# Tiered Service Architecture for Remote Patient Monitoring

Siddharth Chandak1, Isha Thapa2, Nicholas Bambos1,2 and David Scheinker2,3
1 Department of Electrical Engineering, Stanford University, USA.
2 Department of Management Science & Engineering, Stanford University, USA.
3 School of Medicine, Stanford University, USA.
{chandaks, ishadt, bambos<EMAIL_ADDRESS>
###### Abstract

We develop a remote patient monitoring (RPM) service architecture, which has
two tiers of monitoring: ordinary and intensive. The patient’s health state
improves or worsens in each time period according to certain probabilities,
which depend on the monitoring tier. The patient incurs a “loss of quality of
life” cost or an “invasiveness” cost, which is higher under intensive
monitoring than under ordinary. On the other hand, their health improves
faster under intensive monitoring than under ordinary. In each period, the
service decides which monitoring tier to use, based on the health of the
patient. We investigate the optimal policy for making that choice by
formulating the problem using dynamic programming. We first provide analytic
conditions for selecting ordinary vs intensive monitoring in the asymptotic
regime where the number of health states is large. In the general case, we
investigate the optimal policy numerically. We observe a threshold behavior,
that is, when the patient’s health drops below a certain threshold the service
switches them to intensive monitoring, while ordinary monitoring is used
during adequately good health states of the patient. The modeling and analysis
provides a general framework for managing RPM services for various health
conditions with medically/clinically defined system parameters.

##  I Introduction

Remote Patient Monitoring (RPM) is increasingly receiving attention as a
method for monitoring patients with certain medical conditions in their normal
living/working environments to increase their quality of life and the level of
delivered health care [1, 2, 3]. This is becoming feasible via advancements in
wearable medical devices, for example, wearable glucose monitors [4], smart
watches with vital sign monitoring capabilities (e.g., heart rate, ECG, pulse
oximetry) [5, 6], and other such sensors. Further, such devices are
increasingly networked and can transmit and receive data over the Internet and
act as edge devices in communication with computation servers in the Cloud.

Studies have shown the effectiveness of RPM for various medical conditions.
For example, continuous glucose monitoring has been shown to improve glycemic
control in patients with diabetes [7, 8]. Smart watches have been effectively
used to monitor stress, movement disorders, sleep patterns, blood pressure,
heart disease, and COVID-19 [5]. Other RPM devices have also been used to
manage and track cardiac conditions, such as heart failure, arrhythmia, and
hypertension [3]. These studies highlight the potential of RPM to improve
patient outcomes and quality of life by allowing for timely interventions and
personalized care.

However, the question remains of how intensively to monitor patients.
Intensive monitoring schemes could range from remotely collecting more data on
the patient health state and administering more medical intervention remotely
(e.g., alerting the patient to increase medication dosage) to calling the
patient into an urgent care facility.

While aggressive monitoring may provide more comprehensive data, it can be
resource intensive, draining the wearable device battery [6] faster and
requiring the clinicians to review more RPM data. From the patient’s
perspective, under intensive monitoring the patient naturally experiences a
higher “loss of quality of life” cost or “invasiveness” cost, since intensive
monitoring would normally be more invasive to their personal lifestyle. In
contrast, intensive monitoring (and correspondingly elevated medical
intervention) would enable early detection and intervention for adverse
events, and hence the patient’s health is expected to improve faster than
under ordinary one. Therefore, to account for this trade-off between the
invasiveness cost and the possibility of an early intervention, there is an
inherent need for a systematic approach to determining the appropriate level
of monitoring, based on the patient’s health state.

In this paper we develop a RPM service architecture, where the patient is
placed under less or more intensive levels or tiers of monitoring, based on
their health state. We then study the optimal monitoring strategy for this
model and how it varies with different parameters. One would intuitively
expect that a patient would be placed under intensive one when their health
state deteriorates; on the other hand, they would be returned to ordinary
monitoring when their health state improves enough. The decision to switch
from ordinary to intensive monitoring (and/or vice versa) requires a
systematic analysis and depends heavily on the various parameters of the
service in a rather subtle and complicated way, as analyzed in the following
sections.

Of course, the design of an RPM service for a specific medical condition is
highly dependent on the specifics of that condition and requires specialized
medical knowledge. The point of this paper is not to design a particular RPM
service but to provide a general framework and systematic methodology for RPM
services based on tunable parameters (e.g., health improvement or
deterioration probabilities, monitoring options and invasiveness costs) so as
to make justifiable monitoring choices. The parameters will have to be decided
on and tuned by medical/clinical experts for condition-specific RPM services.
While we work with a simplified model here, the intuition gained from the
analysis can help clinicians take more informed monitoring decisions.

In section II, we develop the model of evolution of the patient health state
under ordinary and intensive monitoring and demonstrate how it can be managed,
using the methodology of dynamic programming. In section III, we provide some
analytical results on the optimal management policy in the asymptotic regime
of a large number of health states and provide conditions on the parameters
for choosing ordinary vs intensive monitoring. In section IV, we numerically
investigate the structure of optimal monitoring policies and demonstrate that
they place the patient under intensive monitoring when the health state
deteriorates below a certain threshold; otherwise they use ordinary
monitoring. Finally, in section V, we discuss some extensions. Sketches of
some necessary proofs are presented in the Appendix.

##  II The RPM Service Model

Consider a patient who can be in a health state $h_{t}\in\\{0,1,2,3,...,H\\}$
in each service time period $t\in\\{0,1,2,3,...\\}$. The RPM service places
the patient in a monitoring/intervention state $m_{t}\in\mathcal{M}=\\{o,i\\}$
in each time period $t$, abbreviated to monitoring state, where $o$ denotes
ordinary monitoring and $i$ intensive monitoring. Thus, one can view the
monitoring-patient joint state
$s_{t}=(m_{t},h_{t})\in\mathcal{M}\times\mathcal{H}$ as the system or service
state at time $t$.

