Introduction to

1
Introduction to Epidemology
The Population Dynamics of Disease
2
Lecture Goals

• The S.I.R. model
• background
• conceptual model
• quantitative model
• Herd Immunity
• disease dynamics
• vaccination for disease eradication

3
The S.I.R. Model

• tracks disease through host effects
• three types of host
• Susceptible (S)
• Infected (I)
• Recovered (R)
• some period of immunity after infection)
• instantaneous model
• 3 ordinary differential equations
• no lags, but many diseases have incubation
• a starting point for many current models

4
The S.I.R. Model
Infected
Susceptible
Recovered
5
Change in the Susceptible Componentof the Host
Population
dS/dt b(SIR) - dS bSI gR
b(SIR) new births (lack immunity) dS natural
mortality bSI infection (density
dependent) gR loss of immunity
S, I, and R are the numbers of each population
component (variables). b, d, b, and g are the
rate parameters for each process.
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Change in the Infected Componentof the Host
Population
dI/dt bSI (adn) I
bSI new infections (adn) I loss from -
disease induced mortality (a)
(virulence) - natural mortality (d) -
recovery (n)
Recovered individuals are immune.
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Change in the Recovered Componentof the Host
Population
dR/dt nI (dg) R
nI newly recovered from infection (dg)
R loss from - natural mortality (d) - loss of
immunity (g)
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The Classic S.I.R. Model
dS/dt b(SIR) - dS bSI gR
dI/dt bSI (adn) I
dR/dt nI (dg) R
bd
bgtd
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Population Dynamics of the Disease
R0 net reproductive rate of the disease bSL
S number of susceptible hosts b infection
rate as in SIR model L life expectancy of
infected host (adn) 1 from the SIR model
representing the lose of infected individuals
by mortality and recovery
What can we do with such a simple model?
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When R0 gt 1, disease is increasing (epidemic)
When R0 1, disease is in equilibrium (endemic)
S (bL)-1
When R0 lt 1, disease is in decline (eradication)
S (bL)-1 is the threshold value of the
infection.
The threshold value indicates the number of
susceptible hosts required for the disease to
spread.
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Herd Immunity
R0 bSL S R0/(bL)
Initially (t0) S0 R0/(bL) After successful
immunization R0 1 Endemic (te) Se
1/(bL) Proportion to immunize
PV (S0 Se) / S0 1 Se/S0 1 1/R0
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1.0
Eradication
measles
0.8
rubella
0.6
smallpox
Persistence
PV
0.0
0
30
10
20
R0