Haven’t read the book, so I’m working off of a half-remembered watching of the early 90s miniseries. Assuming that the virus had some of the properties that the series showed (a high likelihood of contagion and a quick incubation), the mathematics works out fairly well.
This is actually a very highly studied branch of mathematics. (This link is good for a laugh, even if you don’t understand the math.) A good first order (although highly flawed) approximation is exponential growth. Assume, for the purposes of argument, that each person contracts the disease on one day, infects two people the next, then dies. On day one, 1 person will contract the disease. On day two, there will be 2 newly infected. On day three, 22 = 4. On day four, 42 = 8. On day n, there will be 2^(n-1) new cases. That means the total number of infected on day n will be 1 + 2 + 4 + 8 + … + 2^(n-1) = 2^n - 1 total cases. Since 2^28 is approximately 270 million, it would take less than a month to infect the entire country under the stated hypotheses. If each new person infected three more, it would actually reduce the time to about 2 1/2 weeks.
The flaws come from that the further we are into an epidemic, the tougher it is to find unexposed victims. The hypotheses become unsustainable. Under the original assumptions, it took about 28 days to infect almost everyone in the US. Another 12 days would put the total number of infected up to 1 trillion (which is obviously absurd). A logistic model is likely to fit better.
That said, my memory of the spread in the series would indicate that, at least initially, the assumption of each person infecting two (or three) more victims is a substantial underestimate. I would expect that the biology shows much less likelihood than the math.