# American Institute of Mathematical Sciences

2012, 9(3): 553-576. doi: 10.3934/mbe.2012.9.553

## Parameter estimation and uncertainty quantification for an epidemic model

 1 Center for Quantitative Sciences in Biomedicine and Department of Mathematics, North Carolina State University, Raleigh, NC 27695, and Department of Mathematics & Computer Science, Valparaiso University, 1900 Chapel Drive, Valparaiso, IN 46383, United States 2 Department of Mathematics, University of North Carolina, Chapel Hill, CB #3250, Chapel Hill, NC 27599, United States 3 Program in Applied Mathematics, University of Arizona, 617 N. Santa Rita Ave., PO Box 210089, Tucson, AZ 85721-0089, United States 4 Department of Mathematics, Morehouse College, 830 Westview Drive SW Unit 142133, Atlanta, GA 30314, United States 5 Department of Mathematics, North Carolina State University, Raleigh, NC 27695, United States 6 Biomathematics Graduate Program and Department of Mathematics, North Carolina State University, Raleigh NC, 27695, USA and Fogarty International Center, National Institutes of Health, Bethesda, MD 20892, United States

Received  December 2009 Revised  April 2012 Published  July 2012

We examine estimation of the parameters of Susceptible-Infective-Recovered (SIR) models in the context of least squares. We review the use of asymptotic statistical theory and sensitivity analysis to obtain measures of uncertainty for estimates of the model parameters and the basic reproductive number ($R_0$)---an epidemiologically significant parameter grouping. We find that estimates of different parameters, such as the transmission parameter and recovery rate, are correlated, with the magnitude and sign of this correlation depending on the value of $R_0$. Situations are highlighted in which this correlation allows $R_0$ to be estimated with greater ease than its constituent parameters. Implications of correlation for parameter identifiability are discussed. Uncertainty estimates and sensitivity analysis are used to investigate how the frequency at which data is sampled affects the estimation process and how the accuracy and uncertainty of estimates improves as data is collected over the course of an outbreak. We assess the informativeness of individual data points in a given time series to determine when more frequent sampling (if possible) would prove to be most beneficial to the estimation process. This technique can be used to design data sampling schemes in more general contexts.
Citation: Alex Capaldi, Samuel Behrend, Benjamin Berman, Jason Smith, Justin Wright, Alun L. Lloyd. Parameter estimation and uncertainty quantification for an epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (3) : 553-576. doi: 10.3934/mbe.2012.9.553
##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases of Humans," Oxford University Press, Oxford, 1991. [2] D. T. Anh, M. P. Bonnet, G. Vachaud, C. V. Minh, N. Prieur, L. V. Duc and L. L. Anh, Biochemical modeling of the Nhue River (Hanoi, Vietnam): Practical identifiability analysis and parameters estimation, Ecol. Model., 193 (2006), 182-204. doi: 10.1016/j.ecolmodel.2005.08.029. [3] H. T. Banks, M. Davidian, J. R. Samuels Jr. and K. L. Sutton, An inverse problem statistical methodology summary, in "Mathematical and Statistical Estimation Approaches in Epidemiology" (eds. G. Chowell, J. M. Hyman, L. M. A. Bettencourt and C. Castillo-Chávez), Springer, New York, (2009), 249-302. [4] H. T. Banks, S. Dediu and S. L. Ernstberger, Sensitivity functions and their uses in inverse problems, Tech. Report CRSC-TR07-12, Center for Research in Scientific Computation, North Carolina State Unversity, July 2007. [5] H. T. Banks, S. L. Ernstberger and S. L. Grove, Standard errors and confidence intervals in inverse problems: Sensitivity and associated pitfalls, J. Inverse Ill-Posed