The dynamics of infectious disease outbreaks Jonathan Dushoff - - PowerPoint PPT Presentation

The dynamics of infectious disease outbreaks

Jonathan Dushoff McMaster University Department of Biology Seminar Series Feb 2020

Novel Coronavirus: What do we need to know?

◮ How deadly is the disease? ◮ Can spread be stopped?

◮ What resources will be needed?

◮ How much time do we have to prepare? ◮ Can virus evolution be affected?

How can modelers help?

◮ Analysis of quantitative information ◮ Propagating uncertainty ◮ Linking local and global phenomena

Outline

How deadly? Dynamical modeling Speed and strength Epidemic Epidemic strength Linking Propagating error in novel coronavirus A false dichotomy Measuring generation intervals

200 400 600 800 1000 1200 1400 1600 1800 2000 1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000 Deaths per 100,000 per year US Annual Mortality Rate All causes Infectious Disease

Dushoff, from Armstrong et al.

Case fatality proportion

◮ Worst-case scenario; most of us get the infection ◮ Fatalities per case

◮ We know what a fatality is, but what is a case?

◮ Denominators!

◮ People with (detected) severe disease ◮ people with (detected) recognizable disease ◮ people who develop antibodies

Case-fatality proportion

◮ Currently estimated at 2–4% ◮ Denominators not reported clearly ◮ As time goes on (and we focus on general public) this number should go down

1918 Age distribution

Gagnon et al. 10.1371/journal.pone.0069586

Influenza Age distribution

Ma et al. 10.1016/j.jtbi.2011.08.003

What do we know?

Huang et al. 10.1016/S0140-6736(20)30183-5

What do we know?

◮ 80% of reported deaths age > 60 ◮ Life expectancy, harvesting and attributable risk

◮ The older the profile, the smaller the overall impact

Will everyone get nCoV?

◮ Why did everyone get the flu?

◮ Fast generations ◮ Pre-symptomatic and sub-clinical transmission ◮ Effective antigenic evolution

◮ Can we control nCoV? ◮ How will nCoV evolve

Outline

How deadly? Dynamical modeling Speed and strength Epidemic Epidemic strength Linking Propagating error in novel coronavirus A false dichotomy Measuring generation intervals

Dynamical modeling connects scales

1950 1955 1960 1965 10000 30000

Measles reports from England and Wales

date cases

◮ Start with rules about how things change in short time steps

◮ Usually based on individuals

◮ Calculate results over longer time periods

◮ Usually about populations

Compartmental models

Divide people into categories:

S I R

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

Differential equation 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

Lessons

◮ Exponential invasion potential ◮ Tendency to oscillate ◮ Thresholds

Coronavirus forecasting

• 1e+02

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

date Incidence type

• forecast

reported

Coronavirus forecasting

◮ Counterfactual forecasting ◮ Relationship between forecasts and cases

• 1e+02

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

date Incidence type

• forecast

reported

Outline

How deadly? Dynamical modeling Speed and strength Epidemic Epidemic strength Linking Propagating error in novel coronavirus A false dichotomy Measuring generation intervals

Speed and strength

◮ Current coronavirus modeling is largely focused on inferring R0.

◮ The “basic reproductive number”

◮ Modelers are essentially trying to infer the strength of the epidemic ◮ By observing the speed of the epidemic

◮ And making explicit or implicit assumptions about generation intervals

Outline

How deadly? Dynamical modeling Speed and strength Epidemic Epidemic strength Linking Propagating error in novel coronavirus A false dichotomy Measuring generation intervals

Epidemic

◮ Diseases have a tendency 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?

• ● ●
• ●
• ●
• ● ● ● ● ● ● ●

1990 2000 2010 R0 = 5.66

Year HIV prevalence

0.0 0.1 0.2 0.3

West African Ebola

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

little r

◮ We measure epidemic speed using little r:

◮ Units: [1/time] ◮ Disease increases like ert

◮ Time scale is C = 1/r

◮ Ebola, C ≈ 1month ◮ HIV in SSA, C ≈ 18month

Coronavirus speed

• 5000

10000 15000 20000 Jan 20 Jan 27 Feb 03

date Cumulative Cases

Coronavirus speed

• 100

1000 10000 Jan 20 Jan 27 Feb 03

date Cumulative Cases

Coronavirus speed

• 1000

2000 3000 Jan 20 Jan 27 Feb 03

date New cases

Coronavirus speed

• 10

100 1000 Jan 20 Jan 27 Feb 03

date New cases

Coronavirus speed

500 1000 1500 2000 2500 3000 Days Cases 1 3 5 7 9 11 13 15 17 19 21 23 25

• logistic

nbinom dates: 1−01−01 − 26−01−01 gof = 0.959922 window = 8:26 peak = 25 r = 0.305 (0.252,0.352) Total deaths: 19624 in all 19603 in window

