Spatial Evolutionary Epidemiology of Drug Resistance
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Talk given at the R0 conference in 2008. Based on the article http://dx.doi.org/10.1371/journal.pcbi.1000337 Evolutionary Epidemiology of Drug-Resistance in Space Debarre, Lenormand and Gandon, 2009, PLOS Computational Biology
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Spatial Evolutionary Epidemiology of Drug Resistance
- 1. Evolutionary epidemiology of drug resistance A spatial model Florence D´ebarre, Thomas Lenormand, Sylvain Gandon CNRS UMR 5175, CEFE, Montpellier, France October 30, 2008
- 2. Introduction Models Results Conclusion Supp Introduction Antimicrobial drugs and life expectancy
- 3. Introduction Models Results Conclusion Supp Introduction Antimicrobial drugs and life expectancy But emergence of drug resistance Penicillin consumption in 2000 %ofdrug-resistantSpneumoniae Goossens (2005)
- 4. Introduction Models Results Conclusion Supp Introduction Antimicrobial drugs and life expectancy But emergence of drug resistance Penicillin consumption in 2000 %ofdrug-resistantSpneumoniae Goossens (2005)
- 5. Introduction Models Results Conclusion Supp Introduction Antimicrobial drugs and life expectancy But emergence of drug resistance Treatment is not applied uniformly . . . Penicillin consumption in 2000 %ofdrug-resistantSpneumoniae Goossens (2005)
- 6. Introduction Models Results Conclusion Supp Introduction Questions: What is the impact of the heterogeneity of treatment?
- 7. Introduction Models Results Conclusion Supp Introduction Questions: What is the impact of the heterogeneity of treatment? on the persistence of drug-sensitive parasites (WT)
- 8. Introduction Models Results Conclusion Supp Introduction Questions: What is the impact of the heterogeneity of treatment? on the persistence of drug-sensitive parasites (WT) on the spread of drug-resistance
- 9. Introduction Models Results Conclusion Supp Introduction Questions: What is the impact of the heterogeneity of treatment? on the persistence of drug-sensitive parasites (WT) on the spread of drug-resistance Which spatial pattern best prevents the propagation of drug resistant parasites?
- 10. Introduction Models Results Conclusion Supp Model: local changes Epidemiology S IWT Drug-sensitive IR Drug-resistant Susceptible individuals Infected individuals
- 11. Introduction Models Results Conclusion Supp Model: local changes Epidemiology S IWT Drug-sensitive IR Drug-resistant Susceptible individuals Infected individuals β transmission
- 12. Introduction Models Results Conclusion Supp Model: local changes Epidemiology S IWT Drug-sensitive IR Drug-resistant Susceptible individuals Infected individuals β transmission γ recovery
- 13. Introduction Models Results Conclusion Supp Model: local changes Epidemiology S IWT Drug-sensitive IR Drug-resistant Susceptible individuals Infected individuals β transmission γ recovery In a well-mixed population: R0 = Nβ γ
- 14. Introduction Models Results Conclusion Supp Model: local changes Epidemiology S IWT Drug-sensitive IR Drug-resistant Susceptible individuals Infected individuals β transmission γ recovery In a well-mixed population: R0 = Nβ γ Heterogeneous environment TREATED UNTREATED x 0 -qT L (1-qT) L L qT proportion of treated areas L environmental grain
