AUTHORS
Gerardo Chowell Lone Simonsen Cécile Viboud Yang Kuang
ABSTRACT
An unprecedented epidemic of Zaire ebolavirus (EBOV) has affected West Africa since approximately December 2013, with intense transmission on-going in Guinea, Sierra Leone and Liberia and increasingly important international repercussions. Mathematical models are proving instrumental to forecast the expected number of infections and deaths and quantify the intensity of interventions required to control transmission; however, calibrating mechanistic transmission models to an on-going outbreak is a challenging task owing to limited availability of epidemiological data and rapidly changing interventions. Here we project the trajectory of the EBOV epidemic in Liberia by fitting logistic growth models to the cumulative number of cases. Our model predictions align well with the latest epidemiological reports available as of October 23, and indicates that the exponential growth phase is over in Liberia, with an expected final attack rate of ~0.1-0.12%. Our results indicate that simple phenomenological models can provide complementary insights into the dynamics of an outbreak and capture early signs of changes in population behavior and interventions. In particular, our results underscore the need to treat the effective size of the susceptible population as a dynamic variable rather than a fixed quantity, due to reactive changes in transmission throughout the outbreak. We show that predictions from the logistic model are more variable in the earlier stages of an epidemic (such as the EBOV epidemics in Sierra Leone and Guinea). More research is warranted to compare the performances of mechanistic and phenomenological approaches for disease forecasts, before such predictions can be fully used by public health authorities.
FUNDING STATEMENT
CV and GC acknowledge the financial support from the Division of International Epidemiology and Population Studies, The Fogarty International Center, United States National Institutes of Health, funded in part by the Office of Pandemics and Emerging Threats at the United States Department of Health and Human Services. GC also acknowledges support from grant NSF grant 1414374 as part of the joint NSF-NIH-USDA Ecology and Evolution of Infectious Diseases programme, United Kingdom Biotechnology and Biological Sciences Research Council grant BB/M008894/1. We would like to acknowledge the Lundbeck Foundation and the Research And Policy for Infectious Disease Dynamics program (RAPIDD) of the United States Department of Homeland Security for sponsorship of LS.
COMMENTARY
An unprecedented epidemic of Zaire ebolavirus (EBOV) has affected West Africa since approximately December 2013, with intense transmission on-going in Guinea, Sierra Leone and Liberia and increasingly important international repercussions 1. As in previous international health crises such as the SARS epidemic and the 2009 influenza pandemic, mathematical models are proving instrumental to forecast the expected number of infections and deaths, quantify the intensity of interventions needed to achieve control (e.g., Ebola treatment centers, protective kits) and monitor the benefits of these interventions once they have been initiated. Calibrating mechanistic transmission models to the current Ebola outbreak is a challenging task however, complicated by underreporting of infections and deaths, population heterogeneity in infection risk 2, changes in population behavior and intervention strategies that blunt the transmission rate of the virus after the initial phase of the epidemic, and stochastic effects 3.
Here we build on a recent modeling study published by Lewnard et al. to contrast predictions derived from traditional mechanistic models and phenomenological approaches 4. Lewnard et al calibrated a mechanistic model of EBOV transmission to epidemiological reports of cases and deaths in the greater Monrovia area, the capital city of Liberia, during the early phase of the epidemic, June 14 to September 23, 2014. This study has produced by far the most dire outlook yet on the Ebola outbreak, suggesting that if nothing is done above and beyond current interventions, about 27% of the Monrovia population will be infected with EBOV before the epidemic runs its course towards the end of 2015. Lewnard et al’s model is based on six estimated parameters, including the basic reproduction number Ro, estimated to be particularly high (R0=2.4). The model showed signs of overestimating the trajectory of the epidemic by mid-October 2014 (Figure 1 in 4). In parallel, the Liberia national response team has reported that the EBOV outbreak has slowed considerably in recent weeks 10, before the heightened international response kicked in, probably as a result of social distancing measures, patient isolation and changes in funeral practices 5. Public health experts have confirmed the epidemic slowdown in at least two counties of Liberia in a series of recent reports10,14,15.
