2
2
)(
I
−
I
∗
)
2
−
(
µ
−
σ
2
3
)(
Q
−
Q
∗
)
2
−
2
µ
(
S
−
S
∗
)(
I
−
I
∗
)
−
2
µ
(
S
−
S
∗
)(
Q
−
Q
∗
)
−
2
µ
(
I
−
I
∗
)(
Q
−
Q
∗
)
+
σ
2
1
(
S
∗
)
2
+
σ
2
2
(
I
∗
)
2
+
σ
2
3
(
Q
∗
)
2
,
(4.1)
LV
2
=
c
1
(
I
−
I
∗
)(
βSI
1+
αI
−
βS
∗
1+
αI
∗
)+
1
2
c
1
σ
2
2
I
∗
=
c
1
(
I
−
I
∗
)[
βS
(
1
1+
αI
)
−
1
1+
αI
∗
+
β
1+
αI
∗
(
S
−
S
∗
)]+
1
2
c
1
σ
2
2
I
∗
≤
c
1
β
(
S
−
S
∗
)(
I
−
I
∗
)+
1
2
c
1
σ
2
2
I
∗
,
(4.2)
LV
3
=
c
2
(
S
−
S
∗
+
I
−
I
∗
)[
−
µ
(
S
−
S
∗
)
−
(
µ
+
δ
)(
I
−
I
∗
)+
ε
(
Q
−
Q
∗
)]
+
1
2
c
2
σ
2
1
S
2
+
1
2
c
2
σ
2
2
I
2
=
−
c
2
µ
(
S
−
S
∗
)
2
−
c
2
(
µ
+
δ
)(
I
−
I
∗
)
2
−
c
2
(
δ
+2
µ
)(
S
−
S
∗
)(
I
−
I
∗
)
+
c
2
ε
(
S
−
S
∗
)(
Q
−
Q
∗
)+
c
2
ε
(
I
−
I
∗
)(
Q
−
Q
∗
)+
1
2
c
2
σ
2
1
S
2
+
1
2
c
2
σ
2
2
I
2
≤−
c
2
(
µ
−
σ
2
1
)(
S
−
S
∗
)
2
−
c
2
(
µ
+
δ
−
σ
2
2
)(
I
−
I
∗
)
2
−
c
2
(
δ
+2
µ
)(
S
−
S
∗
)(
I
−
I
∗
)+
c
2
ε
(
I
−
I
∗
)(
Q
−
Q
∗
)+
σ
2
1
(
S
∗
)
2
+
σ
2
2
(
I
∗
)
2
,
(4.3)
qd
V
=
V
1
+
V
2
+
V
3
,
Œ
:
LV
≤−
(1+
c
2
)(
µ
−
σ
2
1
)(
S
−
S
∗
)
2
−
[(1+
c
2
)(
µ
−
σ
2
2
)+
c
2
δ
](
I
−
I
∗
)
2
−
(
µ
−
σ
2
3
)(
Q
−
Q
∗
)
2
−
[
c
1
β
−
2
µ
−
c
2
(
δ
+2
µ
)](
S
−
S
∗
)(
I
−
I
∗
)
+(
c
2
ε
−
2
µ
)(
S
−
S
∗
)(
Q
−
Q
∗
)+(
c
2
ε
−
2
µ
)(
I
−
I
∗
)(
Q
−
Q
∗
)
+2
σ
2
1
(
S
∗
)
2
+2
σ
2
2
(
I
∗
)
2
+
σ
2
3
(
Q
∗
)
2
+
c
1
2
I
∗
σ
2
2
,
(4.4)
d
½
n
3
^
‡
n
Œ
:
LV
≤−
B
1
(
S
−
S
∗
)
2
−
B
2
(
I
−
I
∗
)
2
−
B
3
(
Q
−
Q
∗
)
2
+
K
2
,
(4.5)
Ù
¥
B
1
= (1+
c
3
)(
µ
−
σ
2
1
)
,B
2
= (1+
c
2
)(
µ
−
σ
2
1
)+
δc
2
,B
3
=
µ
−
σ
2
3
,
DOI:10.12677/aam.2021.1072472366
A^
ê
Æ
?
Ð
4
•
J
K
2
= 2
σ
2
1
(
S
∗
)
2
+2
σ
2
2
(
I
∗
)
2
+
σ
2
3
(
Q
∗
)
2
+
c
2
2
I
∗
σ
2
2
.
Ï
d
k
dV
=
LV
dt+(
S
−
S
∗
+
I
−
I
∗
+
Q
−
Q
∗
)[
σ
1
S
dB
1
(
t
)+
σ
2
I
dB
2
(
t
)+
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