ESS of minimal mutation rate in an evo-epidemiological model
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Introduction Results Conclusions References
Evolutionary stability of minimal mutation rates inan evo-epidemiological model
Ben Bolker and Michael D. Birch, McMaster UniversityDepartments of Mathematics & Statistics and Biology
ESA
August 2014
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Introduction Results Conclusions References
Outline
1 IntroductionBackgroundModel
2 Results
3 Conclusions
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Introduction Results Conclusions References
Acknowledgements
People Colleen Webb
Support NSERC Discovery grant
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Introduction Results Conclusions References
Outline
1 IntroductionBackgroundModel
2 Results
3 Conclusions
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Introduction Results Conclusions References
Evolution of evolvability
Genetic variation + selection → evolution by natural selection
Mutation (+ selection) → genetic variation
How do mutation rates evolve?
Stable environment: minimal is bestVariable environment:tradeo� between environment-tracking and deleterious e�ects(Kimura, 1967; Ishii et al., 1989)
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Introduction Results Conclusions References
Evolution of evolvability
Genetic variation + selection → evolution by natural selection
Mutation (+ selection) → genetic variation
How do mutation rates evolve?
Stable environment: minimal is bestVariable environment:tradeo� between environment-tracking and deleterious e�ects(Kimura, 1967; Ishii et al., 1989)
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Introduction Results Conclusions References
Eco-evolutionary dynamics
feedback between ecological and evolutionary processes(Lenski and May, 1994; Luo and Koelle, 2013)
e.g. density-dependent selection
no separation of time scales;typical emphasis on short-term dynamics(cf. adaptive dynamics)
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Introduction Results Conclusions References
Host-parasite dynamics
basic compartmental model (SIR)
density-dependent transmission
seasonally varying transmission
vital dynamics (constant total birth rate,constant per capita mortality)
consider evolution of virulence (exploitation)
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Introduction Results Conclusions References
Tradeo�s
assumetransmission-virulence
tradeo� . . .
simplest deceleratingcurve:β(α) ∝ α1/γ(Frank, 1996; Bolkeret al., 2010)
Virulence
Fitness(w
)
w = Sβ(α)− (α+ µ)
S=0.7
S=0.9
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Introduction Results Conclusions References
Question
What is the ESS for mutation rate?
Why?
How does it depend on parameters?
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Introduction Results Conclusions References
Model structure
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Introduction Results Conclusions References
Model equations
dS
dt= ν︸︷︷︸
birth
− S
∫ ∞0
β(α, t)i(α, t) dα︸ ︷︷ ︸infection
− µS︸︷︷︸death
∂i
∂t= [Sβ(α, t)︸ ︷︷ ︸
infection
− (α+ µ)︸ ︷︷ ︸vir+natural mort
]i(α, t) + D∂2i
∂α2︸ ︷︷ ︸mutation
β(α, t) = cα1/γ︸ ︷︷ ︸tradeo� curve
·[1+ δ sin
(2πt
τ
)]︸ ︷︷ ︸
seasonal forcing
No-�ux boundary conditions at 0and in�nity ( ∂i
∂α → 0 as α→∞)
(play example solution)
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Introduction Results Conclusions References
Model equations
dS
dt= ν︸︷︷︸
birth
− S
∫ ∞0
β(α, t)i(α, t) dα︸ ︷︷ ︸infection
− µS︸︷︷︸death
∂i
∂t= [Sβ(α, t)︸ ︷︷ ︸
infection
− (α+ µ)︸ ︷︷ ︸vir+natural mort
]i(α, t) + D∂2i
∂α2︸ ︷︷ ︸mutation
β(α, t) = cα1/γ︸ ︷︷ ︸tradeo� curve
·[1+ δ sin
(2πt
τ
)]︸ ︷︷ ︸
seasonal forcing
No-�ux boundary conditions at 0and in�nity ( ∂i
∂α → 0 as α→∞)
(play example solution)
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Introduction Results Conclusions References
