(1) It is me that was confused ("lazy thinking" might be the better term for it). You have this right. The caseload prediction formula:
caseload = N * P * K * C
has:
K = 1 + (t / 7.5) = 2.6
This uses the estimate of the average duration of an untreated SAM episode (i.e. 7.5 months) from two African populations from a study published in 2009 using older data. All times should be expressed in the same units. For one year (twelve months) we have:
K = 1 + (12 / 7.5) = 2.6
For six months we have:
K = 1 + (6 / 7.5) = 1.8
Recent work indicates that the value of K varies between locations. It seems that K = 2.6 will often be an underestimate.
(2) Here is a worked example ...
The program estimated caseload to be:
Estimated caseload (E) = 58,000
The program admitted:
Caseload (L) = 84,000
We have:
L = CNP(K + 1) = 84,000
where:
L = Observed caseload
N = Population
P = Prevalence
K = Incidence (expressed as a correction to prevalene)
The process to get at a new K is:
L = CNP(K + 1)
K + 1 = L / CNP
K = (L / CNP) - 1
With some numbers:
C = 50% (0.5)
N = 1,000,000
P = 2% (0.02)
L = 84,000
CNP(K + 1) = 84000
0.5 * 1000000 * 0.02 * (K + 1) = 84000
10000 * (K + 1) = 84000
K + 1 = 84000 / 10000
K = (84000 / 10000) - 1
K = 7.4
Another (simpler) way of expressing this is:
New.K = (Observed caseload / Expected caseload) - 1
This holds if we don't expect a change in coverage. We can include a change in coverage using something like:
New.K = ((new coverage / old coverage) * observed caseload)
/ expected coverage) - 1
This is a way of calibrating K from program experience.
(3) This article and this article might be of use. The is a "current topic" in our field and other reports are in press. An international technical working group is being established.
(4) Use what is appropriate. The example above is ... "The available evidence indicates that a single value for K does not exist. There is a need to use a local value for K. I think the only thing to do is to choose a known local K. If there is no known local K then we can pick a value that seems appropriate (e.g. in Southern Niger K hae been estimated to be K = 9 so we might use K = 9 for a program in Northern Nigeria) and then calibrate K using estimates of population, prevalence, coverage, and known caseload". In the absence of useful information then the default value (2.6) could be used.
I hope this is of some use.