- 15. Introduction Models Results Conclusion Supp Model: local changes Epidemiology S IWT Drug-sensitive IR Drug-resistant Susceptible individuals Infected individuals β transmission γ recovery In a well-mixed population: R0 = Nβ γ Heterogeneous environment TREATED UNTREATED x 0 -qT L (1-qT) L L TREATED x UNTREATED UNTREATED L qT proportion of treated areas L environmental grain
- 16. Introduction Models Results Conclusion Supp Treatment and Resistance Effects on parameters UNTREATEDTREATED Treatment's effect x Basic Reproductive Numbers (R0) WT WT R0 WT,U R0 WT,T WT: drug-sensitive R: drug-resistant
- 17. Introduction Models Results Conclusion Supp Treatment and Resistance Effects on parameters Cost of resistance R UNTREATEDTREATED R Treatment's effect x Basic Reproductive Numbers (R0) WT WT R0 WT,U R0 R R0 WT,T WT: drug-sensitive R: drug-resistant
- 18. Introduction Models Results Conclusion Supp The full model Reaction-diffusion ∂IWT ∂t = βWT(x)IWT(N − IWT − IR) − γWT(x)IWT + σ2 2 ∂2IWT ∂x2 ∂IR ∂t = βRIR(N − IWT − IR) − γRIR + σ2 2 ∂2IR ∂x2
- 19. Introduction Models Results Conclusion Supp The full model Reaction-diffusion ∂IWT ∂t = βWT(x)IWT(N − IWT − IR) − γWT(x)IWT + σ2 2 ∂2IWT ∂x2 ∂IR ∂t = βRIR(N − IWT − IR) − γRIR + σ2 2 ∂2IR ∂x2
- 20. Introduction Models Results Conclusion Supp The full model Reaction-diffusion ∂IWT ∂t = βWT(x)IWT(N − IWT − IR) − γWT(x)IWT + σ2 2 ∂2IWT ∂x2 ∂IR ∂t = βRIR(N − IWT − IR) − γRIR + σ2 2 ∂2IR ∂x2
- 21. Introduction Models Results Conclusion Supp The full model Reaction-diffusion ∂IWT ∂t = βWT(x)IWT(N − IWT − IR) − γWT(x)IWT + σ2 2 ∂2IWT ∂x2 ∂IR ∂t = βRIR(N − IWT − IR) − γRIR + σ2 2 ∂2IR ∂x2
- 22. Introduction Models Results Conclusion Supp The full model Reaction-diffusion ∂IWT ∂t = βWT(x)IWT(N − IWT ) − γWT(x)IWT + σ2 2 ∂2IWT ∂x2
- 23. Introduction Models Results Conclusion Supp Eradication of the drug-sensitive strain (WT) Sasaki (2004)
- 24. Introduction Models Results Conclusion Supp Eradication of the drug-sensitive strain (WT) Sasaki (2004) Trivial when RWT,T 0 > 1 (always persists) UNTREATEDTREATED x Basic Reproductive Numbers (R0) WT WT R0 WT,U R0 WT,T 1
- 25. Introduction Models Results Conclusion Supp Eradication of the drug-sensitive strain (WT) Sasaki (2004) Trivial when RWT,T 0 > 1 (always persists) When RWT,T 0 < 1 . . . UNTREATEDTREATED x Basic Reproductive Numbers (R0) WT WT R0 WT,U R0 WT,T 1
- 26. Introduction Models Results Conclusion Supp Eradication of the drug-sensitive strain (WT) Sasaki (2004) Trivial when RWT,T 0 > 1 (always persists) When RWT,T 0 < 1 . . . UNTREATEDTREATED x Basic Reproductive Numbers (R0) WT WT R0 WT,U R0 WT,T 1 How to? Study the stability of the IWT = 0 equilibrium
- 27. Introduction Models Results Conclusion Supp Eradication of the drug-sensitive strain (WT) Condition for the eradication (1 − qT) L √ 2 σ < 1 γU WT R WT,U 0 − 1 arctan − γT WT R WT,T 0 − 1 γU WT R WT,U 0 − 1 tanh qTL √ 2 σ −γT WT RT 0 − 1
- 28. Introduction Models Results Conclusion Supp Eradication of the drug-sensitive strain (WT) Condition for the eradication (1 − qT) L √ 2 σ < 1 γU WT R WT,U 0 − 1 arctan − γT WT R WT,T 0 − 1 γU WT R WT,U 0 − 1 tanh qTL √ 2 σ −γT WT RT 0 − 1
- 29. Introduction Models Results Conclusion Supp Eradication of the drug-sensitive strain (WT) Condition for the eradication (1 − qT) L √ 2 σ < 1 γU WT R WT,U 0 − 1 arctan − γT WT R WT,T 0 − 1 γU WT R WT,U 0 − 1 tanh qTL √ 2 σ −γT WT RT 0 − 1
