the interplay of infectivity that decreases with virulence with limited cross-immunity (toy) models...

The interplay of infectivity that decreases with virulence

with limited cross-immunity

(toy) models for respiratory disease evolution

Hans (= J A J *) Metz

(formerly ADN) IIASA

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VEOLIA-Ecole Poly-technique



&Mathematical Institute, Leiden University

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prelude: the Anderson-May framework

Starting point:dS

dt dN0 dS (x)IS,

dI

dt (x)SI d (x) I ,

dR

dtI dR

Equilibrium: S d (x)

(x)

R0 (x) Q(x)N0Q(x) :(x)

d (x)

Initial per capita growth rate of mutant with trait value y:

(y)S d (y)

( Implicit assumption: full cross-immunity.)

(y)d (x)

(x) d (y)

† † ††

R0:

expected disease-lifetime number of infections produced by freshly infected individuals

present in negligible numbers in an otherwise infection-free community

x: trait

Susceptibles Infecteds Recovereds



(y)d (x)

(x) d (y)

This growth rate is positive, and hence y can invade, if

Q(y) Q(x) R0 (y) R0 (x), or equivalently

R0 (x) Q(x)N0Q(x) :(x)

d (x)

Evolution maximises R0.


(y)S d (y) (y)d (x)

(x) d (y)


R0 (x) Q(x)N0Q(x) :(x)

d (x)


R0

x

To get interesting conclusions a negative trade-off is assumed:

x

x



Limitations:(i) Usually will depend on x as well. (ii) The assumed life cycle of the disease is extremely simple. (iii) This is even worse for the host. (The optimising property is

lost when host death rates are made density dependent !)(iv) The assumed negative trade-off is based on an

oversimplified view of the body as one well-mixed compartment: faster growing agent populations do both more harm and produce a larger infective output.

In reality the death toll often depends less on the growth capacity of the agent than on its location in the body.

(v) Diseases that specialise on different body parts carry different antigens, and hence have limited cross-immunity.

The simplest model forthe dynamics of

diseases with limited cross-immunity

c.f. Viggo Andreasen, Juan Lin, Simon A. Levin (1997)

The dynamics of cocirculating influenza strains conferring partial cross-immunityJ. Math. Biol. 35: 825-842

immune and disease states of individuals

: # of disease strains present in the population at time t,k(t) •

X X1,K , Xk , Xi ° n: their trait values.

p PK : immune status of an individual, with the power set (i.e., set of all subsets) of .

PK

K :{1,K ,k}: probability of a p-host becoming ill from an encounter with an i-inoculum:

i, p

i, p Xi;{X j | j p} .

Assumptions: (i) immunity only affects initial infection, (ii) immunity profile does not wane, (iii) infection order does not matter.

cross-immunity: ‘all doors should be open’

A more specific assumption is:

Y ;{X j | j p} 1 (Y , X j jp ,

Y , X 1

with the ‘crossimmunity profile’ a smooth function satisfying

Y X

Y X

1

for ,for .

population state

: density of healthy individuals (or hosts) with immune status p,

H p

Ii, p : density of individuals suffering from disease i with immune status .

p PK \{i}

: force of i-infection.

i :i Ii, ppPK \{i}

: primary infection rate constant,i (Xi )

i (Xi ) : disease related mortality rate,

i (Xi ) : recovery rate.

& disease parameters

uninfected population dynamics

: total birth rate,dN0

: per capita death rated

endemic dynamics

dH

dtdN0 d i

i1

k

H

dIi, p

dt ii, pH p d i i Ii, p

for i K , p PK \ {i}

dH p

dt i Ii, p \ {i}

ip d ii, p

i1

k

H p

for p PK \

neglect multiple infections

: force of i-infection.

i :i Ii, ppPK \{i}

Interlude:the adaptive dynamics toolbox

community dynamical background

Populations are considered as measures over a space ot i(ndividual)-states (e.g. spanned by age and size).

