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PureMathematicsnØêÆ,2022,12(6),938-951
PublishedOnlineJune2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.126103
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SIRS.§;¾«ý§)•3•˜5
TheThresholdDynamicBehaviorofa
StochasticInfectiousDiseaseModel
withVaryingTotalPopulationSize
RuyiRen
SchoolofScience,LanzhouUniversityofTechnology,LanzhouGansu
Received:May4
th
,2022;accepted:Jun.8
th
,2022;published:Jun.15
th
,2022
©ÙÚ^:?T¤.˜ao<•Cz‘ÅD/¾.KŠÄåÆ1•[J].nØêÆ,2022,12(6):938-951.
DOI:10.12677/pm.2022.126103
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Abstract
Tostudytheeffectsofmortality,infectionratesofrandomdisturbancesandrandom
perturbationofthesystemonthespreadofdisease,inthispaper,weconsidera
stochasticinfectiousdiseasemodelinwhich themortality,transmissionandsystemof
the diseasearedisturbed withchanging populationsize.The existenceanduniqueness
ofknowledgeisprovedbyconstructingappropriateLyapunovfunctions.Established
thresholdsR
s
0
fordeterminingdiseaseextinction,applyingMartingale’stheoremof
largenumbersandItˆoformulas,sufficientandalmostnecessaryconditionshavebeen
obtainedfortheextinctionofinfectiousdiseases.Morespecifically,ifR
s
0
<1,the
diseasewilldieout.Finally,byconstructingthresholdR
s
0
,wefindthatthestochastic
perturbations ofthe deathrate and randomperturbationof the systemfor susceptible
populationcanenhancethespreadofdisease,whilethestochasticperturbationsof
thedeathrateandrandomperturbationofthesystemforinfectiouspopulation,as
wellasthetransmissionrateofthediseasecansuppressthespreadofthedisease.
Keywords
SIRSModel,ExtinctionofDisease,ExistenceandUniquenessofSolutions
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CC BY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2022.126103940nØêÆ
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2
2
x
2
−σ
2
4
z(1−z)]
−
1
2
[σ
2
1
y
2
+σ
2
2
(1−x)
2
+σ
2
3
y
2
+σ
2
4
z
2
]
−
1
2
[σ
2
1
x
2
+σ
2
2
x
2
+σ
2
3
(1−y)
2
+σ
2
4
z
2
]
−
1
2
[σ
2
2
x
2
+σ
2
3
y
2
+σ
2
4
(1−z)
2
]
DOI:10.12677/pm.2022.126103943nØêÆ
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=[
b
x
−2b−δ+(α−β)y+
δz
x
+3σ
2
3
y
2
−σ
2
2
x+3σ
2
4
z
2
+βx−(b+α+γ)+αy−σ
2
3
y+3σ
2
2
x
2
+
γy
z
+αy−σ
2
4
z]
−
1
2
[(σ
2
1
+2σ
2
2
)x
2
+(σ
2
1
+2σ
2
3
)y
2
+2σ
2
4
z
2
+σ
2
2
(1−x)
2
+σ
2
3
(1−y)
2
+σ
2
4
(1−z)
2
]
≥−2b−δ−βy−(σ
2
3
+σ
2
2
)x−(b+α+γ)−σ
2
3
y−σ
2
4
z
−
1
2
[(σ
2
1
+2σ
2
2
)x
2
+(σ
2
1
+2σ
2
3
)y
2
+2σ
2
4
z
2
+σ
2
2
(1−x)
2
+σ
2
3
(1−y)
2
+σ
2
4
)(1−z)
2
].
