Nash epidemics

TL;DR

The paper analyzes endogenous social distancing in an epidemic by embedding a Nash equilibrium among individuals choosing personal contact rates within a standard SIR framework. Using a Pontryagin maximum-principle formulation, it derives an analytic solution that links the equilibrium social activity to the current infection level via $k = R_0 - \frac{\alpha s_f}{2} i$ and provides closed-form expressions for the final susceptible fraction and peak incidence. The work reveals scaling laws: in the non-behavioural limit, excess cases and peak height follow classical expressions; in the high-cost regime, they scale as $\\varepsilon \\sim \frac{2 R_0^2}{\alpha}$ and $\\hat i \\sim \\frac{2 R_0(R_0-1)}{\alpha}$, with crossover costs $\\alpha_s^*$ and $\\alpha_i^*$ demarcating behavioural regimes. The results offer intuitive heuristics and policy insights, including a simple message to bootstrap rational behaviour and potential pathways to align Nash outcomes with social optima via incentives, making the analytic framework useful for policymakers and public communication.

Abstract

Faced with a dangerous epidemic humans will spontaneously social distance to reduce their risk of infection at a socio-economic cost. Compartmentalised epidemic models have been extended to include this endogenous decision making: Individuals choose their behaviour to optimise a utility function, self-consistently giving rise to population behaviour. Here we study the properties of the resulting Nash equilibria, in which no member of the population can gain an advantage by unilaterally adopting different behaviour. We leverage a new analytic solution to obtain, (1) a simple relationship between rational social distancing behaviour and the current number of infections; (2) new scaling results for how the infection peak and number of total cases depend on the cost of contracting the disease; (3) characteristic infection costs that divide regimes of strong and weak behavioural response and depend only on the basic reproduction number of the disease; (4) a closed form expression for the value of the utility. We discuss how these analytic results provide a deep and intuitive understanding into the disease dynamics, useful for both individuals and policymakers. In particular the relationship between social distancing and infections represents a heuristic that could be communicated to the population to encourage, or "bootstrap", rational behaviour.

Figures (5)

• Figure 1: Direct plots of the analytic solution. (a) The analytic solution of the Nash equilibrium social distancing problem as obtained in \ref{['eq:analytic_solution']} as a function of the recovered $r$ for an exemplary range of infection costs $\alpha$ and $R_0 = 4$. Initial conditions here and in all following figures are set to $r_0 = 10^{-6}$ and $i_0 = 3\cdot 10^{-6}$. (b) The fraction of infectious $i$ as a function of the susceptible $s$ for the same range of $\alpha$. (c) Deviation of the social distancing behaviour $k$ from the pre-epidemic default $R_0$ as a function of $i$, emphasising their linear relationship as established in \ref{['eq:k_vs_i']}.
• Figure 2: Analytic solution as a function of time. (a) Equilibrium social activity behaviour of the population $k(t)$ and corresponding dynamics of the disease (b) $s$ and (c) $i$ for an exemplary range of infection costs $\alpha$ and $R_0 = 4$. Since infections incur a cost, the equilibrium behaviour seeks to avoid excessive infections by self-organised social distancing. The higher the cost, the more reduced social activity $k$ becomes.
• Figure 3: Scaling. (a) Excess cases $\varepsilon(\alpha,R_0)$ vs. infection cost $\alpha$ for a range of basic reproduction numbers $R_0$. The high infection cost asymptotes, see \ref{['eq:excess_cases_high_alpha']}, are shown as dashed lines and the crossover costs $\alpha^\star_s$, see \ref{['eq:excess_cases_crossover']}, as black stars. Inset: The data collapses onto the low-$\alpha$ and high infection cost asymptotes by rescaling the cost $\alpha$ with the crossover cost $\alpha_s^\star$, see \ref{['eq:excess_cases_crossover']}, while rescaling $\varepsilon(\alpha,R_0)$ with its non-behavioural limit, see \ref{['eq:excess_cases_nonbehavioural']}. (b) The infection peak $\hat{i}$ vs. $\alpha$ for a range of $R_0$. The high infection cost asymptotes, see \ref{['eq:peak_high_alpha']}, are shown as dashed lines and the crossover costs $\alpha^\star_i$, see \ref{['eq:peaks_crossover']}, as grey stars. Inset: The data collapses onto the low-$\alpha$ and high infection cost asymptotes by rescaling the cost $\alpha$ with the crossover cost $\alpha^\star_i$, \ref{['eq:peaks_crossover']}, while rescaling the peak height with its non-behavioural limit, see \ref{['eq:peak_nonbehavioural']}.
• Figure 4: Behavioural response. Characterisation of the Nash equilibrium response in the $R_0$ -- $\alpha$ parameter space. On the high $R_0$ -- low-$\alpha$ side of the line, the behaviour is well represented by the non-behavioural limit, in which it is not rational to significantly modify one's behaviour. On the low $R_0$ -- high infection cost side, it is rational to strongly modify one's behaviour. The lines describing the crossover are given by the critical costs $\alpha_s^\star$ for the transition in the excess cases, see \ref{['eq:excess_cases_crossover']}, and/or $\alpha_i^\star$ for the transition in the infection peak, see \ref{['eq:peaks_crossover']}. The parameter values used for some of the curves in \ref{['fig:Fig1', 'fig:Fig2']} are marked by analogously coloured dots.
• Figure 5: Cost of the epidemic. Total epidemic cost relative to the cost of an infection, $-U/\alpha$, as a function of infection cost $\alpha$ under equilibrium social distancing. The corresponding non-behavioural, \ref{['eq:utility_lowAlpha']}, and high-infection-cost asymptotes, \ref{['eq:utility_highAlpha']} are indicated by dotted and dashed lines, respectively.