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1) Migration of APCs to lymph node
4) APC-driven proliferation of CD4+ cells.
IL-2-driven proliferation of CD8+ cells &
iTregs.
s A
a(t)A 0
kA 1 H
A 0
A 1
x 2
H
H
delay =
ρ 1
A 1
2) Initial T cell activation
kPK
s H
x 2
P
K
K
kA 1 H 0
delay = ρ 2
x 2 m 1
H 0
H
delay = σ 1
kPR
x 2
P
R
R
A 1
delay =
ρ 1
s K
kA 1 K 0
5) CD4+ T cells differentiate into iTregs
rH
x 2 m 2
K 0
K
delay =
σ 2
H
R
A 1
6) iTregs suppress effector T cells and
consume positive growth signal.
kRH
3) CD4+ and CD8+ T cells secrete
positive growth signal (IL-2)
H
H
R
kRK
r 1 H
K
K
H
P
r 2 H
kPR
R
K
P
PR
Fig. 3 Diagram of the extended adaptive iTreg model. (1) Immature APCs pick up antigen at the
site of infection at a time-dependent rate a
. These APCs mature and migrate to the lymph node.
(2) Mature antigen-bearing APCs present antigen to naıve CD4+ and CD8+ T cells causing them
to activate and enter the minimal developmental program of m 1 and m 2 divisions, respectively. (3)
Effector CD4+ and CD8+ T secrete positive growth signals at different rates. (4) CD4+ and CD8+
T cells that completed the minimal program become effector cells and continue to divide. CD4+
T cells proliferate upon further interaction with mature APCs. CD8+ T cells and iTregs proliferate
after consuming positive growth signal. (5) Effector CD4+ T cells differentiate into iTregs at a
constant rate. (6) The iTregs suppress effector CD4+ and CD8+ T cells. Although not indicated,
each cell in the diagram has a natural death rate
(
t
)
H
2 m 1 kA 1 (
H 0
(
t
)=
t
− σ 1 )
(
t
− σ 1 ) −
kA 1 (
t
)
H
(
t
)+
2 kA 1 (
t
− ρ 1 )
H
(
t
− ρ 1 )
(10)
− ( δ H +
r
)
H
(
t
) −
kR
(
t
)
H
(
t
) ,
K 0
s K − δ 0 K 0
K 0
(
t
)=
(
t
) −
kA 1 (
t
)
(
t
) ,
(11)
K
2 m 2 kA 1
K 0
(
)=
(
− σ
)
(
− σ
) −
(
)
(
)+
(
− ρ
)
(
− ρ
)
t
t
t
kP
t
K
t
2 kP
t
K
t
2
2
2
2
(12)
− δ
(
) −
(
)
(
) ,
K K
t
kR
t
K
t
P
(
t
)=
r 1 H
(
t
)+
r 2 K
(
t
) − δ
P P
(
t
) −
kP
(
t
)
K
(
t
) −
kP
(
t
)
R
(
t
) ,
(13)
R
(
t
)=
rH
(
t
) −
kP
(
t
)
R
(
t
)+
2 kP
(
t
− ρ
)
R
(
t
− ρ
) − δ
H R
(
t
) .
(14)
1
1
As in Sect. 2.1 , A 0 is the concentration of APCs at the site of infection and A 1 is
the concentration of APCs that have matured, started to present target antigen, and
 
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