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FICHA CAT ALOGRÁFICA ELABORADA PELA

BIBLIOTECA 00 IMECC

OA

UNJCAMP

Bibliotecária: Maria F abiana

B~zerra

Müller- CRB8

f

6162

Galante, Elia.• Tayar

G 131 d

Dinâmica populacional do sistema imune aplicada au sarampuiElias

Tayar Galante-- Campinas, {S.P.: s.n.), 2010.

Orientador: Hyun

Mo

Yang

Dissertação

(mestrado) ·

Universidade

Estadual de

Campinas,

Instituto de Matemática, Estatlstka

e

Computação Científica.

LBiomatemiltiça.

2.1munologia.

3. Modelos

matomãti'cus

,

4.Sistcmas

dinâmicos. 5.Simulaç!\o. 6.VIrus do sarampo.

I. Vang, llyun

Mo.

11. Universidade

.

Estadual

de

Campinas.

lnstitu

'

t<,>

de Matemática,

Estatística

.

• Computação Cientltica. 11

I. Titulo.

Titulo em

inglêS:

J>opulation

dynamics o

f

the inunune system applied

to measles

Pala\ll'as·chave

em

inglês

(Keywords):

I. Biomatllematks. 2. lmmunolugy. 3. Mathematical

models.

4,

Dynamical systems.

5. Simulation

· .

6. Measies vírus.

Área

de

concentração: Bío

.

matemática

TiiUiação:

Mestrad!)

Banca

examinadol'!l: Pro

f.

Dr. Hyun Mo Yang (IM

ECC- UNICAMP)

Ptof.

Dr. Luciano

Barbanti (IM E-

USP)

l'rof'.

Dr'.

Maria Silvia

Vicari

Gaui (IB · UNICAMP)

Data

da

defesa:

2710112010

(4)

Dissertação de Mestrado defendida

em

27 de jan

e

iro

de 2010

e

aprovada

Pe

la Banca Examinadora

composta

pelo

s Profs.

Drs.

.

-

.,.

..

:

..

~

..

...

....

. .

/.-

.. .

c"

.

/

-' J

Prof.(a).

Dr(a).

H~tiN

MO

V ANG

.

· ~

.···.·'_)-;;

Q

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.

ii

.

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!

µ

′

D

9 #!)7! %&+(/!"%!2 D(/;1/ !+!/.) .")& !#(#,-! () 1*-*() !" #$%& ()&"0! %( ())4. !)%#-$ ()7!*. ."%(%. ./

() 1*-*()D (%&+( ()2 8))&/9 )!" .

γ

( %(:( !)%! ."%(%.9 %!/.)<

dD

∗

dt

= βSD − (µ

D

+ µ

′

D

)D

∗

− γT

a

D

∗

,

." !

T

a

#!7#!)!"%( ( ." !"%#(54. ! 1*-*() D &%.%E:& () (%&+( () !

γ

1 ( %(:( ! !)%#-&54. ! 1*-*() !" #$%& () &"0! %( ()7.# 1*-*()D &%.%E:& ()(%&+( ()2

F(!,-(54. 7(#( &"@/& ( 7.7-*( &."(* () 1*-*() !" #$%& () *&+#!) ( &/(9 )! ."C

)& !#(#/.)

S = 0

%!/.)( !) #&54. ( G./!.)%()&( !))() 1*-*() "..#=("&)/.G-/(". HI℄ HK℄2 L)%. 19 ( ." !"%#(54. !)%() 1*-*() #!) ! ./ -/( %(:( ! #!7.)&54. "(%-#(*

.")%("%!

k

D

9/() ! #!) ! 7#.7.# &."(*/!"%!A)-( ." !"%#(54. ./-/(%(:( ! /.#C %(*& ( !"(%-#(*

µ

D

2 3"%#!%("%.9 .")& !#(" .(7#!)!"5( .+$#-)"( .##!"%!)("=-$"!(9 &)%. 19

S > 0

9 ( ." !"%#(54. !)%() 1*-*() !+! &/&"-&#9 %(/;1/9 !+& . A &"0! 54.2 M./. ( &"0! 54. () 1*-*() !" #$%& () *&+#!) )! ? 7!*. ."%(%. !)%() ./ . +$#-)9 (

." !"%#(54. !*() ! #!) ! 7#.7.# &."(*/!"%! (. 7#. -%. () ." !"%#(56!)9 . ,-! 1

#!7#!)!"%( . 7!*. %!#/.

βSD

2 3))! /!)/. %!#/. !+! !)%(# 7#!)!"%! "( !,-(54. 7(#( 1*-*() !" #$%& () &"0! %( ()9"!)%! ().9 ."%#&;-&" . 7(#( . (-/!"%. ( ." !"%#(54.

