Understanding Conditional Probability Density for Pathogen Exposure and Disease Contracting

 An individual whose level of exposure to a certain pathogen is x will contract the

disease caused by this pathogen with probability P(x). If the exposure level of a

randomly chosen member of the population has probability density function f,

determine the conditional probability density of the exposure level of that member

given that he or she

(a) has the disease.

(b) does not have the disease.

(c) Show that when P(x) increases in x, then the ratio of the density of part (a) to

that of part (b) also increases in x.

A pathogen is a microorganism that can cause disease. The probability that an individual will contract the disease caused by a pathogen is dependent on the level of exposure to the pathogen, represented by x. This probability is denoted as P(x). The exposure level of a randomly chosen member of the population has a probability density function (pdf) f.

In this post, we will determine the conditional probability density of the exposure level of a person given that they have or do not have the disease caused by the pathogen.

(a) Conditional Probability Density for Contracting the Disease

The conditional probability density of the exposure level given that the person has the disease is given by:

f_A(x) = f(x) * P(x) / Φ

where Φ is the total probability of contracting the disease:

Φ = ∫f(x) * P(x)dx

This represents the density of exposure levels for individuals who have contracted the disease.

(b) Conditional Probability Density for Not Contracting the Disease

The conditional probability density of the exposure level given that the person does not have the disease is given by:

f_B(x) = f(x) * (1 - P(x)) / (1 - Φ)

where (1 - Φ) represents the total probability of not contracting the disease.

This represents the density of exposure levels for individuals who have not contracted the disease.

(c) Increasing P(x) and the Ratio of Densities

As P(x) increases in x, the ratio of the density of part (a) to that of part (b) also increases in x. This means that as exposure level increases, the likelihood of contracting the disease also increases relative to not contracting the disease.

In conclusion, understanding the conditional probability density of exposure level and disease contracting can provide valuable information for disease prevention and control efforts. By examining the relationship between exposure level and disease probability, we can identify populations that are at higher risk and target interventions to reduce the spread of the disease.