,H(x,y,z).
duk^‡x+ y+z=1…¼êH(x,y,z)´ëY§K∃C<0§¦é?¿(x,y,z)∈Γ§
kH(x,y,z) ≥C¶Ïd
dD(x,y,z)≥Cdt+σ
1
(x−y)dB
1
(t)+σ
2
(3x−1)dB
2
(t)
+σ
3
(3y−1)dB
3
(t)+σ
4
(3z−1)dB
4
(t).(7)
2镧(7)ü>l0τ
n
∧TÈ©Œ
Z
τ
n
∧T
0
dD(x,y,z)≥
Z
τ
n
∧T
0
Cdt+
Z
τ
n
∧T
0
σ
1
(x−y)dB
1
(t)+
Z
τ
n
∧T
0
σ
2
(3x−1)dB
2
(t)
+
Z
τ
n
∧T
0
σ
3
(3y−1)dB
3
(t)+
Z
τ
n
∧T
0
σ
4
(3z−1)dB
4
(t).
éþãØªÏ"§Œ±
ED(x(τ
n
∧T),y(τ
n
∧T),z(τ
n
∧T)) ≥CT+D(x
0
,y
0
,z
0
) >−∞.
éun≥n
1
§-8ÜΩ
n
={ω∈Ω:τ
n
=τ
n
(ω)≤T}"d(6)ª§kP(Ω
n
)≥ε¤á"d§é?
¿ω∈Ω
n
§3x(τ
n
,ω),y(τ
n
,ω),z(τ
n
,ω)¥–k˜‡u
1
n
"Ïd§
D(x(τ
n
,ω),y(τ
n
,ω),z(τ
n
,ω)) ≤ln
1
n
.
Œ±
E[D(x(τ
n
∧T),y(τ
n
∧T),z(τ
n
∧T))]≤E[lny(T∧τ
n
)]
≤E[1
Ω
n
lny(τ
n
)]
≤εln
1
n
.
DOI:10.12677/pm.2022.126103944nØêÆ
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Ù¥1
Ω
k
´¼êΩ
k
A¼ê§Ï•n→∞ž§−∞<CT<−∞´gñ§Ïdkτ
∞
=∞§
•Ò´τ
n
= ∞§Kτ
e
= ∞a.s"Ún3.1y²¤"2
3Ún3.1Ä:þ§Œ±wÑé?¿t≥0§kx(t)>0,y(t) >0,z(t)>0…x+y+z=1§½
n3.1(J²w´¤á"
½n3.1.PΛ = {(x,y) : x>0,y>0,x+y<1}§KΛ´.(4)ØC8"d§éu?ÛЩ
Š(x
0
,y
0
) ∈Λ§.(4)•3˜‡•˜)(x(t),y(t))…)±VÇ1•3uΛ¥§=é?¿t≥0§
(x(t),y(t)) ∈Λa.s"
4.;¾«ý
3!¥§ò?Ø.(4)'u;¾«ý¿©ÚA7‡^‡"Äk§••B¡y
²§JÑÚn4.1ÚIP4.1¥‰Ñ˜ÎÒ"
Ún4.1.镧
dx= [b(1−x)+δ(1−x)−σ
2
2
x
2
(1−x)+σ
2
4
x(1−x)]dt+σ
2
x(1−x)dB
2
−σ
4
x(1−x)dB
4
.
b^‡b+δ>
1
2
σ
2
2
¤á§Ké?¿x∈(0,1)§K•§(8))x˜—Âñu1"
y².éu•§(8)§A^Itˆoúª§Œ±
dln(1−x)=[−b−δ+
1
2
σ
2
2
x
2
−σ
2
4
x−
1
2
σ
2
4
x
2
]dt+σ
2
xdB
2
−σ
4
xdB
4
≤[−b−δ+
1
2
σ
2
2
−σ
2
4
x−
1
2
σ
2
4
x
2
]dt+σ
2
xdB
2
−σ
4
xdB
4
,Υ
1
(x)dt+σ
2
xdB
2
−σ
4
xdB
4
.(8)
Ù¥Υ
1
(x) = −b−δ+
1
2
σ
2
2
−σ
2
4
x−
1
2
σ
2
4
x
2
"Ï•b^‡b+δ>
1
2
σ
2
2
¤áÚ¼êΥ
1
(x)ëY5§
éu?¿x∈(1 −%
1
,1)§ÀJ¿©~ê%
1
¦Υ
1
(x)<0"éª(8)ü>l0tÈ©¿Ø
±t§Œ±Ñ
ln(1−x(t))
t
≤
ln(1−x(0))
t
+
R
t
0
Υ
1
(x)dt
t
+
R
t
0
σ
2
xdB
2
(t)
t
−
R
t
0
σ
4
xdB
4
(t)
t
.