Ma et al., 10.1007/s11538-013-9918-2

Outline

How deadly? Dynamical modeling Speed and strength Epidemic Epidemic strength Linking Propagating error in novel coronavirus A false dichotomy Measuring generation intervals

Epidemic strength

◮ We estimate epidemic strength using R. ◮ R is the number of people who would be infected by an infectious individual in a fully susceptible population. ◮ R = β/γ = βD = (cp)D

◮ c: Contact Rate ◮ p: Probability of transmission (infectivity) ◮ D: Average duration of infection

Big Rx

◮ A disease can invade a population if and only if R > 1. ◮ In a purely “naive” population R is called R0

Homogeneous endemic curve

0.1 0.5 2.0 5.0

endemic equilibrium

R0 Proportion affected 0.0 0.5 1.0

homogeneous

◮ Threshold value ◮ Sharp response to changes in factors underlying transmission ◮ Works – sometimes ◮ Sometimes predicts unrealistic sensitivity

Yellow fever in Panama

0.1 0.5 2.0 5.0

endemic equilibrium

R0 Proportion affected 0.0 0.5 1.0

Outline

How deadly? Dynamical modeling Speed and strength Epidemic Epidemic strength Linking Propagating error in novel coronavirus A false dichotomy Measuring generation intervals

Linking

◮ We’re very interested in the relationship between little r and R. ◮ We might have good estimates of r and want to know more about equilibrium burden or expected outbreak size

◮ e.g., West African Ebola outbreak, HIV in Africa

◮ Or we might have good estimates of R and want to know how fast disease would spread if introduced to a new population

◮ Measles, influenza

◮ Much coronavirus modeling has explicitly or implicitly estimated R from r.

How long is a disease generation? (present)

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

Generations and R

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

• Reproduction number: 1.65

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

• Reproduction number: 1.4

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.

Conditional effect of generation time

◮ Given the reproductive number R

◮ faster generation time G means higher r ◮ More danger

◮ Given r

◮ faster generation time G means smaller R ◮ Less danger

Linking framework

◮ Epidemic speed r is a product:

◮ (something to do with) generation speed ◮ × (something to do with) epidemic strength

◮ Epidemic strength R is therefore (approximately) a quotient

◮ Epidemic speed ◮ ÷ (something to do with) generation speed

Effect of variation in generation time

◮ For a given value of mean generation time, what happens if we have more variation in generation time?

◮ Events that happen earlier than expected compound through time ◮ If R is fixed then r will be higher = ⇒ ◮ If r is fixed then R will be lower

Approximations

Approximate generation intervals

Generation interval (days) Density (1/day) 10 20 30 40 50 0.00 0.02 0.04 0.5 1.0 1.5 2.0 1 2 3 4 5 Exponential growth rate (per generation) Effective reproductive number R

Moment approximation

Approximate generation intervals

Generation interval (days) Density (1/day) 10 20 30 40 50 0.00 0.02 0.04 0.06 0.5 1.0 1.5 2.0 1 2 3 4 5 Exponential growth rate (per generation) Effective reproductive number R

Moment approximation

Approximate generation intervals

Generation interval (days) Density (1/day) 10 20 30 40 50 0.00 0.02 0.04 0.06 0.08 0.5 1.0 1.5 2.0 1 2 3 4 5 Exponential growth rate (per generation) Effective reproductive number R

Moment approximation

Approximate generation intervals

Generation interval (days) Density (1/day) 10 20 30 40 50 0.00 0.04 0.08 0.5 1.0 1.5 2.0 1 2 3 4 5 Exponential growth rate (per generation) Effective reproductive number R

Approximation framework

◮ R ≈ X(r ¯ G; 1/κ)

◮ κ is the dispersion parameter of the generation-interval distribution (measures the effective amount of variation

◮ X is the compound-interest function

◮ R ≈ 1 + r ¯ G when variation is large ◮ R ≈ exp(r ¯ G) when variation is small

◮ Key quantity is r ¯ G: the relative length of the generation interval compared to the characteristic time scale of spread

Intuition

◮ Longer generation times mean less speed

◮ = ⇒ more strength, when speed is fixed

◮ What about more variation in generation times?