Probl., 15 (2007), 1-18. [6] R. Bellman and K. J. Åström, On structural identifiability, Math. Biosci., 7 (1970), 329-339. doi: 10.1016/0025-5564(70)90132-X. [7] R. Brun, M. Kühni, H. Siegrist, W. Gujer and P. Reichert, Practical identifiability of ASM2d parameters-systematic selection and tuning of parameter subsets, Water Res., 36 (2002), 4113-4127. doi: 10.1016/S0043-1354(02)00104-5. [8] M. Burth, G. C. Verghese and M. Vélez-Reyes, Subset selection for improved parameter estimation in on-line identification of a synchronous generator, IEEE Trans. Power Syst., 14 (1999), 218-225. doi: 10.1109/59.744536. [9] A. Capaldi, S. Behrend, B. Berman, J. Smith, J. Wright and A. L. Lloyd, Parameter estimation and uncertainty quantification for an epidemic model, Tech. Report CRSC-TR09-18, Center for Research in Scientific Computation, North Carolina State Unversity, August 2009. [10] S. Cauchemez, P.-Y. Böelle, G. Thomas and A.-J. Valleron, Estimating in real time the efficacy of measures to control emerging communicable diseases, Am. J. Epidemiol., 164 (2006), 591-597. doi: 10.1093/aje/kwj274. [11] S. Cauchemez and N. M. Ferguson, Likelihood-based estimation of continuous-time epidemic models from time-series data: Application to measles transmission in London, J. R. Soc. Interface, 5 (2008), 885-897. doi: 10.1098/rsif.2007.1292. [12] G. Chowell, C. E. Ammon, N. W. Hengartner and J. M. Hyman, Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland, Math. Biosci. Eng., 4 (2007), 457-470. doi: 10.3934/mbe.2007.4.457. [13] G. Chowell, N. W. Hengartner, C. Castillo-Chávez, P. W. Fenimore and J. M. Hyman, The basic reproductive number of ebola and the effects of public health measures: The cases of Congo and Uganda, J. Theor. Biol., 229 (2004), 119-126. doi: 10.1016/j.jtbi.2004.03.006. [14] G. Chowell, M. A. Miller and C. Viboud, Seasonal influenza in the United States, France and Australia: Transmission and prospects for control, Epidemiol. Infect., 136 (2007), 852-864. [15] A. Cintrón-Arias, H. T. Banks, A. Capaldi and A. L. Lloyd, A sensitivity matrix based methodology for inverse problem formulation, J. Inv. Ill-Posed Problems, 17 (2009), 545-564. doi: 10.1515/JIIP.2009.034. [16] A. Cintrón-Arias, C. Castillo-Chávez, L. M. A. Bettencourt, A. L. Lloyd and H. T. Banks, The estimation of the effective reproductive number from disease outbreak data, Math. Biosci. Eng., 6 (2009), 261-282. doi: 10.3934/mbe.2009.6.261. [17] C. Cobelli and J. J. {DiStefano, III}, Parameter and structural identifiability concepts and ambiguities: A critical review and analysis, Am. J. Physiol. (Regulatory Integrative Comp. Physiol. 8), 239 (1980), R7-R24. [18] M. Davidian and D. M. Giltinan, "Nonlinear Models for Repeated Measurement Data," Chapman & Hall, 1996. [19] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000. [20] K. Dietz, The estimation of the basic reproduction number for infectious diseases, Stat. Meth. Med. Res., 2 (1993), 23-41. doi: 10.1177/096228029300200103. [21] M. Eslami, "Theory of Sensitivity in Dynamic Systems. An Introduction," Springer-Verlag, Berlin, 1994. [22] N. D. Evans, L. J. White, M. J. Chapman, K. R. Godfrey and M. J. Chappell, The structural identifiability of the susceptible infected recovered model with seasonal forcing, Math. Biosci., 194 (2005), 175-197. doi: 10.1016/j.mbs.2004.10.011. [23] B. Finkenstädt and B. Grenfell, Empirical determinants of measles metapopulation dynamics in England and Wales, Proc. R. Soc. Lond. B, 265 (1998), 211-220. doi: 10.1098/rspb.1998.0284. [24] K. Glover and J. C. Willems, Parametrizations of linear dynamical systems: Canonical forms and identifiability, IEEE Trans. Auto. Contr., AC-19 (1974), 640-646. [25] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [26] T. D. Hollingsworth, N. M. Ferguson and R. M. Anderson, Will travel