- 30. Introduction Models Results Conclusion Supp Eradication of the drug-sensitive strain (WT) Condition for the eradication (1 − qT) L √ 2 σ < 1 γU WT R WT,U 0 − 1 arctan − γT WT R WT,T 0 − 1 γU WT R WT,U 0 − 1 tanh qTL √ 2 σ −γT WT RT 0 − 1
- 31. Introduction Models Results Conclusion Supp Eradication of the drug-sensitive strain (WT) Condition for the eradication (1 − qT) L √ 2 σ < 1 γU WT R WT,U 0 − 1 arctan − γT WT R WT,T 0 − 1 γU WT R WT,U 0 − 1 tanh qTL √ 2 σ −γT WT RT 0 − 1 Proportion of treated area qT Localdensityof infectedindividuals prediction σ = 1
- 32. Introduction Models Results Conclusion Supp Eradication of the drug-sensitive strain (WT) Condition for the eradication (1 − qT) L √ 2 σ < 1 γU WT R WT,U 0 − 1 arctan − γT WT R WT,T 0 − 1 γU WT R WT,U 0 − 1 tanh qTL √ 2 σ −γT WT RT 0 − 1 Proportion of treated area qT Localdensityof infectedindividuals σ = 5
- 33. Introduction Models Results Conclusion Supp Eradication of the drug-sensitive strain (WT) Condition for the eradication (1 − qT) L √ 2 σ < 1 γU WT R WT,U 0 − 1 arctan − γT WT R WT,T 0 − 1 γU WT R WT,U 0 − 1 tanh qTL √ 2 σ −γT WT RT 0 − 1 Conclusion Migration and local selection have antagonistic effects
- 34. Introduction Models Results Conclusion Supp Evolution of drug-resistance Suppose now that there are also drug-resistant parasites. How does the heterogeneity of treatment affect the evolution of drug-resistance?
- 35. Introduction Models Results Conclusion Supp Invasion of the drug-resistant parasites How to Study the stability of the equilibrium with the drug-sensitive strain only
- 36. Introduction Models Results Conclusion Supp Invasion of the drug-resistant parasites How to Study the stability of the equilibrium with the drug-sensitive strain only When RWT,T 0 > 1 R UNTREATEDTREATED R x Basic Reproductive Numbers (R0) WT WT R0 WT,U R0 R R0 WT,T
- 37. Introduction Models Results Conclusion Supp Invasion of the drug-resistant parasites How to Study the stability of the equilibrium with the drug-sensitive strain only When RWT,T 0 > 1 R UNTREATEDTREATED R x Basic Reproductive Numbers (R0) WT WT R0 WT,U R0 R R0 WT,T 1
- 38. Introduction Models Results Conclusion Supp Invasion of the drug-resistant parasites How to Study the stability of the equilibrium with the drug-sensitive strain only Approximation Real solution qT L L IWT/N x
- 39. Introduction Models Results Conclusion Supp Invasion of the drug-resistant parasites How to Study the stability of the equilibrium with the drug-sensitive strain only Approximation Real solution qT L L IWT/N 1 - 1/R0 WT,T 1 - 1/R0 WT,U x
- 40. Introduction Models Results Conclusion Supp Invasion of the drug-resistant parasites How to Study the stability of the equilibrium with the drug-sensitive strain only Approximation Real solution qT L L IWT/N 1 - 1/R0 WT,T 1 - 1/R0 WT,U x Low migration approximation qT L L IWT/N 1 - 1/R0 WT,T 1 - 1/R0 WT,U x
- 41. Introduction Models Results Conclusion Supp Invasion of the drug-resistant parasites How to Study the stability of the equilibrium with the drug-sensitive strain only Approximation Real solution qT L L IWT/N 1 - 1/R0 WT,T 1 - 1/R0 WT,U x Low migration approximation qT L L IWT/N 1 - 1/R0 WT,T 1 - 1/R0 WT,U x τ = 1 − 1/RWT,U 0 1 − 1/RWT,T 0
- 42. Introduction Models Results Conclusion Supp Invasion of the drug-resistant parasites τ = 1 − 1/RWT,U 0 1 − 1/RWT,T 0
