Models of Host Immune Response, and the(co)Evolution of Virulence: - - PowerPoint PPT Presentation

Models of Host Immune Response, and the(co)Evolution of Virulence: limited and preliminary extensions on Gilchrist-Sasaki

Andrea Pugliese

• Dept. of Mathematics, Univ. of Trento

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.1/26

Introduction

Gilchrist and Sasaki (2002) introduced a very nice framework to discuss co-evolution of virulence and resistance without invoking hypothetical trade-offs. Aspects I aimed at addressing:

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.2/26

Introduction

Gilchrist and Sasaki (2002) introduced a very nice framework to discuss co-evolution of virulence and resistance without invoking hypothetical trade-offs. Aspects I aimed at addressing: (Slightly) more complex models of virus–immune system interactions not limited to short-term after infection.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.2/26

Introduction

Gilchrist and Sasaki (2002) introduced a very nice framework to discuss co-evolution of virulence and resistance without invoking hypothetical trade-offs. Aspects I aimed at addressing: (Slightly) more complex models of virus–immune system interactions not limited to short-term after infection. Reinfection of already infected hosts (to deal with issues like super-infection.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.2/26

Introduction

Gilchrist and Sasaki (2002) introduced a very nice framework to discuss co-evolution of virulence and resistance without invoking hypothetical trade-offs. Aspects I aimed at addressing: (Slightly) more complex models of virus–immune system interactions not limited to short-term after infection. Reinfection of already infected hosts (to deal with issues like super-infection. Variability of hosts (not genetically determined).

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.2/26

Models for virus-immune system, 1

P pathogen load I specific immunity level

• P ′

= rP − cIP I′ = aIP

(Gilchrist-Sasaki, 2002) with I(0) = I0 > 0, P(0) = P0 > 0.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.3/26

Models for virus-immune system, 1

P pathogen load I specific immunity level

• P ′

= rP − cIP I′ = aIP

(Gilchrist-Sasaki, 2002) with I(0) = I0 > 0, P(0) = P0 > 0. Infection grows (if r > cI0) and then is cleared by immune system. Some computations are easier since it is Kermack-McKendrick model disguised. Hence one obtains

P = Φ(I) := r a log(I) − I + I0 + P0.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.3/26

Models for virus-immune system, 2

• P ′

= rP − cIP I′ = aIP

(Gilchrist-Sasaki, 2002)

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.4/26

Models for virus-immune system, 2

• P ′

= rP − cIP I′ = aIP

(Gilchrist-Sasaki, 2002)

• P ′

= rP − cIP I′ = βI

(André-Gandon, 2006)

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.4/26

Models for virus-immune system, 2

• P ′

= rP − cIP I′ = aIP

(Gilchrist-Sasaki, 2002)

• P ′

= rP − cIP I′ = βI

(André-Gandon, 2006) Equations can be solved to have

P(t) = P0 exp

• rt + cI0

β (1 − eβt)

• .

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.4/26

Models for virus-immune system, 3

• P ′

= rP − cIP I′ = aIP

(Gilchrist-Sasaki, 2002)

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.5/26

Models for virus-immune system, 3

• P ′

= rP − cIP I′ = aIP

(Gilchrist-Sasaki, 2002)

• P ′

= rP − cIP I′ = βI

(André-Gandon, 2006) All infections are eventually cleared.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.5/26

Models for virus-immune system, 3

• P ′

= rP − cIP I′ = aIP

(Gilchrist-Sasaki, 2002)

• P ′

= rP − cIP I′ = βI

(André-Gandon, 2006) All infections are eventually cleared.

• P ′

= rP − cIP I′ = kP − δI + h

(Mohtashemi-Levins, 2002)

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.5/26

Models for virus-immune system, 3

• P ′

= rP − cIP I′ = aIP

(Gilchrist-Sasaki, 2002)

• P ′

= rP − cIP I′ = βI

(André-Gandon, 2006) All infections are eventually cleared.

