Social distancing game and insurance investment in a pandemic

Abstract

We consider a heterogeneous Susceptible–Infected–Recovered epidemic model, calibrated to the COVID-19 pandemic characteristics. We study the equilibrium of a voluntary social distancing game on a network of individuals subject to epidemic risk. We quantify the absolute and relative utility gaps across age cohorts. We further introduce life insurance in the model, which serves to mitigate the loss in the severe individual outcomes. We find that in most cases, insurance decreases the risk of contagion in the network because more individuals can be incentivized to self-isolate. On the other hand, in the case when the insurer does not have sufficient information on the self-isolation strategy of the individual, insurance can introduce moral hazard. We find that when premiums cannot be sufficiently differentiated, individuals may choose not to socially distance. This illustrates the importance of disclosing self-isolation strategies to the insurer, or inferring these strategies for the various types. Partial monitoring of social distancing strategy can mitigate the two problems of moral hazard and adverse selection.

A Final outcome of heterogeneous SIR epidemics

In this appendix we state some results on the final outcome of heterogeneous SIR epidemic model, for a given individuals social distancing strategies profile across individuals, when contact takes place on general networks. We also provide an iterative algorithm for our baseline model to compute both the asymptotic Nash equilibrium, as well as the social optimum on random networks with given vertex degrees. We assume that the degrees are bounded from above by $\Delta$. We are interested in ${\mathcal {S}}^{({\textbf{s}})}_f$ and ${\mathcal {R}}^{({\textbf{s}})}_f$ the final set of susceptible and removed individuals, respectively, when the individuals follow the social distancing strategy ${\textbf{s}}$.

1.1 A.1 Cohort targeting to bound epidemics in general networks

We first state some general conditions on the adjacency matrix of the interaction graph and epidemics parameters for the size of the epidemics to be small compared to the size of the network. This can be used in the case where the entire or part of the realized network is known, in order to understand the potential for a large epidemic. In particular, it is essential to choose a social distancing strategy, imposed by policy makers or by setting various incentives, so that the epidemic dies out. For example, if one population type is prone to a large epidemic, then the incentive to socially distance may come from higher costs of insurance for that population type. In particular, the suggestion is that in the context of a pandemic, one needs to take into account not only the individual risk but the risk at the level of an entire population that engages in a certain (type-dependent) social distancing strategy.

Let A denote the adjacency matrix of the social contact graph. The probability that an infected individual makes infectious contact with any susceptible neighbour with type t and social activity $s=1$ is given by $1-e^{-\beta _t}$. Given the social distancing profile ${{\textbf{s}}}$, we define the infection matrix $B^{({{\textbf{s}}})}$ as

$$\begin{aligned} B^{({{\textbf{s}}})}_{ij}:= \left( 1-e^{-\beta _{t_j}s_j}\right) A_{ij}, \end{aligned}$$

(31)

for all $i,j \in [n]$. Note that the infection rates are not necessarily symmetric and, in general, the matrix $B^{({{\textbf{s}}})}$ might not be symmetric even (if) the adjacency matrix A is symmetric.

We first give a condition on the maximum row sum of the matrix B, which gives us an upper bound for the expected amplification of infected individuals $(|{\mathcal {R}}^{({\textbf{s}})}_f| - |{\mathcal {R}}(0)| ) /|{\mathcal {I}}(0)|$.

The set of removed individuals at the end of the epidemic ${\mathcal {R}}^{({\textbf{s}})}_f \setminus {\mathcal {R}}(0)$ is the set of recovered or dead during the epidemic and is same set of all individuals who have ever been infected starting from the initial seed.

Proposition A.1

Let $B^{({{\textbf{s}}})}_i=\sum _{j=1}^n B^{({{\textbf{s}}})}_{ij}$ and $B^{({{\textbf{s}}})}_{\max }=\max _i(B^{({{\textbf{s}}})}_i)$. If $B^{({{\textbf{s}}})}_{\max }<1$, then

$$\begin{aligned} {\mathbb {E}}[|{\mathcal {R}}^{({{\textbf{s}}})}_f|]\le |{\mathcal {R}}(0)|+\frac{1}{1-B^{({{\textbf{s}}})}_{\max }} |{\mathcal {I}}(0)|, \end{aligned}$$

(32)

which in particular implies that for all $k>0$,

$$\begin{aligned} {\mathbb {P}}\left( |{\mathcal {R}}^{({{\textbf{s}}})}_f|-|{\mathcal {R}}(0)|\ge \frac{k}{1-B^{({{\textbf{s}}})}_{\max }} |{\mathcal {I}}(0)| \right) \le \frac{1}{k}. \end{aligned}$$

