The Behavioral SIR Model, with Applications to the Swine Flu and … · 2020. 6. 30. · The...
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The Behavioral SIR Model,with Applications to the Swine Flu and
COVID-19 Pandemics
Samuel Engle Jussi Keppo Marianna Kudlyak∗
Elena Quercioli Lones Smith Andrea WilsonNY Fed Seminar, 4PM on 6/30/2020
(∗Views expressed are not those of the Federal Reserve Bank of San Francisco or the Federal Reserve System.)
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Plan of Talk: Deriving and Using a New Contagion Model▶ The influential SIR contagion model
1. is linear, and so tractable2. makes extreme predictions, especially later on in a contagion3. ignores human behavior
▶ We create a Behavioral SIR (BSIR) model that1. accounts for optimal avoidance behavior in a Nash equilibrium2. is log-linear, and so still tractable for analysis3. makes less extreme predictions (consistent with COVID so far)4. subsumes the SIR model as a special case for low infectiousness
or small disease losses (crucial for statistical tests)▶ For COVID19 and Swine Flu (2009), we reject the SIR model
▶ For COVID19, our BSIR model make sense of time seriesproperties in countries and states, pre- and post-lockdown
▶ Data from the Swine Flu allows us to evaluate the BSIR modelthrough the entire course of the contagion to herd immunity
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
▶ Contagion math in the best of times depends on1. Biology: how infectious is the infection?
2. Sociology: networks, segregation, “Super spreaders”
3. Geography: meeting rates are higher in dense cities
4. Culture: in Italy, the kiss sometimes replaces the handshake
5. Game theory: how we react to payoffs and each other
6. Political economy: Do lockdowns or stay-in-place work? Arepeople responsive?
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
SI / SIS / SIR
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
The SIR Model (1927)▶ The model takes place in continuous time t ∈ [0, ∞)
▶ Population is the continuum [0, 1] (no aggregate randomness)
▶ State transition process of people in the SIR model
▶ Mass σ(t) of individuals are susceptible to a disease
▶ prevalence πt ∈ (0, 1) is the mass of contagious individuals▶ Given: seed mass π0 > 0, with σ0 = 1 − π0
▶ Incidence is the inflow of new infections
▶ The passing rate is the mean number β > 0 of susceptiblepeople per unit time each contagious person infects▶ β increases in disease contagiousness, population density▶ β reflects culture and social networks.
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
The SIR Model▶ A mass ρ is recovered/removed and immune
▶ Anyone infected gets better (or dies) at recovery rate r > 0.
▶ We ignore ρ(t), as it does not impact dynamics: ρ̇(t) = rπ(t)
▶ random and independent meetings ⇒ incidence is βσπ
σ̇(t) = −incidence = −βπ(t)σ(t)π̇(t) = incidence − recoveries = βπ(t)σ(t) − rπ(t)
LemmaThe susceptible mass σ(t) monotonically falls, and prevalence π(t)first rises and then falls.
▶ Proof: π̇(t) = [βσ(t) − r ]π(t)
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Herd Immunity
▶ Herd immunity: Epidemic dies out when enough of thepopulation is immune (high ρ) that its spread stops naturallybecause too few people can transmit it (low σ)
▶ tipping point ⇔ π̇(t) ≤ 0 ⇔ βσ̂π̂ = r π̂.
⇒ basic reproduction number R0 ≡ β/r .
LemmaHerd immunity happens if βσπ ≤ rπ ⇔ σ · R0 ≤ 1.
▶ Published COVID estimates R0 = 2.3 ⇒ ρt >1 − 1/2.3≈0.56
▶ “Newsom projection: 56% of California would be infected in 8weeks without mitigation effort” (2020/03/19)
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Goal: Marry Economics and Epidemiology
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Incentives Matter in Contagions
▶ A disease does not pass the same1. among humans or animals in the SIR model.
2. among chill people as alert▶ Example: Measles outbreaks have much higher infection rate
than measles pandemics.▶ We will focus on optimizing strategic behavior, since it can
change very rapidly in the contagion
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Incentives Matter in Contagions
▶ A huge and longstanding literature in epidemiology (includingsome economists lately!) posits exogenous ways that peoplemodify reduce the passing rate as the contagion worsens.
▶ This is like the adaptive expectations literature of the 1960s.▶ The Lucas Critique: must close the loop with equilibrium
▶ disease prevalence rises ⇒ more vigilant ⇒ realize others aremore vigilant ⇒ relax (strategic substitutes)
▶ equilibrium fully accounts for this (infinite) feedback cycle.