• P ′

= rP − cIP I′ = kP − δI + h

(Mohtashemi-Levins, 2002) If an infection can occur (r > ch

δ ), then system always goes

to an equilibrium, generally after several infection cycles.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.5/26

Proposed model for within-host dynamics

Several other models in Nowak-May (2002) share this feature: If an infection is possible, it is never cleared completely (at least, in the deterministic model).

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.6/26

Proposed model for within-host dynamics

Several other models in Nowak-May (2002) share this feature: If an infection is possible, it is never cleared completely (at least, in the deterministic model). An extension with functional response in immune cells-virus interaction:

• P ′

= rP −

cI 1+kcP P − m 1+kmP P

I′ =

aP 1+kaP I − δI + h

m level (activity) of aspecific immunity. kc and km modulate functional response. ka allows for different rules of immune response.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.6/26

Proposed model for within-host dynamics

Several other models in Nowak-May (2002) share this feature: If an infection is possible, it is never cleared completely (at least, in the deterministic model). An extension with functional response in immune cells-virus interaction:

• P ′

= rP −

cI 1+kcP P − m 1+kmP P

I′ =

aP 1+kaP I − δI + h

m level (activity) of aspecific immunity. kc and km modulate functional response. ka allows for different rules of immune response.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.6/26

Behaviour of within-host model

If r small, no internal equilibria. Infection is cleared completely.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.7/26

Behaviour of within-host model

If r small, no internal equilibria. Infection is cleared completely.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.7/26

Behaviour of within-host model

If r small, no internal equilibria. Infection is cleared completely.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.8/26

Behaviour of within-host model

If r small, no internal equilibria. Infection is cleared completely. If r intermediate, 2 internal equilibria. Infection is either cleared, or it goes to equilibrium (or limit cycle).

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.8/26

Behaviour of within-host model

If r small, no internal equilibria. Infection is cleared completely. If r intermediate, 2 internal equilibria. Infection is either cleared, or it goes to equilibrium (or limit cycle).

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.8/26

Behaviour of within-host model

If r small, no internal equilibria. Infection is cleared completely.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.9/26

Behaviour of within-host model

If r small, no internal equilibria. Infection is cleared completely. If r intermediate, 2 internal equilibria. Infection is either cleared, or it goes to equilibrium (or limit cycle).

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.9/26

Behaviour of within-host model

If r small, no internal equilibria. Infection is cleared completely. If r intermediate, 2 internal equilibria. Infection is either cleared, or it goes to equilibrium (or limit cycle). If r large (r > m + ch/δ), 1 internal equilibrium. Infection always goes to equilibrium (or limit cycle).

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.9/26

Behaviour of within-host model

If r small, no internal equilibria. Infection is cleared completely. If r intermediate, 2 internal equilibria. Infection is either cleared, or it goes to equilibrium (or limit cycle). If r large (r > m + ch/δ), 1 internal equilibrium. Infection always goes to equilibrium (or limit cycle).

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.9/26

Behaviour of within-host model

If r small, no internal equilibria. Infection is cleared completely.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.10/26

Behaviour of within-host model

If r small, no internal equilibria. Infection is cleared completely. If r intermediate, 2 internal equilibria. Infection is either cleared, or it goes to equilibrium (or limit cycle).

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.10/26

Behaviour of within-host model

If r small, no internal equilibria. Infection is cleared completely. If r intermediate, 2 internal equilibria. Infection is either cleared, or it goes to equilibrium (or limit cycle). If r large (r > m + ch/δ), 1 internal equilibrium. Infection always goes to equilibrium (or limit cycle).

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.10/26

Behaviour of within-host model

If r small, no internal equilibria. Infection is cleared completely. If r intermediate, 2 internal equilibria. Infection is either cleared, or it goes to equilibrium (or limit cycle). If r large (r > m + ch/δ), 1 internal equilibrium. Infection always goes to equilibrium (or limit cycle). Moreover, for r + δ > a/ka, solutions may diverge to infinity (immune system does not control infection)

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.10/26

Question

Into which qualitative regime will the parameters (especially the replication rate r) evolve?

