Theoretical perspectives on the infectiousness of Ebola virus disease
© Nishiura and Chowell; licensee BioMed Central. 2015
Received: 20 November 2014
Accepted: 22 December 2014
Published: 6 January 2015
Ebola virus disease (EVD) has generated a large epidemic in West Africa since December 2013. This mini-review is aimed to clarify and illustrate different theoretical concepts of infectiousness in order to compare the infectiousness across different communicable diseases including EVD.
We employed a transmission model that rests on the renewal process in order to clarify theoretical concepts on infectiousness, namely the basic reproduction number, R 0, which measures the infectiousness per generation of cases, the force of infection (i.e. the hazard rate of infection), the intrinsic growth rate (i.e. infectiousness per unit time) and the per-contact probability of infection (i.e. infectiousness per effective contact).
Whereas R 0 of EVD is similar to that of influenza, the growth rate (i.e. the measure of infectiousness per unit time) for EVD was shown to be comparatively lower than that for influenza. Moreover, EVD and influenza differ in mode of transmission whereby the probability of transmission per contact is lower for EVD compared to that of influenza.
The slow spread of EVD associated with the need for physical contact with body fluids supports social distancing measures including contact tracing and case isolation. Descriptions and interpretations of different variables quantifying infectiousness need to be used clearly and objectively in the scientific community and for risk communication.
An epidemic of Ebola virus disease (EVD) centred in three West African countries has been ongoing since December 2013, with limited international spread to other countries in Africa, Europe and the USA . It is likely that the duration of this EVD epidemic, associated with a high case fatality risk (CFR) estimated at ~70% [2, 3], will extend well into 2015. To investigate the ongoing EVD transmission dynamics and consider a range of possible countermeasures, it is vital to understand the natural history and epidemiological dynamics of this disease.
Owing to the rapid progression of the EVD epidemic in West Africa, attempts have been made to clarify the fundamental epidemiological characteristics of EVD [1, 2, 4]. For instance, several studies have reported statistical estimates of the reproduction number, i.e., the average number of secondary cases generated by a single primary case, as a measure of the transmission potential of EVD [2, 5–12]. Despite substantial progress, it remains unclear how measures of infectiousness (or the transmissibility) of EVD should be communicated to the public and interpreted in light of the set of control interventions that could be considered in practical settings. Hence, the purpose of this mini-review is to comprehensively classify different theoretical aspects of infectiousness using a basic transmission model formulated in terms of a renewal process. This approach allows us to compare different measures of infectiousness across different communicable diseases and design possible countermeasures.
and it can be interpreted as the number of secondary cases produced by a single primary case throughout its entire course of infection in a completely susceptible population. Although the concept of R 0 is well-known, it is important to note from (2) that R 0 results from the integration over all infection-ages. It is well known that the mathematical definition of R 0 in a heterogeneously mixing population is described by using the multivariate version of (1) and the next-generation matrix that maps secondary transmissions between and within sub-populations. R 0 is defined as the largest eigenvalue of this matrix [15, 16]. Similarly, the definition of R 0 can be adapted to the situation of periodic infectious diseases by handling the seasonal dynamics using a vector and employing Floquet theory (see e.g., ).
Although R 0 is clearly a dimensionless quantity, the conceptual interpretation from the renewal process (1) permits us to regard R 0 as the average number of infected cases produced “per generation”. For this reason, R 0 could also be referred to as the basic reproductive ratio, as it could be calculated as the ratio of secondary to primary cases.
which yields a measure of the risk of infection in a susceptible population. The force of infection can be interpreted as the hazard of infection in statistical sense -- the rate at which susceptible individuals are infected . In the classical Kermack and McKendrick epidemic model, λ(t) is modelled as proportional to the disease prevalence . The force of infection is useful for the analysis of incidence data.
Comparison of three communicable diseases
The mode of transmission greatly differs for three diseases considered (Table 1). Measles is transmitted efficiently through the air while the transmission of influenza mostly occurs via droplet although airborne transmission is also possible in a confined setting . In contrast, transmission of EVD is greatly constrained to physical contacts via body fluids . Despite the differences in the mode of transmission for these diseases, it is important to note that the estimates of R 0 for H1N1-2009 and EVD are not too different (Table 1). Does that indicate that influenza (H1N1-2009) and Ebola are similarly infectious?
