FRACTIONAL SIR MODEL OF COVID-19 WAVES WITH TIME-CHANGING PARAMETERS AND IDENTIFICATION BASED ON NEURAL NETWORK METHODS

Abstract

The paper considers the task of describing individual waves of the COVID-19 epidemic using SIR-type models on real data. It is shown that the classical SIR model with constant parameters of  transmission and  extraction is not able to adequately reproduce the form of the epidemic wave, since on real data these parameters are significantly time-dependent and reflect the impact of anti-epidemic measures, changes in population behavior, vaccination, etc. Additionally, the standard SIR model is memoryless (Markovian), while empirical morbidity series show the effects of long memory.

The aim of the work is to combine fractional SIR models with time-varying parameters and neural network identification methods to describe COVID-19 waves in three European countries (Germany, Italy, Ukraine). A hierarchy of five models is built with increasing flexibility and physical structure. As the simplest basic option, the classic SIR model with constant parameters is considered. Next, a window SIR model with piecewise time-constant  and  is introduced, which are evaluated by gradient methods on sliding windows. The next step is the window fractional FSIR model, in which the system's memory is taken into account using a power-law kernel with a fractional order of . The fourth model is the window FSIR-PINN, in which a multilayer neural network approximates the hidden trajectories  taking into account discrete FSIR equations in the error functional. Finally, the fifth model is the windowless global FSIR-PINN, which learns on the whole wave at once.

Numerical experiments show that in all three countries, the window FSIR model with  provides a better balance between the error on the training and test parts of the series than the window classic SIR model, and gives smooth and interpreted trajectories of effective transfer and extraction ratios. Window FSIR-PINN achieves comparable accuracy in incidence and additionally provides model-aligned  estimates. On the other hand, the global FSIR-PINN, in the presence of only one observed series of incidents, is not able to stably reproduce the waveform, which indicates significant limitations in the identification of such global models on noisy real data. The results obtained confirm the feasibility of using fractional SIR models with time-varying parameters and local PINN architectures as a practical compromise between interpretation and flexibility for retrospective analysis of epidemic waves.