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.11/26

Epidemic dynamics

i(t, P, I) infectives of immune-level I and pathogen load P S susceptibles

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.12/26

Epidemic dynamics

i(t, P, I) infectives of immune-level I and pathogen load P S susceptibles ∂ ∂ti(t, P, I) + ∂ ∂P ( f(P, I) i(t, P, I)) + ∂ ∂I ( g(P, I) i(t, P, I)) = −(µ + α(P, I))i(t, P, I)

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.12/26

Epidemic dynamics

i(t, P, I) infectives of immune-level I and pathogen load P S susceptibles ∂ ∂ti(t, P, I) + ∂ ∂P ( f(P, I) i(t, P, I)) + ∂ ∂I ( g(P, I) i(t, P, I)) = −(µ + α(P, I))i(t, P, I)

where

P ′ = f(P, I)

and

I′ = g(P, I)

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.12/26

Epidemic dynamics

i(t, P, I) infectives of immune-level I and pathogen load P S susceptibles ∂ ∂ti(t, P, I) + ∂ ∂P ( f(P, I) i(t, P, I)) + ∂ ∂I ( g(P, I) i(t, P, I)) = −(µ + α(P, I))i(t, P, I) α disease-induced mortality

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.12/26

Epidemic dynamics

i(t, P, I) infectives of immune-level I and pathogen load P S susceptibles ∂ ∂ti + ∂ ∂P (fi) + ∂ ∂I (gi) = −(µ + α(P, I))i(t, P, I)

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.13/26

Epidemic dynamics

i(t, P, I) infectives of immune-level I and pathogen load P S susceptibles ∂ ∂ti + ∂ ∂P (fi) + ∂ ∂I (gi) = −(µ + α(P, I))i(t, P, I)

and

S′(t) = Λ − (µ + λ(t))S(t) aP0i(t, 1) = λ(t)S(t) λ(t) = β

• Pi(t, P, I) dP dI

α(P, I) = k1aIP + k2rP

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.13/26

Epidemic dynamics

i(t, P, I) infectives of immune-level I and pathogen load P S susceptibles ∂ ∂ti + ∂ ∂P (fi) + ∂ ∂I (gi) = −(µ + α(P, I))i(t, P, I) λ infection rate α disease-induced mortality S′(t) = Λ − (µ + λ(t))S(t) aP0i(t, 1) = λ(t)S(t) λ(t) = β

• Pi(t, P, I) dP dI

α(P, I) = k1aIP + k2rP

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.13/26

Age-of-infection setting

Since P and I are deterministic function P(θ), I(θ) of time since infection, one can rewrite it as

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.14/26

Age-of-infection setting

Since P and I are deterministic function P(θ), I(θ) of time since infection, one can rewrite it as

        

∂ ∂tu(t, θ) + ∂ ∂θu(t, θ)

= −(µ + α(P(θ), I(θ))u(t, θ) u(t, 0) = λ(t)S(t) λ(t) = β ∞

0 P(θ)u(t, θ) dθ

S′(t) = Λ − (µ + λ(t))S(t)

with u(t, θ) related to i(t, P(θ), B(θ))

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.14/26

Age-of-infection setting

Since P and I are deterministic function P(θ), I(θ) of time since infection, one can rewrite it as

        

∂ ∂tu(t, θ) + ∂ ∂θu(t, θ)

= −(µ + α(P(θ), I(θ))u(t, θ) u(t, 0) = λ(t)S(t) λ(t) = β ∞

0 P(θ)u(t, θ) dθ

S′(t) = Λ − (µ + λ(t))S(t)

with u(t, θ) related to i(t, P(θ), B(θ)) This system is in the class considered by Thieme and Castillo-Chavez for AIDS.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.14/26

R0

The behaviour of the system is mainly determined by R0:

R0 =

Λ µ

β ∞ P(θ) × exp

• −(µθ + k1

θ

• k1aI(s)P(s) + k2rP(s) ds
• dθ.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.15/26

R0

The behaviour of the system is mainly determined by R0:

R0 =

Λ µ

β ∞ P(θ) × exp

• −(µθ + k1

θ

0 k1aI(s)P(s) + k2rP(s) ds

• dθ.

