Optimal illness policy for an unethical daycare center

TL;DR

This work analyzes whether daycare centers have a financial incentive to keep sick children in attendance by framing a modified SIR model that splits infected individuals into those who attend and those who stay home, with attendance fraction $a$ determining infectious contact. The center’s profit proxy is the cumulative time children spend at home, $T_h(\infty)=(1-a)R_\infty/\gamma$, maximized by solving $\frac{d T_h(\infty)}{d a}=0$, with $R_\infty$ linked to $a$ and $R_0=\beta/\gamma$ through a Lambert $W$ relation. Results show that the optimal attendance $a^*$ decreases toward zero as disease infectiousness increases, and higher $R_0$ diseases yield larger potential staffing savings, illustrating a structural incentive to propagate illness under certain assumptions. The paper discusses policy implications, including removing under-staffing and implementing paid sick leave to mitigate these incentives, and suggests model extensions (SEIR, stochasticity) for more realism. Code and data are provided to reproduce the analysis.

Abstract

While businesses are typically more profitable if their workers and communities are minimally exposed to diseases, the same is not true for daycare centers. Here it is shown that a daycare center could maximize its profits by maintaining a population of sick children within the center, with the intention to infect more children who then do not attend. Through a modification of the Susceptible-Infected-Recovered (SIR) model for disease spread we find the optimal number of sick children who should be kept within the center to maximize profits. We show that as disease infectiousness increases, the optimal attendance rate of sick children approaches zero, while the potential profit increases.

Figures (3)

• Figure 1: Schematic of the modified SIR compartment model. The total population is divided into susceptible ($S$), infected ($I$) and recovered ($R$) individuals, and the infected population is further divided into those who attend daycare ($I_a$) and those who are sent home ($I_h$). Susceptible individuals become infected due to interactions with individuals who are infected and attending. Infected individuals gradually recover.
• Figure 2: (a-d) Time evolution of the components $S$ (dot-dashed), $I$ (solid), $I_a$ (dotted) and $R$ (dashed) of the modified SIR model (\ref{['eq:SIR']}) with $N=100$ children, infectivity $\beta = 0.5$, recovery rate $\gamma = 0.1$, basic reproduction number $R_0 = \beta/\gamma = 5$, initial populations $S(0) = 99$, $I(0) = 1$, $R(0) = 0$, and varied attendance rate $a$. (a) $a=0.3$, (b) $a=a^*\approx 0.40$, (c) $a=0.6$, (d) $a=0.8$. (e) The cumulative time that children are kept home $T_h(t)$ (\ref{['eq:Th_t']}) for the same parameters as in (a-d), with corresponding colors. The dashed horizontal lines show respective long-term theoretical values $T_h(\infty)$ based on (\ref{['eq:Th_infty']}) and (\ref{['eq:r_infty']}). (f) Dependence of the total time at home over the course of the disease $T_h(\infty)$ on the attendance parameter $a$ for a range of $R_0$ values (all other parameters the same as in (a-e)). The cases directly corresponding to the parameters in (a-e) are highlighted with stars in the respective colors from (a-d).
• Figure 3: Long-term time at home $T_h(\infty)$ across the $(a,R_0)$ parameter space, with $\gamma= 0.1$ and $N=100$ kept fixed. The black curve shows the attendance rates $a^*$ with maximal $T_h(\infty)$ for fixed $R_0$.