Reconstructing influenza incidence by deconvolution of daily mortality time series

Abstract

We propose a mathematically straightforward method to infer the incidence curve of an epidemic from a recorded daily death curve and time-to-death distribution; the method is based on the Richardson-Lucy deconvolution scheme from optics. We apply the method to reconstruct the incidence curves for the 1918 influenza epidemic in Philadelphia and New York State. The incidence curves are then used to estimate epidemiological quantities, such as daily reproductive numbers and infectivity ratios. We found that during a brief period before the official control measures were implemented in Philadelphia, the drop in the daily number of new infections due to an average infector was much larger than expected from the depletion of susceptibles during that period; this finding was subjected to extensive sensitivity analysis. Combining this with recorded evidence about public behavior, we conclude that public awareness and change in behavior is likely to have had a major role in the slowdown of the epidemic even in a city whose response to the 1918 influenza epidemic is considered to have been among the worst in the U.S.

Fig. 1.

The influenza epidemic in Philadelphia (Upper) and New York State (Lower) in 1918. Black is death curve. Blue is the initial condition for RL iterations (death curve, shifted back by nine days). Red is the estimated incidence (scaled by a factor of p, where p is the case fatality ratio), obtained from the first iteration for which the χ² value from Eq. 3 is below 1 (sixth iteration for Philadelphia and fourth iteration for New York State). Green are results of RL after 30 iterations.

Fig. 2.

Epidemic progression in Philadelphia (Upper) and New York State (Lower), for certain time periods. Red is deconvolved incidence (scaled), and black represents the estimated reproductive numbers.

Fig. 3.

Estimated infectivity ratios (black) vs. (scaled) incidence (red) in Philadephia, September 14–October 19, 1918.

Fig. 4.

Natural logarithm of the recorded daily number of deaths from pneumonia and influenza in Philadelphia, 1918.

Fig. 5.

Assumed proportions of deaths on each day since infection. Based on the Cook County Hospital data from ref. for time from symptom onset to death and the latent period distribution from ref. .