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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