Early efforts in modeling the incubation period of infectious diseases with an acute course of illness - PubMed (original) (raw)

Early efforts in modeling the incubation period of infectious diseases with an acute course of illness

Hiroshi Nishiura. Emerg Themes Epidemiol. 2007.

Abstract

The incubation period of infectious diseases, the time from infection with a microorganism to onset of disease, is directly relevant to prevention and control. Since explicit models of the incubation period enhance our understanding of the spread of disease, previous classic studies were revisited, focusing on the modeling methods employed and paying particular attention to relatively unknown historical efforts. The earliest study on the incubation period of pandemic influenza was published in 1919, providing estimates of the incubation period of Spanish flu using the daily incidence on ships departing from several ports in Australia. Although the study explicitly dealt with an unknown time of exposure, the assumed periods of exposure, which had an equal probability of infection, were too long, and thus, likely resulted in slight underestimates of the incubation period. After the suggestion that the incubation period follows lognormal distribution, Japanese epidemiologists extended this assumption to estimates of the time of exposure during a point source outbreak. Although the reason why the incubation period of acute infectious diseases tends to reveal a right-skewed distribution has been explored several times, the validity of the lognormal assumption is yet to be fully clarified. At present, various different distributions are assumed, and the lack of validity in assuming lognormal distribution is particularly apparent in the case of slowly progressing diseases. The present paper indicates that (1) analysis using well-defined short periods of exposure with appropriate statistical methods is critical when the exact time of exposure is unknown, and (2) when assuming a specific distribution for the incubation period, comparisons using different distributions are needed in addition to estimations using different datasets, analyses of the determinants of incubation period, and an understanding of the underlying disease mechanisms.

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Figures

Figure 1

Figure 1

The relationship between the incubation period and observed onset of influenza after departure from Australia, 1918–19. Daily frequencies of influenza onset were observed after departure. Those who developed symptoms on board were assumed to have experienced exposure before departure. Since the time of exposure was difficult to identify explicitly, it was necessary to consider all possible times of exposure before departure. Asymptomatic infections and potential secondary transmissions on board were ignored. See supporting material for original descriptions and original data [36, 42, 43].

Figure 2

Figure 2

The incubation period distribution of typhoid fever in Old Salem Chautauqua, 1916, fitted to Pearson's Type I distribution. The incubation period started at an assumed time of exposure due to a flood that occurred 4 days before closing the water supply to Chautauqua. Since there were 4 possible days of exposure to contaminated water, the original study used the mid-point as a single time point of exposure. See [46] for the original descriptions.

Figure 3

Figure 3

The incubation period distributions of measles (A and B) and poliomyelitis (C and D) fitted to lognormal distributions. A &C) Observed frequencies (bars) are compared with predicted frequencies (solid line) based on the maximum likelihood method assuming lognormal distribution. The ends of the box represent the 25th and 75th quantiles (i.e., quartiles), and the line across the middle of the box identifies the median sample value. The means diamond indicates the sample mean and 95% confidence interval. The whiskers extending from both ends show additional quantiles (5th, 10th, 90th and 95th) on the response axis (note: for poliomyelitis (C), some quantiles are overlapping, and therefore only the 90th quantile is displayed). B & D) Lognormal quantile plots of the incubation periods. The diagonal reference lines show the line of fit and the two dashed lines denote confidence limits of 95% equal precision bounds with a = 0.001 and b = 0.99. See [54,55] for original data.

Figure 4

Figure 4

A method for estimating the time of exposure during a point source outbreak. The horizontal axis shows the time since exposure and the distribution the frequency of cases according to the time of onset. The vertical dashed line is the median incubation period observed x days after exposure. The remaining two vertical lines indicate the times when fractions α and 1-α of cases developed the disease. The intervals between the median and other two vertical lines represent a and b, respectively. The illustration was drawn by the author with reference to [59].

Figure 5

Figure 5

Comparison of the quantile plots for the incubation period distributions of smallpox assuming (A) lognormal and (B) gamma distribution. The diagonal reference lines show the line of fit. The maximum likelihood estimates of the mean (μ) and standard deviation (σ) for lognormal distribution were 2.47 (95% CI: 29.1, 38.6) and 0.36 (0.31, 0.41), respectively. The shape (α) and scale (β) parameters for gamma distribution were estimated as 33.6 (95% CI: 29.1, 38.6) and 0.36 (0.31, 0.41), respectively. See [96] for detailed descriptions.

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