The impact of e-visits on patient access to primary care

Abstract

To improve patient access to primary care, many healthcare organizations have introduced electronic visits (e-visits) to provide patient-physician communication through secure messages. However, it remains unclear how e-visit affects physicians’ operations on a daily basis and whether it would increase physicians’ panel size. In this study, we consider a primary care physician who has a steady patient panel and manages patients’ office and e-visits, as well as other indirect care tasks. We use queueing-based performance outcomes to evaluate the performance of care delivery. The results suggest that improved operational efficiency is achieved only when the service time of e-visits is smaller enough to compensate the effectiveness loss due to online communications. A simple approximation formula of the relationship between e-visit service time and e-visit to office visit referral ratio is provided serving as a guideline for evaluating the performance of e-visit implementation. Furthermore, based on the analysis of the impact of e-visits on physician’s capacity, we conclude that it is not the more e-visits the better, and the condition for maximal panel size is investigated. Finally, the expected outcomes of implementing e-visits at Dean East Clinic are discussed.

Appendix: Proofs

Due to space limitation, we omit the majority of algebraic operations and only provide the sketch of proofs.

Proof of Theorems 1 and 2

To model physician’s office and e-visit services, consider an M/G/1 queue with server vacations. According to the Pollaczek-Khinchin transformequation ([57], Sec. 5.8), let W ^∗(s), S ^∗(s), and V ^∗(s)denote the Laplace-Stieltjes transform (LST) of waiting time W , service time S, and vacation time V .Then,

$$W^{*}(s) = \frac{1-V^{*}(s)}{sE(V)} \cdot \frac{(1-\rho)s}{s-\lambda + \lambda S^{*}(s)}, $$

where ρ = λ E[S], λ is the arrival rate, and E[S]is the expectation of service timeS. Taking the first derivative of W ^∗(s)at s = 0and applying L’Hôspital’srule, the mean waiting time E(W)can be achieved:

$$E(W) = \frac{\lambda E(S^{2})}{2(1-\rho)}+ \frac{ E(V^{2})}{2E(V)}. $$

In the case of the FCFS policy, let the effective total arrival rate be$\lambda _{e} =\lambda _{ov}^{\prime }+\lambda _{ev}$, anddefine

$$\begin{array}{@{}rcl@{}} p_{ev} &=& \lambda_{ev}/\lambda_{e}, \\ p_{ov} &=& \lambda_{ov}^{\prime}/\lambda_{e}. \end{array} $$

Then, the first and the second moments of the service time S can be evaluated as

$$\begin{array}{@{}rcl@{}} E(S) &=& \sum\limits_{i=ev,ov} p_{i} \frac{1}{\mu_{i}} =(\frac{\lambda_{ov}^{\prime}}{\mu_{ov}} + \frac{\lambda_{ev}}{\mu_{ev}}) \frac{1}{\lambda_{e}} = \frac{\rho}{\lambda_{e}}, \\ E(S^{2}) &=& \sum\limits_{i=ev,ov} p_{i} E({S_{i}^{2}})= \frac{2}{\lambda_{e}}\left( \frac{\lambda_{ov}^{\prime}\delta_{ov}}{\mu_{ov}^{2}}+ \frac{\lambda_{ev}\delta_{ev}}{\mu_{ev}^{2}}\right)\\ &=& \frac{2}{\lambda_{e}}(\rho_{ov}\omega_{ov}+ \rho_{ev}\omega_{ev}). \end{array} $$

Consequently, the mean waiting time can be calculated as

$$E(W) = (\rho_{ov}\omega_{ov}+ \rho_{ev}\omega_{ev})\frac{1}{(1-\rho)}+ \omega_{v}. $$

Finally,the average time each type of patients spent in the system are obtained:

$$T_{i} = E(W) +E(S_{i}),\quad i = ev, ov. $$

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Proof of Proposition 1

Recall (8),

$$\rho = \rho_{ov} + \rho_{ev} = (1-\alpha(1-\gamma-\beta))\rho_{t}. $$

Since 0 < α < 1, then, ρ < ρ _t if andonly if 1 − γ − β > 0. □

Proof of Proposition 2

Compare the change in the average cycle time T _{o v} − T _t :

$$ T_{ov} - T_{t} \,=\, \frac{\alpha \lambda \left( \gamma \delta_{ev} \left( -\lambda+\gamma \mu_{ev}\right)\!+\delta_{ov} \left( \lambda -(1-\beta) \mu_{ev}\right)\right)}{\mu_{ev} \left( \lambda-\gamma \mu_{ev} \right) \left( (1-\alpha(1-\gamma-\beta )) \lambda -\gamma \mu_{ev}\right)}. $$

