Beyond R0: heterogeneity in secondary infections and probabilistic epidemic forecasting

Abstract

The basic reproductive number, R₀, is one of the most common and most commonly misapplied numbers in public health. Often used to compare outbreaks and forecast pandemic risk, this single number belies the complexity that different epidemics can exhibit, even when they have the same R₀. Here, we reformulate and extend a classic result from random network theory to forecast the size of an epidemic using estimates of the distribution of secondary infections, leveraging both its average R₀ and the underlying heterogeneity. Importantly, epidemics with lower R₀ can be larger if they spread more homogeneously (and are therefore more robust to stochastic fluctuations). We illustrate the potential of this approach using different real epidemics with known estimates for R₀, heterogeneity and epidemic size in the absence of significant intervention. Further, we discuss the different ways in which this framework can be implemented in the data-scarce reality of emerging pathogens. Lastly, we demonstrate that without data on the heterogeneity in secondary infections for emerging infectious diseases like COVID-19 the uncertainty in outbreak size ranges dramatically. Taken together, our work highlights the critical need for contact tracing during emerging infectious disease outbreaks and the need to look beyond R₀.

Keywords: branching processes; complex networks; epidemiology.

Figure 1.

Final size of outbreaks with different average R₀ and heterogeneity k in the distribution of secondary cases. We use a negative binomial distribution of secondary cases and scan a realistic range of parameters. The range of parameters corresponding to estimates for COVID-19 based on a binomial negative distribution in large populations is highlighted by a red box (see [25] and table 1). Most importantly, with fixed average, the dispersion parameter is inversely proportional to the variance of the underlying distribution of secondary cases. The degree of freedom, p₀, is here set by setting the average number of infections around patient zero to be less than or equal to R₀. The Kermack–McKendrick solution would correspond to the limit k → ∞, and could be more appropriate in some dense and well-mixed settings.

Figure 2.

Using published estimates of R₀ and the dispersion parameter k, we estimated the total outbreak size for six different diseases using three versions of the network approach and compared them with the classic Kermack–McKendrick solution. The confidence intervals span the range of uncertainty reported for R₀ and k. The black markers show reported total outbreak sizes (total proportion of susceptible individuals infected) for each disease. For influenza, we report the estimated proportion of school-aged children infected. For COVID-19, we use tentative markers showing the range of attack rates measured in different contexts as there is currently no consensus for what constitutes a typical COVID-19 outbreak. We highlight though the differences between the final size estimates for COVID-19: most typify the observed over-dispersed nature of transmission, except for the outbreak on a fishing vessel (right side point) where contacts are more well mixed and thus better characterized by a Kermack–McKendrick transmission process. The red circles are the estimated proportion infected using the method developed by Kermack and McKendrick, i.e. equation (1.1). The other markers show the estimated proportion infected obtained with equation (3.17) under different assumptions about patient zero: the model described in the main text, which ensures that the expected number of secondary infections caused by patient zero is at most R₀ (blue squares); the same model but assuming p₀ = 0 such that no individuals have exactly zero contact (cyan stars); and a network version of [13], where G₀(x) ≡ G₁(x) such that patient zero is no different from subsequent patients (green triangles). See table 1 for data and additional information.