éþªü>þ(.¿$^Œê½nŒ
limsup
t→∞
ln(1−x(t))
t
≤limsup
t→∞
R
t
0
Υ
1
(x)dt
t
<0, ∀x∈(1−%
1
,1).
Ïd§é?¿x∈(1−%,1)§Œ±Ñx˜—ªu1"X§y²•§(8)?¿)x§l(0,1)m©
•ªŒ?\«•(1−%
1
,1)"½Â¼êV
∗
= −(2x+1)
k
∗
§Ù¥k
∗
•~ê§3¡y²L§¥¬
DOI:10.12677/pm.2022.126103945nØêÆ
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(½§dItˆoúªŒ
LV
∗
=−2k
∗
(2x+1)
k
∗
−1
[b(1−x)+δ(1−x)−σ
2
2
x
2
(1−x)+σ
2
4
x(1−x)]
−
4k
∗
(k
∗
−1)(2x+1)
k
∗
−2
2
[σ
2
2
x
2
(1−x)
2
+σ
2
4
(1−x)
2
]
=−2k
∗
(2x+1)
k
∗
−2
[(2x+1)(1−x)(b+δ−(σ
2
1
+σ
2
2
)x
2
+σ
2
4
x)
+(k
∗
−1)x
2
(1−x)
2
(σ
2
2
+σ
2
4
)]
,−2k
∗
(2x+1)
k
∗
−2
·Π
Ù¥Π = (2x+1)(1−x)(b+δ−(σ
2
1
+σ
2
2
)x
2
+σ
2
4
x)+(k
∗
−1)x
2
(1−x)
2
(σ
2
2
+σ
2
4
)"
x∈(0,1−%
1
]§Øª(2x+1)(1−x)(b+σ
2
4
x) >(b+σ
2
4
)(1−1+%
1
) = %
1
b¤á§KŒ±év
Œ~êk
∗
¦
Π>%
1
b−(2x+1)(σ
2
1
+σ
2
2
)x
2
+(k
∗
−1)(σ
2
2
+σ
2
4
)x
2
(1−x)
2
>
1
4
%
1
b, ∀x∈(0,1−%
1
].
Ïd§
LV
∗
= −2k
∗
(2x+1)
k
∗
−2
·Π <−2k
∗
·
1
4
%
1
b<−
1
2
k
∗
%
1
b, for allx∈(0,1−%
1
].
éþªl0τ
%
1
∧tÈ©…dDynkinúª§Œ±
EV
∗
(x(τ
%
1
∧t))=V
∗
(x(0))+E
Z
τ
%
1
∧t
0
LV
∗
(Ψ(s))ds
<V
∗
(x(0))−
1
2
k
∗
%
1
b·E(τ
%
1
∧t).
¦t→∞…A^FatouÚn§U
EV
∗
(x(τ
%
1
)) <V
∗
(x(0))−
1
2
k
∗
%
1
b·E(τ
%
1
).
