Evolution of pathogens: a within-host approach Vitaly V. Ganusov - - PowerPoint PPT Presentation

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II

Evolution of pathogens: a within-host approach

Vitaly V. Ganusov

Theoretical Biology Utrecht University, Utrecht, The Netherlands

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II

Outline

1

Introduction evolution of virulence

2

Evolution of infectious diseases a “within-host” approach changing model details imperfect vaccines

3

Conclusions implications for immuno-epidemiology

4

Appendix I heterogeneity details vaccines

5

Appendix II modelling mortality

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II virulence

Emergence of infectious diseases

reservoir

New host population R0 in the new host population

R0=2/3 rate of introduction l † † † † R0 equals the average number of secondary infections causes by an infected host introduced into a wholly susceptible population. 3 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II virulence

Evolution of pathogens in new hosts

Once a pathogen has emerged (R0 > 1), the important question is whether it is going to evolve to be benign or virulent. The evolution of pathogens is generally considered in terms of the basic reproductive number R0.

Pathogens evolve to maximize R0 (i.e., their total transmission). Pathogens evolve their virulence, defined as the reduction in host fitness due to infection with the pathogen. In models, virulence is measured by host mortality rate or case mortality.

Anderson and May (1982); Bremermann and Thieme (1989) 4 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II virulence

The basic reproductive number R0

For directly transmitted diseases R0 = βN

infection rate

× 1 α + d + ν

• duration of infection

= β(α)N α + d + ν(α)

α

β ν

α R0

Anderson and May (1982) 5 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II virulence

Introduced virus strain: 1950

I II III IV V Grade of the virus 20 40 60 80 100 % rabbits infected 1950

most virulent least virulent case mortality > 0.99 case mortality ≈ 0.23

Fenner and Fantini (1999); virulence was measured in laboratory (standard) rabbits. 6 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II virulence

Virus prevalence: 1952

I II III IV V Grade of the virus 20 40 60 80 100 % rabbits infected 1952

most virulent least virulent case mortality > 0.99 case mortality ≈ 0.23

Fenner and Fantini (1999); virulence was measured in laboratory (standard) rabbits. 6 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II virulence

Virus prevalence: 1970

I II III IV V Grade of the virus 20 40 60 80 100 % rabbits infected 1970

most virulent least virulent case mortality > 0.99 case mortality ≈ 0.23

Fenner and Fantini (1999); virulence was measured in laboratory (standard) rabbits. 6 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II virulence

Virus prevalence: 1984

I II III IV V Grade of the virus 20 40 60 80 100 % rabbits infected 1984

most virulent least virulent case mortality > 0.99 case mortality ≈ 0.23

Fenner and Fantini (1999); virulence was measured in laboratory (standard) rabbits. 6 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II virulence

Trade-offs for the myxoma virus infection of rabbits

0.02 0.04 0.06 0.08 virulence, day-1 0.005 0.01 0.015 0.02 0.025 0.03 host recovery rate, day-1 0.02 0.04 0.06 0.08 virulence, day-1 20 40 60 80 100 % infected vectors 0.02 0.04 0.06 0.08 virulence, day-1 1 2 3 4 5 R0, relative value

α* = 0.0400183 Fenner et al. 1956; Mead-Briggs et al. 1975; Anderson and May 1982* 7 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Within-host dynamics of pathogens

P X

Parasite Immune response

rP

hXP

sPX k+P

uP

+ + − +

transmission

8 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Dynamics of the pathogen and the immune response

˙ P = rP − hPX, P = 0, if P(t) ≥ D, ˙ X = sXP k + P , l(r) = u ∆ P(t) dt. Pathogen kills the host if it reaches a lethal density D; There is no transmission from a dead host; Pathogens evolve to maximize their total transmission.

P – pathogen, X – immune response, l – total transmission, ∆ – duration of infection. Parameters: P (0) = 1, X(0) = 1, h = 10−3, k = 103, s = 1, D = 109, r = 2.08. Antia et al. 1994 9 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Dynamics of the pathogen and the immune response

˙ P = rP − hPX, P = 0, if P(t) ≥ D, ˙ X = sXP k + P , l(r) = u ∆ P(t) dt.