.+$#-)2 8 &/&"-&54. ( ." !"%#(54. () 1*-*() !" #$%& ()&"0! %( ())! ?7!*(/.#%!

(20)

#1 71"6&9:# *7 5/3/&'*71*6;"! &'/!)67' 87/&*7 5/3/&' < !"#"$%! &' &"!)&*&',

=&6& #08/7"&60#' # 0#*7/#> &867'71"&0#' &' *3&' 7?3&9@7' *!A7671 !&!' ?37 *7'B

67)70 & *!1C0! & 8#83/& !#1&/ *&' 5/3/&' < !"#"$%! &' (

T

+ 7 5/3/&' < !"#"$%! &' &"!)&*&'(

T

a

+, D''!0 #0#&' 5/3/&'*71*6;"! &'/!)67'>&' 5/3/&'< !"#"$%! &'(

T

+7'":# '3E07"!*&' F G#07#'"&'!& *# #6H&1!'0#, D/50 *!''#> ':# &"!)&*&' 87/# #1"&"# #0

5/3/&' *71*6;"! &' !1A7 "&*&' (

D

∗

+, I71*#

δ

& "&%&*7 #1"&"# 71"67 7'"&' *3&' 5/3/&'> )70 ?37J

dT

dt

= k

T

− µ

T

T − δT D

∗

_,

#1*7

T

67867'71"& & #1 71"6&9:# *7 5/3/&' < !"#"$%! &' (&!1*& 1:# &"!)&*&'+>

k

T

67B 867'71"& & "&%& #1'"&1"7 *7 678#'!9:# 1&"36&/ *&' 5/3/&' < !"#"$%! &' 87/# #6H&1!'0#>

µ

T

67867'71"& &"&%&*7 0#6"&/!*&*7*7''&' 5/3/&'7

δ

5 &"&%&*7 &"!)&9:# *&' 5/3/&'<

!"#"$%! &' 87/&' 5/3/&' *71*6;"! &'!1A7 "&*&',

=&6& & *!1C0! &8#83/& !#1&/*&' 5/3/&' <&"!)&*&' "70#'J

dT

a

dt

= δT D

∗

_{− µ}

T

a

,

#1*7

T

a

67867'71"&& #1 71"6&9:#*7 5/3/&'< !"#"$%! &'&"!)&*&', D?3! #1'!*76&B'7&' 5/3/&' *71*6;"! &' ?37 A#6&0 !1A7 "&*&' 87/# );63' *# '&6&08# #0# 5/3/&' &867'71"&B

*#6&'*7&1";H71#'8&6&& &"!)&9:#*&' 5/3/&'< !"#"$%! &', K:#'7 #1'!*76&# 86# 7''#

(21)











dS

dt

= α(µ

D

+ µ

′

D

)D

∗

− µ

S

dD

dt

= k

D

− µ

D

D − βSD

dD

∗

dt

= βSD − (µ

D

+ µ

′

D

)D

∗

− γT

a

D

∗

(I)

dT

dt

= k

T

− µ

T

T − δT D

∗

dT

a

dt

= δT D

∗

_{− µ}

T

a

.

4&+)5!) #&!26-6./)3

Ω = {(S, D, D

∗

_{, T, T}

a

)|S ≥ 0, D ≤ D

0 , D

∗

≥ 0, T ≤ T

0 , T

a

≥ 0},

&*+)3

D

0 =

k

D

µ

D

)

T

0 =

k

T

µ

T

,

6)'6)!)*2-- 6)1"7& +) "*2)6)!!)8"&,91" &:

;7& &*!"+)6-6)#&! - +"*<#" - '&'/,- "&*-, +-! =,/,-! > )# *&!!& #&+),&$ '&"!$

*& -!& +& ?@6/! +& !-6-#'&$ - 6)!'&!2- "#/*&,91" - A/#&6-, 2)# '&/ -

"#'&62<* "-)# &#'-6-07& &# -6)!'&!2- "#/*&,91" - "2&29B" -CDD℄ *)!2-F-!) "*" "-,+- "*F) 07&:

G)#86-#&! ./)*)!2) #&+),& +)! 6)?)#&! -')*-!- F-!) "*" "-,+-"*F) 07&:

H-#&!-1&6-)!2/+-6&!'&*2&!+))./",@86"&+&!"!2)#-IJK: L!'&*2&!+))./",@86"&+&

!"!2)#-IJK$ +-+&!'),-! &&6+)*-+-!

