An Analytical Model of Behavior and Policy in an Epidemic

Layer 1: Overview

This paper builds a tractable, fully analytical version of the workhorse macro-epidemiology (“econ-epi”) model and uses it to characterize how susceptible individuals behave during a deadly epidemic, how a social planner would have them behave, and the externality that separates the two. The motivation is that prior macro-SIR results came almost entirely from numerical simulation; a closed-form treatment can expose general insights those simulations missed and provide a transparent benchmark for any future epidemic. The model appends the standard Kermack-McKendrick SIR system (susceptible S, infected I, recovered R, deceased D, with transmission rate β, recovery rate γr, death rate γd, and γ := γr + γd) with forward-looking agents who choose an activity level λ ∈ [0,1] that scales transmission via β = βa·λ + βo. The single key modeling departure is LINEAR (rather than convex) costs of mitigation, microfounded by indivisible activity choices in the spirit of Rogerson (1988); this makes the optimal control bang-bang or singular and yields closed-form solutions. Three constants organize the analysis: the herd immunity threshold S̄ := γ/β, the basic reproduction number R0 := 1/S̄, and the infection fatality rate IFR := γd/γ. A central composite statistic is the cost-benefit ratio of mitigation κ := (uW − uL)/(βa·IFR·VSL), where VSL := uW/ρ is the value of statistical life in utility terms.\n\nMain results. (1) Decentralized equilibrium (Proposition 1): there is no mitigation at the very start and the very end of the epidemic; mitigation occurs only over an interval [t0, t1). Susceptibles begin mitigating just below full susceptibility, the infection rate peaks exactly at t0 (when precautions are greatest), and from then on the effective reproduction number sits slightly below one, producing a gently declining infection path — a pattern the author notes is broadly consistent with first-wave Covid-19 data. The equilibrium infection trajectory is approximated by the simple ray I(t) ≈ (S(t)/S̄)·κ, and the equilibrium steady-state susceptibility is S∞ ≈ S̄ − S̄·√(2κR0). A higher κ and lower S̄ both reduce mitigation and raise infections (a “fatalism effect”). (2) Socially optimal behavior (Propositions 2-3): optimal policy is bang-bang (λ* ∈ {0,1}) — no mitigation at start and end, full mitigation in a single intermediate interval. The planner “holds fire,” lets infections climb high, then imposes maximal restrictions late, driving the system quickly to herd immunity. The optimal long-run susceptibility is S∞* ≈ S̄ − S̄·2κR0/(κR0 − 1)². (3) The externality: contrary to the conventional view, susceptibles’ privately optimal behavior is EXCESSIVELY cautious — the equilibrium infection rate lies below the optimal infection rate for any S above herd immunity — yet cumulative deaths are HIGHER in equilibrium than under the planner. Mitigation by susceptibles mostly substitutes infection risk intertemporally (“flattening the curve also makes it fatter”); beyond eliminating epidemic overshoot it cannot prevent the inevitable share 1 − S̄ from being infected. The planner’s late-strong-short lockdown comes close to implementing a lottery that randomly selects who gets sick.\n\nImplications. Because the externality runs in the opposite direction to standard intuition, optimal policy can call for the government to INCREASE interaction (the paper cites the UK’s 2020 “Eat Out To Help Out” subsidy as an analogue). Results are framed as technical/foundational insights, not direct prescriptions: the benchmark abstracts from reinfection, variants, vaccines/cures, healthcare capacity limits, and endogenous IFR, all of which can shift specific recommendations while leaving the underlying forces intact.

Layer 2: Deep Dive

What is the ‘identification’ or solution strategy, and what makes the analytical characterization possible?

This is a theory paper, so the relevant strategy is solving the dynamic optimization analytically rather than empirically. The enabling assumption is LINEAR costs of mitigation (instantaneous utility u = λ·uW + (1−λ)·uL), microfounded by indivisible activity choices as in Rogerson (1988), where λ is the probability of being active in a mixed-strategy equilibrium. Linearity makes the current-value Hamiltonian linear in the control λ, so the optimal control is bang-bang or singular with switching function ψ(t) := uW − uL − (ηs(t) − ηi)·βa·I(t). This permits closed-form characterization of switching points and trajectories. The main ’threat’ the author addresses is generality: does linearity drive the conclusions? Section VI shows numerically that convex costs (U = uL + λ^(1−α)·(uW − uL), with α the convexity degree) merely smooth out the kinks and corners without changing qualitative features — passing what the author calls the ‘Solow test.’

What is the core economic mechanism behind ’excessive caution,’ and the two ways the paper frames the externality?