A higher patient health state $h\in\\{0,1,2,3,...,H\\}$ corresponds to the
patient having better health. In particular, the lowest health state $0$ is
critical in the sense that, when the patient drifts into that state, they go
beyond the scope of the current service model; at that point other emergency
and/or more severe medical interventions are required, which are outside the
scope of this service. Because of that, the states $(i,0)$ and $(o,0)$ are
absorbing for the Markovian evolution of the health state, as explained below.
Indeed, when the patient enters heath state $0$ under any monitoring state $i$
or $o$, the service evolution stops, as other medical measures/interventions
are initiated.

As seen below, in defining costs incurred at the various states, we first take
the patient’s quality of life point of view. Under ordinary monitoring, the
patient incurs a constant cost $C_{o}\geq 0$ at any state $(o,h)$ with
$h\in\\{1,2,...,H\\}$. Correspondingly, under intensive monitoring, the
patient incurs a constant cost $C_{i}\geq 0$ at any state $(i,h)$ with
$h\in\\{1,2,..,H\\}$. These costs reflect the invasiveness loss for the
patient. One may argue about the costs in more elaborate ways, for example,
including patient risk factors and operational considerations of the service.
For simplicity, we focus here on the invasiveness argument mentioned above.

Of special interest is the critical health state $h=0$, where this model
ceases to apply. On either state $(o,0)$ or $(i,0)$ where the patients health
is critical a cost $C_{c}$ is incurred.

We finally define the transition costs $C_{o\to i}$ (and $C_{i\to o}$), which
is associated with the service transitioning the patient from ordinary to
intensive monitoring (and vice versa, respectively).

We model the system as a controlled Markov chain, trying to stay as simple as
possible, yet still capture the essence of the problem and get insights into
its solution. We discuss the more general case in the final section. At the
beginning of every time period $t$, the service takes the decision/action
(control) to either keep the monitoring state the same (as in the previous
time period) or switch it to the alternate monitoring state. Formally, the
decision/action space is $\mathcal{A}=\\{o,i\\}$ and each (state, action) pair
is associated with a cost given by the function
$c:\mathcal{S}\times\mathcal{A}\mapsto\mathbb{R}^{+}$. The transition
probabilities are given by $p(s^{\prime}|s,a)$ where
$s^{\prime},s\in\mathcal{S}$ and $a\in\mathcal{A}$. The cost functions and
transition probabilities are defined as follows.

1\. At health state $\mathbf{h=0}$:

> No action is taken with the the service ceasing operation. A cost of $C_{c}$
> is incurred.

2\. At health states $\mathbf{1\leq h\leq H}$:

  1. a)

Ordinary Monitoring ($m=o$), no Switching ($a=o$): Does not induce a
monitoring change, and the system state transitions as follows:

    1. i)

$(o,h)\xrightarrow{a=o}(o,\min\\{h+1,H\\})$ with prob. $\lambda_{o}$

    2. ii)

$(o,h)\xrightarrow{a=o}(o,h-1)$ with prob. $\mu_{o}=1-\lambda_{o}$,

and a cost $c\big{(}(o,h),o\big{)}=C_{o}$ is incurred. Note that
$\min\\{h+1,H\\}$ above is used to account for $(o,H)\to(o,H)$ with prob.
$\lambda_{o}$ since $H$ is the highest health state.

  2. b)

Ordinary Monitoring ($m=o$) with Switching ($a=i$): Induces a switch to
intensive monitoring $m=i$, and the system state transitions as follows:

    1. i)

$(o,h)\xrightarrow{a=i}(i,\min\\{h+1,H\\})$ with prob. $\lambda_{i}$

    2. ii)

$(o,h)\xrightarrow{a=i}(i,h-1)$ with prob. $\mu_{i}=1-\lambda_{i}$,

and a cost $c\big{(}(o,h),i\big{)}=C_{o\to i}+C_{i}$ is incurred.

  3. c)

Intensive Monitoring ($m=i$), no Switching ($a=i$): Does not induce a
monitoring change, and the system state transitions as follows:

    1. i)

$(i,h)\xrightarrow{a=i}(i,\min\\{h+1,H\\})$ with prob. $\lambda_{i}$

    2. ii)

$(i,h)\xrightarrow{a=i}(i,h-1)$ with prob. $\mu_{i}=1-\lambda_{i}$,

and a cost $c\big{(}(i,h),i\big{)}=C_{i}$ is incurred.

  4. d)

Intensive Monitoring ($m=i$) with Switching ($a=o$): Induces a switch to
ordinary monitoring $m=o$, and the system state transitions as follows:

    1. i)

$(i,h)\xrightarrow{a=o}(o,\max\\{h+1,H\\})$ with prob. $\lambda_{o}$

    2. ii)

$(i,h)\xrightarrow{a=o}(o,h-1)$ with prob. $\mu_{o}=1-\lambda_{o}$,

and a cost $c\big{(}(i,h),o\big{)}=C_{i\to o}+C_{o}$ is incurred.

We can easily incorporate health state dependent costs and transition
probabilities, but for simplicity we assume constant ones here. We make the
following natural assumptions.

######  Assumption 1.

  1. (a)

The transition probabilities satisfy: $\lambda_{i}\geq\lambda_{o}$.

  2. (b)

The costs satisfy: $0\leq C_{o}\leq C_{i}\leq C_{c}$.

The first assumption 1.a intuitively states that the patient’s health improves
faster under intensive monitoring, rather than under ordinary. Regarding
assumption 1.b, it is naturally expected that $C_{o}\leq C_{i}$, as the
patient’s “annoyance” is higher under intensive monitoring/intervention than
under ordinary. Further, given the severity of entering the critical state
$h=0$, it is naturally expected that $C_{o}\leq C_{i}\leq C_{c}$, and
practically $C_{c}$ may be much larger than $C_{i}$.

