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Dynamics of Infectious Diseases
A module in Phys 7654 (Spring 2010):
Basic Training in Condensed Matter Physics
Feb 24 - Mar 19
Chris Myers
[email protected]
Clark 517 / Rhodes 626 / Plant Sci 321
Wednesday, February 24, 2010
Overview of module
•
Introduction to models of infectious disease dynamics
- some basic biology of infectious diseases (not much)
- standard classes of models
‣ compartmental (fully-mixed)
‣ spatial (metapopulations & network-based)
- phenomenology of disease dynamics & control
‣ epidemic thresholds, herd immunity, critical component
‣
Wednesday, February 24, 2010
size, percolation, role of contact network structure,
stochastic vs. deterministic models, control strategies, etc.
case studies (FMD, SARS, measles, H1N1?)
Tentative schedule
•
•
•
•
•
•
•
•
•
Wed 2/24 : Lecture
Fri 2/26✧: Lecture
Wed 3/3
: Lecture; Homework #1 due (see website)
Fri 3/5
: No Class (Physics Prospective Grad Visit Day)
Wed 3/10 : Myers away - possibly a guest lecture
Fri 3/12✧: Lecture
Wed 3/17*: Lecture
Fri 3/19*: Lecture
3/20-3/28: Spring break; module finished
*APS March Meeting (Myers here. Who is away?)
✧Overlap with CAM colloquium (Fri 3:30)
Wednesday, February 24, 2010
Resources
•
Course website
- www.physics.cornell.edu/~myers/InfectiousDiseases
- also accessible via Basic Training website:
‣ people.ccmr.cornell.edu/~emueller/
-
Basic_Training_Spring_2010/Infectious_Diseases.html
contains links to relevant reading materials (some requiring
institutional subscription*), lecture slides, class schedule,
homeworks, etc. (continually updated)
* use Passkey from CIT:
https://confluence.cornell.edu/display/CULLABS/Passkey+Bookmarklet
Wednesday, February 24, 2010
Wednesday, February 24, 2010
Resources
•
Books
-
M. Keeling & P. Rohmani, Modeling Infectious Diseases in Humans and
Animals [K&R]; programs online at www.modelinginfectiousdiseases.org
-
O. Diekmann & J.A.P. Heesterbeek, Mathematical Epidemiology of Infectious
Diseases [D&H]
-
R.M. Anderson & R.M. May, Infectious Diseases of Humans [A&M]
•
D.J. Daley & J. Gani, Epidemic Modeling: An Introduction [D&G]
S. Ellner & J. Guckenheimer, Dynamic Models in Biology (Ch. 6) [E&G]
Local activity
-
EEID - Ecology and Evolution of Infections and Disease at Cornell
‣
-
website at www.eeid.cornell.edu, mailing list: [email protected]
8th annual EEID conference: http://www.eeidconference.org/
‣
Wednesday, February 24, 2010
to be held at Cornell in early June 2010
Simulations: Milling about at a conference
•
Individuals executing
random walk on a 2D
square lattice
•
Two individuals in
“contact” if they occupy
the same lattice site
•
RED = susceptible (not
infectious)
•
•
YELLOW = infectious
•
Infectious individuals
can infect susceptibles
with probability β
•
Infectious individuals
can recover with
probability γ
BLUE = recovered
(and immune)
Wednesday, February 24, 2010
Simulations: Milling about at a conference
•
Individuals executing
random walk on a 2D
square lattice
•
Two individuals in
“contact” if they occupy
the same lattice site
•
RED = susceptible (not
infectious)
•
•
YELLOW = infectious
•
Infectious individuals
can infect susceptibles
with probability β
•
Infectious individuals
can recover with
probability γ
BLUE = recovered
(and immune)
Wednesday, February 24, 2010
Dynamics
Wednesday, February 24, 2010
Tracing
distribution of
infectious contacts
network of infectious spread
Wednesday, February 24, 2010
Increased infectiousness
•
Same as before,
except two individuals
in “contact” if they
occupy the same
lattice site or are on
neighboring sites
•
RED = susceptible (not
infectious)
•
•
YELLOW = infectious
BLUE = recovered
(and immune)
Wednesday, February 24, 2010
Increased infectiousness
•
Same as before,
except two individuals
in “contact” if they
occupy the same
lattice site or are on
neighboring sites
•
RED = susceptible (not
infectious)
•
•
YELLOW = infectious
BLUE = recovered
(and immune)
Wednesday, February 24, 2010
Dynamics
Wednesday, February 24, 2010
Vaccination
•
Same as original
simulation, except 40%
of the population has
been vaccinated
•
RED = susceptible (not
infectious)
•
•
YELLOW = infectious
•
CYAN = vaccinated
(and immune)
BLUE = recovered
(and immune)
Wednesday, February 24, 2010
Vaccination
•
Same as original
simulation, except 40%
of the population has
been vaccinated
•
RED = susceptible (not
infectious)
•
•
YELLOW = infectious
•
CYAN = vaccinated
(and immune)
BLUE = recovered
(and immune)
Wednesday, February 24, 2010
Dynamics
Wednesday, February 24, 2010
Culling
•
Same as original
simulation, except
instead of recovery,
infecteds - and their
nearest neighbors - are
culled at rate c
-
ORIGIN Middle English : from
Old French coillier, based on
Latin colligere (see collect).
