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

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doi:10.3934/mbe.2018009

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2018, Volume 15, Issue 1: 209-232. Doi: 10.3934/mbe.2018009

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

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

* Corresponding author: vnleonenko@yandex.ru
* Corresponding author: vnleonenko@yandex.ru

Received: November 07, 2016

Accepted: April 05, 2017

Published: January 2018

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Abstract

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.

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Mathematics Subject Classification: 37N25, 65C20, 92C60.

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Figure 1. An example of the epidemic curve extraction from the interpolated ARI incidence

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Figure 2. ARI incidence curve showing the discrepancy between different curve allocation algorithms

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Figure 3. The fitting quality of the algorithms in the case of complete outbreak data

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Figure 4. An example of epidemic peak prediction by Baroyan-Rvachev model

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Figure 5. 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

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Figure 6. The dependence of the percentage of accurate predictions from the number of days before the peak, in overall for the three cities

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Figure 7. The dependence of the percentage of accurate predictions from the number of days before the peak, horizontal stripe

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Figure 8. The dependence of the percentage of accurate predictions from the number of days before the peak, vertical stripe

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Figure 9. The dependence of the percentage of accurate predictions from the number of days before the peak, square

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Figure 10. The comparison of the statistical and modeling forecasting methods

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Table 1. 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

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Table 2. 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

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Table 3. 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

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