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SI model

08 May 2020 - viridi - Sparisoma Viridi

Try to understand the SI model from Albert-László Barabási’s book.

Introduction

From book chapter - 10 Spreading Phenomena, it states that epidemic models

where for SI model only S dan I variables are considered.

Differential equation

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>(1i)i,

since

s+i=1.

Solution

Eqns (5) can further written as

di(1i)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(1i)i=di1i+dii=d(1i)1i+dii.

Substitution Eqn (8) into (7) and performing integration on both sides will produce

ln(1i)+ln(1i0)+lnilni0=β<k>(tt0),

which can be simplified into

ln[(1i0)i(1i)i0]=β<k>(tt0).

Eqn (10) dan further written as

(1i0)i=(1i)i0eβ<k>(tt0)

and then

(1i0)i=i0eβ<k>(tt0)i0ieβ<k>(tt0),

i{1+i0[eβ<k>(tt0)1]}=i0eβ<k>(tt0),

which leads to

i=i0eβ<k>(tt0)1+i0[eβ<k>(tt0)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>t1).

Case of t

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.

Case of t=0

If t=0 is put to Eqn (15) then

i(0)=i01+i0(11)=i0

will be obtained.

Case of small t

Using MacLaurin series, Eqn (15) can be written as

i=i0+β<k>i0(1i0)t+,

which depends on the number of terms. Uncomplete derivation can be found here, a future post.