Early Real-Time Estimation of the Basic Reproduction Number of Emerging Infectious Diseases

When an infectious disease strikes a population, the number of newly reported cases is often the only available information during the early stages of the outbreak. An important goal of early outbreak analysis is to obtain a reliable estimate for the basic reproduction number, ${\mathcal{R}}_0$. Over the past few years, infectious disease epidemic processes have gained attention from the physics community. Much of the work to date, however, has focused on the analysis of an epidemic process in which the disease has already spread widely within a population; conversely, very little attention has been paid, in the physics literature or elsewhere, to formulating the initial phase of an outbreak. Careful analysis of this phase is especially important as it could provide policymakers with insight on how to effectively control an epidemic in its initial stage. We present a novel method, based on the principles of network theory, that enables us to obtain a reliable real-time estimate of the basic reproduction number at an early stage of an outbreak. Our method takes into account the possibility that the infectious period has a wide distribution and that the degree distribution of the underlying contact network is heterogeneous. We validate our analytical framework with numerical simulations.

Popular Summary

Over the past few years, physicists have contributed to modeling infectious disease epidemics, resulting in innovative public health measures for confronting and controlling the threat. Many of these studies, however, have focused on epidemic processes in which the disease has already spread widely within a population. Little attention has been paid in the physics literature or elsewhere to modeling the initial phase of an outbreak. We present a new methodology to estimate the key parameter that controls disease spread during the early stages of an epidemic, using case-notification data and the structure of contact networks in a population.

Successful control of an epidemic at an early stage hinges on the timely knowledge of the basic reproduction number $R_0$. This quantity is generally defined as the expected number of newly infected cases arising from one infectious individual during the entire period of his or her infection, in a totally susceptible population. Establishing a theoretical framework capable of estimating this value, in real time, using real-life disease-notification data has been a major challenge for scientists in recent years.

Drawing upon concepts from network theory and stochastic processes, our paper details an analytical framework that can be used to directly confront these issues. We have carried out simulations of the spread of an infectious disease in an idealized contact network, and show that the methodology correctly estimates $R_0$ even when only limited information about the disease and the structure of the underlying contact network is available. We further show that the methodology can be used to estimate certain properties of a contact network if some information about the biological characteristics of the disease are available. Our hope is that this contribution from the physics community is another tool that can be used to optimally mitigate the spread of infectious diseases.

Figure 1

Hypothetical infectivity functions ${\lambda }_i(\tau )$. They show the general infectivity patterns that can occur, varying by complexity of the disease. The top left panel shows the infectivity function of a very generic disease, the top right panel shows the infectivity function of an HIV type disease, the bottom left displays the infectivity function of any recurrent disease such as chicken pox, and finally, the bottom right panel exhibits the infectivity function for an influenza type disease.

Figure 2

Two hypothetical realizations of an epidemic process on a network. The left, middle, and right sections represent stochastic, exponential, and declining regimes, respectively.

Figure 3

The logarithm (base $e$) of the rate of new infections $\ln \tilde{J}(t)$ for a binomial network with $z_1=5$ (left panel) and for an exponential network with $\kappa =4$ (right panel), with ${\lambda }_i=0.12771$ and ${\lambda }_r=0.25$. Two independent epidemic realizations are shown in each panel (green and blue). The solid red line shows the tangent of $\ln [\tilde{J}(t)]$ in the exponential regime.

Figure 4

This figure illustrates schematically the dependency of the rate of new infections, $\tilde{J}(t)$ (blue curve), on its past values. Only a fraction of the cases infected at time $t-\tau$, $\tilde{J}(t-\tau )\delta t$, contributes to the infections at time $t$, ${\tilde{J}}^i(\tau ,t)\delta t=\tilde{J}(t-\tau )\delta t\Psi (\tau )$ (red curve); the rest, ${\tilde{J}}^r(\tau ,t)\delta t=\tilde{J}(t-\tau )\delta t[1-\Psi (\tau )]$, have already been removed.

