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AdvancesinAppliedMathematicsA^êÆ?Ð,2022,11(7),4142-4161
PublishedOnlineJuly2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.117441
˜aäkØëY››üÑä¾Ó.
ÄåÆ©Û
MMM§§§ààà
•ânóŒÆ§êƆÚOÆ§H•â
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Á‡
©ïÄ˜aäkSCàÓ^‡Ú-CXÚüÑ©ãëYä¾Ó.§l››¤•Ä§
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'…c
©ãëY§ä¾Ó§Poincar´eN§ÛÄåÆ
DynamicsofaNetworkVirus
ModelwithDiscontinuous
ControlStrategies
ZhongqiXu
SchoolofMathematicsandStatistics,ChangshaUniversityofScienceandTechnology,Changsha
Hunan
Received:Jun.1
st
,2021;accepted:Jun.27
th
,2022;published:Jul.4
th
,2022
©ÙÚ^:M§à.˜aäkØëY››üÑä¾Ó.ÄåÆ©Û[J].A^êÆ?Ð,2022,11(7):
4142-4161.DOI:10.12677/aam.2022.117441
M§à
Abstract
This paper studiesakind ofpiecewisecontinuous network virusmodelwith anti-virus
softwareinstallationandsystemreinstallationstrategy.Consideringthecontrolcost,
wewillconsiderwhethertostartthecontrolstrategiesaccordingtothenumberof
networkusers.TheglobaldynamicsofthemodelareanalyzedbyusingBendixson-
Dulac criterion,Green’sformula, ΩlimitsetandPoincar´e mapand soon.In theend,
numericalsimulationsarecarriedout.
Keywords
PiecewiseContinuous,NetworkVirus,Poincar´eMap,GlobalDynamics
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2022.1174414144A^êÆ?Ð
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DOI:10.12677/aam.2022.1174414145A^êÆ?Ð
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DOI:10.12677/aam.2022.1174414146A^êÆ?Ð
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, I
3
<I
T
<I
1
ž, –²ï:E
p
2
∈Σ
S
2
ž´-½.
y².S
1
<S
P
2
<S
3
ž,•3–²ï:E
P
2
=(S
P
2
,I
T
).ddëY¼ê":•3½n•,
g(S
1
) >0,…÷vg(S
3
) <0 ž,=I
3
<I
T
<I
1
ž, –²ï:E
P
2
•3.•Ò´E
P
2
•3, d
uI
3
<I
T
<I
1
.,d·K2.3 –²ï:E
P
2
•3ž´-½.2
duS
T
<S
1
<S
2
<S
3
,éu?¿I
T
,/•¾²ï:E
2
Ñ´J²ï:.qI
3
<I
1
,©
•n«œ/.
œ/1.a:I
T
<I
3
<I
1
dž/•¾²ï:E
1
•J²ï:,/•¾²ï:E
3
•¢²ï:.Šâ·K2.1®²üØ
G
i
(i= 1,2,3) S4;‚•35,e5üØXÚ(2.2) w4•‚•35.
Ún3.2.S
T
<S
1
…I
T
<I
3
<I
1
ž, XÚ(2.2) Ø•3ã1(a)w4•‚.
y².|^‡y{[21],Ø”˜„5,éuã1(a),b•3•¹²ï:E
3
˜‡w4•‚Γ,K
;‚Γ ²Lƒ:(S
3
,I
T
) ¿…2gˆ
Σ
S
2
. duE
3
3G
3
SÛìC-½5,G
3
S);‚ò
ªuE
3
, u´4;ΓÜ;‚òªuE
3
,ù†)•˜5gñ,¤±l(S
3
,I
T
) Ñu;‚ج
2gˆΣ
S
2
,=Ø•3w4•‚.ã1(b),ã1(c)üØw4•‚œ¹aq,Ø2Kã.2
e5üØB4•‚•35,ŒU•3B4•‚Xã1(d)∼(f)¤«.
Ún3.3.S
T
<S
1
…I
T
<I
3
<I
1
ž, XÚ(2.2) Ø•3ã1(d)B4•‚.
y².Ø”˜„5,b•3˜‡B4•‚Γ,Xã2¤«,-Γ=Γ
1
+ Γ
2
+ Γ
3
,Ù¥Γ
i
=
Γ ∩G
i
,i=1,2,3,-U•Γ.½«•,¿…U
i
=U∩G
i
,i=1,2,3.:D
1
=(D
11
,I
T
),
DOI:10.12677/aam.2022.1174414147A^êÆ?Ð
M§à
Figure1.NonexistenceoflimitcyclesinCase1.a
ã1.œ¹1.a4•‚Ø•35
D
2
= (D
21
,I
T
),D
3
= (S
T
,D
32
),‚ãM
21
M
22
,†Γ
1
Œ¤«••
¯
U
1
,M
11
M
32
,M
32
M
31
†Γ
2
Œ
¤«••
¯
U
2
,M
41
M
42
,M
42
M
12
†Γ
3
Œ¤«••
¯
U
3
,…M
11
M
12
= I
T
+ε,M
21
M
22
= I
T
−ε,
M
31
M
32
= S
T
−ε,M
42
M
42
= S
T
+ε,Dulac ¼êB(S,I) =
1
SI
,KdBendixson-Dulac OK,·
‚k
ZZ
U