ŠâV
∗
3(0,1)k.5§Œ±E(τ
%
1
) <∞"Ïd§éux∈(0,1)§t→∞ž§kx→1"Ú
ny²¤"2
IP4.1.·‚PXÚ(4))•Ψ(t) = (x(t),y(t))"AO/§L«ÐŠ•ψ= (x
0
,y
0
) ∈Λ"P¼ê
f(x,y)=βx−(b+α+γ)+αy−σ
2
3
y(1−y)+
1
2
(σ
2
2
−σ
2
1
)x
2
+
1
2
σ
2
4
(1−x−y)
2
−
1
2
σ
2
3
(1−y)
2
.(9)
ò²ï:(1,0)“\þ㕧§Œ±(½;¾«ýKŠ^‡§=
f(1,0)=[β−(b+α+γ)+
σ
2
2
−σ
2
1
−σ
2
3
2
]
=(b+α+γ)[R
s
0
−1].(10)
DOI:10.12677/pm.2022.126103946nØêÆ
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Ù¥R
s
0
=
β
b+α+γ
+
σ
2
2
−σ
2
1
−σ
2
3
2(b+α+γ)
"
e¡òy²§3,^‡e§Éa/<꥕êeü–"§=;¾«ý"
½n4.1.éЊψ= (x
0
,y
0
)§PΨ(t) = (x(t),y(t))•.(4))"XJR
s
0
<1…b+δ>
1
2
σ
2
2
§é
?¿ÐŠ(x
0
,y
0
) ∈Λ§t→∞§K(x(t),y(t)) →(1,0)a.s";¾«ý"•õ/§
P

lim
t→∞
lny(t)
t
= (b+α+γ)(R
s
0
−1) <0

= 1,∀(x
0
,y
0
) ∈Λ,y>0.(11)
y².Ï•R
s
0
<1§Kl(10)ª¥Œ±wÑf(1,0) <0"é(x,y) ∈Λ§.(4)$^ItˆoúªŒ
Llny=
1
y
[βxy−(b+α+γ)y+αy
2
−σ
2
3
y
2
(1−y)+σ
2
2
x
2
y+σ
2
4
y(1−x−y)
2
]
−
1
2y
2
[σ
2
1
x
2
y
2
+σ
2
2
x
2
y
2
+σ
2
3
y
2
(1−y)
2
+σ
2
4
y
2
(1−x−y)
2
]
=βx−(b+α+γ)+αy−σ
2
3
y(1−y)+
1
2
(σ
2
2
−σ
2
1
)x
2
+
1
2
σ
2
4
(1−x−y)
2
−
1
2
σ
2
3
(1−y)
2
=f(x,y)
dlny=f(x,y)+σ
1
xdB
1
(t)−σ
2
xdB
2
+σ
3
(1−y)dB
3
−σ
4
(1−x−y)dB
4
.
Ï•¼êf(x,y)´ëY¿…f(1,0)<0§é?¿(x,y)∈U
%
2
={(x,y),x∈(1−%
2
,1),y∈
(0,%
2
)}"À~ê%
2
∈(0,1)¿©¦f(x,y) <0¤á"
éþªü>l0tÈ©¿Ø±t§Œ±
lny
t
=
lny
0
t
+
1
t
Z
t
0
f(x,s)ds+
1
t
Z
t
0
σ
1
xdB
1
(t)−
1
t
Z
t
0
σ
2
xdB
2
+
1
t
Z
t
0
σ
3
(1−y)dB
3
−
1
t
Z
t
0
σ
4
(1−x−y)dB
4
.
éþªü>þ(.¿…ŠâŒê½nŒ
limsup
t→∞
lny
t
= limsup
t→∞
f(x,y) <0,∀(x,y) ∈U
%
2
.(12)
Ïd§Œ±wÑé?¿(x,y) ∈U
%
2
§y˜—ªu0"
DOI:10.12677/pm.2022.126103947nØêÆ
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e5§·‚òy²é?¿(x,y) ∈U
%
2
§x→1"é?¿%
2
>0¦y<%
2
§Œ±
Lln(1−x)=−b−
α−β
1−x
xy−
δ
1−x
(1−x−y)−σ
2
3
xy
2
1−x
−σ
2
4
x(1−x−y)
2
1−x
+
σ
2
2
2
x
2
−σ
2
4
x
2
(1−x−y)
2
2(1−x)
2
−(σ
2
1
+σ
2
3
)
x
2
y
2
2(1−x)
2
≤−b+β
xρ
2
1−x
−
δ
1−x
(1−x−ρ
2
)−σ
2
4
x(1−x−ρ
2
)
2
1−x
+
σ
2
2
2
x
2
−σ
2
4
x
2
(1−x−ρ
2
)
2
2(1−x)
2
,Υ(x).
