Since 2020, when the Covid-19 pandemic became one of the key health, economic and daily life challenges, more than a hundred different models have been developed to predict the further spread of the disease. The problem that arises here is that, unlike other diseases, SARS-Cov-2 has a number of significant features. For example, the dynamics of the incidence varies depending on the region under consideration and is extremely unstable due to the emergence of new strains or the applying of antiviral measures. Coronavirus also has a number of symptomatic differences from other viruses, often (depending on the strain) is mild or asymptomatic, and a cured person does not receive stable immunity and can become infected again.
To predict the dynamics of the spread of infectious diseases in mathematical epidemiology, there are many models. Most of them are based on the use of differential equations and population clustering depending on the immune status of its representatives. Such models are commonly referred to as compartmental or SIR-type models, where the original SIR model was proposed by Kermack and Mackendrick in 1927, and since the 1920s a huge number of modifications have been proposed. However, despite their computational simplicity, SIR-type models become unsuitable for long-term forecasting when it is necessary to take into account the heterogeneity of the population, the temporal variability of the virus, or the impact of antiviral measures. This led to the fact that with the development of computer technologies, more complex agent models, machine learning models and neural networks began to appear. At present, an approach is also gaining popularity that makes it possible to computationally simplify agent-based models using the theory of "mean field games", which makes it possible to describe the dynamics of the system's behavior using a small number of partial differential equations, taking into account the heterogeneity of the population using spatial variables.
This report is devoted to an overview of several "mean field" models as applied to modeling the spread of Covid-19 using the example of the cities of Novosibirsk and Krasnoyarsk, as well as to the specifics of the numerical implementation of such models.
This work was supported by the Russian Science Foundation (project no. 18-71-10044).