# American Institute of Mathematical Sciences

February  2018, 15(1): 209-232. doi: 10.3934/mbe.2018009

## Prediction of influenza peaks in Russian cities: Comparing the accuracy of two SEIR models

 ITMO University, 49 Kronverksky Pr, 197101, St. Petersburg, Russia

* Corresponding author: vnleonenko@yandex.ru

Received  November 08, 2016 Accepted  April 06, 2017 Published  May 2017

This paper is dedicated to the application of two types of SEIR models to the influenza outbreak peak prediction in Russian cities. The first one is a continuous SEIR model described by a system of ordinary differential equations. The second one is a discrete model formulated as a set of difference equations, which was used in the Baroyan-Rvachev modeling framework for the influenza outbreak prediction in the Soviet Union. The outbreak peak day and height predictions were performed by calibrating both models to varied-size samples of long-term data on ARI incidence in Moscow, Saint Petersburg, and Novosibirsk. The accuracy of the modeling predictions on incomplete data was compared with a number of other peak forecasting methods tested on the same dataset. The drawbacks of the described prediction approach and possible ways to overcome them are discussed.

Citation: Vasiliy N. Leonenko, Sergey V. Ivanov. Prediction of influenza peaks in Russian cities: Comparing the accuracy of two SEIR models. Mathematical Biosciences & Engineering, 2018, 15 (1) : 209-232. doi: 10.3934/mbe.2018009
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##### References:
An example of the epidemic curve extraction from the interpolated ARI incidence
ARI incidence curve showing the discrepancy between different curve allocation algorithms
The fitting quality of the algorithms in the case of complete outbreak data
An example of epidemic peak prediction by Baroyan-Rvachev model
An example of estimating forecast accuracies. The height of the green bars corresponds to the duration of the outbreak before reaching the peak, the markers indicate the day when the accuracy criterion was reached by the particular model. The absent bars correspond to the years without an epidemic outbreak
The dependence of the percentage of accurate predictions from the number of days before the peak, in overall for the three cities
The dependence of the percentage of accurate predictions from the number of days before the peak, horizontal stripe
The dependence of the percentage of accurate predictions from the number of days before the peak, vertical stripe
The dependence of the percentage of accurate predictions from the number of days before the peak, square
The comparison of the statistical and modeling forecasting methods
Parameters of the fitting algorithm for the continuous SEIR model
 Definition Description Value Unit Epidemiological parameters $\alpha$ Initial ratio of susceptible individuals in the population Estimated - $\beta$ Intensity of infection Estimated 1/(person$\cdot$day) $\gamma$ Intensity of transition to infective form of the disease 0.39 1/day $\delta$ Intensity of recovery 0.133 1/day $I_{0}$ Initial ratio of the infected 0.0001 - Curve positioning parameters $k_{inc}$ Relative vertical bias of the modeled incidence curve position Estimated - $\Delta_s$ Absolute horizontal bias of the modeled incidence curve epidemic start position compared to the data Estimated day
 Definition Description Value Unit Epidemiological parameters $\alpha$ Initial ratio of susceptible individuals in the population Estimated - $\beta$ Intensity of infection Estimated 1/(person$\cdot$day) $\gamma$ Intensity of transition to infective form of the disease 0.39 1/day $\delta$ Intensity of recovery 0.133 1/day $I_{0}$ Initial ratio of the infected 0.0001 - Curve positioning parameters $k_{inc}$ Relative vertical bias of the modeled incidence curve position Estimated - $\Delta_s$ Absolute horizontal bias of the modeled incidence curve epidemic start position compared to the data Estimated day
Parameters of the fitting algorithm for Baroyan-Rvachev model
 Definition Description Value Unit Model parameters $\alpha$ Initial ratio of susceptible individuals in the population Estimated - $\beta$ Intensity of infection Estimated - $I_{0}$ Initial ratio of infected in the population Estimated - $T$ Duration of infection Fixed day $g_\tau$ A fraction of infectious individuals among those who were infected $\tau$ days before the current moment Fixed - $\rho$ Population size Fixed persons Curve positioning parameters $\Delta_p$ Absolute horizontal bias of the modeled incidence curve peak position compared to the data Estimated day
 Definition Description Value Unit Model parameters $\alpha$ Initial ratio of susceptible individuals in the population Estimated - $\beta$ Intensity of infection Estimated - $I_{0}$ Initial ratio of infected in the population Estimated - $T$ Duration of infection Fixed day $g_\tau$ A fraction of infectious individuals among those who were infected $\tau$ days before the current moment Fixed - $\rho$ Population size Fixed persons Curve positioning parameters $\Delta_p$ Absolute horizontal bias of the modeled incidence curve peak position compared to the data Estimated day
Varied model parameters
 Definition Description Value Variation type Continuous SEIR model $\alpha$ Initial ratio of susceptible individuals in the population $[10^{-2}; 1.0]$ BFGS optimization $\beta$ Intensity of infection $[10^{-7}; 50.0]$ BFGS optimization $k_{inc}$ Relative vertical bias of the modeled incidence curve position $[0.8; 1.0]$ BFGS optimization $\Delta_s$ Absolute horizontal bias of the modeled incidence curve epidemic start position compared to the data $5, \dots, 54$ Iteration Baroyan-Rvachev model $k$ The service parameter defining the product of $\alpha$ and $\beta$ $[1.02; 1.6]$ BFGS optimization $I_{0}$ Initial ratio of infected in the population $[10^{-1}; 50.0]$ BFGS optimization $\Delta_p$* Absolute horizontal bias of the modeled incidence curve peak position compared to the data $-3, \dots, 3$ Iteration $\theta^{(dat)}_{peak}$** Prospected incidence curve peak day $17, \dots, 83$ Iteration * For complete incidence data ** For incomplete incidence data
 Definition Description Value Variation type Continuous SEIR model $\alpha$ Initial ratio of susceptible individuals in the population $[10^{-2}; 1.0]$ BFGS optimization $\beta$ Intensity of infection $[10^{-7}; 50.0]$ BFGS optimization $k_{inc}$ Relative vertical bias of the modeled incidence curve position $[0.8; 1.0]$ BFGS optimization $\Delta_s$ Absolute horizontal bias of the modeled incidence curve epidemic start position compared to the data $5, \dots, 54$ Iteration Baroyan-Rvachev model $k$ The service parameter defining the product of $\alpha$ and $\beta$ $[1.02; 1.6]$ BFGS optimization $I_{0}$ Initial ratio of infected in the population $[10^{-1}; 50.0]$ BFGS optimization $\Delta_p$* Absolute horizontal bias of the modeled incidence curve peak position compared to the data $-3, \dots, 3$ Iteration $\theta^{(dat)}_{peak}$** Prospected incidence curve peak day $17, \dots, 83$ Iteration * For complete incidence data ** For incomplete incidence data
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