How do you calculate the growth rate when there is a population increase?

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I have a population of let's say 5000 in this population, I have 10 sick people and every 5 days the number of sick people grow by 180%.

Let's say now I have introduced another 3000 people to my population at the end of the first 5th day period.

How many sick people will I have in the next 5 days?

I am trying to understand how to include the population growth in this kind of increase.

Also, what is the area of maths that is concerned with this sort of topic, please?