•
RED = susceptible (not
infectious)
•
•
YELLOW = infectious
•
Finding an “optimal”
culling strategy is a
politically and
economically sensitive
issue
GREY = culled (and
dead)
Wednesday, February 24, 2010
Culling
•
Same as original
simulation, except
instead of recovery,
infecteds - and their
nearest neighbors - are
culled at rate c
-
ORIGIN Middle English : from
Old French coillier, based on
Latin colligere (see collect).
•
RED = susceptible (not
infectious)
•
•
YELLOW = infectious
•
Finding an “optimal”
culling strategy is a
politically and
economically sensitive
issue
GREY = culled (and
dead)
Wednesday, February 24, 2010
Culling (take 2)
•
Same as last
simulation, with a
higher cull rate
•
RED = susceptible (not
infectious)
•
•
YELLOW = infectious
GREY = culled (and
dead)
Wednesday, February 24, 2010
Culling (take 2)
•
Same as last
simulation, with a
higher cull rate
•
RED = susceptible (not
infectious)
•
•
YELLOW = infectious
GREY = culled (and
dead)
Wednesday, February 24, 2010
Inhomogeneous mixing
•
Two segregated
subpopulations, with a
small percentage of
mixers who flow freely
back and forth
•
RED = susceptible (not
infectious)
•
•
YELLOW = infectious
BLUE = recovered
(and immune)
Wednesday, February 24, 2010
Inhomogeneous mixing
•
Two segregated
subpopulations, with a
small percentage of
mixers who flow freely
back and forth
•
RED = susceptible (not
infectious)
•
•
YELLOW = infectious
BLUE = recovered
(and immune)
Wednesday, February 24, 2010
Dynamics
Wednesday, February 24, 2010
Scales, foci & the multidisciplinary nature of
infectious disease modeling & control
response
- control strategies
- epidemiology
- public health & logistics
- economic impacts
between hosts
- disease ecology
- demography
- vectors, water, etc.
- zoonoses
- weather & climate
transmission
within-host
Wednesday, February 24, 2010
- virology, bacteriology, mycology, etc.
- immunology
Scales, foci & the multidisciplinary nature of
infectious disease modeling & control
Wednesday, February 24, 2010
Scales, foci & the multidisciplinary nature of
infectious disease modeling & control
Wednesday, February 24, 2010
Infection Timeline
K&R, Fig. 1.2
Wednesday, February 24, 2010
Compartmental models
•
Assumptions:
-
•
population is well-mixed: all contacts equally likely
only need to keep track of number (or concentration) of hosts in different states
or compartments
Typical states
-
Susceptible: not exposed, not sick, can become infected
Infectious: capable of spreading disease
Recovered (or Removed): immune (or dead), not capable of spreading disease
Exposed: “infected”, but not infectious
Carrier: “infected” (although perhaps asymptomatic), and capable of spreading
disease, but with a different probability
Wednesday, February 24, 2010
Compartmental models
SIR: lifelong immunity
S
I
SIS: no immunity
S
I
SEIR: SIR with latent (exposed) period
S
SIR with waning immunity
S
SIR with carrier state
E
R
I
I
R
R
C
S
I
R
Wednesday, February 24, 2010
adapted from K&R
An aside on graphical notations
S
I
state transitions
influence
R
adapted from K&R
S
I
infection
R
recovery
Petri Net: bipartite graph of places
(states) and transitions (reactions)
Wednesday, February 24, 2010
A&M, Fig. 2.1
Susceptible-Infected-Recovered (SIR)
•
•
Dates back to Kermack & McKendrick (1927), if
not earlier
Assume initially no demography
•
Let:
•
•
disease moving quickly through population of
fixed size N
X = # of susceptibles; proportion S = X/N
Y = # of infectives; proportion I = Y/N
Z = # of recovereds; proportion R = Z/N
note X+Y+Z = N, S+I+R=1
average infectious period = 1/γ
force of infection λ
-
per capita rate at which susceptibles become
infected
Wednesday, February 24, 2010
S
I
infection
dS/dt
dI/dt
dR/dt
R
recovery
= −βSI
= βSI − γI
= γI
Transmission & mixing
•
Must make assumption regarding form of transmission rate
- N = population size, Y = number of infectives, and β = product of contact
rates and transmission probability
•
•
