Disease Population Dynamics

Question 1

Identify the components of N in the SIR model and describe the differential growth equations for each component

Answer 1

N = host population density
S = susceptible host density, dS/dt = bN – mS – βSI + 𝛾R
I = infected host density, dI/dt = βSI – (m + 𝛼 + 𝜈)I
R = recovered host density, dR/dt = 𝜈I – mR – 𝛾R

Question 2

Describe the conditions, according to the model, under which a disease will spread through a host population, with reference to D, R0 and ST.

Answer 2

BSD = R0 = the mean number of new infections caused by a single infected individual
R0 > 1 -> disease will spread -> epidemic
R0 = 1 -> disease will persist at constant frequency of infection
R0 < 1 -> disease will eventually disappear

Question 3

Describe the relationship between S, R0, and the average age of infection, A

Answer 3

S is inversely related to the average age of infected individuals, A
↑ S > ↑ βSI > ↑ R0 > ↓ A

Question 4

Define the term “mass action” and compare its importance in the spread of previous and recent pandemics

Answer 4

The rate of mixing of I and S. Mass action was more prevalent in cities and less common in rural when travel between communities was less common, now there is less of a difference

Question 5

Explain why diseases have historically exhibited cyclical dynamics

Answer 5

↑ I > ↑ R, ↓ S > ↓ I… negative feedback loop

…until new births ↑ S to the point where S > ST

Question 6

How does immunization impact the spread of disease in a population? with reference to p, R0 and the attainment of “herd immunity”

Answer 6

Yes. Immunization stops the spread of a disease if it reduces S below ST,
Let p = the fraction of a population that has been immunized
Let R0* = R0 after immunization
R0* = (1-p)R0
When R0* = 1, (1-p)R0 = 1 and p = 1 - 1/𝑅_0
Therefore for a disease to decline, p > 1 - 1/𝑅_0
Herd immunity is reached when p, due to vaccination and/or immunity following recovery, exceeds 1 - 1/𝑅_0

Question 7

Describe how mask-wearing and social distancing can impact the spread of COVID-19 and other airborne diseases, with reference to the components of Meff

Answer 7

Meff = Mava * Mcov * Mred
R with mask use = Rint = (1- Mava * Mcov * Mred)R0

Mask wearing and social distancing can reduce the number of indv. that need to be vaccinated to reach herd immunity.

Question 8

For a metapopulation, describe under what circumstance ease of movement between patches could result in a decline in the proportion of occupied patches due to the spread of disease

Answer 8

Hess (1996): high migration rates can actually reduce patch occupancy and increase metapopulation extinction risk in presence of disease
Increasing movement rate between patches (m) > increasing likelihood of infection
As spillover infection rate increases, movement rate between patches has less impact on proportion infected  benefit of movement outweighs cost