▶ no arbitrary adjustment rule works▶ We build on the model of “Contagious Matching Games”
(2006 Quercioli and Smith), where people best reply to aprevalence, which acts like a price in an “implicit market”
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Passing Games
▶ Counterfeit money vs disease: unwitting sharing of a rival“bad” vs unwitting sharing nonrival “bad”
▶ We build on the model of “Contagious Matching Games”(2006 Quercioli and Smith), where people best reply to aprevalence, which acts like a price in an “implicit market”
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
The Contagious Matching Game
▶ World’s biggest game: Everybody is a player [0, 1]
▶ The highest stake game: life of death (or sickness): loss L
▶ Action: Vigilance v ≥ 0 costs v and reduces the passing rate
▶ Players minimize expected total losses
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Some Motivation for Our Model
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Some Motivation for Our Model
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
How Vigilance Reduces Passing: the Filter function▶ Filter function f (v) ∈ [0, 1] linearly scales down passing rates
⇒ Passage rate is βf (v)f (w) if vigilance v contagious personjust meets vigilance w susceptible people
⇒ diminishing returns: f (0)=1>0= f (∞) & f ′ < 0 < f ′′.
▶ A symmetric function is a simplifying assumption▶ Intensive margin: a mask is equally protective of both parties.▶ Extensive margin: Not meeting also symmetrically protects
both parties — f (v) = fraction of meetings one keeps▶ This multiplicative (log-modular) form is for simplicity.
▶ A vaccination is easy vigilance: one jab ⇒ nearly perfect filter
▶ Posit hyperbolic filter function f (v) = (1 + v)−γ , for γ >0▶ γ = filter elasticity in terms of “total vigilance” V = 1 + v .
▶ 1% more total vigilance leads to γ% infection risk reduction
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Vigilance Optimization
▶ People are first obliviously contagious, and next knowingly so.
▶ π = mass of unaware contagious individuals
▶ A potentially susceptible if infected with chance q(π) = σσ+π .
▶ Potentially susceptible people minimize selfish expected totallosses:
βf (v)E [f (W )]q(π)πL + v▶ f ′ <0< f ′′ ⇒ ∃ a corner solution or a unique interior optimum.
▶ Since everyone makes the same choice, only pure strategysymmetric Nash equilibria exist, with W = v∗ ≥ 0.
▶ flow disease loss as δ(v , v∗, π) = βf (v)f (v∗)q(π)πL
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Individual Optimality in Equilibrium
zero vigilance expected flow disease loss δ(0, 0, π) = βπL
Strategic substitutes: Equilibrium vigilance is v∗ < v̂
δ(v , 0, π)
δ(v , v∗, π)
v + δ = c2v + δ = c1
0 Flow vigilance cost vv∗ v̂
flow
dise
ase
loss
δ
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Two Regime Nash Equilibrium▶ Vigilance vanishes for low prevalence π ≤ π, where
π ≈ [βL(1 − φ)/(2φ)]−1.
where φ ≡ 1/(2γ + 1) does not depend on L, β
TheoremThere is a unique Nash equilibrium for any prevalence π ≥ 0.Equilibrium vigilance v∗(π) vanishes for π ∈ [0, π], and isincreasing for π ≥ π, for some prevalence threshold π > 0 that isfalling in L and β, but rising in φ.
▶ Note: Any dynamic equilibrium of a continuum agent gamerequires static Nash play every period
⇒ The only assumption here is a constant loss L, which holds if▶ people are motivated by current losses, or▶ people are forward-looking but act as if in a steady-state.▶ Dynamics impossibly hard to forecast — even experts disagree
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Behavioral Passing Rate▶ The behavioral passing rate B(π|φ) = βf (v∗)2 is the innate
passing rate β times any two individuals’ equilibrium filter.
Theorem
The behavioral passing rate has two regimes:
B(π|φ) ={
β π ≤ π (chill)q(π)β(π/π)1−φ ≈ β(π/π)1−φ π > π (vigilant)
Given our filter, we have
incidence rate = SI meeting rate × passing chance
⇒ incidence-prevalence elasticity = 1 + passing rate elasticity= 1 + (φ − 1)= φ
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Incidence-Prevalence Elasticityconstant SIR passing rate
BSIR passing rate (less deadly)
BSIR passing rate (more deadly)0 1prevalence
pass
ing
rate
SIR: incidence linear in prevalence(less deadly)
BSIR: incidence shifts to log-linear in prevalenceBSIR: incidence linear in prevalence initially
(more deadly)
1prevalence
incid
ence
0
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Incidence-Prevalence Elasticity
SIR: βπ
BSIR: min(βπ, π1−φ0 βπφ)
BSIR: min(βπ, π1−φ1 βπφ)
Lower loss L0
Higher loss L1
1ππ1 π00 Chill Vigilant
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Prevalence Elasticity of Incidence
▶ Near breakout, almost everyone is susceptible.