(S

0 , D

∗

0 , T

a0

) ∈ R

5

$!7& &82"+&! "#'&*+&5!)3

dS

dt

=

dD

dt

=

dD

∗

dt

=

dT

dt

=

dT

a

dt

= 0.

4-6- &82)6#&! 2&+&! &! '&!!@?)"! '&*2&! +) )./",@86"& +)!!- F&6#-$ &*!"+)6-#&! &

(22)











α(µ

D

+ µ

′

D

)D

∗

− µ

S

S = 0

(i)

k

D

− µ

D

D − βSD = 0

(ii)

βSD − (µ

D

+ µ

′

D

)D

∗

− γT

a

D

∗

= 0 (iii)

k

T

− µ

T

T − δT D

∗

= 0

(iv)

δT D

∗

_{− µ}

T

a

= 0.

(v)

23435 !" #$%&$ '( )*+,-./0,$ 10,2,3-

P

0

6,7/%8%$7 !$97/8%+7/:#/7 % ($%7($%! 8%/!;8#! # 7."#/7#'(!+<./!7 7//# 97=> &#="#1

34 ?%:#=&7

S = 0

@8#$!$#&!%"%$#="#&%#'(%)A7

1.1(i)

'(#

D

∗

_{= 0}

4 B%<@&%#'(%)A7

1.1(v)

8#$ '(#

T

a

= 0

4 B% #'(%)A7

1.1(iv)

8#$ '(#

T =

k

T

µ

T

# &% #'(%)A7

1.1(ii)

8#$ '(#

D =

k

D

µ

D

4 C."#$7 % !$7 97="7&# #'(!+<./!7

P

0 =

0,

k

D

µ

D

, 0,

k

T

µ

T

, 0

4

D4 ?%:#=&7

D = 0

@"#$7 &% #'(%)A7

1.1(ii)

'(#

k

D

= 0

@7 '(# -($ %. (/&7E; '(# % "%F% 7= "%="# &# /#97 !)A7 &# -+(+% &#=&/<"! % &#8# #/ 97 !"!8%4 G7/"%="7@ =A7

97&#$7 "#/ ($# "%&7 &# #'(!+<./!7 7$

D = 0

4

H4 ?%:#=&7

D

∗

_{= 0}

@ "#$7 &% #'(%)A7

1.1(i)

'(#

S = 0

4 B%<@ &% #'(%)A7

1.1(ii)

D =

k

D

µ

D

4 B%#'(%)A7

1.1(iv)

8#$ '(#

T =

k

T

µ

T

4 B% #'(%)A7

1.1(v)

8#$ '(#

T

a

= 0

4 G7/"%="7

D

∗

_{= 0 ⇒ P}

0 =

0,

k

D

µ

D

, 0,

k

T

µ

T

, 0

4

I4 ?%:#=&7

T = 0

@"#$7 &% #'(%)A7

1.1(iv)

'(#

k

T

= 0

@7 '(#- ($ %. (/&7E; '(# % "%F% 7= "%="#&# /#97 !)A7 &# -+(+% J &#8# #/ 97 !"!8%4 G7/"%="7@ =A797&#$7

(23)

! "#$%&'(

T

a

= 0

) *+# ,%$ -*% '%,%+(. /%0

T 6= 0

) /%+(. '# %-*#12(

1.1(v)

-*%

D

∗

_{= 0}

! 3#4) '# %-*#12(

1.1(i)

,%+ -*%

S = 0

! 3# %-*#12(

1.1(ii)

,%+ -*%

D =

k

D

µ

D

! 3#%-*#12(

1.1(iv)

,%+-*%

T =

k

T

µ

T

! 5(0/#&/()(6/7,%+(.(+%.+(8(&/( '% %-*794607(#&/%07(0

P

0 =

0,

k

D

µ

D

, 0,

k

T

µ

T

, 0

! :(;()(<&7 ( 8(&/('% %-*794607(/07,7#9'( .7./%+#>?@AB

P

0 =

0,

k

D

µ

D

, 0,

k

T

µ

T

, 0

.