Model parameters
Symbol Meaning Baseline
ν birth rate (nondim.)µ birth rate (nondim.)c virulence slope 5γ virulence curvature 2δ seasonal amplitude 0.3 (< 1)τ seasonal period 10D mutation varies
w(α, t) �tness ≡ (di/dt)/i =Sβ(α, t)− (α+ µ)
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Introduction Results Conclusions References
Moment approximation: justi�cation
derive coupled equations for moments of virulence distribution
then close (truncate) the series
vanishing higher moments → Gaussian(Turelli and Barton, 1994)
→ coupled ODEs for S , I , 〈α〉, σ2α. . . depending on state variables, parameters, gradient andcurvature of β(α)
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Introduction Results Conclusions References
Outline
1 IntroductionBackgroundModel
2 Results
3 Conclusions
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Introduction Results Conclusions References
Maximum viable mutation rate
in a constant environment (δ = 0)
invasion from disease-free equilibrium
separable solution
parasite cannot invade if
D > −2 [w (α0)]2
w ′′ (α0)≡ Dmax
perturbation analysis extends the result to the seasonal(δ > 0) case
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Introduction Results Conclusions References
ESS
Strain 2 can't invade strain 1 if its time-averaged �tness is < 0:
〈β(α, t)〉2· S1,∗(t)−
(〈α〉
2+ µ
)< 0
(· ≡ time-average over one period)If time-averaged equilibrium of full model ≈ unforced equilibrium ofmoment equations then this implies
S1,∗ < S2,∗ (unforced mom eq)
Conjecture: strain 1 is ESS if
S1,∗(t) < S2,∗(t)
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Introduction Results Conclusions References
ESS
Strain 2 can't invade strain 1 if its time-averaged �tness is < 0:
〈β(α, t)〉2· S1,∗(t)−
(〈α〉
2+ µ
)< 0
(· ≡ time-average over one period)If time-averaged equilibrium of full model ≈ unforced equilibrium ofmoment equations then this implies
S1,∗ < S2,∗ (unforced mom eq)
Conjecture: strain 1 is ESS if
S1,∗(t) < S2,∗(t)
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Introduction Results Conclusions References
ESS
Strain 2 can't invade strain 1 if its time-averaged �tness is < 0:
〈β(α, t)〉2· S1,∗(t)−
(〈α〉
2+ µ
)< 0
(· ≡ time-average over one period)If time-averaged equilibrium of full model ≈ unforced equilibrium ofmoment equations then this implies
S1,∗ < S2,∗ (unforced mom eq)
Conjecture: strain 1 is ESS if
S1,∗(t) < S2,∗(t)
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Introduction Results Conclusions References
Resource-depletion result
Multi-strain competition: strain with lowest S∗ wins
Echoes ecological (Tilman's R∗), epidemiological results (Dayand Gandon, 2007)
In other systems coexistence can occur in periodic systems(Cushing, 1980)
Allows strain-by-strain analysis
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Introduction Results Conclusions References
Time-average vs unforced moment equation results
0.9
1.0
1.1
1.2
0.1 0.2 0.3 0.4 0.5d
S*(t)
/ S
uf*
Moment Approx.
Full Model
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Introduction Results Conclusions References
Mutation vs time-averaged susceptibles
10.5
51020
0.050.10.2
0.3
0.4
0.5
1311
9
7
5
3
1.5
1.75
2
2.252.5
Seasonal Period ( t ) Seasonal Amplitude ( d )
Trade-off Multiplier ( c ) Trade-off Power ( g )
0.40
0.42
0.44
0.46
0.42
0.45
0.48
0.51
0.2
0.4
0.6
0.40
0.42
0.44
0.46
0.48
0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6D
S*(t)
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Introduction Results Conclusions References
Why doesn't mutation increase �tness?
net �tness e�ect = (improved tracking) - (increased spreadaround optimum)
〈w(α, t)〉 = w0(t)︸ ︷︷ ︸max �tness
− wmut(t)︸ ︷︷ ︸mutation load
− wtrack(t)︸ ︷︷ ︸distance from optimum
wmut(t) ≈ 1
2Sβαα(α0, t)σ
2α
wtrack(t) ≈ 1
2Sβαα(α0, t) (〈α〉 − α0)2
Expect wmut(t) ↑, wtrack(t) ↓ as D increases . . .