Environments are delimited such that given their environment individuals are independent,

and hence their mean numbers have linear dynamics.Resident populations are assumed to be so large that

we can approximate their dynamics deterministically.These resident populations influence the environment

so that they do not grow out of bounds.The resulting dynamical systems therefore have

attractors, which are assumed to produce ergodic environments.

mutant population dynamics: fitness

Mutants enter the population singly.Therefore, initially their impact on the environment can

be neglected. The initial growth of a mutant population can be

approximated with a branching process. Invasion fitness is the (generalised) Malthusian parameter

(= averaged long term exponential growth rate of the mean) of this proces.(Existence guaranteed by the multiplicative ergodic theorem.)

Residents have fitness zero.

as dominant transversal eigenvalue

resident population size

population sizes of

other species

mutantpopulationsize

resident population size

population sizes of

other species

mutantpopulationsize

or transversal natural Lyapunov exponent

trait substitutions

C := {X1,..,Xk}: trait values of the residents

Environment: Eattr(C) Y: trait value of mutant

Fitness (rate of exponential growth in numbers) of mutant:

s (Y | C) := (Y | Eattr(C))

* Y has a positive probability to invade into a C community iff s (Y|C) > 0.

* After invasion, Xi can be ousted by Y only if s(Xi | X1,..,Y,.., Xk) ≤ 0.

* For small mutational steps Y takes over, except near so-called “ess”es.

Evolution proceeds through uphill movements in a fitness landscape that keeps changing so as to keep the fitness of the resident types at exactly zero.

resident trait value(s) x

evol

utio

nary

tim

e

fitness landscape picture of evolution

0

0

0

fitness landscape: (Y,E(t))

mutant trait value y

0

0

resident trait value(s) x

evol

utio

nary

tim

e

0

0

0

fitness landscape: (Y,E(t))

mutant trait value y

0

0

i.a. branching points and Evolutionarily Steady States

evolutionarily singular strategies (ess-es),

fitness landscape picture of evolution

Organisers of landscape change:

essential formost conclusions

i.e., separated population dynamical and mutational time scales:the population dynamics relaxes before the next mutant comes

1. mutation limited evolution

2. clonal reproduction

3. good local mixing4. largish system sizes

5. “good” c(ommunity)-attractors6. interior c-attractors unique

7. fitness smooth in traits8. small mutational steps

essential conceptuallly

essential

underlying simplifying assumptions

t

, rescale time, only consider traits

rescale numbers to densities

= system size, = mutations / birth

the associated limit

x

adaptive dynamicslimit

individual-basedsimulation

classical largenumber limit

t

trait valuex

classical largenumber limit

individual-basedsimulation

adaptive dynamicslimit

For comparison:

y

x

+

+

-

--

fitness contour plotx: residenty: potential mutant

xx0x1

x1

x2

Pairwise Invasibility Plot

PIP

trait substitution sequences

x

+

-

+-

.

- +

+ -

Trait Evolution Plot

TEP


PIP

y

x

X

X1

2

x

x2


+

+

-

-


PIP

X2

X1x

Trait Evolution Plot

TEP

x2


y

x

+

+

-

-

monomorphicconvergenceto x0

dimorphicconvergencefrom x0

yes

no

evolutionary"branching"evolutionary"branching"

Evolutionary AttractorsEvolutionary Attractors

Evolutionary RepellersEvolutionary Repellers Žy2 y=x=x0

Ž2s(y|x)Žy2 y=x=x0

Ž2s(y|x)

Ž2s(y|x)Žx2 y=x=x0

Ž2s(y|x)Žx2 y=x=x0

noyes

noyes

the evolutionarily singular strategies

trait valuex

evol

utio

nary

tim

e

t i m e t r a

i t

fitne

ssfitness

minimum

population

geometry of adaptive branching

x1

x

x2

Exploring parameter space

= 1/3: = 2: = 3:

interrupted: branching prone ( trimorphically repelling)

In a thought experiment where the other type does not mutatean isocline corresponds to a locus of monomorphic singular points

matryoska principle

x2

x1

y

x

+

-

+

-

consistency conditions

There also exist various global consistency relations:

The classification of the singular points was based on just a smoothness assumption and some ecologically reasonable consistency conditions.

