Modelling Epidemics

Two viruses have been particularly in the news this year, the Influenza A(H1N1)v virus ("swine flu") and measles. The rapid increase in the number of cases was evident in news reports, and is typical of epidemics. For instance, 7 cases of measles were reported in NZ in June, jumping to 93 notified cases in July, as the disease spread from Dunedin and Christchurch up to Auckland. Although measles is a common childhood infection that many in NZ would probably not think of as serious, it is highly contagious and can be fatal. 7 people died in the 1991 epidemic here. Globally, mortalities from measles exceed that of the flu, although are less than maleria and HIV.

Mathematics and Measles - A Case Study in the Value of a Mathematical Model

Up until about a decade ago, New Zealand faced regular measles epidemics, but a mathematical model successfully predicted the outbreak in 1997, and was instrumental in the decision to carry out an extensive vaccination campaign that year. Subsequent work on the mathematical model led to the recommendation that changes be made to the immunisation schedule in order to prevent further epidemics. First a little background:

Measles first came to NZ in the 1800's, via ships from England. The first instance seems to be in the mid 1830's, when a ship brought the disease from Sydney, and South Island Maori were severely affected. Before a vaccine was available, almost all children in NZ caught measles.
Immunisation was introduced in NZ in 1969, but there was still a pattern of 2- to 3-yearly epidemics, so in 1978 the Department of Health instituted a 5-year measles epidemic elimination programme. The next epidemics after 1980 were in 1985 and 1991, so the programme had only partial success. In Nov 1990, the measles vaccine was replaced by the measles-mumps-rubella vaccine (MMR) generally administered to babies when they were about 15 months old. In 1992, a second dose of MMR, scheduled at age 11 years, was added in response to the 1991 epidemic.

Which brings us to 1996 when Mick Roberts (pictured above), then working for AgReasearch, produced a mathematical model of the dynamics of measles in NZ for the Ministry of Health.

SIR Model

The model that Mick Roberts used was an extension of a standard deterministic model introduced by Kermack and McKendrick in 1927, which uses a set of linked differential equations. A population, taken to be constant for the duration of the epidemic, is divided into 3 groups:

1. Susceptibles (S) - those who are uninfected but at risk of getting the disease


2. Infectious (I) - those who are infected and pass on the disease to susceptibles


3. Removed/recovered (R) - those who have had the disease but are now immune (or dead or isolated or otherwise removed from the population)

As time passes, susceptible people become infected and infected recover and become immune.

The differential equations describe how the numbers of susceptible, infectious and removed change during the epidemic.

These equations state that the rate at which susceptibles become infected is proportional to the rate of interaction between the groups (proportional to the product of the numbers in each group - the Law of Mass Action), but the number of infected decreases at a rate proportional to their number (i.e. the infected recover at an exponential rate).

The progress of the epidemic depends on the two parameters (infection and recovery rates) and initial numbers of susceptibles, which combined form the basic reproduction ratio (or epidemic threshold) Ro, which gives the number of secondary infections caused by a single primary case in a fully susceptible population. For measles, Ro ranges from 12 to 18. If it was less than 1, an epidemic would not occur.

The assumptions of the above simple SIR model are:
•Fixed population size (no births, deaths)
•Incubation period instantaneous
•Duration of infectivity = length of disease
•Completely homogeneous population (no age, spatial or social structure)

Mick Roberts' SIR Model for Measles in NZ (1996)

Assumptions:

•Those aged less than 6 months or more than 25 years take no part in the epidemic.
•4 active age classes coinciding with potential ages at vaccination.
•Different contact rates within and between the age classes
•Disease transmission is seasonal - high between Feb 28 and 1 Dec, low in summer.
•Annual birth rate constant at 57435 births/year.
•Model to incorporate vaccination strategies and historical vaccination rates (around 80%) from 1969-96 and assume 1996 strategy continues until 2000.
•Primary vaccine failure rate taken as 10%

There were differential equations for the change in susceptibles and infective for each of the four age groups (6 months to 15 months, 15months to 4 years, 4 - 11 years and 11 - 25 yrs)

The model was solved numerically for the period 1962 to 2000, using the combination of parameters that gave best match to historical timing of epidemics for the period 1970-92. It predicted there would be an epidemic the following year. The model was consistent with a basic reproduction ratio of 12.8 (2.85 with vaccination).

The 1997 Epidemic

The epidemic began a few months earlier than predicted, and was contained by a mass-vaccination effort. Just under 2000 reported cases resulted.

The mathematical model predicted another epidemic in 2003 - 2004. Was there a better immunisation strategy that might prevent further epidemics?

An extended model to test strategies:

•Further subdivision of age classes into eight classes
•Model run with 4 different immunisation schedules (each with 2 vaccinations)
•4 different coverage rates (from 80% to 95%)
•‘catch-up’ opportunities included (at entry to preschool and primary school)
•Model solved numerically over 20 years for selected strategies

The result of this modelling was a set of recommendations to the Ministry of Health, including moving the second vaccination at 11 years forward to 3 years or 6 years of age. In addition, coverage at 15 months needs to be at least 90% if further outbreaks are to be prevented.

In 2003, the immunisation schedule was amended to have the second vaccination at 4 years. The coverage is still too low, perhaps not helped by a (now discredited) report some years back linking the MMR vacine to Autism. Hence, we seem to have another outbreak in 2009.

Acknowledgements:

Roberts, M. G. & Tobias, M. I. Predicting and preventing measles epidemics in New Zealand: Application of a mathematical model. (Epidemiology and Infection 124, 279-287.)

With thanks to Professor Mick Roberts, Massey University, Auckland

To see the paper mentioned above and other recent research of Mick Roberts:
http://tur-www1.massey.ac.nz/~mgrobert/Research.html