restrictions control the international spread of pandemic influenza?, Nature Med., 12 (2006), 497-499. doi: 10.1038/nm0506-497. [27] A. Holmberg, On the practical identifiability of microbial growth models incorporating Michaelis-Menten type nonlinearities, Math. Biosci., 62 (1982), 23-43. doi: 10.1016/0025-5564(82)90061-X. [28] J. A. Jacquez and P. Greif, Numerical parameter identifiability and estimability: Integrating identifiability, estimability and optimal sampling design, Math. Biosci., 77 (1985), 201-227. doi: 10.1016/0025-5564(85)90098-7. [29] S. Kotz, N. Balakrishnan, C. Read and B. Vidakovic, eds., "Encyclopedia of Statistics," 2nd edition, Wiley-Interscience, Hoboken, New Jersey, 2006. [30] M. Kretzschmar, S. van den Hof, J. Wallinga and J. van Wijngaarden, Ring vaccination and smallpox control, Emerg. Inf. Dis., 10 (2004), 832-841. doi: 10.3201/eid1005.030419. [31] 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. [32] A. L. Lloyd, The dependence of viral parameter estimates on the assumed viral life cycle: Limitations of studies of viral load data, Proc. R. Soc. Lond. B, 268 (2001), 847-854. doi: 10.1098/rspb.2000.1572. [33] —, Sensitivity of model-based epidemiological parameter estimation to model assumptions, in "Mathematical and Statistical Estimation Approaches in Epidemiology" (eds. G. Chowell, J. M. Hyman, L. M. A. Bettencourt and C. Castillo-Chavez), Springer, New York, (2009), 123-141. [34] C. D. Meyer, "Matrix Analysis and Applied Linear Algebra," SIAM, Hoboken, New Jersey, 2000. doi: 10.1137/1.9780898719512. [35] M. A. Nowak, A. L. Lloyd, G. M. Vasquez, T. A. Wiltrout, L. M. Wahl, N. Bischofberger, J. Williams, A. Kinter, A. S. Fauci, V. M. Hirsch and J. D. Lifson, Viral dynamics of primary viremia and antiretroviral therapy in simian immunodeficiency virus infection, J. Virol., 71 (1997), 7518-7525. [36] J. G. Reid, Structural identifiability in linear time-invariant systems, IEEE Trans. Auto. Contr., AC-22 (1977), 242-246. doi: 10.1109/TAC.1977.1101474. [37] A. Saltelli, K. Chan and E. M. Scott, eds., "Sensitivity Analysis," Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 2000. [38] M. A. Sanchez and S. M. Blower, Uncertainty and sensitivity analysis of the basic reproductive rate, Am. J. Epidemiol., 145 (1997), 1127-1137. doi: 10.1093/oxfordjournals.aje.a009076. [39] G. A. F. Seber and C. J. Wild, "Nonlinear Regression," John Wiley & Sons, Hoboken, New Jersey, 2003. [40] K. Thomaseth and C. Cobelli, Generalized sensitivity functions in physiological system identification, Ann. Biomed. Eng., 27 (1999), 607-616. doi: 10.1114/1.207. [41] J. Wallinga and M. Lipsitch, How generation intervals shape the relationship between growth rates and reproductive numbers, Proc. R. Soc. Lond. B, 274 (2007), 599-604. [42] H. J. Wearing, P. Rohani and M. Keeling, Appropriate models for the management of infectious diseases, PLoS Med., 2 (2005), e174. doi: 10.1371/journal.pmed.0020174. [43] L. J. White, N. D. Evans, T. J. G. M. Lam, Y. H. Schukken, G. F. Medley, K. R. Godfrey and M. J. Chappell, The structural identifiability and parameter estimation of a multispecies model for the transmission of mastitis in dairy cows, Math. Biosci., 174 (2001), 77-90. doi: 10.1016/S0025-5564(01)00080-3. [44] H. Wu, H. Zhu, H. Miao and A. S. Perelson, Parameter identifiability and estimation of HIV/AIDS dynamic models, Bull. Math. Biol., 70 (2008), 785-799. doi: 10.1007/s11538-007-9279-9. [45] X. Xia and C. H. Moog, Identifiability of nonlinear systems with application to HIV/AIDS models, IEEE Trans. Auto. Contr., 48 (2003), 330-336. doi: 10.1109/TAC.2002.808494. [46] H. Yue, M. Brown, F. He, J. Jia and D. B. Kell, Sensitivity analysis and robust experimental design of a signal transduction pathway system, Int. J. Chem. Kinet., 40 (2008), 730-741. doi: 10.1002/kin.20369.

