On the usefulness of set-membership estimation in the epidemiology of infectious diseases

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doi:10.3934/mbe.2018006

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2018, Volume 15, Issue 1: 141-152. Doi: 10.3934/mbe.2018006

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On the usefulness of set-membership estimation in the epidemiology of infectious diseases

ORCOS, Institute of Statistics and Mathematical Methods in Economics, Vienna University of Technology, Wiedner Hauptstrasse 8-10, A-1040 Vienna, Austria

Received: September 11, 2016

Accepted: March 13, 2017

Published: February 2018

The author is supported by FWF grant P 24125-N13

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Abstract

We present a method, known in control theory, to give set-membership estimates for the states of a population in which an infectious disease is spreading. An estimation is reasonable due to the fact that the parameters of the equations describing the dynamics of the disease are not known with certainty. We discuss the properties of the resulting estimations. These include the possibility to determine best-or worst-case-scenarios and identify under which circumstances they occur, as well as a method to calculate confidence intervals for disease trajectories under sparse data. We give numerical examples of the technique using data from the 2014 outbreak of the Ebola virus in Africa. We conclude that the method presented here can be used to extract additional information from epidemiological data.

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Mathematics Subject Classification: 92D30, 49J53, 34H99, 90C30.

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Figure 1. A comparison with the set membership estimation with the data. The dotted line is calculated using the maximum likelihood estimate. The solid lines are the set-membership estimation calculated by using the $95\%$ confidence intervals. The dashed lines are calculated from a $95\%$ confidence region using Bonferroni's method

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Figure 2. A long term calculation of the maximum likelihood estimate (dotted line), and estimations using the $95\%$ confidence intervals (dashed lines), and a $95\%$ confidence region (solid lines)

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Figure 3. Set-membership estimates for $I(t)$ along with the parameter combinations that maximise and minimise $I(t)$. Even in the simple model presented here this combination may change with time

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