(A.1)

According to assumption (vii), the denominator of Eq. A.1 is larger than zero. Then, a closer look is taken at thenumerator:

$$\begin{array}{@{}rcl@{}} && \gamma\delta_{ev} (-\lambda+\gamma\mu_{ev})+\delta_{ov} (\lambda -(1-\beta)\mu_{ev}) \\ &&\quad \leq \gamma\delta_{ov} (-\lambda+\gamma\mu_{ev})+\delta_{ov} (\lambda -(1-\beta)\mu_{ev}) \\ && \quad = ((1-\gamma)\lambda - (1-\gamma^{2}- \beta)\mu_{ev})\delta_{ov}\\ &&\quad < ((1-\gamma)\gamma\mu_{ev} - (1-\gamma^{2}- \beta)\mu_{ev})\delta_{ov} \\ && \quad = -(1-\gamma-\beta )\mu_{ev}\delta_{ov}. \end{array} $$

The above inequalities indicate 1 − γ − β ≥ 0is asufficient but not necessary condition for T _{o v} − T _t < 0. □

Proof of Proposition 3

Under the same total external arrival rate λ,if 1 − γ − β = 0,

$$\begin{array}{@{}rcl@{}} {\Phi}_{\beta = 1-\gamma} &=& - \alpha \lambda (1-\gamma)\omega_{v}\\ && + \alpha \lambda^{2}\frac{(-1+2\gamma \,+\, (1-\gamma)\gamma\alpha)\delta_{ov} \!- \gamma^{2}(1\,+\,(1-\gamma)\alpha)\delta_{ev}}{\gamma\mu_{ev}(-\lambda +\gamma\mu_{ev} )}. \end{array} $$

Thus, a necessary and sufficient condition for Φ _β=1−γ > 0is

$$ \omega_{v} \!<\! \lambda\frac{(-1\,+\,2\gamma \,+\, (1\,-\,\gamma)\gamma\alpha)\delta_{ov} \,-\, \gamma^{2}(1\!+(1-\gamma)\alpha)\delta_{ev}}{ \gamma(1-\gamma)\mu_{ev}(-\lambda +\gamma\mu_{ev} )}. $$

(A.2)

Then, according to the monotonicity of Φwith respect to β, if condition (A.2) is satisfied, and β < 1 − γ,we have Φ > 0. □

Proof of Proposition 4

Take the partial derivatives of Φwith respect to e-visit service rate μ _{e v} and the variation factor δ _{e v} :

$$\begin{array}{@{}rcl@{}} \frac{\partial {\Phi}}{\partial \mu_{ev}} &=& \frac{\rho_{ev}}{\mu_{ev}}+(1+ \alpha\beta)\lambda \left( \frac{2\rho_{ev}\omega_{ev}}{(1-\rho)\mu_{ev}} \right.\\ &&+\left. \frac{(\rho_{ev}\omega_{ev} + \rho_{ov}\omega_{ov})\rho_{ev}}{(1-\rho)^{2}\mu_{ev}}\right)> 0, \\ \frac{\partial {\Phi} }{\partial \delta_{ev}} &=& -\frac{\rho_{ev}(1+ \alpha\beta)\lambda}{(1-\rho)\mu_{ev}} < 0. \end{array} $$

Therefore, Φis monotonically increasingwith respect to e-visit service rate μ _{e v} ,and is monotonically decreasing with respect to e-visit variation factor δ _{e v} . □