Thus, based on the infectiousness as measured by the growth rate of cases per unit time, it is very encouraging that EVD is far less dispersible than influenza. Although static countermeasures (e.g. mass vaccination at a certain age) can be planned using R 0, the feasibility to deploy dynamic countermeasures, such as contact tracing and case isolation rests on the competition between the growth of cases and public health control, and in this context, the key parameter of infectiousness to assess the feasibility of control interventions is the intrinsic growth rate of cases.
Per contact risk of infection
As mentioned above, R 0 for EVD is similar to that of influenza. Nevertheless, the infectious period, modelled by Γ(s) for EVD is longer than that of influenza. Assuming an identical contact rate, c, between EVD and influenza, equation (11) indicates that the per-contact probability of infection for EVD is smaller than that for influenza.
We have comparatively discussed concepts of infectiousness for EVD in relation to other communicable diseases from a mathematical modelling point of view. The measure of infectiousness per generation of cases is R 0. R 0 offers a threshold principle and we have discussed that this measure is important for planning some static countermeasures such as mass vaccination. Based on R 0, the overall infectiousness of EVD may be perceived to be similar to that of influenza. Nevertheless, the infectiousness per unit time for EVD was shown to be comparatively lower than influenza. The slow spread of EVD supports social distancing measures including contact tracing and case isolation. Moreover, the per-contact probability of infection for EVD is lower than that for influenza, and the mode of transmission also differs. These findings should also be regarded as encouraging news for healthcare workers who would have to have unavoidable and protected contact with EVD cases. In summary, there is a need for the use of clear and objective descriptions and interpretations of different variables quantifying infectiousness among the scientific community and for risk communication.
- R 0 :
The basic reproduction number
Ebola virus disease
Case fatality risk.
HN received funding support from the Japan Science and Technology Agency (JST) CREST program, RISTEX program for Science of Science, Technology and Innovation Policy, and St Luke’s Life Science Institute Research Grant for Clinical Epidemiology Research 2014. GC acknowledges financial support from the NSF grant 1414374 as part of the joint NSF-NIH-USDA Ecology and Evolution of Infectious Diseases program, UK Biotechnology and Biological Sciences Research Council grant BB/M008894/1, and the Division of International Epidemiology and Population Studies, The Fogarty International Center, US National Institutes of Health.
- Chowell G, Nishiura H: Transmission dynamics and control of Ebola virus disease (EVD): a review. BMC Med 2014, 12:196.View ArticlePubMed CentralPubMedGoogle Scholar
- WHO Ebola Response Team: Ebola virus disease in West Africa–the first 9 months of the epidemic and forward projections. N Engl J Med 2014, 371:1481–1495.View ArticlePubMed CentralGoogle Scholar
- Kucharski AJ, Edmunds WJ: Case fatality rate for Ebola virus disease in west Africa. Lancet 2014, 384:1260.View ArticlePubMedGoogle Scholar
- Incident Management System Ebola Epidemiology Team, CDC; Ministries of Health of Guinea, Sierra Leone, Liberia, Nigeria, and Senegal; Viral Special Pathogens Branch, National Center for Emerging and Zoonotic Infectious Diseases, CDC: Ebola virus disease outbreak - West Africa, September 2014. Morb Mortal Wkly Rep 2014, 63:865–866.Google Scholar
- Nishiura H, Chowell G: Early transmission dynamics of Ebola virus disease (EVD), West Africa, March to August 2014. Euro Surveill 2014., 19: Available online: http://www.eurosurveillance.org/ViewArticle.aspx?ArticleId=20894 Google Scholar
- Althaus CL: Estimating the reproduction number of Zaire ebolavirus (EBOV) during the 2014 outbreak in West Africa. PLOS Curr Outbreaks 2014. Sep 2. Edition 1. doi:10.1371/currents.outbreaks.91afb5e0f279e7f29e7056095255b288Google Scholar
- Fisman D, Khoo E, Tuite A: Early epidemic dynamics of the West African 2014 Ebola outbreak: estimates derived with a simple two-parameter model. PLOS Curr Outbreaks 2014. Sep 8. Edition 1. doi:10.1371/currents.outbreaks.89c0d3783f36958d96ebbae97348d571Google Scholar