5 10 15 20 days 1 102 104 106 108 density lethal density D

X P

P – pathogen, X – immune response, l – total transmission, ∆ – duration of infection. Parameters: P (0) = 1, X(0) = 1, h = 10−3, k = 103, s = 1, D = 109, r = 2.08. Antia et al. 1994 9 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Total transmission of pathogens

5 10 15 20 days 0.2 0.4 0.6 0.8 1 parasite density, 109 r=1.9 r=2.2 lethal density D r=2.08 1.6 1.8 2 2.2 2.4 2.6 2.8 3 growth rate r 20 40 60 80 100 % maximal transmission r=r*

where total transmission l(r) = ∆ P(t) dt.

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Stochastic heterogeneity in r

average growth rate

 r

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Stochastic heterogeneity in r

average growth rate host 1

 rr1  r

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Stochastic heterogeneity in r

average growth rate host 1

 rr1

host 2

 r−r2  r

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Stochastic heterogeneity in r

average growth rate host 1

 rr1

host 2

 r−r2

host i ...

 r±ri  r

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Stochastic heterogeneity in r

growth rate r prob density function f σ=0.1 σ=0.2

σ r

__

f(r, r) = r/σ2 Γ(r2/σ2) rr σ2 r2/σ2−1 × exp

• − rr

σ2

• ,

L(r) = ∞ l(r)f(r, r)dr.

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Optimal growth rate and total transmission

L(r) = ∞ l(r, x) f(x) dx.

1.6 1.8 2 2.2 2.4 2.6 2.8 3

the average growth rate r

__

20 40 60 80 100 % total transmission

σ=0.02 σ=0.05

where f(x) is given by a gamma distribution of r with standard deviation σ. 13 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Changes in virulence

M(r) = ∞

r∗ f(r, r)dr

0.05 0.1 0.15 0.2 0.25 0.3 Standard deviation σ 0.2 0.4 0.6 0.8 1 case mortatality 14 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Changes in virulence

M(r) = ∞

r∗ f(r, r)dr

0.05 0.1 0.15 0.2 0.25 0.3 Standard deviation σ 0.2 0.4 0.6 0.8 1 case mortatality

LD50(r) = P0 : ∞

r∗[P0]

f(r, r)dr = 0.5

0.05 0.1 0.15 0.2 Standard deviation σ 100 200 300 400 LD50 2.04 2.05 2.06 2.07 2.08 r 14 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Estimating epidemiological parameters and trade-offs

time P

∆

total transmission duration of infection time P

ˆ β(r) = l(r) ∆(r) β(r) = ∞ ˆ β(r)f(r, r) dr

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Estimating epidemiological parameters and trade-offs

time P

∆

total transmission duration of infection time P

ˆ β(r) = l(r) ∆(r) β(r) = ∞ ˆ β(r)f(r, r) dr α(r) = ∞ m(r) ∆(r)f(r, r) dr ν(r) = ∞ 1 − m(r) ∆(r) f(r, r) dr

where m(r) is the probability of host’s death following infection. 15 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Trade-offs emerging from the within-host dynamics

0.02 0.04 0.06 0.08 0.1 0.12 0.14 host mortality rate α, day1 0.02 0.04 0.06 0.08 recovery rate ν, day1

ν vs. α

0.02 0.04 0.06 0.08 0.1 0.12 0.14 host mortality rate α, day1 0.05 0.1 0.15 0.2 transmission rate β103

β vs. α

0.02 0.04 0.06 0.08 0.1 0.12 0.14 host mortality rate α, day1 0.5 1 1.5 2 2.5 3 Ro

Ro vs. α

α To myxoma trade-offs Ganusov et al 2002 16 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach

Short summary

Within-host and between host dynamics of pathogens are inherently linked. Trade-offs for the myxoma virus infection can be originated from simple properties of the within-host dynamics.

but: other explanations may work too.

Prediction on the evolution of pathogen virulence may depend on the definition of virulence used.