###  II-A Optimal Monitoring Control

We study this problem under the discounted cost setting of the dynamic
programming methodology [9], hence, costs incurred $t$ time periods into the
future (with respect to present) are discounted by a factor of $\gamma^{t}$
with $0<\gamma<1$. Starting from state
$s_{0}=s\in\mathcal{M}\times\mathcal{H}$, the total expected (discounted) cost
to be incurred is

$\mathbb{E}\Big{[}\sum_{t=0}^{T-1}\gamma^{t}c(s_{t},a_{t})+\gamma^{T}C_{c}\Big{|}\
s_{o}=s\Big{]}$

when control action $a_{t}$ is taken by the service, at cost $c(s_{t},a_{t})$
introduced above, when its state is $s_{t}=(m_{t},h_{t})$ at time $t$, until
the patient enters the critical state $h=0$ at time $T$ and the service ceases
operation. Hence, $T$ is the time the patient spends in the service, and at
time $T$ the critical cost $C_{c}$ is incurred, however, discounted to
$\gamma^{T}C_{C}$. Thus, discounting by $\gamma$ implicitly reflects the
patient’s desire to stay longer in service, hence, incur the critical cost
deeper in the future and discounted to $\gamma^{T}C_{c}$.

A (stationary) monitoring policy $\pi(s)$ is a rule mapping each state
$s=(m,h)\in\mathcal{M}\times\mathcal{H}$ to a control action
$a\in\mathcal{A}=\\{o,i\\}$ to be taken at that state. The value function
$V_{\pi}(s)$ of a policy $\pi(s)$ is the total expected (discounted) cost the
system will incur until reaching the critical state $0$ and stop, when it
starts from state $s$ at time $t=0$. That is,

$V_{\pi}(s)=\mathbb{E}\left[\sum_{t=0}^{T-1}\gamma^{t}c\Big{(}s_{t},\pi(s_{t})\Big{)}+\gamma^{T}C_{c}\
\Big{|}\ s_{0}=s\right]$

and satisfies the dynamic programming equation [9].

$V_{\pi}(s)=c\Big{(}s,\pi(s)\Big{)}+\gamma\sum_{s^{\prime}\in\mathcal{S}}\mathbb{P}\Big{(}s^{\prime}\mid
s,\pi(s)\Big{)}V_{\pi}(s^{\prime}),$

for all $s\in\mathcal{S}=\mathcal{M}\times\mathcal{H}$, given the Markovian
evolution dynamics of the system, specified by the state transition
probabilities defined above.

The goal is to find an optimal policy $\pi^{*}$ minimizing the $V_{\pi}$ over
all policies $\pi$, that is, $V_{\pi^{*}}(s)\leq V_{\pi}(s)$ for every
$s\in\mathcal{S}$ over all policies $\pi$. For simplicity we write
$V^{*}(s)=V_{\pi^{*}}(s)$ and the following dynamic programming equation is
satisfied [9].

$V^{*}(s)=\min_{a\in\\{o,i\\}}\Bigg{\\{}c(s,a)+\gamma\sum_{s^{\prime}\in\mathcal{S}}\mathbb{P}\Big{(}s^{\prime}|s,a\Big{)}V^{*}(s^{\prime})\Bigg{\\}},$

and can be solved numerically to yield the optimal policy
$\pi^{*}(s)=\pi^{*}(m,h)$, that is, what optimal decision to take when the
patient is in health state $h$ under monitoring $m$. For the state transition
probabilities and costs defined before, this dynamic programming equation
unfolds into:

  1. (i)

For health states $1\leq h\leq H-1$

$\displaystyle
V^{*}(i,h)=\min\bigg{\\{}C_{i}+\gamma\Big{[}\lambda_{i}V^{*}(i,h{+}1)+\mu_{i}V^{*}(i,h{-}1)\Big{]},$
$\displaystyle\;\;\;\;\;\;\;\;C_{i\to
o}+C_{o}+\gamma\Big{[}\lambda_{o}V^{*}(o,h{+}1)+\mu_{o}V^{*}(o,h{-}1)\Big{]}\bigg{\\}},$
$\displaystyle
V^{*}(o,h)=\min\bigg{\\{}C_{o}+\gamma\Big{[}\lambda_{o}V^{*}(o,h{+}1)+\mu_{o}V^{*}(o,h{-}1)\Big{]},$
$\displaystyle\;\;\;\;\;\;\;\;C_{o\to
i}+C_{i}+\gamma\Big{[}\lambda_{i}V^{*}(i,h{+}1)+\mu_{i}V^{*}(i,h{-}1)\Big{]}\bigg{\\}}.$

  2. (ii)

At health state $H$,

$\displaystyle
V^{*}(i,H)=\min\bigg{\\{}C_{i}+\gamma\Big{[}\lambda_{i}V^{*}(i,H)+\mu_{i}V^{*}(i,H{-}1)\Big{]},$
$\displaystyle\;\;\;\;\;\;\;\;C_{i\to
o}+C_{o}+\gamma\Big{[}\lambda_{o}V^{*}(o,H)+\mu_{o}V^{*}(o,H{-}1)\Big{]}\bigg{\\}},$
$\displaystyle
V^{*}(o,H)=\min\bigg{\\{}C_{o}+\gamma\Big{[}\lambda_{o}V^{*}(o,H)+\mu_{o}V^{*}(o,H{-}1)\Big{]},$
$\displaystyle\;\;\;\;\;\;\;\;C_{o\to
i}+C_{i}+\gamma\Big{[}\lambda_{i}V^{*}(i,H)+\mu_{i}V^{*}(i,H{-}1)\Big{]}\bigg{\\}}.$

  3. (iii)

At the critical health state $h=0$,

$V^{*}(i,0)=V^{*}(o,0)=C_{c}.$

Note that in every $\min\\{\cdot,\cdot\\}$ above, the first term corresponds
to the control keeping the existing monitoring state, while the second
corresponds to the control switching to the alternate monitoring state and
incurring the switching cost.