◮ More action (both before and after the mean time) ◮ But what happens early is more important in a growing system

◮ More variation means more speed

◮ = ⇒ less strength, when speed is fixed

Test the approximations

◮ Simulate realistic generation intervals for various diseases ◮ Compare approximate rR relationship with known exact relationship

◮ Known because we are testing ourselves with simulated data

Ebola distribution

Lognormal SEIR

Generation interval (days) Density 20 40 60 80 0.00 0.02 0.04 0.06

Single−gamma approximation

Generation interval (days) Density 20 40 60 80 0.00 0.02 0.04 0.06

Ebola curve

0.5 1.0 1.5 2.0 1 2 3 4 5 Exponential growth rate (per generation) Effective reproductive number R

Measles curve

Biologically realistic range (12.5 − 18)

• 10

20 1 2 3 Relative length of generation interval (ρ) Reproduction number empirical approximation theory (moment)

Rabies curve

Ngorongoro Serengeti

• ●

1.0 1.5 2.0 2.5 0.00 0.25 0.50 0.75 1.00 Relative length of generation interval (ρ) Reproduction number empirical approximation theory (moment) approximation theory (MLE)

100 200

Generation interval (days)

Outline

How deadly? Dynamical modeling Speed and strength Epidemic Epidemic strength Linking Propagating error in novel coronavirus A false dichotomy Measuring generation intervals

Assumptions

Assumptions

Assumptions

Propagating error

• 2.0

2.5 3.0 3.5 4.0 4.5 base growth rate GI mean growth rate + GI mean all

Uncertainty type Basic reproductive number

• A. Baseline

Propagating error

• 2.6

3.0 3.4 base growth rate GI mean growth rate + GI mean all

Uncertainty type Basic reproductive number

• B. Reduced uncertainty in the growth rate

Outline

How deadly? Dynamical modeling Speed and strength Epidemic Epidemic strength Linking Propagating error in novel coronavirus A false dichotomy Measuring generation intervals

A false dichotomy

◮ Why are people scrambling to estimate R and mostly ignoring r?

◮ History ◮ Modelers gotta model

The strength paradigm

◮ R > 1 is a threshold ◮ If we can reduce transmission by a constant factor of θ > R, disease can be controlled ◮ In general, we can define θ as a (harmonic) mean of the reduction factor over the course of an infection

◮ weighted by the intrinsic generation interval

◮ Epidemic is controlled if θ > R ◮ More useful in long term (tells us about final size, equilibrium)

The speed paradigm

◮ r > 0 is a threshold ◮ If we can reduce transmission at a constant hazard rate of φ > r, disease can be controlled ◮ In general, we can define φ as a (very weird) mean of the reduction factor over the course of an infection

◮ weighted by the backward generation interval

◮ Epidemic is controlled if φ > r ◮ More useful in short term (tells us about, um, speed)

Epidemic strength (present)

10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Ebola

time (days) Density (1/days)

◮ R, the epidemic strength, is the area under the curve.

Strength of intervention

10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Ebola

time (days) Density (1/days)

◮ . . . by what factor do I need to reduce this curve to eliminate the epidemic?

Different interventions (present)

10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Ebola

time (days) Density (1/days)

◮ idealized vaccination ◮ removes a fixed proportion of people

Different interventions (present)

10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Ebola

time (days) Density (1/days)

◮ idealized quarantine ◮ removes people at a fixed rate

Epidemic speed

10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Ebola

time (days) Density (1/days)

◮ r, the epidemic speed, is the “discount” rate required to balance the tendency to grow

Epidemic speed

10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Ebola

time (days) Density (1/days)

◮ k(τ) = exp(rτ)b(τ), where b(τ) is the initial backward generation interval

Speed of intervention

10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Ebola

time (days) Density (1/days)

◮ . . . how quickly do I need to reduce this curve to eliminate the epidemic?

