Davis eco-evo virulence

Overview Emerging disease Seasonal disease Theory vs. data References

Eco-evolutionary virulence of pathogens:models and speculations

Ben Bolker, McMaster UniversityDepartments of Mathematics & Statistics and Biology

UC Davis

16 April 2014


Outline

1 OverviewThe evolution of host-pathogen theoryToy models

2 Transient virulence and emerging diseasesOverviewToy modelMyxomatosis data

3 Transient virulence and seasonalityOverviewToy modelWNV data

4 More on theory vs. dataTradeo� curvesConclusions


Acknowledgements

People Arjun Nanda and Dharmini Shah; Christophe Fraser;Marm Kilpatrick; Anson Wong

Support NSF IRCEB grant 9977063; QSE3 IGERT; NSERCDiscovery grant


Outline






Host-pathogen evolutionary biology

Why is it interesting?

Intellectual merit

Coevolutionary loopsCryptic e�ectsEco-evolutionary dynamics (Luo and Koelle, 2013)Cool storiesLots of data (sometimes)

Broader applications

MedicalConservation and managementOutreach


Host-pathogen evolutionary biology

Why is it interesting?

Intellectual merit

Coevolutionary loopsCryptic e�ectsEco-evolutionary dynamics (Luo and Koelle, 2013)Cool storiesLots of data (sometimes)

Broader applications

MedicalConservation and managementOutreach


Virulence: de�nitions

General public: badness

Plant biologists: infectivity

Evolutionists: loss of host �tness

Theoreticians: rate of host mortality(mortality rate vs. case mortality vs. clearance)


Evolution of virulence evolution theory

Classical dogma monotonic trend toward avirulence

Ewald era virulence as an evolved (adaptive) trait. Tradeo�theory, modes of transmission.

post-Ewald more formal tradeo� models, mostly based on R0

optimization. Adaptive dynamics

Now tradeo� backlashwithin-host dynamics/multi-level modelseco-evolutionary dynamicshost e�ects: resistance vs tolerance vs virulence























Basic tradeo� theory: assumptions

Homogeneous, non-evolving hosts

No superinfection/within-host competition

Horizontal, direct transmission

Tradeo� between rate of transmissionand length of infectious period

Infectious period ∝ 1/clearance(= recovery+disease-induced mortality+natural mortality)


Tradeo�s, R0, and r

Clearance+disease−induced mort.

Transmissionrate

mu 0 1 2 3 4 5




Transmissionrate

mu 0 1 2 3 4 5




Transmissionrate

mu 0 1 2 3 4 5


Outline






Epidemiological model

SIR(Susceptible / Infected /Recovered ) model

Constant population size(birth=death)

Ignore recovery

Rescale: µ = 1, N = 1(time units of host lifespan)

I

S

disease−

mortality(α)induced

mortality(µ)

birth

R

infection (β)

recovery





Ignore recovery


I

S

disease−


mortality(µ)

birth

R

infection (β)

recovery





Ignore recovery


I

S

disease−


mortality(µ)

birth

R

infection (β)

recovery





Ignore recovery


I

S

disease−


mortality(µ)

birth

R

infection (β)

recovery


The model (2): evolutionary dynamics

Put trait dynamics in the model.Standard quantitative genetics model (Abrams, 2001):

Fitness depends on mean trait value (α) and context (numberof susceptibles)

Constant, small additive genetic variance Vg

Trait evolves toward increased �tness:rate proportional to ∆�tness/∆trait

Alternatives: multi-strain, adaptive dynamics, PDEs, agent-basedmodels . . .






























Evolutionary dynamics, cont.

Virulence

Fitn

ess

(w)

frac inf=0.3

frac inf=0.1


Power-law tradeo� curves

Virulence

Tran

smis

sion

β(α) = cα1 γ

c = 2, γ = 2

c = 1, γ = 2

c = 1, γ = 3


Outline






(Why) are emerging pathogens more virulent?

What explains initially high, but rapidly decreasing, virulence ofemerging pathogens?