∂BF
1
∂S
+
∂BF
2
∂I

dSdI=
3
X
i=1
ZZ
U
i

∂BF
i1
∂S
+
∂BF
i2
∂I

dSdI
=
3
X
i=1
ZZ
Ui
−
1
S
2
I
dSdI<0,
d‚úªk:
ZZ
¯
U
1

∂BF
11
∂S
+
∂BF
12
∂I

dSdI=
I
∂
¯
U
1
BF
11
dI−BF
12
dS
=∫
Γ
1
BF
11
dI−BF
12
dS+
M
21
∫
M
22
BF
11
dI−BF
12
dS
= −
M
21
∫
M
22
BF
12
dS,
DOI:10.12677/aam.2022.1174414148A^êÆ?Ð
M§à
Figure2.WhenE
3
istherealequilibrium,excludethe
crossinglimitcycleinFigure1(d)
ã2.E
3
•¢²ï:ž§üØã1(d)B‚
ZZ
¯
U
2

∂BF
21
∂S
+
∂BF
22
∂I

dSdI=
I
∂
¯
U
2
BF
21
dI−BF
22
dS=
M
31
Z
M
32
BF
21
dI−
M
32
Z
M
11
BF
22
dS,
ZZ
¯
U
3

∂BF
31
∂S
+
∂BF
32
∂I

dSdI=
I
∂
¯
U
3
BF
31
dI−BF
32
dS=
M
32
Z
M
41
BF
31
dI−
M
12
Z
M
42
BF
32
dS,
3
X
i=1
ZZ
U
i

