Upload
dangxuyen
View
219
Download
4
Embed Size (px)
Citation preview
Dynamics of Infectious Diseases
Chris [email protected]
Clark 517 / Rhodes 626 / Plant Sci 321
A module in Phys 7654 (Spring 2010):Basic Training in Condensed Matter Physics
Feb 24 - Mar 19
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
size, percolation, role of contact network structure, stochastic vs. deterministic models, control strategies, etc.
‣ case studies (FMD, SARS, measles, H1N1?)
Wednesday, February 24, 2010
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
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/
‣ to be held at Cornell in early June 2010
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
• BLUE = recovered (and immune)
• Infectious individuals can infect susceptibles with probability β
• Infectious individuals can recover with probability γ
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
• BLUE = recovered (and immune)
• Infectious individuals can infect susceptibles with probability β
• Infectious individuals can recover with probability γ
Wednesday, February 24, 2010
Tracing
network of infectious spread
distribution of infectious contacts
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
Vaccination
• Same as original simulation, except 40% of the population has been vaccinated
• RED = susceptible (not infectious)
• YELLOW = infectious
• BLUE = recovered (and immune)
• CYAN = vaccinated (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
• BLUE = recovered (and immune)
• CYAN = vaccinated (and immune)
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
• GREY = culled (and dead)
• Finding an “optimal” culling strategy is a politically and economically sensitive issue
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
• GREY = culled (and dead)
• Finding an “optimal” culling strategy is a politically and economically sensitive issue
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
Scales, foci & the multidisciplinary nature of infectious disease modeling & control
within-host
between hosts
- virology, bacteriology, mycology, etc.- immunology
- disease ecology- demography- vectors, water, etc.- zoonoses- weather & climate
response
- control strategies- epidemiology- public health & logistics- economic impacts
transmission
Wednesday, February 24, 2010
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
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
S I R
S I
S I RE
S I R
S IR
C
SIR: lifelong immunity
SIS: no immunity
SEIR: SIR with latent (exposed) period
SIR with waning immunity
SIR with carrier state
adapted from K&R
Wednesday, February 24, 2010
An aside on graphical notations
S I R
A&M, Fig. 2.1
adapted from K&R
Petri Net: bipartite graph of places (states) and transitions (reactions)
state transitionsinfluence
S I R
infection recovery
Wednesday, February 24, 2010
Susceptible-Infected-Recovered (SIR)
S I R
infection recovery
• Dates back to Kermack & McKendrick (1927), if not earlier
• Assume initially no demography
- disease moving quickly through population of fixed size N
• Let:
- 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
dS/dt = −βSI
dI/dt = βSI − γI
dR/dt = γI
Wednesday, February 24, 2010
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
1− δq = (1− c)(κY/N)δt
δq = 1− e−βY δt/N where β = −κlog(1− c)
limδt→0
δq/δt = dq/dt = βY/N
mass action:κ = # contacts / unit time; c = prob. of transmission upon contact; 1-δq = prob. that a susceptible escapes infection in time δt
Wednesday, February 24, 2010
SIR dynamics
S I R
infection recovery
dS/dt = −βSI
dI/dt = βSI − γI
dR/dt = γI
γ= 1.0
• Outbreak dies out if transmission rate is sufficiently low
• Outbreak takes off if transmission rate is sufficiently high
Wednesday, February 24, 2010
R0 and the epidemic threshold
dS/dt = −βSI
dI/dt = βSI − γI
dR/dt = γI
• 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
dI/dt = I(β − γ)> 0 if β/γ > 1< 0 if β/γ < 1
(grows)(dies out)
Introduction into fully susceptible populationS I R
infection recovery
dS/dτ = −R0SI
dI/dτ = R0SI − I
dR/dτ = I
τ = γt
Wednesday, February 24, 2010
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
S I R
infection recovery
dS/dt = −βSI
dI/dt = βSI − γI
dR/dt = γI
• integrate with respect to R:
dS/dR = −βS/γ = −R0S
S(t) = S(0)e−R(t)R0
R ≤ 1 =⇒ S(t) ≥ e−R0 > 0
• 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 infectedS(t) = S(0)e−R(t)R0
• solve this equation (numerically) for R(∞) = total proportion of population infected
S(∞) = 1−R(∞) = S(0)e−R(∞)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
initial slope = R0
Wednesday, February 24, 2010
SIR with demography
• Allow for births and deaths
- assume each happen at a constant rate µ
- R0 reduced to account for both recovery and mortality
S I R
infection recovery
birth
death death death
dS/dt = µ− βSI − µS
dI/dt = βSI − γI − µI
dR/dt = γI − µR
R0 =β
γ + µ
Wednesday, February 24, 2010
Equilibria
S I R
infection recovery
birth
death death death
dS/dt = µ− βSI − µS
dI/dt = βSI − γI − µI
dR/dt = γI − µR
dS/dt = dI/dt = dR/dt = 0
I (βS − (γ + µ)) = 0 =⇒I = 0 orS = (γ + µ)/β = 1/R0
(S∗, I∗, R∗) = (1
R0,µ
β(R0 − 1), 1− 1
R0− µ
β(R0 − 1))
• Disease-free equilibrium
• Endemic equilibrium (only possible for R0>1):
(S∗, I∗, R∗) = (1, 0, 0)
Wednesday, February 24, 2010
Endemic equilibrium
R0 = 5
• 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
(S∗, I∗, R∗) = (1
R0,µ
β(R0 − 1), 1− 1
R0− µ
β(R0 − 1))
Wednesday, February 24, 2010
Vaccination• minimum size of susceptible population needed to sustain epidemic
• vaccination reduces the size of the susceptible population
• immunizing a fraction p reduces R0 to:
• critical vaccination fraction is that required to reduce R0 < 1
ST = γ/β =⇒ R0 = S/ST
Ri0 =
(1− p)SST
= (1− p)R0
pc = 1− 1R0
“herd immunity”
K&R, Fig. 8.1
alternatively, pc needed to drive endemic equilibrium to I*=0:
(S∗, I∗, R∗) = (1
R0,µ
β(R0 − 1), 1− 1
R0− µ
β(R0 − 1))
Wednesday, February 24, 2010