Observation and model error effects on parameter estimates in susceptible-infected-recovered epidemiological models

Recently, confidence intervals (CIs) associated with parameter estimates in the susceptibleinfected- recovered (SIR) epidemiological model have been developed (Chowell et. al. [4]). When model assumptions are met and the observation error is relatively small, these CIs are relatively short, as we will illustrate. This work describes the behavior of CIs for parameters as observation and/or equation or model error becomes larger, and includes a comparison of two estimation procedures. The first procedure fits a simple linear regression relating the per-timestep response and predictors. This procedure demonstrates significant bias as observation error increases. In general, observation error in predictors leads to bias to varying degrees, as has been illustrated in the “errors-in-variables” literature (Carroll et al. [3]).

For example, bias arising from using measurements of species abundance rather than true species abundance has recently been reported in the context of a biological random walk extinction model (Buonaccorsi et al. [2]). The other procedure evaluated here solves the nonlinear differential equations to produce parameter estimates, thereby relying heavily on the shape of the observed epidemic curve, and mitigating the effects of any errors, such as observation errors, that do not distort the curve’s shape. This method demonstrates significant bias if model error increases sufficiently to distort the curve’s shape.

References:

• G. Chowell, H. Nishiura. Quantifying the transmission potential of pandemic influenza. Physics of Life Reviews, doi:10.1016/j.plrev.2007.12.001 (pdf)
• G. Chowell, L.M.A. Bettencourt, N. Johnson, W.J. Alonso, C. Viboud.The 1918-1919 influenza pandemic in England and Wales: Spatial patterns in transmissibility and mortality impact. Proc. Roy. Soc. B, doi:10.1098/rspb.2007.1477 (pdf)
• H. Nishiura and G. Chowell. Household and community transmission of the Asian influenza A (H2N2) and influenza B viruses in 1957 and 1961. Southeast Asian J. Trop. Med. Pub. Health. 38(6), 1075-1083 (2007) (pdf)
• G. Chowell, C. E. Ammon, N. W. Hengartner, J. M. Hyman. Estimating the reproduction number from the initial phase of the Spanish flu pandemic waves in Geneva, Switzerland. Math Biosci Eng. 2007 Jul;4(3):457-70.(pdf)
• M. Nuno, G. Chowell, A. Gumel. Assessing the role of basic control measures, antivirals and vaccine in curtailing pandemic infuenza: Scenarios for the US, UK, and the Netherlands. J R Soc Interface 2007 Jun 22;4(14):505-21. (pdf)
• G. Chowell, C. E. Ammon, N. W. Hengartner, J. M. Hyman. Estimation of the Reproductive number of the Spanish Flu Epidemic in Geneva, Switzerland. Proceedings of the Second European Influenza Conference (St-Julians, Malta, September, 2005). Vaccine 24, 6747-6750 (2006) (pdf)
• G. Chowell, C. E. Ammon, N. W. Hengartner, J. M. Hyman. Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland: Assessing the effects of hypothetical interventions. J. Theor. Biol. 241(2), 193-204 (2006) (pdf)
• T.L. Burr, G. Chowell. Observation and model error effects on parameter estimates in susceptible-infected-recovered epidemic model. Far East J. Theor. Stat. 19(2), 163-183 (2006) (pdf)
• G. Chowell, H. Nishiura, L.M.A. Bettencourt. Comparative estimation of the reproduction number for pandemic influenza from daily case notification data. J R Soc Interface 4, 155-166 (2007) (pdf)
• A.B. Gumel, M. Nuno, G. Chowell. Mathematical assessment of Canada's pandemic influenza preparedness plan. Can. J. Inf. Dis. & Med. Microb. (in press).

Figure 1. Representative simulated epidemic curves (dots) and their corresponding fit (solid line) using the standard SIR model with nonlinear differential equation fitting to the cumulative number of recovered cases R(t) in log scale with N = 1000, β = 1/2, γ = 1/4, and I(0) = 5.

Figure 2. The average scaled bias for case D for estimates of β, γ, and R0 = β/γ as a function of the population size N from SIR epidemics (β = 1/2, γ = 1/4, N = 1000, and I(0) = 5) using two different estimation methods with observation error through a Gamma error structure with variance σ2 = kμ for two values of k = 2 (solid) and 5(dashed) plus model error such that β varies deterministically, equal to a constant until day 15, then exponentially decaying to a new value, so β = β1 for t <= 15 and β = β2 + (β1 − β2) exp (−0.2 × (t − 15)) for t > 15.