∂BF
i1
∂S
+
∂BF
i2
∂I

dSdI=lim
∈→0
3
X
i=1
ZZ
¯
U
i

∂BF
i1
∂S
+
∂BF
i2
∂I

dSdI
= µ
1
ln
D
21
D
11
+µ
2
ln
D
21
S
T
−µ
2
ln
I
T
D
32
>0.
Ñygñ.Ïd,üØ•Œw«•Ú²ï:E
3
B4•‚.e5ã1(e),ã1(f)œ¹üØ
B‚aq,Ø2`².2
½n3.4.S
T
<S
1
…I
T
<I
3
<I
1
ž,/•¾²ï:E
3
´ÛìC-½.
y².S
T
<S
1
…I
T
<I
3
<I
1
ž,E
3
´¢²ï:…´ÛÜìC-½,E
1
ÚE
2
•J²ï:.
džØëY> .Σ
1
þvkw« •,E
p
2
/∈Σ
S
2
. du/•¾ ²ï:E
1
, E
2
3fXÚG
1
, G
2
S´
ìC-½,¤±lG
1
, G
2
Ñu;‚3ˆƒ†‚ƒc´ªu/•¾²ï:E
1
ÚE
2
.lG
1
Ñu;‚½lw«•½lB«•ˆG
3
,lG
2
Ñu;‚½lB«•†ˆG
3
½l
B«•²G
1
2²w«•½öB«•ˆG
3
,ŠâÚn3.2 ÚÚn3.3ŒXÚ(2.2) Ø•
34•‚,KdΩ4•8nØ[20],lR
2
+
Ñu;‚•ªªu/•¾²ï:E
3
,=E
3
´ÛìC
-½.2
DOI:10.12677/aam.2022.1174414149A^êÆ?Ð
M§à
œ/1.b:I
3
<I
T
<I
1
dž/•¾²ï:E
1
,E
2
,E
3
•J²ï:.
½n3.5.S
T
<S
1
…I
3
<I
T
<I
1
ž,–²ï:E
P
2
´ÛìC-½.
y².d·K3.1,dž3w«•Σ
S
2
þ–²ï:E
P
2
•3…´-½,ÏdXÚ(2.2) Ø•3•
¹w«•Σ
S
2
ãw4•‚,ÄK†–²ï:E
P
2
3w«•þÛÜ-½5gñ,e5ü
ØB4•‚•35.
Figure3.NonexistenceofcrossinglimitcyclesinCase1.b
ã3.œ¹1.bB‚Ø•35
aqÚn3.3Œ±üØã3(a),ã3(b)B4•‚,e5ò‰ÑüØã34•‚y
².Ø”˜„5,|^‡y{,b•3ã3(c)4•‚.Xã3(d)¤«,bXÚ;‚l:
B=(b,I
T
)∈{(S,I) ∈Σ
2
|S>S
3
}?Ñu,²Q
3
=(S
T
,a
3
)∈{(S,I) ∈Σ
3
|I>I
T
}ˆQ
2
=
(S
T
,a
2
) ∈{(S,I) ∈Σ
3
|I>I
T
},Ù¥a
3
>a
2
.b•3:Q
1
= (S
T
,a
1
) ∈{(S,I) ∈Σ
3
|I>I
T
},
¦lù:Ñu;‚ˆA= (a,I
T
) ∈{(S,I) ∈Σ
2
|S
T
<S<S
1
}.e5·‚òy²a
2
>a
1
.
E¼ê[22]
g
1
(S,I) = V
6
(S,I)−V
4
(S,I),
Kk
∂g
1
(S,I)
∂I
=
I
2
−I
3
I
>0,?˜Úk
V
6
(S
T
,a
3
)−V
4
(S
T
,a
3
) <V
6
(S
T
,a
2
)−V
4
(S
T
,a
2
).
DOI:10.12677/aam.2022.1174414150A^êÆ?Ð
M§à
dLyapunov ¼ê5Ÿ•,
V
4
(S
T
,a
3
) = V
4
(S
T
,a
2
),
V
6
(S
T
,a
3
) = V
6
(S
T
,a
1
),
u´kV
6
(S
T
,a
1
) <V
6
(S
T
,a
2
).
q
∂V
6
(S,I)
∂I
=
I−I
3
I
, …a
1
>I
T
>I
3
, a
2
>I
T
>I
3
, u´ka
1
<a
2
,ÏdØ•3ã3(c)4•
‚. üØw4•‚ÚB4•‚ƒ, XÚ(2.2) Ω 4•8´{E
P
2
}, =XÚ(2.2) ?ÛlR
2
+
S
Ñu)‘XtO\•ªÑ¬ªu–²ï:E
P
2
.2
œ/1.c:I
3
<I
1
<I
T
dž/•¾²ï:E
1
•¢²ï:,/•¾²ï:E
3
•J²ï:.
½n3.6.S
T
<S
1
…I
3
<I
1
<I
T
ž,¢²ï:E
1
´ÛìC-½.
y².ŒU•3B4•‚„œ /1.a,üØ•{aq,ùpòØ2‰Ñ.ŒU•3w4•
‚Xã4¤«,|^Ún3.2•{,b•3Xã4¤«w4•‚,@olƒ:(S
1
,I
T
) Ñ
u;‚,½²G
1
ˆ
Σ
S
2
,Xã4(a)¤«,½´²G
1
«• ˆG
2
«• ƒ2ˆΣ
S
2
,Xã
4(b)¤«,¢²ï:E
1
3G
1
S´ÛìC-½,@o3G
1
S);‚Ѭªu¢²ï:E
1
,
K;‚7†w4•‚ƒ,†)•˜5gñ,u´Ø•3w4•‚. üØw4•‚±9B
4•‚ƒ, /•¾²ï:E
1
3R
2
+
S´ÛìC-½.2
e5&ÄÙ¦œ/ÛÄåÆ1•,·‚•òæ^üØ4•‚g´,2|^XÚΩ 4
•8,y²•{ƒÓÜ©¡ÒØ2Kã.
Figure4.NonexistenceofslidingmodelimitcyclesinCase1.c
ã4.œ¹1.cw‚Ø•35
œ/2:S
1
<S
T
<S
2
ùpk5?Øw«•ÄåÆ, dc¡?Ø•3ØëY>.Σ
3
þØ•3w«•, =
3ƒ†‚Σ
3
þÑ´B«•,3Yœ/¥Ø2?1`².e5?؃†‚þΣ
1
3œ/
S
1
<S
T
<S
2
ewÄå Æ. Šâ´aKŠS
T
†S
1
,S
2
ƒéŒ, džΣ
1
þw«• A•
DOI:10.12677/aam.2022.1174414151A^êÆ?Ð
M§à
Σ
S
1
=

