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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(7),2359-2368
PublishedOnlineJuly2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.107247
˜aäkš‚5a/Ç‘ÅSIQSD/¾
.)ìC1•
444•••JJJ
¥‰EÆ§àHx²
ÂvFϵ2021c612F¶¹^Fϵ2021c71F¶uÙFϵ2021c714F
Á‡
©ïÄ˜a‘kš‚5a/Ç‘ÅSIQSD/¾."Äky²T‘ÅSIQSD/¾.
éЩ^‡•3X•˜Û)",§ÏLE·Lyapunov¼ê¿(ÜžBúª
A^§éT‘ÅSIQSD/¾.)3þ²ï:±9/•¾²ï:NCì?1•?1©
Û?Ø"
'…c
š‚5a/ǧ‘ÅSIQSD/¾.§žBúª§ìC1•
AsymptoticBehavioroftheSolutionof
aRandomSIQSEpidemicModelwith
NonlinearInfectionRate
XiangrongLiu
ZhongyuanInstituteofTechnology,ZhengzhouHenan
Received:Jun.12
th
,2021;accepted:Jul.1
st
,2021;published:Jul.14
th
,2021
©ÙÚ^:4•J.˜aäkš‚5a/Ç‘ÅSIQSD/¾.)ìC1•[J].A^êÆ?Ð,2021,10(7):
2359-2368.DOI:10.12677/aam.2021.107247
4•J
Abstract
Thispap erstudiesakindofrandomSIQSinfectiousdiseasemodelwithnonlinear
infectionrate.First,itisprovedthattherandomSIQSinfectiousdiseasemodelhas
auniqueglobalpositivesolutiontotheinitialconditionsofthepositive.Then,by
constructinganappropriateLyapunovFunctionandcombinedwiththeapplication
ofIto’sformula,thegradualbehaviorofthesolutionoftherandomSIQSinfectious
diseasemodelnearthedisease-freebalancepointandtheendemicdiseasebalance
pointisanalyzedanddiscussed.
Keywords
Non-LinearInfectionRate,RandomSIQSEpidemicModel,ItˆoFormula,Asymptotic
Behavior
Copyright
c
2021byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
1.Úó
3)ÔêÆïÄ+•¥,éD/¾ÄåÆ.ïĪ٘-‡©|,cÙ´2020c#.)
G¾Ó3¥lç,‰¥‘5î-/J,•ÏdD/¾.52•ÚåIS¯õ)Ô
êÆÆöéÙ'5[1][2][3][4][5].©3©z[6]¥D/¾.Ä:þ•Ä±e˜‡‘k
š‚5a/ÇD/¾.:















dS(t)
dt
= Λ−
βS(t)I(t)
1+αI(t)
−µS(t)+rI(t)+εQ(t),
dI(t)
dt
=
βS(t)I(t)
1+αI(t)
−(r+δ+µ)I(t),
dQ(t)
dt
= δI(t)−(µ+ε)Q(t),
(1)
Ù ¥S(t),I(t),Q(t)©OL«3tž•´aö!/¾öÚ…löêþ.ΛL«<•Ñ\Ç,… •
~ê;µ•´aö!/¾öÚ…lög,kÇ;r!ε!δþL«G=£Ç;
βS(t)I(t)
1+αI(t)
•a/Ǽ
DOI:10.12677/aam.2021.1072472360A^êÆ?Ð
4•J
ê;Λ,µ,r,ε,δ,þ•ê.
´•XÚ(1)Ä2)ê´R
0
=
βΛ
µ(γ+δ+µ)
,§'XX;¾u)†Ä.R
0
≤1 ž,XÚ(1)•
3ÛìC-½þ²ï:E
0
=(S
0
,0,0)= (
Λ
µ
,0,0);R
0
>1ž,XÚ(2)•3•˜ÛìC
-½/•¾²ï:E
∗
= (S
∗
,I
∗
,Q
∗
)[6].
d,du)D/¾uЬÉ‚¸ÏƒK•.Ù¥‚¸D(XxD(éD/¾u
ЬÚåØŒÑK•.Ïd,ïÄ‚¸D((XxD()ľ´N5K•D/¾ÄåÆXÚŒ
±¦·‚ïÄ(J•äky¢¿Â.éD/¾.ÄåÆ1•ïÄ,Ú\‘ÅÅÄ´Nõ
)ÔêÆÆö~^•{ƒ˜[7][8][9].3©¥,·‚æ^‘Å6Ä•{aquö[10],Ú\
‘Å6Ä'uS(t),I(t),Q(t).Ïd,dƒA(½.(1)Œ±e¡‘Å6Ä.:















dS(t)
dt
= Λ−
βS(t)I(t)
1+αI(t)
−µS(t)+rI(t)+εQ(t)+σ
1
S(t)dB
1
(t),
dI(t)
dt
=
βS(t)I(t)
1+αI(t)
−(r+δ+µ)I(t)+σ
2
I(t)dB
1
(t),
dQ(t)
dt
= δI(t)−(µ+ε)Q(t)+σ
3
Q(t)dB
1
(t),
(2)
Ù¥(B
1
(t),B
2
(t),B
3
(t)´½Â3Vǘm(Ω,F,{F
t
}
t>0
,P)þn‘IOBrownian$Ä,
σ
2
i
,i= 1,2,3L«xD(rÝ.
••BïÄ.,ØAÏ`²,·‚b(Ω,F,{F
t
}
t>0
,P)´˜‡äkÈf{F
t
}
t>0