###  II-B Simplified RPM Service

In order to reduce the number of parameters for tractability of the analysis,
we make the following simplification.

######  Definition 1.

Simplified RPM Service:.

  1. (a)

The cost for ordinary monitoring is set to zero: $C_{o}=0$.

  2. (b)

The switching costs are set to zero: $C_{i\to o}=C_{o\to i}=0$.

Assumption 1 is still satisfied.

Given the limited space of this short paper, we work with the simplified RPM
service below (analysis and numerical results), where interesting insights
emerge. The zero cost of invasiveness under ordinary monitoring has no
significant impact on the results, and is merely a technical assumption to
make our analysis easier. The non-zero transition cost is true in several
applications where the intensive monitoring just involves a higher rate of
collecting data about the patient’s health. We briefly comment on how non-zero
transition costs affect our results in Section V.

The simplified RPM service is illustrated in Figure 1 below.

Figure 1: The Simplified RPM Service. The blue and red arrows represent the
possible transitions from state $(o,3)$ for actions $o$ and $i$ respectively.
The arrows are labeled with probability of transition and cost incurred.

The dynamic programming equations for $V^{*}$ given above reduce in this case
to the following:

  1. (i)

For health states $1\leq h\leq H-1$,

$\hskip-8.5359ptV^{*}(i,h)=V^{*}(o,h)\\\
\hskip-8.5359pt=\min\Bigg{\\{}C_{i}+\gamma\Big{[}\lambda_{i}V^{*}(i,h+1)+\mu_{i}V^{*}(i,h-1)\Big{]},\\\
\hskip
38.41139ptC_{o}+\gamma\Big{[}\lambda_{o}V^{*}(o,h+1)+\mu_{o}V^{*}(o,h-1)\Big{]}\Bigg{\\}}$
(1)

  2. (ii)

At health state $H$,

$V^{*}(o,H)=V^{*}(i,H)\\\
=\min\Bigg{\\{}C_{i}+\gamma\Big{[}\lambda_{i}V^{*}(i,H)+\mu_{i}V^{*}(i,H-1)\Big{]},\\\
C_{o}+\gamma\Big{[}\lambda_{o}V^{*}(o,H)+\mu_{o}V^{*}(o,H-1)\Big{]}\Bigg{\\}}$
(2)

  3. (iii)

At the critical health state $h=0$,

$V^{*}(i,0)=V^{*}(o,0)=C_{c},$ (3)

Note that, in the absence of switching costs (that is, $C_{o\to i}=C_{i\to
o}=0$), for any health state $h$ the $V^{*}$ is the same under both ordinary
and intensive monitoring.

##  III Asymptotic Analysis

The optimal policy ${\pi^{*}}$ and value function $V^{*}$ can be computed
numerically (in general) from the dynamic programming equations (1)-(3) for
any given set of system parameters. But to develop intuition and characterize
the optimal policy, in this section, we analyse ${\pi^{*}}$ in the asymptotci
regime of a large number of health states $H\gg 1$, i.e., a dynamic range of
health states $H\to\infty$. This allows for analytic tractability of the
optimal policy and closed-form conditions on our policies of interest. We make
the following assumptions in this section.

######  Assumption 2.

Large $H$ Asymptotic Regime.

  1. (a)

The number of health states is very large, i.e., the system operates in the
asymptotic regime of $H\to\infty$.

  2. (b)

Under ordinary monitoring the patient’s health drifts downwards, i.e., the
improvement probability is $\lambda_{o}<0.5$ and the worsening one is
$\mu_{o}=1-\lambda_{o}>0.5$

Assumption 2.a allows us to use tools from random walk analysis, as done in
Lemma 1, making the analysis tractable. Assumption 2.b allows for the Markov
chain to remain stable (positive recurrent) in the asymptotic regime.

The results obtained in this asymptotic regime can be thought of as an
approximation for the RPM service with finite $H$, when $H$ grows large. In
the next section, we numerically demonstrate that this asymptotic
approximation tracks the optimal policy and the value function for our
simplified RPM for a number of health states as low as $H=5$.

We define $\widetilde{\Pi}$ as the the set of policies under which the service
chooses the same action irrespective of the monitoring state, that is,

$\widetilde{\Pi}=\\{\pi\mid\pi(i,h)=\pi(o,h),\;\forall h\geq 1\\}.$

Based on (1), for all $h\geq 1$, we then have

$\displaystyle{\pi^{*}}(o,h)={\pi^{*}}(i,h)$
$\displaystyle=\operatorname*{arg\,min}_{\\{i,o\\}}\\{C_{i}+\gamma\lambda_{i}V^{*}(i,h+1)+\gamma(1-\lambda_{i})V^{*}(i,h-1),$
$\displaystyle\;\;\;\;\;\;C_{o}+\gamma\lambda_{o}V^{*}(o,h+1)+\gamma(1-\lambda_{o})V^{*}(o,h-1)\\}$

This implies that ${\pi^{*}}\in\widetilde{\Pi}$ and we can restrict our
attention to the set $\widetilde{\Pi}$. For simplicity, we introduce notation
$V^{*}(h)=V^{*}(o,h)=V^{*}(i,h)$ for $h\in\mathcal{H}$ and similarly the
notation ${\pi^{*}}(h)$.

