Unified Screening for Multiple Diseases
Authors: Yiğit Narter
Alihan Hüyük
Mihaela van der Schaar
Cem Tekin
Venue: ICML 2025 poster (2025)
Keywords: Unified screening
healthcare
competing risks
constrained optimization
threshold policy
Primary area: applications->health_medicine

Abstract: Current screening programs that focus on improving patient health while minimizing screening costs are tailored for individual diseases. Designing unified screening programs for multiple diseases requires carefully balancing competing disease risks, which is an open problem. In this work, we address this problem by casting unified screening as a referral problem, in which we choose to activate a subset of screening policies for individual diseases by accounting for competing risks that influence patient outcomes. We introduce a novel optimization framework that incorporates disease risks, budget constraints, and diagnostic error limits and characterize the structural properties of the optimal referral policy. For the unified screening of two diseases, we show that the optimal activation threshold for the screening of one disease depends on the risk of the other, resulting in decision boundaries with distinct risk-dependent profiles. We compare our unified model with independent screening programs that apply isolated activation thresholds for screening of each disease. Our approach optimizes screening decisions collectively, improving overall survival outcomes, particularly for patients with high disease risks.

Unified Screening for Multiple Diseases
Yigit Narter' Alihan Hüyük? Mihaela van der Schaar
Cem Tekin
Abstract
(De Mutsert et al., 2009; Weiner et al., 2006; Thayer et al., 2010). This limitation reduces the efficiency of healtheare interventions, as potential synergies between diseases are overlooked, and resources may be misallocated. Screening a patient with high risks for two conditions for only one con- dition might fail to improve their overall health outcomes, whereas a unified screening approach could yield better re- sults (Fan et al., 2024: Huang et al., 2023; Yang et al., 2006) by addressing the competing risks, which refer to events that interfere with or alter the likelihood of observing the event of interest (Pintilie, 2011).
Current screening programs that focus on improv- ing patient health while minimizing screening costs are tailored for individual diseases. De- signing unified screening programs for multiple diseases requires carefully balancing competing disease risks, which is an open problem. In this work, we address this problem by casting uni- fied screening as a referral problem, in which we choose to actívate subset of screening policies for individual discases by accounting for com- peting risks that influence patient outcomes. We introduce a novel optimization framework that in- corporates disease risks, budget constraints, and diagnostic error limits and characterize the struc- tural properties of the optimal referral policy. For the unified screening of two diseases, we show that the optimal activation threshold for the screen- ing of one disease depends on the risk of the other, resulting in decision boundaries with dis- tinct risk-dependent profiles. We compare our unified model with independent screening pro- grams that apply isolated activation thresholds for screening of each discase. Our approach opti- mizes screening decisions collectively, improving overall survival outcomes, particularly for patients with high disease risks.
For instance, consider a patient who has both cardiovascular disease and cancer but only screened for cancer, where the adverse event from cancer happens occur after the adverse event from heart disease. When screening for cancer, if the patient unexpectedly suffers from heart disease, then the cancer screening offers no benefit to the patient in terms of quality-adjusted life years (QALYs). However, if the patient is screened for both diseases, heart disease will be identified carlier. In that case, the adverse event due to heart disease will be prevented, and cancer screening will result in QALY gain by detecting cancer before the adverse event associated with it happens. However, screening the patient for both diseases is costly. Hence, optimal allocation of screening resources to patients with different risk profiles is an important task.
In this paper, we propose a novel framework for design- ing unified screening programs that simultaneously address multiple diseases. Unlike existing programs, our approach accounts for the competing risks associated with each dis- ease when determining whom to screen for which condition while also considering resource constraints, an important factor in screening programs (Cevik et al., 2018; Bansal et al., 2020; Teh et al., 2015). We introduce an optimization formulation for the screening problem, which incorporates budget constraints and diagnostic error limits. Using proper- ties of the Lagrange dual function, we characterize optimal decision boundaries and validate our findings through ex- tensive in-silico experiments. In particular, for the case of unified screening of two diseases, we show that the opti- mal activation threshold for one disease is not static but a function of the risk of the other disease. Instead of sin- gle threshold per discase, our model introduces multiple regions that correspond to distinct combinations of screen-
1. Introduction
The challenge of designing effective screening programs for disease detection is central to improving healthcare out- comes. Traditionally, screening programs developed in isolation for individual diseases, ignoring the interac- tions between diseases and risks for different conditions
'epartmennt of Electrical and Electronics Engineering, Bilkent University, Ankara, Turkey 'Department of Com- puter Science, Harvard University. Cambridge, MA, US Unniversity of Cambridge, Cambridge, UK. Correspondence to: Yigit Narter arggginnaatreeuggglll..t........ bilken...lllttkk.r..r. Cem Tekin <cemtekin @ ce. ww..bikeet..c..rr...
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).

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Unified Screening for Multiple Diseases
ing activations. These regions emerge due the interplay of competing risks, where the decision to screen for one disease depends not only on its individual risk but also its interaction with the risks of other diseases. This also results in decision boundaries with curvy shape, which reflects the interplay between the risks of the diseases.
kmy)
C:
0_(xal
Our experimental results demonstrate that the mathematical characterizations of the optimal policies align closely with the numerical solution of the convex program (see Figure 1). Importantly, we compare the performance of our unified screening model against that of independent screening pro- grams. Independent screening programs operate under the assumption that diseases are unrelated and apply distinct threshold policies for each condition. Specifically, they ac- tivate screening for a disease if the patient's risk for that disease exceeds predetermined threshold, independent of the risks for other conditions. In contrast, our unified model accounts for the interplay between diseases and optimizes the screening policies collectively rather than in isolation. This approach enhances survival times for patients whose risk profiles indicate significant interactions between dis- eases, particularly in scenarios where one disease's 's optimal threshold influences the other's. For instance, in cases of competing risks (Wolbers et al., 2014; Pintilie, 2011; Allig- nol et al., 201 1), addressing one condition indirectly benefits the other, while in cases of individual risks, prioritizing the most pressing condition ensures optimal resource allocation. Our results show that, when compared with the indepen- dent screening model, the unified model prioritizes patients with higher relative risks for specific disease, activating screenings for these individuals in cases where independent screening would not (as shown by the colored regions in Figure I(c)). Conversely, it reduces screenings for patients with moderate risks for both diseases, resulting in grey re- gions in Figure I(c). In essence, the model places greater emphasis on addressing higher relative risks, which leads to better health outcomes for these patients (see Figure 2).
(a)
(h)
(c)
Figure 1: Optimal referral policy for N = 2 under uniform risk distribution for (a) independent screening model, (b) our unified screening model. Grey:
1 (no screening). Pink: a 2 (screening only the first disease). Purple: a = (screening only the second discase), Red: a = 4 (screening both discases), Notice that unlike in the independent case with constant thresholds, the activation threshold for first discase (to the pink and red regions) is curve which is in fact function of 22, the risk of the other disease. The difference between two policies given (c).
Untfled Hotel
Indezerdert Vtdel
Oue al Meen 1D deddi 37.10
Overal Mean (ideepandent)
Ar Taal Use Brral chaness)
Figure 2: Average survival times (in years excess of 50) with respect to 21 (the first disease risk) for both the unified and inde- pendent screening programs computed by taking the mean survival times over all z2 values (the second disease risk), in the case of screeming for two discases, along with the mean survival times for both programs indicated by the horizontal dashed lines. Details are provided in Section 5.
Fernandez et al., 1991; Yu et al., 2021; Xiao et al., 2022; Idrees et al., 2021; Wang et al., 2024b) and screening policy frequency (Hemmati et al., 2024; Wu & Suen, 2022; Wang et al., 2024b; Wu et al., 2024). Most of the existing studies focus on Markov models and processes (Wu et al., 2024; Wu & Suen, 2022; Zhang al., 2012a;b; Maillart et al., 2008). Kamalzadeh et al. (2021) propose framework combining partially observable Markov Decision Processes (POMDP), Hidden Markov Models (HMM), and Predictive Risk Mod- eling (PRM) to design individualized diabetes screening policies. To optimize decision making healthcare, Bertsi- mas et al. (2018) propose a framework for prostate cancer screening that identifies strategies effective across multiple mathematical models using heuristic search.
Our unified screening framework represents significant step forward in improving the design and efficiency of screening programs, ensuring better health outcomes in resource-cone constrriin environments. In silico experiments for the unified screening of two diseases demonstrate that survival times improve compared the optimal individ- ual screening programs, validating the benefit of unified screening (see Figure 2).
2. Related Work
From a medical perspective. prior research has predom- inantly focused on optimizing sampling schedules or decision-making processes for single-disease screenings. These strategies often aim to determine or evaluate inclu- sion criteria (Tomaszewski et al., 2022; Bauer et al., 2015;
Costeeffectiveness analyses have also been a critical com- ponent of the literature to identify the optimal trade-off be- tween screening costs and health benefits (Xia et al., 2024; Qin et al., 2022; 2024; Wang et al., 2024a; Bao et al., 2022).
2

