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A VERY IMPORTANT
CONCEPT
• Disease epidemiology is first and
foremost a population biology problem
• Important proponents: Anderson, May,
Ewald, Day, Grenfell
S.I.R. MODELS
• Susceptible, Infected, Recovered
• Assign each individual to one of the
mutually exclusive categories and then
apply equations to describe rates of
transition to the different categories
TWO KINDS OF
TRANSMISSION
• Direct - through physical contact or
through production of free-living
stages (e.g. measles)
• Indirect - through one or more
obligatory intermediate hosts (e.g.
malaria)
DIRECT TRANSMISSION
BETWEEN HOSTS
•
•
•
•
ASSUMPTIONS:
Hosts do NOT recover
Hosts do NOT die
Host population is stable (static)
DIRECT TRANSMISSION
BETWEEN HOSTS
• X = density of susceptibles
• Y = density of infecteds
• N=X+Y
= transmission coefficient
TRANSMISSION
COEFFICIENT
• Product of 2 constants:
– C = contact rate
– T = probability of transmission when a
susceptible contacts an infected (0,1)
DIRECT TRANSMISSION
BETWEEN HOSTS
• X = density of susceptibles
• Y = density of infecteds
• N=X+Y
= transmission coefficient
dX
= - X bY
dt
dY
Y
= Xb N
dt
N
= X bY
DISEASE DYNAMICS
TWO STABLE EQUILIBRIA
• When Y = 0
• When Y = N
BASIC RATE OF INFECTION
• R0 Def’n - the rate of acquisition of
susceptible individuals that get
converted to infecteds (2º infections)
• If R0 > 1 disease is growing
MEASLES
• Virus-caused disease
• Transmission is primarily by large respiratory
droplets.
• The disease typically consists of a high fever,
cough, runny nose and a generalized maculopapular rash.
• Infants under one year of age have the
highest case fatality rates reaching as
high as 20% in epidemic situations.
INCIDENCE TRENDS
MEASLES ASSUMPTIONS
• Human population is stable. Deaths
immediately lead to birth of a healthy
susceptible individual.
• No age structure.
• Death can arise from natural causes (b) or
from measles ( )
• Immunity once gained is never lost
• No incubation time
BASIC MEASLES MODEL
• X = number of susceptibles
• Y = number of infecteds
• Z = number of immunes
• N=X+Y+Z
β = transmission coefficient
b = death due to natural causes
α= death due to disease
 = immunity gain rate
BASIC MEASLES SIR
MODEL
d X /dt = b N +  Y - β XY -bX
dY /dt = β XY -( + b +  ) Y
dZ /dt =  Y - bZ
WHAT IS R0
R0 = β N/ ( + b +  )
TWO STABLE EQUILIBRIA
If β N/ ( + b +  ) > 1 the Y* adopts
non-zero values
If β N/ ( + b +  ) < 1 then X* = 1, Y*
= 0 and Z* = 0
MINIMUM HOST POP?
Nt > ( + b +  )/ β
for R > 1
ADD REALISM
• Add an incubation class Q that
includes individuals that are infected
but not yet infectious (12 day period)
• Let vary according to season
PHASE PLANE
INFECTEDS
WITH AGE STRUCTURE
INFECTEDS
THE KEYS
• The susceptible class is exhausted by
infection and replenished by birth of
new susceptibles.
• Latency period accentuates oscillatory
behavior.
• Seasonally changing β values
accentuates oscillatory behavior.
DYNAMICS
A CRITICAL ASSUMPTION
• We assumed that decisions to vaccinate are
tactical i.e. a rationale decision was made based
upon costs and benefits independent of the actions
of others in the population
WHY THAT MIGHT BE A
PROBLEM
• The payoffs from vaccination could be
dependent upon the vaccination decisions of
others e.g. if everyone else vaccinated then the
risk to you from not vaccinating should be small
and vice versa
WHAT IS GAME THEORY?
• Def’n – the study of strategic decision making
•In Biology (Evolutionary Game Theory), the
principle utility is fitness
WHAT IS GAME THEORY?
• Example from economics: The Prisoners’
Dilemma:
• Two prisoners are given the option to betray or
remain silent
•If A and B both betray, each gets a 2 year sentence
•If A betrays and B remains silent, and B gets 3
years and vice versa
•If A and B remain silent, each receives 1 year
•Result: both betray
EXAMPLES IN BIOLOGY
• Def’n – the study of strategic decision making
•In Biology (Evolutionary Game Theory), the
principle utility is fitness
•In Biology, we solve for the Evolutionary Stable
Strategy (ESS), which is a Nash Equilibrium,
wherein no one player can gain fitness by
unilaterally changing his/her strategy unilaterally
PARAMETERS
• P – an individual’s strategy that he/she will
vaccinate with probability P
• p - proportion of the population that is
vaccinated
• rv – morbidity risk from vaccination
• ri - morbidity risk from infection
• πp - risk to an unvaccinated individual of
eventual infection given vaccination coverage p
AN IMPORTANT
TRADEOFF
• There are two key risks, (i) the morbidity risk
from the infection and (ii) the morbidity risk from
the vaccination ri vs rv
PAYOFF FROM VACCINATION
• This will always depend upon the proportion of
the population that is vaccinated, p
E(P, p) = P (- rv) + (1 – P) (-ri πp)
•Set r (relative risk) to rv/ri
•This gives: E(P, p) = -rP - πp (1 – P)
TO SOLVE THE ESS
• You need to calculate the risk of infection πp
•This requires that you build an S.I.R. model
dS
= m (1- p ) - b SI - mS
dt
dI
= b SI - g I - m I
dt
dR
= m p + g I - mR
dt
TWO IMPORTANT
PARAMETERS
•Growth rate of the
disease:
•Infectious period:
R0 =
b
m +g
m
f=
g
THE BIG INSIGHT
• It is impossible to eradicate a disease through
voluntary vaccination whenever the perceived risk from
vaccination rv > 0
•For any perceived risk from vaccination (i.e. rv > 0), p*
< pcrit
•If rv > πp then P* = 0 otherwise 0 < P* < 1
•Recall, πp depends upon R0
•If r > 1, parents will not vaccinate
STABLE VACCINATION
ESS coverage as a function of vaccination risk
and risk of infection
RESPONSE TO A
VACCINATION SCARE
• The change in vaccine uptake ▵P depends upon
the new perceived risk from the vaccine and risk
from infection
ANOTHER INSIGHT
• A vaccination scare can cause vaccination rate
to drop (relatively easily) however it will be
relatively more difficult to restore vaccination
rates to pre-scare levels
WHAT’S MISSING?
• Seasonality
•Transient dynamics
•Variance in risk perception
•Social dynamics