We next define an important policy $\pi_{o}$ where the patient stays under
ordinary monitoring at all health states, i.e., $\pi_{o}(i,h)=\pi_{o}(o,h)=o$
for all $h\geq 1$. Note that $\pi_{o}\in\widetilde{\Pi}$ and we define the
corresponding value function $V_{o}(h)$. Our first lemma gives the value
function for this policy and presents an important property about the optimal
policy.

######  Lemma 1.

For the simplified RPM (Definition 1) and under Assumption 2,

  1. (a)

The value function $V_{o}$ for the policy $\pi_{o}$ is given by:

$V_{o}(h)=\phi^{h}C_{c},$

where

$\phi=\frac{1-\sqrt{1-4\lambda_{o}\mu_{o}\gamma^{2}}}{2\lambda_{o}\gamma}.$

Note that $\phi<1$ for $\gamma<1$.

  2. (b)

For any choice of parameters, there exists $h^{\prime}$ such that, under the
optimal policy, the patient prefers to stay in ordinary monitoring above
health state $h^{\prime}$, i.e., ${\pi^{*}}(h)=o$, for all $h\geq h^{\prime}$.

###### Proof.

See Appendix A. ∎

For the simplified RPM, the cost of invasiveness under ordinary monitoring is
zero. Then $V_{o}(h)=\mathbb{E}[\gamma^{T}C_{c}|h_{0}=h]$, where $T$ is the
time taken to reach health state $0$, when started at health state $h$ and
when the patient always stay under ordinary monitoring. $T$ here is precisely
the hitting time of state $0$ for a random walk initiated at state $h$. The
proof for this lemma then follows from the moment generating function of the
hitting time for an $\infty$-state random walk.

(a) Optimal policy is $\pi_{o}$ (always in ordinary monitoring)

(b) Optimal policy is $\pi_{t,\bar{h}}$ (threshold policy with $\bar{h}=3$)

Figure 2: Optimal policies (numerically computed) for $H=6$ and two different
parameter sets. (a) For
$\lambda_{o}=0.2,\lambda_{i}=0.3,C_{c}=20,C_{i}=1,C_{o}=0,\gamma=0.9$ the
optimal policy is $\pi_{o}$ (using ordinary monitoring at all health states).
(b) For $\lambda_{o}=0.2,\lambda_{i}=0.3,C_{c}=60,C_{i}=1,C_{o}=0,\gamma=0.9$
the optimal policy is $\pi_{t,\bar{h}}$ with $\bar{h}=3$, that is, ordinary
monitoring is used for $h>3$ and intensive for $h\leq 3$.

An important implication of the above lemma is that the policy where the
patient chooses to stay in intensive monitoring for all health states is never
optimal. Our next theorem shows that the policy $\pi_{o}$ is actually optimal
for a large choice of parameters.

######  Theorem 1.

Under Assumption 2, the policy $\pi_{o}$ is optimal (${\pi^{*}}=\pi_{o}$) for
the simplified RPM (Definition 1) when the parameters satisfy

$\gamma(\lambda_{i}-\lambda_{o})(1-\phi^{2})\leq\frac{C_{i}}{C_{c}}.$

###### Proof.

See Appendix A. ∎

We next define a threshold-policy $\pi_{t,\bar{h}}$ characterized by the
health state ${\bar{h}}>0$. These are the policies under which there exists a
threshold ${\bar{h}}$ such that the patient stays in intensive monitoring when
the patient’s health is below or at the threshold ${\bar{h}}$ and in ordinary
monitoring when their health is better than ${\bar{h}}$. Note that
$\pi_{t,\bar{h}}\in\widetilde{\Pi}$. So $\pi_{t,\bar{h}}(h)=i$ for $1\leq
h\leq{\bar{h}}$ and $\pi_{t,\bar{h}}(h)=o$ for $h>{\bar{h}}$. Our next theorem
gives a set of conditions under which $\pi_{t,\bar{h}}$ is the optimal policy
for some threshold ${\bar{h}}$.

######  Theorem 2.

Under Assumption 2, the policy $\pi_{t,\bar{h}}$ is optimal
(${\pi^{*}}=\pi_{t,\bar{h}}$) for some threshold ${\bar{h}}$ for the
simplified RPM (Definition 1) when the following two conditions are satisfied:

  1. (a)

$\gamma(\lambda_{i}-\lambda_{o})(1-\phi^{2})>\frac{C_{i}}{C_{c}}$.

  2. (b)

$\frac{\gamma\mu_{o}(1+\gamma\mu_{o})}{1-\gamma^{2}\lambda_{o}\mu_{o}}\leq 1$.

###### Proof.

See Appendix A. ∎

Condition a) here is precisely the complement of the condition in Theorem 1.
Condition b) is an additional condition which our proof requires for the
threshold policy to be optimal. In the asymptotic regime, we strongly believe
that condition b) is not necessary, and condition a) alone is sufficient.
Hence we believe that in the asymptotic regime, condition a) alone dictates
what the optimal policy is and that the optimal policy can only be of two
forms - $\pi_{o}$ and $\pi_{t,\bar{h}}$. This is reinforced by the numerical
analysis, presented in the next section.

##  IV Performance

In this section, we glean insights on the optimal policy by numerically
solving the dynamic programming equations given by (1)-(3) to find the optimal
policy.

Figure 2 depicts the two policies discussed in the last section and a sample
set of parameters under which they are optimal. Figure 2(a) shows the policy
$\pi_{o}$, under which the patient stays in ordinary monitoring at all health
states. Figure 2(b) shows the policy $\pi_{t,\bar{h}}$ with threshold $t=3$,
where the patient stays in intensive monitoring for health states $h\leq 3$
and in ordinary monitoring for health states $h>3$. Let model 2(a) use the set
of parameters
$\lambda_{o}=0.2,\lambda_{i}=0.3,C_{c}=20,C_{i}=1,C_{o}=0,\gamma=0.9$ and
model 2(b) use the set of parameters
$\lambda_{o}=0.2,\lambda_{i}=0.3,C_{c}=60,C_{i}=1,C_{o}=0,\gamma=0.9$. Then
$\pi_{o}$ and $\pi_{t,\bar{h}}$ with $t=3$ are optimal for model 2(a) and
2(b), respectively.