!"! #$%&'() *+ )(,+-'&'*+*) &. +&

C %./#6797'#'% 9( #9 '%./% 8(&/( '% %-*794607( 8('% .%0 %./*'#'# 97&%#07$#&'(D.% #. %-*#1E%. '( +('%9(! 3%..# F(0+#) (&.7'%0%+(.B











F

1 (S, D, D

∗

, T, T

a

) = α(µ

D

+ µ

′

D

)D

∗

− µ

S

F

2 (S, D, D

∗

, T, T

a

) = k

D

− µ

D

D − βSD

F

3 (S, D, D

∗

, T, T

a

) = βSD − (µ

D

+ µ

′

D

)D

∗

− γT

a

D

∗

F

4 (S, D, D

∗

, T, T

a

) = k

T

− µ

T

T − δT D

∗

F

5 (S, D, D

∗

, T, T

a

) = δT D

∗

− µ

T

a

,

*+ .7./%+# '% F*&1E%. &2(D97&%#0%. '#. ,#07G,%7. %./*'#'#.! 5('%+(. (6/%0 # +#/07$ H# (67#&#

J

*.#&'(B

J =







∂F

1 ∂S

· · ·

∂F

1 ∂T

a

! ! ! ! ! ! ! ! !

∂F

5 ∂S

· · ·

∂F

5 ∂T

a







,

% #,#97GD9#&( 8(&/('% %-*794607(/07,7#9

P

0 =

0,

k

D

µ

D

, 0,

k

T

µ

T

, 0

! C..7+) /%+(. -*%B

(24)

J(P

0 ) =







−µ

S

0 α(µ

D

+ µ

′

D

)

0

0 −

k

D

β

µ

D

−µ

D

0

0 k

D

β

µ

D

0 −µ

0

0 −

k

T

δ

µ

T

−µ

T

0

0 −

k

T

δ

µ

T

0 −µ

T







.

!"#$%&$'$()*(*$%&+!$,)+-$%+(*! &*!)$.$)("/&*)*(."#$(0$*!)$1"%"&$&*%+ $%&+

'+#)+ &* *3,"%41("+)("-"$%

P

0

5 '+%"#."+ $($ )*(4!)" +&$ .$)("/

J(P

0 )

78

P (q) = −

(µ

D

+ q)(µ

T

+ q)

2 _R(q)

µ

D

,

!*#&+

R(q)

,.'+%"#."+&+ !*9,#&+9($,5

R(q) = a

0 + a

1 q + a

2 q

2 ,

+#&*











a

0 = (µ

D

+ µ

′

D

)(1 − ρ

0 )

a

1 = µ

D

(µ

D

+ µ

′

D

+ µ

S

)

a

2 = µ

D

,

+.

ρ

0 =

αβk

D

µ

S

µ

D

.

:.*&"$)$.*#)* )*.+! 3,* )(;! &+! "# + $,)+-$%+(*!

q

1 , q

2 , q

3 , q

4 , q

5 ,

!<+ #*9$)"-+!= $

(25)

q

1 = −µ

D

< 0,

q

2 = q

3 = −µ

T

< 0.

& '!$()$ *+ +#- )#.(#

a

0

/

a

1

#

a

2

*+ '+0).2)+

R(q)

/ '+*#2+ .3+ 4 !0 50!$ + +5($+ *+) !5(+6!0+$# #2 (#$2+ *+ '!$72#($+ *+ 2+*#0+/ +2+ (!2"82 '+*#2+

# (!"#0# #$526!0+$ $9() +

ρ

0

'!$!!# (!")0)*!*#*+'+.(+($)6)!0: ;!$!(!.(+).6+ !2+ + (#+$#2!*# <+5(=>?5$@)(AB !"# $%&'"! #CDE℄ DGH℄:

& )2/'!$!I5#+ *+) !5(+6!0+$# *#(#$2).!*+ '+$

R(q)

(#.=!2'!$(#$#!0.#J!()6! #! #J5$#2/*# !K+$2!/!# (!")0)*!*#! ).(4() !*+'+.(+*##I5)09"$)+($)6)!0/*#6#2+ (#$ DLL℄%

a

0 > 0,

a

1 > 0 e a

2 > 0.

M+2+ (+*! ! (!N! *# 2+$(!0)*!*## *# ).!()6!O3+#.6+06)*! .! #N'$# P# '!$!

a

2

# '!$!

a

1

3+ 6!0+$# # ($)(!2#.(# '+ )()6+ / #J5# )2#*)!(!2#.(#I5#

a

1 > 0

#

a

2 > 0

: QR'!$!

a

0

*#6#2+ (#$%

(µ

D

+ µ

′

D

)(1 − ρ

0 ) > 0.

S+J+/*#6#> # (#$%

ρ

0 < 1.

T# !K+$2!/(#2+ I5#% U#

ρ

0 < 1 ⇒ P

0

8 0+ !0 # ! ).(+() !2#.(# # (R6#0: U#

ρ

0 ≥ 1 ⇒ P

0

8 ). (R6#0: V6!0+$ *#

ρ

0

+"()*+ ! )2! *#'#.*# *! +. (!.(#*# '$+'+$ )+.!0)*!*#

α

/ *! (!N!