Pathogen level at time θ since infection

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.15/26

R0

The behaviour of the system is mainly determined by R0:

R0 =

Λ µ

β ∞ P(θ) × exp

• −(µθ + k1

θ

0 k1aI(s)P(s) + k2rP(s) ds

• dθ.

Survival probability to time θ

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.15/26

R0

The behaviour of the system is mainly determined by R0:

R0 =

Λ µ

β ∞ P(θ) × exp

• −(µθ + k1

θ

0 k1aI(s)P(s) + k2rP(s) ds

• dθ.

Population at disease-free equilibrium

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.15/26

R0

The behaviour of the system is mainly determined by R0:

R0 =

Λ µ

β ∞ P(θ) × exp

• −(µθ + k1

θ

0 k1aI(s)P(s) + k2rP(s) ds

• dθ.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.15/26

• R0. II

If R0 < 1, disease-free equilibrium is globally stable

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.16/26

• R0. II

If R0 < 1, disease-free equilibrium is globally stable If R0 > 1, there exists a unique positive equilibrium:

     ¯ S =

Λ µR0

¯ λ = µ(R0 − 1) ¯ u(θ) = ¯ λ ¯ S . . .

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.16/26

• R0. II

If R0 < 1, disease-free equilibrium is globally stable If R0 > 1, there exists a unique positive equilibrium:

     ¯ S =

Λ µR0

¯ λ = µ(R0 − 1) ¯ u(θ) = ¯ λ ¯ S . . .

If two strains compete, with complete cross-immunity, the strain with the highest R0 outcompetes the other (Bremermann-Thieme, 1989).

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.16/26

Graph of R0

20 40 60 80 100 120 0.01 0.1 1 10 100 1000 R0 survival r R0 GS k1=k2=1.e-2 R0 survival

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.17/26

Graph of R0

20 40 60 80 100 120 0.01 0.1 1 10 100 1000 R0 survival r R0 GS k1=k2=1.e-2 R0 survival

Maximum at an intermediate r. All graphs look like this (no proof!).

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.17/26

Optimal r for fixed a

5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 r survival a model GS k1= k2=1e-2

• ptimal r

survival

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.18/26

Optimal r for fixed a

5 10 15 20 25 30 35 40 5 10 15 20 25 30 35 40 r survival a model GS k1= k2=1e-2

• ptimal r

survival

If host evolution of a is slower than pathogen’s, move along the red curve to the maximum of the blue.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.18/26

Lower costs of immune response

2000 4000 6000 8000 10000 12000 20 40 60 80 100 r survival a model GS k1= 1.e-6 k2=1.e-4

• ptimal r

survival

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.19/26

Lower costs of immune response

2000 4000 6000 8000 10000 12000 20 40 60 80 100 r survival a model GS k1= 1.e-6 k2=1.e-4

• ptimal r

survival

Higher survival, but still a rather lethal infection.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.19/26

Effect of innate immunity

2000 4000 6000 8000 10000 12000 14000 16000 20 40 60 80 100 r survival a model M=5 k1= 1.e-6 k2=1e-4

• ptimal r

survival

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.20/26

Effect of innate immunity

2000 4000 6000 8000 10000 12000 14000 16000 20 40 60 80 100 r survival a model M=5 k1= 1.e-6 k2=1e-4

• ptimal r

survival

Survival lower than without.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.20/26

Host variability and pathogen evolution

Assume the values of a in the host population follow some distribution (with average 1):

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.21/26

Host variability and pathogen evolution

Assume the values of a in the host population follow some distribution (with average 1):

5 10 15 20 25 30 35 40 45 50 0.01 0.1 1 10 100 1000 R0 survival r R0 with var. a k1= k2=1.e-2 R0 survival

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.21/26

Host variability and pathogen evolution

Assume the values of a in the host population follow some distribution (with average 1):

5 10 15 20 25 30 35 40 45 50 0.01 0.1 1 10 100 1000 R0 survival r R0 with var. a k1= k2=1.e-2 R0 survival

How does it compare with fixed a?