Note that the parameters for model 2(a) satisfy
$\gamma(\lambda_{i}-\lambda_{o})(1-\phi^{2})\leq C_{i}/C_{c}$, which is
sufficient for Theorem 1 to hold. Similarly the parameters for model 2(b)
satisfy $\gamma(\lambda_{i}-\lambda_{o})(1-\phi^{2})>C_{i}/C_{c}$, which is
condition (a) in Theorem 2. Note, however, that the parameters in model 2(b)
do not satisfy condition (b) of Theorem 2, implying that the condition is not
necessary.

When $H$ is finite, there also exist instances where the optimal policy is
$\pi_{i}$, where the patient chooses to stay in intensive monitoring for all
health states $h$. But we observed that this policy is optimal only in extreme
cases where $H$ is very small or $\gamma$ is very close to $1$. Hence we do
not further analyse this policy here.

Figure 3: The optimal (blue) value function $V^{*}(h)$ (numerically computed)
compared to its asymptotic counterpart (red, $V_{o}(h)$ obtained from Lemma 1)
for various health states $h$ in a system with
$H=5,\lambda_{o}=0.2,\lambda_{i}=0.4,C_{c}=5,C_{i}=1,C_{o}=0,\gamma=0.9$. Note
the close proximity of the two plots.

(a) Variation in optimal policy with $C_{c}/C_{i}$

(b) Variation in optimal policy with $\lambda_{i}$

(c) Variation in optimal policy with $\gamma$

Figure 4: Dependence of the (numerically computed) optimal monitoring policy
on the (a) cost ratio $C_{c}/C_{i}$ (with fixed
$\lambda_{o}=0.2,\lambda_{i}=0.4,C_{i}=1,C_{o}=0,\gamma=0.9$); on (b)
$\lambda_{i}$ (with fixed
$\lambda_{o}=0.2,C_{c}=50,C_{i}=1,C_{o}=0,\gamma=0.9$); and on (c) $\gamma$
(with fixed $\lambda_{o}=0.2,\lambda_{i}=0.4C_{c}=50,C_{i}=1,C_{o}=0$). We set
$H=10$ in all cases. Below the vertical orange dashed line, the optimal policy
is $\pi_{o}$ (ordinary monitoring is used for all health states). This line is
positioned at the point where the condition of Theorem 1 achieves equality.
Above this value the policy changes to $\pi_{t,\bar{h}}$ and each threshold
$\bar{h}$ is marked in blue. Note that the threshold $\bar{h}$ is increasing
as $C_{o}/C_{i}$ gets higher in (a), i.e., the patient stays “longer” under
intensive monitoring. The same behavior is seen in (b) and (c).

Figure 3 shows how closely our asymptotic analysis in Section III relates to
the actual solutions of the dynamic programming equations. The model
parameters here are chosen such that the optimal policy is $\pi_{o}$. We
calculate $V^{*}(h)$ for this model, and compare it with the value function
$V_{o}(h)=\phi^{h}C_{c}$ obtained for the asymptotic case $H\to\infty$ (Lemma
1). As observed in the plot, the value function obtained for $H=5$ is almost
identical to the asymptotic approximation $V_{o}(h)$, and they slightly differ
only near the top health state $H$. The accuracy of the asymptotic
approximation in predicting the optimal policy is further demonstrated by our
next result.

We next study the impact of different parameters on the optimal policy
${\pi^{*}}$. Figure 4(a) shows how the optimal policy (numerically computed)
varies with the cost ratio $C_{c}/C_{i}$. For $C_{c}/C_{i}<20.02$ the optimal
policy is $\pi_{o}$, while for $C_{c}/C_{i}>20.02$ it is $\pi_{t,\bar{h}}$
with varied thresholds. Note that $C_{c}/C_{i}=20.02$ satisfies the condition
in Theorem 1 with equality (the value of $C_{c}/C_{i}$ which satisfies
$\gamma(\lambda_{i}-\lambda_{o})(1-\phi^{2})=C_{i}/C_{C}$). This shows that
the condition obtained under the asymptotic assumption is a good indicator for
our original problem with finite health states. As the ratio $C_{c}/C_{i}$
grows, the cost incurred on reaching the critical health state increases as
compared to the cost of invasiveness, and it gets optimal for the patient to
stay under intensive monitoring till their health significantly improves.

Figure 4(b) shows how the optimal policy varies as the probability
$\lambda_{i}$ increases. The optimal policy is $\pi_{o}$ for
$\lambda_{i}<0.28$ and $\pi_{t,\bar{h}}$ with varied thresholds for
$\lambda_{i}>0.28$. Again, $\lambda_{i}=0.28$ solves the condition in Theorem
1 with equality. As the probability $\lambda_{i}$ increases, the probability
of the patient’s health improving under intensive monitoring improves,
incentivizing the patient to stay under intensive monitoring for longer.
Finally, Figure 4(c) shows the impact of $\gamma$ on the optimal policy. As
$\gamma$ increases, the patient incurs a higher discounted cost on reaching
the critical state, and hence they stay under intensive monitoring for a
broader range of states.

##  V Conclusions and Extensions

We have developed a two-tier service architecture for remote patient
monitoring (RPM), where the service policy decides whether to place the
patient under ordinary or intensive monitoring, given their health state. The
optimal policy is first analyzed in asymptotic regimes and conditions are
established for choosing ordinary vs intensive monitoring. The policy is then
numerically computed and the dependence of its behavior on various key
parameters is investigated.

