Quantification of the resilience of primary care networks by stress testing the health care system

Abstract

There are practically no quantitative tools for understanding how much stress a health care system can absorb before it loses its ability to provide care. We propose to measure the resilience of health care systems with respect to changes in the density of primary care providers. We develop a computational model on a 1-to-1 scale for a countrywide primary care sector based on patient-sharing networks. Nodes represent all primary care providers in a country; links indicate patient flows between them. The removal of providers could cause a cascade of patient displacements, as patients have to find alternative providers. The model is calibrated with nationwide data from Austria that includes almost all primary care contacts over 2 y. We assign 2 properties to every provider: the "CareRank" measures the average number of displacements caused by a provider's removal (systemic risk) as well as the fraction of patients a provider can absorb when others default (systemic benefit). Below a critical number of providers, large-scale cascades of patient displacements occur, and no more providers can be found in a given region. We quantify regional resilience as the maximum fraction of providers that can be removed before cascading events prevent coverage for all patients within a district. We find considerable regional heterogeneity in the critical transition point from resilient to nonresilient behavior. We demonstrate that health care resilience cannot be quantified by physician density alone but must take into account how networked systems respond and restructure in response to shocks. The approach can identify systemically relevant providers.

Keywords: coevolving networks; dynamics of collapse; patient-sharing network; quality of care; robustness.

Fig. 1.

Schematic representation of patient displacement dynamics. (A) Doctors are represented as nodes (size represents the number of patients treated per year). They are linked if they share patients in the patient-sharing network, $A$ (black arrows). The color represents their current capacity; green means that they have capacity, and red means that they can no longer accept new patients. (B) Doctor $a$ retires at time step 1; his/her patients are distributed to other doctors according to the weights of the links from $a$ to $b$ and from $a$ to $c$ (yellow arrows). This, in turn, changes the capacity of the other doctors. (C) As $c$ has reached its capacity limit (red), he/she must send patients to other doctors (blue arrows from $c$ to $b$ and $d$). This creates a cascade of patient displacements of size 2. D–F show the same steps as in A–C in a simulation of a realistic environment. Doctors are localized (due to data protection) at random locations within a district, and a real patient-sharing network is used. (E) A doctor is removed, and his/her patients are shared (yellow). (F) Those doctors who reach their capacity send excess patients to others in a second round (blue). At this point, all patients are cared for, and the model dynamic terminates.

Fig. 2.

Systemic risk profile of Austrian health care providers. (A) The distribution of average quarterly patient numbers ${\mu }_i$ of doctors has a median of 945 patients. (B) Displacements, $D_i$, for every doctor in Austria tend to correlate with patient numbers ${\mu }_i$ of doctors (Pearson’s $R=0.52$, $p<10^{-4}$). The color encodes the number of doctors with a given (${\mu }_i$, $D_i$) pair. (C) Systemic risk contributions of doctors, $D_i$, show only little correlation with their systemic benefit (Pearson’s $R=0.42$, $p<10^{-4}$), $B_i$. The 4 quadrants indicate regions where $D_i$ and $B_i$ are above or below their population medians, respectively.

Fig. 3.

Number of patients, $L_{\mathcal{S}}\left(f,\delta \right)$, who cannot be cared for as a function of the fraction of unavailable PCPs, $f$, in district $\delta$. $L_{\mathcal{S}}\left(f,\delta \right)$ is shown for 2 different scenarios where doctors are removed in a different order: sequence $A$ (green) and sequence $B$ (blue). Labels display indexes of the removed PCPs at each step (i.e., the green sequence first removes PCP 2 followed by 13, 10, and so forth, whereas the blue sequence first removes PCP 4 followed by 7, 3, etc.). The shaded area envelops all observed values of $L_{\mathcal{S}}\left(f,\delta \right)$ (100$\%$ CI). Sequence $B$ gives a scenario where 44% (8pcps) have to be removed before losing patients, whereas 22% (4pcps) are sufficient to put the district in a condition where it cannot care for all patients for sequence $A$. The red arrow marks the position of the critical fraction $f_c$, which is the smallest $f$ such that $L_{\mathcal{S}}\left(f,\delta \right)>0$ holds for each observed sequence.

Fig. 4.

Map of Austria that shows the upper bound of the resilience indicator, $f_c\left(\delta \right)$, for all districts. Districts colored in green (red) have a particularly high (low) resilience: that is, critical removal fractions of $f_c\left(\delta \right)$.

Fig. 5.

Resilience vs. (primary care) provider density. Every circle (size is proportional to the district population) is a district with its provider density (number of PCPs per thousand population) on the $x$ axis and the lower bound of resilience indicator, $f_c\left(\delta \right)$, on the $y$ axis. While there is some correlation between them (Pearson’s $R=0.38$, $p<10^{-4}$; Spearman’s $R=0.37$, $p=0.0002$), physician density can vary by up to 1 order of magnitude for districts of similar resilience.