evolution of virulence: devil in the details

Overview Simple models HIV case study Conclusions References

Eco-evolutionary virulence of pathogens:the devil is (still) in the details

Ben Bolker, McMaster UniversityDepartments of Mathematics & Statistics and Biology

University of Tennessee

15 September 2016

Outline

1 OverviewThe evolution of host-pathogen theoryToy models

2 Transient virulence: simple modelsOverviewToy modelMyxomatosis model

3 Transient virulence: HIV case study

4 Conclusions

Acknowledgements

People Arjun Nanda and Dharmini Shah; Christophe Fraser;Daniel Park

Support NSF IRCEB grant 9977063; QSE3 IGERT; NSERCDiscovery grant

Outline

4 Conclusions

Host-pathogen evolutionary biology

Why?Intellectual merit

Coevolutionary loopsCryptic effectsEco-evolutionary dynamics (Luo and Koelle, 2013)Lots of data (for humans)

Broader applicationsMedicalConservation and managementOutreach

Host-pathogen evolutionary biology

Why?Intellectual merit

Coevolutionary loopsCryptic effectsEco-evolutionary dynamics (Luo and Koelle, 2013)Lots of data (for humans)

Broader applicationsMedicalConservation and managementOutreach

Virulence: definitions

General public: badnessPlant biologists: infectivityEvolutionists: loss of host fitnessTheoreticians: rate of host mortality

Evolution of virulence evolution theory

Classical avirulence theoryAnderson & May/Ewald virulence as an evolved (adaptive) trait.

Tradeoff theory, modes of transmission.post-Ewald mathematical models based on R0

Now tradeoff backlashwithin-host dynamics/multi-level modelseco-evolutionary dynamicshost effects: resistance vs tolerance vs virulence

Basic tradeoff theory: assumptions

Homogeneous, non-evolving hostsNo super- or coinfectionHorizontal, direct transmissionTradeoff between rate of transmissionand length of infectious periodInfectious period ∝ 1/clearance rate

Tradeoffs, R0, and r

Disease−induced mortality

Transmissionrate

µ 0 1 2 3 4 5

Transmissionrate

µ 0 1 2 3 4 5

Transmissionrate

µ 0 1 2 3 4 5

●

Outline

4 Conclusions

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

SIR model

I

S

disease−

mortality(µ)

birth

R

infection (β)

recovery

SIR model

I

S

disease−

mortality(µ)

birth

R

infection (β)

recovery

The model (2): evolutionary dynamics

Incorporate trait dynamicsStandard 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

Evolutionary dynamics, cont.

Virulence

Fitn

ess

(w)

frac inf=0.1

Evolutionary dynamics, cont.

Virulence

Fitn

ess

(w)

frac inf=0.1

frac inf=0.3

Power-law tradeoff curves

Virulence

Tran

smis

sion

β(α) = cα1 γ

c = 2, γ = 2

c = 1, γ = 2

c = 1, γ = 3

Outline

4 Conclusions

Transient virulence

Selection differs between theoutbreak phases(Frank, 1996; Day and Proulx,2004)

endemic phase selection forper-generation offspringproduction: maximize R0,βN/(α + µ)

epidemic phase selectionfor per-unit-time offspringproduction: maximize r ,βN − (α + µ)

Transmissionrate

µ 0 1 2 3 4 5

●

Transient virulence

Transmissionrate

µ 0 1 2 3 4 5

●

Transient virulence

Transmissionrate

µ 0 1 2 3 4 5

●

Transient emerging virulence

When a parasite previously in eco-evolutionary equilibriumemerges in a new host population (at low density) it will showa transient peak in virulence as it spreads

How big is the peak? Does it matter?

Outline

4 Conclusions

Model parameters

Parameterc Transmission

scaleγ Transmission

curvatureI (0) Initial

epidemic sizeVg Genetic variance

AlternativeR∗

0 Equilibrium R0

α∗ Equilibriumvirulence

1/N0 Inversepopulation size

Example

Time

Fra

ctio

n in

fect

ive

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

Response variables

Time

peaktime

peak height(α)

Peak height

Equilibrium transmission (R0*)

Equ

ilibr

ium

viru

lenc

e (α

* )

1

10

100

1000

1.1 2 5 10 50

1.025

CV

g=

0.1

1.025

1.05

1.1 2 5 10 50

1.05

1.075

1.5

CV

g=

0.5

1.5

2.01

10

100

1000

1.5

2.01

10

100

1000

1.5

2.02.5

3.0

I(0) = 10−2 C

Vg

=1

1.1 2 5 10 50

1.5

2.0

2.53.0

3.5

I(0) = 10−3

1.52.0

2.53.0

3.5

4.0

I(0) = 10−4

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Estimates for emerging pathogens

Order of magnitude estimates for some emerging high-virulencepathogens:

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)

Emerging pathogens: where are we?