(S,I) ∈R
2
+
|S
1
<S<S
T
,I= I
T

w•§•(2.6).
·K3.7.S
1
<S
T
<S
2
, H
1
<I
T
<I
1
ž, –²ï:E
p
1
∈Σ
S
1
ž´-½.
y².df(S)5Ÿ, S
1
<S
P
1
<S
T
ž,•3–²ï:E
P
1
= (S
P
1
,I
T
).,,S
1
<S
P
1
<S
T
,
kf(S
1
) >0,f(S
T
) <0,-H
1
=
b
λS
T
−S
T
,dëY¼ê":•3½nK–²ï:E
P
1
•3d
^‡•H
1
<I
T
<I
1
.dd·K2.2,Σ
S
1
þ–²ï:•3ž´-½.2
d4.3!?Ø,Šâ´aKŠS
T
†S
1
,S
3
ƒéŒ,džΣ
2
þw«••Σ
S
2
=

(S,I) ∈R
2
+
|S
T
<S<S
3
,I= I
T

,džΣ
S
2
þw•§•(2.7).
·K3.8.S
1
<S
T
<S
2
ž, …÷vI
3
<I
T
<H
2
ž, k–²ï:E
P
2
∈Σ
S
2
´-½.
y².dg(S)5Ÿ,Šâ":•3½n,g(S
T
)>0…g(S
3
)<0,•Ò´I
3
<I
T
<
b
λS
T
−
1
λ

dµ
2
µ
1
+µ
2
+
λµ
1
µ
1
+µ
2
S
T

ž–²ï:•3,-H
2
=
b
λS
T
−
1
λ

dµ
2
µ
1
+µ
2
+
λµ
1
µ
1
+µ
2
S
T

,=–²ï:
E
P
2
•3…=I
3
<I
T
<H
2
.d·K2.3Σ
S
2
þ–²ï:•3ž´-½.2
••Ð/ä–²ï:E
P
1
,E
P
2
•35,I‡äI
T
,±9H
1
ÚH
2
ƒmŒ'X,
e5k?ØH
1
ÚH
2
Œ'X:
H
1
−H
2
=
1
λ
dµ
2
µ
1
+µ
2
−
µ
2
µ
1
+µ
2
S
T
<0,
=•‡S
T
>S
1
,ÒkH
1
<H
2
.
?˜Ú/,S
1
<S
T
<S
2
<S
3
k
I
3
=
b
λS
3
−S
2
<H
1
=
b
λS
T
−S
T
<
b
λS
1
−S
1
= I
1
,
=I
3
<H
1
<I
1
.
qS
T
>S
1
ž,
I
1
−H
2
=
b
d
−
d
λ
−
b
λS
T
+
1
λ

