Mathematical models of COVID-19 spread

TL;DR

This paper classifies and analyzes various mathematical models of COVID-19 spread across different population groups, covering time-series, differential, imitation, and combined models, along with methods to estimate unknown parameters.

Contribution

It provides a comprehensive classification of COVID-19 spread models and discusses algorithms for solving inverse problems to estimate model parameters.

Findings

Models vary by population group and complexity.

Inverse problem algorithms are essential due to unknown parameters.

Multiple modeling approaches are integrated for comprehensive analysis.

Abstract

The paper presents classification and analysis of the mathematical models of COVID-19 spread in different groups of populations such as the family, school, office (3-100 people), neighborhood (100-5000 people), city, region (0.5-15 million people), country, continent and the world. The classification covers the main types of models including time-series, differential, imitation ones, and their combinations. The time-series models are built from analysis of the time series derived using filtration, regression and network methods (Section 2). The differential models include those derived from systems of ordinary and stochastic differential equations as well as partial-derivative equations (Section 3). The imitation models include cellular automata and agent-based models (Section 4). The fourth group in the classification is combinations of nonlinear Markov chains and optimal control theory, derived within the framework of the mean-field game theory. Due to the novelty of the disease and the difficulties it causes, the parameters of most models are, as a rule, unknown, which necessitates one to solve the inverse problem, so the paper also analyses the main algorithms to solve the inverse problem such as stochastic optimization; nature-like algorithms (genetic; differential evolution; particle swarm, etc.); the understanding method; big-data analysis, and machine learning.