(Ayşe Hümeyra Bilge, Funda Samanlıoğlu and Önder Ergönül)

The susceptible-infected-removed (SIR) and the susceptible-exposed-infected-removed (SEIR) epidemic models with constant parameters are adequate for describing the time evolution of seasonal diseases for which available data usually consist of fatality reports. The problems associated with the determination of system parameters starts with the inference of the number of removed individuals from fatality data, because the infection to death period may depend on health care factors. Then, one encounters numerical sensitivity problems for the determination of the system parameters from a correct but noisy representative of the number of removed individuals. Finally as the available data is necessarily a normalized one, the models fitting this data may not be unique. We prove that the parameters of the (SEIR) model cannot be determined from the knowledge of a normalized curve of “Removed” individuals and we show that the proportion of removed individuals, R(t), is invariant under the interchange of the incubation and infection periods and corresponding scalings of the contact rate. On the other hand we prove that the SIR model fitting a normalized curve of removed individuals is unique and we give an implicit relation for the system parameters in terms of the values of Rm/Rf and R′m/Rf, where Rf is the steady state value of R(t) and Rm and R′m are the values of R(t) and its derivative at the inflection point tm of R(t). We use these implicit relations to provide a robust method for the estimation of the system parameters and we apply this procedure to the fatality data for the H1N1 epidemic in the Czech Republic during 2009. We finally discuss the inference of the number of removed individuals from observational data, using a clinical survey conducted at major hospitals in Istanbul, Turkey, during 2009 H1N1 epidemic.

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(Funda Samanlıoğlu, Ayşe Hümeyra Bilge)

(Ali Demirci, Ayşe Peker Dobie, Ayşe Hümeyra Bilge, Semra Ahmetolan)

The data of the 2009 A(H1N1) epidemic in Istanbul, Turkey is unique in terms of the collected data, which include not only the hospitalization but also the fatality information recorded during the pandemic. The analysis of this data displayed an unexpected time shift between the hospital referrals and fatalities. This time shift, which does not conform to the SIR and SEIR models, was explained by multi-stage SIR and SEIR models [21]. In this study we prove that the delay for these models is half of the infectious period within a quadratic approximation, and we determine the epidemic parameters R0, T and I0 of the 2009 A(H1N1) Istanbul and Netherlands epidemics.These epidemic parameters were estimated by comparing the normalized cumulative fatality data with the solutions of the SIR model. Two different error criteria, the L2 norms of the error over the whole observation period and over the initial portion of the data, were used in order to obtain the best-fitting models. It was observed that, with respect to both criteria, the parameters of "good" models were agglomerated along a line in the T-R0 plane, instead of being scattered uniformly around a "best" model. As this fact indicates the existence of a nearly invariant quantity, interval estimates for the parameters were given. As the initial phase of the epidemics were less influenced by the effects of medical interventions, the error norm based on the initial portion of the data was preferred. However, the presented parameter ranges are well out of the range for the usual influenza epidemic parameter values. To confirm our observations on the Istanbul data, the same error criteria were also used for the 2009 A(H1N1) epidemic for the Netherlands, which has a similar population density as in Istanbul. As in the Istanbul case, the parameter ranges do not match the usual influenza epidemic parameter values.

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