Processes 2018, 6, 134
Many of the initial conditions are set equal to zero in order to help understand if the overall
system response is in agreement with the known biological behaviour and are given by
¯
g h (0)= ¯
g h0 , ¯
g r (0)= ¯
g r0 , ¯
s(0)=¯ s 0 , ¯
s bh (0)=0, ¯
s br (0)=0
¯
m h (0)= ¯
m h0 , ¯
m r (0)= ¯
m r0 , ¯
h(0)= ¯
h 0 , ¯
l E (0)= ¯
l E0 , ¯
l RB (0)=0,
¯
l I (0)=0, ¯
v E (0)= ¯
v E0 , ¯
v RB (0)=0, ¯
v I (0)=0, ¯
r f (0)=¯ r f 0 ,
¯
r I (0)=0, ¯
c(0)=0 and ¯
c b (0)=0.
(21)
2.2. Model Reduction
Our model consists of 18 coupled linear and non-linear ODEs. We now seek to reduce the number
of equations by application of the quasi-steady state approximation and conservation laws as detailed
in Appendix A. This reduces the number of equations to 12 such that
J
d ¯
m h
d ¯
t
=
¯
μ ∗
mh
1 +
¯
κ mh (1+(
¯
c
¯
Kc
) xc )
¯
s 0
x h − ¯
δ mh ¯
m h ,
(22)
J
d ¯
m r
d ¯
t
=
¯
μ ∗
mr
1 +
¯
κmr(1+(
¯
c
¯
Kc
) xc )
¯
s 0
xr − ¯
δ mr ¯
m r ,
(23)
d ¯
h
d ¯
t
= ¯
μ h ¯
m h − ¯
δ h ¯
h,
(24)
W
d ¯
l E
d ¯
t
= −¯ α L ¯
r f ¯
l E + ¯
α −L ¯
l RB + W ¯
χ v ¯
v E ,
(25)
d ¯
l RB
d ¯
t
= ¯
α L ¯
r f ¯
l E − ¯
α −L ¯
l RB − ¯
β L ¯
l RB ,
(26)
d ¯
l I
d ¯
t
= ¯
β L ¯
l RB − ¯
γ L ¯
l I ,
(27)
W
d ¯
v E
d ¯
t
= −¯ α v ¯
r f ¯
v E + ¯
α −v ¯
v RB − W ¯
χ v ¯
v E + W ¯
ω,
(28)
d ¯
v RB
d ¯
t
= ¯
α v ¯
r f ¯
v E − ¯
α −v ¯
v RB − ¯
β v ¯
v RB ,
(29)
d ¯
v I
d ¯
t
= ¯
β v ¯
v RB − ¯
γ v ¯
v I ,
(30)
d¯ r f
dt
= ¯
γ r ¯
r I − P ¯
β 0 ¯
r
P
f − m l ¯
β L
¯
r f ¯
l RB
¯
r f 0 − ¯
r f
− m l ¯
α L ¯
l E ¯
r f + m l ¯
α −L ¯
l RB
−m v ¯
β v
¯
r f ¯
v RB
¯
r f 0 − ¯
r f
− m v ¯
α v ¯
v E ¯
r f + m v ¯
α −v ¯
v RB ,
(31)
d¯ r I
dt
= ¯
μ r ¯
m r − ¯
γ r ¯
r I + Pf ¯
β 0 ¯
r
P
f + fm l ¯
β L ¯
l RB
1 +
¯
r f
¯
r f 0 − ¯
r f
+ fm v ¯
β v ¯
v RB
1 +
¯
r f
¯
r f 0 − ¯
r f
,
(32)
d ¯
c
d ¯
t
= R
chol
L
¯
γ L ¯
l I + R
chol
v
¯
γ v ¯
v I + ¯
μ c ¯
h − ¯
δ c ¯
c,
(33)
with the initial conditions
¯
m h (0)= ¯
m h0 , ¯
m r (0)= ¯
m r0 , ¯
h(0)= ¯
h 0 , ¯
l E (0)= ¯
l E0 ,
¯
l RB (0)=0, ¯
l I (0)=0, ¯
v E (0)= ¯
v E0 , ¯
v RB (0)=0,
(34)
¯
v I (0)=0, ¯
r f (0)=¯ r f 0 , ¯
r I (0)=0 and ¯
c(0)=0.
