Dimensional analysis and the Ebola epidemic

2014 October 18
by Daniel Lakeland

To follow up on my discussion of the Ebola uncertainty. Let’s take a look at some very basic differential equations that we can use to get an idea of the factors that go into making up an epidemic.

First, we’ll model a population as having infected $$I$$ and uninfected $$U$$. Let’s also measure these populations as a fraction of the total population. So initially $$I=\epsilon$$ and $$U=1-\epsilon$$ and $$\epsilon$$ is small (like maybe $$10^{-6}$$ or $$10^{-8}$$). Now, how does the infected population grow?

\[\frac{dI}{dt} = k_{IU} I U = k_{IU} I (1-I)\]

The assumption here is that in a short unit of time, each $$I$$ person becomes in contact with a certain number of $$U$$ people, and for the initial stages at least, this drives the infection. Note that in later stages, $$I$$ population will begin to be reduced as they die off, and there is more going on. We’re interested mainly in the initial stages because we’d like to avoid a major epidemic killing off a few percent of the worlds population etc.

Now, $$I$$ and $$U$$ are unitless (they are the ratios of counts of people), and $$t$$ has units of time, so $$k_{IU}$$ has units of “per time”. It represents the rate at which infected people mix with uninfected people, times the fraction of these mixings which result in transmission. In theory, the fraction of mixing that results in transmission is the definition of $$R_0$$ from my previous post (EDIT: not quite, R_0 is actually the fraction of mixings that result in infection, times the average number of mixings throughout a total epidemic… but we could imagine that’s constant…)… so we can replace $$k_{IU}$$ with $$R_0 r_{IU}$$ where $$r_{IU}$$ is the rate of mixing.

\[\frac{dI}{dt} = R_0 r_{IU}I(1-I)\]

$$I$$ starts out at near zero, and we’re interested in how the infection grows, hopefully we will do something to squash it before it reaches more than 0.005 or 1/2 a percent of the population, so we can assume $$(1-I) \approx 1$$ initially, that is for small $$t$$.

\[\frac{dI}{dt} \approx R_0 r_{IU}I\]

This is the equation for exponential growth, we can make it dimensionless by choosing $$t_0 = 1/r_{IU}$$ to be the unit of time, and we get:

\[\frac{dI}{dt’} \approx R_0I : t’ = O(1)\]

So all epidemics are similar at some time scale, and controlled by $$R_0$$, this reassures only the naivest of mathematicians, because the assumption is only valid for $$t’ = O(1)$$. In a situation in which the mixing time $$t_0$$ is small, this could mean we have only say a few days before $$I \approx 0.02$$ at which point we have a SERIOUS problem (2% of the population actively has Ebola, and that would be devastating). The point is, the equation has to change before $$t$$ gets too big in dimensionless time.

So  $$R_0$$ is useful as an index of how infective the virus is, but NOT how quickly it will spread, since there is also the mixing time to be considered. In western countries we’d have to imagine that the mixing time could be much lower than in West Africa, and so effective response would have to be much faster.

In addition, another dimensionless group is important, namely $$t_rR_0/t_0$$, where $$t_r$$ is the time it takes to effectively institute response measures and $$t_0$$ is the mixing time. The larger this is (the longer the response takes in dimensionless time) the bigger will be the problem.

Fortunately, we also have maybe some suggestion that $$R_0$$ would be smaller in the US, in West Africa many tribal groups wash and prepare their dead, then kiss the bodies to say goodbye… not a good idea with Ebola. Also, there have been attacks on healthcare workers as ignorant people believe Ebola is either a hoax or spread by the government or whatever. Those things probably won’t happen in the US.

All this is to say, there is a lot of uncertainty, with mixing time $$t_0$$ and infectivity $$R_0$$ both having different values in Western countries than in West Africa. So the actual number of days or weeks we will have to effectively respond, and change the equation of growth of the infected population is unknown. One thing we DO know though, is the faster the better. And this is where the CDC and other officials are not driving a lot of confidence in the US population. The general population’s cry of “we need to do something about this NOW” is well justified. Given that Ebola has been around for decades, there should be an established plan and some contingencies that have already been thought out. That this doesn’t seem to be the case is not confidence inspiring.




Comments are closed.