Proof of Proposition 5

Take the partial derivative of Φwith respect to β,

$$\begin{array}{@{}rcl@{}} \frac{\partial {\Phi}}{\partial \beta} &=& -\frac{\alpha \lambda}{\mu_{ov}} -\alpha \lambda \omega_{v} - \frac{\alpha}{\mu_{ov}}\frac{(1+\alpha \beta)\lambda^{2}}{1-\rho}\omega_{ov} \\ &-& (\frac{\alpha \lambda}{1-\rho} \!+ \frac{\alpha }{\mu_{ov}}\frac{(1+\alpha\beta) \lambda^{2}}{(1-\rho)^{2}})(\rho_{ov}\omega_{ov} \!+\rho_{ev}\omega_{ev}) < 0. \end{array} $$

It can be concluded that Φismonotonically decreasing with β. □

Proof of Proposition 6

When the physician’s other indirect care work is not considered, the system can be modeled as a single server queue.

$$\begin{array}{@{}rcl@{}} {\Phi}_{nv}&=& \phi_{t,\omega_{v}=0}-\phi_{e,\omega_{v}=0}\\ &=&(\phi_{t}-\lambda\omega_{v})-(\phi_{e}-\omega_{v}(1+\alpha\beta)\lambda)\\ &=&{\Phi} + \alpha\beta\omega_{v}. \end{array} $$

□

Proof of Corollary 2

Suppose β = 1 − γ and plug-in δ _{o v} = δ _{e v} ,

$$\begin{array}{@{}rcl@{}} {\Phi}_{\beta = 1-\gamma} &=& - \alpha \lambda (1-\gamma)\omega_{v} -\frac{\alpha \lambda^{2}(1-\gamma)^{2}(1-\gamma\alpha)\delta_{ov}}{\gamma\mu_{ev}(-\lambda +\gamma\mu_{ev} )} \\ &<& 0. \end{array} $$

Based on the monotonicity property of Φwith respect to β,when β ≥ 1 − γ,${\Phi }_{\delta _{ov}= \delta _{ev}} < 0$. □

Proof of Proposition 7

According to the physician utilizations defined in Eqs. 21 and 22,

$$\begin{array}{@{}rcl@{}} && \rho- \rho_{t}= \frac{(1-\alpha (1-\gamma-\beta )) \lambda-\lambda_{t}}{\gamma \mu_{ev}} \leq 0, \\ &&\text{iff.} \quad \lambda \leq \frac{\lambda_{t}}{1- \alpha(1-\gamma-\beta )}. \end{array} $$

Thus, the maximum arrival rate λ ^∗the physician can accommodate is $\lambda ^{*} = \frac {\lambda _{t}}{1- \alpha (1-\gamma -\beta )}$. □

Proof of Corollary 3

Plug-in$\lambda = \frac {\lambda _{t}}{1- \alpha (1-\gamma -\beta )}$to Eqs. 9 and 10,

$$\begin{array}{@{}rcl@{}} T_{ov} - T_{t} &=& -\frac{\alpha \left( \delta_{ov}-\gamma\delta_{ev}\right) \lambda_{t}}{\mu_{ev} (1-\alpha(1-\gamma-\beta)) \left( \gamma\mu_{ev} -\lambda_{t}\right)} < 0, \\ T_{ev} - T_{t} &<& T_{ov} - T_{t} < 0. \end{array} $$

□

Proof of Corollary 4

When $\lambda = \frac {\lambda _{t}}{1- \alpha (1-\gamma -\beta )}$and δ _{o v} = δ _{e v} ,

$$\begin{array}{@{}rcl@{}} {\Phi}_{\lambda = \lambda^{*}} &=& -\frac{\alpha(1-\gamma)^{2} \delta_{ov} (1-\alpha (1-\beta)) {\lambda_{t}^{2}}}{\gamma\mu_{ev} (1-\alpha(1-\gamma-\beta))^{2} \left( \gamma\mu_{ev}-\lambda_{t}\right)}\\ &&- \frac{\alpha (1-\gamma)\lambda_{t} \omega_{v}}{1-\alpha(1-\gamma-\beta)} < 0. \end{array} $$

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Proof of Corollary 5

In Eq. 29, the denominator is a quadratic function of α.To find the optimal α ^∗to maximize λ ^∗isequal to maximizing α(1 − γ − k ₁ α − k ₂),which yields

$$\alpha^{*} = \frac{1-\gamma-k_{2}}{2k_{1}}.$$

Notably α ∈ (0, 1]; therefore,$\alpha ^{*} = \min (\frac {1-\gamma -k_{2}}{2k_{1}},1)$. □