Modelling Infectious Diseases
Ayman D’Souza
Joseph Lazzaro
A key use of mathematics that has developed after recent disease outbreaks is the modelling of infectious diseases. This helps in two ways, by informing health workers how much of a population needs to be vaccinated for herd immunity to work, and helps to decide the course of actions taken by NGOs and the WHO in the outbreak of an epi- or pandemic. Herd immunity occurs when a large proportion of a community are vaccinated against or immune to a disease. This protects those who are not immune as it disrupts chains of infection, by acting as a buffer, preventing an epidemic from taking off. The diseases can range from Ebola to swine flu here in the UK, with the models easily adjustable for many different infections. The most basic model is the SIR model. This model has four key factors- the number of people susceptible to infection, number of people infectious, number of people recovered and the timeframe for how these factors change. , defined by “how many people an infectious person will pass on their infection to in a 100% susceptible population”. Naturally this varies from disease to disease- measles is airborne, hence highly infectious, and malaria, due to the number of mosquitoes and humans to act as disease vectors, has an even higher R 0 . The model also has parameter R 0
This is the formula used to predict what percentage of population should be vaccinated to prevent disease (assuming a 100% effective vaccine):
% 𝑣 𝑖 = 1 – 1 0
Using the table of values above, we see that less infectious diseases such as the HIV virus with low R 0 value, a maximum of 80% of the people would need vaccination for herd immunity. More easily transmissible diseases, require a far higher proportion of vaccinated people- i.e. measles would require ~95% vaccinations, and malaria >99% for herd immunity to even have a chance of working.
So what is the actual model?
The equations below show the simplest SIR model – all require differential equations, to reflect the timeframe factor for the development of a disease.
The rate of change of the number susceptible to illness with respect to time:
= −𝛽𝐼
The rate of change of number infected with respect to time:
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