Feb 24, 2010 ... Introduction to models of infectious disease dynamics. - some basic biology of
infectious diseases (not much). - standard classes of models.
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