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Introduction Results Conclusions References
E�ect of mutation on optimum tracking
2010
5
0.51
0.10.050.20.30.4
0.5
1113975
3
1.51.75
2
2.25
2.5
Seasonal Period ( t ) Seasonal Amplitude ( d )
Trade-off Multiplier ( c ) Trade-off Power ( g )0.0
0.1
0.2
0.3
0.4
0.5
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.4
0.1
0.2
0.3
0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6D
( < a
> -
a0)
2
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Introduction Results Conclusions References
Time-variation (γ = 2)
100 110 120 130 140 150
1.2
1.4
1.6
1.8
t (Time)
α (V
irule
nce)
< α >
α0
σα
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Introduction Results Conclusions References
Distribution (γ = 2)
0 2 4 6 8 10
0.00
0.01
0.02
0.03
0.04
α (Virulence)
i (In
fect
ious
Den
sity
)
< α >
α0
σα
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Introduction Results Conclusions References
Outline
1 IntroductionBackgroundModel
2 Results
3 Conclusions
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Introduction Results Conclusions References
Conclusions
Very high mutation is bad
No coexistence via temporal partitioning
Even moderate mutation is bad (in this case)
Lower bound on virulence is important
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Introduction Results Conclusions References
Conclusions
Very high mutation is bad
No coexistence via temporal partitioning
Even moderate mutation is bad (in this case)
Lower bound on virulence is important
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Introduction Results Conclusions References
Conclusions
Very high mutation is bad
No coexistence via temporal partitioning
Even moderate mutation is bad (in this case)
Lower bound on virulence is important
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Introduction Results Conclusions References
Conclusions
Very high mutation is bad
No coexistence via temporal partitioning
Even moderate mutation is bad (in this case)
Lower bound on virulence is important
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Introduction Results Conclusions References
Loose ends
What could overturn our results?
Lower curvature values? (γ = 1.05 doesn't help)
Di�erent tradeo� curve?
Log-scale virulence (geometric Brownian motion)?
Allow mutualism (α < 0)?
Get the paper at http://tinyurl.com/birchbolker!
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Introduction Results Conclusions References
References
Bolker, B.M., Nanda, A., and Shah, D., 2010. J.R. Soc. Interface, 7(46):811�822.
Cushing, J.M., 1980. Journal of MathematicalBiology, 10(4):385�400. ISSN 0303-6812,1432-1416. doi:10.1007/BF00276097.
Day, T. and Gandon, S., 2007. Ecology Letters,10(10):876�888. ISSN 1461-0248.doi:10.1111/j.1461-0248.2007.01091.x.
Frank, S.A., 1996. Q Rev Biol, 71(1):37�78.
Ishii, K., Matsuda, H., et al., 1989. Genetics,121(1):163�174.
Kimura, M., 1967. Genetics Research,9(01):23�34.
Lenski, R.E. and May, R.M., 1994. J Theor Biol,169:253�265.
Luo, S. and Koelle, K., 2013. The AmericanNaturalist, 181(S1):S58�S75. ISSN0003-0147. doi:10.1086/669952.
Turelli, M. and Barton, N.H., 1994. Genetics,138(3):913 �941.
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Introduction Results Conclusions References
Moment approximation: equations
dS
dt= ν − SI
[β(〈α〉 , t) + 1
2σ2αβαα(〈α〉 , t)
]− µS (1a)
dI
dt= [Sβ(〈α〉 , t)− (〈α〉+ µ)] I +
1
2SIσ2αβαα(〈α〉 , t) (1b)
d 〈α〉dt
= σ2α [Sβα(〈α〉 , t)− 1] (1c)
dσ2αdt
=[σ2α]2
Sβαα(〈α〉 , t) + 2D, (1d)