Dynamical modeling of infectious diseases Jonathan Dushoff McMaster - - PowerPoint PPT Presentation

Dynamical modeling of infectious diseases

Jonathan Dushoff McMaster University Global Health Expert Perspectives Webinar May 2020

What is dynamical modeling?

1950 1955 1960 1965 10000 30000

Measles reports from England and Wales

date cases

◮ A way to connect scales ◮ Start with rules about how things change in short time steps

◮ Usually based on individuals

◮ Calculate results over longer time periods

◮ Usually about populations

Example: Post-death transmission and safe burial

◮ How much Ebola spread occurs before vs. after death ◮ Highly context dependent ◮ Funeral practices, disease knowledge ◮ Weitz and Dushoff Scientific Reports 5:8751.

Simple dynamical models use compartments

Divide people into categories:

S I R

◮ Susceptible → Infectious → Recovered ◮ Individuals recover independently ◮ Individuals are infected by infectious people

Deterministic implementation

500 1000 1500 2000 2500 3000 3500 4000 200 400 600 800 1000 Number infected Time (disease generations) Deterministic

Individual-based implementation

500 1000 1500 2000 2500 3000 3500 4000 200 400 600 800 1000 Number infected Time (disease generations) SIR disease, N=100,000 Stochastic Deterministic

Disease tends to grow exponentially at first

◮ I infect three people, they each infect 3 people . . . ◮ How fast does disease grow? ◮ How quickly do we need to respond?

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1990 2000 2010 R0 = 5.66

Year HIV prevalence

0.0 0.1 0.2 0.3

More detailed dynamics

Childs et al., http://covid-measures.stanford.edu/

Exponential growth

• 5000

10000 15000 20000 Jan 13 Jan 20 Jan 27 Feb 03

date Cumulative Cases

Mike Li, https://github.com/wzmli/corona

There are natural thresholds

◮ R is the number of new infections per infection ◮ A disease can invade a population if and only if R > 1. ◮ The value of R in a naive population is called R0

Non-linear response ◮ R = β/γ = βD = (cp)D ◮ c: Contact Rate ◮ p: Probability of transmission (infectivity) ◮ D: Average duration of infection

0.1 0.5 2.0 5.0

endemic equilibrium

R0 Proportion affected 0.0 0.5 1.0

homogeneous

Disease incidence tends to oscillate

500 1000 1500 2000 2500 3000 3500 4000 200 400 600 800 1000 Number infected Time (disease generations) SIR disease, N=100,000 Stochastic Deterministic

What is not dynamical modeling?

https://tinyurl.com/forbes-ihme ◮ Phenomenological modeling uses history and statistics ◮ Does not incorporate mechanistic processes

Coronavirus forecasting

• 1e+02

1e+04 1e+06 Jan 20 Jan 27 Feb 03

date Incidence type

• forecast

reported

Linking

50 100 150 100 200 300 400 R0 = 1.5 I(t) Days, t 100 200 300 400 R0 = 2.0 I(t) 100 200 300 400 R0 = 2.5 I(t) 200 400 600 800 1000 1200 10 10

1

10

2

10

3

10

4

10

5

10

6

Days, t Infected, I(t) R0 = 2.5 R0 = 2.0 R0 = 1.5

Coronavirus speed

• 5000

10000 15000 20000 Jan 13 Jan 20 Jan 27 Feb 03

date Cumulative Cases

How long is a disease generation? (present)

Generation intervals

◮ Sort of the poor relations of disease-modeling world ◮ Ad hoc methods ◮ Error often not propagated

Generation intervals

◮ The generation distribution measures the time between generations of the disease ◮ Interval between “index” infection and resulting infection ◮ Generation intervals provide the link between R and r

Approximate generation intervals

Generation interval (days) Density (1/day) 10 20 30 40 50 0.00 0.02 0.04 0.06 0.08

Generations and R

2 4 6 8 10 20 30 40 50 60 70 Time (weeks) Weekly incidence

• Reproduction number: 1.65

Generations and R

2 4 6 8 10 20 30 40 50 60 70 Time (weeks) Weekly incidence

• Reproduction number: 1.4

Propagating error for coronavirus

2.6 3.0 3.4 none ˆ µr µG µκ all

Uncertainty type Basic reproductive number

• B. Reduced uncertainty in r

Growing epidemics

◮ Generation intervals look shorter at the beginning of an epidemic ◮ A disproportionate number

• f people are infectious right

now ◮ They haven’t finished all of their transmitting

• 1

10 100 2014−01 2014−07 2015−01

cases

Liberia

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1990 2000 2010

Year HIV prevalence

0.0 0.1 0.2 0.3

Backward intervals

Champredon and Dushoff, 2015. DOI:10.1098/rspb.2015.2026

Outbreak estimation

tracing based empirical individual based contact tracing population correction individual correction empirical egocentric intrinsic 2 4 8

Reproductive number

Serial intervals

Flattening the curve

Bolker and Dushoff, https://github.com/bbolker/bbmisc/

Flattening the curve

Bolker and Dushoff, https://github.com/bbolker/bbmisc/

What happens when we open?

200 400 600 Feb 01 Feb 15 Mar 01 Mar 15

Date Reconstructed incidence

• A. Daegu

10 20 30 40 50 Feb 01 Feb 15 Mar 01 Mar 15

Date Reconstructed incidence

• B. Seoul

2 4 6 8 0.00 0.25 0.50 0.75 1.00 1.25 Feb 01 Feb 15 Mar 01 Mar 15

Date Effective reproduction number (Daily traffic, 2020)/(Mean daily traffic, 2017 − 2019)

• C. Daegu

2 4 6 8 0.00 0.25 0.50 0.75 1.00 1.25 Feb 01 Feb 15 Mar 01 Mar 15

Date Effective reproduction number (Daily traffic, 2020)/(Mean daily traffic, 2017 − 2019)

• D. Seoul

Park et al., https://doi.org/10.1101/2020.03.27.20045815

Making use of immunity

Weitz et al., https://www.nature.com/articles/s41591-020-0895-3

Modeling responses

50 100 150 200 100 101 102 103

Weitz et al., https://github.com/jsweitz/covid19-git-plateaus

Modeling responses

China CA Iran WA Italy NY UK GA USA LA

Apr May Mar Apr May Apr May Apr May Apr May Feb Mar Apr May Mar Apr May Mar Apr May Apr May Mar Apr May 1 10 100 1000 1 10 100 1 10 100 1000 1 10 100 1 10 100 1000 10 100 1000 1 10 100 1 3 10 30 1 10 100 1000 1 10 100

Daily number of reported deaths Countries US States

Weitz et al., https://github.com/jsweitz/covid19-git-plateaus

Modeling responses

50 100 150 200 250 300 350 400 25 50 75 100 125

Weitz et al., https://github.com/jsweitz/covid19-git-plateaus

Going forward

◮ Statistical methods for inference and understanding uncertainty ◮ Work with policymakers to evaluate and tune strategies for gradual opening

Thanks

◮ Department ◮ Collaborators

◮ Bolker, Champredon, Earn, Li, Ma, Park, Weitz, many others

◮ Funders: NSERC, CIHR