Use that on the boundaries of the coexistence set one type is extinct.

the bifurcation ESS branching point

21

1

2

Many of the results discussed so far hold good just as well for higher dimensional trait spaces.

Usually more traits evolve simultaneously, i.e., trait spaces have more than one dimension.

more than one trait variable

(Of course, some care is needed in how one generalises !)

The evolutionary model



nose

alveoli

x

0

1

(x) 2 x2 x

(x) 3x3

(x) 1(1 x1 )

R0:

x 01

evolutionary ingredients

primary infection rate constant:

disease induced mortality rate:

recovery rate:

further ingredients

cross-immunity:

y, x e 12 c(y x)2

y-x

monomorphic populations

endemic dynamics:

dH

dt N0 d 1I1, H

dI1,

dt 1I1,H d 1 1 I1,

dH{1}

dt1I1, H{1}

0

0

0

equilibrium:

H d 1 1

1

Q1 1

I1, N0 H

1H

1 1 R0.1 1

H{1} 1I1, 1

1

R0,1 1

I1, 0for :

with Qi :i

d i i

and R0,i :Qi N0

S:

I:

R:

s(Y | X1) Y H 1 (Y , X1) H{1} d (Y ) (Y )

monomorphic ess-s, general theory

X* is an evolutionarily singular strategy (ess) if s(Y | X1)

Y Y X1 X*0

A scalar ess x* is an Evolutionary Steady Strategy (ESS) if 2s(y | x1)

y2

yx1 x*

0

and a branching point if 2s(y | x1)

y2

yx1 x*

0

s(Y | X1) Y H 1 (Y , X1) H{1} d (Y ) (Y )


s(Y | X1)

Y Y H 1 (Y , X1) H{1}

Y 1(Y , X1)H{1} Y Y

with etc.

and : Y : y1 ,K , yh

1(X1, X2 ) :(X1, X2 ) X1

s(Y | X1)

Y Y X1 X

X H X X X

Use and . 1(X, X) 0(X, X) 1

s(Y | X1) Y H 1 (Y , X1) H{1} d (Y ) (Y )

s(Y | X1) Y H 1 (Y , X1) H{1} d (Y ) (Y )


Equation for the monomorphic ess:

X* X*

X* X* d X* X*

s(Y | X1)

Y Y X1 X

X H X X X

ln X* ln d

X*

Q X* 0

H Q1 1 Qi

i

d i i

R0,i :Qi N0

R0 X* 0

Theorem: The maximima of R0 attract, the minima repel.s(Y | X1)

Y Y X1 X

X H X X X


For scalar traits:2s(y | x1)

y2

yx1 x*

(x*)H (x*) (x*) (x*)

(x*) 1,1(x*, x*)H{1}(x*)

Q 1(x*) (x*) (x*) (x*)

(x*) R0 (x*) 1 1,1(x*, x*)

(x*)H (x*) (x*) (x*) (x*)H{1}(x*) 1,1(x*, x*)

Branching point when

c

s(Y | X1)

Y (Y ) H 1 (Y , X1) H{1}

(Y ) 1(Y , X1)H{1} (Y ) (Y )

1 (X, X) 01(X, X) 0

1 (X, X) 01(X, X) 0

polymorphic populations

dH

dtN0 d i

i1

k

H

dIi, p

dt ii, pH p d i i Ii, p

dH p

dt i Ii, p \ {i}

ip d ii, p

i1

k

H p

H N0

d ii1

k

Ii, p ii, pH p

d i i

H p i Ii, p \ {i}

ip

d i, piiK \ p

endemic dynamics:

For given these equations can be solved succesively to give

0

0

0

I , H G1

equilibrium:


For k = 2:

I , H G1

H I H I H † † †

† † †

†† {1,2}

(2,{1})

(1,{2})

(1,) {1}

{2}(2,)

1

2

1

2

2

1

22{1}

11{2}

N0etc.


