Lecture 5. Dynamics 1: Introduction & Simple Models

Question 1

Why do we model?

Answer 1

Models allow us to predict collective behaviour in terms of our understanding of simpler component elements

Question 2

What does modelling disease allow us to do?

Answer 2

Predict the likely size and time-course of an epidemic from early cases (first question when there’s an outbreak)
Consider optimal deployment of control measures
Identify important heterogeneities (e.g groups/regions not conforming to the “norm”)
Explain observed behaviour

Question 3

Infectious diseases are community based, what does this mean?

Answer 3

My risk depends on your infectious status (not individual)

Question 4

Infectious diseases are dynamic, what does this mean?

Answer 4

The risks change with time (normally quite rapidly)
Rate of which infection happens changes over time along with rate people recover

Question 5

Because we are interested in predicting the amount of disease, what do we work with?

Answer 5

Infection (not risk)
You can be infected but not be diseased (infection and disease are different)

Question 6

What does I(t) mean?

Answer 6

The number of infected individuals in a population (t means time)

Question 7

What does S(t) mean?

Answer 7

The people not infected

Question 8

What does R(t) mean?

Answer 8

The people who cannot be infected who are resistant

Question 9

What is the change in S(t) dependent on?

Answer 9

I(t)

Question 10

What is the rate of change of S in time?

Answer 10

dS/dt

Question 11

What does dS/dt define and what does it tells us?

Answer 11

The local gradient (slope) at any point. It tells us how rapidly S is changing over time

Question 12

What is the function of exponential growth/decay and what is the simple differential equation derived from the function?

Answer 12

x(t) = x₀exp(At)
dx/dt =Ax

Question 13

When does exponential growth occur?

Answer 13

When A in the equation dx/dt = Ax is positive

Question 14

When does exponential decay occur?

Answer 14

When A in the equation dx/dt = Ax is negative

Question 15

What happens when both the rates of change of S and I are equal to zero?

Answer 15

The dynamics are at equilibrium or a fixed point - endemicity

Question 16

What are epidemiologists concerned with when it comes to infection?

Answer 16

The ability of the host to acquire, nurture and transmit the pathogen

Question 17

What are medics concerned with when it comes to infection?

Answer 17

The patient, how ill they feel and whether they are showing symptoms

Question 18

What is the SIR model?

Answer 18

S, I and R define three distinct, non-overlapping classes. Every individual in the population must belong to one (and only one) class
S → I → R
Population size is constant as no births or deaths

Question 19

What is the force of infection (λ)?

Answer 19

The rate at which each susceptible individual gets infected – we need to relate this rate to the number of infectious individuals

Question 20

What does γ represent?

Answer 20

The rate at which each infected individual recovers, and1/γ is the average infectious period

Question 21

What does β represent?

Answer 21

Number of contacts * probability of transmission

Question 22

What equation represents the force of infection (λ)?

Answer 22

λ = βI/N

Question 23

What is the equation for dS/dt?

Answer 23

-βSI/N

Question 24

What is the equation for dI/dt?

Answer 24

+βSI/N - γI

Question 25

What is the equation for dR/dt?

Answer 25

+γI

Question 26

When the population size isn’t changing (i.e in a SIR model), what does N equal?

Answer 26

1
dS/dt = -βSI
dI/dt = +βSI - γI
dR/dt = +γI

Question 27

What does S₀ mean?

Answer 27

The number of people susceptible at the beginning of the SIR model

Question 28

When will an infection grow?

Answer 28

When (βS₀ - γ) > 0

Question 29

How cna we calculate the susceptible population when the infection is at its peak (i.e I at maximum)?

Answer 29

γ/β

Question 30

What is R₀?

Answer 30

The average number of second cases produced by (an average) infected individual in a totally susceptible population

Question 31

What equation calculates R₀?

Answer 31

βS x 1/γ (S = 1 as population is susceptible)
Therefore R₀ = β/γ

Question 32

What value of R₀ is required for an infection to be able to take-off in a totally susceptible population?

Answer 32

R₀ > 1

Question 33

What is R∞?

Answer 33

The number of individuals that get infected (the size of the epidemic) is the drop in susceptibles or the increase in recovered

Question 34

What are the uses of the SIR model?

Answer 34

Provides a “first approximation” for modelling a great number of infectious diseases and their likely impact (pandemic influenza, bio-terrorist smallpox, foot-and-mouth disease)
Works well at modelling single epidemics of relatively simple pathogens that occur quickly (so that births and deaths can be ignored)
Always predicts the eventual extinction of the disease. If we want to consider endemic infections, then we need to include births to replenish the susceptible population