The number of people $P$ infected with a virus after $t$ days is modeled using the exponential function

where $k$ is a constant.

1. Explain the significance of the number $k.$

2. It is estimated that the virus first passed to a human on $1^{\text{st}}$ December, and that on ${20}^{\text{th}}$ December there were $50$ people infected. According to the model, how many people will be infected on ${31}^{\text{st}}$ January?

3. On what date does the model predict the number of infected people would exceed one million?

4. Find the rate at which the virus is spreading after $10$ weeks, giving the correct units for your answer.

5. Give one criticism of the model.