An important extension would be to consider the general monitoring model,
which includes non-zero transition costs. Based on numerical experiments
performed in the general case, the optimal policy in this case would be a
threshold policy with two thresholds instead of the one observed in this
paper. A patient under ordinary monitoring would be switched to intensive when
their health state deteriorates below a certain lower health threshold, and a
patient under intensive monitoring would switch to ordinary when their health
state improves above an upper health threshold.

There are many other direct extensions that we could not include in the
limited space of this short paper. One can consider a service architecture
with more than two tiers, where a patient can be under ordinary monitoring
(tier m=0), or intensive monitoring tier m=1, or (more) intensive monitoring
tier m=2 and so on, with corresponding health care features and attributes.
The costs and probabilities of transitions could also be made dependent on the
health state, allowing for a more realistic model.

## References

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##  Appendix A Proofs

###  A-A Proof for Lemma 1

###### Proof.

  1. (a)

For the policy $\pi_{o}$, the value function is

$V_{o}(h)=\mathbb{E}\left[\sum_{t=0}^{T-1}(\gamma^{t}c(s_{t},\pi_{o}(s_{t})))+\gamma^{T}C_{c}|h_{0}=h\right].$

Here $T$ denotes the time at which the patient reaches health state $0$. Since
$\pi_{o}(h)=o$, $c(s_{t},\pi_{o}(s_{t}))=0$ for all $t<T$. This implies that
$V_{o}(h)=C_{c}\mathbb{E}[\gamma^{T}|h_{0}=h]$, where $T$ is the time at which
the patient reaches health state $0$ and hence the hitting time of state $0$
for a random walk initialized at state $h$. Since $\mu_{o}>\lambda_{o}$, the
probability that the patient reaches health state $0$ under this policy is
$1$. Then [10, Chapter XIV, eqn. 4.8] gives us $V_{o}(h)=C_{c}\phi^{h},$ where

$\phi=\frac{1-\sqrt{1-4\lambda_{o}\mu_{o}\gamma^{2}}}{2\lambda_{o}\gamma}.$

  2. (b)

Let $h^{\prime}=\lceil\log(C_{i}/C_{c})/\log(\phi)\rceil+1$. Then for any
$h\geq h^{\prime}$, $V_{o}(h)\leq V_{o}(h^{\prime})\leq\phi C_{i}$. Consider a
policy $\pi^{\prime}$ such that $\pi^{\prime}(h)=i$. Then
$V_{\pi^{\prime}}(h^{\prime})\geq C_{i}$. Now for any $h\geq h^{\prime}$,
$V_{o}(h)<V_{\pi^{\prime}}(h)$ as $\phi<1$ for $\gamma<1$. This implies that
the policy $\pi^{\prime}$ cannot be optimal, and hence under the optimal
policy, the patient prefers to stay in ordinary monitoring for all health
states $h\geq h^{\prime}$. This completes the proof for Lemma 1.

∎

###  A-B Proof for Theorem 1

###### Proof.

We know that a policy $\pi$ is optimal if and only if

$\displaystyle
c(s,\pi(s))+\gamma\sum_{s^{\prime}}p(s^{\prime}|s,\pi(s))V_{\pi}(s^{\prime})$
$\displaystyle=\min_{a}\left(c(s,a)+\gamma\sum_{s^{\prime}}p(s^{\prime}|s,a)V_{\pi}(s^{\prime})\right),$

is true for all states $s$ [9, Proposition 2.2 and 2.3]. Hence policy
$\pi_{o}$ is optimal if and only if

$\displaystyle\gamma(\lambda_{o}V_{o}(h+1)+\mu_{o}V_{o}(h-1))$
$\displaystyle\leq C_{i}+\gamma(\lambda_{i}V_{o}(h+1)+\mu_{i}V_{o}(h-1)),$ (4)

for all $1\geq h$. Using Lemma 1, we have that $\pi_{o}$ is optimal if and
only if, for all $1\leq h$,

$\displaystyle C_{c}\gamma(\lambda_{o}\phi^{h+1}+\mu_{o}\phi^{h-1})\leq
C_{i}+C_{c}\gamma(\lambda_{i}\phi^{h+1}+\mu_{i}\phi^{h-1})$
$\displaystyle\iff$ $\displaystyle
C_{c}\gamma\phi^{h-1}((\lambda_{o}-\lambda_{i})\phi^{2}+(\mu_{o}-\mu_{i}))\leq
C_{i}$ $\displaystyle\stackrel{{\scriptstyle(b)}}{{\iff}}$ $\displaystyle
C_{c}\gamma\phi^{h-1}(\lambda_{i}-\lambda_{o})(1-\phi^{2})\leq C_{i}.$

Here (b) follows from the definition that $\mu_{o}=1-\lambda_{o}$ and
$\mu_{i}=1-\lambda_{i}$. Note that if

$\gamma(\lambda_{i}-\lambda_{o})(1-\phi^{2})\leq\frac{C_{i}}{C_{c}}$

is true, then (A-B) is satisfied for all $h\geq 1$ which implies that the
policy $\pi_{o}$ is optimal. This completes the proof for Theorem 1. ∎