Ganusov et al. 2002; Gilchrist and Sasaki 2002; Andr´ e et al. 2003; Andr´ e and Gandon (2006); Ganusov and Antia 2003, 2006 17 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II changing model details

Can virulence be predicted from a single factor?

Ewald 1983 18 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II changing model details

Changing pathogen transmissibility

0.2 0.4 0.6 0.8 1 PD 0.2 0.4 0.6 0.8 1 ΖP

linear saturated exponential

Ganusov and Antia 2003 19 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II changing model details

Changing pathogen transmissibility

0.2 0.4 0.6 0.8 1 PD 0.2 0.4 0.6 0.8 1 ΖP

linear saturated exponential

1.6 1.8 2 2.2 2.4 2.6 2.8 3 growth rate r 20 40 60 80 100 % transmission

linear saturated exponential

r=r*

0.05 0.1 0.15 0.2 0.25 0.3 Heterogeneity 0.2 0.4 0.6 0.8 1 case mortality

exponential saturated linear Ganusov and Antia 2003 19 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II changing model details

Changing mechanism of pathogenesis

5 10 15 20 days 0.2 0.4 0.6 0.8 1 density r=1.9 r=2.08 r=2.2 Rd

P (0) = 1, R(0) = R0 = 104, X(0) = 1, h = 10−3, k = 103, s = 1, Rd = 3.25 × 103, c = 103, y = 105, d = 0.05. Heterogeneity is modelled by a normal distribution of r. 20 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II changing model details

Changing mechanism of pathogenesis

5 10 15 20 days 0.2 0.4 0.6 0.8 1 density r=1.9 r=2.08 r=2.2 Rd

1.6 1.8 2 2.2 2.4 2.6 2.8 3 growth rate r 20 40 60 80 100 % transmission

l e t h a l d e n s i t y m

• d

e l resource depletion model r=r*

0.05 0.1 0.15 0.2 0.25 0.3 Heterogeneity 0.2 0.4 0.6 0.8 1 case mortality

lethal density model resource depletion model P (0) = 1, R(0) = R0 = 104, X(0) = 1, h = 10−3, k = 103, s = 1, Rd = 3.25 × 103, c = 103, y = 105, d = 0.05. Heterogeneity is modelled by a normal distribution of r. 20 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II changing model details

Details do matter!

Changing the structure of the model may dramatically affect the optimal level of virulence. It seems unlikely that a single factor can determine virulence of diverse pathogens.

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

Vaccines and pathogen evolution

Escape from vaccines

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

Vaccines and pathogen evolution

Escape from vaccines Evolution of pathogen virulence in response to vaccination

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

Imperfect vaccines and evolution of pathogens

Epidemiological approach and R0 R0[α∗, α] = β∗(ˆ x + σˆ y) d + α∗ + ν∗ + σβˆ y

where ∗ and ˆ · denote mutant and resident, and σ is superinfection parameter. 23 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

Imperfect vaccines and evolution of pathogens

Epidemiological approach and R0 R0[α∗, α] = β∗(ˆ x + σˆ y) d + α∗ + ν∗ + σβˆ y

where ∗ and ˆ · denote mutant and resident, and σ is superinfection parameter.

Both virulence α and transmissibility β are reduced in vaccinated hosts. αV = (1 − r2)(1 − r4)αU, βV = (1 − r3)βU[(1 − r2)αU],

where r2, r3, and r4 are the efficacies of vaccines blocking replication, transmission and virulence, respectively. 23 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

Imperfect vaccines and evolution of pathogens

Epidemiological approach and R0 R0[α∗, α] = β∗(ˆ x + σˆ y) d + α∗ + ν∗ + σβˆ y

where ∗ and ˆ · denote mutant and resident, and σ is superinfection parameter.

Both virulence α and transmissibility β are reduced in vaccinated hosts. αV = (1 − r2)(1 − r4)αU, βV = (1 − r3)βU[(1 − r2)αU],

where r2, r3, and r4 are the efficacies of vaccines blocking replication, transmission and virulence, respectively.

Vaccination does not affect trade-offs β = β(α) and ν = ν(α).