Vǘm,Ù¥Èf{F
t
}
t>0
÷vÏ~^‡(3˜‡Vǘmp,XJ˜‡ÈfF
t
÷vF
t
=
T
s>t
F
s
(t≥0)…F
0
•¹¤kP"8).R
3
+
= {(S(t),I(t),Q(t))|S(t) >0,I(t) >0,Q(t) >0}.
…l)ÔÆÝw,D/¾.•kšK)žâäk¢S¿Â.Ïd©312Ü©y²3‰
½Щ^‡e,XÚ•3•˜Û);313Ü©,y²XÚ(3)3þ²ï:ìC1•;
14Ü©y²XÚ(3)3/•¾²ï:ìC1•;•‰Ñ©(Ø.
2.Û)•3•˜5
3©ÛD/¾.ÄåÆ1•ž,·‚ÄkI‡ïÄTXÚ´Ä•3Û).Ïd,3
!,·‚‰ÑXÚ(2)•3Û)y².
½n1é?¿‰½Њ(S(0),I(0),Q(0))∈R
3
+
,3t≥0þ,XÚ(2)•3•˜
)(S(t),I(t),Q(t)),…d)•VÇ1Ê33R
3
+
¥,•=(S(t),I(t),Q(t))∈R
3
+
é¤kt≥0
A7,¤á.
y²´•XÚ(2)Xê÷vÛÜLipschitz^‡,@oé?¿‰½Њ(S(0),I(0),Q(0))∈
R
3
+
,3t∈[0,τ
e
)þ,XÚ(2)Ñ•3•˜ÛÜ)(S(t),I(t),Q(t)),Ù¥τ
e
´»žm[11].•
y²ù‡ÛÜ)´Û,·‚•Iy²τ
e
=∞A7,¤á=Œ.•d,m
0
>0¿©Œ,¦
S(0),E(0)ÚQ(0)á3«m
h
1
m
0
,m
0
i
.?˜Ú,éz‡êm≥m
0
,½ÂÊž
τ
m
= inf

t∈[0,τ
e
) : S(t) 6∈

1
m
,m

½I(t) 6∈

1
m
,m

½Q(t) 6∈

1
m
,m

,
DOI:10.12677/aam.2021.1072472361A^êÆ?Ð
4•J
w,,m→∞ž,τ
m
üN4O.-τ
∞
=lim
m→∞
τ
m
,´•,τ
∞
≤τ
e
.eUy²τ
∞
= ∞,Kkτ
e
= ∞
¤á,ù¯K=Œ±y².
eyτ
∞
= ∞a.s.,æ^‡y{.XeØ,,=bτ
∞
6= ∞,K•3~êT>0Úε∈(0,1),k
P(τ
∞
≤t) >ε,
Ïd,•3êm
1
≥m
0
,¦é?¿m
1
≥m
0
,Ñk
P(τ
m
≤t) ≥ε.
½Â˜‡n¼êV: R
3
+
→R
+
,Xe:
V(S,I,Q) =

S−a−aln
S
a

+(I−1−lnI)+(Q−1−lnQ),
Ù¥a•~ê,u≥0ž,w,÷vu−1−lnu≥0,ddŒT¼ê´šK.
é?¿m≥m
0
ÚT>0.A^žBúª[12],k
dV(S,I,Q) = LV(S,I,Q)dt+σ
1
(S−a)dB
1
(t)+σ
2
(I−1)dB
2
(t)+σ
3
(Q−1)dB
3
(t).
Ù¥
LV(S,I,Q) =(1−
a
S
)(Λ−
βSI
1+αI
−µS+rI+εQ)+(1−
1
I
)(
βSI
1+αI
−(r+δ+µ)I)
+(1−
1
Q
)(δI−(µ+ε)Q)+
1
2
aσ
2
1
+
1
2
σ
2
2
+
1
2
σ
2
3
≤(Λ+aµ+r+δ+2µ+ε+
1
2
aσ
2
1
+
1
2
σ
2
2
+
1
2
σ
2
3
)+aβI−µI
≤Λ+aµ+r+δ+2µ+ε+
1
2
aσ
2
1
+
1
2
σ
2
2
+
1
2
σ
2
3
= K.
Ù¥a=
µ
β
,KkaβI−µI= 0.
Ïd
dV(S,I,Q) ≤Kdt+σ
1
(S−a)dB
1
(t)+σ
2
(I−1)dB
2
+bσ
3
(Q−1)dB
3
(t),
(2.1)
é(2.1)ªüàl0τ
m
∧TÈ©¿Ï"
EV(S(τ
m
∧T),I(τ
m
∧T),Q(τ
m
∧T)) ≤V(S(0),I(0),Q(0))+KE(τ
m
∧T),