Figure 5

Number of infectious (left panel) and removed individuals (right panel) as a function of time for the binomial ($z_1=5$, ${\lambda }_i=0.12771$, and ${\lambda }_r=0.25$) and exponential ($\kappa =4$) networks. The solid curves come from the evaluation of Eqs. (20, 21), and the symbols come from the direct counting of the infectious and removed individuals at any given time for a specific realization of the process.

Figure 6

Dependency of the rate of new infections at time $t^{\prime }$, $\tilde{J}(t^{\prime })$, on the rate of new infections at time $t^{\prime }-\tau$, ${\tilde{J}}^i(\tau ,t^{\prime })$. The blue and red curves show the rates of new infections by time $t^{\prime }$ and $t$, respectively. The green and yellow curves show the fraction of those that remained infectious for at least $\tau$ and $\tau +t-t^{\prime }$ units of time, respectively.

Figure 7

The estimated basic reproduction number of removed individuals for the binomial (left panel: $z_1=5$, ${\lambda }_i=0.12771$, and ${\lambda }_r=0.25$) and exponential (right panel: $\kappa =4$) networks in terms of the number of removed individuals. The right-hand and left-hand sides represent the right- and left-hand sides of Eq. (27), respectively. The solid curve represents the “analytical” calculation of Eq. (27) and the symbols show the “exact” values of the right-hand and left-hand sides for this specific realization of the epidemic process.

Figure 8

Top panels: Estimated basic reproduction number for the binomial (left panel: $z_1=5$, ${\lambda }_i=0.12771$, and ${\lambda }_r=0.25$) and exponential (right panel: $\kappa =4$) networks in terms of the number of removed individuals. The red line corresponds to the real value of the basic reproduction number. The blue line shows the estimated ${\mathcal{R}}_0$ from our methodology. For comparison, the green line shows the ${\mathcal{R}}_0$ estimates obtained from Eq. (18) and the estimation of the growth rate during the exponential phase ($\alpha$). Bottom panels: The number of removed individuals by time $t$ (logarithmic scale, base 10). The red line shows the number of removed individuals from a realization of the epidemic process. For comparison, the black line shows the theoretical exponential phase of the epidemic process.

Figure 9

The estimated basic reproduction number for the binomial (left panel: $z_1=5$, ${\lambda }_i=0.12771$, and ${\lambda }_r=0.25$) and exponential (right panel: $\kappa =4$) networks in terms of the number of removed individuals. The red line corresponds to the real value of the basic reproduction number. The blue area shows the variation of the estimated value for a hundred different realizations.

Figure 10

Estimates for the value of $Z_x$ for the binomial (left) and exponential (right) networks, when ${\lambda }_i$ and ${\lambda }_r$ are known. The red line and green curves correspond to the average and the estimated excess degree.

Figure 11

The estimated basic reproduction number for the binomial [left panel: $z_1=5$, ${\lambda }_i=\tau /(1+{\tau }^2)(1+0.5{\tau }^2)$, and ${\lambda }_r=2\tau /(1+{\tau }^2)$] and exponential (right panel: $\kappa =4$) networks in terms of the number of removed individuals. The red line and green curves correspond to the real and the estimated value. The blue line shows the estimates if we incorrectly assume that ${\lambda }_i$ is constant. The pink line shows the corresponding estimates if we incorrectly assume that both ${\lambda }_i$ and ${\lambda }_r$ are constant.

Figure 12

The estimated basic reproduction number for the binomial (left panel: $z_1=5$, ${\lambda }_i=0.12771$, and ${\lambda }_r=0.25$) and exponential (right panel: $\kappa =4$) networks in terms of the number of removed individuals. The green curve and red line correspond to the estimated (with the correct excess degree) and real value of the basic reproduction number. The blue area shows the variation of the estimated value with respect to change of the excess degree.