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. 2014 Feb;104(2):e32-41.
doi: 10.2105/AJPH.2013.301704. Epub 2013 Dec 12.

Unraveling R0: Considerations for Public Health Applications

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Unraveling R0: Considerations for Public Health Applications

Benjamin Ridenhour et al. Am J Public Health. .
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Abstract

We assessed public health use of R0, the basic reproduction number, which estimates the speed at which a disease is capable of spreading in a population. These estimates are of great public health interest, as evidenced during the 2009 influenza A (H1N1) virus pandemic. We reviewed methods commonly used to estimate R0, examined their practical utility, and assessed how estimates of this epidemiological parameter can inform mitigation strategy decisions. In isolation, R0 is a suboptimal gauge of infectious disease dynamics across populations; other disease parameters may provide more useful information. Nonetheless, estimation of R0 for a particular population is useful for understanding transmission in the study population. Considered in the context of other epidemiologically important parameters, the value of R0 may lie in better understanding an outbreak and in preparing a public health response.

Figures

FIGURE 1—
FIGURE 1—
The number of publications regarding infectious disease and mathematical modeling as reported by Web of Science. Note. The figure was produced by searching Web of Science on the terms “reproduction number” or “reproductive number” and limiting the results to the fields of infectious diseases, mathematical computational biology, and applied mathematics. Clearly, interest in research regarding the basic reproductive number has risen dramatically since the 1990s. The number of publications in this area currently appears to be growing exponentially.
FIGURE 2—
FIGURE 2—
A comparison of the dynamics of (a) an SIR model and (b) an SEIR model. Note. SEIR = susceptible–exposed–infected–recovered; SIR = susceptible–infected–recovered. These 2 models have identical values of R0 (specifically R0 = 1.5). Obviously, the SIR model and the SEIR model produce dramatically different dynamics. The epidemic predicted by the SIR model peaks earlier and has a higher peak incidence as well as shorter duration than the epidemic predicted by the SEIR model. These differences in dynamics simply reflect whether modelers (or other researchers) believe there is a latent period for a virus. The parameters used for these plots were N = 1000; β = 0.1; γ = 0.0667; ν = 0.1.
FIGURE 3—
FIGURE 3—
Attack rate as predicted by R0 based on simple models. Note. At least for simple models—such as the susceptible–infected–recovered (SIR) and susceptible–exposed–infected–recovered (SEIR) models discussed in this article—the basic reproductive number of an epidemic offers insight into the overall attack rate. However, estimation of R0 often results in broad confidence intervals. Shown are the ranges for the 1918 Spanish influenza pandemic as well as the 2009 H1N1 pandemic; these ranges provide little confidence for the predicted attack rate. This problem is exacerbated at lower values of R0 because of the asymptotic dependence of attack rate on formula image near the y-axis (R0 = 1).
FIGURE 4—
FIGURE 4—
The dependence of epidemic duration on (a) transmission rate in an SIR model and (b) latency period in an SEIR model. Note. SEIR = susceptible–exposed–infected–recovered; SIR = susceptible–infected–recovered. The duration of the epidemic was measured as the time period between a 5% cumulative disease incidence and a 95% cumulative disease incidence. The duration of the epidemic does not depend on the population size (not shown) for either an SIR model or an SEIR model. For the SIR model, as the transmission rate and recovery rate (not shown: γ = β/1.5 for R0 = 1.5) increase, the duration of the epidemic decreases. For the SEIR model (R0 = 1.5), the latency period changes the expected duration of an epidemic; as the latency period decreases, the duration of the epidemic also decreases and converges on the SIR model. Note also that as the transmission rate declines, the dependence on the latency period diminishes.
FIGURE 5—
FIGURE 5—
The dependence of system dynamics on N, β, and γ. Note. The results shown are for a susceptible–infected–recovered (SIR) system with a fixed R0 = 1.5. First, by comparing the black (N = 1×106) and gray (N = 1×103) lines we can see that population size has pronounced effects on the time to certain epidemic landmarks. Landmarks chosen here are time to peak incidence (solid lines), time to 5% cumulative incidence (lower dashed gray and black lines), and time to 95% cumulative incidence (upper dashed gray and black lines). We can see that for a population of 1×103, an epidemic would reach its 95% cumulative incidence before an epidemic occurring in a population of 1×106 reaching its 5% cumulative incidence level. This figure also illustrates the dependence of dynamics on the actual values for the transmission rate (x-axis) and the recovery rate (not shown, for the figure γ = β/1.5, thus forcing R0 = 1.5). As the transmission rate and recovery rate increase, the time for the epidemic to run its course is abbreviated.
FIGURE 6—
FIGURE 6—
The dependence of time to peak incidence on the transmission rate, recovery rate, and basic reproductive ratio. Note. This is a contour plot of a 3-dimensional surface based on the time to peak incidence for a susceptible–infected–recovered (SIR) model in a population of 1×106. By following the ray corresponding to a particular R0 value (e.g., R0 = 2.5), we can see how the time to peak incidence varies for that basic reproductive number. We can see that for large values of R0 (e.g., 5), the duration is more variable as β and γ are altered (i.e., its ray crosses more contours). In fact, for R0 = 5, the time to peak incidence varies from a little as e time steps to nearly e time steps (i.e., 5 log orders of magnitude).

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