Mathematical modelling Flashcards
What are mathematical models used for?
For a specific outbreak to answer questions such as:
- Why was the outbreak like that?
- Is the outbreak under control?
- When will it be over?
For general understanding of disease:
- enable us to understand, explain, predict and explore scenarios
What does the risk depend on in non-infectious diseases?
Individual behaviour
What is a mathematical disease model?
Set of equations that describe key processes at work in a system affected by an infectious diseases
What are the 3 diseases statuses in a population?
Susceptible
Infected (and infectious)
Recovered (and immune)
What can be added to an SIR model initially to add more complexity?
Births - all new-borns are susceptible so added to S compartment
Deaths - death can occur in any compartment, may be unrelated to disease
Between S and I in the model, which compartment can be added to increase complexity?
Exposed
- individuals that are latently infected, so aren’t infectious
How does the recovered population link to the susceptible population?
Loss of immunity, so R becomes S
Name all of the processes in a mathematical model
- Transmission of infection (S become E)
- Development of infectiousness (E become I)
- Recovery to temporary immunity (I become R)
- Loss of immunity (R become S)
- Birth (new S added)
- Death (losses from S, E, I and R)
Define the latent and incubation periods
Latent = time from becoming infected to becoming infectious Incubation = time from becoming infected to showing symptoms
Which additional category must be considered in many diseases?
Fatalities - died due to the diseases
Come from the infectious population
The rate of change of S is equal to…?
The transmission rate
The rate of change of I is equal to…?
Transmission rate - recovery rate
The rate of change of R is equal to …?
The recovery rate
The recovery rate is denoted as what symbol?
Gamma.I (γI)
Gamma = 1 / ?
Infectious period
e.g. If people are infectious with measles for 10 days on average, each day 1/10 of people with measles recover
Transmission rate is denoted as?
Lamda S (λS)
What is λS?
Proportion of people that become infectious per unit time
What is a typical SIR output?
S decreases and I increases as susceptible individuals become infected
After a lag, R starts to increase as infectious individuals recover
Eventually the pool of S becomes so small that I plateaus and falls
Because there is immunity infection will fade out eventually
Why does S not drop to 0?
Herd immunity
How will a higher force of infection affect the SIR graph?
Shift and compressed to the left I increases sooner I gets to a higher peak Outbreak finishes faster Nearly all individuals infected
How does a shorter infectious period / Higher recovery rate affect the SIR graph?
- I increases more slowly
- Smaller, later peak in infections
- Many individuals do not become infected
How does adding births and deaths affect the SIR graph?
I increases, then decreases when S gets small, but increases as new S arise by birth
R increases but then begins to decreases because of deaths
What is R0?
Average number of cases (secondary infections) generated by a single infectious individual (primary infection) introduced into a totally susceptible population
What are the R0 defining epidemiological measures?
- When R0 > 1, an epidemic can occur.
- When R0 = 1, endemic
- When R0 < 1, outbreak will reduce and may stop
What are some assumptions of simple models?
- Simple models are deterministic, giving the same results every time. In reality, chance plays a big role in life, death, and disease.
- There is free mixing of all individuals
- There is no heterogeneity - all individuals, regardless of age, sex, type, are equally likely to become infected
- There is no effect of space