Gandon et al. 2001 23 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

Imperfect vaccines: Gandon et al. conclusions

Anti-growth and anti-virulence vaccines are expected to select for pathogens with high virulence. Anti-transmission vaccines are expected to select for pathogens with low virulence.

Gandon et al. 2001 24 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

Within-host approach

P X1

Immune response Pathogen

+

replication

+ + −

transmission

P X1

Immune response Pathogen

+

replication

+ + −

transmission

P X1

Immune response

X2

Immune response Pathogen

+

replication

+ + −

transmission

− + +

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

Model

response X1 reduces the rate of expansion of the pathogen population within the host; response X2 reduces the rate of pathogen transmission from infected hosts.

˙ P = (r − h1X1)P, ˙ Xi = sXiP k + P , i = 1, 2, l(r) = ∆ P(t)dt 1 + h2X2(t). Vaccination results in an increase in the number of pathogen-specific immune cells (precursor numbers) existing prior to infection.

Ganusov and Antia 2006 26 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

Within-host dynamics: anti-growth vaccines

5 10 15 20 days 0.2 0.4 0.6 0.8 1 density P X1 lethal density D l=100% 5 10 15 20 days 0.2 0.4 0.6 0.8 1 density P X1 lethal density D l=24% X10=2

P (0) = 1, h1 = 10−3, h2 = 10−4, k = 103, s = 1, D = 109, r = 2.08, pathogen density is multiplied by 10−9, the immune response densities are multipled by 4 × 10−6. 27 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

Transmission and virulence: anti-growth vaccines

1.6 1.8 2 2.2 2.4 2.6 2.8 3 growth rate r 20 40 60 80 100 % maximal transmission unvacc σ0 M0 σ0.1 M0.31 0.1 0.2 0.3 0.4 0.5 heterogeneity σ 0.2 0.4 0.6 0.8 1 case mortality unvaccinated vaccinated

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

Within-host dynamics: anti-transmission vaccines

5 10 15 20 days 0.2 0.4 0.6 0.8 1 density P X1 lethal density D l=100% 5 10 15 20 days 0.2 0.4 0.6 0.8 1 density P X1 X2 lethal density D l39% X2010

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

Transmission/virulence: anti-transmission vaccines

1.6 1.8 2 2.2 2.4 2.6 2.8 3 growth rate r 20 40 60 80 100 % maximal transmission unvacc σ0 M0 σ0.1 M0.73 0.1 0.2 0.3 0.4 0.5 heterogeneity σ 0.2 0.4 0.6 0.8 1 case mortality unvaccinated vaccinated

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

ES growth rate and virulence: partially vaccinated

0.2 0.4 0.6 0.8 1 fraction vaccinated p 2 2.05 2.1 2.15 2.2 2.25 2.3 Optimal growth rate r anti-growth anti- growth anti- transm 0.2 0.4 0.6 0.8 1 fraction vaccinated 0.2 0.4 0.6 0.8 1 average mortality anti- growth anti- transm anti-growth For anti-growth vaccines, the precursor number increases from X10 = 1 to X10 = 2 (bold red lines) or to X10 = 10 (plain red lines). For anti-transmission vaccines, the precursor number increases from X20 = 0 to X20 = 10 (bold blue lines). 31 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II imperfect vaccines

Do results depend on the model?

0.2 0.4 0.6 0.8 1 fraction vaccinated 0.2 0.4 0.6 0.8 1 average mortality anti- growth anti- transm anti-growth

In these models, the difference arises due to different description

• f pathogenesis and as the result, due to high ES virulence in

unvaccinated hosts in the right panel (at p = 0).

Ganusov and Antia 2006; Andr´ e and Gandon 2006 32 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II implications for immuno-epidemiology

Implications for epidemiology

Epidemiological models can include the within-host dynamics: dS(t) dt = λ − dS(t) − h(t)S(t), ∂I(t, τ) ∂t + ∂I(t, τ) ∂τ = −(d + α(τ) + ν(τ))I(t, τ), dR(t) dt = t I(t, τ)ν(τ) dτ − dR(t), I(t, 0) = h(t)S(t) = S(t) t I(t, τ)β(τ) dτ. Future studies may investigate the role of mutation, co- and super-infection in determining evolution of pathogens using within-host models.