dµ
2
µ
1
+µ
2
+
λµ
1
µ
1
+µ
2
S
T

>
b
d
−
d
λ
−
b
λS
1
+
1
λ

dµ
2
µ
1
+µ
2
+
λµ
1
µ
1
+µ
2
S
1

= 0,
=H
2
<I
1
.
dudžéu?‰I
T
,/•¾²ï:E
2
Ñ´J²ï:, ¿…I
3
<I
1
, ÏdÛÄåÆŠâI
T
†I
3
, I
1
ƒéŒ©•n«œ¹?Ø. Šâa/KŠI
T
Š‰Œ,·‚ò‰ÑS
1
<S
T
<S
2
ž,XÚ(2.2) ÄåÆ(J.
½n3.9.S
1
<S
T
<S
2
ž,ŠâI
T
ØÓŠ,Xe(Ø:
(a)I
T
<I
3
<I
1
ž, ¢²ï:E
3
´ÛìC-½.
DOI:10.12677/aam.2022.1174414152A^êÆ?Ð
M§à
(b)I
3
<I
T
<I
1
ž, ©•n«œ¹:
(b.1)I
3
<I
T
<H
1
ž, –²ï:E
P
2
´ÛìC-½.
(b.2)H
1
<I
T
<H
2
ž, XÚ(2.2) ;‚•ªªu–²ï:E
P
1
½E
P
2
.
(b.3)H
1
<H
2
<I
T
ž, –²ï:E
P
1
´ÛìC-½.
(c)I
3
<I
1
<I
T
ž, /•¾²ï:E
1
´ÛìC-½.
y².éuœ/(a),dž/•¾²ï:E
3
•¢²ï:,E
1
•J²ï:.Šâ·K3.7,–²ï:
E
P
1
/∈Σ
S
1
,Šâ·K3.8, –²ï:E
P
2
/∈Σ
S
2
, džŒU•34•‚Xã5¤«, ŠâÚn3.3 
•{Œ±üØã5(a),ã5(b)B4•‚. ŠâÚn3.2 •{Œ±üØã5(c)∼(e)w4•
‚.b•3˜‡lƒ:(S
2
,I
T
) Ñu,2gˆ:(S,I
T
),ùpS
1
≤S≤S
3
.du/•¾²ï:
E
3
ÛÜ-½5,dž3G
3
«•4;);‚òªuE
3
, †)•˜5gñ,u´Ø•3w
4•‚.lG
2
«•Ñu;‚3ˆØëY>.ƒc´ªuE
2
,½ˆw«•²ƒ:(S
3
,I
T
)
ªuE
3
, ½²«•G
1
ˆØëY>.Σ
2
2ªu/•¾²ï:E
3
. lG
1
«•Ñu;‚, 3ˆ
ØëY>.ƒc´ªuE
1
, ²w«•Σ
S
2
½B«•ªuE
3
, u´/•¾²ï:E
3
´Ûì
C-½.
Figure5.NonexistenceoflimitcyclesinCase2.a
ã5.œ¹2.a4•‚Ø•35
éuœ/(b.1),–²ï:E
P
2
∈Σ
S
2
´-½.dž/•¾²ï:E
2
,E
2
,E
3
•J²ï:,
ŒU•3B4•‚Xã5(a)¤«, ŠâÚn3.3 •{Œ± üØ. Óž·‚•3w«•þ,
DOI:10.12677/aam.2022.1174414153A^êÆ?Ð
M§à
S<S
P
2
ž, Šâ•þ|©Û,XÚ)3w«•þl†–mwÄ,S>S
P
2
ž, XÚ)3
w«•þlm †wÄ, ¤±džØ•3w4•‚,ÄK†–²ï:E
P
2
3w«•þÛÜ
-½5gñ. Šâ/•¾²ï:E
i
3«•G
i
ÛÜ-½5, l«•G
i
Ñu;‚3ˆƒ†‚ƒ
c´ªuE
i
, i=1,2,3, ˆw«•Σ
S
2
ªu–²ï:E
P
2
, XÚΩ 4•8´{E
P
2
},¤
±E
P
2
´ÛìC-½.œ/(b.2), (b.3)aqŒy,ùpØ2‰Ñ.
éuœ/(c), dž–²ï:E
P
1
/∈Σ
S
1
,–²ï:E
P
2
/∈Σ
S
2
,/•¾²ï:E
1
•¢²ï:,
E
1
, E
2
•J²ï:.ŒU•3B4•‚Xã6¤«, Š âÚn3.3 •{Œ±üØã6(a)B
4•‚,éuã5((b)∼(d))w4•‚,ŠâÚn3.2,b•3˜‡w4•‚•Œ/•¾
²ï:E
1
, lƒ:(S
1
,I
T
) Ñu,²«•G
1
ˆΣ
2
, 2²«•G
2
½G
3
•ˆ:(S,I
T
), ùp
S
1
≤S≤S
3
. du/•¾²ï:E
1
ÛÜ-½5, dž3G
1
«•4;);‚òªuE
1
, †)
•˜5gñ,¤±E
1
´ÛìC-½.2
Figure6.NonexistenceoflimitcyclesinCase2.c
ã6.œ/2.c4•‚Ø•35
œ/3:S
2
<S
T
<S
3
ŠâS
T
Š‰Œ,džΣ
1
þw«••Σ
S
1
=

