Exercise 5

Consider the SIR model $$\begin{aligned} \frac{dS}{dt} & = - \alpha SI \\ \frac{dI}{dt} & = \alpha SI - \beta I, \end{aligned}$$ where $S$ is the susceptible population and $I$ is the infected population.

1. Modify the SIR model to account for the situation where the susceptible population is reproducing at a rate proportional to the current population.

2. Modify the SIR model to account for the situation where the susceptible population is decreasing at a constant rate such as susceptibles leaving an infected area or city.

3. Modify the SIR model to account for the situation where the infected population is increasing at a constant rate such as infected individuals entering an infected area or city from the outside.