Lecture 6. Dynamics 2: More Models Flashcards
What does B represent in a SIR model?
Births - now replenish the susceptible class
What does d represent in a SIR model?
Deaths - Remove individuals from the population
What are the differential equations for a SIR model with B and d?
dS/dt = +B - βSI - dS
dI/dt = +βSI - γI - dI
dR/dt = +γI - dR
Why is it usual to make B = d?
To keep population size constant (otherwise we need to include N in the force of infection)
What is generally assumed about birth and death rates when compared to infection and recovery rates?
Birth and death rates are slower than infection and recovery rates
What do the initial dynamics of a SIR model with B and d resemble and why?
Look remarkably similar to simple SIR as infection and recovery dominate
What can occur in the trough phase of infection in a SIR model with B and d?
The number of susceptibles can increase - being replenished by births faster than being infected
What happens during the rest of the SIR model before endemicity is achieved?
Regular damped oscillations occur with S and I being asynchronous (out of phase)
How do you calculate R₀ in a SIR model with B and d?
R₀ = βS * 1/(γ+d) = β/(γ+d)
What do S* and I* mean?
When the rate of change of S and I are zero (endemic state)
How do you calculate S* in a SIR model that includes B and d?
S* = (γ+d)/β = 1/R₀
How do you calculate I* in a SIR model that includes B and d?
I* = B/β * (R₀ - 1)
What is the SIS model?
Only consists of two states, S and I
In this model, recovered individuals are once again susceptible - therefore births aren’t needed to produce an endemic infection
What are the differential equations for a SIS model?
dS/dt = -βSI + γI
dI/dt = +βSI - γI
How do you calculate R₀ in a SIS?
R₀ = β/γ