β

*# ).K# O3+ *! 8050! *#.*$9() ! 0)6$# '#0+ 69$5 *+ !$!2'+/*! (!N!

k

D

*# $#'+ )O3+ +. (!.(#*# 8050! *#.*$9() ! '#0++$J!.) 2+#*! (!N! *#).!()6!O3+#*#2+$(!0)*!*#

µ

S

#

µ

D

/ $# '# ()6!2#.(#/ *+ !$!2'+# *! 8050! *#.*$9() ! :

(26)

!!"#$ %&'()*+!,

µ

S

,

µ

D

$ -.'(/* #'"*0 1*0

α

$

β

*.

k

D

$ #'"*0!,02 * 3'4*0),

ρ

0

,$ *(!,-.,(/,#,(/,$ #'"*0 ' 6*!!"7"4")'), ), -.,

ρ

0 ≥ 1

8 9':$ #'"*0 ' ;'( , ), -., ' "(1, <=* *7/,(;'!. ,!!* (' ,/'6' "(" "'4>),"(1, <=* )'! ?4.4'! ),()0:/" '!@8 A*0 *./0* 4')*$-.'(/* #'"*0

µ

S

*.

µ

D

$#'(/,()*+!,

α

$

β

,

k

D

%&*!$ #,(*0 !,02

ρ

0

,#'"*0'6*!!"7"4")'),), -.,* 3:0.!)* !'0'#6*(=* *(!"B' "(1, /'0!.% ",(/,! ?4.4'! ),()0:/" '!, -.,$ 6*0/'(/*$ ' "(1, <=* !,C'),7,4')'8 D#'#'(,"0'), !,"(/,060,/'0',&60,!!=**7/")'6'0'

ρ

0

?'!,B."(/,E *(!"),0,.#' F(" ' ?4.4' ),()0:/" ' "(1, /')' 6,4* 3:0.! )* !'0'#6*8 G(/=* *(!"),0'#*!

ρ

0

,! 0"/* )* !,B."(/, #*)*E

ρ

0 =

α · β ·

k

D

µ

D

µ

S

.

D#' ?4.4' "(1, /')' 4"7,0' ,# #?)"'

α

(*3*! 3:0.!$ -., "0 .4'# (* (H).4* 4"(+ 12/" * )* ;*!6,),"0*8 I*#*

β

? ' /'&' ), "(1, <=* )* 3:0.! )* !'0'#6* 6*0 .(")'), ), *( ,(/0'<=* ), ?4.4'! ),()0:/" '! 4"30,! ,

k

D

µ

D

? ' *( ,(/0'<=* ;'7"/.'4 ), ?4.4'! ),()0:/" '!4"30,!$ (.#*0B'("!#*!'.)23,4$)"!6*(:3,46'0' * 3:0.! )* !'0'#6* "(1, /'0$ ,(/=*

α · β ·

k

D

µ

D

?'/'&'), (*3'! ?4.4'!"(1, /')'!8 I*#*

1 µ

S

?*/,#6*#?)"*), 3")'), .#3:0.! )* !'0'#6*$/,#*! -.,

ρ

0

0,60,!,(/'* !"#$% "&'(% '# &*+*,- '# '$./(0 ,- 1+# 2%$," ( 2# /,',- 3#*%- 4.$+- '% -,$,"3% , 3,$/($'# +",! ( , &*+*, '# '$./( , ( 2# /,', ( ( (,*"# /# #" +", 3%3+*,56% /%/,*"# /# -+- #/.4#*8 9,!!'1*0#'$?('/.0'4 -.,!, *(F#,0*

ρ

0

1*0#'"*0)* -.,

1

$/,0,#*! ')'3,J #'"! ?4.4'!"(1, /')'!$ *().J"()*'.# 0,! "#,(/*(' *( ,(/0'<=*),3:0.!$-.,6*0!.'3,J '.#,(/'02*0"! *)' #'("1,!/'<=*)')*,(<'(*;*!6,),"0*8 A*0*./0*4')*$!,*(F#,0*

ρ

0

1*0 #,(*0 )* -.,

1

$ * (F#,0* ), ?4.4'! ),()0:/" '! "(1, /')'! ,!/'02 )"#"(."()* ,$ *(!,-.,(/,#,(/,$ * 3:0.! !,02 ,4"#"(')*8

!"!" #$%&'() *+ )(,+-'&'*+*) .&/-+&

K2 3"#*! '(/,0"*0#,(/, -.,$ -.'()*

ρ

0 < 1

$ ,(/=* * 6*(/* ), ,-."4:70"* /0"3"'4

P

0

!,02 4* '4 , '!!"(/*/" '#,(/, ,!/23,48 L*!/0'0,#*! 'B*0' -.,

P

0

/'#7?# ? B4*7'4 , '!!"(/*/" '#,(/,,!/23,4 ),(/0*)* *(C.(/*"(3'0"'(/,

Ω

8

(27)

