a talk given in the EEB department at the University of Tennessee, Knoxville on eco-evolutionary dynamics of virulence

1. Overview Simple models HIV case study Conclusions References
Eco-evolutionary virulence of pathogens:
the devil is (still) in the details
Ben Bolker, McMaster University
Departments of Mathematics & Statistics and Biology
University of Tennessee
15 September 2016

2. Overview Simple models HIV case study Conclusions References
Outline
1 Overview
The evolution of host-pathogen theory
Toy models
2 Transient virulence: simple models
Overview
Toy model
Myxomatosis model
3 Transient virulence: HIV case study
4 Conclusions

3. Overview Simple models HIV case study Conclusions References
Acknowledgements
People Arjun Nanda and Dharmini Shah; Christophe Fraser;
Daniel Park
Support NSF IRCEB grant 9977063; QSE3 IGERT; NSERC
Discovery grant

4. Overview Simple models HIV case study Conclusions References
Outline
1 Overview
The evolution of host-pathogen theory
Toy models
2 Transient virulence: simple models
Overview
Toy model
Myxomatosis model
3 Transient virulence: HIV case study
4 Conclusions

5. Overview Simple models HIV case study Conclusions References
Host-pathogen evolutionary biology
Why?
Intellectual merit
Coevolutionary loops
Cryptic effects
Eco-evolutionary dynamics (Luo and Koelle, 2013)
Lots of data (for humans)
Broader applications
Medical
Conservation and management
Outreach

6. Overview Simple models HIV case study Conclusions References
Host-pathogen evolutionary biology
Why?
Intellectual merit
Coevolutionary loops
Cryptic effects
Eco-evolutionary dynamics (Luo and Koelle, 2013)
Lots of data (for humans)
Broader applications
Medical
Conservation and management
Outreach

7. Overview Simple models HIV case study Conclusions References
Virulence: definitions
General public: badness
Plant biologists: infectivity
Evolutionists: loss of host fitness
Theoreticians: rate of host mortality

8. Overview Simple models HIV case study Conclusions References
Evolution of virulence evolution theory
Classical avirulence theory
Anderson & May/Ewald virulence as an evolved (adaptive) trait.
Tradeoff theory, modes of transmission.
post-Ewald mathematical models based on R0
Now tradeoff backlash
within-host dynamics/multi-level models
eco-evolutionary dynamics
host effects: resistance vs tolerance vs virulence

9. Overview Simple models HIV case study Conclusions References
Evolution of virulence evolution theory
Classical avirulence theory
Anderson & May/Ewald virulence as an evolved (adaptive) trait.
Tradeoff theory, modes of transmission.
post-Ewald mathematical models based on R0
Now tradeoff backlash
within-host dynamics/multi-level models
eco-evolutionary dynamics
host effects: resistance vs tolerance vs virulence

10. Overview Simple models HIV case study Conclusions References
Evolution of virulence evolution theory
Classical avirulence theory
Anderson & May/Ewald virulence as an evolved (adaptive) trait.
Tradeoff theory, modes of transmission.
post-Ewald mathematical models based on R0
Now tradeoff backlash
within-host dynamics/multi-level models
eco-evolutionary dynamics
host effects: resistance vs tolerance vs virulence

11. Overview Simple models HIV case study Conclusions References
Evolution of virulence evolution theory
Classical avirulence theory
Anderson & May/Ewald virulence as an evolved (adaptive) trait.
Tradeoff theory, modes of transmission.
post-Ewald mathematical models based on R0
Now tradeoff backlash
within-host dynamics/multi-level models
eco-evolutionary dynamics
host effects: resistance vs tolerance vs virulence

12. Overview Simple models HIV case study Conclusions References
Basic tradeoff theory: assumptions
Homogeneous, non-evolving hosts
No super- or coinfection
Horizontal, direct transmission
Tradeoff between rate of transmission
and length of infectious period
Infectious period ∝ 1/clearance rate

13. Overview Simple models HIV case study Conclusions References
Tradeoffs, R0, and r
Disease−induced mortality
Transmission
rate
µ 0 1 2 3 4 5

14. Overview Simple models HIV case study Conclusions References
Tradeoffs, R0, and r
Disease−induced mortality
Transmission
rate
µ 0 1 2 3 4 5

15. Overview Simple models HIV case study Conclusions References
Tradeoffs, R0, and r
Disease−induced mortality
Transmission
rate
µ 0 1 2 3 4 5
q
q

16. Overview Simple models HIV case study Conclusions References
Outline
1 Overview
The evolution of host-pathogen theory
Toy models
2 Transient virulence: simple models
Overview
Toy model
Myxomatosis model
3 Transient virulence: HIV case study
4 Conclusions