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Unified Screening for Multiple Diseases
Table I: Comparison of Related Works.
Reference
Joint Screening
Multiple Diseases
Resource Constraints
Competing Risks
Yala et al. (2022)
No
No
Yes
No
Cevik et al. (2018)
No
No
Yes
No
Huang et al. (2023)
Yes
Yes
No
No
Wright et al. (2015)
No
No
No
Yes
Peng & Xiang (2021)
Yes
No
No
Yes
Alaa & Van Der Schaar (2016)
No
No
Yes
No
This Work
Yes
Yes
Yes
Yes
Resource constraints, such as limited budgets, personnel, and equipment (Yadla et al., 2024; Khoza-Shangase et al., 2017), play a fundamental role in these analyses, as they directly impact the feasibility and scalability of sereening programs (Cevik et al., 2018; Bansal et al., 2020). Cevik et al. (2018) tackle resource constraints by proposing constrained POMDP model to optimally allocate limited mammography screening resources, and prioritize patients based on their risk levels and screening capacity. Yala et al. (2022) present a reinforcement learning-basse framework, which optimizes resstrce-constrained breast cancer screen- ing by AI-driven risk models, enhance early detection and minimize excessive screening. However, to the best of our knowledge, none of the existing methods address both resource constraints and competing risks of multiple diseases simultaneously, which leaves gap in optimizing multi-disease screening programs under such conditions.
et al. (2023) discuss that combined cancer screening would enhance cost-effectivenesss and improve early detection through shared risk factors. Regarding the screening for multiple diseases, Yang al. (2006) evaluate the integration of colorectal cancer (CRC) screening into multi-disease program through Markov model. Their study projects that multiple disease screening reduces CRC mortality by 33%, which outperforms single-disease screening (23% reduc- tion). Huang et al. (2023) implement combined cancer screening program for colorectal, breast, lung, liver, and stomach cancers with C-BLAST trial, using a multi-stage design.
Our work on designing disease screening programs, which is about making decisions based on accumulating evidence to optimize outcomes under uncertainty, is closely related to sequential hypothesis testing (Chernoff, 1992; Drugowitsch et al., 2014; 2012; Shvartsman al., 2015). With multiple diseases, the general problem an instance of sequential hy- pothesis testing with multiple hypotheses. Prior work, such as by Alaa & Van Der Schaar (2016) examines resource allo- cation with Bayesian model in single-hypothest (disease) scenarios but does not extend to the complexities of multi- disease contexts. However, in our context, multiple diseases correspond to multiple hypotheses, which transforms our problem into active learning challenge, where actions influence the acquisition of information across discases.
A competing risks problem in our approach refers the assessment of the likelihood of health benefits from in- tervention, considering the presence of other potential out- comes that may interfere with or prevent the occurrence of the event of interest (Varadhan et al., 2010). This ap- proach enables the analysis of both the timing of the first observed event and its type (Wolbers al., 2014). Compet- ing risk analyses have been tackled in the literature (Allignol et al., 2011; Wolbers al., 2014; Satagopan et al., 2004; Pintilie, 2011; Cho et al., 2022), which highlights its im- portance. Stenling et al. (2020) employ competing risks approach to develop lifetime risk models for cardiovaseular events, while Wright et al. (2015) develop a competing risks model for screening preeclampsia by incorporating maternal demographic characteristics and medical history. In semi-competing risks framework, Peng & Xiang (2021) propose joint feature screening method that is based on the correlation ranking of gene features, to address challenges in Sttaaigh--dimensinn breast cancer data analysis.
Different than all of the works listed, the framework de- veloped in this paper integrates the optimization of referral decisions with probabilistic decision-making. The proof techniques illustrate how boundaries in the decision space can be derived, connecting to geometric approaches found in threshold-based medical screening policies. By incor- porating stochastic constraints, approach generalizes these methods into probabilistic framework that adaptively balances performance metrics, such as survival time and cost. Table compares our work with the closely related works. See Appendix for more detail on related works.
Several studies highlight the benefits of joint screening pro- grams. Fan et (2024) evaluate joint screening for prostate, lung, colorectal, and ovarian cancer, and demonstrate that joint cancer screening reduces all-cause and all-neoplasm mortality, which makes it promising strategy. Huang
3

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Unified Screening for Multiple Diseases
3. Problem Formulation
1
T1
Y2
3.1. The Unified Screening Problem
Diseases and Adverse Events. We consider the screening of N diseases. Patients have latent disease states represented by random variables , E {0, 1). n E [N]. The disease state vector is defined Q 1 ........ .....
X
nn
θx T If , 1, the patient experiences adverse event related to disease at random time Tn E N..' If O = O, no such event occurs, and n = co. The pmf for Tn conditioned on O is given as
ON
Tn
YN
Figure 3: Bayesian network of (X, e YY..
1, (r.(1). 0,
if O₂ 0 and I = -0, if 0, 1 and t N+, otherwise.
Pr. 0n)
Ih= PT. (tn (n). PTIx I PT. xx (bnlx)), (r],, 0) = pTo((10)
I PT,,e, (tn 10,) (V)) = py..(t) (Vn). In addition, we assume that given O screening samples taken at different times are independent: (t) TL ((t') ( for t' ft.
Here, (t) is probability distribution (e.g., geomet- ric) defined on N,. The CDF of Tn given θ, = o, is FT. e, (m)))) = P(T, S e = 0.). The random vector T= T represents the adverse event times for all diseases.
Screening and Diagmostic Actions. For each patient, the decision-maker first observers the risk vector x. Then, it needs to determine the disease states θ by observing costly samples of the screening targets Y, before the adverse events associated with the diseases occur. At each t, they take two types of actions. The first is the screening action 6(x, t) E {0, 13. If (x, t) = 1, then the decision-maker observes the nith screening target (t). If 8n (z. t) = O, then they do not observe the nith screening target. We de- (x, t) 1}| the
Risks and Screening Targets. Each patient has a risk vector X = (X Xv]" E 1]", where X; denotes the prior probability of the patient having disease i.2 For n E [M]:
Ens
if fn = 1, In. if 0,
P(O,- = 0n X =x} =
1-Ins
note with # T) l(t [0, 6, total number of observations taken from the nath screen- ing target up to time T. The second is the diagnostic action 0(x. t) E 1)N. The first time point when è, (z. t) = 1 indicates that the ceeiision-maker has diag- nosed the patient with the nth disease. We denote with t (z) = minft E RR, (I, t) = 1} the diagnosis time for the rath disease for patient with risk vector x. When the dependence on is clear from the context, we will drop the argument inside the paranthesis from the notation, and use 8, (t), 6, (t), t etc.
The distribution of risk vector over patient population is given by (z). There are N screening targets, each of which is associated with one disease. The with screening target Y, : I > R is a random function with (t) repre- senting its value at time L. The conditional distribution of (t) given O, is (o)ee, (ym)6) (.).. The vector random target functions is represented as Y = ,Y',..., YN]".
Probabilistic Dependencies. The random elements in the system can be represented as tuple (X, O, T, Y). The prob- abilistic relationships between these random elements are encoded in the Bayesian network given in Figure 3, which implies the following factorization for the joint distribution of (X, O, TYY): Px,e,T,y (z. f, = Px (x)Pe) (# <) O) (t | 0, w). Moreover, the following conditional independence statements hold as a result of d- separation: xeix (0 = 11 Ix (0m12)) (1)0)
Decision-making Policies. The decision-maker takes screening actions and diagnostic actions i according to some screeming policy 8 and some diagnostic pol- icy år respectively, which constitutes a complete pol- icy all of the information available at time T to the screening policy and Iz(T) = Is(T) u {8(T) the diagnostic policy. Clearly, 8(t) = (t) = (t()))((tt) and O(t) = @=(t) = 0- (1, Iz()). Let IT represent the set of all joint screening and diagnostic policies.
(ŌT, Denote with IT') = {X} U(6(() heett))
'A time unit correspond to day, month, year, etc. "One can use off-the-shelf risk prediction models to obtain dis- case risks for particular discases. For instance, the Gail model (for breast cancer), QRISK3 (for cardiovascular discase), (normalized) polygenic risk scores, and even Al-based models can used to provide risk scores.
'y, may represent a biomarker value, clinical metric, sim- ilar quantity related to discase n. The randomness in Y, be attributed to measurement noise or natural biological fluctuations.
Objective. Each sample from the screening target n E [N] has a fixed cost of Cn E R4. For instance, Gn can represent

