S4 File. Computation of the age-dependent incidence rate effect in a progressively varying
environment.
We now consider the progressive environmental change depicted in Figure A. For those
individuals that are born at time t-x (cohort of age x at time t):
A fraction 1-(t) remains under the old environment E1 (denoted group I),
A fraction (t-x) lives already under the new environment E2 (denoted group II),
A fraction (t) – (t-x) transits from E1 to E2 (denoted group III).
Note: the cohort born at time t-x (i.e. during the year t-x) is assumed to be representative of
the whole population, hence the fraction x (t) of individuals that are exposed to the new environment
is the same among this cohort as in the whole population (also assuming that the environmental
change affects individuals notwithstanding their age).
For time t < t1 the environment (denoted E1 ) was supposed constant and uniform (i.e. it did
not change with time and it was homogenously distributed in the population). After the transition was
completed, at time t > t 2 , the new environment (denoted E2 ) was constant and uniform again. For
any time t between t1 and t 2 , we note x ( t ) the fraction (increasing with time t ) of the population
that was exposed to the new environment E2 ( 1- x ( t ) being the fraction of the population still
exposed to environment E1 at time t). Finally, we noted t50 the date at which half of the population
was exposed to the environmental factor (Figure A).
We assumed that the environmental change did not significantly affect the overall stabilization
dynamics and the biological network model, so that the averaged values  and N remained unchanged
with time. In order to model the impact of the environmental change on both parameters  and T we
also assumed:
(i) the MDP Fi of each module M i changes from Fi,1 to Fi,2 as the environment changes
from E1 to E2 so that the geometrical mean of the N modules changes from 1 to 2 .
(ii) the value of the aging lifetime transiently changes form T to T * as the environment
changes from E1 to E2 only for those modules that are yet stabilized in the original environment E1 .
For the others, still immature modules, the aging lifetime is set to the original value T . The rationale
behind hypothesis (ii) is that the network may adapt to the new environment so far as concerns the
immature modules. When these immature modules get stabilized in the new environment E2 they are
as robust to ageing as the former modules were in the original environment E1 .
The fraction of the cohort born at time t x that is ill at time t is written as:
dx
Pr(CD x, t, tE ) dtE
t-x dt
E
Pr(CD x, t) = [1- x (t)] PI (CD x, t) + x (t - x)PII (CD x, t) + ò
t
[S19]
with
ìæ
ü
ö
x
x
- ö
- öï
ïç 1- F1 ÷ æ
æ
1F
1
PI (CD | x) = Pr(CD x,t,t) = íç 11- e t ÷ +
1- e T ÷ ý
ç
t ÷ çè
t
øï
ø 1- è
ïç
1- ÷
è
ø
T
T
î
þ
N1
ìæ
ïç
Y1
íç 1t
ïç 1T
îè
ìæ
ü
ö
x
x
ïç 1- F2 ÷ æ
- ö
- öï
æ
1F
2
PII (CD | x) = Pr(CD x,t,t - x) = íç 11- e t ÷ +
1- e T ÷ ý
ç
t ÷ çè
t
øï
ø 1- è
ïç
1- ÷
è
ø
T
T
î
þ
(
N1
ü
ö
x
x
ï
Y
÷e t +
1
T
e
ý
÷
t
ï
1÷ø
T
þ
ìæ
ïç
Y2
íç 1t
ïç 1T
îè
N -N1
ü
ö
x
x
ï
Y
÷ t
2
T
e
+
e
ý
÷
t
ï
1÷ø
T
þ
[S20]
N -N1
)
and Pr CD x,t,t E is given in Equation S18.
Note that
PI (CD | x) = Pr(CD x,t,t) because t E = t for people of group I, whereas
PII (CD | x) = Pr(CD x,t,t - x) because t E = t - x for people of group II.
The age-specific incidence rate is:
I(x, t) =
Pr(CD x + Dt, t + Dt) - Pr(CD x, t)
Dt
=
¶
¶
Pr(CD x, t) + Pr(CD x, t)
¶x
¶t
[S22]
so that we finally get
dP
dP
dx
I(x, t) = [1- x (t)] I (x) + x (t - x) II (x) + ò
dx
dx
dtE
t-x
t
¶
é¶
ù
ê ¶x Pr(CD x, t, tE ) + ¶t Pr(CD x,t, tE )ú [S23]
ë
û
[S21]
Figure A. Schematic representation of the transition between two environments E1 and E2
with a progressive exposure of the general population. t1 and t2 are the dates of the beginning and the
end of the transition, respectively. Before t1 and after t2, the environmental exposure is supposed to
be stable (constant with time) and uniformly distributed in the population. During the transition, an
increasing proportion x (t) of the population was exposed to the environmental risk factor(s). t50
represents the date when half of the population has been exposed to the risk factor(s). During the
transition, at any time t of exposure, the cohort born at time t-x can be divided into 3 subgroups:
i) A fraction 1 (t) of people who only lived in environment E1
ii) A fraction (t  x) of people who only lived in environment E2
iii) A fraction  (t )   (t  x) of people for whom environment has changed
Environments E1 and E2 were associated with module disease propensities 1 (also referred
as before) and 2 (also referred as after).