17. Overview Simple models HIV case study Conclusions References
Epidemiological model
SIR model
Constant population size
(birth=death)
Rescale: µ = 1, N = 1
(time units of host lifespan)
I
S
disease−
mortality
(α)induced
mortality
(µ)
birth
R
infection (β)
recovery

18. Overview Simple models HIV case study Conclusions References
Epidemiological model
SIR model
Constant population size
(birth=death)
Rescale: µ = 1, N = 1
(time units of host lifespan)
I
S
disease−
mortality
(α)induced
mortality
(µ)
birth
R
infection (β)
recovery

19. Overview Simple models HIV case study Conclusions References
Epidemiological model
SIR model
Constant population size
(birth=death)
Rescale: µ = 1, N = 1
(time units of host lifespan)
I
S
disease−
mortality
(α)induced
mortality
(µ)
birth
R
infection (β)
recovery

20. Overview Simple models HIV case study Conclusions References
The model (2): evolutionary dynamics
Incorporate trait dynamics
Standard quantitative genetics model (Abrams, 2001):
Fitness depends on mean trait value (¯α)
and ecological context (proportion susceptible)
Constant additive genetic variance Vg
Trait evolves toward increased fitness:
rate proportional to ∆fitness/∆trait

21. Overview Simple models HIV case study Conclusions References
The model (2): evolutionary dynamics
Incorporate trait dynamics
Standard quantitative genetics model (Abrams, 2001):
Fitness depends on mean trait value (¯α)
and ecological context (proportion susceptible)
Constant additive genetic variance Vg
Trait evolves toward increased fitness:
rate proportional to ∆fitness/∆trait

22. Overview Simple models HIV case study Conclusions References
The model (2): evolutionary dynamics
Incorporate trait dynamics
Standard quantitative genetics model (Abrams, 2001):
Fitness depends on mean trait value (¯α)
and ecological context (proportion susceptible)
Constant additive genetic variance Vg
Trait evolves toward increased fitness:
rate proportional to ∆fitness/∆trait

23. Overview Simple models HIV case study Conclusions References
The model (2): evolutionary dynamics
Incorporate trait dynamics
Standard quantitative genetics model (Abrams, 2001):
Fitness depends on mean trait value (¯α)
and ecological context (proportion susceptible)
Constant additive genetic variance Vg
Trait evolves toward increased fitness:
rate proportional to ∆fitness/∆trait

24. Overview Simple models HIV case study Conclusions References
Evolutionary dynamics, cont.
Virulence
Fitness(w)
frac inf=0.1

25. Overview Simple models HIV case study Conclusions References
Evolutionary dynamics, cont.
Virulence
Fitness(w)
frac inf=0.1
frac inf=0.3

26. Overview Simple models HIV case study Conclusions References
Power-law tradeoff curves
Virulence
Transmission β(α) = cα1 γ
c = 2, γ = 2
c = 1, γ = 2
c = 1, γ = 3

27. Overview Simple models HIV case study Conclusions References
Outline
1 Overview
The evolution of host-pathogen theory
Toy models
2 Transient virulence: simple models
Overview
Toy model
Myxomatosis model
3 Transient virulence: HIV case study
4 Conclusions

28. Overview Simple models HIV case study Conclusions References
Transient virulence
Selection differs between the
outbreak phases
(Frank, 1996; Day and Proulx,
2004)
endemic phase selection for
per-generation offspring
production: maximize R0,
βN/(α + µ)
epidemic phase selection
for per-unit-time offspring
production: maximize r,
βN − (α + µ)
Disease−induced mortality
Transmission
rate
µ 0 1 2 3 4 5
q
q

29. Overview Simple models HIV case study Conclusions References
Transient virulence
Selection differs between the
outbreak phases
(Frank, 1996; Day and Proulx,
2004)
endemic phase selection for
per-generation offspring
production: maximize R0,
βN/(α + µ)
epidemic phase selection
for per-unit-time offspring
production: maximize r,
βN − (α + µ)
Disease−induced mortality
Transmission
rate
µ 0 1 2 3 4 5
q
q

30. Overview Simple models HIV case study Conclusions References
Transient virulence
Selection differs between the
outbreak phases
(Frank, 1996; Day and Proulx,
2004)
endemic phase selection for
per-generation offspring
production: maximize R0,
βN/(α + µ)
epidemic phase selection
for per-unit-time offspring
production: maximize r,
βN − (α + µ)
Disease−induced mortality
Transmission
rate
µ 0 1 2 3 4 5
q
q

31. Overview Simple models HIV case study Conclusions References
Transient emerging virulence
When a parasite previously in eco-evolutionary equilibrium
emerges in a new host population (at low density) it will show
a transient peak in virulence as it spreads
How big is the peak? Does it matter?

