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Introduction dynamical system modelling Pierre Nouvellet - - PowerPoint PPT Presentation

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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

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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

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Understanding the dynamics of ID

Being prepared and responding promptly require understanding the dynamics of the disease

Karesh et al. (2012) Lancet

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Understanding the dynamics of ID

Being prepared and responding promptly require understanding the dynamics of the disease

Karesh et al. (2012) Lancet

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Objectives

  • Introduction to concepts in quantitative

epidemiology of infectious diseases

  • Understand the dynamics of epidemics
  • Understanding key parameters
  • Modelling control
  • Application to Ebola
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Objectives, details

  • Exponential growth
  • Epidemic curve
  • Flow diagrams – dynamical system
  • Contact rate
  • Model SEI
  • Reproduction number
  • Models for Ebola
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  • One dog is infected.
  • He will infect other.
  • Who will infect more.
  • We obtain a chain reaction:

an epidemic

1 2 3 4 5 6 7 8 1 2 3 4 t

I

Exponential growth

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t=0, I0= 1 t=1, I1= 2 t=2, I2= 4

1 2 3 4 5 6 7 8 1 2 3 4 t

I

Exponential growth

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t=0, I0= 1 t=1, I1= 2 = I0 x 2 t=2, I2= 4 = I1 x 2

1 2 3 4 5 6 7 8 1 2 3 4 t

I

Exponential growth

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SLIDE 9

t=0, I0= 1 t=1, I1= 2 = I0 x 2 t=2, I2= 4 = I1 x 2 Exponential growth: It= I0 x 2t = I0 x er t

1 2 3 4 5 6 7 8 1 2 3 4 t

I

Exponential growth

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Epidemic curve

Healthy Infected Dead or recovered t=0 t=1

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t=0 t=1 t=2 t=3 t=4

Epidemic curve

Healthy Infected Dead or recovered

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t=0 t=1 t=2 t=3 t=4

Epidemic curve

Healthy Infected Dead or recovered

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t=0 t=1 t=2 t=3 t=4

Epidemic curve

Healthy Infected Dead or recovered

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t=0 t=1 t=2 t=3 t=4 Exponential phase Epidemic tailing off Less susceptible

Epidemic curve

Healthy Infected Dead or recovered

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S I

𝛾

Model SI: St = St-1 -

𝛾 𝑂 St-1 It-1

It = It-1 +

𝛾 𝑂 St-1 It-1 -𝛿 It-1

𝛾 : transmission rate

𝛾 𝑂St-1 It-1 : new infections

𝛿: recovery or death rate 𝛿It-1: nb of recoveries/deaths

𝛿

Flow diagram

Healthy Infected Dead or recovered

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Flow diagram

S I

𝛾

𝛾 𝑂St-1 It-1 : large

  • Growing epidemic

𝛾 𝑂St-1 It-1 : small

  • Decreasing epidemic

𝛿

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Flow diagram

S I

𝛾

Model SI, discrete time St = St-1 -

𝛾 𝑂 St-1 It-1

It = It-1 +

𝛾 𝑂 St-1 It-1 -𝛿It-1

𝛿

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Flow diagram

S I

𝛾

Model SI, discrete time St = St-1 -

𝛾 𝑂 St-1 It-1

It = It-1 +

𝛾 𝑂 St-1 It-1 -𝛿It-1

Continuous time 𝑒𝑇 𝑒𝑒 = βˆ’ 𝛾 𝑂 𝑇𝑒𝐽𝑒 𝑒𝐽 𝑒𝑒 = 𝛾 𝑂 𝑇𝑒𝐽𝑒 βˆ’ 𝛿𝐽𝑒

𝛿

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Flow diagram

S I

𝛾

During 𝑒𝑒 𝑒𝑇: change in susceptibles 𝑒𝐽: change in infectious Continuous time 𝑒𝑇 𝑒𝑒 = βˆ’ 𝛾 𝑂 𝑇𝑒𝐽𝑒 𝑒𝐽 𝑒𝑒 = 𝛾 𝑂 𝑇𝑒𝐽𝑒 βˆ’ 𝛿𝐽𝑒