EV(S(τ
m
∧T),I(τ
m
∧T),Q(τ
m
∧T)) ≤V(S(0),I(0),Q(0))+KT,(2.2)
m>m
1
ž,-Ω
m
={τ
m
≤T},KkP(Ω
m
)≥ε.éz‡w ∈Ω
m
,dÊž½ÂŒ
•S(τ
m
,w),I(τ
m
,w),Q(τ
m
,w)¥–k˜‡um½
1
m
,¤±
DOI:10.12677/aam.2021.1072472362A^êÆ?Ð
4•J
V(S(τ
m
,w),E(τ
m
,w),I(τ
m
,w)) ≥(m−1−lnm)½(
1
m
−1−ln
1
m
),
d(2.2)
V(S(0),E(0),I(0))+KT≥E[1
Ω
k
V((S(τ
m
,w),I(τ
m
,w),Q(τ
m
,w))]
≥ε(m−1−lnm)∧

1
m
−1−ln
1
m

,
Ù¥,1
Ω
k
•Ω
k
«5¼ê,-m→∞,k
∞>V(S(0),E(0),I(0))+KT= ∞.
gñ.u´kτ
e
= ∞.
(Øy.
3.÷þ²ï:ìC1•
½n2bR
0
=
βΛ
µ(γ+δ+µ)
≤1,…÷v^‡σ
2
1
<µ,σ
2
2
<2µ,σ
2
3
<µ,Ké?¿‰½Щ
Š(S(0),I(0),Q(0)) ∈R
3
+
,.(2))(S(t),I(t),Q(t)) ∈R
3
+
kXe5Ÿ:
limsup
t→∞
1
t
E
Z
t
0
{(S−
Λ
µ
)
2
+I
2
+Q
2
}dr≤
K
1
M
1
,
Ù¥
c
1
=
2µε+2µ(δ+2µ)
βε
,c
2
=
2µ
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
.
y²ÄkŠCþO†u= S−
Λ
µ
,v= I,w= Q,KXÚ(2)ŒC†•:

