Andr´ e and Gandon 2006 33 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II implications for immuno-epidemiology

More implications for epidemiology

Predictions on the evolution of pathogens may depend on the model used as well on the model parameters, and therefore, building of proper models requires better understanding of the biology of pathogen-host interactions. Other factors may further complicate the picture: within-host evolution of pathogens, co- and super-infection, locality of transmission, host evolution, etc.

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II implications for immuno-epidemiology

Acknowledgements

Rustom Antia and Carl Bergstrom Theoretical Biology group at Utrecht University Marie Curie Incoming International Fellowship (Framework Programme 6)

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II implications for immuno-epidemiology

Testing model predictions?

Higher levels of (host) heterogeneity select for more virulent pathogens.

High mutation rate of Neisseria meningitidis helps escaping immune response. Malaria (P . falciparum) infecting resistant adults and nonimmune infants.

Transmission-blocking vaccines may select for more rapidly growing pathogens

?

Testing both predictions in serial passage experiments?

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II heterogeneity

Coevolution of the myxoma virus and rabbits

2 3 4 5 7 # of epizootics 20 40 60 80 100 % recovered rabbits 37 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II heterogeneity

Coevolution of the myxoma virus and rabbits

2 3 4 5 7 # of epizootics 20 40 60 80 100 % recovered rabbits I II III IV V Grade of the virus 20 40 60 80 100 % rabbits infected 1984 1992 37 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II heterogeneity

Heterogeneity in other parameters

0.1 0.2 0.3 0.4 0.5 Heterogeneity 0.2 0.4 0.6 0.8 1 case mortality

D h P0 k r s

0.1 0.2 0.3 0.4 0.5 Heterogeneity 100 200 300 400 500 LD50

D h k r s

0.1 0.2 0.3 0.4 0.5 Heterogeneity 20 40 60 80 100 % maximal transmission

D h P0 k r s

0.1 0.2 0.3 0.4 0.5 Heterogeneity 2 2.02 2.04 2.06 2.08 2.1

• ptimal growth rate

D h P0 k r s heterogeneity = σ/mean. 38 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II details

Changing pathogen transmissibility

2 4 6 8 10 parasite density, 108 0.2 0.4 0.6 0.8 1 transmission rate

linear saturated squared

1.6 1.8 2 2.2 2.4 2.6 2.8 3 growth rate r 20 40 60 80 100 % maximal transmission

linear saturated squared r=r*

0.1 0.2 0.3 0.4 0.5 Heterogeneity 0.2 0.4 0.6 0.8 1 The case mortality

squared saturated linear

Heterogeneity (CV = σ/D) is modelled by a gamma distribution of the lethal density D. Ganusov and Antia 2003 39 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II details

Dynamics of the pathogen, resource and the immune response

˙ P = rPR c + R − hPX, ˙ R = d(R0 − R) − y−1 rPR c + R, ˙ X = sXP k + P ,

5 10 15 20 days 1 10 102 103 104 105 106 107 108 109 parasite

X P R

Rd

P – pathogen, R – resource, X – immune response. Parameters: P (0) = 1, R(0) = R0 = 104, X(0) = 1, h = 10−3, k = 103, s = 1, Rd = 2.7 × 103, c = 103, y = 105, d = 0, r = 2.08. 40 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II details

Changing mechanism of pathogenesis

5 10 15 20 days 0.2 0.4 0.6 0.8 1 normalized density r=1.9 r=2.08 r=2.2 Rd

Heterogeneity (CV = σ/Rd) is modelled by a gamma distribution in the minimal resource density Rd. P (0) = 1, R(0) = R0 = 104, X(0) = 1, h = 10−3, k = 103, s = 1, Rd = 2.7 × 103, c = 103, y = 105, d = 0. 41 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II details

