Dynamic models of epidemiology in discrete time taking into account processes with lag

Models of epidemic dynamics in the form of systems of differential equations of the type SIR and its generalizations, for example SEIR and SIRS, have become widespread in epidemiology. Their coefficients are averages of some epidemic indicators, for example the time when a person is contagious. Statistical data about spreading of the epidemic are known in discrete periods of time, for example twenty-four hours. Therefore, adjustment of the differential equations system under such data comes across cleanly calculable difficulties. They can be avoided, initially to build a model in discrete time as a system of difference equations. Such initial consideration allows, as it shown in the article, to get a general model. On its basis, the models of development of epidemics can be built taking into account their specific. There is another way to obtain a model in discrete time. It consists in discretizing the original model in continuous time. The model obtained in this way is inaccurate, and it is only an approximation to the original one, which makes it possible to simplify calculations and increase the stability of the calculation process. This model is inappropriate, for example, for fitting the model to statistical data. Another argument against the use of systems of differential equations is that the coefficients of such a model may not be the same during a day. For example, the number of contacts of an infected person with susceptible people during a day differs from that at night. However, there is no such difference for daily data. It is possible depending on the day of the week.