Yes and no.
The effect of total population doesn't change anything until you get closer to the total population. Let's lower the numbers so it's easier to consider. Let's say you have two high schools, one is Allen with 5,000 students. The other is Lampasas High School with 1,000 students. Let's say they both get it on the same day and students of each school are all trapped together, and that the spread of an infectious disease is based on e^(day/3).
Day 0: Lampasas - 1, Allen - 1
Day 1: Lampasas - 1, Allen - 1
Day 2: Lampasas - 2, Allen - 2
Day 3: Lampasas - 3, Allen - 3
Day 4: Lampasas - 4, Allen - 4
Day 5: Lampasas - 5, Allen - 5
Day 6: Lampasas - 7, Allen - 7
Day 7: Lampasas - 10, Allen - 10
Day 8: Lampasas - 14, Allen - 14
Day 9: Lampasas - 20, Allen - 20
Day 10: Lampasas - 28, Allen - 28
Day 11: Lampasas - 39, Allen - 39
Day 12: Lampasas - 55, Allen - 55
Day 13: Lampasas - 76, Allen - 76
Day 14: Lampasas - 106, Allen - 106
Day 15: Lampasas - 148, Allen - 148
Day 16: Lampasas - 207, Allen - 207
Day 17: Lampasas - 289, Allen - 289
Day 18: Lampasas - 403, Allen - 403
Day 19: Lampasas - 563, Allen - 563
Day 20: Lampasas - 786, Allen - 786
Day 21: Lampasas - 1000, Allen - 1097
Day 22: Lampasas - 1000, Allen - 1530
Day 23: Lampasas - 1000, Allen - 2136
Day 24: Lampasas - 1000, Allen - 2981
Day 25: Lampasas - 1000, Allen - 4160
Day 26: Lampasas - 1000, Allen - 5000
Now in reality there are a whole lot more factors than just population size. Population density, distance between population centers (matters more if we limit travel), etc. But the raw numbers of an exponential spread don't give a shit about total population until it infects the total population.