Question

(a) Construct the appropriate SIR model for a

disease that keeps people sick for 20 days. It is also known that a typical
susceptible person meets only about 0.01% of the infected population each
day (because he meets 0.01% of the entire population each day), and that it
takes an average of 8 contacts with an infected person before a susceptible
becomes infected.
(b) How large does the susceptible population need to be for an epidemic
to grow?

Answer 1

Step-by-step explanation + Answer: The SIR model is a mathematical model that describes the spread of a disease through a population. The model divides the population into three compartments: susceptibles (S), infected (I), and recovered (R). Susceptibles are individuals who are not infected with the disease and can be infected. Infected individuals are individuals who have the disease and can spread it to others. Recovered individuals are individuals who have had the disease and are now immune to it.

The SIR model is described by the following system of differential equations:

dS/dt = -βSI

dI/dt = βSI - γI

dR/dt = γI

where:

β is the transmission rate of the disease, which is the rate at which susceptibles become infected when they come into contact with infected individuals.

γ is the recovery rate of the disease, which is the rate at which infected individuals recover and become immune.

For a disease that keeps people sick for 20 days, the recovery rate would be γ = 1/20. This means that on average, infected individuals recover after 20 days.

The transmission rate, β, can be calculated using the following equation:

β = 8 * 0.01% = 0.0008

This means that on average, a susceptible individual needs to come into contact with 8 infected individuals before becoming infected.

Therefore, the SIR model for a disease that keeps people sick for 20 days and has a transmission rate of 0.0008 is as follows:

dS/dt = -0.0008SI

dI/dt = 0.0008SI - 1/20I

dR/dt = 1/20I

(b) How large does the susceptible population need to be for an epidemic to grow?

An epidemic will grow if the number of new infections each day is greater than the number of recoveries each day. This can be expressed mathematically as follows:

βSI > γI

Dividing both sides of the equation by SI, we get:

β > γ

Therefore, for an epidemic to grow, the transmission rate must be greater than the recovery rate.

In the case of the SIR model we constructed, the transmission rate is 0.0008 and the recovery rate is 1/20. Therefore, an epidemic will grow if the susceptible population is greater than 2500.

In other words, if the susceptible population is less than 2500, the disease will eventually die out. However, if the susceptible population is greater than 2500, the disease will spread through the population and cause an epidemic.