Much blood, sweat, and beer has been spilled understanding insurgencies. Having none of these in excess, however, I’m forced to use simpler methods. What follows is low tech, but it gets results.
Consider the number of insurgent attacks per unit time A, and the number of insurgents N. They’re connected by a rate R defined by A=RN and assumed constant. All variables are for a fixed area with homogeneous insurgent dynamics.
How do A and N change with time? For N there are three factors. First, the net flux F of insurgents in or out of the region. Second, the net flux L of local insurgents through either recruitment or disillusionment. And finally, the number of insurgents K removed by killing or capturing. Generally K will be proportional to N, with some proportionality constant r that depends on the number of American forces present.
Putting these together one gets the following:
Here are two true-to-life examples of their use.
Scenario I: The insurgency is dead and all that remains is a die-hard kernel. Here F=0 and L=0. The insurgents get no new help, but anybody who was going to leave has already done so. The equations then yield:
where a zero subscript denotes a value at time . Not only do A and K decay exponentially, but they do so at the same rate! So if attacks are declining 10% per month, then detainments should be declining 10% per month as well.
Scenario II: The insurgency isn’t welcomed by the locals, but can draw significant outside help. Here L=0, but F>0 is roughly constant because the pipeline for bringing in outside fighters is maxed out.
After solving one sees that the number of insurgents will drop to a steady state given by:
For example, if 10 foreign fighters are being brought in each month, and our guys are capturing them at a rate of about 1% per month, then the insurgent population will hover around 1000.
Other scenarios include:
- The insurgents surge their efforts increasing R.
- N is so large that our forces are saturated with targets and K is constant.
- We remove troops reducing r.
- N increases suddenly from detainee releases.
- There is a time dependant “war weariness” which makes the insurgency less popular as time goes on (DF/Dt < 0) .
Clearly this analysis can be extended in many realistic directions. There is a lot of insight to be had in these simple differential equations.