I , H G1 Consider the recurrence:

n1 F(n )

Theorem: (i) The recurrence has the same equilibria as the full

population dynamics. (ii) The equilibria of the full population dynamics are invasible

by a new type Y if and only if this is the case for the recurrence.

Observation: The recurrence is similar to an ecological competition model.

G2 I F

I , H G1


n1 F(n )Theorem: For k = 2: For all all solutions of the recurrence converge to the same stable equilibrium.This equilibrium is internal if and only if both boundary equilibria are invadable.

The latter equilibria are unique and cannot both be uninvadable.

0 ° 02

some numerical results

increasing c




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increasing c

R0

x




increasing c

aside: Lotka Volterra resource competition models

resource distribution, expressed as the resulting

carrying capacity, K

scaled trait: x

K

-1 x 1

K

-1 x 1

competition kernel, a, derived from the capacities

to exploit and deplete

(scaled) trait differencexi - xj xi - xj

aa

c

dni

nidt1

a(xi , x j )n jj

K(xi )

x1

x

x2

Exploring parameter space

= 1/3: = 2: = 3:

increasing c

aside: Lotka Volterra resource competition models

more trait variables, some speculations

x2

x1

x1

R0 0

1

1

1

the further x2 is from its optimal value the worse the disease performs:

R0=1

The end

Thank you for your attention !

Kevin Kleine

Juan KeymerVergara

Prelude:

classical theory of virulence evolution

I1,

H

B d 1 1

1

For mutants the environment is set by the population dynamics of the resident types {X1,..,Xk} =: C.

The fitness of a given type Y in a given stationary environment E can be defined as the exponential growth rate of a clone of individuals of that type in that environment.

short adaptive dynamics refresher

Invasion fitness:

(asymptotic )(hypothetical)

Note that (1) as fitness is measured here on a logarithmic scale, zero is neutral, (2) residents have fitness zero.

~ dominant Lyapunov exponent(Furstenberg & Kesten, Oseledets)

s(Y|X) := (Y|EC)

, average

Fitnesses are not given quantities, but depend on (1) the traits of the individuals, X, Y, (2) the environment in which they live, E :

(Y,E) | (Y | E)E is set by the resident community:

E = Eattr(C), C={X1,...,Xk)

fitnesses cause and change with evolution

Evolutionary change is mainly determined by the fitnesses of potential mutants.

endemic dynamics

dH

dtB H, I d H, I i

i1

k

H

dIi, p

dt ii, pH p d H, I i i Ii, p

for i K , p PK \ {i}

dH p

dt i Ii, p \ {i}

ip d H, I ii, p

i1

k

H p

for p PK \

i :i Ii, ppPK \{i}

H : H p

pPk

I : Ii, p

pPK \{ i}

i1

k

, ,

neglect multiple infections

etc.

†

†

† †


increasing c




















































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x QuickTime™ and a





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Q u ic k T im e ™ a n d a d e c o mp re s s o r


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Q ui ckTi me™ and a decompressorare needed to see thi s pi cture.Q ui ckTi me™ and a decompressorare needed to see thi s pi cture.

Q

d

Q d

d 2

Q d d 2

d 3

ln Q

d

ln Q 2

2

d 2

d 2

Q * d 2

d 2

the interplay of infectivity that decreases with virulence with limited cross-immunity (toy) models...

Documents

mutant population dynamics

linear dynamics

limited crossimmunityc

numbers of mutant

disease states of individuals

partial crossimmunityj

mean numbers

trait values