CVg = 0.5, I (0) = 10−3 (middle panel):

R0

Equ

ilibr

ium

viru

lenc

e (α

* )

1

10

100

1000

1.1 2 5 10 501.5

2.0

SARS

HIV

WNV

MYXO

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

Outline

4 Conclusions

Overview

Mosquito-borne viral disease of rabbitsBenign in South American rabbits,quickly fatal in European rabbitsWell characterized (Fenner et al., 1956; Dwyer et al., 1990)

Myxomatosis tradeoff curve

Scaled virulence

Tota

ltra

nsm

ission

0 2 4 6 8 10 12

0.0

0.2

0.4

0.6

eq epi

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

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

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

1.0Virulence grade

I II III IV V

Myxomatosis variance over time

Date

Gen

etic

varia

nce(

Vg)

0

10

20

30

40

1950 1960 1970

Vg= 10

Vg= 2.5

Vg= 40

Myxomatosis virulence dynamics: power-law tradeoff

Date

Scal

edvi

rule

nce

0

5

10

15

20

25

1950 1960 1970

h=2.5

h=10

h=40

Myxomatosis virulence dynamics: realistic tradeoff

Date

Scal

edvi

rule

nce

0

5

10

15

20

25

1950 1960 1970

h=40

h=10

h=2.5

Myxo virulence: equilibrium start, power-law tradeoff

Date

Scal

edvi

rule

nce

0

5

10

15

1950 1955

h=40h=10h=2.5

Myxo virulence: equilibrium start, realistic tradeoff

Date

Scal

edvi

rule

nce

0

5

10

15

1950 1955

h=40h=10h=2.5

Simple models: conclusions

basic intuition about transient peak is correctrough estimates justify that eco-evo dynamics can matterdetails matter: shape of tradeoff curve, variance dynamics

Phage dynamics (Berngruber et al., 2013)

Experimental evolution: phages started as endemic or epidemic

HIV: background

Set-point viral load quantifies virus exploitation of host

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 7.5log10 set-point viral load

HIV tradeoff curve

0 10 20 30 40 50 60

0.0

0.1

0.2

0.3

0.4

0.5

Scaled virulence

Tran

smission

rate

eq epi

SPVL dynamics

(Shirreff et al., 2011; Herbeck et al., 2014)

Pair-formation dynamics (Champredon et al., 2013)

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)

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)

popu

latio

n m

ean

set−

poin

t vira

l loa

d (lo

g 10)

Univariate summaries

peak SPVL

peak time (years)

equilibriumSPVL

relativepeakSPVL

time

viru

slo

ad

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 1.5value

Bivariate summaries

peak

tim

epe

ak S

PV

Lre

lativ

e pe

ak

equilibrium SPVL peak time peak SPVL

0

100

200

300 ●

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Conclusions

HIV: extra-pair contact washes out structural effectsmore 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?

Crome (1997) on theory

Real research is best for toy problems, and toy research isoften all one can do on real problems . . . Never do toyresearch on toy problems, and recognize the extraordinarylimitations in attempting real research on real problems.. . .Recognize that someone might take your advice based onyour research. Try to estimate what the costs will be ifyou are wrong.

ReferencesAbrams, P.A., 2001. Ecol Lett, 4:166–175.Anderson, R.M., Fraser, C., et al., 2004. Phil Trans R Soc London B, 359(1447):1091–1105.Berngruber, T.W., Froissart, R., et al., 2013. PLoS Pathog, 9(3):e1003209.

doi:10.1371/journal.ppat.1003209.Champredon, D., Bellan, S., and Dushoff, J., 2013. PLoS ONE, 8(12):e82906.

doi:10.1371/journal.pone.0082906.Crome, F.H.J., 1997. In W.F. Laurance and J. Richard O. Bierregard, editors, Tropical Forest

Remnants: Ecology, Management and Conservation of Fragmented Communities, chapter 31, pages485–501. University of Chicago Press, Chicago.

Day, T. and Proulx, S.R., 2004. Amer Nat, 163(4):E40–E63.Dwyer, G., Levin, S., and Buttel, L., 1990. Ecol Monog, 60:423–447.Ebert, D., 1998. Science, 282(5393):1432–1435.Fenner, F., Day, M.F., and Woodroofe, G.M., 1956. J Hyg (London), 54(2):284–302.Frank, S.A., 1996. Q Rev Biol, 71(1):37–78.Herbeck, J., Mittler, J., et al., 2016. bioRxiv, page 039560.Herbeck, J.T., Mittler, J.E., et al., 2014. PLoS Computational Biology, 10(6):e1003673.

doi:10.1371/journal.pcbi.1003673.Knell, R.J., 2004. Proc R Soc London B, 271:S174–S176.Luo, S. and Koelle, K., 2013. The American Naturalist, 181(S1):S58–S75. ISSN 0003-0147.

doi:10.1086/669952.Mackinnon, M.J. and Read, A.F., 1999. Evolution, 53(3):689–703.Payne, R., Muenchhoff, M., et al., 2014. Proceedings of the National Academy of Sciences,

111(50):E5393–E5400. ISSN 0027-8424, 1091-6490. doi:10.1073/pnas.1413339111.Roberts, H.E., Goulder, P.J., and McLean, A.R., 2015. Journal of The Royal Society Interface,

12(113):20150888.Shirreff, G., Pellis, L., et al., 2011. PLoS Computational Biology, 7(10). ISSN 1553-734X.

doi:10.1371/journal.pcbi.1002185. WOS:000297262700019.Velasco-Hernandez, J.X., Gershgorn, H.B., and Blower, S.M., 2002. Lancet, 2:487–493.Wonham, M.J., de Camino-Beck, T., and Lewis, M.A., 2004. Proc R Soc London B, 271:501–507.

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