show all references

##### References:
 [1] R. M. Anderson and R. M. May, "Infectious Diseases of Humans," Oxford University Press, Oxford, 1991. [2] D. T. Anh, M. P. Bonnet, G. Vachaud, C. V. Minh, N. Prieur, L. V. Duc and L. L. Anh, Biochemical modeling of the Nhue River (Hanoi, Vietnam): Practical identifiability analysis and parameters estimation, Ecol. Model., 193 (2006), 182-204. doi: 10.1016/j.ecolmodel.2005.08.029. [3] H. T. Banks, M. Davidian, J. R. Samuels Jr. and K. L. Sutton, An inverse problem statistical methodology summary, in "Mathematical and Statistical Estimation Approaches in Epidemiology" (eds. G. Chowell, J. M. Hyman, L. M. A. Bettencourt and C. Castillo-Chávez), Springer, New York, (2009), 249-302. [4] H. T. Banks, S. Dediu and S. L. Ernstberger, Sensitivity functions and their uses in inverse problems, Tech. Report CRSC-TR07-12, Center for Research in Scientific Computation, North Carolina State Unversity, July 2007. [5] H. T. Banks, S. L. Ernstberger and S. L. Grove, Standard errors and confidence intervals in inverse problems: Sensitivity and associated pitfalls, J. Inverse Ill-Posed Probl., 15 (2007), 1-18. [6] R. Bellman and K. J. Åström, On structural identifiability, Math. Biosci., 7 (1970), 329-339. doi: 10.1016/0025-5564(70)90132-X. [7] R. Brun, M. Kühni, H. Siegrist, W. Gujer and P. Reichert, Practical identifiability of ASM2d parameters-systematic selection and tuning of parameter subsets, Water Res., 36 (2002), 4113-4127. doi: 10.1016/S0043-1354(02)00104-5. [8] M. Burth, G. C. Verghese and M. Vélez-Reyes, Subset selection for improved parameter estimation in on-line identification of a synchronous generator, IEEE Trans. Power Syst., 14 (1999), 218-225. doi: 10.1109/59.744536. [9] A. Capaldi, S. Behrend, B. Berman, J. Smith, J. Wright and A. L. Lloyd, Parameter estimation and uncertainty quantification for an epidemic model, Tech. Report CRSC-TR09-18, Center for Research in Scientific Computation, North Carolina State Unversity, August 2009. [10] S. Cauchemez, P.-Y. Böelle, G. Thomas and A.-J. Valleron, Estimating in real time the efficacy of measures to control emerging communicable diseases, Am. J. Epidemiol., 164 (2006), 591-597. doi: 10.1093/aje/kwj274. [11] S. Cauchemez and N. M. Ferguson, Likelihood-based estimation of continuous-time epidemic models from time-series data: Application to measles transmission in London, J. R. Soc. Interface, 5 (2008), 885-897. doi: 10.1098/rsif.2007.1292. [12] G. Chowell, C. E. Ammon, N. W. Hengartner and J. M. Hyman, Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland, Math. Biosci. Eng., 4 (2007), 457-470. doi: 10.3934/mbe.2007.4.457. [13] G. Chowell, N. W. Hengartner, C. Castillo-Chávez, P. W. Fenimore and J. M. Hyman, The basic reproductive number of ebola and the effects of public health measures: The cases of Congo and Uganda, J. Theor. Biol., 229 (2004), 119-126. doi: 10.1016/j.jtbi.2004.03.006. [14] G. Chowell, M. A. Miller and C. Viboud, Seasonal influenza in the United States, France and Australia: