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? Measles reports from England and Wales 30000 cases 10000 0 1950 1955 1960 1965 date ◮ 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 4000 Deterministic 3500 3000 Number infected 2500 2000 1500 1000 500 0 0 200 400 600 800 1000 Time (disease generations)

Individual-based implementation SIR disease, N=100,000 4000 Stochastic Deterministic 3500 3000 Number infected 2500 2000 1500 1000 500 0 0 200 400 600 800 1000 Time (disease generations)

Disease tends to grow exponentially at first R0 = 5.66 0.3 ● ● ● ● ● ● ● ● ◮ I infect three people, they HIV prevalence ● ● ● ● each infect 3 people . . . ● ● 0.2 ● ● ◮ How fast does disease grow? 0.1 ● ● ● ◮ How quickly do we need to ● ● ● 0.0 respond? 1990 2000 2010 Year

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

Exponential growth 20000 ● ● 15000 ● Cumulative Cases ● 10000 ● ● ● 5000 ● ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● Jan 13 Jan 20 Jan 27 Feb 03 date 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 R 0

endemic equilibrium Non-linear response 1.0 homogeneous ◮ R = β/γ = β D = ( cp ) D Proportion affected ◮ c : Contact Rate ◮ p : Probability of 0.5 transmission (infectivity) 0.0 ◮ D : Average duration of 0.1 0.5 2.0 5.0 infection R 0

Disease incidence tends to oscillate SIR disease, N=100,000 4000 Stochastic Deterministic 3500 3000 Number infected 2500 2000 1500 1000 500 0 0 200 400 600 800 1000 Time (disease generations)

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

Coronavirus forecasting 1e+06 1e+04 ● ● ● Incidence type ● ● ● ● ● ● ● ● forecast ● ● ● ● reported ● ● ● ● ● ● ● ● ● ● ● ● ● 1e+02 ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● Jan 20 Jan 27 Feb 03 date

Linking 6 10 R 0 = 2 . 5 400 R 0 = 2 . 5 R 0 = 2 . 0 R 0 = 1 . 5 300 I ( t ) 5 10 200 100 0 4 Infected, I ( t ) 10 400 R 0 = 2 . 0 300 I ( t ) 3 10 200 100 2 0 10 R 0 = 1 . 5 400 I ( t ) 300 1 10 200 100 0 0 10 0 50 100 150 0 200 400 600 800 1000 1200 Days, t Days, t

Coronavirus speed 20000 ● ● 15000 ● Cumulative Cases ● 10000 ● ● ● 5000 ● ● ● ● ● ● ● ● ● ● 0 ● ● ● ● ● ● ● ● ● ● Jan 13 Jan 20 Jan 27 Feb 03 date

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 Approximate generation intervals ◮ The generation distribution 0.08 measures the time between generations of the disease 0.06 Density (1/day) ◮ Interval between 0.04 “index” infection and 0.02 resulting infection 0.00 ◮ Generation intervals provide 0 10 20 30 40 50 the link between R and r Generation interval (days)

Generations and R 70 Reproduction number: 1.65 60 Weekly incidence ● 50 40 30 20 ● 10 0 2 4 6 8 Time (weeks)

Generations and R 70 Reproduction number: 1.4 60 Weekly incidence ● 50 40 30 20 ● 10 0 2 4 6 8 Time (weeks)

Propagating error for coronavirus B. Reduced uncertainty in r 3.4 Basic reproductive number 3.0 2.6 none ˆ all µ r µ G µ κ Uncertainty type

Growing epidemics Liberia ● ● ● ● ● ● ● ● ● ● ● 100 ● ● ● ● ● ● ● ● ● cases ● ● ● ● ● 10 ◮ Generation intervals look shorter ● ● ● at the beginning of an epidemic 1 ● ● ● ● ● ● ● ● ● ◮ A disproportionate number 2014−01 2014−07 2015−01 of people are infectious right now 0.3 ● ● ● ● ● ● ● ● ◮ They haven’t finished all of HIV prevalence ● ● ● ● their transmitting ● ● 0.2 ● ● ● 0.1 ● ● ● ● ● 0.0 1990 2000 2010 Year

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

Outbreak estimation tracing based empirical individual based Reproductive number 8 4 2 contact population individual empirical egocentric intrinsic tracing correction correction

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? A. Daegu B. Seoul 50 600 40 Reconstructed incidence Reconstructed incidence 30 400 20 200 10 0 0 Feb 01 Feb 15 Mar 01 Mar 15 Feb 01 Feb 15 Mar 01 Mar 15 Date Date C. Daegu D. Seoul (Daily traffic, 2020)/(Mean daily traffic, 2017 − 2019) (Daily traffic, 2020)/(Mean daily traffic, 2017 − 2019) 8 8 1.25 1.25 Effective reproduction number Effective reproduction number 6 1.00 6 1.00 0.75 0.75 4 4 0.50 0.50 2 2 0.25 0.25 0 0.00 0 0.00 Feb 01 Feb 15 Mar 01 Mar 15 Feb 01 Feb 15 Mar 01 Mar 15 Date Date 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 10 3 10 2 10 1 10 0 0 50 100 150 200 Weitz et al., https://github.com/jsweitz/covid19-git-plateaus

Modeling responses China Iran 1000 Italy 1000 UK USA 1000 100 1000 Countries Daily number of reported deaths 100 100 100 100 10 10 10 10 10 1 1 1 1 1 Feb Mar Apr May Mar Apr May Mar Apr May Apr May Mar Apr May 1000 100 CA WA NY GA LA 100 100 30 US States 100 10 10 10 10 3 10 1 1 1 1 Apr May Mar Apr May Apr May Apr May Apr May Weitz et al., https://github.com/jsweitz/covid19-git-plateaus

Modeling responses 125 100 75 50 25 0 0 50 100 150 200 250 300 350 400 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