Quantifying Ebola – I

With the ongoing Ebola outbreak, what seems to me missing in the discourse is some quantification.  There have been a number of detailed epidemiologic models in the literature and on the web.  For example:




These models all use some version of the SEIR model, depicted in the graphic below 






In this approach, a human population is considered to be either susceptible, exposed, infectious or recovered, with a progression as indicated by the arrows (with each arrow representing a rate of “flow” between one state and another.  There may be other paths as well (for example, conversion of some of the recovered to once again susceptible).

If we focus on the first two boxes, susceptible (S) and exposed (E), then using a deterministic form and homogenous populations, two differential equations for S and E can be written following Daley and Gani (1)








The initial conversion to exposed is therefore the term “beta” x S x I, where “beta” is the rate at which new exposures occur from interactions between infectious (I) and susceptible (S) populations.

It seems to me that it is useful and interesting to regard “beta” as being the product of two terms:

beta = b1 x b2

where b1 is the frequency (#/unit time) of contacts between susceptible and infectious individuals and b2 is the probability that a single contact would result in conversion to an “exposed” state.  Note that in standard epidemiological parlance, the term “exposed” refers to an individual who will become but is not yet infectious.  The term b1 would presumably be driven principally by the nature and velocity of population mixing (which could be reduced by isolation of the exposed from the susceptible).

In this terminology, b2 is essentially the risk probability (frequently termed the probability of infection) which is widely used in quantitative microbial risk assessment (2).  To obtain b2, we essentially need two sets of information:

  • the average dose (d) of infectious agent transferred in the interaction between susceptible and infectious individuals during an interaction
  • a dose-response relationship giving the relation between dose and probability of infection.  the following two forms are widely used (see reference 2):







The first equation is the exponential dose response relationship (with an unknown parameter k) and the second is the approximate beta Poisson with unknown parameters N50 and alpha.  N50 is the median infectious dose.  As alpha in the second equation goes to infinity, the beta Poisson equation becomes the exponential.  

This analysis points to two types of data that are in need of quantification in the current Ebola outbreak.  First, what is the average dose transferred between an infectious person and a susceptible person during their contact?  This clearly will be a distribution depending on the nature and extent of contact.  Risk assessors are accustomed to incorporating and modeling various sources of uncertainty and variability (3).

Second, the dose response curve (parameters k or alpha and N50) need to be known.  I have not yet seen data (either in animal systems or humans) needed to construct such a curve, although we have many relationships for a variety of pathogens (4) .  In the context of weaponized aerosolized Ebola, an infectious dose (my interpretation of this is the median infectious dose) of 1-10 organisms is widely reported, e.g. (5), however the infectivity by what are thought to be the most relevant large droplet routes (6) (which might have different portals of entry) does not appear to have been established.

So it seems to me that there are two important research needs identified for quantification of the outbreak: (1) quantification or estimation of the transferred dose upon different types of contacts between infectious and susceptible individuals; and (2) estimation of the dose response parameters for the most important routes of exposure.


1.Daley, D. and J. Gani, Epidemic Modeling: An Introduction NY, New York1999: Cambridge University Press.

2.Haas, C.N., J.B. Rose and C.P. Gerba, Quantitative Microbial Risk Assessment. 2nd ed2014, New York: John Wiley.

3. http://www.fao.org/docrep/008/ae922e/ae922e08.htm

4. http://qmrawiki.msu.edu/index.php?title=Dose_Response

5. http://www.phac-aspc.gc.ca/lab-bio/res/psds-ftss/ebola-eng.php

6. http://virologydownunder.blogspot.com/2014/08/ebola-virus-may-be-spread-by-droplets.html