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

Introduction Results Conclusions References

Outline

1 IntroductionBackgroundModel

2 Results

3 Conclusions

Introduction Results Conclusions References

Acknowledgements

People Colleen Webb

Support NSERC Discovery grant

Introduction Results Conclusions References

Outline

1 IntroductionBackgroundModel

2 Results

3 Conclusions

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)

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)

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)

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)

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

Introduction Results Conclusions References

Question

What is the ESS for mutation rate?

Why?

How does it depend on parameters?

Introduction Results Conclusions References

Model structure

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)

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)

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)

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)

Introduction Results Conclusions References

Outline

1 IntroductionBackgroundModel

2 Results

3 Conclusions

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

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)

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)

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)

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

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

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)

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 . . .

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

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

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

Introduction Results Conclusions References

Outline

1 IntroductionBackgroundModel

2 Results

3 Conclusions

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

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

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

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

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!

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.

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)