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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
References:
[1]

C. L. Althaus, Estimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in West Africa, PLOS Currents Outbreaks, (2014). doi: doi10.1371/currents.outbreaks.91afb5e0f279e7f29e7056095255b288.

[2]

C. L. AlthausN. LowE. O. MusaF. Shuaib and S. Gsteiger, Ebola virus disease outbreak in Nigeria: Transmission dynamics and rapid control, Epidemics, 11 (2015), 80-84. doi: 10.1016/j.epidem.2015.03.001.

[3]

H. Andersson and T. Britton, Stochastic Epidemic Models and Their Statistical Analysis, Lecture Notes in Statistics, 151. Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1158-7.

[4]

R. Baier and M. Gerdts, A computational method for non-convex reachable sets using optimal control, in 2009 European Control Conference (ECC), IEEE, (2009), 97-102.

[5]

D. P. Bertsekas, Dynamic Programming and Optimal Control, Athena Scientific, 1995.

[6]

A. Bhargava and F. Docquier, HIV pandemic, medical brain drain, and economic development in Sub-Saharan Africa, The World Bank Economic Review, 22 (2008), 345-366. doi: 10.1093/wber/lhn005.

[7]

A. BlakeM. T. Sinclair and G. Sugiyarto, Quantifying the impact of foot and mouth disease on tourism and the UK economy, Tourism Economics, 9 (2003), 449-465. doi: 10.5367/000000003322663221.

[8]

A. CapaldiS. BehrendB. BermanJ. SmithJ. Wright and A. L. Lloyd, Parameter estimation and uncertainty quantication for an epidemic model, Mathematical Biosciences and Engineering, 9 (2012), 553-576. doi: 10.3934/mbe.2012.9.553.

[9]

O. Diekmann, H. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton University Press, 2013.

[10]

J. Hsu, Multiple Comparisons: Theory and Methods, CRC Press, 1996. doi: 10.1007/978-1-4899-7180-7.

[11]

M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008.

[12]

A. B. Kurzhanski and P. Varaiya, Dynamics and Control of Trajectory Tubes, Springer, 2014. doi: 10.1007/978-3-319-10277-1.

[13]

P. E. Lekone and B. F. Finkenstädt, Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study, Biometrics, 62 (2006), 1170-1177. doi: 10.1111/j.1541-0420.2006.00609.x.

[14]

N.-A. M. MolinariI. R. Ortega-SanchezM. L. MessonnierW. W. ThompsonP. M. WortleyE. Weintraub and C. B. Bridges, The annual impact of seasonal influenza in the US: measuring disease burden and costs, Vaccine, 25 (2007), 5086-5096. doi: 10.1016/j.vaccine.2007.03.046.

[15]

E. Polak, Computational Methods in Optimization: A Unified Approach, vol. 77, Academic Press, New York-London, 1971.

[16]

E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Second edition. Texts in Applied Mathematics, 6. Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.

[17]

M. R. Thomas, G. Smith, F. H. Ferreira, D. Evans, M. Maliszewska, M. Cruz, K. Himelein and M. Over, The economic impact of Ebola on Sub-Saharan Africa: Updated estimates for 2015, World Bank Group.

[18]

T. ToniD. WelchN. StrelkowaA. Ipsen and M. P. Stumpf, Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems, Journal of the Royal Society Interface, 6 (2009), 187-202. doi: 10.1098/rsif.2008.0172.

[19]

T. TsachevV. M. Veliov and A. Widder, Set-membership estimation for the evolution of infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 74 (2017), 1081-1106. doi: 10.1007/s00285-016-1050-0.

[20]

V. Veliov and A. Widder, Modelling and estimation of infectious diseases in a population with heterogeneous dynamic immunity, Journal of Biological Dynamics, 10 (2016), 457-476. doi: 10.1080/17513758.2016.1221474.

[21]

A. Widder, Matlab files for set-membership estimation, Zenodo, 2016. doi: 10.5281/zenodo.192853.

show all references

References:
[1]

C. L. Althaus, Estimating the reproduction number of Ebola virus (EBOV) during the 2014 outbreak in West Africa, PLOS Currents Outbreaks, (2014). doi: doi10.1371/currents.outbreaks.91afb5e0f279e7f29e7056095255b288.

