# Modeling viral growth

I'm trying to forecast the viral growth for a website I'm developing. I've arrived at some monthly growth rates I can expect per month, the difficulty is I'd like to model my growth per week instead.

Lets say I have these figures

            Growth Rate
Month 1             0.2
Month 2             0.1
Month 3            0.05
Month 4           0.025


What this means is that for every 5 new users to the site; they will invite another 1 new user in the first month. For every 10 new users, they will invite another user in their second month, etc. Looking at the pattern you can see that the growth rate decline is 50% each month. For the sake of simplicity we'll say the viral growth rate is 0 from month 5 onwards therefor a lifetime growth rate would be the sum of the first 4 months.

How does that translate to weekly growth rates, it's ok to make the assumption there's 4 weeks in a month.

Short version of the question (wrong)

I think what I need to calculate is (could be wrong), using month 1 as an example, how can I split 0.2 into four values where each is half (50%) the value of the previous but the sum of all four totals 0.2?

Short version findings

My theory towards approaching it in the short version is flawed, using that method I end up with results that look like this.

Growth           Week 1    Week 2    Week 3    Week 4     Total
Month 1            0.11      0.05      0.03      0.01       0.2
Month 2            0.05      0.03      0.01      0.01       0.1
Month 3            0.03      0.01      0.01         0      0.05
Month 4            0.01      0.01         0         0     0.025


(The values are rounded to 2dp)

This says that in week 4 there would be a viral growth rate of 0.01 whereas in week 5 there would be a growth rate of 0.05. Logically it shouldn't increase, each week should slowly decrease as if on a curve.

Would much appreciate any help on this, thanks.

If the growth rate falls by half every month, you don't want it to fall by half every week. As David Mitra says, that leads to $\frac {0.2}{15}=.0013333$ the last week of the first month. It would also give $8\cdot \frac {0.1}{15}=.005333333$ the first week of the second month, which doesn't make sense.
A more reasonable approach would be to multiply by $\frac 1{2^{1/4}}\approx 0.84$ every week. This retains the drop of a factor $2$ after four weeks. We follow the same approach. If the first week is $a$, we want $a(1+0.84+0.84^2+0.84^3)=0.2$, which gives $a=0.0637$, so you have $0.0637$ the first week, $0.0535$ the second week, $0.0450$ the third week, and $0.0378$ the fourth week, which nicely total to $0.2$ Now for the next month, divide each by $2$