The salmonella inequality

The salmonella inequality
(Copyright (C) Hee Cheong Lee All Rights Reserved.)

Let's think of a logistic curve:

Let x be a population group, let t be a time, let r be a natural increase rate, and let k be a population which can sustain in an environment.

dx/dt = rx(1- x/k)

This equation is called logistic equation. Then, let's consider an infectious disease with this equation.
Well, suppose a man who ate an egg was brought to the hospital with Salmonella Gastroenteritis.

Logistic Equation : dx/dt = rx(1- x/k)
Isolate the variable : dx/x(1-x/k)=rdt
Resolve into partial fractions : dx((1/x) + (1/k-x))=rdt
Integrate : log|x| - log|k-x| + (C1) = rt + (C2)
log|x/(k-x)| = rt + (C3) ((C3)=(C2)-(C1))
Let be exponents of the base e : e^(rt +( C3))=|x/(k-x)|

<-> |x/(k-x)|=exp(C3)e^rt

<-> x/(k-x)=(C4)exp(rt) ((C4)=±exp(C3))

<-> (k-x)/x=(C4)exp(-rt)

<-> k/x = (C4)exp(-rt) + 1

<-> x = k/((C4)exp(-rt) + 1)

Replace (C4) with (C) : x = k/(C*exp(-rt) + 1)

Substitute t = 0 : x = k/(C + 1)

<-> C = (k/x) - 1

In the case of t = 0, the value of x means the early population. Let "sal" be a population of Salmonella included in the egg which the man ate,

C = (k/sal) - 1

Then, the equation mentioned above is x = k/(((k/sal) - 1)*exp(-rt) + 1). Because the latency period of the Salmonella Gastroenteritis it's about 8-48,
you should substitute this latency period for "t". You can measure the value of "k" by an experiment as well the value of "r" because its rate increases naturally.

You can estimate the number of the bacteria which are necessary for the onset of the Salmonella Gastroenteritis if you substitute the result of experiment.

A certain medical book said that the number of the bacteria which are necessary for the onset of the Salmonella Gastroenteritis is more than 10^8.

This means, a Salmonella Gastroenteritis can be said to satisfy the following inequalities:

100000000 < k/(((k/sal) - 1)*exp(-rt) + 1)

Therefore, the probability of Salmonella inequality when eating eggs, meat, and dairy products is really severe... :-()