[2]

C. L. AlthausN. LowE. O. MusaF. Shuaib and S. Gsteiger, Ebola virus disease outbreak in Nigeria: Transmission dynamics and rapid control, Epidemics, 11 (2015), 80-84. doi: 10.1016/j.epidem.2015.03.001.

[3]

H. Andersson and T. Britton, Stochastic Epidemic Models and Their Statistical Analysis, Lecture Notes in Statistics, 151. Springer-Verlag, New York, 2000. doi: 10.1007/978-1-4612-1158-7.

[4]

R. Baier and M. Gerdts, A computational method for non-convex reachable sets using optimal control, in 2009 European Control Conference (ECC), IEEE, (2009), 97-102.

[5]

D. P. Bertsekas, Dynamic Programming and Optimal Control, Athena Scientific, 1995.

[6]

A. Bhargava and F. Docquier, HIV pandemic, medical brain drain, and economic development in Sub-Saharan Africa, The World Bank Economic Review, 22 (2008), 345-366. doi: 10.1093/wber/lhn005.

[7]

A. BlakeM. T. Sinclair and G. Sugiyarto, Quantifying the impact of foot and mouth disease on tourism and the UK economy, Tourism Economics, 9 (2003), 449-465. doi: 10.5367/000000003322663221.

[8]

A. CapaldiS. BehrendB. BermanJ. SmithJ. Wright and A. L. Lloyd, Parameter estimation and uncertainty quantication for an epidemic model, Mathematical Biosciences and Engineering, 9 (2012), 553-576. doi: 10.3934/mbe.2012.9.553.

[9]

O. Diekmann, H. Heesterbeek and T. Britton, Mathematical Tools for Understanding Infectious Disease Dynamics, Princeton University Press, 2013.

[10]

J. Hsu, Multiple Comparisons: Theory and Methods, CRC Press, 1996. doi: 10.1007/978-1-4899-7180-7.

[11]

M. J. Keeling and P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008.

[12]

A. B. Kurzhanski and P. Varaiya, Dynamics and Control of Trajectory Tubes, Springer, 2014. doi: 10.1007/978-3-319-10277-1.

[13]

P. E. Lekone and B. F. Finkenstädt, Statistical inference in a stochastic epidemic SEIR model with control intervention: Ebola as a case study, Biometrics, 62 (2006), 1170-1177. doi: 10.1111/j.1541-0420.2006.00609.x.

[14]

N.-A. M. MolinariI. R. Ortega-SanchezM. L. MessonnierW. W. ThompsonP. M. WortleyE. Weintraub and C. B. Bridges, The annual impact of seasonal influenza in the US: measuring disease burden and costs, Vaccine, 25 (2007), 5086-5096. doi: 10.1016/j.vaccine.2007.03.046.

[15]

E. Polak, Computational Methods in Optimization: A Unified Approach, vol. 77, Academic Press, New York-London, 1971.

[16]

E. D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems, Second edition. Texts in Applied Mathematics, 6. Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.

[17]

M. R. Thomas, G. Smith, F. H. Ferreira, D. Evans, M. Maliszewska, M. Cruz, K. Himelein and M. Over, The economic impact of Ebola on Sub-Saharan Africa: Updated estimates for 2015, World Bank Group.

[18]

T. ToniD. WelchN. StrelkowaA. Ipsen and M. P. Stumpf, Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems, Journal of the Royal Society Interface, 6 (2009), 187-202. doi: 10.1098/rsif.2008.0172.

[19]

T. TsachevV. M. Veliov and A. Widder, Set-membership estimation for the evolution of infectious diseases in heterogeneous populations, Journal of Mathematical Biology, 74 (2017), 1081-1106. doi: 10.1007/s00285-016-1050-0.

[20]

V. Veliov and A. Widder, Modelling and estimation of infectious diseases in a population with heterogeneous dynamic immunity, Journal of Biological Dynamics, 10 (2016), 457-476. doi: 10.1080/17513758.2016.1221474.

[21]

A. Widder, Matlab files for set-membership estimation, Zenodo, 2016. doi: 10.5281/zenodo.192853.

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
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)
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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