- Gomes MF, Piontti AP, Rossi L, Chao D, Longini I, Halloran ME, et al.: Assessing the international spreading risk associated with the 2014 West African Ebola outbreak. PLOS Curr Outbreaks 2014. Sep 2. Edition 1. doi:10.1371/currents.outbreaks.cd818f63d40e24aef769dda7df9e0da5Google Scholar
- Towers S, Patterson-Lomba O, Castillo-Chavez C: Temporal variations in the effective reproduction number of the 2014 West Africa Ebola outbreak. PLOS Curr Outbreaks 2014. Sep 18. Edition 1. doi:10.1371/currents.outbreaks.9e4c4294ec8ce1adad283172b16bc908Google Scholar
- Yamin D, Gertler S, Ndeffo-Mbah ML, Skrip LA, Fallah M, Nyenswah TG, et al.: Effect of Ebola progression on transmission and control in Liberia. Ann Intern Med 2014. doi:10.7326/M14–2255. in pressGoogle Scholar
- Fasina F, Shittu A, Lazarus D, Tomori O, Simonsen L, Viboud C, et al.: Transmission dynamics and control of Ebola virus disease outbreak in Nigeria, July to September 2014. Eur Surveill 2014., 19: Available online: http://www.eurosurveillance.org/ViewArticle.aspx?ArticleId=20920 Google Scholar
- Althause CL, Gsteiger S, Low N: Ebola virus disease outbreak in Nigeria: lessons to learn. Peer J PrePrints 2014, 2:e569v1.Google Scholar
- Diekmann O, Heesterbeek JAP: Mathematical epidemiology of infectious diseases: model building, analysis and interpretation. Chichester: Wiley; 2000.Google Scholar
- Nishiura H: Time variations in the generation time of an infectious disease: implications for sampling to appropriately quantify transmission potential. Math Biosci Eng 2010, 7:851–869.View ArticlePubMedGoogle Scholar
- Diekmann O, Heesterbeek JA, Metz JA: On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J Math Biol 1990, 28:365–382.View ArticlePubMedGoogle Scholar
- Nishiura H, Chowell G, Safan M, Castillo-Chavez C: Pros and cons of estimating the reproduction number from early epidemic growth rate of influenza A (H1N1) 2009. Theor Biol Med Model 2010, 7:1. 10.1186/1742-4682-7-1View ArticlePubMed CentralPubMedGoogle Scholar
- Bacaër N, Ait Dads el H: On the biological interpretation of a definition for the parameter R 0 in periodic population models. J Math Biol 2012, 65:601–621. 10.1007/s00285-011-0479-4View ArticlePubMedGoogle Scholar
- Farrington CP: Modelling forces of infection for measles, mumps and rubella. Stat Med 1990, 9:953–967. 10.1002/sim.4780090811View ArticlePubMedGoogle Scholar
- Fine PE: Herd immunity: history, theory, practice. Epidemiol Rev 1993, 15:265–302.PubMedGoogle Scholar
- Boëlle PY, Ansart S, Cori A, Valleron AJ: Transmission parameters of the A/H1N1 (2009) influenza virus pandemic: a review. Influenza Other Respir Viruses 2011, 5:306–316. 10.1111/j.1750-2659.2011.00234.xView ArticlePubMedGoogle Scholar
- Inaba H, Nishiura H: The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model. Math Biosci 2008, 216:77–89. 10.1016/j.mbs.2008.08.005View ArticlePubMedGoogle Scholar
- Eichner M, Dowell SF, Firese N: Incubation period of Ebola hemorrhagic virus subtype Zaire. Osong Public Health Res Persptect 2011, 2:3–7. 10.1016/j.phrp.2011.04.001View ArticleGoogle Scholar
- Cowling BJ, Ip DK, Fang VJ, Suntarattiwong P, Olsen SJ, Levy J, et al.: Aerosol transmission is an important mode of influenza A virus spread. Nat Commun 2013, 4:1935.View ArticlePubMed CentralPubMedGoogle Scholar
- Keyfitz BL, Keyfitz N: The McKendrick partial differential equation and its uses in epidemiology and population study. Math Comp Model 1997, 26:1–9.View ArticleGoogle Scholar
- Wallinga J, Lipsitch M: How generation intervals shape the relationship between growth rates and reproductive numbers. Proc R Soc Lond Ser B 2007, 274:599–604. 10.1098/rspb.2006.3754View ArticleGoogle Scholar
- Roberts MG, Heesterbeek JA: Model-consistent estimation of the basic reproduction number from the incidence of an emerging infection. J Math Biol 2007, 55:803–816. 10.1007/s00285-007-0112-8View ArticlePubMed CentralPubMedGoogle Scholar
- Eichner M, Dietz K: Transmission potential of smallpox: estimates based on detailed data from an outbreak. Am J Epidemiol 2003, 158:110–117. 10.1093/aje/kwg103View ArticlePubMedGoogle Scholar
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