Introduction dynamical system modelling Pierre Nouvellet - PowerPoint PPT Presentation

Introduction dynamical system modelling Pierre Nouvellet pierre.nouvellet@sussex.ac.uk Modelling infectious disease epidemics, analysis and response Short course. Bogota. 11-15th December 2017

Understanding the dynamics of ID Being prepared and responding promptly require understanding the dynamics of the disease Karesh et al. (2012) Lancet

Understanding the dynamics of ID Being prepared and responding promptly require understanding the dynamics of the disease Karesh et al. (2012) Lancet

Objectives • Introduction to concepts in quantitative epidemiology of infectious diseases • Understand the dynamics of epidemics • Understanding key parameters • Modelling control • Application to Ebola

Objectives, details • Exponential growth • Epidemic curve • Flow diagrams – dynamical system • Contact rate • Model SEI • Reproduction number • Models for Ebola

Exponential growth  One dog is infected. 8  He will infect other. 7 6 5  Who will infect more. I 4 3  We obtain a chain reaction: 2 1 0 an epidemic 1 2 3 4 t

Exponential growth t=0, I 0 = 1 8 7 6 5 t=1, I 1 = 2 I 4 3 2 1 t=2, I 2 = 4 0 1 2 3 4 t

Exponential growth t=0, I 0 = 1 8 7 6 5 t=1, I 1 = 2 = I 0 x 2 I 4 3 2 1 t=2, I 2 = 4 = I 1 x 2 0 1 2 3 4 t

Exponential growth t=0, I 0 = 1 8 7 6 5 t=1, I 1 = 2 = I 0 x 2 I 4 3 2 1 t=2, I 2 = 4 = I 1 x 2 0 1 2 3 4 t Exponential growth: I t = I 0 x 2 t = I 0 x e r t

Epidemic curve Healthy Infected Dead or recovered t=0 t=1

Epidemic curve Healthy Infected Dead or recovered t=0 t=1 t=2 t=3 t=4

Epidemic curve Healthy Infected Dead or recovered t=0 t=1 t=2 t=3 t=4

Epidemic curve Healthy Infected Dead or recovered t=0 t=1 t=2 t=3 t=4

Epidemic curve Healthy Infected Dead or recovered t=0 t=1 t=2 t=3 t=4 Less susceptible Exponential phase Epidemic tailing off

Flow diagram Healthy Infected Dead or recovered 𝛾 𝛿 S I 𝛾 : transmission rate Model SI: 𝛾 𝛾 𝑂 S t-1 I t-1 : new infections S t = S t-1 - 𝑂 S t-1 I t-1 𝛿 : recovery or death rate 𝛾 𝑂 S t-1 I t-1 - 𝛿 I t-1 I t = I t-1 + 𝛿 I t-1 : nb of recoveries/deaths

Flow diagram 𝛾 𝛿 S I 𝛾 𝛾 𝑂 S t-1 I t-1 : large 𝑂 S t-1 I t-1 : small  Growing epidemic  Decreasing epidemic

Flow diagram 𝛾 𝛿 S I Model SI, discrete time 𝛾 S t = S t-1 - 𝑂 S t-1 I t-1 𝛾 𝑂 S t-1 I t-1 - 𝛿 I t-1 I t = I t-1 +

Flow diagram 𝛾 𝛿 S I Continuous time Model SI, discrete time 𝑒𝑇 𝑒𝑢 = − 𝛾 𝛾 𝑂 𝑇 𝑢 𝐽 𝑢 S t = S t-1 - 𝑂 S t-1 I t-1 𝛾 𝑒𝐽 𝑒𝑢 = 𝛾 𝑂 S t-1 I t-1 - 𝛿 I t-1 I t = I t-1 + 𝑂 𝑇 𝑢 𝐽 𝑢 − 𝛿𝐽 𝑢

Flow diagram 𝛾 𝛿 S I During 𝑒𝑢 Continuous time 𝑒𝑇 𝑒𝑢 = − 𝛾 𝑒𝑇 : change in susceptibles 𝑂 𝑇 𝑢 𝐽 𝑢 𝑒𝐽 𝑒𝑢 = 𝛾 𝑒𝐽 : change in infectious 𝑂 𝑇 𝑢 𝐽 𝑢 − 𝛿𝐽 𝑢

Flow diagram 𝛾 𝛿 S I Model SI 𝑒𝑇 𝑒𝑢 = − 𝛾 𝑂 𝑇 𝑢 𝐽 𝑢 𝑒𝐽 𝑒𝑢 = 𝛾 𝑂 𝑇 𝑢 𝐽 𝑢 − 𝛿𝐽 𝑢

Characterise contacts 𝛾 𝛿 S I 1. The number of contact is fixed, regardless of density  Frequency dependent contacts 2. The number of contact increase with density  Density dependent contact ‘Frequency dependent’ ‘Density dependent’

Characterise contacts 𝛾 𝛿 S I ‘Frequency dependent’ ‘Density dependent’ Model Model 𝑒𝑇 𝑒𝑢 = − 𝛾 𝑒𝑇 𝑂 𝑇 𝑢 𝐽 𝑢 𝑒𝑢 = −𝛾 𝑇 𝑢 𝐽 𝑢