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Unified Screening for Multiple Diseases
the cost of mammography or chest X-ray. We say the nith disease is diagnosed on time if and only if (iff)
This leads to the following composite policy:
1 if - = (2r, 1,
On (O, = 1 < T.
(x,11,)
(0 otherwise.
Let T be the time point when the first adverse event that is not diagnosed on time occurs such that T = min(Tn n E -O Then, the objective of the decision-maker is to maximize T up to some survival time To E N+ subject to constraints on budget and diagnostic error rates, which is formalized a joint screening and diagnosis problem:
[1 if Pn(x))= = 0 (ar, (1, =1,
(x(,t)
otherwise.
This composite policy samples the nith target when screen- ing for disease n is active and the policy for that disease asks to screen for that target.
maximize E[min (To,
Let 11 C II represent the set of all referral deci- sions given individual screening and diagnostic policies for all diseases. As a subset of the referral decisions, we define the independent referral decisions as p(X) [ár (21), (2(2)), ................................ PN xx)) where P. 's are indepen- dent from 24, Vi * n. In this case, unlike the general re- ferral decisions where the activation of disease screening depends on the risk vector x, in the independent referral decision, the activation of the nith disease depends on only Ine Let Ilind C I1ref denote the set of independent referral decisions.
S.t. E ME Gn x (min Tn, [ & B,
p(O,- 0, 8, (t) = 1} S an. Vt [To], n E [M]
π E II,
where B is the total screening budget and a, is the maxi- mum allowed false positive rate for disease n. The expecta- tions are taken over the randomness induced by interaction of π with (X, O.Y,T).
3.2. The Referral Problem
To represent all possible combinations of screening de- cisions for N diseases as integers, we define the action a(p) : {0, 13N - AA, where A = 21, 2 mapping referral decisions to integer indices. In the case of N 2, the actions a 1, 2, 3, 4, correspond to the following re- ferral decisions respectively: nothing is screened, only the first disease is screened, only the second disease is screened, and both discases are screened. Lastly, we define g(a (z) as the probability of selecting action (and consequently its corresponding referral decision p), given a patient's 's risk vector x. Hence for any x, we have q(alz) 2 0, Va E A and
The referral problem is about determining the optimal screening and diagnostic policy tor given a set of existing policies, rather than creating completely new policy from scratch. This is crucial because the joint optimization approach in (I) can yield complex policies, which are difficult to implement in real-world clin- ical practice. Moreover, new policies are hard to justify in clinical guidelines, especially if they are entirely novel, and implementing novel clinical guidelines is another challenge due to many factors, including legislation (Wang et al., 2023; Francke et al., 2008; Correa et al., 2020). On the other hand, combining existing policies in an optimal way, as we do by solving the referral problem, is feasible and easier to support in practice (Stoll & Norton, 2018). Our approach, which optimizes the referral process, provides a lower bound on the performance of (1).
under each risk vec-
=
We formalize the problem of finding the optimal referral decision p", which provides lower bound on the optimal value of (1) as follows:*
for each disease E [N] is already provided. Each of these policies only ever diagnose their corresponding diseases such that 0% (x, t) = n'' # n for all æ e [0, 1]x8 and t E R.. To represent which policies are active for a given risk vector x, define the referral decision p: [0, 1] - {0, 13 such that p(x) = (p((x) ... PN (x)) where
Suppose individual policy n.
/ px(x)) ZEX
Σ [(alr))- -a,z dx AeE
maximizes
(2)
'If the goal were to learn the optimal referral policy directly from dataset of patient screening trajectories, the problem could indeed be framed a policy learning task with appropriately defined cost function. However, our work takes a complementary, orthogonal approach. We begin with a well defined optimization problem and focus on analytically characterizing the structure of its optimal solution. Rather than learning from data, our emphasis is understanding the geometry and properties of the optimal referral rule under known probabilistic model.
{
1 if the policy for disease n is active, otherwise.
Pa(x)
5

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Unified Screening for Multiple Diseases
time point that the vth sample is taken (Tn1 = 0), and S, is the maximum sampling number. Since we do not assume adaptive scheduling, Sn is deterministic. We have:
px(
Σ ((a/z) ma,e
dæ S B,
xE
WEEA
((0 2 0, Ya E A, Vo E.
s.t.
if (t) A E [0,t). - (t') = =0,
Σ g(a/xr) 1 Vx E X
5 (t) = 0 otherwise
(3)
aEA
P(O,- = 0, (t) = 1] 1 an Vt E [To) [M]
where (1) == Ei E (S.) :t= Tni-
where
Ta, = [ Ha, z]. ma,e
The diagnostic policy is to issue diagnosis only if the posterior probability of patient having the disease exceeds some threshold "/n E (0, 1) such that
Σn C, ( a, x and is the set of all actions.
4. The Optimal Referral Decision
if
In
LR,, 2
Yn
In order to characterize the structure of the optimal referral decision, we focus on the case of N = 2. representing two diseases and two screening targets corresponding to each disease. This choice motivated by the fact that screening for two diseases would be common and easier to adapt in clinical practice, whereas screening for three or more diseases simultaneously is much less frequent and often impractical (Ruhwald et al., 2022; Davis et al., 2015; Deverka et al., 2022; Castle, 2022). Additionally, the case of N = allows for a clear visual inspection of the results and provides insights that can be intuitively generalized to higher dimensions. On the other hand, characterizing the optimal referral policy becomes significantly more complex and far less intuitive when considering more than two diseases. One would expect the number of boundaries of interest (e.g., as shown in Figure 1(b)) to grow combinatorial with N, as one must account for boundaries that separate screening decisions across different subsets of discases. We assume that px is a uniform distribution over the risk vector domain [0, 1]
0T (t) =
1- In
1 Pn
(4)
otherwise
where (Rn = II [I::1. Plattm)l00,1100lmm (yn((re)) For (1, (T..))
this be well defined, assume [r, absolutely con- tinuous with respect pr, In this model, the diag- nostic decision is based on the disease probability reaching a certain level. Although it is hard to create a model, given the difficulty of capturing all the factors clinicians consider, a probabilistic approach is applicable for a diagnosis decision (Agency for Healthcare Research and Quality, 2022; Sox et al., 1989). This model maintains the false positive rate be- low a specific level, and essential aspect of this diagnosis policy is that patients are never diagnosed as "disease-free" given how there may exist medical conditions that may remain undiagnosed (National Institutes of Health (NIH), 2023; Tifft & Adams, 2014) due to their difficulty in being identified (Macnamara et al., 2020; Global Genes, 2023). Once screening begins, deterministic sampling continues until either the disease detected or an adverse event related to the disease occurs.
We restrict our attention to deterministic screening policies where the screening frequency remains constant over time. This includes periodic screening as special case. This choice is supported by the literature (Curry et al., 2018; Wolf et al., 2024; Canadian Task Force on Preventive Health Care, 2011; Eddy, 1989; Bhatia al., 2022; Shapiro, 1992), for example annual screening (Perros et al., 1995; Nagao & Warnakulasuriya, 2003), which highlights that periodic screening is (or should be) widely adopted in practice due to its health outcomes, simplicity and case of implementation. Adaptive screening policies, while potentially more flexible, are often more complex and require continuous monitoring and adjustments, which may not be feasible or cost-effective in many clinical settings.
First, note that the error constraint in (2) is satisfied for a suitable choice of 7/n- We can rewrite (2) as:
maximize, Σ ((alo)ra dr
(5)
ZEN'
do S B,
ZEX
s.t.
((alx)) 2 0, Va [4], E X Σ d(aa)) ( X.
For each disease (1,2), the screening policy provided is to take samples from the nith target (one target corre- sponding to each disease) with a fixed schedule until the patient is diagnosed with the disease (or an adverse event occurs, interrupting screening). Let (fri...., Tns.) } denote the sampling schedule for the rath disease, where Tue is the
Let p" represent the optimal value of this problem. The Lagrange dual function associated with the above problem takes the form
Σ ((ap))) A(ma. - B3) da
g(A) = max
ZE
6