Pathogens with low virulence go unnoticed

Hosts less resistant to / tolerant of novel parasites

High transmission → frequent coinfection → selection forvirulence

Disease-induced drop in population density decreases selectionfor virulence (Lenski and May, 1994)























Transient virulence

Selection di�ers between the epidemic and endemic phases of anoutbreak (Frank, 1996; Day and Proulx, 2004)

endemic phase selection for per-generation o�spring production:maximize R0, βN/(α + µ)

epidemic phase selection for per-unit-time o�spring production:maximize r , βN − (α + µ)


Transient virulence





Transient virulence





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






Model parameters

Parameter De�nition

c Transmission scaleγ Transmission curvature

Vg (CVg ) Additive genetic varianceI (0) Initial epidemic size

R∗0

Equilibrium intrinsic reproductive numberα∗ Equilibrium virulence


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)

1.01.21.41.61.82.0

α


Analysis?

Brute force simulation:

Vary

I (0) (≈ 1/pop. size),genetic variance, CVg (expressed as coef of variation)equilibrium transmission (R∗

0)

equilibrium virulence (α∗)

Peak height (virulence):scaled by equilibrium virulence (no change=1)

Peak time:ratio of disease prevalence at peak α to maximum prevalence


Peak height and timing

Time

peaktime

peak 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

I(0) = 10−2

CV

g=

0.1

1.0251.05

I(0) = 10−3

CV

g=

0.1

1.1 2 5 10 50

1.05

1.075

I(0) = 10−4

CV

g=

0.1

1.5

I(0) = 10−2

CV

g=

0.5

1.5

2.0

I(0) = 10−3

CV

g=

0.5

1

10

100

1000

1.5

2.0 I(0) = 10−4

CV

g=

0.51

10

100

1000

1.5

2.02.5

3.0

I(0) = 10−2 C

Vg

=1

1.1 2 5 10 50

1.52.0

2.5

3.0

3.5

I(0) = 10−3

CV

g=

1

1.52.02.5

3.03.5

4.0

I(0) = 10−4

CV

g=

11.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5


Relationships among parameters

Assuming a power-law tradeo� curve, we can estimate the tradeo�curve from:

R∗0: (equilibrium) intrinsic reproductive numberhard to measure (but frequently attempted)

α∗ : equilibrium virulenceget a crude guess from the observed time to death(may be overestimated)


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)HIV 1.43 6.36 Velasco-Hernandez et al. (2002)

West Nile 1.61�3.24 639 Wonham et al. (2004)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 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


Outline






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)


Myxomatosis tradeo� curve

Scaled virulence

Tota

l tra

nsm

issi

on

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, 1999a)

we rarely have enough information to estimate Vg

Only (?) for myxomatosis do we know the variation invirulence among circulating strains
















Myxomatosis grades vs. time

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 vs. time

Date

Gen

etic

var

ianc

e (V

g)

0

10

20

30

40

1950 1960 1970

Vg= 10Vg= 2.5

Vg= 40


Myxomatosis virulence dynamics: power-law tradeo�

Date

Sca

led

viru

lenc

e

0

5

10

15

20

25

1950 1960 1970

h=2.5

h=10

h=40


Myxomatosis virulence dynamics: realistic tradeo�

Date

Sca

led

viru

lenc

e

0

5

10

15

20

25

1950 1960 1970

h=40

h=10h=2.5


Myxo virulence: equilibrium start, power-law tradeo�

Date

Sca

led

viru

lenc

e

0

5

10

15

1950 1955

h=40h=10h=2.5


Myxo virulence: equilibrium start, realistic tradeo�

Date

Sca

led

viru

lenc

e

0

5

10

15

1950 1955

h=40h=10h=2.5


Outline






Seasonality

Many pathogens �uctuate annually

Host contact/aggregation patternsHost (or vector) demographyClimatic e�ects on transmissibility

Fluctuating incidence = �uctuating selection

Seasonal variation or latitudinal variation?


Outline






Toy model

Basic Ross-MacDonaldvector-host model

Simple vector (mosquito)demography

No host demography

Two pathogen strains

I

S

disease−


S

I

birth

infection (β)

recovery

R


Parameters

Table: Summary parameters for simulation cases

s.tab r1 r2 R0,1 R0,2

Case I 0.11 0.09 10.8 10.8Case II 0.09 0.09 16.2 10.8Case III 0.09 0.11 16.2 10.8


