Lecture 5. Dynamics 1: Introduction & Simple Models Flashcards
Why do we model?
Models allow us to predict collective behaviour in terms of our understanding of simpler component elements
What does modelling disease allow us to do?
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
Infectious diseases are community based, what does this mean?
My risk depends on your infectious status (not individual)
Infectious diseases are dynamic, what does this mean?
The risks change with time (normally quite rapidly)
Rate of which infection happens changes over time along with rate people recover
Because we are interested in predicting the amount of disease, what do we work with?
Infection (not risk)
You can be infected but not be diseased (infection and disease are different)
What does I(t) mean?
The number of infected individuals in a population (t means time)
What does S(t) mean?
The people not infected
What does R(t) mean?
The people who cannot be infected who are resistant
What is the change in S(t) dependent on?
I(t)
What is the rate of change of S in time?
dS/dt
What does dS/dt define and what does it tells us?
The local gradient (slope) at any point. It tells us how rapidly S is changing over time
What is the function of exponential growth/decay and what is the simple differential equation derived from the function?
x(t) = x₀exp(At)
dx/dt =Ax
When does exponential growth occur?
When A in the equation dx/dt = Ax is positive
When does exponential decay occur?
When A in the equation dx/dt = Ax is negative
What happens when both the rates of change of S and I are equal to zero?
The dynamics are at equilibrium or a fixed point - endemicity
What are epidemiologists concerned with when it comes to infection?
The ability of the host to acquire, nurture and transmit the pathogen
What are medics concerned with when it comes to infection?
The patient, how ill they feel and whether they are showing symptoms
What is the SIR model?
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
What is the force of infection (λ)?
The rate at which each susceptible individual gets infected – we need to relate this rate to the number of infectious individuals
What does γ represent?
The rate at which each infected individual recovers, and1/γ is the average infectious period
What does β represent?
Number of contacts * probability of transmission
What equation represents the force of infection (λ)?
λ = βI/N
What is the equation for dS/dt?
-βSI/N
What is the equation for dI/dt?
+βSI/N - γI
What is the equation for dR/dt?
+γI
When the population size isn’t changing (i.e in a SIR model), what does N equal?
1
dS/dt = -βSI
dI/dt = +βSI - γI
dR/dt = +γI
What does S₀ mean?
The number of people susceptible at the beginning of the SIR model
When will an infection grow?
When (βS₀ - γ) > 0
How cna we calculate the susceptible population when the infection is at its peak (i.e I at maximum)?
γ/β
What is R₀?
The average number of second cases produced by (an average) infected individual in a totally susceptible population
What equation calculates R₀?
βS x 1/γ (S = 1 as population is susceptible)
Therefore R₀ = β/γ
What value of R₀ is required for an infection to be able to take-off in a totally susceptible population?
R₀ > 1
What is R∞?
The number of individuals that get infected (the size of the epidemic) is the drop in susceptibles or the increase in recovered
What are the uses of the SIR model?
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