87
Many of the initial conditions are set equal to zero in order to help understand if the overall
system response is in agreement with the known biological behaviour and are given by
¯
g h (0)= ¯
g h0 , ¯
g r (0)= ¯
g r0 , ¯
s(0)=¯ s 0 , ¯
s bh (0)=0, ¯
s br (0)=0
¯
m h (0)= ¯
m h0 , ¯
m r (0)= ¯
m r0 , ¯
h(0)= ¯
h 0 , ¯
l E (0)= ¯
l E0 , ¯
l RB (0)=0,
¯
l I (0)=0, ¯
v E (0)= ¯
v E0 , ¯
v RB (0)=0, ¯
v I (0)=0, ¯
r f (0)=¯ r f 0 ,
¯
r I (0)=0, ¯
c(0)=0 and ¯
c b (0)=0.
(21)
2.2. Model Reduction
Our model consists of 18 coupled linear and non-linear ODEs. We now seek to reduce the number
of equations by application of the quasi-steady state approximation and conservation laws as detailed
in Appendix A. This reduces the number of equations to 12 such that
J
d ¯
m h
d ¯
t
=
¯
μ ∗
mh
1 +
¯
κ mh (1+(
¯
c
¯
Kc
) xc )
¯
s 0
x h − ¯
δ mh ¯
m h ,
(22)
J
d ¯
m r
d ¯
t
=
¯
μ ∗
mr
1 +
¯
κmr(1+(
¯
c
¯
Kc
) xc )
¯
s 0
xr − ¯
δ mr ¯
m r ,
(23)
d ¯
h
d ¯
t
= ¯
μ h ¯
m h − ¯
δ h ¯
h,
(24)
W
d ¯
l E
d ¯
t
= −¯ α L ¯
r f ¯
l E + ¯
α −L ¯
l RB + W ¯
χ v ¯
v E ,
(25)
d ¯
l RB
d ¯
t
= ¯
α L ¯
r f ¯
l E − ¯
α −L ¯
l RB − ¯
β L ¯
l RB ,
(26)
d ¯
l I
d ¯
t
= ¯
β L ¯
l RB − ¯
γ L ¯
l I ,
(27)
W
d ¯
v E
d ¯
t
= −¯ α v ¯
r f ¯
v E + ¯
α −v ¯
v RB − W ¯
χ v ¯
v E + W ¯
ω,
(28)
d ¯
v RB
d ¯
t
= ¯
α v ¯
r f ¯
v E − ¯
α −v ¯
v RB − ¯
β v ¯
v RB ,
(29)
d ¯
v I
d ¯
t
= ¯
β v ¯
v RB − ¯
γ v ¯
v I ,
(30)
d¯ r f
dt
= ¯
γ r ¯
r I − P ¯
β 0 ¯
r
P
f − m l ¯
β L
¯
r f ¯
l RB
¯
r f 0 − ¯
r f
− m l ¯
α L ¯
l E ¯
r f + m l ¯
α −L ¯
l RB
−m v ¯
β v
¯
r f ¯
v RB
¯
r f 0 − ¯
r f
− m v ¯
α v ¯
v E ¯
r f + m v ¯
α −v ¯
v RB ,
(31)
d¯ r I
dt
= ¯
μ r ¯
m r − ¯
γ r ¯
r I + Pf ¯
β 0 ¯
r
P
f + fm l ¯
β L ¯
l RB
1 +
¯
r f
¯
r f 0 − ¯
r f
+ fm v ¯
β v ¯
v RB
1 +
¯
r f
¯
r f 0 − ¯
r f
,
(32)
d ¯
c
d ¯
t
= R
chol
L
¯
γ L ¯
l I + R
chol
v
¯
γ v ¯
v I + ¯
μ c ¯
h − ¯
δ c ¯
c,
(33)
with the initial conditions
¯
m h (0)= ¯
m h0 , ¯
m r (0)= ¯
m r0 , ¯
h(0)= ¯
h 0 , ¯
l E (0)= ¯
l E0 ,
¯
l RB (0)=0, ¯
l I (0)=0, ¯
v E (0)= ¯
v E0 , ¯
v RB (0)=0,
(34)
¯
v I (0)=0, ¯
r f (0)=¯ r f 0 , ¯
r I (0)=0 and ¯
c(0)=0.
87