Case I: r1 > r2, equal R0

0.000

0.025

0.050

0.075

0 25 50 75 100 125time

dens

ity

variable

I1

I2

0.0

0.2

0.4

0 25 50 75 100 125time

frac

tion

of s

trai

n 1


Case II: R0,1 > R0,2, equal r

0.00

0.05

0.10

0.15

0.20

0 25 50 75 100 125time

dens

ity

variable

I1

I2

0.5

0.6

0.7

0.8

0.9

1.0

0 25 50 75 100 125time

frac

tion

of s

trai

n 1


Case III: R0,1 > R0,2, r2 > r1

0.00

0.05

0.10

0.15

0 25 50 75 100 125time

dens

ity

variable

I1

I2

0.5

0.6

0.7

0.8

0.9

1.0

0 25 50 75 100 125time

frac

tion

of s

trai

n 1


Outline






Titer vs infectiousness

● ● ●

●

●

●

●

●

●

●

●

●

●

●

●

●

0.0

0.2

0.4

0.6

4 6 8titer

Tran

smis

sion

pro

babi

lity

source

●

●

●

●

Dohm

Tiawsirisup_2005_VBZD

Turell_altjmh

Turell_JME


Titer curves (American crows)

1e−04

1e−02

1e+00

2 4 6 8day

tran

smis

sion

pro

babi

lity

strain

BIRD1153

KEN

KENsub

NY99

P991

P991sub

TM171−03−pp5

TM173−03−pp1

TWN301


Transmission vs clearance for WNV

BIRD1153BIRD1461

NY99

TM171−03−pp5BIRD1153KEN

KENsub

NY99P991

TM171−03−pp5

TWN301

0.0

0.2

0.4

0.6

0.00 0.25 0.50 0.75 1.00Clearance rate (1/infectious period)

Ave

rage

tran

smis

sion

rat

e

species

aa

sparrow

amcrow


Outline






Estimating tradeo� curves

Usually assume a tradeo� between virulence and transmission

Positive correlation virulence and transmissibility (or proxies)known from many systems (Lipsitch and Moxon, 1997)

the shape of tradeo� curves is largely unknown


Malaria (Mackinnon and Read, 1999b; Paul et al., 2004)

●

●●●

0 500 1000 1500

0

20

40

60

80

100

Scaled virulence

% m

osqu

itoes

infe

cted

Plasmodium gallinaceum

●

low−dose mixedhigh−dose mixedSLThai

●

●

●

●

●●

●

●

10 15 20 25

10

15

20

25

Maximum parasitemia

Ove

rall

infe

ctio

n (%

)

Plasmodium chabaudi


Pasteuria ramosa (Jensen et al., 2006)

●●●●●

●

●

●

●●

●

●

●

●●●

●●

●

●

●

●

●

●●●●

●

●●

●

0 1 2 3 4 50.00

0.02

0.04

0.06

0.08

0.10

0.12

Scaled virulence

Spo

res/

day

(×10

6 )

●●●●●

●

●

●

●●

●

●

●

●●●

●●

●

●

●

●

●

●●●●

●

●●

●


HIV (Fraser et al., 2007)

0 10 20 30 40 50 60

0.0

0.1

0.2

0.3

0.4

0.5

Scaled virulence

Tran

smis

sion

rat

e

eq epi


HIV dynamics (Shirre� et al., 2011)


Phage dynamics (Berngruber et al., 2013)


Outline






Conclusions

Eco-evolutionary dynamics of virulence are still plausible(Alizon et al., 2009; Luo and Koelle, 2013)

Especially sensitive to genetic variance and shape of tradeo�curve

Theory meets molecular biology:mutations of large e�ect vs. quantitative variability


Conclusions

Eco-evolutionary dynamics of virulence are still plausible(Alizon et al., 2009; Luo and Koelle, 2013)

Especially sensitive to genetic variance and shape of tradeo�curve

Theory meets molecular biology:mutations of large e�ect vs. quantitative variability


Crome 1997 on theory (part 1)

When we regard theories as tight, real entities and devote

ourselves to their analysis, we can limit our horizons and,

worse, attempt to make the world �t them. A lot of

ecological discussion is not about nature, but about

theories, generalizations, or models supposed to represent

nature . . .


Crome 1997 on theory (part 2)

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.


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Alizon, S., Hurford, A., et al., 2009. J. Evol. Biol., 22:245�259.doi:10.1111/j.1420-9101.2008.01658.x.

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.

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.

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�, 1999b. Proc R Soc London B, 266(1420):741�748.

Paul, R.E.L., Lafond, T., et al., 2004. BMC Evol Biol, 4:30.

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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.