Theoretical perspectives on the infectiousness of Ebola virus disease
 Hiroshi Nishiura^{1, 2}Email author and
 Gerardo Chowell^{3, 4, 5}
DOI: 10.1186/17424682121
© Nishiura and Chowell; licensee BioMed Central. 2015
Received: 20 November 2014
Accepted: 22 December 2014
Published: 6 January 2015
Abstract
Background
Ebola virus disease (EVD) has generated a large epidemic in West Africa since December 2013. This minireview is aimed to clarify and illustrate different theoretical concepts of infectiousness in order to compare the infectiousness across different communicable diseases including EVD.
Methods
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 percontact probability of infection (i.e. infectiousness per effective contact).
Results
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.
Conclusions
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.
Background
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 [1]. 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 minireview 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.
Discussion
Renewal process
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 wellknown, it is important to note from (2) that R _{0} results from the integration over all infectionages. 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 nextgeneration matrix that maps secondary transmissions between and within subpopulations. 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., [17]).
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 [18]. In the classical Kermack and McKendrick epidemic model, λ(t) is modelled as proportional to the disease prevalence [13]. 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 [23]. In contrast, transmission of EVD is greatly constrained to physical contacts via body fluids [1]. Despite the differences in the mode of transmission for these diseases, it is important to note that the estimates of R _{0} for H1N12009 and EVD are not too different (Table 1). Does that indicate that influenza (H1N12009) 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 percontact probability of infection for EVD is smaller than that for influenza.
Conclusion
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 percontact 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.
Abbreviations
 R _{0} :

The basic reproduction number
 EVD:

Ebola virus disease
 CFR:

Case fatality risk.
Declarations
Acknowledgements
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 NSFNIHUSDA 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.
Authors’ Affiliations
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