In equilibrium, the singular-control optimality condition equates a constant marginal cost of mitigation (uW − uL) to a marginal benefit (ηs(t) − ηi)·βa·I(t). The shadow value of being susceptible ηs(t) rises over time (cumulative future infection risk and cumulative future mitigation effort both decline as the epidemic progresses), while ηi is constant. To keep the equation balanced, βa·I(t) must fall, so agents become more cautious over time. First framing of the externality: the planner recognizes that at least 1 − S̄ of the population must eventually be infected (and a share IFR of those die); individuals recognize this too (perfect foresight) but each wants to avoid being in the infected group, so they over-mitigate, merely delaying rather than preventing infections. Second framing: stronger mitigation today lowers near-term infections but raises later infections — ‘flattening the curve also makes it fatter’ — so beyond removing overshoot, mitigation only substitutes infection risk intertemporally. The planner internalizes the whole time path; individuals take the aggregate infection rate as given.

Why is the optimal lockdown ’late, strong, and short’ rather than gradual?

From the planner’s law of motion, the velocity Ṡ/S is proportional to I. An interior λ would lower instantaneous costs proportionately but increase the duration of mitigation more than proportionately (since both λ and I are lower), so gradualism is dominated. This makes optimal policy bang-bang with a single interval of maximal restriction. The planner therefore holds fire, lets I climb high (where the system moves fast), then imposes λ=0 to drive the trajectory quickly to herd immunity — minimizing cumulative deaths at minimum cost rather than flattening the curve.

How do equilibrium and optimal cumulative deaths compare, and why does the more cautious equilibrium produce MORE deaths?

Cumulative deaths equal IFR·(1 − S∞). The equilibrium steady-state susceptibility S∞ ≈ S̄ − S̄·√(2κR0) lies below the planner’s S∞* ≈ S̄ − S̄·2κR0/(κR0 − 1)², meaning the equilibrium overshoots herd immunity by more, so 1 − S∞ (cumulative infections) and hence deaths are higher in equilibrium. The equilibrium’s caution lowers the infection rate at each S above herd immunity and stretches the epidemic out (raising economic cost), but does not prevent the inevitable infections and in fact allows more overshoot than the planner’s quick-to-herd-immunity strategy. Cumulative death toll is increasing in R0 and in κ.

What is the role of the cost-benefit ratio κ and the ‘fatalism effect’?

κ := (uW − uL)/(βa·IFR·VSL) combines preferences, epidemiology, and policy effectiveness: the numerator is the utility cost of mitigation; the denominator is the benefit (lower activity reduces transmission by βa, preventing deaths by IFR, each life worth VSL = uW/ρ). A higher κ lowers mitigation and raises the equilibrium infection rate, starts mitigation later (lower S(t0)), and raises cumulative deaths. The ‘fatalism effect’ has two parts: a lower S̄ (greater lifetime chance of falling ill) dissuades mitigation today; and the high expected cumulative future mitigation effort at the epidemic’s start lowers the value of staying alive, further tempering precaution. The simple approximation I(t) ≈ (S(t)/S̄)·κ captures the first part but omits the second.

What is the practical ‘back-of-the-envelope’ contribution?

The paper provides a recipe to trace the equilibrium epidemic path without solving the full dynamic model: (1) compute the thresholds S(t0) ≈ 1 − κ/(√(2κR0)·(1−S̄))·S̄(1−S̄), S(t1) ≈ S̄ − ρ/(βo + βa), and S∞ ≈ S̄ − S̄·√(2κR0); (2) plot the ray I = (S/S̄)·κ between the thresholds; (3) splice it on both sides with the no-mitigation (λ=1) trajectory I = −S + S̄·log S + C0. This rivals running the naive SIR model in simplicity but is grounded in optimizing behavior, giving a more plausible benchmark for human populations. The author intends it for forecasting any future epidemic.

How do the results relate to and differ from prior numerical econ-epi work?

The equilibrium characterization is qualitatively consistent with Farboodi et al. (2021) — little mitigation at the start, then a jump keeping the effective reproduction number just below 1 — the only difference being their path is smoother due to convex costs. Eichenbaum-Rebelo-Trabandt (2021) get a qualitatively different, still hump-shaped equilibrium infection path because in their calibration mitigation is too weak to push the effective reproduction number below 1 (so βo is not ‘sufficiently low’). For the planner, the paper’s late-strong-short lockdown differs from work finding early/strong responses (Farboodi et al.) or intermediate restrictions (Alvarez et al. 2021; Eichenbaum et al. 2021), for two reasons: (1) this model rules out suppression/vaccine arrival as a feasible endgame, whereas papers allowing vaccine arrival find early strong suppression optimal; (2) the planner here controls only susceptibles’ behavior with linear costs, whereas broader instruments and convex costs make intermediate restrictions more attractive. The paper is, to the author’s knowledge, the first to derive equilibrium and optimal behavior fully analytically and to show the susceptibles’ externality makes the infection rate too LOW socially.

What do the costate (shadow-value) dynamics reveal?