Characterise contacts Implications for the epidemic curve ‘Frequency dependent’ ‘Density dependent’ 𝑒𝑇 𝑒𝑢 = − 𝛾 𝑒𝑇 𝑂 𝑇 𝑢 𝐽 𝑢 𝑒𝑢 = −𝛾 𝑇 𝑢 𝐽 𝑢

Characterise contacts 𝛾 𝛿 S I ‘ Frequency dependent ’ ‘ Density dependent ’ Ebola Model Model 𝑒𝑇 𝑒𝑇 𝑒𝑢 = − 𝛾 𝑒𝑢 = −𝛾 𝑇 𝑢 𝐽 𝑢 𝑂 𝑇 𝑢 𝐽 𝑢

Reproduction number Definition: Average number of secondary cases generated by an index case in a large entirely susceptible population

Reproduction number Duration of infectiousness: context dependent Time from onset to recovery (or death) Time from onset to isolation (SARS)

Reproduction number Transmission rate: very difficult to estimate

Reproduction number Contact rate: needs clear definition of infectious contact

Reproduction number Deriving R0 from compartmental models

Reproduction number Deriving R0 from compartmental models Model SI Overall transmission rate: 𝛾 𝑒𝑇 𝑒𝑢 = − 𝛾 𝑂 𝑇 𝑢 𝐽 𝑢 𝑂 𝑇 𝑢 𝐽 𝑢 Duration of infectiousness: 𝑒𝐽 𝑒𝑢 = 𝛾 1 𝑂 𝑇 𝑢 𝐽 𝑢 − 𝛿𝐽 𝑢 𝛿

Reproduction number Deriving R0 from compartmental models Model SI With S=N and I=1 Overall transmission rate: 𝑒𝑇 𝑒𝑢 = − 𝛾 𝑂 𝑇 𝑢 𝐽 𝑢 𝛾 So 𝑒𝐽 𝑒𝑢 = 𝛾 𝑆 0 = 𝛾 𝑂 𝑇 𝑢 𝐽 𝑢 − 𝛿𝐽 𝑢 𝛿

Reproduction number Deriving R0 from compartmental models ! If we change the model, we (usually) change the formula for R0! Model SI With S=N and I=1 Overall transmission rate: 𝑒𝑇 𝑒𝑢 = − 𝛾 𝑂 𝑇 𝑢 𝐽 𝑢 𝛾 So 𝑒𝐽 𝑒𝑢 = 𝛾 𝑆 0 = 𝛾 𝑂 𝑇 𝑢 𝐽 𝑢 − 𝛿𝐽 𝑢 𝛿

Ebola model Natural history of the disease: 1. A susceptible person becomes infected ( 𝛾 ) 1 𝜏 ) – or virus incubation period 2. Latency period ( Τ 3. Infectious period ( Τ 1 𝛿 ): symptomatic, associated with large mortality and high viral load 4. Case fatality ratio ( 𝜈 ): proportion of death 1 − 𝜈 R 𝛿 𝛾 𝜏 S E I 𝜈 D

Ebola model 1 − 𝜈 R 𝛿 𝛾 𝜏 E I S 𝜈 D 𝑒𝑇 𝑒𝑢 = −𝛾 𝑇 𝑢 𝐽 𝑢 𝑂 𝑢 𝑒𝐹 𝑒𝑢 = +𝛾 𝑇 𝑢 𝐽 𝑢 − 𝜏𝐹 𝑢 Model for contacts: 𝑂 𝑢 ‘frequency dependent’ 𝑒𝐽 𝑒𝑢 = +𝜏𝐹 𝑢 − 𝛿 𝐽 𝑢 𝑒𝑆 𝑒𝑢 = + 1 − 𝜈 𝛿𝐽 𝑢

Ebola model 1 − 𝜈 R 𝛿 𝛾 𝜏 E I S 𝜈 D 𝑒𝑇 𝑒𝑢 = −𝛾 𝑇 𝑢 𝐽 𝑢 𝑂 𝑢 𝑒𝐹 𝑒𝑢 = +𝛾 𝑇 𝑢 𝐽 𝑢 Reproduction number: − 𝜏𝐹 𝑢 𝑂 𝑢 1 𝛿 𝑒𝐽 𝑆 0 = 𝛾 × ൗ 𝑒𝑢 = +𝜏𝐹 𝑢 − 𝛿 𝐽 𝑢 𝑒𝑆 𝑒𝑢 = + 1 − 𝜈 𝛿𝐽 𝑢

Ebola model Increasing model complexity: • delay onset/death ≠ delay onset/recovery 𝛿 𝑠 1 − 𝜈 I r R 𝛾 𝜏 𝛿 𝑒 E S 𝜈 D I d

