Suppose that a given population can be divided into parts, those who have a given disease and can infect others and those who do not have it, but are susceptibles. Let x be the proportion of susceptible individuals and y the proportion of infectious individuals, then x + y = 1. Assume that the disease spreads by contact between sick and well members of the population and the rate of spread is proportional to the number of such contacts. If yo is the initial proportion of infectious individuals then 

i) Formulate a mathematical model for the given problem and write a differential equation governing it.

ii) Find the equilibrium points of the equation obtained in (i).

iii) Solve the given problem. What happens to the spread of the disease as t→∞?

i) To model the disease spread, we can use the SIR model (Susceptible-Infectious-Recovered) where:

- \( x \) represents the proportion of susceptible individuals.
- \( y \) represents the proportion of infectious individuals.
- \( x + y = 1 \), since every individual in the population is either susceptible or infectious.

The rate of change of susceptible individuals is proportional to the number of contacts between susceptible and infectious individuals. Hence, the rate of change of \( x \) is:

\[
\frac{dx}{dt} = -\beta x y
\]

where \( \beta \) is the rate constant of transmission, and \( x \) and \( y \) are the fractions of susceptible and infectious individuals, respectively.

From \( x + y = 1 \), we have \( y = 1 - x \). Substituting this into ___ ____ ___ ____ ___ _______ _________ ________ _______ ________ ______.
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