Lecture 7. Dynamics 3: Heterogeneities

Question 1

Risk of disease is multi-dimensional, what does this mean?

Answer 1

Risk consists of both (host) behaviours and the risk of disease (e.g. health, genetics)

Question 2

For non-communicable disease, what is risk determined by?

Answer 2

Each individual (my behaviour and my genetics)

Question 3

For communicable disease, what is risk determined by?

Answer 3

Our behaviour and our genetics

Question 4

How can heterogeneity be ‘created’?

Answer 4

By infection and immunity (e.g susceptibility and/with ageing)

Question 5

What infection and immunity is created by heterogeneity?

Answer 5

Who acquires infection from whom
Risk and sexually-transmitted diseases
Other forms of structure

Question 6

What equation can be used to model a cohort of newly-born susceptibles as they are exposed to an equilibrium level of infection (I*)?

Answer 6

dS/da = -β(I*)S = -β(B/β * (R₀ - 1))S ≈ -B(R₀ - 1)S (a is the same as time)
-B(R₀ - 1)S - actually a simple exponential differential equation

Question 7

What equation shows the proportion of the population that are susceptible decays exponentially with age (amount of S at age a)?

Answer 7

S(a) = exp(-B(R₀ - 1)a)

Question 8

What formula can be used to calculate an average age of infection?

Answer 8

A = L/(R₀ - 1) where L = life expectancy

Question 9

When does the average age of infection increase?

Answer 9

When life expectancy is higher (high life expectancy means lower birth and death rate and therefore less infection at equilibrium)

Question 10

When does the average age of infection decrease?

Answer 10

When R₀ is higher (as this means an increased force of infection and so increases the rate at which a susceptible is likely to be infected)

Question 11

Why doesn’t exponential decrease theory math the available data?

Answer 11

This is due to strong assortative mixing - people interact most often with others of the same age - and school children are more ‘mixy’ than adults

Question 12

How can we capture the effect caused by assortative mixing?

Answer 12

By splitting the [population into groups according to heterogeneity

Question 13

What differential equations are used for the children population?

Answer 13

dSc/dt = -(λc * Sc)
dIc/dt = +(λc * Sc) - (γc * Ic)
dRc/dt = +(γc * Ic)

Question 14

What differential equations are used for the adult population?

Answer 14

dSA/dt = -(λA * SA)
dIA/dt = +(λA * SA) - (γA * IA)
dRA/dt = +(γA * IA)

Question 15

What does λc represent?

Answer 15

λc = βccIc + βcAIA

Question 16

What does λA represent?

Answer 16

λA = βAcIc + βAAIA

Question 17

What is the name of the matrix of transmission rates?

Answer 17

Who Acquires Infection from Whom matrix (WAIFW)

Question 18

The WAIFW matrix is usually assortative, what does this mean?

Answer 18

(The diagonal terms dominate) such that individuals mix most with their own age

Question 19

The WAIFW matrix is usually symmetric, what does this mean?

Answer 19

(βXY = βYX) As long as all groups have the same epidemiological response, and β just measures the interaction between them

Question 20

By convention we define the transmission terms in the matrix how?

Answer 20

β(to)(from)
βcA = β to children from adults

Question 21

What is the matrix described by WAIFW?

Answer 21

[βcc βcA]
[βAc βAA]

Question 22

How can R₀ be calculated from WAIFW?

Answer 22

[Ncβcc/γc NcβcA/γ]
[NAβAc/γc NAβAA/γA]

Question 23

What do NA and NC represent?

Answer 23

The number of individuals in each class

Question 24

If the R₀ matrix has values of [4 2], what does that mean?
[3 2]

Answer 24

This basically says that (when everyone is susceptible):
each infected child produces 4 cases in children and 3 cases in adults;
while each infected adult produces 2 cases in children and 2 cases in adults.

Question 25

What does mimicking the opening and closing of schools allow us to capture?

Answer 25

The oscillatory dynamics of infections such as measles

Question 26

What are the two things that distinguish the spread od sexually-transmitted diseases?

Answer 26

They are spread through the network of sexual contacts - which are easier to define and trace than contacts for airborne infections
There is huge variability in the number of sexual contacts

Question 27

The variability in the number of sexual contacts means that we need to consider risk groups based on what?

Answer 27

Number of sexual partners

Question 28

What two ways are risk groups based on the number os sexual partners considered?

Answer 28

Either directly from a sexual contact network; or indirectly from the distribution of sexual partners

Question 29

What are the limitations of making risk-structured models from networks?

Answer 29

Predictions are limited by the size and accuracy of the network
‘Snowball sample’ - gets bigger and bigger

Question 30

What are the notices of making risk-structured models from distributions?

Answer 30

Extreme variability
Relatively high means
Extremely high maxima

Question 31

Depending on the values of the WAIFW matrix and the initial conditions, what unusual behaviours can be observed?

Answer 31

R₀ from the low risk group can be less than 1, so if the infection starts in the low risk group the total level of infection may initially decrease, even though overall R₀ >1.
Alternatively, if R₀ from the high risk group is large and the infection starts in the high-risk group the total level of infection may
initially increase, even if overall R₀ <1.
Assuming that overall R₀ > 1, eventually low-risk group slaved to the high-risk group.

Question 32

What does the initial growth rate depend on?

Answer 32

Initial conditions

Question 33

What is another common form of structure?

Answer 33

To account for the time since infection (slightly more realistic dynamic)

Question 34

What happens in a structure that accounts for time since infection?

Answer 34

The standard model assumes that infectious individuals have a constant level of transmission, and that infectious individuals recover at a constant rate
This leads to a exponential distribution of the infectious period
To get more realistic behaviour we need to add more structure to the model