(

D > D

0 ⇒

D < 0

˙

T > T

0 ⇒

T < 0

˙

!"#$%# & '( )"!( * +











S = 0

⇒

S ≥ 0

˙

D = 0

⇒

D ≥ 0

˙

D

∗

_{= 0 ⇒}

_D

˙

∗

_{≥ 0}

T = 0

⇒

T ≥ 0

˙

T

a

= 0

⇒

T

˙

a

≥ 0.

,"-" &-(."- " /!"$0'0*"* 1'($"'*

P

0

* 230#(/ " )4356(

V : Ω → R

*" / 1403! #"3 0-"+

V (S, D

∗

_{) = S + αD}

∗

_.

7 //")(-#"8" * -0."*"(-$0!"' *

V

%*"*"

&(-˙

V = −µ

S

S + αβSD − αγT

a

D

∗

.

,(* #(/ - / - . -

V

˙

*" #"3 0-""$"0:(+

˙

V = µ

S

−1 + ρ

0 D

D

0 − αγT

a

D

∗

.

;//0#8 . #(/ 0# *0"!"# 3! <4 8 /

ρ

0 ≤ 1,

! #(/ <4

D ≤ D

0

8 ( <4 0#&'0 " #

˙

V ≤ 0

= ,(-!"3!(8 &(* #(/*0> - <4

V ≤ 0

˙

#

Ω

8 &"-"

ρ

0 ≤ 1

= ;1(-"3(! <4 8 (#(

˙

V = 0 ⇔











S = 0

D

∗

_{= 0}

ou

S = 0

T

a

= 0,

3!6(8( /4$ (3?43!(

Γ

(3!0*( #

Ω

!"' <4

V = 0

˙

% *"*(&(-+

Γ = Γ

1 ∪ Γ

2 ,

(28)

Γ

1 = {(S, D, D

∗

, T, T

a

)|S = 0, D ≤ D

0 , D

∗

= 0, T ≤ T

0 , T

a

≥ 0}

Γ

2 = {(S, D, D

∗

, T, T

a

)|S = 0, D ≤ D

0 , D

∗

≥ 0, T ≤ T

0 , T

a

= 0}.

$%

Γ

1

&'&(#%)&# *#"+,

)-









dS

dt

= 0

dD

dt

= k

D

− µ

D

dD

∗

dt

= 0

dT

dt

= k

T

− µ

T

dT

a

dt

= −µ

T

a

.

.#&&)& #/+)01#& (#%2&#/+#

S(t) = 0

3

D(t) → D

0

3

D

∗

_{(t) = 0}

3

T (t) → T

0

#

T

a

(t) → 0

3 /+)!"

t → ∞

4 $%

Γ

2

&'&(#%)&# *#"+,

)-









dS

dt

= α(µ

D

+ µ

′

D

)D

∗

dD

dt

= k

D

− µ

D

dD

∗

dt

= −(µ

D

+ µ

′

D

)D

∗

dT

dt

= k

T

− µ

T

T − δT D

∗

dT

a

dt

= δT D

∗

_.

.#&&)& #/+)01#& (#%2&#/+#

S(t) = 0

3

D(t) → D

0

3

D

∗

_{(t) → 0}

(29)

!"#$%

t → ∞

&

'(()*+,*%( !, % *")%-(!. %#0!#1%)#+"-)"#1,$,

Γ = Γ

1 ∪ Γ

2

M = {P

0 }

&

3,(!*)#$%4"((!*)#$% !,

ρ

0 < 1

1,*%("((,5!)#1,(6)781,(,($%1,%-,*"$,9":";;,< 9="7!#%+> !"# $%&'"! #?@A℄ @CD℄@CE℄F

C&

V

2 ;)*)1"$",*

Ω

4 G&

V < 0

˙

,*

Ω

4, H& I*")%- %#0!#1%)#+"-)"#1,,*

Ω

2

M

& J%-1"#1%4 7,;% 1,%-,*" $, 9" :";;,<9="7!#%+4 1%$" 1-"0,18-)" !, %*,K" ,*

Ω

, 7,-*"#, , ;)*)1"$" "7-%L)*" 7"-" % 7%#1%

P

0

& M%(1-"*%( !, % 7%#1% 1-)+)";