(S,I) ∈R
2
+
|S
1
<S<S
2
,I= I
T

,d
c¡?Ø,Σ
S
1
þw•§•(2.6),–²ï:E
P
1
•3…=S
1
<S
P
1
<S
2
.
džΣ
2
þw«••Σ
S
2
=

(S,I) ∈R
2
+
|S
T
<S<S
3
,I= I
T

.Σ
S
2
þw•§•
(2.7),–²ï:E
P
2
•3…=S
T
<S
P
2
<S
3
.
duI
3
<I
2
<I
1
,ÏdÛÄåÆ©•4 «œ¹,3‰Ñ½n ƒc·‚k‰Ñü‡·KÚ˜‡
Ún.
DOI:10.12677/aam.2022.1174414154A^êÆ?Ð
M§à
·K3.10.S
2
<S
T
<S
3
ž, …÷vI
2
<I
T
<I
1
ž,kE
P
1
∈Σ
S
1
, ¿…–²ï:E
P
1
´-½
; S
2
<S
T
<S
3
ž, …÷vI
3
<I
T
<H
2
ž, kE
P
2
∈Σ
S
2
, ¿…–²ï:E
P
2
´-½.
y².dëY¼ê":•3½n,f(S
1
)>0 …f(S
2
)<0 ž–²ï:E
P
1
•3,•Ò´÷v
I
2
<I
T
<I
1
ž,–²ï:•3, d·K2.2, –²ï:E
P
1
3w«•þ´-½.
dg(S)5Ÿ,g(S
T
)>0…g(S
3
)<0ž–²ï:•3,•Ò´I
3
<I
T
<
b
λS
T
−
1
λ

dµ
2
µ
1
+µ
2
+
λµ
1
µ
1
+µ
2
S
T

ž–²ï:E
P
2
•3,=–²ï:•3…=I
3
<I
T
<H
2
,d·K2.3,
–²ï:E
P
2
3w«•þ´-½.2
·K3.11.I
3
<I
T
<I
2
<I
1
ž,ØëY>.:E
T
=(S
T
,I
T
) ?•þ|,|^‡©•¹½
Â[20]ØëY>.?:(S
T
,I
T
) •þ|•§•
F(S
T
,I
T
) = l
1
F
1
(S
T
,I
T
)+l
2
F
2
(S
T
,I
T
)+l
3
F
3
(S
T
,I
T
),l
1
,l
2
,l
3
∈(0,1),
…kl
1
+l
2
+l
3
= 1.(S
T
,I
T
) •XÚ(2.2) ²ï:ž,=(0,0) ∈F(S
T
,I
T
),









S
T
=
1
λ
[d+(l
2
+l
3
)µ
1
+l
3
µ
2
],
I
T
=
1
λ

b
S
T
−d−(l
2
+l
3
)µ
1

.
Ún3.12.S
2
<S
T
<S
3
, …I
3
<H
2
<I
T
ž,E
T
Ø´XÚ(2.2) ²ï:.
y².^‡y{y²,Ø”˜„5,bE
T
´XÚ(2.2) ²ï:.(0,0)∈F(S
T
,I
T
) ž,¿›
X
1
λ