du
dt
= −µu−
β(u+
Λ
µ
)v
1+αv
+rv+εw+σ
1
(u+
Λ
µ
)dB
1
(t),
dv
dt
=
β(u+
Λ
µ
)v
1+αv
−(r+δ+ε)v+σ
2
vdB
2
(t),
dw
dt
= δv−(µ+ε)w+σ
3
wdB
1
(t),
(3)
½Â¼ê:
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.2021.1072472363A^êÆ?Ð
4•J
Ù¥c
1
,c
2
•?¿~ê,A^žBúª[12],·‚ŒXe(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
−2µuv−2µuw−2µvw+
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+2µ)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)
¼êV= V
1
+V
2
+V
3
,=ò(3.1),(3.2),(3.3)ªƒ\,
LV=−(µ+c
2
µ)u
2
−[µ+c
2
(δ+µ)v
2
]−µw
2
+[c
1
β−2µ−c
2
(2µ+δ)uv]
+(c
2
ε−2µ)uw+(c
2
ε−2µ)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
=
2µε+2µ(δ+2µ)
βε
,c
2
=
2µ
ε
,
Kkc
2
ε−2µ= 0,c
1
β−2µ−c
2
(2µ+δ),…÷vR
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
,
qd½n2^‡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.2021.1072472364A^êÆ?Ð
4•J
é(3.5)ªüàl0tÈ©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
)dr,
líÑ
E
Z
t
0
(−A
1
u
2
−A
2
v
2
−A
3
w
2
)dr≤E[V(S(0),I(0),Q(0))]+K
1
)+K
1
t,
þªüàӞرt,2-t→∞,Œ
limsup
t→∞
1
t
E
Z
t
0
(−A
1
u
2
−A
2
v
2
−A
3
w
2
)dr≤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
}dr≤
K
1
M
1
.
(Øy.
4.÷/•¾²ï:ìC1•
½n3bR
0
=
βΛ
µ(γ+δ+µ)
>1,…÷v^‡σ
2
1
<µ,σ
2
2
<µ,σ
2
3
<µ,Ké?¿‰½Щ
Š(S(0),I(0),Q(0)) ∈R
3
+
,.(2))(S(t),I(t),Q(t)) ∈R
3
+
kXe5Ÿ:
limsup
t→∞
1
t
E
Z
t
0
{(S−S
∗
)
2
+(I−I
∗
)
2
+(Q−Q
∗
)
2
}dr≤
K
2
M
2
,
Ù¥
c
1
=
2µε+2µ(δ+2µ)
βε
,c
2
=
2µ
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
= 2σ
2
1
(S
∗
)
2
+2σ
2
2
(I
∗
)
2
+σ
2
3
(Q
∗
)
2
+
c
2
2
I
∗
σ
2
2
.
y²½Â¼ê:
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.2021.1072472365A^êÆ?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^žBúª[12],·‚ŒXe(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
−2µ(S−S
∗
)(I−I
∗
)−2µ(S−S
∗
)(Q−Q
∗
)−2µ(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+2µ)(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+2µ)(S−S
∗
)(I−I
∗
)+c
2
ε(I−I
∗
)(Q−Q
∗
)+σ
2
1
(S
∗
)
2
+σ
2
2
(I
∗
)
2
,
(4.3)
qdV= 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−2µ−c
2
(δ+2µ)](S−S
∗
)(I−I
∗
)
+(c
2
ε−2µ)(S−S
∗
)(Q−Q
∗
)+(c
2
ε−2µ)(I−I
∗
)(Q−Q
∗
)
+2σ
2
1
(S
∗
)
2
+2σ
2
2
(I
∗
)
2
+σ
2
3
(Q
∗
)
2
+
c
1
2
I
∗
σ
2
2
,
(4.4)
d½n3^‡nŒ:
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.2021.1072472366A^êÆ?Ð
4•J
K
2
= 2σ
2
1
(S
∗
)
2
+2σ
2
2
(I
∗
)
2
+σ
2
3
(Q
∗
)
2
+
c
2
2
I
∗
σ
2
2
.
Ïdk
dV=LVdt+(S−S
∗
+I−I
∗
+Q−Q
∗
)[σ
1
SdB
1
(t)+σ
2
IdB
2
(t)+σ
3
QdB
3
(t)]
+c
1
σ
2
(I−I
∗
)dB
2
(t)+c
2
(S−S
∗
+I−I
∗
)[σ
1
SdB
1
(t)+σ
2
IdB
2
(t)],
(4.6)
e5,·‚y²•{†½n2y²L§ƒq,ŽÑ.
(Øy.
5.(Ø
©¤?Ø‘ÅD/¾.´3®‰Ñš‚5a/ÇD/¾.Ä: þ,•Ä‘Å
ÅÄïá.ÄuT‘ÅD/¾.,·‚éÙÄåÆ1•?1ïÄ: Äk©ÛTXÚ3‰½
Њ^‡e,•3•˜Û), ÙgÏLE·Lapunov¼ê,y²TXÚ)3þ
²ï:†/•¾²ï:ìC1•,ù(Øé‘kš‚5a/ÇD/¾.ïÄ•äky¢
¿Â.
ë•©z
[1]±,½B—.äk<D<€9D /¾.ÄåÆ©Û[J].-ŸnóŒÆÆ:g,‰Æ
‡,2021,35(2):258-267.
[2]Liu,Q. andJiang,D. (2016)The Threshold ofa Stochastic Delayed SIR Epidemic Model with
Vaccination.PhysicaA:StatisticalMechanicsandItsApplications,461,140-147.
https://doi.org/10.1016/j.physa.2016.05.036
[3]Liu,Q.,Jiang,D.,Hayat,T.,etal.(2018)AnalysisofaDelayedVaccinatedSIREpidemic
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[4]Wen, B.Y., Teng, Z.D., Li,Z.M., etal.(2018)TheThreshold ofaPeriodic StochasticSIVSEpi-
demicModelwithNonlinear Incidence.PhysicaA:StatisticalMechanicsandItsApplications,
508,532-549.https://doi.org/10.1016/j.physa.2018.05.056
[5]Wang,L.,Zhang,X.andLiu,Z.(2018)AnSEIREpidemicModelwithRelapseandGeneral
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Systems,17,1309-329.
[6]dl,Mw.˜aäkš‚5u)ÇÚž¢SIQSD/¾.Û-½5[J].ìÀŒÆÆ
nƇ,2014,49(5):67-74.
[7]Xiang,H.,Tang,Y.L.andHuo,H.F.(2016)AViralModelwithIntracellularDelayand
HumoralImmunity.BulletinoftheMalaysianMathematicalSciencesSociety,40,1011-1023.
https://doi.org/10.1007/s40840-016-0326-2
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4•J
[8]Jiang,D.,Yu,J.,Ji,C.,etal.(2011)AsymptoticBehaviorofGlobalPositiveSolutiontoa
StochasticSIRModel.MathematicalandComputerModelling,54,221-232.
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