P

0

2 "(()#1%1) "*,#1, ,(1N+,; ,4 %*% % $%*O#)% $, ,(1".);)$"$, "(()#181) " 2 1%$% % ,(7"K% %#$, % ()(1,*"$)#P*) %,(1N $,Q#)$%4 (,5!, !, % 7%#1%

P

0

2 5;%.";, "(()#1%1) "*,#1, ,(1N+,; (,

ρ

0 < 1

& !" #$%&$ '( ()*+,-./+$ %0$1&/+2+3, R*7%#1%$, , !);O.-)%#S%<1-)+)";$% ()(1,*"

(I)

4

P

∗

_{= ( ¯}

_{S, ¯}

_{D, ¯}

_D

∗

, ¯

T , ¯

T

a

)

42 1"; !,

S

¯

2 #S%<#!;">)(1% 24

S > 0

¯

?& '5%-"4+"*%(4"1-"+2($, *"#)7!;"KT,(";52.-) "($"(, !"KT,($%*%$,;%4%.1,-!*"

, !"KS%7%;)#%*)";7"-"

S

¯

& '(-"OU,( $,(1,7%;)#*)%(,-S%%(7%((O+,)( +";%-,($,

S

¯

4)(1% 24%( 7%((O+,)( +";%-,( 7"-"

S

#!*7%#1%$, , !);O.-)%#S%<1-)+)";&

W%*,K"*%( %*"(, !"KT,( %--,(7%#$,#1,(X($,-)+"$"($, 1%$"("(+"-)N+,)()5!";<

"$"( " U,-%

(1.1)

& '5%-" #%1, !, (%*"#$% 1,-*% " 1,-*% "( , !"KT,(

1.1(ii)

,

1.1(iii)

%.1,*%(" , !"KS%F

k

D

− µ

D

D − (µ

D

+ µ

′

D

)D

∗

− γT

a

D

∗

= 0,

>C&G?

(,* % 1,-*% #S%<;)#,"-

βSD

& '(()*4 (!.(1)1!O*%( " , !"KS%

1.1(iii)

7%- ,(1" #%+" , !"KS% >C&G?&

Y%1, 1"*.2* !,4 (%*"#$% 1,-*% " 1,-*% "( , !"KT,(

1.1(iv)

,

1.1(v)

%.1,*%( " , !"KS%F

(30)

k

T

− µ

T

T − µ

T

a

= 0,

!"#$

%&' (&()*' +&,'- (.-/01(&2," 3%%1'4 %*5%+1+*6'-% 2 &7*28.-

1.1(v)

9-, &%+2 (-:2 &7*28.- !"#$"

;&%%2<-,'249-=&'-%2>-,2+,2520)2, -' 2% %&>*1(+&% &7*28@&%A











α(µ

D

+ µ

′

D

)D

∗

− µ

S

S = 0

k

D

− µ

D

D − βSD = 0

k

D

− µ

D

D − (µ

D

+ µ

′

D

)D

∗

− γT

a

D

∗

= 0

k

T

− µ

T

T − δT D

∗

= 0

k

T

− µ

T

T − µ

T

a

= 0.

;2&7*28.- !"#$:&' 7*&A

¯

T =

k

T

− µ

T

¯

a

µ

T

.

!"B$

C-, -*+,-02=-4=2 &7*28.-

1.1(iv)

+&'-% 7*&A

¯

T =

k

T

µ

T

+ δ ¯

D

∗

.

!"D$

E>*202(=-2% &7*28@&% F"G$ & F"H$& 1%-02(=-2 :2,1I:&0

T

¯

a

4:&' 7*&A

¯

T

a

=

δk

T

D

¯

∗

δµ

T

D

¯

∗

+ µ

T

2 .

!"G$

3>-,2(-+&7*& =2 &7*28.- !"F$:&'A

¯

T

a

=

k

D

− µ

D

D − (µ

¯

D

+ µ

′

D

) ¯

D

∗

γ ¯

D

∗

.

!"H$

(31)

¯

D =

k

D

µ

T

2 _{+ µ}

T

D

¯

∗

[δk

D

− µ

T

(µ

D

+ µ

′

D

)] − δ ¯

D

∗

2 [γk

T

+ µ

T

(µ

D

+ µ

′

D

)]

µ

D

µ

T

(µ

T

+ δ ¯

D

∗

)

.

!"#$

%&'( )*+,'-& *./&01,

1.1(ii)

./*2

¯

D =

k

D

µ

D

+ β ¯

S

.