b
S
T
−d−(l
2
+l
3
)µ
1

>
b
λS
T
−
1
λ

dµ
2
µ
1
+µ
2
+
λµ
1
µ
1
+µ
2
S
T

,
u´k
d+(l
2
+l
3
)µ
1
<
dµ
2
µ
1
+µ
2
+
λµ
1
µ
1
+µ
2
S
T
= d+
µ
1
µ
1
+µ
2
[(l
2
+l
3
)µ
1
+l
3
µ
2
],
?k
(l
2
+l
3
)µ
2
<l
3
µ
2
,
)gñ, =`²E
T
džØ´XÚ(2.2)²ï:.2
½n3.13.S
2
<S
T
<S
3
ž,ŠâI
T
Š, ±e(Ø:
(a)I
T
<I
3
<I
2
<I
1
ž, XÚ(2.2) ;‚‘XtO\òªu/•¾²ï:E
2
½E
3
.
(b)I
3
<I
T
<I
2
<I
1
ž, ©•n«œ¹:
(b.1)(0,0)/∈F(S
T
,I
T
), …I
3
<I
T
<H
2
<I
2
ž, XÚ(2.2) )‘XžmtO\½ªu
¢²ï:E
2
½ªu–²ï:E
P
2
.
DOI:10.12677/aam.2022.1174414155A^êÆ?Ð
M§à
(b.2)I
3
<H
2
<I
T
<I
2
ž, /•¾²ï:E
2
´ÛìC-½.
(b.3)I
3
<I
T
<I
2
<H
2
ž, XÚ(2.2) )‘XžmtO\•ªªu/•¾²ï:E
2
½
–²ï:E
P
2
.
(c)I
3
<I
2
<I
T
<I
1
ž, ©•n«œ¹:
(c.1)(0,0)/∈F(S
T
,I
T
),I
2
<I
T
<I
1
ž, –²ï:E
p
1
´ÛìC-½.
(c.2)I
3
<I
2
<I
T
<H
2
<I
1
ž,XÚ(2.2) )‘XžmtO\•ª½ªu–²ï:
E
P
1
, ½ªu–²ï:E
P
2
.
(c.3)I
3
<I
2
<H
2
<I
T
<I
1
ž, –²ï:E
P
1
´ÛìC-½.
y².ùp·‚•‰Ñ(b.2) Ú(c.1)y², Ù¦œ/y²†c¡aq.
éuœ/(b.2), dž,E
3
•J²ï:,E
2
•¢²ï:,E
1
•J²ï:. Šâ·K3.10 dž–²
ï:E
P
1
/∈Σ
S
1
, E
P
2
∈Σ
S
2
.ŠâÚn3.12, •E
T
Ø´XÚ(2.2) ²ï:.ŒU•34•‚X
ã7¤«,|^Ún3.2 Œ±üØã7(a),ã7(b), ã7(e)B4•‚. du–² ï:E
P
1
3w
«•Σ
S
2
þÛÜ-½5, Ø•3•¹w«•Σ
S
2
ãw4•‚, A^Ún3.3 •{,b•
3lƒ:(S
2
,I
T
)²«•G
2
†ˆ
Σ
S
1
, ½²G
2
ˆG
1
ƒ2ˆ«•Σ
S
1
,†E
2
3G
2
S
ÛÜ-½5gñ, u´Œ±üØXÚ(2.2)4•‚•35, /•¾²ï:E
2
´ÛìC-½.
Figure7.NonexistenceoflimitcyclesinCase3.b
ã7.œ/3.b4•‚Ø•35
éu œ/(c.1),dž/•¾²ï:E
1
,E
2
,E
3
•J²ï:.Xãã8¤«,‰ ÑŒU•3
4•‚. dÚn3.3 •{Œ±üØB4•‚•35,†c¡y²aq,ùpØ2Kã, ·‚
DOI:10.12677/aam.2022.1174414156A^êÆ?Ð
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ò‰ÑüØw4•‚y².(0,0)/∈F(S
T
,I
T
) ž,b•3Xã8(c),ã8(d)ü«w4•
‚,ùp·‚ò|^‡y{[23]•{üØã8(c)w4•‚,ã8(d)y²aq.
Figure8.NonexistenceoflimitcyclesinCase3.c
ã8.œ/3.c4•‚Ø•35
|^‡y{,Xã8(c)¤«,Ø”˜„5,b•3˜‡4;lA(S
T
,I
T
)Ñu,ˆ:
B(S
B
,I
T
),Ù¥S
B
>S
T
. Šâ·K3.10,džE
P
1
∈Σ
S
1
, ¿…–²ï:E
P
1
´-½, ¿…÷v
S
1
<S
P
1
<S
2
,N´•3˜ ‡:C(S
C
,I
T
),S
2
<S
C
<S
T
, ¦l:CÑu;‚•ªˆ
–²ï:E
P
1
(S
P
1
,I
T
),u´éu?¿:(S
0
,I
T
), Ù¥S
C
<S
0
<S
T
Ñu;‚, ²L«•G
2
Ñ
¬2gˆƒ†‚þ:(S,I
T
), ÷vS
P
1
<S<S
B
,u´Œ±E˜‡Poincar´e N:
P(S
0
) = S,
·‚òU¼ê½Â•:
d(S
0
) = P(S
0
)−S
0
.
N´•U¼ê3[S
C
,S
T
]´˜‡ëY¼ê, …dP(S
C
) = S
P
1
, P(S
T
) = S
B
, kd(S
C
) = S
P
1
−
S
C
<0,d(S
T
) = S
B
−S
T
>0,u´Šâ4«mþëY¼ê":•3½n, •3S
∗
∈(S
C
,S
T
),¦
d(S
∗
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∗
) = S
∗
, ùÒ`²XÚ•3B4• ‚,A^Ún3.2 •{ŒüØB4
•‚•35, )gñ,u´üØã8(c)w4•‚. u´(0,0)/∈F(S
T
,I
T
)ž,–²ï:
E
p
1
´ÛìC-½.2
DOI:10.12677/aam.2022.1174414157A^êÆ?Ð
M§à
œ/4:S
T
>S
3
džÃØI
T
?ÛŠ,/•¾²ï:E
3
•J²ï:,džΣ
2
þvkw«•,Σ
1
þw
«••Σ
S
1
=

