COMPLEX DYNAMICS EMERGING FROM INTERACTING DYNAMICAL SYSTEMS - An Invitation to Biomathematics - page 96

We assume:

1. The probability that a particular susceptible is infected by a
particular infective is p (again implying the groups are uniformly
mixed).

2. The number of susceptibles is denoted by S and the number
of infectives by I.

With these assumptions, the expected number of susceptibles that do not
interact with an infective is S

I .

ð 1 p Þ

C. Idea 3

There are some facts from calculus we now apply:

1. If x is close to 0, then e x 1 þ x;

2. If p is close to 0, then ln ð 1 p Þ p;

0, then x Y e ln ð x Y

Þ ¼

e Y ln ð x Þ .

3. If x > ¼

I

We want to find an expression for S ð 1 p Þ that is easier to work with.

From 3 above:

I

e I ln ð 1 p Þ : ð 1 p Þ ¼

Now, if p is close to 0, then ln ð 1 p Þ is close to p. Thus:

e I ln ð 1 p Þ e Ip

If, in addition to p being close to 0, Ip is also close to 0, then we would
have:

e Ip 1 Ip

Therefore, under these assumptions,

I

ð Þ ð Þ:

S 1 p S 1 pI

So, where does the differential equation come from? We explain this
through a discrete model. To construct our difference equation, we let:

S n ¼ the number of susceptibles at the n-th stage :

I n ¼ the number of infectives at the n-th stage :

The disease spreads when a susceptible interacts with an infective. So
S n þ 1 is the number of susceptibles in the n-th stage who did not interact
with an infective. According to our work above:

S n þ 1 ¼ S n ð 1 pI n Þ¼ S n pS n I n ;