!"3$

4&5(67/&8&9-,&*:;<*''1,./*,=)6>*+,';&<&

D

¯

,+&*./&01, !"#$* !"3$,=)*+,' & '*7/69)**./&01, 9& >&<6@>*8

D

¯

∗

2

f

0 ( ¯

S) + f

1 ( ¯

S) ¯

D

∗

+ f

2 ( ¯

S) ¯

D

∗

2 = 0,

!"!A$

,9-*

f

0

(

f

1

*

f

2

'1, B/90C*' -& >&<6@>*8

S

¯

" D'), E( ;,-*+,' ,=)*< ,' >&8,<*' -*

D

¯

∗

*+

B/901, -*

S

¯

& ;&<)6< -* /+& *./&01,;,869,+6&8-, '*7/9-,7<&/ ,+ ,*F 6*9)*'

f

0

(

f

1

*

f

2

( ,'./&6' -*;*9-*+ -*

S

¯

* -,' ;&<G+*)<,' -, +,-*8," H''&' B/90C*' '1, ,+6)6-&'( ;,6'91, '*<1,&9&86'&-&'"

I7,<&(9,)* ./* -& *./&01,

1.1(i)

-, '6')*+&9J' )*+,'2

¯

D

∗

=

µ

S

¯

S

α(µ

D

+ µ

′

D

)

.

!"!!$ K/=')6)/69-,L'* !"!!$ *+ !"!A$)*+L'*2

b

1 S + b

¯

2 S

¯

2 + b

3 S

¯

3 + b

4 S

¯

4 = 0,

!"!M$

./* ;,-* '*< <*L*' <6)&9& B,<+&

¯

S · P ol( ¯

S) = 0,

,9-*

P ol( ¯

S)

E /+;,869+6,-, O 7<&/ *+

S

¯

" P*+L'* &''6+

S = 0

¯

,+,/+& ',8/01," I''6+( ;&''&+,' & )<&=&8Q&< ,+ & *./&01, ;,869,+6&8-, )*< *6<, 7<&/

P ol( ¯

S) = 0

( ,/2

(32)

b

1 + b

2 S + b

¯

3 S

¯

2 + b

4 S

¯

3 = 0,

!"!#$

&' &( &)* +),-)(

b

1

.

b

2

.

b

3

)

b

4

/0/&( 1&23

b

1 = αµ

D

µ

2 T

(µ

D

+ µ

′

D

)

2 (µ

D

µ

S

− αβk

D

)

b

2 = −{α

2 β

2 µ

2 T

k

D

(µ

D

+ µ

′

D

)

2 _{+ αβµ}

D

µ

T

µ

S

(µ

D

+ µ

′

D

)[δk

D

− 2µ

T

(µ

D

+ µ

′

D

)]−

−δµ

2 D

µ

2 S

[γk

T

+ µ

T

(µ

D

+ µ

′

D

)]}

b

3 = βµ

S

{αβµ

T

(µ

D

+ µ

′

D

)[µ

T

(µ

D

+ µ

′

D

) − δk

D

] + 2δµ

D

µ

S

[γk

T

+ µ

T

(µ

D

+ µ

′

D

)]}

b

4 = β

2 δµ

2 S

[γk

T

+ µ

T

(µ

D

+ µ

′

D

)].

4),/& 05&20 )' '6&( & 1&7+,'+& 0 +'0 &9-+/&. :-+7+;02)'&( 0 <)520 /&( =+,0+( /)

>)( 02-)( !"#$%&'"! #$1020'&(-202'&(0)?+(-@, +0):,+ +/0/)/&1&,-&/))A:+7B92+&

,6& -2+C+07 DE!℄" G0'&( /+C+/+2 0 ,&((0 0,H7+() )' # 102-)(3

0 ≤ ρ

0 ≤ 1

.

1 < ρ

0 ≤ 2

)

ρ

0 > 2

"

I020 ')7J&2 C+(:07+;02'&( &( (+,0+( A:) &( &)* +),-)( 0((:')'. &( 2)K)( 2)C)'&(

&'&3

b

1 = αµ

D

µ

2 T

η

2 (1 − ρ

0 )

b

2 = C

1 

−ρ

2

0 η

2 µ

2 T

+ k

D

k

T

γ +

ηµ

T

(1 − ρ

0 )

k

T



δ +

2αβµ

2 T

η

2 k

T

µ

D

µ

S

γ +

ηµ

T

(1−ρ

0 )

k

T









b

3 = C

2 γ +

ηµ

T

(2 − ρ

0 )

2k

T



δ +

αβµ

2 T

η

2 2k

T

µ

D

µ

S

γ +

ηµ

T

(2−ρ

0 )

2k

T





b

4 = β

2 δµ

2 S

[γk

T

+ µ

T

η],