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Unified Screening for Multiple Diseases
addition, bt satisfy the following properties: (i) (1) <
s.t. ((((a)) 2 0, Σ ((al)) = 1, E (4).
(6)
by (12) I by (x2) for all x2 [0, (ū) by (0) bt (0) b- (1); (iii) (0 = (1) S (x for all 22 E [0, 1].
where A 2 0. The Lagrange dual function provides an up- per bound for the problem, i.e., g(A) 2 p². Given x, we pay attention to the term Σg=1 (((a)z))n (zr) in (6), where (z(z) == Ta,z am This term is maximized when q(a" [z:) = 1 for some a" arg m tha (x).. The La- grange dual problem associated with (5) takes the following form: minimize (A) s.t. À 2 O. Let X represent the dual optimum. Since (5) is a feasible LP, strong duality holds. Therefore, g(>") = p". This implies that the optimal policy for (5) takes the form a" (ar) E arg max Ta. X Ma,2r X.
Combining the lemmata above we characterize the form of the optimal referral decision as follows.
Proposition 4.5. When b= (I2) is decreasing, and b- (x2) and bt (x2) have a unique intercept 22T, then the optimal referral decision pi has the form (i(x) = 1fx1 > Ci (122) where
bi (x2) if
12 E [0, [2]]
(1(22) = b=((z)
if
x2 E (121,
(7)
bt (122) if
02 E 1..
To determine the optimal referral decision, we will consider the referral decision pi (ar) of the first disease. Intuitively, (x) = for x's such that k's associated with actions where the first disease is screened (a = 2 or 4) exceed the ones associated with actions where the first disease is not screened (a- or a = 3). A similar intuition holds for (ar) as well. Define sub-rule pi)(x) == ((<((z)= > (.))) ) The following proposition characterizes (ix) in terms of sub-rules (t)ss)..
Moreover, (0) = (1)), (1(0) S (1(x2) all x2 E 10.
Similarly, (2 = 1f22 > (2(11) where
6 (21) if x [0,011]
(2(x1)=
b=(21) if x (IT1)
(8)
(x() if x E [=IT,]]
such that (2 = C2))). <2(0)= S (e(x2) for all 21 E [0,
As can be seen from Figure 1 (bb), our characterization in Proposition 4.5 matches with the empirical observation ob- tained by solving the LP in (5).
Proposition 4.1. pi (x) p2)(2) V (pr)(x) A pL.(())
and (1 = P3.1 (x) V (p))(z) A paz(z)).
Proposition 1.1 gives complete characterization of when to screen for diseases 1 and 2. Next, we characterize a special case when each sub-rule in Proposition 4.1 takes an intuitive form. Given Ø2 E [0, 1]. for any (i. let
The assumptions that b=(x)) is decreasing in 22, and bg (x2) and (x:2) have a unique intercept are hard to prove due to the complicated nature of the gaps (x( - (x) be- tween actions 4-1 and 4-3 in terms of z. This stems from the fact that diagnosis depends on the posterior, not just the likelihood. We provide empirical evidence for these assumptions. We show monotonicity of b=(22) and unique intersection of b (12) and b- (a22) across wide range of parameters in Appendix C. If these assumptions are violated, the decision boundaries may not be as simple as the ones stated in Proposition 4.5. However, this does not diminish the practical applicability of our proposed joint screening program. Another intriguing task is to understand the unified effect of the risk vector on the optimal referral decision. To this end, we also characterize the optimal referral decision when the screening activation of each disease is restricted to depend only on the risk of that disease.
+o0, if (x) - (x) < Vz, E [0, 1] 0,
(1,( =
-00, if (e(x)) ky(x) > 0, Von E [0, 1]
{ E [0, 1] <x(x)) - (x) 0}. else.
The following three lemmas characterize the forms of the de- cision boundaries بT, b,, bt : [0, 1] + [0, [] } that correspond to sub-rules P2,1 P4,1, and P4.3 used to de- fine pi (x) The form of the decision boundaries for the sub-rules of (1() follows from the symmetry.
Lemma 4.2. For each I2 E [0. 1]. there is unique b1(22) E (0, 1). Thus, P2,(2)) = 1 (21 > by (x2)). More- over, b1 is an increasing function, i.e., (x2) < (1(2)) if x2 < z/2-
Proposition 4.6. Restricted to the set of independent re- ferral decisions pro (x) = po (x)) which depend solely on the variables related to disease n when deciding whether to screen, the optimal referral decision has the form (1
Lemma 4.3. Assume that if root of (x) - Ki (x) exists
for a given Ø2 E [0, 1), then it is unique. (0) < b= (0), (1) > (1). Thus, (z,( = 1fx1 > b (x2)) If (22) is decreasing in 22, then, bi (x2) and b (x2) have unique intercept 121 E (0, 1).
1 1[x1 > [11 and (1 1 > for some thresholds E E 10. When the budget allocated for screening of each disease under independent referral decision matches the budget allocations in the optimal unified referral deci-
Lemma 4.4. For each 32 E [0, 1]. there is a unique (x2) E (0, Thus, p1,(z)) = 1x > (x2)) In
sion, the thresholds satisfy (1( = min 10.1] Si (122) S
TTcchnica assumptions and proofs are given in the Appendix.
G S max Si (12) and So (0) S &a.
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Unified Screening for Multiple Diseases
As can be seen from Figure ((a), our characterization in Proposition 4.6 matches with the empirical observation for the optimal independent referral decision. The differences between the actions induced by the optimal referral deci- sions in Propositions 4.5 and 4.6 are illustrated in Figure I()). If the prior probability of having the other disease is zero (e.g., #2 0 in the case of N = 2), then the activation threshold of the current disease in the unified case is less than the value in independent case (Gn, (0) S Cn).
(a)
(b)
(c)
(d)
Figure 4: Varying the screening costs (C₂) for the discases with
other parameters fixed. (a) ci = 6, = 1, (b) c = 2, = (c)
e1 1, = 2, (d) ei = 1,02 = 6.
5. In-Silico Experiments
5.1. Setup
default budget B 10, B 5 and the individual screening costs c C2 1.
We consider N = 2 diseases and assume that the earliest start time of screening is 50 years of age which in line with the most common recommended age for screening pro- grams in the US, such as for prostate, breast and colorectal cancer (Smith et al., 2001; Zoorob et al., 2001). As another example, Ma et al. (2023) calculate life expectancy starting at age 50 for cardiovascular disease. For each iteration, sur- vival time To (in years) is given as 40, where we round the value of 36.5-year average life expectancy of people with high cardiovascular health (CVH) score starting from age 50, as determined by (Ma et al., 2023). Adverse event times Tn, n E [N] (in years) are generated from (M,, o²). In (Ma et al., 2023), the average life expectancy with low and moderate CVH is estimated as 27.3 and 32.9 years respec- tively, hence we take the average of these two as un = 30.1 for the mean adverse event times, and we set 0² = 1.6
At each Monte Carlo iteration, we simultaneously run M 200 independent simulations for each of the N = 11,000 pa- tient feature vectors x = (II, to estimate the expected outcomes under various screening actions. Specifically, for each x and action o, we compute empirical expecta- tions of the form (TT(TT) a, + M |CT (xr, a)), where h(T*) {Ti T" #n (min T [)), and TP (xr a) denotes the ith simulated outcome under action a. We fix the diagnosis threshold to ho 0.95 throughout.
5.2. Results
Optimal actions/referral decisions are presented in Figure 1(b) (refer to Figure 11 for a detailed view). In the plot, the characterizations can be observed, and therefore be ver- ified. We observe a symmetry for the activation bound- aries of the first and second disease, the non-increasing or non-decreasing properties of the sub-referral decisions as derived, also that (1()) = (1).
We set px (ar) x fieu( at, (3) (12; ar (3) where [Bem a, 33 the pdf of the Beta distribution. Here, we assume that X2 and X2 are independent, however, they are distributed with the same coefficients. When experi- menting with other joint distributions where X1 and X2 are not necessarily independent, we obtain nearly identi- cal results with similar boundary shapes. This distribu- tion is chosen for ease of implementation and to see the effects of varying a and B. The samples (,,)) are gener- ated by adding Gaussian noise to true measurement &n $x t) which is uniform distribution between 0 and 1, given that O, = 1,, and if O, = O, we set (z, t) = 0. The noise is sampled from a normal distribution with a standard devia- tion of a = 0.5, such that Y, (z. t)= = sn $,, (x, t) + (x(x, t),
A series of experiments was conducted with systematic variations in key parameters B, B,Ca, Hr, E[T,], a, 3, as presented in Appendix D.I. As an example, we observe that when we vary the individual screeming costs (c,) for diseases, an increase in the cost of screening for a particular disease leads the model show preference for referral decisions targeting only the alternative, lower-cost disease (see Figure 4). This behavior highlights the model's sensitiv- ity to cost constraints, as it prioritizes affordable screening options under limited resources.
where 8n (zx 1) ~ Uniform(O, 1) and en ~ NO, o²). We sample once and use s,, (t) = sn for all t. but the er- en (I., t) is sampled independently for each leading to (t) T1 (t') θ for t' * t.
Next, we examined the values of k; across the four policies, selecting the policy with the highest n, at each point. We generated decision boundaries by selecting policies with the largest values, and recreated the decision points found in our mathematical proof by locating intersections where one K, surpasses the others (see Figure 16b). A comparison of these boundaries, derived by selecting the maximum K; (per our theoretical framework) and those from LP solution, confirms that the boundary shapes are nearly identical (see
We assume that the screening policy for each disease when active screens periodically every 1 year, until an adverse event happens or a positive diagnosis is made. We take the
*Code is available at https: htt////ttttt www.ttt....ooeeeeeeeeeeeeeeeteee /nart/// Uni Screen.
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Unified Screening for Multiple Diseases
Figure 17).
To find the optimal independent referral decision, we modi- fied our framework to separately optimize the referral poli- cies for the two diseases. The budget is distributed equally to the two diseases such that B1 = B2 = 5, while the bud- get for our model is B = 10. The independent policies are then combined into a single referral decision using bitwise operations. In contrast, our model determines a joint referral decision by solving LP that simultaneously considers both diseases. The resulting plot can be seen in Figure I(a). As expected, the thresholds are constant as we had shown in the proof, and they lie between the minimum and maximum value of the unified screening thresholds, which leads to both gray (no screening) and colored (screening) points in the policy difference plot (Figure ((c)).
34
04
96
83 100
02
04
10
(a)
(b)
Figure 5: (a) Difference of survival times (EB [min 1" Ha, 2))
between unified and independent screening (unified indepen- dent) (h) points at which the survival time for unified screening exceeds independent screening.
considering these interactions, leading better resource allocation and survival/health outcomes.
We determine the survival times ra, for both models (see Appendix D.3 for details) to compare differences in survival outcomes (Figure 5). We also plot the average survival times for each 21 over all 22 's for better visualization, and deter- mine the overall mean of the survival times for both models, as displayed in Figure 2. We observe that the overall mean for the unified model is 37.70, whereas for the independent model, this is 37.47, which indicates overall better perfor- mance. Moreover, as detailed in Appendix E, we verify the statistical significance of our results through further Monte Carlo simulations, demonstrating that the observed survival benefits of the unified approach are robust and do not result from random variation.
6. Conclusion, Limitations, Future Research
We proposed a unified screening model that accounts for the interactions between multiple diseases, to address limi- tations of traditional independent screening programs. By incorporating multiple disease risks into single optimiza- tion framework, our approach enables efficient resource allocation under budget limits. Our experimental results demonstrate that the mathematical characterizations of opti- mal policies align with the numerical solution of the convex program. Additionally, the unified model increases the num- ber of joint screenings, improving scheduling efficiency and enhancing survival outcomes, particularly for patients with higher relative risks for one disease.
As observed in Figures 2 and 5, our model performs signif- icantly better than independent screening at points where the risk of one disease outweighs the other, particularly around the optimal activation threshold of the disease in the independent case. This demonstrates the effectiveness of prioritizing screenings based on individual risks while allowing for joint screenings when appropriate, as previ- ously discussed in the introduction. In the middle points of the grid where the two risks are moderate and close to each other, independent screening performs better. This is caused by the emphasis we put on the relatively high risks of diseases, where we allow for screenings as opposed to independent screening (see Figure 1(c)). Joint screenings (a = 4) also occur more often in the unified model (2363 vs. 1847), suggesting more efficient scheduling and improved survival for high-risk patients.
A limitation of our work is that it provided the theoretical characterization of the optimal referral decision under N = and uniform risk profile. Nevertheless, we studied the effect of non-uniform risk profiles in the experiments. Our work underscores the importance of considering competing risks and disease interactions in screening policy design. For future work, we aim to develop a detailed roadmap for implementing our model in clinical practice.
Our claims are based on the premise that screening is benefi- cial, and should be adapted when there are enough resources. In our case, the only detrimental effect of screening is a false positive, which can be controlled by adjusting the threshold for the likelihood ratio test. We agree that screening does not always offer benefits and sometimes can even be harmful, as in the case of overdiagnosis. A recent review paper for cancer screening (Bretthauer et al., 2023) argues that many common cancer screening programs do not significantly prolong life. Investigating optimal resource allocation by considering detrimental effects of screening left as future research.
Our results display the limitations of independent policies in addressing overlaps and interactions between diseases. Independent screening programs often miss opportunities to improve patient care by failing to consider how screening for one disease can indirectly benefit another. For example, treating one condition, such as heart disease, can enhance the effectiveness of screening for another, such lung cancer. The unified model overcomes these limitations by
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Unified Screening for Multiple Diseases
Acknowledgments
observable Markov decision processes. In Transforming Research into Action, pp. 75-89. INFORMS, 2011.
The work of Cem Tekin was supported by Turkish National Academy of Sciences Distinguished Young Scientist Award Program TÜBA- GEEA--EBBP-2023, and the Scientific and Tech- nological Research Council of Türkiye (TÜBÍTAK) 2024 Incentive Award.
Allignol, A., Schumacher, M., Wanner, C., Drechsler, C.,, and Beyersmann, J. Understanding competing risks: simulation point of view. BMC Medical Research Method- ology, 11:1-13, 2011.
Ampavathi, A. and Saradhi, T. V. Multi discase-pres peeeass-prediction framework using hybrid deep learning: an optimal pre- diction model. Computer Methods in Biomechanics and Biomedical Engineering, 24(10): 146-1168, 2021.
Impact Statement
This work aims to advance the field of optimization by addressing the problem of unified screening for multiple dis- eases. By developing an optimization model that integrates competing risks and proposes optimal referral decision, the research has the potential to improve healthcare decision- making and resource allocation. Once an implementation of these methods in clinical practice is achieved, healthcare systems could significantly improve early disease detection and treatment allocation, which can ultimately lead to better patient outcomes and reduced healthcare costs. The soci- etal benefits of this work include enhancing public health outcomes by enabling more accurate and efficient screen- ing strategies, particularly in ssource-constrainee settings. Moreover, the methodology could contribute to reducing health disparities by tailoring screening programs to patient- specific characteristics.
Areia, M., Mori, Y., Correale, L., Repici, A., Bretthauer, M., Sharma, P., Taveira, F,, Spadaccini, M., Antonelli, G., Ebigbo, A., et al. Cost-effectiveness of artificial intelligence for screening colonoscopy: a modelling study. The Lancet Digital Health, 4(6))e436 e444, 2022.
Arumugam, K., Naved, M., Shinde, P. P., Leiva-Chaucs O., Mitaman-Osorio A., and Gonzales Yanac, T. Multiple disease prediction using machine learning algorithms. Materials Today: Proceedings, 80:3682-366 3685, 2023.
Association, A. D. 2. classification and diagnosis of diabetes: standards of medical care in diabetes- 2018. Diabetes Care, 4 (
2018.
Ayer, T. Alagoz, O., and Stout, N. K. OR Forum-A POMDP approach to personalize mammography screen- ing decisions. Operations Research, 6015):1019-1044, 2012.
Ethical considerations include ensuring faimess and equity in the application of the proposed screening policies. Care must be taken to prevent unintended biases, particularly when implementing these models in diverse populations. The algorithms rely on patient data, emphasizing the need for secure data handling and adherence to privacy regula- tions to safeguard sensitive information. While the focus of this work is primarily methodological, the potential societal consequences highlight the importance of interdisciplinary collaboration with healthcare professionals and policymak- ers to ensure ethical and equitable implementation in real- world settings.
Bansal, S., Deshpande, V., Zhao, X., Lauer, J. A., Meheus, F., Ilbawi, A., and Gopalappa, C. Analysis of mam- mography screening schedules under varying resource constraints for planning breast cancer control programs in low-and middle-income countries: a mathematical study. Medical Decision Making, 40(3):364-378, 2020.
Bao, H., Zhang, X., Bi, H., Zhao, Y., Fang, L.. Wang, H., and Wang. L. How can sigh-eerformance screening strategy be determined for cervical cancer prevention? evidence from hierarchical clustering analysis of mul- ticentric clinical study. Frontiers in Oncology, 12:816788 2022.
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Unified Screening for Multiple Diseases
A. Additional Discussion on Related Work
Prior research has predominantly focused on optimizing sampling schedules or srcision-making processes for single-disease sereenings. These strategies often involve fixed thresholds like age and weight (Association, 2018; Handelsman, 2015), to determine or evaluate inclusion criteria (Tomaszewski al., 2022; Bauer al., 2015; Fernandez et al.. 1991; Yu et al., 2021; Xiao et al., 2022; Idrees et al., 2021; Wang al., 2024b) and screening policy frequency (Hemmati et al., 2024; Wu & Suen, 2022; Wang et al., 2024b; Wu et al., 2024). Most of the existing studies focus on Markov models and processes (Wu et al., 2024; Wu & Suen, 2022: Alagoz, 2011; Ayer et al., 2012; Zhang et al., 20:2a;;; Maillart et al., 2008; Kamalzadeh et al.,
2021; Steimle & Denton, 2017). For example, Maillart et al. (2008) evaluate partially observed Markov chain model to balance breast cancer mortality risk and mammogram frequency.
Cost-effectiveness analyses have also been a critical component of the literature (Toumazis et al., 2021; Fritzen et al., 2018; Cusick et al., 2023; Meng et al., 2024; Sharma al., 2022; Areia et al., 2022; Huang et al., 2022), particularly for identifying the optimal trade-off between screening costs and health benefits (Xia al., 2024; Qin et al., 2022; 2024; Wang et al., 2024a; Bao et al., 2022). Resource constraints, such as limited budgets, personnel, and equipment (Yadla et al., 2024; Khoza-Shangase et al., 2017), play a fundamental role in these analyses, as they directly impact the feasibility and scalability of screening programs (Cevik et al., 2018; Bansal et al., 2020). For instance, constrained budgets may require prioritizing patients or conditions (Cevik et al., 2018), while limited personnel and equipment can restrict the number of screenings performed in a given timeframe. Teh et al. (2015) analyze opportunistic mammography screening healthcare systems. They demonstrate that their approach is effective, particularly for high-risk individuals, and recommend population-based screening for women aged 50 and above if resources permit. Bansal et al. (2020) address this by applying a Markov decision process model to determine optimal breast cancer screening schedules under varying infrastructure capacities. However, to the best of our knowledge, none of the existing methods address both resource constraints and competing risks of multiple diseases simultaneously, which leaves a gap in optimizing multi-disease screening programs under such conditions.
essorce-constrainge
competing risks problem in our approach refers the assessment of the likelihood of health benefits from an intervention, considering the presence of other potential outcomes that may interfere with prevent the occurrence of the event of interest (Varadhan et al., 2010). This approach enables the analysis of both the timing of the first observed event and its type (Wolbers et al., 2014). Competing risk analyses have been tackled in the literature (Allignol et al., 2011; Wolbers et al., 2014: Satagopan et al., 2004; Pintilic, 2011; Cho et al., 2022), which highlights its importance.
Several studies highlight the benefits of joint screening programs (Fan et al., 2024; Huang et al., 2023). Liang et al. (2016) assess joint detection using 3D-CPA, HR-HPV, and TCT, which improves accuracy and sensitivity in cervical disease screening. Regarding the screening for multiple diseases, Wang et al. (2022) propose a novel multi-view attention network for screening multiple retinal diseases from optical coherence tomography (OCT) images. Alternating from the "single disease" approach, Hajjar& & Alagoz (2023) develop a stochastic modeling framework that personalizes screening decisions for an index disease by accounting for the dynamic and stochastic nature of chronic conditions. Moreover, for the detection of multiple diseases, frameworks leveraging deep learning methods exist (Ampavathi & Saradhi, 2021; George et al., 2024; Satheeskumaran et al., 2024; Arumugam et al., 2023; Men et al., 2021; Khadir et al., 2023; Sudha et al., 2024).
Our work on designing disease screening programs, which about making decisions based on accumulating evidence to optimize outcomes under uncertainty, is closely related sequential hypothesis testing (Chernoff, 1992; Alaa & Van Der Schaar, 2016; Drugowitsch et al., 2014; 2012; Shvartsman et al., 2015; Shiryaev, 2007; Peskir & Shiryaev, 2006). With multiple diseases, the general problem is an instance of sequential hypothesis testing with multiple hypotheses. Prior work, such as by Alaa & Van Der Schaar (2016) examines resource allocation with a Bayesian model in single-hypothesis (disease) scenarios but does not extend to the complexities of multi-disease contexts. However, in our context, multiple diseases correspond to multiple hypotheses, which transforms our problem into an active learning challenge, where actions influence the acquisition of information across diseases.
B. Proofs of the Statements in Section 4
Recall that when Ix, 7n. disease is immediately diagnosed (no need for screening). A similar case happens for xn 0 (no screening necessary). Hence, in practice, whether to screen or not is only a question when Ir E (0, 7n). Nevertheless, to characterize differences between the benefits of different screening actions, we will compare their rewards and costs for all v, E [0, 1]. In comparing the effectiveness of different screening policies, we assume that a patient is admitted for
15