When few epidemiological data are available to fit mechanistic models, and health behavior and interventions are rapidly changing, standard model theory can quickly overshoot the trajectory of an epidemic. In this commentary, we propose a complementary method to project the temporal course of an epidemic, which is based on fitting a logistic growth curve to time series of cumulative case counts7 — a phenomenological approach that is less data hungry and does not make complex mechanistic assumptions about the transmission process. We illustrate this approach with data from the rapidly changing epidemic in Liberia, and provide additional estimates for Sierra Leone and Guinea.
The logistic growth model provides a statistical description of the outbreak trajectory, inspired from population biology7 . This model has two fitted parameters and follows the equation C’(t) = rC(1-C/K) where C’(t) is the rate of change in the cumulative number of total Ebola cases comprising suspected, probable and confirmed cases, “r” is the intrinsic growth rate ( per unit of time) and K is the final epidemic size (total no. of cases) 7,11.The logistic growth model assumes a saturation effect, so that as more EBOV cases accumulate, the size of the at-risk susceptible population decreases, implicitly capturing behavior changes (e.g., campaigns to educate the population on how to avoid contracting the disease), initiation of public health interventions, and increasing use of diagnostic tests to quickly identify and isolate EBOV cases 6,13. As a case in point of a rapidly changing outbreak landscape, empirical estimates of the reproduction number decreased from 12 during the first generation of the EBOV outbreak in Nigeria to less than one during the second and third generations due to rapid initiation of public health measures 16.
The logistic growth model fitted to the early phase of the Ebola epidemic in Liberia yields modest predictions of the final epidemic size of around 5,000 (95%CI: 4,574; 5,445) reported cases, corresponding to population attack rates of ~0.10%-0.12% (Figure 1, Liberia population size=4.4 millions). Specifically, the model is fitted by least-squares to 26 daily observations of cumulative case counts, starting from June-18, 2014 (red dots) and provides a good approximation of the course of the epidemic thus far in Liberia, including the 14 most recent daily case reports (blue dots in Figure 1). These predictions provide a more optimistic outlook on the epidemic in Liberia, which aligns better with the latest WHO epidemiological reports available as of October 23, 2014 (9 and Figure 1).
Although the logistic growth model is phenomenologic, it can provide implicit information about the transmission process as estimates of the reproduction number can be derived. The basic reproduction number Ro is given by Ro = exp(r*T), where r is the intrinsic growth rate (estimated as before) and T is the mean generation interval. Further, the effective reproduction number at time t is given by R(t)= (1-(C(t)/K)))*exp(r*T), where K is the mean final epidemic size (estimated as before). Logistic growth model estimates for the EBOV outbreak in Liberia support a decrease in transmission around September 6, 2014, with the reproduction number declining from Ro=2.4 at the beginning of the epidemic in August, to R=1.7 around September 6, to R=1.3 by October 1, assuming a generation interval of 15 days 8,12.
To check the validity of the logistic model predictions in different settings, we fitted the model to data from Sierra Leone and Guinea, tow countries with active Ebola transmission. Figures 2 and 3 show that final size predictions are consistently lower than those of a simple SEIR-type model in exponential growth mode, as the logistic curve captures a slowdown in transmission in both countries. We note however that final size predictions are more variable for Sierra Leone and Guinea as estimates keep increasing with inclusion of additional observations, suggesting that EBOV transmission is still high in these countries and the outbreak is far from over. Our estimates for the effective reproduction number on October 1, 2014 are 1.2 in Sierra Leone and 1.4 in Guinea. We also note that the logistic curve is unable to capture the biphasic nature of the outbreak in Guinea, resulting from the asynchronous dissemination of EBOV in different areas of the country. Hence here we only fit the most recent phase of the epidemic in Guinea