Proof

Let $p_i = {\mathbb {P}}(i \in {\mathcal {R}}_f/{\mathcal {R}}(0))$. Hence, $p_i = 1$ if $i \in {\mathcal {I}}(0)$ and otherwise $p_i \le \sum _{j=1}^n B_{ji}^{({\textbf{s}})} p_j$, which writes for all $i=1, 2, \dots , n$ as

$$\begin{aligned} p_i \le \mathrm{1\hspace{-0.90ex}1}\left( i \in {\mathcal {I}}(0)\right) + \sum _{j=1}^n B^{({\textbf{s}})}_{ji} p_j. \end{aligned}$$

(33)

We thus obtain

$$\begin{aligned} \sum _{i=1}^n p_i\le & {} \sum _{i=1}^n \mathrm{1\hspace{-0.90ex}1}(i \in {\mathcal {I}}(0)) + \sum _{i=1}^n \sum _{j=1}^n B^{({\textbf{s}})}_{ji} p_j \\= & {} |{\mathcal {I}}(0)| + \sum _{j=1}^n \left( \sum _{i=1}^n B^{({\textbf{s}})}_{ji}\right) p_j \le |{\mathcal {I}}(0)| + B^{({\textbf{s}})}_{\max } \sum _{j=1}^n p_j. \end{aligned}$$

We conclude ${\mathbb {E}}\left[ |{\mathcal {R}}_f|\right] - |{\mathcal {R}}(0)|= \sum _{i=1}^n p_i \le \frac{1}{1-B^{({\textbf{s}})}_{\max }} |{\mathcal {I}}(0)|.$ The second statement follows using the Markov inequality. Namely, for any $k>0$ we have

$$\begin{aligned} {\mathbb {P}}\left( |{\mathcal {R}}^{({{\textbf{s}}})}_f|-|{\mathcal {R}}(0)|\ge \frac{k}{1-B^{({{\textbf{s}}})}_{\max }} |{\mathcal {I}}(0)| \right) \le \frac{1}{k}. \end{aligned}$$

by the Markov inequality. $\square$

We now consider the $L_2$ norm of the matrix $B^{({{\textbf{s}}})}$. Let $\lambda _{\max }(B)=||B||_2$ be the largest singular value of B, which is the square root of the largest eigenvalue of the positive-semidefinite matrix $B^TB$. The following proposition shows that the expected amplification is $O(\sqrt{n})$ whenever the largest singular value is smaller than 1.

Proposition A.2

If $\lambda _{\max }(B^{({{\textbf{s}}})})<1$, then

$$\begin{aligned} {\mathbb {E}}[|{\mathcal {R}}^{({{\textbf{s}}})}_f|]\le |{\mathcal {R}}(0)|+\frac{1}{1-\lambda _{\max }(B^{({{\textbf{s}}})})} \sqrt{n|{\mathcal {I}}(0)|}, \end{aligned}$$

(34)

which in particular implies that for all $k>0$,

$$\begin{aligned} {\mathbb {P}}\left( |{\mathcal {R}}^{({{\textbf{s}}})}_f|-|{\mathcal {R}}(0)|\ge \frac{k}{1-\lambda _{\max }(B^{({{\textbf{s}}})})} \sqrt{n|{\mathcal {I}}(0)|} \right) \le \frac{1}{k}. \end{aligned}$$

Proof

Recall that from (33) we have

$$\begin{aligned} p_i \le \mathrm{1\hspace{-0.90ex}1}(i \in {\mathcal {I}}(0)) + \sum _{j=1}^n B^{({\textbf{s}})}_{ji} p_j. \end{aligned}$$

Let ${\textbf{p}} = [p_1, p_2, \dots , p_n]$ denote the vector with components $p_i, {{\textbf{1}}}$ be the vector with all components equal to 1 and ${{\textbf{1}}}_{{\mathcal {I}}(0)}$ be the vector with component 1 for $i\in {\mathcal {I}}(0)$ and 0 for $i \notin {\mathcal {I}}(0)$. By Equation (33), we have

$$\begin{aligned} {{\textbf{p}}} \le {{\textbf{1}}}_{{\mathcal {I}}(0)} + B^{({\textbf{s}})}{{\textbf{p}}}. \end{aligned}$$