Ebola model 𝛿 𝑠 1 − 𝜈 I r R 𝛾 𝜏 𝛿 𝑒 S E 𝜈 D I d 𝑒𝑇 𝑇 𝑢 𝐽 𝑒,𝑢 𝑇 𝑢 𝐽 𝑠,𝑢 𝑒𝑢 = −𝛾 𝑒 − 𝛾 𝑠 𝑂 𝑢 𝑂 𝑢 𝑒𝐹 𝑒𝑢 = + 𝑇 𝑢 𝛾 𝑒 𝐽 𝑒,𝑢 + 𝛾 𝑠 𝐽 𝑠,𝑢 − 𝜏𝐹 𝑢 𝑂 𝑢 𝑒𝐽 𝑠 𝑒𝑢 = + 1 − 𝜈 𝜏𝐹 𝑢 − 𝛿 𝑠 𝐽 𝑒,𝑢 Reproduction number: 𝑒𝐽 𝑒 𝑒𝑢 = +𝜈𝜏𝐹 𝑢 − 𝛿 𝑒 𝐽 𝑒,𝑢 𝑆 0 = 1 − 𝜈 𝛾 𝑠 + 𝜈 𝛾 𝑒 𝑒𝑆 𝛿 𝑠 𝛿 𝑒 𝑒𝑢 = +𝛿 𝑠 𝐽 𝑒,𝑢

Ebola model Increasing model complexity: • delay onset/death ≠ delay onset/recovery • once hospitalised/isolated, no further transmission 𝛿 𝑠 1 − 𝜌 𝑑 𝐽 𝑠 R c 𝛿 𝑒 1 − 𝜈 𝑑 𝐽 𝑒 D c 𝛾 𝐽 𝑠 𝜏 S E 𝛿 ℎ 𝛿 𝑠 𝐽 𝑒 𝜈 ℎ 𝐽 𝑠 R h 𝛿 𝑒 ℎ 𝐽 𝑒 D h 𝜌

Ebola model Stay in community Pre-hospital 𝑑 𝑒𝐽 𝑠 𝑒𝑇 𝑑 𝑒𝑢 = 1 − 𝜌 𝛿 ℎ 𝐽 𝑠,𝑢 − 𝛿 𝑠 𝐽 𝑠,𝑢 𝑒𝑢 = −𝜇 𝑢 𝑇 𝑢 𝑑 𝑒𝐽 𝑒 𝑒𝐹 𝑑 𝑒𝑢 = 1 − 𝜌 𝛿 ℎ 𝐽 𝑒,𝑢 − 𝛿 𝑒 𝐽 𝑒,𝑢 𝑒𝑢 = 𝜇 𝑢 𝑇 𝑢 − 𝜏𝐹 𝑢 𝑒𝑆 𝑑 𝑒𝐽 𝑠 𝑑 𝑒𝑢 = 𝛿 𝑠 𝐽 𝑠,𝑢 𝑒𝑢 = 1 − 𝜈 𝜏𝐹 𝑢 − 𝛿 ℎ 𝐽 𝑠,𝑢 𝑒𝐽 𝑒 𝑒𝑢 = 𝜈𝜏𝐹 𝑢 − 𝛿 ℎ 𝐽 𝑒,𝑢 In hospital ℎ 𝑒𝐽 𝑠 ℎ 𝑒𝑢 = 𝜌𝛿 ℎ 𝐽 𝑠,𝑢 − 𝛿 𝑠 𝐽 𝑠,𝑢 with ℎ 𝑒𝐽 𝑒 𝑑 𝑑 𝐽 𝑒,𝑢 + 𝐽 𝑒,𝑢 𝐽 𝑠,𝑢 + 𝐽 𝑠,𝑢 ℎ 𝑒𝑢 = 𝜌𝛿 ℎ 𝐽 𝑒,𝑢 − 𝛿 𝑒 𝐽 𝑒,𝑢 𝜇 𝑢 = 𝛾 𝑒 + 𝛾 𝑠 𝑂 𝑢 𝑂 𝑢 𝑒𝑆 ℎ ℎ 𝑒𝑢 = 𝛿 𝑠 𝐽 𝑠,𝑢

Ebola model Reproduction number: 1 1 • Someone who will die in community: 𝛾 𝑒 × 𝛿 ℎ + 𝛿 𝑒 1 1 • Someone who will recover in community: 𝛾 𝑠 × 𝛿 ℎ + 𝛿 𝑠 1 • Someone who will die in hospital: 𝛾 𝑒 × 𝛿 ℎ 1 • Someone who will recover in hospital: 𝛾 ℎ × 𝛿 ℎ Weighting to obtain reproduction number: 1 1 1 1 1 1 𝑆 0 = 𝜈 1 − 𝜌 𝛾 𝑒 𝛿 ℎ + 𝛿 𝑒 + 1 − 𝜈 1 − 𝜌 𝛾 𝑠 𝛿 ℎ + 𝛿 𝑠 + 𝜈𝜌𝛾 𝑒 𝛿 ℎ + 1 − 𝜈 𝜌𝛾 𝑠 𝛿 ℎ 1 1 1 1 = 𝜈 𝛾 𝑒 𝛿 ℎ + 1 − 𝜌 𝛿 𝑒 + 1 − 𝜈 𝛾 𝑠 𝛿 ℎ + 1 − 𝜌 𝛿 𝑠