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Unified Screening for Multiple Diseases
screening, the screening policy collects at least one screening sample before the diagnosis.
B.1. Technical Results Related to Forms of Ta,= and Ma,a
Let Un == (o,, 1} A f 2 T represent the event that disease n is present and its adverse event occurs before correctly diagnosed. Let ST- (t, 0) P(T 2), We first provide decomposition lemma for ST (1, 2;, a).
Lemma B.1. For any & N4, [0, 11 .EE.
(1,2,,
II
P(T 4a.22 adverse until time t
P[T, S 1, On(a,,
(9)
[N[N]
adtverse has occurred diagnosed time
Proof.
ST- (t, 2,
P(T t/a, x)
P(min(Tr= n E [N], 1(4)) 1} > 1a,x)
PTmin(Ta::2 E [N], (1((m) = 1] (O, = 0}} > 1}
(10)
= (T, [N], 1(0,)) = > =/a,2}
P(OnE(M):(0.)--017, f)]a.2)
P' (I, 1 [0(()))))
(11)
II P[T, > t ((0,)t(a,z)
(12)
nE[N]
II (1-P(T, <t 1(On)t|a,x))
=
nE[N]
II (1 [T,, <t-1(10,) 1(O.),, PTT, St 1t-11(1)) 10,,), =O,(a,
nE[N]
II nE[N]
(1 P(T, 50,0nla, - P(T,, 1 -Onla,
=
II nE[N]
(1- (TT 21, -0,aa
(13)
II
P[T, > [1, + P(T, S (,00,,,
(14)
where (10) is due to T, = 0o iff θ, 0, (12) holds since the random variables T, + (0,)) are conditionally independent of each other given r, (11) and (13) is due to the fact that Tú is positive random variable, lastly, (14) holds given the partition (T,, < f, - (a, } + P(T, < t, On Ja, x) + ((T > t/a,x) = 1. Note that if screening is not performed for a disease with index n based on the given action a, the term PTT, S t, On aa x) simply zero since there will be no diagnosis, and T will simply denote the first adverse event that occurs (T,).
Next, we make some assumptions under which strict monotonicity of functions P(T, t|a,2) and P[T, st, Onla,, in hold.
Assumption B.2. (i). The support of T, [To). for all n E [N]: (ti) When action a screens for disease n, for all xn E (0, 1}. P(C, < T, = tla, Fn, O,, = 1} > 0.
The next lemma characterizes how P(T, t|a, x) in (9) and P(T, tla, x) change with x. Lemma B.3. Let Fr, (t) = Fr. (t) For any t E N4, E [0, IN e A. P(T, > tla,z} = - FT, (t) is
non-increasing in In, PTT, < t/a, 1 Fr, (t) is non-decreasing in x.. Moreover, if the support of is [Tol. then (TT, > tla, x is strictly decreasing in 2n, and (T, I tla, x is strictly increasing in 2n for E [Tg - 1]. Both functions are constant with respect to Tm, m # n.
16