- 43. Introduction Models Results Conclusion Supp Invasion of the drug-resistant parasites τ = 1 − 1/RWT,U 0 1 − 1/RWT,T 0 Invasion Condition qTL √ 2 σ > 1 √ γR RR 0 R WT,T 0 − 1 arctan τ 2 − RR 0 R WT,U 0 − 1 RR 0 R WT,T 0 − 1 tanh (1 − qT) L √ 2 σ √ γR − RR 0 R WT,U 0 − 1
- 44. Introduction Models Results Conclusion Supp Invasion of the drug-resistant parasites τ = 1 − 1/RWT,U 0 1 − 1/RWT,T 0 Invasion Condition qTL √ 2 σ > 1 √ γR RR 0 R WT,T 0 − 1 arctan τ 2 − RR 0 R WT,U 0 − 1 RR 0 R WT,T 0 − 1 tanh (1 − qT) L √ 2 σ √ γR − RR 0 R WT,U 0 − 1
- 45. Introduction Models Results Conclusion Supp Invasion of the drug-resistant parasites τ = 1 − 1/RWT,U 0 1 − 1/RWT,T 0 Invasion Condition qTL √ 2 σ > 1 √ γR RR 0 R WT,T 0 − 1 arctan τ 2 − RR 0 R WT,U 0 − 1 RR 0 R WT,T 0 − 1 tanh (1 − qT) L √ 2 σ √ γR − RR 0 R WT,U 0 − 1 Proportion of treated area qT Localdensityofdrug- resistantindividuals prediction
- 46. Introduction Models Results Conclusion Supp Conditions for coexistence Find the criteria for the invasion of drug-sensitive parasites in a population of drug-resistant parasites (exact)
- 47. Introduction Models Results Conclusion Supp Conditions for coexistence Find the criteria for the invasion of drug-sensitive parasites in a population of drug-resistant parasites (exact) Coexistence when reciprocal invasion
- 48. Introduction Models Results Conclusion Supp Conditions for coexistence Find the criteria for the invasion of drug-sensitive parasites in a population of drug-resistant parasites (exact) Coexistence when reciprocal invasion Proportion of treated area qT ScaledsizeoftheenvironmentL√2/σ COEXISTENCE DRUG-RESISTANT ONLY DRUG-SENSITIVE ONLY Invasion of the drug- resistant parasites Invasion of the drug- sensitive parasitesdecreasingmigration
- 49. Introduction Models Results Conclusion Supp Beyond R0 R0 is a ratio . . . Illustration with the drug-resistant parasites.
- 50. Introduction Models Results Conclusion Supp Beyond R0 R0 is a ratio . . . Illustration with the drug-resistant parasites. RR 0 = N βR γR = 1.36
- 51. Introduction Models Results Conclusion Supp Beyond R0 R0 is a ratio . . . Illustration with the drug-resistant parasites. RR 0 = N βR γR = 1.36 Costs on recovery γ S IWT IR Fast dynamics
- 52. Introduction Models Results Conclusion Supp Beyond R0 R0 is a ratio . . . Illustration with the drug-resistant parasites. RR 0 = N βR γR = 1.36 Costs on recovery γ S IWT IR Fast dynamics Costs on transmission β S IWT IR Slow dynamics
- 53. Introduction Models Results Conclusion Supp Beyond R0 R0 is a ratio . . . Illustration with the drug-resistant parasites. RR 0 = N βR γR = 1.36 Costs on recovery γ qTL L x Fast dynamics Costs on transmission β S IWT IR Slow dynamics Total prevalence IWT+IR N Proportion of drug-resistance IR IWT+IR
- 54. Introduction Models Results Conclusion Supp Beyond R0 R0 is a ratio . . . Illustration with the drug-resistant parasites. RR 0 = N βR γR = 1.36 Costs on recovery γ qTL L x Fast dynamics Costs on transmission β qTL L x Slow dynamics Total prevalence IWT+IR N Proportion of drug-resistance IR IWT+IR
- 55. Introduction Models Results Conclusion Supp Beyond R0 RR 0 = N βR γR = 1.36 Costs on recovery γ qTL L x Fast dynamics Costs on transmission β qTL L x Slow dynamics Conclusion The R0s do not tell everything; the repartition of the effects between transmission (β) and recovery (γ) rates also matters !