The private value of infection ηi = (uI + (γr/ρ)·uW)/(ρ+γ) is time-invariant (payoffs while ill/recovered/dead don’t depend on timing). The social value of an infected person ηi is time-varying because the planner internalizes onward transmission via a (ηi − ηs)(βaλ + βo)S* term. ηi is deeply negative at the epidemic’s start (diverging as I→0, because an infinitesimal seed inflicts unboundedly large relative damage), rises sharply and roughly tracks the private value during the bulk of the epidemic (e.g. when S ∈ [0.5, 0.9]), and settles just above zero in the long run. In the long run the social value of an additional infected person can even be negative when γd is high, because the value of that person’s life is below the welfare loss from infections they spread. The social value of a susceptible ηs is always below the private value (except converging to uW/ρ in the long run), reflecting unpriced future contagion.

What robustness/extension checks does the paper run?

Section VI: (1) Convex costs (numerical, α=0.3) smooth kinks but preserve qualitative features. (2) Broader planner instruments — controlling susceptibles AND infected (without distinguishing them), or restricting everyone identically — are ‘double-edged’: more costly (especially late when many are recovered) but more effective because they also restrict the infected; effectiveness gains peak at intermediate restrictions (around λ=1/2) due to the quadratic contact function, which makes intermediate restrictions and earlier/longer lockdowns more attractive, moving results toward Alvarez et al. (2021). Section VII discusses healthcare/ICU capacity constraints (optimal to hold infections at the capacity level until near herd immunity; endogenous IFR brings equilibrium and optimal paths closer but doesn’t change the externality’s nature), feasible suppression (optimal policy becomes a discrete choice between herd-immunity and best suppression strategy; equilibrium behavior is largely insensitive to suppression feasibility), and temporary immunity/endemicity (strengthens the fatalism effect, raising equilibrium infections; optimal policy still rushes to steady state, now also to avoid costly multiple waves).

What is the calibration used for the figures, and is it meant to be quantitatively serious?

The calibration resembles Covid-19 but is explicitly illustrative, not a serious quantitative calibration. A model period is a week. Epidemiological parameters: βo = 0.7, βa = 1.24, γr = 0.77, γd = 0.0078, implying R0 = 2.5, S̄ = 0.4, IFR = 1%, and average disease duration of 9 days; under full mitigation (λ=0) R0 falls to 0.9. Annual discount rate is 4% (weekly ρ = 0.96^(−1/52) − 1). Utility is logarithmic; weekly consumption is $60,000/52 ≈ $1,250 so uW = log(1250) ≈ 7; full lockdown cuts consumption 20%, giving uL = 6.6, (uW − uL)/uL = 3.2%. With VSL = $10 million, κ = 0.002 (0.2%).

What are the key caveats and the scope of the policy implications?

The author stresses the model is a stripped-down BENCHMARK: no reinfection, no variants, constant IFR, no cure or vaccine (so herd immunity pins down minimum feasible deaths). Specific results are ’technical contributions, not direct normative prescriptions.’ The striking implication that a planner might subsidize interaction (forcing susceptibles to interact, since optimal activity sometimes exceeds equilibrium activity) faces an implementability problem — restricting activity is easier than increasing it. The herd-immunity-quick strategy ceases to be optimal once suppression is feasible (vaccine/cure expected), ICU constraints bind with endogenous IFR, or immunity is only temporary; but the underlying forces (the susceptibles’ intertemporal infection-substitution externality) continue to operate in all these richer settings.

Key Concepts

Herd immunity threshold (S̄): S̄ := γ/β, the level of susceptibility below which the infected pool shrinks; in this model, because there is no cure or vaccine, it pins down the minimum feasible deaths and is the endgame both equilibrium and planner converge toward.

Cost-benefit ratio of mitigation (κ): κ := (uW − uL)/(βa·IFR·VSL), a composite statistic combining preferences, epidemiology, and policy effectiveness; the numerator is the utility cost of mitigation and the denominator the benefit (transmission reduction βa times deaths averted IFR times value of statistical life). Higher κ means less mitigation and more infections.

Excessive caution / susceptibles’ externality: The paper’s central finding that privately optimal mitigation by susceptibles is too cautious socially — the equilibrium infection rate lies below the optimal rate for any S above herd immunity — because each individual wants to avoid being in the inevitable infected share, merely substituting infection risk intertemporally rather than preventing it; the conventional one-way infected-spreader externality view is therefore incomplete.

Linear costs of mitigation / singular control: The assumption (microfounded by indivisible activity choices à la Rogerson 1988) that utility is linear in activity λ, making the Hamiltonian linear in the control so the optimum is bang-bang or singular; this delivers sharp closed-form solutions whose intuitions survive under convex costs (the ‘Solow test’).

Late-strong-short lockdown: The socially optimal policy in this benchmark: hold fire while infections climb high, then impose maximal restrictions (λ=0) in a single intermediate interval that quickly drives the system to herd immunity — minimizing cumulative deaths at minimum cost rather than flattening the curve.

Costates (ηs, ηi): Shadow values of being in the susceptible and infected states. ηi (private) is constant since the payoffs of being ill are timing-independent; the planner’s η*i is time-varying because it internalizes onward transmission and can even be negative in the long run when the death rate is high.