From
http://errorstatistics.com/2014/07/26/s-senn-responder-despondency-myths-of-personalized-medicine-guest-post/

omaclaren
Well, it might be a little old but still applicable I think –

‘The population went up and down, and so did the model’’ is not totally unjustified as a caricature of the way in which ecological [and mathematical biology] models are often compared to data.’

[Wood, 2001; see http://www.esajournals.org/doi/abs/10.1890/0012-9615%282001%29071%5B0001:PSEM%5D2.0.CO;2 ]

A nice paper presenting one method for relating differential equations to data, including a classical mathematical biology/physiology example (Fitzhugh-Nagumo/Hodgkin-Huxley model) and with lots of discussion at the end from a roundtable of statisticians and others, is

[Ramsay et al. 2007; see http://onlinelibrary.wiley.com/doi/10.1111/j.1467-9868.2007.00610.x/full ]

A general perspective on mathematical biology that might be interesting is

http://wcmb-oxford.blogspot.co.uk/2013/12/perspective-on-mathematical-biology.html

To reemphasise it’s not that it can’t be done, even with relatively straightforward methods in principle, it’s the difference in emphasis/attention that is striking I think. In the first year or two of undergrad maths/physics/engineering one seems to start dealing with models that are already difficult to tackle in practice with what is taught in the first year or three of many standard undergrad stats courses.

An example – the simplest/dumbest approach to do parameter estimation for an ODE that doesn’t have an analytical solution would be to just use what amounts to nonlinear regression. Assuming you have observations for all state variables and your ODE solver outputs a solution ys(t;theta) for a given parameter set theta, you can just take yobs(t)= Normal(ys(t; theta), sigma^2) and away you go with least squares. This quickly throws up all sorts of difficulties though – local minima, the numerical cost of computing a solution for every parameter set, dealing with unobserved state variables, how to deal with model mis-specification, violations of assumptions on errors etc etc. I don’t remember a stats course that covered these problems in depth in this context despite the ubiquity of these models in science/engineering/math (could be my bad memory/lack of exposure).