Corollary (Breakout Incidence)Equilibrium incidence B(π)π is log-linear in prevalence π ≥π,
log (incidence) = log[B(π)π] = b + φ log π
where the incidence-prevalence elasticity is φ ≡ 1/(2γ + 1) < 1,and the intercept b increases in φ and β, and falls in L.
CorollaryFor the same number of cases, the passing rate rises in population.
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Sweden, Pre- and Post-Mitigation
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UK, Pre- and Post-Lockdown
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France, Pre- and Post-Lockdown
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Germany, Pre-Lockdown
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Austria, Pre- and Post-Lockdown
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Mexico, Pre- and Post-Mitigation
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Brazil, Pre- and Post-Mitigation
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Chile, Pre- and Post-Mitigation
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New Zealand, Pre- and Post-Lockdown
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New York, Pre- and Post-Lockdown
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Upwardly Biased Slope φ and Improving Testing
▶ Falling NY Positive-to-Test Ratio induces an omitted variablebias, that inflates the slope estimate φ
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New Jersey, Pre- and Post-Lockdown
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Michigan, Pre- and Post-Lockdown
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California, Pre- and Post-Lockdown/Mitigation
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Florida, Pre- and Post-Mitigation (Riots!)
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Texas, Pre- and Post-Mitigation (Riots!)
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
General Behavioral SIR Dynamics Nest the SIR Dynamics
▶ If π0 ≤ π, SIR dynamics apply
▶ If π0 > π, then the vigilant regime starts. At this point:
σ̇(t) = −βq(π)σ(t)π1−φπ(t)φ
π̇(t) = βq(π)σ(t)π1−φπ(t)φ − rπ(t)
Theorem (Prevalence is Hump-Shaped)
In the BSIR, the susceptible share σ(t) monotonically falls, whileprevalence π(t) either starts falling, or rises and then falls.
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Breakout Theory, when σ ≈ 1 and π ≈ 0: Heterogeneity?
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Breakout Theory
▶ SIR model: only immunity chokes off infections, and so bearbreakout, log-linearity prevails
▶ For times t < τ , the SIR dynamics apply:
π̇(t) ≈ βπ(t) − rπ(t) ⇒ π(t) ≈ π0e(β−r)t
▶ For times t < τ , we have a Bernoulli differential equation:
π′(t) = βπ1−φπ(t)φ−rπ(t) ⇒ π(t) = π
(β
r(1 − ke−r(1−φ)t
)) 11−φ
for the constant k = (β/r − 1) (π/π0)r(1−φ)/(β−r).
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
National Breakout Case Plots Over Time▶ In the SIR model, these are log-linear.▶ Assume a fraction α of non-spreading asymptomatics.
TheoremAssume β(1 − α) > r .▶ In the chill regime (SIR model), π is increasing and log-linear.▶ In the vigilant regime, prevalence π(t) is increasing and
logconcave, and is initially convex, eventually concave.Concavity happens sooner the lower is β(1 − α) or ϕ.
If β(1 − α) < r , then π is decreasing, logconcave, and convex.
CorollaryThe sum of all past cases Υ is logconcave in time. It is convexwhen π is increasing, and concave when π is decreasing.
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Mitigation or Lockdowns▶ Think of mitigation or lockdown as a fall in the passing rate β.▶ Here is a plot of π(t) after β falls from 0.7 to 0.4.
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Herd Immunity
▶ Herd immunity tipping point:
B(t)σ̌φπ̌φφ = r π̌φ ⇔ σ̌φ = (r/B(t))π̌1−φ
φ > r/β
TheoremAs the prevalence elasticity φ ≤ 1 falls, (i) the herd immunity timeτφ advances, (ii) the peak prevalence πφ falls, (iii) the herdimmunity infection share 1 − σφ falls, and (iv) its ratio to theeventual infection share (1 − σφ)/(1 − σφ(∞)) rises.
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
The Road Ahead: SIR versus BSIR▶ SIR Model: immunity chokes off contagions
▶ BSIR Model: immunity and vigilance choke off contagions
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Herd Immunity — Behavioral SIR “Flattens the Curve”
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Herd Immunity Cases ≪ Eventual Total Cases
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Eventual Infections
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Herd Immunity Infections as a Share of Eventual Infections
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Swine Flu Herd Immunity
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Swine Flu Herd Immunity▶ Herd immunity on 10/31, 2009, with about 20% Immunity
▶ Lesson: about half of the sicknesses postdate herd immunity
▶ Lesson: the vaccine arrival in October was critical
▶ Lesson: seasonal component leads to “waves”
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The Behavioral SIR Model COVID Regressions BSIR Dynamics Herd Immunity Swine Flu, 2009
Swine Flu Herd Immunity
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