DISEASE E&E (Models) Flashcards
Purpose of epidemiological models:
-tools that help scientists understand whether and how infectious disease will spread through the host population
-can lead to surprising insights
-that about what features or aspects of infectious disease we need to know more about
Epidemiological models allow us to:
-compare the outcome of different control strategies
-make predictions that help policy makers make decisions
Deterministic compartment models:
-different compartments
-arrows show transitions between compartments
-SIR, SIRS, SEIR models
Compartments:
-host individuals in different states
>susceptible
>infectious
>recovered
>pre-infectious
SIRS model:
-S become I
-I become R
-R become S again
SEIR model:
-pre-infectious (latent) stage where individuals have been exposed but are not yet infectious
SIR model:
-S become I via transmission
I become R via clearance, immunity
*closed population (host population size remains constant over time)
S:
-susceptible individuals
-can acquire the infection from infected individuals
I:
-infected individuals
-can transmit infection to susceptible individuals
-can become recovered individuals that are resistant to future infection
R:
-resistant individuals
-individuals have developed resistance (ie. Immunity) against future infections
N:
-total population size
N=S+I+R
Beta:
-proportionality constant for infection
-transmission coefficient
v (mu):
-rate of recovery of infected hosts
When would SIR model be appropriate?
-for a pathogen with a short incubation (ex. virus) that spreads quickly through the host population in a matter of weeks
S to I, closed population (ordinary differential equation):
=(-beta)(S)(I)
-S can only be lost which is why there is a negative
Rate of I produced, closed population (ordinary differential equation)
=(beta)(S)(I) – (vI)
-produced at a rate of BSI
-lost once they recover and develop resistance (why there is a negative)
Rate of R produced, closed population (ordinary differential equation):
=vl
-loss of the I population are gains for the R population
R0, basic reproductive number of disease:
=(beta x S) / (v)
-average number of new infections caused by a single infection over its duration
*needs to be greater than 1 for the disease to invade a host population
How does beta influence disease invasion?
-increase beta=probability of disease invasion increases
How does S influence disease invasion?
-increase S=probability of disease invasion increases
How does v influence disease invasion?
-increase v=probability of disease invasion decreases
>decrease the average duration of an infection
What does SIR model show about population size?
-shows that population size determines R0 and disease invasion!
*if population is too small, disease can’t invade!
Dynamics of SIR model:
-S decreases over time
-I originally increases and then decreases over time
-R increases over time
*at equilibrium the disease has died out and there are no infected individuals in the population
>the host population now consists of susceptible and recovered individuals
Epidemic of influenza B in Midwest region (2007-2008):
-showed number of observed cases and the predicted number of cases from the SIR model
-can use the data to estimate parameters that are difficult to measure (ex. beta)
-knowledge of the parameters is important for public health strategies
SIR model with births and deaths:
-open population
-2 parameters: birth rate (b) and death rate (mu)
-birth rate = death rate so N is constant in size
-S, I, R all have the same birth and death rates
With the introduction of births:
-now a constant input of naïve susceptible individuals
With the introduction of death:
-R individuals die and leave the population
Under what condition will the disease invade the host population (SIR open population):
-if the rate of infected individuals is GREATER than 0
R0 SIR open population:
-definition of R0 depends on the structure of the model
-still must be greater than 1 for the level of infection to increase
Minimum human population size to maintain measles:
*many infectious diseases can only persist if the host population passes a critical threshold
-measles needs 200,000 or more
-it couldn’t have existed before agricultural revolution
Evidence that measles needs a minimum population size to persist:
-compared oceanic islands over a 16 year period
-can persist 100% when population size is 500,000
-islands with smaller population sizes, measles goes extinct and it must be RE-INTRODUCED from the outside world
3000 years of urban growth:
-many historically important infectious diseases (measles, pertussis, scarlet fever, diphtheria) need relatively large human population sizes to persist
*most directly transmitted infectious diseases have emerged recently (within last 2000 years)
Vaccination and disease invasion:
-disease invasion and persistence depends on S
-vaccination converts S individuals into R individuals
-decrease S=vaccination can prevent disease invasion
Reff:
-only use R0 when population is 100% susceptible
-depends on fraction of S individuals
=(S/N)*R0
-disease wont invade if it is less than 1
*essentially the product of R0 and the fraction of unvaccinated individuals
Susceptible and vaccinated individuals
-host population consists of S and vaccinated (Q) (this is before a breakout of a disease)
-s=proportion of S individuals
-q=proportion of Q individuals
-s+q=1
Critical vaccination threshold:
-proportion of host population that must be vaccinated to prevent the infectious disease from invading the host population
-depends on R0
Qcrit=1-(1/R0)
R0 and critical vaccination threshold:
-higher R0 value=higher critical proportion of hosts that must be vaccinated
-R0=2, qcrit=0.50
-R0=10=0.90
R0 of human infectious disease:
-number of people that one sick person will infect
-measles=18, so about 94% of population must be vaccinated
Why disease persist in developing and developed countries:
-developing: limited access to vaccines
-developed: vaccine hesitancy among a substantial fraction of population
Herd immunity threshold (HIT):
-% of individuals that must become infected to prevent the infection from spreading
-HIT=qcrit
-natural immunity and vaccination both reduce S and prevent spread of disease
-with natural immunity, people have to contract the disease
Herd immunity vs. vaccination:
-more mortality and morbidity (suffering) with herd immunity
>also caring for all the sick individuals which takes a toll on health care professionals and is costly for society
Herd immunity and public health strategy:
-UK government considered herd immunity to fight COVID-19
-assumed R0=2,5
-qcrit=0.6 (60%)
-40million people would get ill
-case fatality rate (CFR) was 0.6% (240,000 would die)
*unacceptable level of mortality and decided to abandon the idea of her immunity
