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 1 / 52
Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II Outline Introduction 1 evolution of virulence Evolution of infectious diseases 2 a “within-host” approach changing model details imperfect vaccines Conclusions 3 implications for immuno-epidemiology Appendix I 4 heterogeneity details vaccines Appendix II 5 modelling mortality 2 / 52
Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II virulence Emergence of infectious diseases New host population † R 0 in the new † host population rate of reservoir introduction l † † R 0 =2/3 R 0 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 ( R 0 > 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 R 0 . Pathogens evolve to maximize R 0 (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 R 0 For directly transmitted diseases 1 β ( α ) N R 0 = βN = × α + d + ν α + d + ν ( α ) � �� � � �� � infection rate duration of infection R 0 β ν α α Anderson and May (1982) 5 / 52
Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II virulence Introduced virus strain: 1950 100 1950 80 % rabbits infected 60 40 20 0 I II III IV V most virulent Grade of the virus 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 100 1952 80 % rabbits infected 60 40 20 0 I II III IV V most virulent Grade of the virus 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 100 1970 80 % rabbits infected 60 40 20 0 I II III IV V most virulent Grade of the virus 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 100 1984 80 % rabbits infected 60 40 20 0 I II III IV V most virulent Grade of the virus 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 100 0.03 host recovery rate, day -1 % infected vectors 80 0.025 0.02 60 0.015 40 0.01 20 0.005 0 0 0 0.02 0.04 0.06 0.08 0 0.02 0.04 0.06 0.08 virulence, day -1 virulence, day -1 5 α * = 0.0400183 R0, relative value 4 3 2 1 0 0 0.02 0.04 0.06 0.08 virulence, day -1 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 + + + sPX rP k+P P X Parasite Immune response hXP − transmission uP 8 / 52
Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach Dynamics of the pathogen and the immune response Pathogen kills the host if it reaches a lethal density D ; ˙ P = rP − hPX, There is no transmission from a P = 0 , if P ( t ) ≥ D, dead host; sXP ˙ X = k + P , Pathogens evolve to maximize � ∆ their total transmission. l ( r ) = u P ( t ) dt. 0 P – pathogen, X – immune response, l – total transmission, ∆ – duration of infection. Parameters: P (0) = 1 , X (0) = 1 , h = 10 − 3 , k = 10 3 , s = 1 , D = 10 9 , 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 lethal density D 10 8 P ˙ P = rP − hPX, density 10 6 P = 0 , if P ( t ) ≥ D, X 10 4 sXP ˙ X = k + P , 10 2 � ∆ 1 l ( r ) = u P ( t ) dt. 0 5 10 15 20 days 0 P – pathogen, X – immune response, l – total transmission, ∆ – duration of infection. Parameters: P (0) = 1 , X (0) = 1 , h = 10 − 3 , k = 10 3 , s = 1 , D = 10 9 , 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 lethal density D 1 100 r = r * % maximal transmission parasite density, 10 9 r = 2.08 r = 2.2 0.8 80 0.6 60 0.4 40 r = 1.9 0.2 20 0 0 0 5 10 15 20 1.6 1.8 2 2.2 2.4 2.6 2.8 3 days growth rate r where total transmission � ∆ l ( r ) = P ( t ) dt. 0 10 / 52
Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach Stochastic heterogeneity in r average growth rate r 11 / 52
Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach Stochastic heterogeneity in r average growth rate r host 1 r r 1 11 / 52
Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach Stochastic heterogeneity in r average growth rate r host 1 host 2 r − r 2 r r 1 11 / 52
Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach Stochastic heterogeneity in r average growth rate r ... host i host 1 host 2 r ± r i r − r 2 r r 1 11 / 52
Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach Stochastic heterogeneity in r prob density function f σ = 0.1 σ σ = 0.2 __ r growth rate r � rr � r 2 /σ 2 − 1 � � r/σ 2 − rr f ( r, r ) = × exp , Γ( r 2 /σ 2 ) σ 2 σ 2 � ∞ L ( r ) = l ( r ) f ( r, r ) dr. 0 12 / 52
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. 0 100 % total transmission 80 σ = 0.02 60 σ = 0.05 40 20 0 1.6 1.8 2 2.2 2.4 2.6 2.8 3 __ the average growth rate r 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 1 case mortatality 0.8 0.6 0.4 0.2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Standard deviation σ 14 / 52
Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach Changes in virulence � ∞ � ∞ LD 50 ( r ) = P 0 : f ( r, r ) dr = 0 . 5 M ( r ) = r ∗ f ( r, r ) dr r ∗ [ P 0 ] 1 400 case mortatality 0.8 2.08 300 2.07 0.6 LD50 r 2.06 200 0.4 2.05 100 0.2 2.04 0 0 0.05 0.1 0.15 0.2 0 0 0.05 0.1 0.15 0.2 0.25 0.3 Standard deviation σ Standard deviation σ 14 / 52
Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach Estimating epidemiological parameters and trade-offs P P l ( r ) ˆ total β ( r ) = transmission ∆( r ) � ∞ duration of ˆ β ( r ) = β ( r ) f ( r, r ) dr infection ∆ 0 time time 15 / 52
Introduction Evolution of infectious diseases Conclusions Appendix I Appendix II a “within-host” approach Estimating epidemiological parameters and trade-offs P P l ( r ) ˆ total β ( r ) = transmission ∆( r ) � ∞ duration of ˆ β ( r ) = β ( r ) f ( r, r ) dr infection ∆ 0 time time � ∞ m ( r ) α ( r ) = ∆( r ) f ( r, r ) dr 0 � ∞ 1 − m ( r ) ν ( r ) = f ( r, r ) dr ∆( r ) 0 where m ( r ) is the probability of host’s death following infection. 15 / 52
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