# Epidemic theory (effective and basic reproduction numbers, epidemic thresholds) and techniques for infectious disease data (construction and use of epidemic curves, generation numbers, exceptional reporting and identification of significant clusters)

### Epidemiology: Epidemic Theory

Communicable disease control is considered in detail in a separate section of the MFPH Part A syllabus – See module 2g: Communicable Disease. This page covers the basic principles of epidemic theory.

**Basic reproduction rate - (R0)**

The basic reproduction rate (R0) is used to measure the transmission potential of a disease. It is thought of as the number of secondary infections produced by a typical case of an infection in a population that is totally susceptible.^{1} It can therefore be measured by counting the number of secondary cases following the introduction of an infection into a totally susceptible population.

For example, if the R0 for measles in a population is 15, then we would expect it to spread rapidly because each new case of measles would produce 15 new secondary cases.

R0 excludes new cases produced by the secondary cases etc.

The basic reproductive rate is affected by several factors:

- The rate of contacts in the host population
- The probability of infection being transmitted during contact
- The duration of infectiousness

In general, for an epidemic to occur in a susceptible population R0 must be >1, so the number of cases is increasing.^{1} If R

In many circumstances not all contacts will be susceptible to infection. That is, some contacts will be immune, for example due to prior infection which has conferred life-long immunity, or as a result of previous immunisation. Therefore, not all contacts will become infected and the average number of secondary cases per infectious case will decrease.

This is measured by the effective reproductive rate (R)

**Effective reproductive rate (R)**

A population will rarely be totally susceptible to an infection in the real world. The effective reproductive rate (R) estimates the average number of secondary cases per infectious case in a population made up of both susceptible and non-susceptible hosts. It can be thought of as the number of secondary infections produced by a typical infective.

**R = R0x**

It is the basic reproductive rate discounted by the fraction of the host population that is susceptible (x).

For example, if R0 for influenza is 12 in a population where half of the population is immune, the effective reproductive rate for influenza is 12 x 0.5 = 6. Therefore under these circumstances a single case of influenza would produce an average of 6 new secondary cases.^{1}

To successfully eliminate a disease from a population, R needs to be maintained

**Herd immunity**

Herd immunity occurs when a significant proportion of the population (or the herd) have been vaccinated, and this provides protection for unprotected individuals. The larger the number of people who are vaccinated in a population, the lower the likelihood that a susceptible (unvaccinated) person will come into contact with the infection. It is more difficult for diseases to spread between individuals if large numbers are already immune, and the chain of infection is broken.

The herd immunity threshold is the proportion of a population that need to be immune in order for an infectious disease to become stable in that community. If this is reached, for example due to immunisation, then each case leads to a single new case and the infection will become stable within the population. That is R=1.

If the threshold is surpassed, then R

This is an important measure used in infectious disease control and immunisation and eradication programmes.

**Epidemics**

An epidemic is defined as an increase in the frequency of occurrence of a disease in a population above its baseline or expected level in a given period.^{2} The term is used broadly and the number of cases and time period are often unspecified. It is generally more widespread than an outbreak, which usually implies two or more epidemiologically linked cases, although the two terms have been used interchangeably. Additionally, the term has also been used to describe increasing levels of non-communicable disease, such as an ‘epidemic of cardiovascular disease.’

The definition above is very general, but the term has been defined quantitatively for certain infections and a threshold is selected above which the term ‘epidemic’ is applied. For example, the Health Protection Agency (HPA) monitors levels of influenza in England and Wales in the flu season, from October to May, drawing on data from GP consultations and lab diagnoses. The HPA has defined the baseline threshold for ‘normal seasonal activity’ in England as 50 GP consultations per week/ 100,000 population. The epidemic threshold would be reached if the number of consultations surpassed 200 per week/100,000.^{3}

**Outbreak management**

More information on both routine surveillance and reporting, as well as the management of significant outbreaks, is available in the communicable disease section of this website. However, the key steps in managing an outbreak are:^{4}

- Establish that an outbreak has truly occurred
- Confirm the diagnosis
- Create a case definition
- Find and count cases
- Draw an epidemic curve
- Determine who is at risk
- Generate and test hypothesis
- Implement control measures (as early as possible)
- Write up your findings

**Epidemic curves**

An epidemic curve is simply a graph that illustrates the distribution of the onset of cases of an infectious disease in relation to the onset of illness. The time interval for the onset of illness used will be determined by the incubation period.

Epidemic curves are a useful tool in outbreak investigations and are used to:

- Determine the type of epidemic (common source, point source, propagated)
- Determine the difference between the maximum and minimum incubation period
- Estimate the likely time of exposure
- Determine incubation period in cases where the time of exposure is known

**References**

- Rothman KJ, Modern Epidemiology, Lippincott Williams & Wilkins, 1998.
- Donaldson L, Donaldson R. Essential Public Health (2nd ed.) Petroc Press, 2003.
- http://www.hpa.org.uk/web - Accessed 01/02/09
- Pencheon D, et al. Oxford Handbook of Public Health Practice (2nd ed). Oxford University Press, 2006.

© Helen Barratt, Maria Kirwan 2009