Simple Mathematical Models
Calibration is the activity of verifying that a model of a given problem in a specified domain correctly describes the phenomena that takes place in that domain. During model calibration, values of various relevant coefficients are adjusted in order to minimize the differences between model predictions and actual observed measurements in the field. Verification is performed to ensure that the model does what it is intended to do.
Validation is performed using some other dataset, which has not been used as dataset for calibration. It is the task of demonstrating that the model is a reasonable representation of the actual system so that it reproduces system behaviour with enough fidelity to satisfy analysis objectives. For most models, there are separate aspects, to consider during model validation: assumptions; input parameter values and distributions; output values and conclusions.
One important feature of mathematical modelling is possibility of simulated experimentation. Often the realistic experimenting may be impossible or too expensive. E.g. experiments with infectious disease spread in human populations are impossible, unethical or expensive. We cannot manage endangered species by trial and error. We dare not set dosage for clinical trials of new drugs on humans or set safe limits for exposure to toxic substances without proper knowledge of the consequences. Sometimes a purely experimental approach is not feasible because the data requirements for estimating a model grow rapidly in the number of variables. Modelling using computer programs is cheap and legal at the time of each government.
“Hence, our truth is the intersection of independent lies.” Levins, Richard. 1966. The strategy of model building in population biology. Am. Sci. 54, 4: 421-431. https://mechanism.ucsd.edu/teaching/models/levins.modelbuilding.pdf
"All models are wrong but some are useful". Box, G. E. P. (1979), "Robustness in the strategy of scientific model building", in Launer, R. L.; Wilkinson, G. N., Robustness in Statistics, Academic Press, pp. 201–236. http://oai.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA070213