(S,I) ∈R
2
+
|S
1
<S<S
2
,I= I
T

,w•§•(2.6), –²ï:E
P
1
•3…-½…
=I
2
<I
T
<I
1
.e¡±½n/ª‰Ñ¤kÄåÆœ¹.
½n3.14.S
T
>S
3
ž, ŠâI
T
Š, d©•±en«œ¹:
(a)I
T
<I
2
<I
1
ž, /•¾²ï:E
2
´ÛìC-½.
(b)I
2
<I
T
<I
1
ž, –²ï:E
P
1
´ÛìC-½.
(c)I
2
<I
1
<I
T
ž, /•¾²ï:E
1
´ÛìC-½.
y².éuœ/(a),dž/•¾²ï:E
2
•¢²ï:,E
1
•J²ï:,–²ï:E
P1
/∈Σ
1
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34•‚Xã9¤«. éuã9(a),ã9(b)B4•‚,ŠâÚn3.3 •{, $^Bendixson-
Dulac OKŒ±üØ. du/•¾²ï:E
2
ÛÜ-½5, b•3Xã9(c), ã9(d)w4•
‚, lƒ:(S
2
,I
T
)Ñu);‚, ²«•G
2
†ˆ
¯
Σ
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1
, ½²G
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ˆG
1
ƒ2ˆ«•
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1
Ú
E
3
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3
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3
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w«•Σ
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2
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Figure9.NonexistenceoflimitcyclesinCase4.a
ã9.œ/4.a4•‚Ø•35
DOI:10.12677/aam.2022.1174414158A^êÆ?Ð
M§à
Table1.Parametervalues
L1.ëêŠ
œ/
ëê
bλdµ
1
µ
2
S
T
I
T
œ/1.a50.20.40.150.81.80.6
œ/1.b50.20.40.20.818
œ/1.c60.20.40.40.81.815
œ/2.a50.20.40.150.82.50.8
œ/2.b.150.20.40.150.82.55
œ/2.b.260.30.20.60.82.56
œ/2.b.360.30.20.60.82.510
œ/2.c40.30.30.30.81.515
œ/3.a60.30.40.40.641.2
œ/3.b.160.30.40.40.633
œ/3.b.2100.30.40.40.63.59.2
œ/3.b.360.30.40.40.634
œ/3.c.160.30.40.40.6310
œ/3.c.29.40.30.40.80.63.85
œ/3.c.39.40.30.40.80.646
œ/3.d60.30.40.40.6315
œ/4.a50.20.40.20.854
œ/4.b50.20.40.20.888
œ/4.c50.20.40.20.87.515
4.(J?Ø
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ÄåÆ¤kœ¹. 3!¥, ·‚òÏLäNëêŠ5 yþ˜!¥½n(Ø,äNëê
Š„L1. l¢S5`,´vk7‡$–ØU‰žØ¾Ó. ¯¢þ,ò´aÅÚa/Å››3
KŠY²S
T
ÚI
T
S´Œ±É. I‡rN´, SCàÓ^‡Ú-CXÚò>f=z••
¼ÅVǵ
1
Úµ
2
é·‚››¾ÓD–'-‡, ·ÀŠ¬ˆ·‚››8I. 3a
/Åêþ‡LKŠI
T
, ´a/Å™‡LKŠS
T
ƒc, SCàÓ^‡åÌŠ^,üöêþþ‡L
KŠž, dž•SCàÓ^‡®²ØU››¾ÓDÂ,Aéa/Åk-CXÚ,2é§?1SCà
Ó^‡?n.
DOI:10.12677/aam.2022.1174414159A^êÆ?Ð
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ë•©z
[1]_¬.&EXÚS†—[M].®:˜uŒÆч,1999.
[2]Howard,J.D.(1998)AnAnalysisofSecurityIncidentsontheInternet1989-1995.Carnegie
MellonUniversity,Pittsburgh,PA.
[3]Û¥².“OÁ”¾Ó©Û9àÓ[J].>M?§E|†‘o,2000(8):92-94.
[4]Cohen,F.(1984)ComputerViruses:TheoryandExperiments.ComputersandSecurity,6,
22-35.https://doi.org/10.1016/0167-4048(87)90122-2
[5]Kephart,J.O.,White,S.R.andChess,D.M.(1993)ComputersandEpidemiology.Spectrum
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