Different interventions (present)

10 20 30 40 0.00 0.02 0.04 0.06 0.08 0.10 0.12

Ebola

time (days) Density (1/days)

◮ Sometimes it’s easier to estimate strength, sometimes speed

Measuring the intervention

HIV

◮ The importance of transmission speed to HIV control is easier to understand using the speed paradigm

◮ We know the speed of invasion

◮ ≈ 0.7/yr ◮ Characteristic scale ≈ 1.4yr

◮ And can hypothesize the speed of intervention

◮ Or aim to go fast enough

HIV test and treat

0.10 0.20 0.30 0.40 2 4 6 8 Early Proportion Strength epidemic intervention

HIV test and treat

0.10 0.20 0.30 0.40 0.00 0.02 0.04 0.06 0.08 Early Proportion Speed epidemic intervention

Paradigms are complementary

◮ HIV

◮ Information and current intervention are both “speed-like”

◮ Measles

◮ Information (long-term) is strength-like ◮ Intervention (vaccine) also strength-like

◮ Ebola outbreak

◮ Information is speed-like ◮ Pre-emptive vaccination is strength-like ◮ Quarantine or reactive vaccination may be more speed-like

Outline

How deadly? Dynamical modeling Speed and strength Epidemic Epidemic strength Linking Propagating error in novel coronavirus A false dichotomy Measuring generation intervals

Measuring generation intervals

◮ Ad hoc methods ◮ Error often not propagated ◮ Importance of heterogeneity

Generations through time

◮ Generation intervals can be estimated by:

◮ Observing patients:

◮ How long does it take to become infectious? ◮ How long does it take to recover? ◮ What is the time profile of infectiousness/activity?

◮ Contact tracing

◮ Who (probably) infected whom? ◮ When did each become infected? ◮ — or ill (serial interval)?

Which is the real interval?

◮ Contact-tracing intervals look systematically different, depending on when you observe them. ◮ Observed in:

◮ Real data, detailed simulations, simple model

◮ Also differ from intrinsic (infector centered) estimates

Types of interval

◮ Define:

◮ Intrinsic interval: How infectious is a patient at time τ after infection? ◮ Forward interval: When will the people infected today infect

• thers?

◮ Backward interval: When did the people who infected people today themselves become infected? ◮ Censored interval: What do all the intervals observed up until a particular time look like?

◮ Like backward intervals, if it’s early in the epidemic

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 ◮ We are biased towards

• bserving faster events
• 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

Generations in space

◮ How do local interactions affect realized generation intervals?

Surprising results

◮ We tend to think that heterogeneity leads to underestimates

• f R, whican can be dangerous.

◮ R on networks generally smaller than values estimated using r.

◮ Trapman et al., 2016. JRS Interface DOI:10.1098/rsif.2016.0288

Generation-interval perspective

◮ Modelers don’t usually question the intrinsic generation interval ◮ But spatial network structure does change generation intervals:

◮ Local interactions ◮ = ⇒ wasted contacts ◮ = ⇒ shorter generation intervals ◮ = ⇒ smaller estimates of R.

Observed and estimated intervals

Locally corrected GI

• based on degree distribution

and contact rate [3]

• depends on between-individual

variation

Intrinsic GI

• patient-based
• infectiousness profile of an

infected individual

local spatial correction

(discount by survival probability)

Effective GI

• reflects network structure, but

corrects for time censoring

• gives the correct link between r

and R

Observed GI in early epidemic

• contact-tracing based
• censored at observation time

Temporal correction

(weight observed periods by exp(rτ))

homogeneous assumption temporal correction network structure

Outbreak estimation

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

Reproductive number

Serial intervals

Serial intervals

◮ Do serial intervals and generation intervals have the same distribution? ◮ It seems that they should: they describe generations of the same process

◮ But serial intervals can even be very different ◮ Even negative! You might report to the clinic with flu before me, even though I infected you

◮ For rabies, we thought that serial intervals and generation intervals should be the same

◮ Symptoms are closely correlated with infectiousness

Rabies

◮ If symptoms always start before infectiousness happens, then serial interval should equal generation interval:

◮ incubation time + extra latent time + waiting time ◮ extra latent time + waiting time + incubation time

Serial Generation 25 50 75 100 50 100 150 200 50 100 150 200

Days count

Incubation Period: Non−Biter Incubation Period: Biter 25 50 75 100 50 100 150 50 100 150

dateInc count

Thanks

◮ Department ◮ Collaborators ◮ Funders: NSERC, CIHR

Linking framework

◮ Epidemic speed (r) is a product:

◮ (something to do with) generation speed × ◮ (something to do with) epidemic strength

◮ In particular:

◮ r ≈ (1/¯ G) × ℓ(R; κg) ◮ ℓ is the inverse of X