Annual epidemics and natural selection in host-pathogen systems - - PowerPoint PPT Presentation

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Annual epidemics and natural selection in host-pathogen systems - - PowerPoint PPT Presentation

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Annual epidemics and natural selection in host-pathogen systems Viggo Andreasen Roskilde University November 26, 2003 Annual epidemics and selection Annual epidemics Onset of epidemic season If susceptible population exceeds threshold an

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Annual epidemics and natural selection in host-pathogen systems

Viggo Andreasen Roskilde University November 26, 2003

Annual epidemics and selection

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Annual epidemics

Onset of epidemic season

If susceptible population exceeds threshold an epidemic occurs

↓ During epidemic season

SIR-type epidemic

↓ Between epidemic seasons

Other processes add to size of susceptible population

↓

Applications:

  • Disease-induced selection (Gillespie, 1975)
  • Disease regulation of hosts (May, 1985; Dwyer et al 2000)
  • Influenza drift (Andreasen,2003)
  • Influenza drift and epidemic size (Boni et al, submitted)
  • Pruning of influenza phylogeny (Andreasen & Sasaki,in prep)

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Outline

  • Annual epidemics
  • Annual epidemics as a way to model disease-

induced selection in diploids

  • Annual epidemics in the description of influenza

epidemiology

  • Virus competition in annual epidemics

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Disease-induced selection in diploids Challenges for the modeller:

  • Host lifespan ≫ infection period
  • Good genetic models for:
  • generation-to-generation
  • slow selection
  • Good epidemic models for:
  • transmission dynamics during an epidemic
  • endemic diseases with constant pop size

Idea: assume one epidemic in each host generation

Annual epidemics and selection 3/29

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The Gillespie model

  • One autosomal locus with two alleles and random mating

Example: resistance is dominant

  • AA susceptible to disease
  • AB and BB resistant

Fitness of uninfected AA 1 Fitness of infected AA 1 − u Fitness of AB and BB 1 − σ p = frequency af A-allele q = 1 − p frequency of B-allele

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The epidemic season

dSAA dt = −τAAΛSAA dIAA dt = τAAΛSAA − µAAIAA Λ = βAAIAA+βABIAB + βBBIBB SAA(0) = p2N IAA(0) ≈ Λ ≪ 1 Fraction infected during the epidemic z z = 1 − e−zp2R0 Effect of disease on fitness of AA: WAA = 1 − z + (1 − u)z = 1 − uz.

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Long term dynamics

At onset of epidemic season frequency of A is p. After

  • epidemic
  • other selective factors
  • perfect regulation of populatrion size !

Frequency of A at onset of next season: p′ = p2WAA + pqWAB ¯ W = (1 − uz)p2 + (1 − σ)pq (1 − uz)p2 + (1 − σ)q(1 + p) (Stable) equilibrium at z = σ/u p =

  • − log(1 − z)/zR0

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Annual epidemics and influenza epidemiology

  • Influenza’s natural history
  • The epidemiology of a drifting virus
  • Drift length and epidemic size
  • Pruning of flu phylogeny

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Earn et al (2002)

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Influenza A subtypes

Cox & Fukuda, 1998

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Deaths caused by P&I in USA

Ferguson et al 2003

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Phylogeny of Influenza A

Fitch et al, 1997

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Reinfection after natural infection H3N2 Houston Family Study

Frank & Taber, 1983

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Reinfection after natural infection H1N1 Houston Family Study

Frank & Taber, 1983

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Reinfection of vaccinees

Pease, 1987 after Gill & Murphy 1976

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Cross-immunity in vitro

Levine, 1992

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Epidemiology of a drifting virus discrete version of model by Pease 1987

  • In each season one new strain appears
  • Prior to each season the strain drifts a fixed

amount

  • If possible an epidemic occurs
  • Epidemic burns out before season is over
  • Susceptibility and infectivity depends of number
  • f seasons since last infection
  • SIR-type dynamics
  • No vital dymanics

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Annual model for flu drift Si : # of hosts who have not been infected in this season and whoes most recent infection occurred i seasons ago Ii : # of hosts who are currently infected and whoes most recent infection occurred i seasons ago Sn, In n or more seasons ago At start of season Si(0) = 1 Ii(0) ≪ 1

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During epidemic

˙ Si = −τiΛSi ˙ Ii = τiΛSi − νIi Λ = β

  • σiIi

Outcome of epidemic φ = Sn(∞)

Sn(0)

Re = β ν

  • σiτiSi(0)

If Re> 1 then 0 < φ < 1 solves 0 = log φ + β/ν σiSi(0)(1 − φτi) and φτi = Si(∞)/Si(0) If Re< 1 No epidemic φ = 1

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Year-to-year dynamics (onset → onset)

F :      S1 S2 . . . Sn−1      →      (1 − φτi)Si φτ1S1 . . . φτn−2Sn−2      Sn = 1 − Si is redundant Γ = { S | Si ≤ 1, si ≥ 0 } F : Γ → Γ Cases n = 2, 3, τi = 1, i.e. infectivity reduction only; ⇒ φ-eqn simplifies 0 = log φ + q(1 − φ) q = R0 σiSi(0)

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Dynamics for Annual flu epidemics, n = 2

Andreasen 2003

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Bifurcation diagram for annual flu epidemics, n = 3

Andreasen 2003

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Attractor in annual flu model, n = 3

Andreasen 2003

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Conclusions flu-drift model

  • Focus on host immune structure
  • Explicit rule for introduction of susceptible
  • Recognizes seasonality and pronounced epidemics
  • Epidemic levels as observed in nature
  • Not a word on time within season
  • Not a word about persistence or causes of drift

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Aminoacid substitutions in HA1 (H3N2) Fitch et al, 1997

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Drift speed and epidemic size Boni et al submitted

  • Seasonal dynamics as before; infectivity reduction
  • X-immunity decays with ”distance”

σ = 1 − exp(−d)

  • Distance is additive over years
  • Distance grows linearly with size of epidemic I, d = κ + λI
  • S = σiSi weighted susceptibility
  • Outcome of epidemic in terms of S

f(S) = 1 − κφλ(1 − φS) where φ prob of not being infected

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Dynamics of size-dependent drift

Boni et al ms

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Invasion and persistence of drifting virus

Boni et al ms

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Virus selection in annual epidemics

  • In haploids competition ≈ selection
  • Assume two virus types I and Y
  • Epidemics within a season

˙ S = −βIIS − βY Y S ˙ I = βIIS − νII ˙ Y = βY Y S − νY Y

  • Only the viral type with the highest R0 will

produce an epidemic

Saunders, 1981

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