# American Institute of Mathematical Sciences

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February  2018, 15(1): 141-152. doi: 10.3934/mbe.2018006

## 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 12, 2016 Accepted  March 14, 2017 Published  May 2017

Fund Project: The author is supported by FWF grant P 24125-N13.

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.

Citation: Andreas Widder. On the usefulness of set-membership estimation in the epidemiology of infectious diseases. Mathematical Biosciences & Engineering, 2018, 15 (1) : 141-152. doi: 10.3934/mbe.2018006
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##### References:
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
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)
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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