Denoting by $|| \cdot ||$ the Euclidean norm, we have

$$\begin{aligned} ||{{\textbf{p}}}|| \le ||{{\textbf{1}}}_{{\mathcal {I}}(0)} + B{{\textbf{p}}} || \le ||{{\textbf{1}}}_{{\mathcal {I}}(0)}|| + ||B {{\textbf{p}}}|| \le \sqrt{|{\mathcal {I}}(0)|} + \lambda _{\max }(B^{({\textbf{s}})}) ||{{\textbf{p}}}||. \end{aligned}$$

We thus have for $\lambda _{\max }(B^{({\textbf{s}})}) < 1$ that $||{{\textbf{p}}}|| \le \frac{\sqrt{|{\mathcal {I}}(0)|}}{1-\lambda _{\max }(B^{({\textbf{s}})})}$. Furthermore by the Cauchy-Schwarz inequality,

$$\begin{aligned} {\mathbb {E}}\left[ |{\mathcal {R}}^{({\textbf{s}})}_f|\right] -|{\mathcal {R}}(0)| = \sum _{i \in [n]} p_i = ||{{\textbf{1}}}^T {{\textbf{p}}}|| \le ||{{\textbf{1}}}^T|| \ ||{{\textbf{p}}}|| = \sqrt{n} ||{{\textbf{p}}}||. \end{aligned}$$

We conclude (if $\lambda _{\max }(B^{({\textbf{s}})}) < 1$)

$$\begin{aligned} {\mathbb {E}}\left[ |{\mathcal {R}}^{({\textbf{s}})}_f|\right] -|{\mathcal {R}}(0)| \le \frac{1}{1-\lambda _{\max }(B)} \sqrt{n|{\mathcal {I}}(0)|}, \end{aligned}$$

and the second statement follows using the Markov inequality.

1.2 A.2 Algorithms to find the Asymptotic Nash equilibrium

In this section, we provide an iterative algorithm for our baseline model to compute both the asymptotic Nash equilibrium, as well as the social optimum. We assume that the degrees are bounded from above by $\Delta$. Remark that for all $\epsilon >0$, by Condition $(C_2)$, one can always choose $\Delta =\Delta _\epsilon$ such that $\sum _{t\in {\mathcal {T}}}\sum _{d=\Delta _\epsilon }^{\infty }d\mu _{t,d}<\epsilon$.

Under the assumptions of the following algorithm is guaranteed to converge to the unique fixed point of the mapping

$$\begin{aligned} \gamma _{t,d} = 1-G_t\left( \frac{\pi _{t,d}}{\kappa _t(x ^*_\gamma )\left( 1- \left( x^*_\gamma +(1-x^*_\gamma )e^{-\beta _t}\right) ^d \right) }\right) , \end{aligned}$$

where $x^*_\gamma$ is the unique fixed point in [0, 1] of equation

$$\begin{aligned} f_{\gamma }(x):= \frac{\lambda _R}{\lambda } + \alpha _S \sum _{t\in {\mathcal {T}}}\sum \limits _{d=0}^{\infty } \frac{d}{\lambda } \mu _{t,d} \left[ \gamma _{t,d}+(1-\gamma _{t,d}) \left( x+(1-x)e^{-\beta _t}\right) ^{d-1}\right] . \end{aligned}$$

${\textbf {Step 0:}}$:

Fix the error tolerance $\epsilon >0$. Set $k=0$ and the initial fraction of individuals with degree $d=0, \dots , \Delta$ and type $t\in {\mathcal {T}}$ who self-isolate as $\gamma _{t,d}^{(0)}=0$. Let $x_0$ be the smallest solution $x\in [0,1]$ of (fixed-point iteration algorithm with error tolerance $\epsilon$)

$$\begin{aligned} f_{0}(x):= \frac{\lambda _R}{\lambda } + \alpha _S \sum _{t\in {\mathcal {T}}}\sum \limits _{d=0}^{\Delta } \frac{d}{\lambda } \mu _{t,d} \left[ \left( x+(1-x)e^{-\beta _t}\right) ^{d-1}\right] . \end{aligned}$$

This is the equilibrium under no self-isolation.