- 56. Introduction Models Results Conclusion Supp Generalization: two-host system Invasion of drug-resistant parasites With a two-host system (e.g. vector-borne disease): Drug-sensitive Drug-resistant S IWT IR VWT Vs VR HOST 1 HOST 2
- 57. Introduction Models Results Conclusion Supp Generalization: two-host system Invasion of drug-resistant parasites One host qTL √ 2 σ > 1 √ γR RR 0 R WT,T 0 − 1 arctan τ 2 − RR 0 R WT,U 0 − 1 RR 0 R WT,T 0 − 1 tanh (1 − qT) L √ 2 σ √ γR − RR 0 R WT,U 0 − 1
- 58. Introduction Models Results Conclusion Supp Generalization: two-host system Invasion of drug-resistant parasites One host qTL √ 2 σ > 1 √ γR RR 0 R WT,T 0 − 1 arctan τ 2 − RR 0 R WT,U 0 − 1 RR 0 R WT,T 0 − 1 tanh (1 − qT) L √ 2 σ √ γR − RR 0 R WT,U 0 − 1 Two hosts qTL √ 2 σe > 1 ΓR RR 0 R WT,T 0 − 1 arctan τk 2 − RR 0 R WT,U 0 − 1 RR 0 R WT,T 0 − 1 tanh (1 − qT) L √ 2 σe ΓR − RR 0 R WT,U 0 − 1
- 59. Introduction Models Results Conclusion Supp Generalization: two-host system Invasion of drug-resistant parasites One host qTL √ 2 σ > 1 √ γR RR 0 R WT,T 0 − 1 arctan τ 2 − RR 0 R WT,U 0 − 1 RR 0 R WT,T 0 − 1 tanh (1 − qT) L √ 2 σ √ γR − RR 0 R WT,U 0 − 1 Two hosts qTL √ 2 σe > 1 ΓR RR 0 R WT,T 0 − 1 arctan τk 2 − RR 0 R WT,U 0 − 1 RR 0 R WT,T 0 − 1 tanh (1 − qT) L √ 2 σe ΓR − RR 0 R WT,U 0 − 1 σ2 e = σ2 V/νR + σ2 H/γR 1/νR + 1/γR
- 60. Introduction Models Results Conclusion Supp Generalization: two-host system Invasion of drug-resistant parasites One host qTL √ 2 σ > 1 √ γR RR 0 R WT,T 0 − 1 arctan τ 2 − RR 0 R WT,U 0 − 1 RR 0 R WT,T 0 − 1 tanh (1 − qT) L √ 2 σ √ γR − RR 0 R WT,U 0 − 1 Two hosts qTL √ 2 σe > 1 ΓR RR 0 R WT,T 0 − 1 arctan τk 2 − RR 0 R WT,U 0 − 1 RR 0 R WT,T 0 − 1 tanh (1 − qT) L √ 2 σe ΓR − RR 0 R WT,U 0 − 1 σ2 e = σ2 V/νR + σ2 H/γR 1/νR + 1/γR ΓR = 1 1/γR + 1/νR
- 61. Introduction Models Results Conclusion Supp Generalization: two-host system Invasion of drug-resistant parasites One host qTL √ 2 σ > 1 √ γR RR 0 R WT,T 0 − 1 arctan τ 2 − RR 0 R WT,U 0 − 1 RR 0 R WT,T 0 − 1 tanh (1 − qT) L √ 2 σ √ γR − RR 0 R WT,U 0 − 1 Two hosts qTL √ 2 σe > 1 ΓR RR 0 R WT,T 0 − 1 arctan τk 2 − RR 0 R WT,U 0 − 1 RR 0 R WT,T 0 − 1 tanh (1 − qT) L √ 2 σe ΓR − RR 0 R WT,U 0 − 1 σ2 e = σ2 V/νR + σ2 H/γR 1/νR + 1/γR ΓR = 1 1/γR + 1/νR
- 62. Introduction Models Results Conclusion Supp Take Home Message Beyond R0: Costs repartition between transmission and recovery The local R0 are not enough ! What is the R0 at the whole population scale ?