Changing mechanism of pathogenesis

5 10 15 20 days 0.2 0.4 0.6 0.8 1 normalized density r=1.9 r=2.08 r=2.2 Rd 1.25 1.5 1.75 2 2.25 2.5 2.75 3 growth rate r 20 40 60 80 100 % maximal transmission r=r* lethal density resource depletion 0.1 0.2 0.3 0.4 0.5 Heterogeneity 0.2 0.4 0.6 0.8 1 The case mortality lethal density resource depletion

Heterogeneity (CV = σ/Rd) is modelled by a gamma distribution in the minimal resource density Rd. P (0) = 1, R(0) = R0 = 104, X(0) = 1, h = 10−3, k = 103, s = 1, Rd = 2.7 × 103, c = 103, y = 105, d = 0. 41 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II vaccines

Changes in trade-offs with vaccination

ANTI-GROWTH ANTI-TRANSMISSION

0.025 0.05 0.075 0.1 0.125 0.15 host mortality rate α 0.1 0.2 0.3 0.4 transmissibility β X10=1 X10=2 X10=10 0.025 0.05 0.075 0.1 0.125 0.15 host mortality rate α 0.1 0.2 0.3 0.4 transmissibility β X20=0 X20=10 0.025 0.05 0.075 0.1 0.125 0.15 host mortality rate α 0.02 0.04 0.06 0.08 0.1 recovery rate ν X10=1 X10=2 X10=10 0.025 0.05 0.075 0.1 0.125 0.15 host mortality rate α 0.02 0.04 0.06 0.08 0.1 recovery rate ν X20=0 X20=10 42 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II vaccines

Changes in R0 with vaccination

ANTI-GROWTH ANTI-TRANSMISSION

0.025 0.05 0.075 0.1 0.125 0.15 host mortality rate α, day-1 1 2 3 4 R0 X10=1 X10=2 X10=10 0.025 0.05 0.075 0.1 0.125 0.15 host mortality rate α, day-1 1 2 3 4 R0 X20=0 X20=10

Trade-offs do change with vaccination although changes may be small at low efficacy of vaccines (small increase in X10 and X20). Anti-transmission vaccines may select for more virulent pathogens.

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II vaccines

Virulence in unvaccinated and vaccinated hosts

0.05 0.1 0.15 0.2 mortality rate αU 0.05 0.1 0.15 0.2 mortality rate αV X10=2 X10=1 X10=10

The relationship between virulence of a pathogen with a fixed growth rate r in vaccinated αV and unvaccinated αU hosts is nonlinear. Note that in the study by Gandon et al. 2001, αV = (1 − r2)(1 − r4)αU.

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II vaccines

Total transmission vs. r

1.5 2 2.5 3 3.5 growth rate r 10 20 30 40 % transmission p0.1

L

__

1pLu pLv

1.5 2 2.5 3 3.5 growth rate r 10 20 30 40 % transmission p0.1

L

__

1pLu pLv

1.5 2 2.5 3 3.5 growth rate r 10 20 30 40 % transmission p=0.5 1.5 2 2.5 3 3.5 growth rate r 10 20 30 40 % transmission p=0.5 45 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II vaccines

Total transmission and virulence vs vaccine efficacy

1 10 102 precursor number after vaccination 10 20 30 40 % transmission anti- growth anti-growth anti- transm anti-transm 1 10 102 precursor number after vaccination 0.2 0.4 0.6 0.8 1 case mortality anti- growth anti-growth anti- transm anti-transm 46 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II vaccines

Two stages X

1

Immune response

X

2

Immune response

P

1

Pathogen

P

2

Pathogen

+

replication transmission

− −

47 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Pathogenesis and transmission

The probability of host survival S(t) until time t is the solution of ˙ S(t) = −π[r, P(t)]S(t) where π(r, P) is the rate of host’s death due to pathogen.

π[r, P] ∼ (P/D)n

0.2 0.4 0.6 0.8 1 PD 0.2 0.4 0.6 0.8 1 ΠP

n1 n2 n10 n100

48 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Pathogenesis and transmission

The probability of host survival S(t) until time t is the solution of ˙ S(t) = −π[r, P(t)]S(t) where π(r, P) is the rate of host’s death due to pathogen.