mass action (frequency dependent, or proportional mixing)
- force of infection λ = βY/N; # contacts is independent of the population size
pseudo mass action (density dependent)
- force of infection λ = βY; # contacts is proportional to the population size
mass action:
κ = # contacts / unit time; c = prob. of transmission upon contact;
1-δq = prob. that a susceptible escapes infection in time δt
1 − δq
δq
=
=
(1 − c)
1 − e−βY δt/N where β = −κlog(1 − c)
(κY /N )δt
lim δq/δt = dq/dt = βY /N
δt→0
Wednesday, February 24, 2010
SIR dynamics
S
I
infection
R
recovery
dS/dt = −βSI
dI/dt = βSI − γI
γ= 1.0
dR/dt = γI
Wednesday, February 24, 2010
•
Outbreak dies out if transmission
rate is sufficiently low
•
Outbreak takes off if transmission
rate is sufficiently high
R0 and the epidemic threshold
Introduction into fully susceptible population
dI/dt = I(β − γ)
> 0 if β/γ > 1
< 0 if β/γ < 1
I
infection
(grows)
(dies out)
• define basic reproductive ratio:
R0 = β/γ
= average number of secondary cases
arising from an average primary case
in an entirely susceptible population
• epidemic threshold at R0 = 1
Wednesday, February 24, 2010
S
R
recovery
dS/dt
dI/dt
=
=
dR/dt
=
dS/dτ
dI/dτ
=
=
dR/dτ
τ
=
=
−βSI
βSI − γI
γI
−R0 SI
R0 SI − I
I
γt
R0 and the epidemic threshold
R0 = β/γ
= average number of secondary cases arising from an average primary
case in an entirely susceptible population
≈ transmission rate / recovery rate
• epidemic threshold at R0 = 1
- fraction of susceptibles must exceed γ/β
- R0-1 [relative removal (recovery) rate] must be small enough
to allow disease to spread
• estimating R0 from incidence data is a major goal when
confronted with new outbreak
Wednesday, February 24, 2010
Epidemic burnout
dS/dR = −βS/γ = −R0 S
S
I
infection
• integrate with respect to R:
S(t) = S(0)e
dS/dt
dI/dt
R ≤ 1 =⇒ S(t) ≥ e−R0 > 0
dR/dt
−R(t)R0
R
recovery
= −βSI
= βSI − γI
= γI
• there will always be some susceptibles who escape infection
• the chain of transmission eventually breaks due to the decline in
infectives, not due to the lack of susceptibles
Wednesday, February 24, 2010
Fraction of population infected
S(t) = S(0)e−R(t)R0
S(∞) = 1 − R(∞) = S(0)e
−R(∞)R0
• solve this equation (numerically) for R(∞) = total proportion of population infected
initial slope
= R0
R0 = 2
• outbreak: any sudden onset of infectious disease
• epidemic: outbreak involving non-zero fraction of population (in limit N→∞), or
which is limited by the population size
Wednesday, February 24, 2010
SIR with demography
birth
•
Allow for births and deaths
-
assume each happen at a
constant rate µ
R0 reduced to account for both
recovery and mortality
β
R0 =
γ+µ
S
infection
death
dS/dt
dI/dt
dR/dt
Wednesday, February 24, 2010
I
R
recovery
death
death
= µ − βSI − µS
= βSI − γI − µI
= γI − µR
Equilibria
birth
dS/dt = dI/dt = dR/dt = 0
S
I
infection
•
I
I
(βS − (γ + µ)) = 0 =⇒
= 0 or
S
= (γ + µ)/β = 1/R0
Disease-free equilibrium
death
R
recovery
death
death
dS/dt = µ − βSI − µS
dI/dt = βSI − γI − µI
dR/dt = γI − µR
(S ∗ , I ∗ , R∗ ) = (1, 0, 0)
•
Endemic equilibrium (only possible for R0>1):
1 µ
1
µ
(S , I , R ) = ( , (R0 − 1), 1 −
− (R0 − 1))
R0 β
R0
β
∗
∗
Wednesday, February 24, 2010
∗
Endemic equilibrium
1 µ
1
µ
(S , I , R ) = ( , (R0 − 1), 1 −
− (R0 − 1))
R0 β
R0
β
∗
∗
∗
R0 = 5
Wednesday, February 24, 2010
•
Pool of fresh susceptibles
enables infection to be
sustained
•
To establish equilibrium, must
have each infective productive
one new infective to replace
itself
•
S = 1/R0
Vaccination
• minimum size of susceptible population needed to sustain epidemic
ST = γ/β =⇒ R0 = S/ST
• vaccination reduces the size of the susceptible population
• immunizing a fraction p reduces R0 to:
i
R0
(1 − p)S
=
= (1 − p)R0
ST
• critical vaccination fraction is that required to reduce R0 < 1
1
pc = 1 −
R0
“herd immunity”
alternatively, pc needed to drive endemic equilibrium to I*=0:
1 µ
1
µ
∗ ∗
∗
(S , I , R ) = ( , (R0 − 1), 1 −
− (R0 − 1))
R0 β
R0
β
Wednesday, February 24, 2010
K&R, Fig. 8.1