32. Overview Simple models HIV case study Conclusions References
Outline
1 Overview
The evolution of host-pathogen theory
Toy models
2 Transient virulence: simple models
Overview
Toy model
Myxomatosis model
3 Transient virulence: HIV case study
4 Conclusions

33. Overview Simple models HIV case study Conclusions References
Model parameters
Parameter
c Transmission
scale
γ Transmission
curvature
I(0) Initial
epidemic size
Vg Genetic variance
Alternative
R∗
0 Equilibrium R0
α∗ Equilibrium
virulence
1/N0 Inverse
population size

34. Overview Simple models HIV case study Conclusions References
Example
Time
Fractioninfective
0.00
0.05
0.10
0.15
0 10 20 30
Vg = 5, c = 3, I(0) = 0.001, γ = 2
(R0
*
= 1.5, α*
= 1, N = 1000)
1.0
1.2
1.4
1.6
1.8
2.0
α

35. Overview Simple models HIV case study Conclusions References
Response variables
Time
peak
time
peak height(α)

36. Overview Simple models HIV case study Conclusions References
Peak height
Equilibrium transmission (R0
*
)
Equilibriumvirulence(α*
)
1
10
100
1000
1.1 2 5 10 50
1.025
CVg=0.1
1.025
1.05
1.1 2 5 10 50
1.05
1.075
1.5
CVg=0.5
1.5
2.0
1
10
100
1000
1.5
2.0
1
10
100
1000
1.5
2.0
2.5
3.0
I(0) = 10−2
CVg=1
1.1 2 5 10 50
1.5
2.0
2.5
3.0
3.5
I(0) = 10−3
1.5
2.0
2.5
3.0
3.5
4.0
I(0) = 10−4
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5

37. Overview Simple models HIV case study Conclusions References
Estimates for emerging pathogens
Order of magnitude estimates for some emerging high-virulence
pathogens:
Pathogen R∗
0 α∗
Reference
SARS 3 640 Anderson et al. (2004)
West Nile 1.61–3.24 639 Wonham et al. (2004)
HIV 1.43 6.36 Velasco-Hernandez et al. (2002)
myxomatosis 3 5 Dwyer et al. (1990)

38. Overview Simple models HIV case study Conclusions References
Emerging pathogens: where are we?
CVg = 0.5, I(0) = 10−3 (middle panel):
R0
Equilibriumvirulence(α*
)
1
10
100
1000
1.1 2 5 10 50
1.5
2.0
SARS
HIV
WNV
MYXO
1.0
1.5
2.0
2.5
3.0
3.5
4.0
4.5

39. Overview Simple models HIV case study Conclusions References
Outline
1 Overview
The evolution of host-pathogen theory
Toy models
2 Transient virulence: simple models
Overview
Toy model
Myxomatosis model
3 Transient virulence: HIV case study
4 Conclusions

40. Overview Simple models HIV case study Conclusions References
Overview
Mosquito-borne viral disease of rabbits
Benign in South American rabbits,
quickly fatal in European rabbits
Well characterized (Fenner et al., 1956; Dwyer et al., 1990)

41. Overview Simple models HIV case study Conclusions References
Myxomatosis tradeoff curve
Scaled virulence
Totaltransmission
0 2 4 6 8 10 12
0.0
0.2
0.4
0.6
eq epi

42. Overview Simple models HIV case study Conclusions References
Estimating evolvability (Vg )
Key parameter: genetic variance in virulence (evolvability)
Despite case studies of rapid pathogen evolution:
myxomatosis (Dwyer et al., 1990)
syphilis (Knell, 2004)
serial passage experiments (Ebert, 1998)
Plasmodium chabaudi (Mackinnon and Read, 1999)
we can rarely estimate Vg reliably

43. Overview Simple models HIV case study Conclusions References
Estimating evolvability (Vg )
Key parameter: genetic variance in virulence (evolvability)
Despite case studies of rapid pathogen evolution:
myxomatosis (Dwyer et al., 1990)
syphilis (Knell, 2004)
serial passage experiments (Ebert, 1998)
Plasmodium chabaudi (Mackinnon and Read, 1999)
we can rarely estimate Vg reliably

44. Overview Simple models HIV case study Conclusions References
Myxomatosis grades over time (Fenner et al., 1956)
1950 1954 1956 1961 1965 1968 1972 1978
Proportion
0.0
0.2
0.4
0.6
0.8
1.0
Virulence grade
I II III IV V

