08 May 2020 - viridi - Sparisoma Viridi
Try to understand the SI model from Albert-László Barabási’s book.
From book chapter - 10 Spreading Phenomena, it states that epidemic models
where for SI model only S dan I variables are considered.
In a population of N individuals, each typical individual has <k> contacts and disease will be transmitted from an infected individual to a susceptible one in a unit time is β, then probability, with the homogenous mixing hypothesis, the infected individual encounters susceptible one is S(t)/N. In a unit time there are <k>S(t)/N infected individual in contact with susceptible one. Infected individual I(t) are transmitting the pathogen at rate β, which make the average number of new infection dI during a time frame dt is
dIdt=β<k>(SN)I.
It s move convenient to work with normalized variables in a close population
s=SN
and
i=IN.
Eqns (2) and (3) will turn (1) into
didt=β<k>si,
or simply
didt=β<k>(1−i)i,
since
s+i=1.
Eqns (5) can further written as
di(1−i)i=β<k>dt,
where each side are already function of one variable. Left side of Eqn (7) can written in the form of
di(1−i)i=di1−i+dii=−d(1−i)1−i+dii.
Substitution Eqn (8) into (7) and performing integration on both sides will produce
−ln(1−i)+ln(1−i0)+lni−lni0=β<k>(t−t0),
which can be simplified into
ln[(1−i0)i(1−i)i0]=β<k>(t−t0).
Eqn (10) dan further written as
(1−i0)i=(1−i)i0eβ<k>(t−t0)
and then
(1−i0)i=i0eβ<k>(t−t0)−i0ieβ<k>(t−t0),
i{1+i0[eβ<k>(t−t0)−1]}=i0eβ<k>(t−t0),
which leads to
i=i0eβ<k>(t−t0)1+i0[eβ<k>(t−t0)−1]
as the final form, where i(t0)=i0. We can choose that t0=0 and this will turn (14) into
i=i0eβ<k>t1+i0(eβ<k>t−1).
For very large value of t, Eqn (15) becomes
i(∞)≈i0eβ<k>t1+i0eβ<k>t
since eβ<k>t>>−1 and then also
i(∞)≈i0eβ<k>ti0eβ<k>t=1
since i0eβ<k>t>>1.
If t=0 is put to Eqn (15) then
i(0)=i01+i0(1−1)=i0
will be obtained.
Using MacLaurin series, Eqn (15) can be written as
i=i0+β<k>i0(1−i0)t+…,
which depends on the number of terms. Uncomplete derivation can be found here, a future post.