𝛿

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Flow diagram

S I

𝛾 𝛿

Model SI 𝑒𝑇 𝑒𝑒 = βˆ’ 𝛾 𝑂 𝑇𝑒𝐽𝑒 𝑒𝐽 𝑒𝑒 = 𝛾 𝑂 𝑇𝑒𝐽𝑒 βˆ’ 𝛿𝐽𝑒

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Characterise contacts

S I

𝛾 𝛿

β€˜Frequency dependent’ β€˜Density dependent’

  • 1. The number of contact is fixed, regardless of density
  • Frequency dependent contacts
  • 2. The number of contact increase with density
  • Density dependent contact
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S I

𝛾 𝛿

β€˜Frequency dependent’ β€˜Density dependent’

Model 𝑒𝑇 𝑒𝑒 = βˆ’ 𝛾 𝑂 𝑇𝑒𝐽𝑒 Model 𝑒𝑇 𝑒𝑒 = βˆ’π›Ύ 𝑇𝑒𝐽𝑒

Characterise contacts

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β€˜Frequency dependent’ β€˜Density dependent’

𝑒𝑇 𝑒𝑒 = βˆ’ 𝛾 𝑂 𝑇𝑒𝐽𝑒 𝑒𝑇 𝑒𝑒 = βˆ’π›Ύ 𝑇𝑒𝐽𝑒

Implications for the epidemic curve

Characterise contacts

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Characterise contacts

S I

𝛾 𝛿

β€˜Frequency dependent’ β€˜Density dependent’

Model 𝑒𝑇 𝑒𝑒 = βˆ’ 𝛾 𝑂 𝑇𝑒𝐽𝑒 Model 𝑒𝑇 𝑒𝑒 = βˆ’π›Ύ 𝑇𝑒𝐽𝑒

Ebola

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Reproduction number

Definition:

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

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Reproduction number

Duration of infectiousness: context dependent

Time from onset to recovery (or death) Time from onset to isolation (SARS)

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Reproduction number

Transmission rate: very difficult to estimate

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Reproduction number

Contact rate: needs clear definition of infectious contact

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Reproduction number

Deriving R0 from compartmental models

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Reproduction number

Deriving R0 from compartmental models

Model SI 𝑒𝑇 𝑒𝑒 = βˆ’ 𝛾 𝑂 𝑇𝑒𝐽𝑒 𝑒𝐽 𝑒𝑒 = 𝛾 𝑂 𝑇𝑒𝐽𝑒 βˆ’ 𝛿𝐽𝑒 Overall transmission rate:

𝛾 𝑂 𝑇𝑒𝐽𝑒

Duration of infectiousness: 1 𝛿

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Reproduction number

Deriving R0 from compartmental models

Model SI 𝑒𝑇 𝑒𝑒 = βˆ’ 𝛾 𝑂 𝑇𝑒𝐽𝑒 𝑒𝐽 𝑒𝑒 = 𝛾 𝑂 𝑇𝑒𝐽𝑒 βˆ’ 𝛿𝐽𝑒 With S=N and I=1 Overall transmission rate: 𝛾 So 𝑆0 = 𝛾 𝛿

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Reproduction number

Deriving R0 from compartmental models ! If we change the model, we (usually) change the formula for R0!

Model SI 𝑒𝑇 𝑒𝑒 = βˆ’ 𝛾 𝑂 𝑇𝑒𝐽𝑒 𝑒𝐽 𝑒𝑒 = 𝛾 𝑂 𝑇𝑒𝐽𝑒 βˆ’ 𝛿𝐽𝑒 With S=N and I=1 Overall transmission rate: 𝛾 So 𝑆0 = 𝛾 𝛿

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Ebola model

Natural history of the disease:

  • 1. A susceptible person becomes infected (𝛾)
  • 2. Latency period ( Ξ€

1 𝜏) – or virus incubation period

  • 3. Infectious period ( Ξ€

1 𝛿): symptomatic, associated with large mortality and high viral load

  • 4. Case fatality ratio (𝜈): proportion of death

S E

𝛾 𝜏

I

1 βˆ’ 𝜈 𝛿 𝜈

R D

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Ebola model

Model for contacts: β€˜frequency dependent’ 𝑒𝑇 𝑒𝑒 = βˆ’π›Ύ 𝑇𝑒𝐽𝑒 𝑂𝑒 𝑒𝐹 𝑒𝑒 = +𝛾 𝑇𝑒𝐽𝑒 𝑂𝑒 βˆ’ πœπΉπ‘’ 𝑒𝐽 𝑒𝑒 = +πœπΉπ‘’ βˆ’ 𝛿 𝐽𝑒 𝑒𝑆 𝑒𝑒 = + 1 βˆ’ 𝜈 𝛿𝐽𝑒

S E

𝛾 𝜏

I

1 βˆ’ 𝜈 𝛿 𝜈

R D

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Ebola model

Reproduction number: 𝑆0 = 𝛾 Γ— ΰ΅— 1 𝛿 𝑒𝑇 𝑒𝑒 = βˆ’π›Ύ 𝑇𝑒𝐽𝑒 𝑂𝑒 𝑒𝐹 𝑒𝑒 = +𝛾 𝑇𝑒𝐽𝑒 𝑂𝑒 βˆ’ πœπΉπ‘’ 𝑒𝐽 𝑒𝑒 = +πœπΉπ‘’ βˆ’ 𝛿 𝐽𝑒 𝑒𝑆 𝑒𝑒 = + 1 βˆ’ 𝜈 𝛿𝐽𝑒

S E

𝛾 𝜏

I

1 βˆ’ 𝜈 𝛿 𝜈

R D

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Ebola model

Increasing model complexity:

  • delay onset/death β‰  delay onset/recovery

𝛿𝑠

R D S E

𝛾 𝜏

Ir Id

1 βˆ’ 𝜈 𝜈 𝛿𝑒

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Ebola model

𝛿𝑠

R D S E

𝛾 𝜏

Ir Id

1 βˆ’ 𝜈 𝜈 𝛿𝑒

𝑒𝑇 𝑒𝑒 = βˆ’π›Ύπ‘’ 𝑇𝑒𝐽𝑒,𝑒 𝑂𝑒 βˆ’ 𝛾𝑠 𝑇𝑒𝐽𝑠,𝑒 𝑂𝑒 𝑒𝐹 𝑒𝑒 = + 𝑇𝑒 𝑂𝑒 𝛾𝑒𝐽𝑒,𝑒 + 𝛾𝑠𝐽𝑠,𝑒 βˆ’ πœπΉπ‘’ 𝑒𝐽𝑠 𝑒𝑒 = + 1 βˆ’ 𝜈 πœπΉπ‘’ βˆ’ 𝛿𝑠𝐽𝑒,𝑒 𝑒𝐽𝑒 𝑒𝑒 = +πœˆπœπΉπ‘’ βˆ’ 𝛿𝑒𝐽𝑒,𝑒 𝑒𝑆 𝑒𝑒 = +𝛿𝑠𝐽𝑒,𝑒 Reproduction number: 𝑆0 = 1 βˆ’ 𝜈 𝛾𝑠 𝛿𝑠 + 𝜈 𝛾𝑒 𝛿𝑒

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Ebola model

Increasing model complexity:

  • delay onset/death β‰  delay onset/recovery
  • nce hospitalised/isolated, no further transmission

