p 11, I think these back-of-the-envelope calculations to relate Ne to epidemic size are illuminating, however I would do some things differently:
A 10 day generation interval is assumed based on a 20 day infectious period, however this is strongly dependent on assumptions about how infectiousness varies over the course of infection. Note that I prefer the term 'generation interval' for this particular situation, see eg https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2365921/. In a simple SIR model, the mean generation interval will equal the mean infectious period, and the choice of 10 days seems a little strange. There is certainly considerable uncertainty about parameters here, so at least the authors could compute this over a range of values.
Other theoretical work suggests a slightly different relationship between Ne tau and I(t), specifically that it will depend more on transmission rate than generation time. However if making the approximation that I(t) is constant (which the authors are) , the formulas will be the same up to a factor of 2x. See Koelle, 2011, DOI: 10.1098/rsif.2011.0495 & Volz and Frost 2013, DOI: 10.1098/rstb.2010.0060. Specifically, I would use (Ne tau ) = I(t) / 2 * [per capita transmission rate], but some of the formulas in Koelle's paper may be more realistic for MERS.
and equating beta = 1 / [generation time] gives 2x the authors' estimate if solving for I.
Note that if using a 20 day generation interval and the above formula, you would have I = 127, so this doesn't make any practical difference, but is more elegant in my opinion, since we are exact about the epidemic model that the estimate is based on. If some estimate of the dispersion parameter for R0 could be obtained from the cluster size distribution, a better estimate for I could be obtained using equations in Koelle 2011.
p 11, I think these back-of-the-envelope calculations to relate Ne to epidemic size are illuminating, however I would do some things differently: