How does Herd Immunity work?

A few days ago, I said that during October I would be posting at least once a week on Vaccines. This post is about Herd Immunity.

Many years ago, Medical Researchers found that even if not everybody in a population was immunised against a disease (either through vaccination or getting and surviving an infection), if a certain percentage were, the disease would stop occurring in that area. From this came the concept of Herd Immunity. What is more, they found they could mathematically model it.

A disease normally infects a person and then, through contagion, is passed to those with whom the person comes in contact within a certain timeframe. If the person to whom the illness is passed is immune, then the disease in that new person is contagious for a much shorter period, or not contagious at all. If enough people are immune, then the disease can’t get to a new host in time, and dies out.

The equation to determine the necessary level of inoculation for Herd Immunity is dependent on a disease’s virulence (how many unimmune people will a host infect) and what portion of a population is immune, susceptible or infected. The former is dependent on how long a disease lasts and how contagious it is. The proportion of immune people is dependent on the number of people who gained immunity from being previously infected (NB: not everyone who gets a disease naturally will become permanently immune) and the number who were inoculated multiplied by the effectiveness of the vaccine (just as with an infection, not all vaccination confer permanent immunity, hence the recommendation to get booster shots after a few years for some vaccines).

The ratio between the immune, infected, and susceptible is dependent on the birth and death rates as newborns will not be immune and immune people will die from non-disease related causes.

In 1927 the work of Kermack and McKendrick was published. They created the SIR (Susceptible, Immune, Recovered) Model.

  •  S(t) is the number of susceptible individuals at time t;
  • I(t) is the number of infected individuals at time t;
  • R(t) is the number of recovered individuals at time t.
  •  R0 (the basic reproduction number) is the average number of secondary infectious cases produced by a single Index Case.
  •  N is the total population size, so S(t) + I(t) + R(t) = N.

From that we get a bunch of very complex equations. I did calculus in university, but that was years ago and I’m afraid I couldn’t follow the equations. If you wish, you can find them here.

The Herd Immunity Threshold for various diseases is as follows:

  •  Diphtheria (R0 6-7): 85%.
  • Measles (R0 12-18): 83-94%.
  • Mumps (R0 4-7): 75-86%.
  • Pertussis (R0 12-17): 92-94%.
  • Polio (R0 5-7): 80-86%.
  • Rubella (R0 5-7): 80-85%.
  • Smallpox (R0 6-7): 83-85%.

Smallpox is extinct because enough people in enough populations were vaccinated, ending its spread everywhere. Polio is down to a few isolated outbreaks in Asia and Africa. On the downside, when Herd Immunity is eroded, the diseases can stage a recovery. This has happened with Pertussis, which is now returning and has started killing children again.

Herd Immunity works. It eliminated Smallpox and Rinderpest. Hopefully, in the not-too distant future, it will also bring Polio, Measles and Pertussis to an end.

About autismjungle

I am a Software Test Analyst. Shortly before I turned 21 I was officially diagnosed, although I had long suspected I was autistic. Welcome to my blog
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