**Published on** March 25th, 2013 |
*by Jo*

# The Mathematics of Immunisation Targets

*Dr Andrew Black is an ARC Research Associate in the School of Mathematical Sciences at the University of Adelaide. *Dr Joshua Ross is a Senior Lecturer in Applied Mathematics, within the same School. In blog, Andrew and Josh will tell us about applications of mathematics and statistics to infectious disease epidemiology.

In November 2012 The Australian Academy of Science a published a document titled “The Science of Immunisation: Questions and Answers’’ (http://www.science.org.au/policy/immunisation.html). This publication covers a number of topics related to what a vaccine is and how we know that vaccines are safe. One of the main motivations for the production of this document is concern that the percentage of the population vaccinated against various diseases is dropping to *risky* levels. So what is a risky level, and what has it got to do with mathematics?

The answer to this question can be found in a paper published almost 90 years ago. “A Contribution to the Mathematical Theory of Epidemics” by William Kermack and Colonel Anderson McKendrick was one of the first studies to analyse the spread of a disease through a population using mathematical models. Along with a handful of other papers, this laid the foundation for the discipline we now know as mathematical epidemiology. Remarkably, the original models they proposed are still in use today.

In this paper, amongst several other important results, was a discovery that is now called the *Threshold Condition. *This result established that for a simple model of the spread of a disease amongst a population of individuals, the initial proportion of the population susceptible to the infection must be larger than a particular threshold in order for the disease to spread, instead of dying out. The consequences of this are that if a fraction (1 – 1/R_{0}) of the population is immune to infection, then effectively everyone in the population is safe from the disease. This is the idea of *herd immunity *– if the majority of a community is vaccinated against a disease then they provide protection to those that aren’t immune.

The number R_{0, }that single-handedly determines the required proportion for herd immunity, is known as the *basic reproduction number*. This number corresponds to how infectious, or transmissible, a disease is in comparison to how long you are infectious for. Hence, this number is different for each disease, and hence the threshold to create herd immunity also changes from disease to disease.

Vaccination targets are determined to ensure that the proportion of the population vaccination is above the threshold that creates herd immunity. There exists state and national registries that track the vaccination coverage, and when this coverage gets too close to the threshold, campaigns are required to increase vaccination rates.

Examples of the consequences of dropping below the threshold are unfortunately becoming more common. In the United Kingdom, measles was under control for decades due to an effective vaccination policy, but due to a fraudulent report that raised concerns over the safety of the vaccine, vaccination rates decreased. Measles is a highly infectious disease, with a value of R_{0} approximately equal to 15, so the resulting drop in coverage was enough to go below the threshold for herd immunity. The consequences were a large increase in the number of measles cases, and even a death.

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