###  A-C Proof for Theorem 2

###### Proof.

Since $\gamma(\lambda_{i}-\lambda_{o})(1-\phi^{2})>\frac{C_{i}}{C_{c}}$, eqn.
(A-B) is not satisfied for $h=1$, and hence $\pi_{o}$ is not optimal, which
implies that staying under intensive monitoring is optimal for some state.
Define $Q^{*}(h,i)=C_{i}+\gamma(\lambda_{i}V^{*}(h{+}1)+\mu_{i}V^{*}(h{-}1))$
and $Q^{*}(h,o)=\gamma(\lambda_{o}V^{*}(h{+}1)+\mu_{o}V^{*}(h{-}1))$. This
denotes the Q-functions, where action $i$ (or $o$, respectively) is taken when
initialized at health state $h$ and then actions are taken using the optimal
policy. Note that ${\pi^{*}}(h)=o$ if $Q^{*}(h,o)<Q^{*}(h,i)$, and
${\pi^{*}}(h)=i$ otherwise. If we show that $Q^{*}(h,o)-Q^{*}(h,i)$ is
monotonically decreasing as $h$ increases, then if action $o$ is optimal at
some $h$, then it will also be optimal at $h+1$ and so on. As condition (a)
already enforces that policy $\pi_{o}$ is not optimal and we have shown in
Lemma 1 part (b) that there exists $h^{\prime}$ such that the optimal policy
for health states above $h^{\prime}$ is $o$, the optimal policy has to be
$\pi_{t,\bar{h}}$ for some $\bar{h}$. We can show that

$Q^{*}(h,o)-Q^{*}(h,i)=\gamma(\lambda_{i}-\lambda_{o})(V^{*}(h-1)-V^{*}(h+1))-C_{i}.$

Now,

$\displaystyle\big{(}Q^{*}(h+1,o)-Q^{*}(h+1,i)\big{)}-\big{(}Q^{*}(h,o)-Q^{*}(h,i)\big{)}$
$\displaystyle=\gamma(\lambda_{i}-\lambda_{o})\Big{(}V^{*}(h)-V^{*}(h+2)$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-V^{*}(h-1)+V^{*}(h+1)\Big{)}$

Hence if $V^{*}(h)+V^{*}(h{+}1)\leq V^{*}(h{-}1)+V^{*}(h{+}2)$ is true for all
$h\geq 1$ then $Q^{*}(h,o){-}Q^{*}(h,i)$ is monotonically decreasing with $h$.
Now we know using [9, Proposition 2.2] that

$V^{*}(h)\leq\gamma\lambda_{o}V^{*}(h+1)+\gamma\mu_{o}V^{*}(h-1),$

and

$V^{*}(h+1)\leq\gamma\lambda_{o}V^{*}(h+2)+\gamma\mu_{o}V^{*}(h).$

With some further manipulation, we can show that

$\displaystyle V^{*}(h)+V^{*}(h+1)$
$\displaystyle\leq\frac{\gamma\lambda_{o}(1+\gamma\lambda_{o})}{1-\gamma^{2}\lambda_{o}\mu_{o}}V^{*}(h+2)$
$\displaystyle+\frac{\gamma\mu_{o}(1+\gamma\mu_{o})}{1-\gamma^{2}\lambda_{o}\mu_{o}}V^{*}(h-1).$

We can show that
$\frac{\gamma\lambda_{o}(1+\gamma\lambda_{o})}{1-\gamma^{2}\lambda_{o}\mu_{o}}$
is always less than $1$ for $\lambda_{o}\leq 0.5$. Hence if
$\frac{\gamma\mu_{o}(1+\gamma\mu_{o})}{1-\gamma^{2}\lambda_{o}\mu_{o}}\leq 1$,
then $V^{*}(h)+V^{*}(h{+}1)\leq V^{*}(h{-}1)+V^{*}(h{+}2)$ and hence
$Q^{*}(h,o){-}Q^{*}(h,i)$ is monotonically decreasing with $h$. Under the
additional condition that
$\gamma(\lambda_{i}-\lambda_{o})(1-\phi^{2})>\frac{C_{i}}{C_{c}}$, this
implies that $\pi_{t,\bar{h}}$ is the optimal policy for some threshold
$\bar{h}$. ∎


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## Appendix

##  Appendix A Geographic Composition of the Indian Subcontinent

Scholars have different opinions about the geographic demarcation of the
Indian subcontinent (McLeod, 2015; Bhopal, 2007). Scholars agree that the
region consists “at least of India, Pakistan, and Bangladesh” (Zeidan and
Britannica, 2022). However, in that case, the term becomes subject to the
politics of nomenclature. While some sources describe the region using the
phrase “Greater India” (Vivekanandan, 2018), Pakistani national historiography
often champions the phrase “Indo-Pakistan subcontinent” (Mann, 2014). Again,
scholars who use the term “South Asia” interchangeably with the term “Indian
subcontinent” view Afghanistan, Bhutan, Nepal, Sri Lanka, and the Maldives as
parts of this region (Norwine and Gonzalez, 1988). However, some scholars have
argued against such views. According to them, Afghanistan is a central Asian
country (Masson et al., 1992; McLeod, 2015), and while Sri Lanka and the
Maldives have cultural similarities with the countries in the Indian
subcontinent, these island countries lack geographic contiguity with the
region (McLeod, 2015). In the case of Bhutan and Nepal, these countries used
to be British protectorates and not British colonies (Onley, 2009). Therefore,
in our study, we focused on Bangladesh, India, and Pakistan–three countries
that, scholars unanimously agree, are parts of the Indian subcontinent and
were subjugated together under British colonial rule till 1947.

##  Appendix B Local and Indigenous Identities in the Indian Subcontinent

Ways to conceptualize indigeneity and identify indigenous people vary across
different regions and are subject to emancipatory politics (Baviskar, 2006;
Eubanks and Sherpa, 2018). For example, in the Indian subcontinent, a region
with long history of human migration, though “indigenous” is usually used to
identify the tribal adivasi groups (e.g., Bhil, Munda, Santhal, etc.)
(Britannica, 2021), non-tribal groups like the Bengalis, Gujratis, and Oriyas,
also have a long history of settlement (Xaxa, 1999). To inclusively recognize
these communities who lived in the region since long before the British
colonization, following prior work (Das and Semaan, 2022a), we are using the
phrase “local and indigenous” communities.