- 63. Introduction Models Results Conclusion Supp Take Home Message Beyond R0: Costs repartition between transmission and recovery The local R0 are not enough ! What is the R0 at the whole population scale ? Critical width of the treatment area Migration selection balance Wiping out the resistant strains → cline models in population genetics
- 64. Introduction Models Results Conclusion Supp Acknowledgements UMR 5175 Centre d’´Ecologie Fonctionnelle et ´Evolutive Montpellier, France UMR 2724 G´en´etique et ´Evolution des maladies infectieuses Montpellier, France Minus van Baalen, Mark Kirkpatrick, Guillaume Martin, Andrew Park, Oph´elie Ronce, Fran¸cois Rousset
- 65. Introduction Models Results Conclusion Supp and thank you for your attention
- 66. Introduction Models Results Conclusion Supp
- 67. Introduction Models Results Conclusion Supp Supplementary Materials R WT = (1 − qT) L √ 2 σ γU WT R WT,U 0 − 1 1 arctan − γT WT R WT,T 0 −1 γU WT R WT,U 0 −1 tanh qTL √ 2 σ −γT WT RT 0 − 1
- 68. Introduction Models Results Conclusion Supp Supplementary Materials R WT = (1 − qT) L √ 2 σ γU WT R WT,U 0 − 1 1 arctan − γT WT R WT,T 0 −1 γU WT R WT,U 0 −1 tanh qTL √ 2 σ −γT WT RT 0 − 1
- 69. Introduction Models Results Conclusion Supp Supplementary Materials R WT = (1 − qT) L √ 2 σ γU WT R WT,U 0 − 1 1 arctan − γT WT R WT,T 0 −1 γU WT R WT,U 0 −1 tanh qTL √ 2 σ −γT WT RT 0 − 1 RWT > 1
- 70. Introduction Models Results Conclusion Supp Two-host model Equivalent migration σe σ2 e = σ2 V/νR + σ2 H/γR 1/νR + 1/γR
- 71. Introduction Models Results Conclusion Supp Two-host model Equivalent migration σe σ2 e = σ2 V/νR + σ2 H/γR 1/νR + 1/γR Initial density (WT) Calculated either in the human (τH) or vector (τV) compartments
- 72. Introduction Models Results Conclusion Supp Two-host model Critical Size With k =H or k =V: qTL √ 2 σe > 1 s arctan τ2 k α tanh α √ s (1 − qT)L √ 2 σe
- 73. Introduction Models Results Conclusion Supp Two-host model Critical Size With k =H or k =V: qTL √ 2 σe > 1 s arctan τ2 k α tanh α √ s (1 − qT)L √ 2 σe s ≈ RR 0 RWT,T 0 − 1 1 1/γR + 1/νR α ≈ − RR 0 RWT,U 0 − 1 RR 0 RWT,T 0 − 1
- 74. Introduction Models Results Conclusion Supp Cost and evolution of resistance γU WT βU WT Recovery rate γR TransmissionβR
- 75. Introduction Models Results Conclusion Supp Cost and evolution of resistance γU WT βU WT Recovery rate γR TransmissionβR
- 76. Introduction Models Results Conclusion Supp Cost and evolution of resistance γU WT βU WT Recovery rate γR TransmissionβR Critical size easierinvasion
- 77. Introduction Models Results Conclusion Supp Cost and evolution of resistance γU WT βU WT Recovery rate γR TransmissionβR Critical size easierinvasion
- 78. Introduction Models Results Conclusion Supp Cost and evolution of resistance γU WT βU WT Recovery rate γR TransmissionβR Critical size easierinvasion