Transmission and prospects for control, Epidemiol. Infect., 136 (2007), 852-864. [15] A. Cintrón-Arias, H. T. Banks, A. Capaldi and A. L. Lloyd, A sensitivity matrix based methodology for inverse problem formulation, J. Inv. Ill-Posed Problems, 17 (2009), 545-564. doi: 10.1515/JIIP.2009.034. [16] A. Cintrón-Arias, C. Castillo-Chávez, L. M. A. Bettencourt, A. L. Lloyd and H. T. Banks, The estimation of the effective reproductive number from disease outbreak data, Math. Biosci. Eng., 6 (2009), 261-282. doi: 10.3934/mbe.2009.6.261. [17] C. Cobelli and J. J. {DiStefano, III}, Parameter and structural identifiability concepts and ambiguities: A critical review and analysis, Am. J. Physiol. (Regulatory Integrative Comp. Physiol. 8), 239 (1980), R7-R24. [18] M. Davidian and D. M. Giltinan, "Nonlinear Models for Repeated Measurement Data," Chapman & Hall, 1996. [19] O. Diekmann and J. A. P. Heesterbeek, "Mathematical Epidemiology of Infectious Diseases. Model Building, Analysis and Interpretation," Wiley Series in Mathematical and Computational Biology, John Wiley & Sons, Ltd., Chichester, 2000. [20] K. Dietz, The estimation of the basic reproduction number for infectious diseases, Stat. Meth. Med. Res., 2 (1993), 23-41. doi: 10.1177/096228029300200103. [21] M. Eslami, "Theory of Sensitivity in Dynamic Systems. An Introduction," Springer-Verlag, Berlin, 1994. [22] N. D. Evans, L. J. White, M. J. Chapman, K. R. Godfrey and M. J. Chappell, The structural identifiability of the susceptible infected recovered model with seasonal forcing, Math. Biosci., 194 (2005), 175-197. doi: 10.1016/j.mbs.2004.10.011. [23] B. Finkenstädt and B. Grenfell, Empirical determinants of measles metapopulation dynamics in England and Wales, Proc. R. Soc. Lond. B, 265 (1998), 211-220. doi: 10.1098/rspb.1998.0284. [24] K. Glover and J. C. Willems, Parametrizations of linear dynamical systems: Canonical forms and identifiability, IEEE Trans. Auto. Contr., AC-19 (1974), 640-646. [25] H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653. doi: 10.1137/S0036144500371907. [26] T. D. Hollingsworth, N. M. Ferguson and R. M. Anderson, Will travel restrictions control the international spread of pandemic influenza?, Nature Med., 12 (2006), 497-499. doi: 10.1038/nm0506-497. [27] A. Holmberg, On the practical identifiability of microbial growth models incorporating Michaelis-Menten type nonlinearities, Math. Biosci., 62 (1982), 23-43. doi: 10.1016/0025-5564(82)90061-X. [28] J. A. Jacquez and P. Greif, Numerical parameter identifiability and estimability: Integrating identifiability, estimability and optimal sampling design, Math. Biosci., 77 (1985), 201-227. doi: 10.1016/0025-5564(85)90098-7. [29] S. Kotz, N. Balakrishnan, C. Read and B. Vidakovic, eds., "Encyclopedia of Statistics," 2nd edition, Wiley-Interscience, Hoboken, New Jersey, 2006. [30] M. Kretzschmar, S. van den Hof, J. Wallinga and J. van Wijngaarden, Ring vaccination and smallpox control, Emerg. Inf. Dis., 10 (2004), 832-841. doi: 10.3201/eid1005.030419. [31] 