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.21/26

Comparison of fixed vs. distributed a

50 100 150 200 250 300 0.1 1 10 100 1000 R0 survival r R0 fixed vs. variable a k1=1.e-6 k2=1.e-4 R0 a var. survival a var. R0 fixed a survival fixed a

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.22/26

Comparison of fixed vs. distributed a

50 100 150 200 250 300 0.1 1 10 100 1000 R0 survival r R0 fixed vs. variable a k1=1.e-6 k2=1.e-4 R0 a var. survival a var. R0 fixed a survival fixed a

Selection for much lower r

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.22/26

Adding innate immunity

50 100 150 200 250 0.1 1 10 100 1000 R0 survival r R0 fixed vs. variable a (M=5) k1=1.e-6 k2=1.e-4 R0 a var. survival a var. R0 fixed a survival fixed a

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.23/26

Adding innate immunity

50 100 150 200 250 0.1 1 10 100 1000 R0 survival r R0 fixed vs. variable a (M=5) k1=1.e-6 k2=1.e-4 R0 a var. survival a var. R0 fixed a survival fixed a

Similar picture

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.23/26

Host variation, and pathogen virulence

Should variation in host immunity levels select for lower virulence? why?

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.24/26

Host variation, and pathogen virulence

Should variation in host immunity levels select for lower virulence? why?

100 200 300 400 500 600 0.01 0.1 1 10 100 1000 R0 survival r R0 for different a (k1= k2=1.e-2) R0 a=0.1 survival a=0.1 R0 fixed a survival fixed a R0 a=5 survival a=5

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.24/26

Conclusions

The nested approach (Gilchrist-Sasaki, 2002) provides a very nice framework. Some complications can be introduced, since one has to resort to numericals, anyway.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.25/26

Conclusions

The nested approach (Gilchrist-Sasaki, 2002) provides a very nice framework. Some complications can be introduced, since one has to resort to numericals, anyway. It seems that pathogen selection on replication rate always brings to the level in which host survivorship is affected (and in the parameter region where the within-host dynamics has one positive equilibrium).

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.25/26

Conclusions

The nested approach (Gilchrist-Sasaki, 2002) provides a very nice framework. Some complications can be introduced, since one has to resort to numericals, anyway. It seems that pathogen selection on replication rate always brings to the level in which host survivorship is affected (and in the parameter region where the within-host dynamics has one positive equilibrium). Host variability (which could be due to age, nutritional status, . . . ) seems always to select for lower pathogen

• virulence. What does that mean for host evolution?

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.25/26

Conclusions

The nested approach (Gilchrist-Sasaki, 2002) provides a very nice framework. Some complications can be introduced, since one has to resort to numericals, anyway. It seems that pathogen selection on replication rate always brings to the level in which host survivorship is affected (and in the parameter region where the within-host dynamics has one positive equilibrium). Host variability (which could be due to age, nutritional status, . . . ) seems always to select for lower pathogen

• virulence. What does that mean for host evolution?

Modelling superinfection within this framework is rather complex, and perhaps pointless. There may be better ways of tackling within-host competition.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.25/26

Conclusions

The nested approach (Gilchrist-Sasaki, 2002) provides a very nice framework. Some complications can be introduced, since one has to resort to numericals, anyway. It seems that pathogen selection on replication rate always brings to the level in which host survivorship is affected (and in the parameter region where the within-host dynamics has one positive equilibrium). Host variability (which could be due to age, nutritional status, . . . ) seems always to select for lower pathogen

• virulence. What does that mean for host evolution?

Modelling superinfection within this framework is rather complex, and perhaps pointless. There may be better ways of tackling within-host competition.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.25/26

Thanks

The within-host model was set up and analysed with Alberto Gandolfi (IASI - Roma). Thanks to DIMACS for providing the support and the smooth organization for this workshop... ... and to you for your attention.

DIMACS workshop on Host-Parasite Coevolution, October 9-11 2006 – p.26/26