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In this book we construct and analyze some new specific differential models with applications to Population Dynamics, Biostatistics, Debugging and Test Theory and Theory of Computer Viruses Propagation.We prove upper and lower estimates for the one-sided Hausdorff approximation of the shifted Heaviside function by means of the general solutions of these differential equations.The task is important in the treatment of questions related to the study of the "supersaturation" - the object of the research in various fields.We illustrate the advances of the solutions of approximating and modelling of some specific datasets from the radioactive exponential decay, tumor growth, epidemics, population dynamics and reliability analysis usually concentrated in unexpected intervals.We will note that in some cases the parameters of the modeling solutions can be used as ''limiters of specifically located data''.The specialists working in the field of ''reaction-kinetic mechanisms'' have the word.Numerical examples using CAS Mathematica, illustrating our results are given.