Mathematical modelling

Question 1

What are mathematical models used for?

Answer 1

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

Question 2

What does the risk depend on in non-infectious diseases?

Answer 2

Individual behaviour

Question 3

What is a mathematical disease model?

Answer 3

Set of equations that describe key processes at work in a system affected by an infectious diseases

Question 4

What are the 3 diseases statuses in a population?

Answer 4

Susceptible
Infected (and infectious)
Recovered (and immune)

Question 5

What can be added to an SIR model initially to add more complexity?

Answer 5

Births - all new-borns are susceptible so added to S compartment
Deaths - death can occur in any compartment, may be unrelated to disease

Question 6

Between S and I in the model, which compartment can be added to increase complexity?

Answer 6

Exposed

- individuals that are latently infected, so aren’t infectious

Question 7

How does the recovered population link to the susceptible population?

Answer 7

Loss of immunity, so R becomes S

Question 8

Name all of the processes in a mathematical model

Answer 8

• 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)

Question 9

Define the latent and incubation periods

Answer 9

Latent = time from becoming infected to becoming infectious 
Incubation = time from becoming infected to showing symptoms

Question 10

Which additional category must be considered in many diseases?

Answer 10

Fatalities - died due to the diseases

Come from the infectious population

Question 11

The rate of change of S is equal to…?

Answer 11

The transmission rate

Question 12

The rate of change of I is equal to…?

Answer 12

Transmission rate - recovery rate

Question 13

The rate of change of R is equal to …?

Answer 13

The recovery rate

Question 14

The recovery rate is denoted as what symbol?

Answer 14

Gamma.I (γI)

Question 15

Gamma = 1 / ?

Answer 15

Infectious period

e.g. If people are infectious with measles for 10 days on average, each day 1/10 of people with measles recover

Question 16

Transmission rate is denoted as?

Answer 16

Lamda S (λS)

Question 17

What is λS?

Answer 17

Proportion of people that become infectious per unit time

Question 18

What is a typical SIR output?

Answer 18

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

Question 19

Why does S not drop to 0?

Answer 19

Herd immunity

Question 20

How will a higher force of infection affect the SIR graph?

Answer 20

Shift and compressed to the left
I increases sooner
I gets to a higher peak 
Outbreak finishes faster 
Nearly all individuals infected

Question 21

How does a shorter infectious period / Higher recovery rate affect the SIR graph?

Answer 21

• I increases more slowly
• Smaller, later peak in infections
• Many individuals do not become infected

Question 22

How does adding births and deaths affect the SIR graph?

Answer 22

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

Question 23

What is R0?

Answer 23

Average number of cases (secondary infections) generated by a single infectious individual (primary infection) introduced into a totally susceptible population

Question 24

What are the R0 defining epidemiological measures?

Answer 24

• When R0 > 1, an epidemic can occur.
• When R0 = 1, endemic
• When R0 < 1, outbreak will reduce and may stop

Question 25

What are some assumptions of simple models?

Answer 25

• 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