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Unified Screening for Multiple Diseases
Proof. Since IT, 1|a, is independent of the screening action a, we can drop the conditioning on a. Also, if the disease is not present, i.e., , 0, then т, co. Using this, obtain:
P(T, > xla,)) P(T, } = P(T,, O, 0|2) + P(T, > t, 0, =1
P(O,. = (T, > :/0, =0,a) + P = 1|T > 10, = =1,0}
= (1 - Ix) + +,, > =/0, =1,}}
= (1 - xm. + xn(1 Fr, (t[1))
= 1 FT, (1)za
(15)
Since the CDF Fr, (1) 2 0, Vt, we conclude that (T > 1/2} non-increasing in In.
We also have
PTT, S = PT,, x/a, Fr,, (t)
= 1-
which is non-decreasing in n--
Let S, (Tal, Tni..) represent the set of all screening times for disease n. Let (i) = {Tnj E S, : Tnj i} represent the set of screening times of disease n prior to time step i. The next lemma characterizes how P(T, S t, tn,Onla,z in (9) changes with x.
Lemma B.4. For any E such that does not screen for disease n, P[T, S t,0n,a, = f all t E N+ and E [0. 1] For any E such that screens for disease n, for all t E N4. P(T, 1t, st,Onla,a is (i) non-decreasing in Vu, (ii) strictly increasing in In if there exists E [0] such that [!= < i, Ta = ila, x, On = 1} > 0, (iā) constant with respect to Ima m # n,, and (iv) 0 when In = 0 and Fr, (t) when In
Proof. First, consider the case when action a does not screen for disease n. Note that a referral policy never makes a diagnosis if screening is not active for disease n. Hence, PTT, < t, (0,aa,z) = 0 for all x, since disease n will never be diagnosed on time by the referral policy is present. Next, consider the case when action screens for disease n. Observe that
P(T, S xn)aa,, P(T, St, < Tn, O, 1/a,x}
P(T, St, <Tala, x, O, 1)2nn
P(T, 51, < <Tn, Tw = i, O, = =130n
Σ
P(T, St, < Tn,, T, = 11 On = 1)
i=1
P(t, < i,, = i|0, x. On 1)
(16)
Next, we will investigate how P(t; < T, i/a, x, θ 1) in (16) varies with Note that
P(t < i, = i O, 1} = P(t < iTT, i, O, 1)P(T, ila, x, O, = 1} = Pft < i|T, = i,a, H, 1)P(T, = i0,, = =1}.
(17)
Next, we further investigate P(t; < i = i, i, x, 0 = (17).
P(C < i = i, a, X =x,0, 1}
Pmmin((t E Sn (, = 1}
i|T = i,a, X = = O, 1]
= =P(3t (i) (z, = LT,, = i, X =,,0, =1}
= =P 3t e S,,
Xx
LR, (t) >
Tn = i X = x, O, =1}
1-X, X
1 Yn
17