π[r, P] = limn→∞(P/D)n

0.2 0.4 0.6 0.8 1 PD 0.2 0.4 0.6 0.8 1 ΠP

n

48 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Pathogenesis and transmission

The probability of host survival S(t) until time t is the solution of ˙ S(t) = −π[r, P(t)]S(t) where π(r, P) is the rate of host’s death due to pathogen. The total transmission of the pathogen during the infection is l = ∆ ζ[P(t)]S(t) dt where ζ(P) is the rate of pathogen transmission.

π[r, P] = limn→∞(P/D)n

0.2 0.4 0.6 0.8 1 PD 0.2 0.4 0.6 0.8 1 ΠP

n

0.2 0.4 0.6 0.8 1 PD 0.2 0.4 0.6 0.8 1 ΖP

P

• P
• Θ P

P2

48 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Pathogenesis and transmission

The probability of host survival S(t) until time t is the solution of ˙ S(t) = −π[r, P(t)]S(t) where π(r, P) is the rate of host’s death due to pathogen. The total transmission of the pathogen during the infection is l = ∆ ζ[P(t)]S(t) dt where ζ(P) is the rate of pathogen transmission.

π[r, P] = limn→∞(P/D)n

0.2 0.4 0.6 0.8 1 PD 0.2 0.4 0.6 0.8 1 ΠP

n

0.2 0.4 0.6 0.8 1 PD 0.2 0.4 0.6 0.8 1 ΖP 48 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Back to a more general “stochastic” approach

Host survival during an infection is a stochastic process ˙ S(t) = −π[r, P(t)]S(t) l(r) = ∆ ζ[P(t)]S(t) dt

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Back to a more general “stochastic” approach

Host survival during an infection is a stochastic process ˙ S(t) = −π[r, P(t)]S(t) l(r) = ∆ ζ[P(t)]S(t) dt Consider a particular case when π ∼ P n and ζ ∼ P:

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Back to a more general “stochastic” approach

Host survival during an infection is a stochastic process ˙ S(t) = −π[r, P(t)]S(t) l(r) = ∆ ζ[P(t)]S(t) dt Consider a particular case when π ∼ P n and ζ ∼ P: π(r, P) = P D 1

1.5 2 2.5 3 3.5 growth rate r 0.5 1 1.5 2 transmission109

n1

1.5 2 2.5 3 3.5 growth rate r 0.2 0.4 0.6 0.8 1 case mortality

n1

49 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Back to a more general “stochastic” approach

Host survival during an infection is a stochastic process ˙ S(t) = −π[r, P(t)]S(t) l(r) = ∆ ζ[P(t)]S(t) dt Consider a particular case when π ∼ P n and ζ ∼ P: π(r, P) = P D 2

1.5 2 2.5 3 3.5 growth rate r 0.5 1 1.5 2 transmission109

n2

1.5 2 2.5 3 3.5 growth rate r 0.2 0.4 0.6 0.8 1 case mortality

n2

49 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Back to a more general “stochastic” approach

Host survival during an infection is a stochastic process ˙ S(t) = −π[r, P(t)]S(t) l(r) = ∆ ζ[P(t)]S(t) dt Consider a particular case when π ∼ P n and ζ ∼ P: π(r, P) = P D 5

1.5 2 2.5 3 3.5 growth rate r 0.5 1 1.5 2 transmission109

n5

1.5 2 2.5 3 3.5 growth rate r 0.2 0.4 0.6 0.8 1 case mortality

n5

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Back to a more general “stochastic” approach

Host survival during an infection is a stochastic process ˙ S(t) = −π[r, P(t)]S(t) l(r) = ∆ ζ[P(t)]S(t) dt Consider a particular case when π ∼ P n and ζ ∼ P: π(r, P) = P D 10

1.5 2 2.5 3 3.5 growth rate r 0.5 1 1.5 2 transmission109

n10

1.5 2 2.5 3 3.5 growth rate r 0.2 0.4 0.6 0.8 1 case mortality

n10

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Back to a more general “stochastic” approach: II

Basic formulas ˙ S(t) = −π[r, P(t)]S(t) l(r) = ∆ P(t)S(t) dt M(r) = 1 − S(∆) Traditionally, stochastic host survival is modelled differently, π = λrmP.