${\textbf {Step 1:}}$:

Set $k \leftarrow k+1$. For all $d=0, \dots , \Delta$ and type $t\in {\mathcal {T}}$, set the fraction of individuals with degree d and type t who self-isolate at step k as

$$\begin{aligned} \gamma _{t,d}^{(k)}:= 1-G_t\left( \frac{\pi _{t,d}}{\kappa _t(x_{k-1})\left( 1- \left( x_{k-1}+(1-x_{k-1})e^{-\beta _t}\right) ^d \right) }\right) , \end{aligned}$$

and let $x_k$ be the unique fixed-point $x\in [0,1]$ of

$$\begin{aligned} f_{k}(x) := \frac{\lambda _R}{\lambda } + \alpha _S \sum _{t\in {\mathcal {T}}}\sum \limits _{d=0}^{\Delta } \frac{d}{\lambda } \mu _{t,d} \left[ \gamma ^{(k)}_{t,d}+(1-\gamma ^{(k)}_{t,d}) \left( x+(1-x)e^{-\beta _t}\right) ^{d-1}\right] . \end{aligned}$$

${\textbf {Step 2:}}$:

If $x_{k}-x_{k-1}<\epsilon$, terminate the algorithm. Otherwise, return to Step 1.

Note that since F is a strictly increasing cumulative distribution function, the fixed point solutions sequence $x_k$ will be increasing, converging to $x^*_\gamma$ which is unique as proven in Amini and Minca (2022), Theorem 3.1.

Further, for individuals with type t and degree d, $\pi _{t,d}(1-\gamma _{t,d})$ is the total payoff from social activity and $\kappa _t(x^*_\gamma ) \int _\gamma ^1 G_t^{-1}(1-u)du$ is the average loss faced by the $(1-\gamma _{t,d})$-fraction of individuals who are not following isolation and therefore are subject to epidemic risk.

It follows that the social utility averaged over the population converges to

$$\begin{aligned} \frac{1}{n} \sum _{i=1}^n u_i ({\varvec{\ell }}, \textbf{s}) {\mathop {\longrightarrow }\limits ^{p}} \bar{u}_\textrm{social}(\gamma ):=\sum _{t\in {\mathcal {T}}}\sum \limits _{d=0}^{\infty } \mu _{t,d}\bar{u}_{t,d}(\gamma ), \end{aligned}$$

with

$$\begin{aligned} \bar{u}_{t,d}(\gamma ):= \pi _{t,d}(1-\gamma _{t,d}) -\kappa _t(x^*_\gamma )\left( 1- \left( x^*_\gamma +(1-x^*_\gamma )e^{-\beta _t}\right) ^d \right) \int _{\gamma _{t,d}}^1 G_t^{-1}(1-u)du. \end{aligned}$$

The following algorithm is used to compute the social optimal network immunity fixed-point solution, as well as for social optimal isolation policy over different individuals:

${\textbf {Step 0:}}$:

Set again $k=0$, $\gamma _{t,d}^{(0)}=0$ and let $x_0$ be the smallest solution $x\in [0,1]$ of

$$\begin{aligned} f_{0}(x):= \frac{\lambda _R}{\lambda } + \alpha _S \sum _{t\in {\mathcal {T}}}\sum \limits _{d=0}^{\Delta } \frac{d}{\lambda } \mu _{t,d} \left[ \left( x+(1-x)e^{-\beta _t}\right) ^{d-1}\right] . \end{aligned}$$

${\textbf {Step 1}}:$:

Set $k \leftarrow k+1$. For all $d=0, \dots , \Delta$ and type $t\in {\mathcal {T}}$, set the fraction of individuals with degree d and type t who self-isolate at step k as

$$\begin{aligned} \gamma _{t,d}^{(k)}:= & {} \underset{\gamma \in [0,1]}{\arg \max } \pi _{t,d}(1-\gamma ) -\kappa _t(x_{k-1})\left( 1- \left( x_{k-1}+(1-x_{k-1})e^{-\beta _t}\right) ^d \right) \\ {}{} & {} \times \int _{\gamma }^1 G_t^{-1}(1-u)du, \end{aligned}$$

and let $x_k$ be the unique fixed-point $x\in [0,1]$ of

$$\begin{aligned} f_{k}(x) := \frac{\lambda _R}{\lambda } + \alpha _S \sum _{t\in {\mathcal {T}}}\sum \limits _{d=0}^{\Delta } \frac{d}{\lambda } \mu _{t,d} \left[ \gamma ^{(k)}_{t,d}+(1-\gamma ^{(k)}_{t,d}) \left( x+(1-x)e^{-\beta _t}\right) ^{d-1}\right] . \end{aligned}$$

${\textbf {Step 2:}}$:

If $x_{k}-x_{k-1}<\epsilon$, terminate the algorithm. Otherwise, return to Step 1.