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Unified Screening for Multiple Diseases
P 31 E Sn)
In (Rn(1) 2
nn
Tn = i, On =
(18)
1
1
Yn
= P lL E Sn (1) 2 (1 - In I'n
Tn = i, on =1
In (1 7m)
(t) 2 In In
P
max (E)
Tn in 100=1
(19)
(1 7n
where (18) holds due to observations (Tss). E Si( being independent X given On and Tn. From (19), we conclude that P( T = i, a, Ptt < i, n i/a, z, O 1}. By (17), P(T, S t, Onla, #) is sum of non-negative non non-decreasing functions multiplied by Znt it is non-decreasing in Zin. Moreover, it is increasing in In if least one of these functions is positive. Note that none of the terms above depend 2m. m # n, so P(TR t, Onla,z) does not vary with When In 0. (TT,, < t, On ] 0 by (16). When In = 1,
x, O 13 non-decreasing in In- By (17), this property also applies to
Σ PlC <,, = ila, 20 O₂ = =1
P(T, = ila, On = 132n.
P(T, = 110, = =1J0n = P(T, S 110, = 1Ja, = Fr, (1)
Using Lemmas (B.1), (B.3) and (B.4), ST (1, a E [4] can be written as
(t, 1) P(T; > [ 87- (1, (P(T) [|x1) 2,,00 (12),
(20)
(21)
(t,x,3, = P(T1 >
(22)
(t, = (P(T) > + ((002))))) =2 PRT s1,,,130)))
(23)
We define the following gaps:
(t, == St, 1) t,0,1,,,,,,221p111111) (
> f|22].
(24)
(f,,(r, is non-decreasing in and non-increasing in æ2. Under Assumption B.2, it is strictly increasing in æ1 and strictly decreasing in z2. Moreover, AT, (t, 21 = 0, xz2) = 0.
(t, = (t, x, - xt, 1, x, [2[P > P t|21}.
(25)
AT, (f) is non-increasing in æ and non-decreasing in 22. Under Assumption B.2, it is strictly decreasing in xı and strictly increasing in 222 Moreover, 47, 21, = 0) 0.
LT x) = Sr- xt, (t, x) = (P(T1 > +|11) + & t, x, S 0213, (26)
(t, is non-decressing in x2 Under Assumption B.2, it is strictly increasing in ø2. Moreover, (T, (t, -1, 0) 0.
(t, = ST- (t, (t, = P(Ti <
> [|22] + P(T2 S 02/3 x2)))
(27)
AT, (t, .)) is non- dendeecea in Ø1. Under Assumption B.2, it is strictly increasing in TT. Moreover, AT .1 = 0, = 0.
Define the last gap as
(t, = (T 1| +P st,0, > +P
(28)
- PPi (/ > f|x2]
=P(7) 51,0,12.0))P111 > + P(T2 S1,0, P(T: A PT2 51,0,1,,22
18

---

Unified Screening for Multiple Diseases
= P 54, + P(T] (1001,,))))) + PP2p)) ,1,0,1, (29)
The following display lists the relations between these gaps.
(t, (T, (t, (1, (1, 2 ((, 2 23, (1,2))
As the next step, we will derive an expression for [To T } Ha, x]]. Since min T } is non-negative, we write
To
Eimin[To, Ha, x|
Σ
P(min T" (a, x}
Σ
P(To > 1, 21,
PTT 1|,
1=0
1=
To
ST- (t,
t=
We define the survival gaps as (x) := [To, x] Emmin (To, 7") = ri.z - Fj.a. Note that using the
expression in the above display, these can be written in terms of gaps (T, x) as (x) = Eio
To-1 (t, z). The
characteristics of L x)s directly apply to ATj(r) (r) due to summation preserving monotonicity.
Lemma B.5. Under Assumption B.2: (i) A (z) is strictly increasing in HT, strictly decreasing in zg, and A (2) 0, 02) = 0; (ii) (x) is stricily decreasing Ti, strictly increasing in 22, and (511 22 = 0) = 0; (ii) A ())(2))
strictly increasing in 22, and (1, T2 = 0) = 0; (iv) (x) is strictly increasing in Ij, and ( = 0, 0,æ2) = 0;
(v) The following relations hold.
(x)
3, x 2 (z), (1) (T,(() 2 (z).
(30)
Next, we focus on the expected costs of the actions. Recall that ma, e, c a, x|| + e2 a, x). Under (i) no
screening (a 1), m1, = 0; (ii) screening only for disease 1 (a 2), m2.5 e₁ # la = 2, 2]: (iii) screening only for
disease 2 (a- 3), ma.5. c2B[#2le 3, ar). screening for both diseases M4,5. ci B(# la = 1, c2 4, []
e, 1 2, + og # 3, xr). for all 1].
We analyze (min Tn, ll.
Lemma B.6. When action a screens for disease n, B[in To, Tn, (a)), a] is non-increasing in on and constant in Im. m7 # n. Otherwise, it is constant in In- H[#, (min(To. Tn, t, Ha, In = 1] = 1, E # (To T, t [a, In Sm-
Proof. Let S.. represent the indicator variable that disease n is screened for the ith time.
(Tp) Σ
Sw
E #minn (To, Tn, 2))
P(S.. 1|a,,) =
Σ
P(T,. Tni rmla,,),
(31)
where
IP [T > Tnis f > -nilaa x} = PPTT > Tnis f > Tni O, = 0|a, + P(T, > t, > Tris On = 1|a, x}
= [P(Ta > Tni: f > Tui On, = 0, a, (1 xn) - -P(T, > Tni, 1o > Tni O = 1, a, x )
[P > Tai O,, = 0, 0, x] - n,) + pTT > Tui 1 Tni O₂ 1,0, ).
Itt > Tna O₂, 0, a, a, (1 xx, + P[T, Tri t > Tnd On L, } In. (32)
f(xn)
g(a.)
We have
Pt > Tan 0r 0, = =x,} = PP(mnnf E Sn (x t) = 1} > Tni = 0, Xx = xnn}
= =P(YjE
[i]
Xn
LR, (Tnz) <
In
O, = 0, X, I
i- Xn
X_
1 nn
19

