Epidemic threshold

The epidemic threshold refers to a threshold above which agents can spread explosively and cause epidemics. It is usually expressed in terms of the ratio between birth rate and death rate. If the ratio is larger than one, then an epidemic may result. If it is less than one then an epidemic becomes extremely unlikely.

The epidemic threshold is an important concept in epidemiology. It has previously been applied to cultural evolution - for example by Seth Godin in Unleashing the Ideavirus (p.77) - though he called it the magic-number.

Many factors influence the birth rates - including their level of virulence, their lifespan, whether they are defeated by attacks from the host immune system - and so on.

Being above the epidemic threshold does not guarantee that an epidemic will result. Statistical fluctuations may result in extinction before an epidemic forms. Being below the epidemic threshold makes an epidemic extremely unlikely, though.

The epidemic threshold is a pretty important idea in viral marketing. If your shareable item is below the threshold, and you get exponential decay - with negligibe viral effect. If it is above it, you could see exponential growth - as your marketing work is done for you by the masses.

Diagram

Epidemic failure and success.

This diagram shows the output from a very simple model of an epidemic. Infected hosts are plotted against time, from a fixed initial "seed" population. Different plot lines reflect different birth and death rates for the parasite.

Economic significance

The result of this is typically a non-linear R.O.I. on the marketing budget:

• Below the epidemic threshold the effects are so small that you might as well not have bothered.
  
• Above the epidemic threshold you see explosive growth - then it is more likely that the marketing budget has been well spent.

Much the same model applies equally well to mutualists as well as pathogens.

Big seed marketing

A related idea is big seeding - which is another strategy for avoiding extinction.

Tipping points

Malcom Gladwell once wrote a book called The Tipping Point - which referred to essentially the same concept. For more about that, see our post on that topic.

Note that the term epidemic threshold is also - rather confusingly - used to describe a number of hosts which need to be infected before an outbreak is classified as an epidemic. This post is not about that concept.

References

• How Viruses Spread
  
• Some simple epidemic models
  
• Discrete epidemic models
  
• Discussion: the Kermack-McKendrick epidemic threshold theorem
  
• The Kermack-McKendrick epidemic model revisited
  
• A threshold theorem for the general epidemic in discrete time
  
• Viral Math: R-Naught and Zarrella’s Hierarchy of Contagiousness