45. Overview Simple models HIV case study Conclusions References
Myxomatosis variance over time
Date
Geneticvariance(Vg)
0
10
20
30
40
1950 1960 1970
Vg= 10
Vg= 2.5
Vg= 40

46. Overview Simple models HIV case study Conclusions References
Myxomatosis virulence dynamics: power-law tradeoff
Date
Scaledvirulence
0
5
10
15
20
25
1950 1960 1970
h=2.5
h=10
h=40

47. Overview Simple models HIV case study Conclusions References
Myxomatosis virulence dynamics: realistic tradeoff
Date
Scaledvirulence
0
5
10
15
20
25
1950 1960 1970
h=40
h=10
h=2.5

48. Overview Simple models HIV case study Conclusions References
Myxo virulence: equilibrium start, power-law tradeoff
Date
Scaledvirulence
0
5
10
15
1950 1955
h=40
h=10
h=2.5

49. Overview Simple models HIV case study Conclusions References
Myxo virulence: equilibrium start, realistic tradeoff
Date
Scaledvirulence
0
5
10
15
1950 1955
h=40
h=10
h=2.5

50. Overview Simple models HIV case study Conclusions References
Simple models: conclusions
basic intuition about transient peak is correct
rough estimates justify that eco-evo dynamics can matter
details matter: shape of tradeoff curve, variance dynamics

51. Overview Simple models HIV case study Conclusions References
Phage dynamics (Berngruber et al., 2013)
Experimental evolution: phages started as endemic or epidemic

52. Overview Simple models HIV case study Conclusions References
HIV: background
Set-point viral load quantifies virus exploitation of host

53. Overview Simple models HIV case study Conclusions References
SPVL mediates a transmission/virulence tradeoff
transmission probability duration (years)
0.0
0.1
0.2
0.3
0
5
10
15
20
25
0.0 2.5 5.0 7.5 0.0 2.5 5.0 7.5
log10 set-point viral load

54. Overview Simple models HIV case study Conclusions References
HIV tradeoff curve
0 10 20 30 40 50 60
0.0
0.1
0.2
0.3
0.4
0.5
Scaled virulence
Transmissionrate
eq epi

55. Overview Simple models HIV case study Conclusions References
SPVL dynamics
(Shirreff et al., 2011; Herbeck et al., 2014)

56. Overview Simple models HIV case study Conclusions References
Pair-formation dynamics (Champredon et al., 2013)

57. Overview Simple models HIV case study Conclusions References
Model structure
Six models:
extra-pair contact [“epc”] ×
instantaneous pair-formation [“instswitch”]
“implicit” model: approx. (Shirreff et al., 2011)
random-mixing (no structure)
Latin hypercube sampling over epidemic parameters
(Champredon et al., 2013)

58. Overview Simple models HIV case study Conclusions References
SPVL trajectories
random pairform+epc pairform
instswitch+epc instswitch implicit
3
4
5
3
4
5
0 200 400 600 0 200 400 600 0 200 400 600
time (years)
populationmeanset−pointviralload(log10)

59. Overview Simple models HIV case study Conclusions References
Univariate summaries
peak SPVL
peak time (years)
equilibrium
SPVL
relative
peak
SPVL
time
virusload

60. Overview Simple models HIV case study Conclusions References
Results: univariate summaries
peak time peak SPVL
equilibrium SPVL relative peak
random
pairform+epc
pairform
instswitch+epc
instswitch
implicit
random
pairform+epc
pairform
instswitch+epc
instswitch
implicit
0 100 200 300 3.0 3.5 4.0 4.5 5.0
3 4 1.1 1.2 1.3 1.4 1.5
value

61. Overview Simple models HIV case study Conclusions References
Bivariate summaries
peaktimepeakSPVLrelativepeak
equilibrium SPVL peak time peak SPVL
0
100
200
300 q
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model
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random
pairform+epc
pairform
instswitch+epc
instswitch
implicit
3.0
3.5
4.0
4.5
5.0
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62. Overview Simple models HIV case study Conclusions References
Conclusions
HIV: extra-pair contact washes out structural effects
more complexity isn’t always better —
need the right complexity
(the modeling question)
models purport to predict HIV evolution
(Payne et al., 2014; Roberts et al., 2015; Herbeck et al., 2016):
can we trust them?

63. Overview Simple models HIV case study Conclusions References
Crome (1997) on theory
Real research is best for toy problems, and toy research is
often all one can do on real problems . . . Never do toy
research on toy problems, and recognize the extraordinary
limitations in attempting real research on real problems.
. . .
Recognize that someone might take your advice based on
your research. Try to estimate what the costs will be if
you are wrong.

64. Overview Simple models HIV case study Conclusions References
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