Rational social distancing policy during epidemics with limited healthcare capacity

TL;DR

It is shown how costly interventions, such as taxes or subsidies on behaviour, can be used to exactly align individuals’ decision making with government preferences even when these are not aligned.

Abstract

Epidemics of infectious diseases posing a serious risk to human health have occurred throughout history. During recent epidemics there has been much debate about policy, including how and when to impose restrictions on behaviour. Policymakers must balance a complex spectrum of objectives, suggesting a need for quantitative tools. Whether health services might be `overwhelmed' has emerged as a key consideration. Here we show how costly interventions, such as taxes or subsidies on behaviour, can be used to exactly align individuals' decision making with government preferences even when these are not aligned. In order to achieve this, we develop a nested optimisation algorithm of both the government intervention strategy and the resulting equilibrium behaviour of individuals. We focus on a situation in which the capacity of the healthcare system to treat patients is limited and identify conditions under which the disease dynamics respect the capacity limit. We find an extremely sharp drop in peak infections at a critical maximum infection cost in the government's objective function. This is in marked contrast to the gradual reduction of infections if individuals make decisions without government intervention. We find optimal interventions vary less strongly in time when interventions are costly to the government and that the critical cost of the policy switch depends on how costly interventions are.

Figures (5)

• Figure 1: Causal hierarchy of the model. Epidemic dynamics are modelled using a simple Susceptible-Infected-Recovered compartmental model. This informs all decision making (black arrows). The progress of the disease depends only on the behaviour of individuals, who adopt a behaviour consistent with an infectiousness $k(t)$ at time $t$ (gold arrow). Individuals may receive government incentives $\varepsilon(t)$ (brown arrow) to modify their behaviour. They then adopt a rational strategy $k(t)$, corresponding to a Nash equilibrium, based on some utility functional. The government maximises its own value functional and intervenes with incentives for individuals to realise this. This intervention process will, in general, itself carry costs.
• Figure 2: Comparison of social distancing behaviour. (A) Population behaviour $k(t)$, (B) government intervention $\varepsilon(t)$ and (C, D) dynamics of the disease $s$, $i$ for a range of scenarios with $i_0=3\cdot 10^{-8}$, $f=1$, and $\kappa^* = 4$ throughout: a baseline where there is no behavioural modification (corresponding to equilibrium behaviour at an infection cost $\alpha=0$, grey lines); the Nash equilibrium for $\alpha=400$ (black lines), calculated numerically via forward-backward sweep, see section C in S1 Text (In order to demonstrate that the numerical solution is accurate, we also show the analytical solution of the same equations SchnyderTurnerAnalytic as black dots); the utilitarian maximum for $\alpha=400$ (gold dashes); and finally the population behaviour for two optimal government policies, one being without cost to the government, $\gamma_g = 0$ (gold lines), and one being costly, $\gamma_g = 0.5$, with $\alpha=400$ (cyan lines). When government interventions are cost-free, they enable the population to reach the utilitarian maximum.
• Figure 3: The rational behaviour in the presence of a healthcare threshold depends on the maximum cost of infection. (A) The infection cost $\alpha(i)$ for a range of healthcare thresholds $i_{hc}$, see Eq \ref{['eq:healthcare_threshold']} with steepness $\sigma=300$. The colours encode the position of the healthcare threshold $i_{hc}$ for the whole figure. For comparison, two scenarios where $\alpha(i) = \alpha_0$ (grey line) and $\alpha(i) = \alpha_1$ (black) are considered as well. The base infection cost is kept constant throughout, $\alpha_0 = 100$, whereas $\alpha_1$ is varied in the following panels. (B) Typical example for the equilibrium behaviour of population $k(t)$ (B1) and the corresponding infectious cases $i$ (B2) over time for low maximum infection cost $\alpha_1$. We compare the behaviour for a healthcare threshold, $\alpha_0 = 100$, $\alpha_1= 1.75\alpha_2$ with $i_{hc} = 0.1$ (yellow) to the behaviour for constant infection costs $\alpha(i) = 100$ (grey) and $\alpha(i) = 1.75\alpha_0$ (black) These cases are also marked in panel (C) by correspondingly coloured circles beneath the letters B. (C) The peak of the epidemic $\max(i)$ as a function of the maximum cost of being infectious $\alpha_1$ corresponding to the infection cost scenarios shown in (A). We also mark the data points corresponding to the examples shown in panels (B1-2) and (D1-2) with circles and corresponding labels above. (D) Typical example for the equilibrium behaviour of population $k(t)$ (D1) and the corresponding infectious cases $i$ (D2) over time for high maximum infection cost $\alpha_1$. We compare the behaviour for a healthcare threshold, $\alpha_0 = 100$, $\alpha_1= 4\alpha_2$ with $i_{hc} = 0.1$ (yellow) to the behaviour for constant infection costs $\alpha(i) = 100$ (grey) and $\alpha(i) = 4\alpha_0$ (black) These cases are also marked in panel (C) by correspondingly coloured circles beneath the letters D. For the same data as in panel (C): (E) Total number of cases after the epidemic has run its course. (F) Duration of the epidemic as defined by the time interval for which $i > 10^{-4}$. (G) Total cost of the epidemic $-U$ for equilibrium behaviour in units of the minimal infection cost $\alpha_0$. In the inset, the epidemic cost is shown in units of the maximum infection cost. Lines in (D-G) serve as guides to the eye.
• Figure 4: Optimal government policy. (A) Peak of the epidemic as a function of the maximum cost of infection for a range of scenarios where the (maximum) infection cost for either population or government is varied: Nash equilibrium behaviour of the population, without government intervention for a constant infection cost (black line, replotted from Fig \ref{['fig:hct']}) and with a healthcare threshold at $i_{hc}=0.01$ (red, replotted from Fig \ref{['fig:hct']}); with government intervention for a constant infection cost $\alpha_g$ (cost-free $\gamma_g=0$: gold, costly $\gamma_g = 0.5$: cyan) and with a healthcare threshold at $i_{hc}=0.01$ (cost-free $\gamma_g=0$: green, costly $\gamma_g = 0.5$: purple). The circles mark the scenarios shown in Fig \ref{['fig:hct_gov_2']}. For these scenarios, we also show (B) the total number of cases, (C) the duration of the epidemic, as well as (D) the total cost of the epidemic in units of the minimal infection cost $\alpha_0$. In the cases without government intervention, the total cost is calculated as $-U$, whereas in the cases with government intervention, we report $-U_g$. In the inset, the epidemic cost is shown in units of the maximum infection cost. Lines serve as guides to the eye.
• Figure 5: Course of the epidemic with government intervention. Government intervention $\varepsilon$ assuming (A) constant infection cost $\alpha_g$ and cost-free intervention $\gamma_g=0$, (B) a healthcare threshold (HT) and cost-free intervention $\gamma_g=0$, and (C) a healthcare threshold and costly intervention $\gamma_g = 0.5$; all for a range of $a_{g}$ or $a_{1g}$, respectively, as marked by circles in Fig \ref{['fig:hct_gov']}A and listed in the legends of (G-I), with individuals assuming that $\alpha_0 = 100$. (D-F) Equilibrium population behaviour $k$ in response to $\varepsilon$ of (A-C), respectively. (G-I) Infectious $i$ over time, corresponding to the behaviour shown in (D-F), respectively. Here, the $y$-axis has linear scale between $0$ and $10^{-2}$ and logarithmic scale above that.