Sasaki and Iwasa 1991; Gilchrist and Sasaki 2002; Andr´ e et al. 2003. 50 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Back to a more general “stochastic” approach: II

Basic formulas ˙ S(t) = −π[r, P(t)]S(t) l(r) = ∆ P(t)S(t) dt M(r) = 1 − S(∆) Traditionally, stochastic host survival is modelled differently, π = λrmP. π(r, P) = λr1P

Sasaki and Iwasa 1991; Gilchrist and Sasaki 2002; Andr´ e et al. 2003. 50 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Back to a more general “stochastic” approach: II

Basic formulas ˙ S(t) = −π[r, P(t)]S(t) l(r) = ∆ P(t)S(t) dt M(r) = 1 − S(∆) Traditionally, stochastic host survival is modelled differently, π = λrmP. π(r, P) = λr1P

1 1.5 2 2.5 3 growth rate r 1 2 3 4 5 transmission108

m1

1 1.5 2 2.5 3 growth rate r 0.2 0.4 0.6 0.8 1 case mortality

m1

Sasaki and Iwasa 1991; Gilchrist and Sasaki 2002; Andr´ e et al. 2003. 50 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Back to a more general “stochastic” approach: II

Basic formulas ˙ S(t) = −π[r, P(t)]S(t) l(r) = ∆ P(t)S(t) dt M(r) = 1 − S(∆) Traditionally, stochastic host survival is modelled differently, π = λrmP. π(r, P) = λr2P

1 1.5 2 2.5 3 growth rate r 1 2 3 4 5 transmission108

m2

1 1.5 2 2.5 3 growth rate r 0.2 0.4 0.6 0.8 1 case mortality

m2

Sasaki and Iwasa 1991; Gilchrist and Sasaki 2002; Andr´ e et al. 2003. 50 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Back to a more general “stochastic” approach: II

Basic formulas ˙ S(t) = −π[r, P(t)]S(t) l(r) = ∆ P(t)S(t) dt M(r) = 1 − S(∆) Traditionally, stochastic host survival is modelled differently, π = λrmP. π(r, P) = λr5P

1 1.5 2 2.5 3 growth rate r 1 2 3 4 5 transmission108

m5

1 1.5 2 2.5 3 growth rate r 0.2 0.4 0.6 0.8 1 case mortality

m5

Sasaki and Iwasa 1991; Gilchrist and Sasaki 2002; Andr´ e et al. 2003. 50 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Back to a more general “stochastic” approach: II

Basic formulas ˙ S(t) = −π[r, P(t)]S(t) l(r) = ∆ P(t)S(t) dt M(r) = 1 − S(∆) Traditionally, stochastic host survival is modelled differently, π = λrmP. π(r, P) = λr10P

1 1.5 2 2.5 3 growth rate r 1 2 3 4 5 transmission108

m10

1 1.5 2 2.5 3 growth rate r 0.2 0.4 0.6 0.8 1 case mortality

m10

Sasaki and Iwasa 1991; Gilchrist and Sasaki 2002; Andr´ e et al. 2003. 50 / 52

Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Summary

Even in simple “within-host” models, a variety of methods exist to describe pathogenesis. Moderate levels of virulence (case mortality) can evolve if rate of pathogenesis π ∼ P n. When π ∼ rmP, saturation in the transmission rate may help to reduce the case mortality.

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Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II modelling mortality

Introducing heterogeneity in parameters

For a parameter X, f(x) dx is the probability that a during a given infection, the parameter X will be in the range (x, x + dx). Then total transmission of the pathogen with the growth rate r in a heterogeneous population is calculated as L(r) = ∞ l(r, x) f(x) dx. Thus, such heterogeneity may arise due to stochasticity in pathogen-host interactions. We illustrate the results with heterogeneity in the growth rate r described by a gamma distribution.

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