S E

𝛾 𝜏 1 βˆ’ 𝜌 𝜌 𝐽𝑠 𝐽𝑒 𝛿𝑠

Rh Dh

𝛿𝑒 𝐽𝑠

β„Ž

𝐽𝑒

β„Ž

𝛿𝑠

Rc Dc

𝛿𝑒 𝐽𝑠

𝑑

𝐽𝑒

𝑑

1 βˆ’ 𝜈 𝜈 π›Ώβ„Ž

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Ebola model

Pre-hospital 𝑒𝑇 𝑒𝑒 = βˆ’πœ‡π‘’ 𝑇𝑒 𝑒𝐹 𝑒𝑒 = πœ‡π‘’ 𝑇𝑒 βˆ’ πœπΉπ‘’ 𝑒𝐽𝑠 𝑒𝑒 = 1 βˆ’ 𝜈 πœπΉπ‘’ βˆ’ π›Ώβ„Žπ½π‘ ,𝑒 𝑒𝐽𝑒 𝑒𝑒 = πœˆπœπΉπ‘’ βˆ’ π›Ώβ„Žπ½π‘’,𝑒 with πœ‡π‘’ = 𝛾𝑒 𝐽𝑒,𝑒 + 𝐽𝑒,𝑒

𝑑

𝑂𝑒 + 𝛾𝑠 𝐽𝑠,𝑒 + 𝐽𝑠,𝑒

𝑑

𝑂𝑒 Stay in community 𝑒𝐽𝑠

𝑑

𝑒𝑒 = 1 βˆ’ 𝜌 π›Ώβ„Žπ½π‘ ,𝑒 βˆ’ 𝛿𝑠𝐽𝑠,𝑒

𝑑

𝑒𝐽𝑒

𝑑

𝑒𝑒 = 1 βˆ’ 𝜌 π›Ώβ„Žπ½π‘’,𝑒 βˆ’ 𝛿𝑒𝐽𝑒,𝑒

𝑑

𝑒𝑆𝑑 𝑒𝑒 = 𝛿𝑠𝐽𝑠,𝑒

𝑑

In hospital 𝑒𝐽𝑠

β„Ž

𝑒𝑒 = πœŒπ›Ώβ„Žπ½π‘ ,𝑒 βˆ’ 𝛿𝑠𝐽𝑠,𝑒

β„Ž

𝑒𝐽𝑒

β„Ž

𝑒𝑒 = πœŒπ›Ώβ„Žπ½π‘’,𝑒 βˆ’ 𝛿𝑒𝐽𝑒,𝑒

β„Ž

π‘’π‘†β„Ž 𝑒𝑒 = 𝛿𝑠𝐽𝑠,𝑒

β„Ž

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Ebola model

Reproduction number:

  • Someone who will die in community: 𝛾𝑒 Γ—

1 π›Ώβ„Ž + 1 𝛿𝑒

  • Someone who will recover in community: 𝛾𝑠 Γ—

1 π›Ώβ„Ž + 1 𝛿𝑠

  • Someone who will die in hospital: 𝛾𝑒 Γ—

1 π›Ώβ„Ž

  • Someone who will recover in hospital: π›Ύβ„Ž Γ—

1 π›Ώβ„Ž

Weighting to obtain reproduction number:

𝑆0 = 𝜈 1 βˆ’ 𝜌 𝛾𝑒

1 π›Ώβ„Ž + 1 𝛿𝑒 + 1 βˆ’ 𝜈

1 βˆ’ 𝜌 𝛾𝑠

1 π›Ώβ„Ž + 1 𝛿𝑠 +πœˆπœŒπ›Ύπ‘’ 1 π›Ώβ„Ž + 1 βˆ’ 𝜈 πœŒπ›Ύπ‘  1 π›Ώβ„Ž

= 𝜈 𝛾𝑒

1 π›Ώβ„Ž + 1 βˆ’ 𝜌 1 𝛿𝑒 + 1 βˆ’ 𝜈 𝛾𝑠 1 π›Ώβ„Ž + 1 βˆ’ 𝜌 1 𝛿𝑠

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Increase complexity

1. Impact of unsafe funeral - vaccination 2. Stochastic Model 3. Spatial Model 4. Individual based Model Warning: β€˜To explain a complex and poorly understood reality with a complex poorly understood model is not progress’ John Maynard Smith

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Ebola model

practical

S E

𝛾 𝛿

I

𝜏 1 βˆ’Pcfr Pcfr

R D Ih

πœβ„Ž

S E

𝛾 𝜏

I

1 βˆ’Pcfr 𝛿 Pcfr

R D