dynamics of infectious diseases -...

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?)


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)


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


http://www.physics.cornell.edu/~myers/InfectiousDiseases

http://www.physics.cornell.edu/~myers/InfectiousDiseases

https://confluence.cornell.edu/display/CULLABS/Passkey+Bookmarklet

https://confluence.cornell.edu/display/CULLABS/Passkey+Bookmarklet

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


http://www.modelinginfectiousdiseases.org

http://www.modelinginfectiousdiseases.org

http://www.eeid.cornell.edu

http://www.eeid.cornell.edu

mailto:[email protected]

mailto:[email protected]

http://www.eeidconference.org

http://www.eeidconference.org

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


Dynamics


Tracing

network of infectious spread

distribution of infectious contacts


Increased infectiousness

• Same as before, except two individuals in “contact” if they occupy the same lattice site or are on neighboring sites





Dynamics


Vaccination

• Same as original simulation, except 40% of the population has been vaccinated




• CYAN = vaccinated (and immune)


Dynamics


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).



• GREY = culled (and dead)

• Finding an “optimal” culling strategy is a politically and economically sensitive issue


Culling (take 2)

• Same as last simulation, with a higher cull rate



• GREY = culled (and dead)


Inhomogeneous mixing

• Two segregated subpopulations, with a small percentage of mixers who flow freely back and forth





Dynamics


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


Scales, foci & the multidisciplinary nature of infectious disease modeling & control


Infection Timeline

K&R, Fig. 1.2


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


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


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


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


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


SIR dynamics

S I R

infection recovery

dS/dt = −βSI


dR/dt = γI

γ= 1.0

• Outbreak dies out if transmission rate is sufficiently low

• Outbreak takes off if transmission rate is sufficiently high


R0 and the epidemic threshold

dS/dt = −βSI


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


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


Epidemic burnout

S I R

infection recovery

dS/dt = −βSI


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


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


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


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)


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))


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))


dynamics of infectious diseases -...

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