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. [32] A. L. Lloyd, The dependence of viral parameter estimates on the assumed viral life cycle: Limitations of studies of viral load data, Proc. R. Soc. Lond. B, 268 (2001), 847-854. doi: 10.1098/rspb.2000.1572. [33] —, Sensitivity of model-based epidemiological parameter estimation to model assumptions, in "Mathematical and Statistical Estimation Approaches in Epidemiology" (eds. G. Chowell, J. M. Hyman, L. M. A. Bettencourt and C. Castillo-Chavez), Springer, New York, (2009), 123-141. [34] C. D. Meyer, "Matrix Analysis and Applied Linear Algebra," SIAM, Hoboken, New Jersey, 2000. doi: 10.1137/1.9780898719512. [35] M. A. Nowak, A. L. Lloyd, G. M. Vasquez, T. A. Wiltrout, L. M. Wahl, N. Bischofberger, J. Williams, A. Kinter, A. S. Fauci, V. M. Hirsch and J. D. Lifson, Viral dynamics of primary viremia and antiretroviral therapy in simian immunodeficiency virus infection, J. Virol., 71 (1997), 7518-7525. [36] J. G. Reid, Structural identifiability in linear time-invariant systems, IEEE Trans. Auto. Contr., AC-22 (1977), 242-246. doi: 10.1109/TAC.1977.1101474. [37] A. Saltelli, K. Chan and E. M. Scott, eds., "Sensitivity Analysis," Wiley Series in Probability and Statistics, John Wiley & Sons, Ltd., Chichester, 2000. [38] M. A. Sanchez and S. M. Blower, Uncertainty and sensitivity analysis of the basic reproductive rate, Am. J. Epidemiol., 145 (1997), 1127-1137. doi: 10.1093/oxfordjournals.aje.a009076. [39] G. A. F. Seber and C. J. Wild, "Nonlinear Regression," John Wiley & Sons, Hoboken, New Jersey, 2003. [40] K. Thomaseth and C. Cobelli, Generalized sensitivity functions in physiological system identification, Ann. Biomed. Eng., 27 (1999), 607-616. doi: 10.1114/1.207. [41] J. Wallinga and M. Lipsitch, How generation intervals shape the relationship between growth rates and reproductive numbers, Proc. R. Soc. Lond. B, 274 (2007), 599-604. [42] H. J. Wearing, P. Rohani and M. Keeling, Appropriate models for the management of infectious diseases, PLoS Med., 2 (2005), e174. doi: 10.1371/journal.pmed.0020174. [43] L. J. White, N. D. Evans, T. J. G. M. Lam, Y. H. Schukken, G. F. Medley, K. R. Godfrey and M. J. Chappell, The structural identifiability and parameter estimation of a multispecies model for the transmission of mastitis in dairy cows, Math. Biosci., 174 (2001), 77-90. doi: 10.1016/S0025-5564(01)00080-3. [44] H. Wu, H. Zhu, H. Miao and A. S. Perelson, Parameter identifiability and estimation of HIV/AIDS dynamic models, Bull. Math. Biol., 70 (2008), 785-799. doi: 10.1007/s11538-007-9279-9. [45] X. Xia and C. H. Moog, Identifiability of nonlinear systems with application to HIV/AIDS models, IEEE Trans. Auto. Contr., 48 (2003), 330-336. doi: 10.1109/TAC.2002.808494. [46] H. Yue, M. Brown, F. He, J. Jia and D. B. Kell, Sensitivity analysis and robust experimental design of a signal transduction pathway system, Int. J. Chem. Kinet., 40 (2008), 730-741. doi: 10.1002/kin.20369.
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