Ù¥Υ(x) = −b+β
xρ
2
1−x
−
δ
1−x
(1−x−ρ
2
)−σ
2
4
x(1−x−ρ
2
)
2
1−x
+
σ
2
2
2
x
2
−σ
2
4
x
2
(1−x−ρ
2
)
2
2(1−x)
2
"Šâ¼êΥ(x)
ëY5ÚÚn8(ا¿›Xé?¿x∈(1−%
2
,1)§kΥ(x) <0¤á"Ïd
dln(1−x)≤Υ(x)+σ
1
xy
1−x
dB
1
(t)−σ
2
xdB
2
(t)+σ
3
xy
1−x
dB
3
(t)
+σ
4
x(1−x−y)
1−x
dB
4
(t)(13)
é(13)ªü>l0tÈ©¿…رt§Œ±
ln(1−x)
t
≤
ln(1−x
0
)
t
+
Υ(x)
t
+
1
t
Z
t
0
σ
1
xy
1−x
dB
1
(t)−
1
t
Z
t
0
σ
2
xdB
2
(t)
+
1
t
Z
t
0
σ
3
xy
1−x
dB
3
(t)+
1
t
Z
t
0
σ
4
x(1−x−y)
1−x
dB
4
(t).(14)
(14)ªü>þ4•¿…ŠâŒê½nk
limsup
t→∞
ln(1−x)
t
≤limsup
t→∞
Υ(x)
t
<0∀x∈(1−%
2
,1).
Ïd§
limsup
t→∞
ln(1−x)
t
<0∀(x,y) ∈U
%
2
.(15)
KŠâØª(15)Ú(12)§Œ±Ñé?¿(x,y)∈U
%
2
§(x,y)˜—Âñu(1,0)"Ïdé?¿
¯ε>0§Œ±uy0 <%<%
2
k
P
n
lim
t→∞
(x(t),y(t)) = (1,0)
o
≥1−¯ε,(x
0
,y
0
) ∈U
%
.(16)
2
5.(Ø
©3©z[23]Ä:þ§•Ä;¾kÇÚ¡EǬÉ‚¸6ħïá˜‡ä
DOI:10.12677/pm.2022.126103948nØêÆ
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k<•5Cz‘ÅSIRS."y².(4)•3•˜Û)"XÏLïáû½;¾
«ýKŠR
s
0
=
β
b+α+γ
+
σ
2
2
−σ
2
1
−σ
2
3
2(b+α+γ)
§Ñ;¾«ý¿©A7‡^‡§=R
s
0
<1…^
‡b+δ>
1
2
σ
2
2
¤á§;¾«ý"
3YïÄ󊥧„Œ±•Ä©™¤óŠÚ&ÄÙ¦ïgŽ"3JÑ.
Ä:þ§„Œ±•Ääk<•logisticO•‘ÅD/¾.",§3XÚ(4)¥Ú\Ornstein-
UhlenbeckL§½\\>D(•´Š•Ä"ù¯KŒ±3e5óŠ¥?1?Ø"
ë•©z
[1]Pruss-Ustun,A.,Wolf,J.,Corvalan,C.,Bos,R.andNeira,M.(2016)PreventingDisease
throughHealthy Environments:AGlobal Assessment oftheBurden ofDisease fromEnviron-
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[2]May,R.M.(1973)Stability andComplexity inModel Ecosystems.PrincetonUniversityPress,
Princeton.
[3]Du,N.H.andNhu,N.N.(2020)PermanenceandExtinctionfortheStochasticSIREpidemic
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https://doi.org/10.1016/j.jde.2020.06.049
[4]Gray,A.,Greenhalgh,D.,Hu,L.,Mao, X.andPan,J.(2011)AStochasticDifferentialEqua-
tionsSISEpidemicModel.SIAMJournalonAppliedMathematics,71,876-902.
https://doi.org/10.1137/10081856X
[5]Du,N.H.andNhu,N.N.(2017)Permanence andExtinctionofCertain Stochastic SIRModels
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