---

Unified Screening for Multiple Diseases
= PP max LR, (Tng) < (1 - un) [EE
θ - =0 0
Fn 7v))
which is non-increasing in Fn. Similarly,
Pl Tni Tn > Tns a, = In
P Tras Tn Tnis Tn = i Xn =In}
Σ
P > mmi In > Tni = = = =1,0, X, = In
Σ
P(G nns Tn 0 a, Xn x
T
Σ
P(C > Tna TTa = i, O = 1, a, Xn = In JP = i - =1}
(1- In mm In (1 7n)
= = p(T, = =10,, = =}},
Σ
P
max Lm, (Tnj)
Tn 0n =
[E(
which is non-increasing in Fn- Also observe that
P
{ max (Tn))))) JE(]]
(1- 1--nn) Tn
On = - 2
max LR,,, (Tnz) jC(1]
(1 In 7n
On = [
Fn (1 7n)
In - 7/n)
Therefore, Pft; > Taa Or = 0, )n 2 P(t, > On = 11,, 2 P(t > Tri Ta > Tra mn = 1, a, In
By the above properties and the decomposition in (32), the following holds for In < r/y::
f(xm)(1 (2)] gz 2 (I - 2',1) + {Z 2 f(z 2',)) + g(xr))='n
by which we conclude that IPTT,, > Tnis 17 > Tui [a, x non-increasing in Vy. Moreover, it does not depend on Tms m A n. By (31), El#,,. (To, 1 a) x] is also non-increasing 2n, (since it is the sum of non-increasing functions) and constant in Fm, m + n.
Let (x) mı.x Mj.x represent the expected cost gaps of the actions. The following lemma characterizes the dependence of these gaps to x. For the sake of simplicity of the analysis, assume that the costs of the first screening samples are excluded from the cost calculation. This makes sense when the goal is to manage the excess cost due to multiple screenings (when a patient is admitted to screening program, usually many screenings take place over a long time horizon).
Lemma B.7. (i) (z)(2) = A6,1 (z) = e (Bi#i Ti, la, x] - 1) is non-increasing in Iι and constant in (Ei T2, t5 |0, - 1) is constant in and non non-increasing in 22; (iii) AC (x: ci [E[# (Tin(To. T1, (1))))))) - 1) + 02 c2 (mi+ T2, la, - 1) is non-increasing in Ia and x22
Ty; (ii) (z) (x)
Let A.,( == K₁,z Kjzz- Then, (x)(x) (=) - X
- (z). We assume that the gap functions are continuous in 21,
32, which will hold under mild assumptions the distributions of Y, and Tap n E [2]. Assumption B.8 The functions (r) and (x) i,j E [4] are continuous in æ. Therefore, Ai5(x), i,j [4] is continuous in x.
B.2. Proof of Lemma 4.2
Consider P2,1 (x2(2z) > kz()). Under Assumption B.2, (z) is strictly increasing in 21 and strictly decreasing in J22
- 1)c1 < 0 (screening programs are set to screen more than once) and (2) 1, 12, -h- Fr. (T) [[>2] > 0. Since 22,1 is strictly increasing in xı, by (x2) is the unique point where 22) crosses O. Thus, we obtain
Fix 22 E 10. Observe that = 0- O
(x) [2 x2)) > xs (x), 22)) = 1 (( x2) > 0} = 1 > by (x2))
20

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Unified Screening for Multiple Diseases
For any 222 22 12,1 (1))) az)) 0. 22,1 (12)) 0 and 2,,1 is increasing in #1 implies that b1 (1/2 bj (x2).
The analysis of Pa.1 (z) follows similar steps. The only difference is that the role of and 22 are reversed since the gap terms in A3.1 (t, z) in (25) involve PTT22 & t, O2 x2 [TT > t|21} J instead of P(Ti < t, O, PP x and cost incurred from screening for disease 2. Therefore, we conclude that
P3,1(2) {2² > 81 (x1));
(33)
where 8 (x1) is increasing in T1.
B.3. Proof Lemma 4.3
Recall that
To -1
(x, =
Σ
(P(T2 1, 02 > x/1] + < 0,011,2))) + P(T, > 1|22) st, 0112,z1))
1=
de ( 71,4)) - (min (,, 1).
Fix 22 E [0, 1]. When z₁ =
To --
(11(11=00 0, =
Σ
P[Z2 S t, 02/3,,222 Ac1 ( Tr, (i),,, = =0] 1)
( T2, (2, (2, (5)))a
-).
When 21 1,
To-1
A11(21=1,71) = 1,
Σ 1=0
(P < 2t,00 + Fr, (t)) (mii T2, (,)))a,2)))) - 1).
Observe that A1,1 = 0, 22) < =1,121==110)))
When 32 = 0,
To-1
=2
Σ P(T t, 211,,12,11
Aci (R(#) (min To. Tı, T1,41))),, 1) Acz(S2 - 1);
which is strictly increasing in x1. ( 0, = 0) - 1) Ac2(S2 1) < 0, 44,1
1,32 = =0) Ele Fr, (t) Xc2(S2 - 1). Therefore, when ITo-1 Fr, - Acz(S2 - 1) 2 0, b=( E (0. Else, b= (0) = +oc.
Proof of by (0) b= (0): Fix 52 = 0. Since A3, (I1, = 0) = AJ, (21, 0) and A (21, = 0) >
A x2 = 0). we have (I, 22 = 0) < 2 (0 x2 = 0) for all 21. Since 2 is strictly increasing in 3t
this implies that 2 will cross zero no later than A1,- Hence, (0) < bz)
Proof of b1 (1) > b= (1): Fix x2 = 1. Since TT (01, = 1) ETe (x1, 22 1) showed that b (1) < Thus, b= (1) < co. Since A2, is strictly increasing in æT, this implies that 2 will cross zero no earlier than A1. Hence, (1) > (1).
To-1 Fa, (f) - Fr, (t) S AT, (21. 22 1) and 12 1), have 14,1(11,55 1) > A7, (21, 1) for all 21- In Lemma B.3,
Proof of b= (x2) and by (x2) have unique intercept in Ø21 E [0, 1]:
Note that A4, 0. = E (t) (St 1) and A4, (x) 1.22 = 1) -tT + (1 (t) (t)) > 0.
(a) Assume that (11 = 0, وم = 1) < 0. Then, (1) E (0, 1), and by (1) < (1). Since b= (x2) is decreasing and continuous on [0, 1). one of the following two is possible:
21

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Unified Screening for Multiple Diseases
(a.i) There exists 0 S 2y < 1 such that 0= (21) We also know from Lemma 4.2 that by (x1) E (0, 1). Since by (x2) is increasing, under the assumption that b= (222 is decreasing, since both functions dp (22) and b (22) continuous (follows from the continuity of 1 and 44,3), 6 (12) and bj (22) have unique intercept in 22 E (2), 1), say at #2.1.
(a.ii) b= (0) S 1. Then, uniqueness holds due to the same reasoning as in (i).
(b) Assume that (1, x2 = 1) > for all 21 E [0, 1]. Then, b= (1) = -OG. Assume that (=) < co. Since bm (0) E (0, 1]. and b- (22) is decreasing and continuous, there exists Iu E (0, 1) such that bm (wiu) = O. We also know that bp (Ix) > Then, using the same argument in part a, we conclude that b= (x2) and by (x2) have unique intercept in
22 (0, say 2211
(c) Assume that b- (1) = -o and b (0) +o0. Then, since bm (222 is decreasing and continuous there must be and in [0, 1] such that b= (z1) = 1 and (Ix) = 0. Since (xz) < 1 and by (zw) > 0, we conclude that (x2) and bj (x2)
have a unique intercept in 22 E (xy, n.), say at 2211
B.4. Proof of Lemma 4.4
Consider (1,)(z) (x) > Ka (r)). Under Assumption B.2, A15(() is strictly increasing in .1.
Fix 22 E 1]. ( = x2) = 0, A4 = 0, > 0. Hence, 44,3(11 = 0,02) < 0. ( = 1, > 0, (Z = 1, x2) = 0. Hence, 4 1 = 1,12) > 0. Since (4,3(11,22) is strictly increasing in 21, there is a unique zero
crossing, which is denoted by 2 = bt (x2).
Proof of (0) = bt (0) = bt (1): Note that
To-1 PÍTi 2t,0,12,z1
( = 0) = 22 =
4 =0)
Σ 1=0
(34)
(1, = 0) =
= 2 (F1, 0),
which implies that A4,3(11,12 same zero crossing, i.e.. b_ (0) = b- (0) = by (1).
(4, 1) 2 (II, 0). Therefore, conclude that they have the
Proof of by (0) S bt (x2) - by (x2): AT x22 2 A!, (I1, x2) and AL x22) -6, (21, 22). which implies that A4,((11,42) 2 2 (2), Hence, b (12) & bi (I2). From (34), note that (21,, = 0) > (3, and
A6, (21, = 0) - Af, (11, which implies that 2 x2 0) 44,(111 22). Hence, bt (x2) (0).
Proof of by (0) (1) < (x()): This again follows from the fact that (1, 0) 11, = 1) 2
(1,3(Z1, 22), so the zero crossing of the latter cannot be earlier than the others.
B.5. Proof of Proposition 4.1
To construct p] (z), we consider all cases separately, noting that the values of corresponding to actions where the first
disease is screened (a = 2 or 4) should be greater than those corresponding to actions where the first disease is not screened (a or 3):
K2 ()) >
Kog (x)
K1( > Kg (x)
(1(x) > (z(
(x:
0
O
0
1
1
0
O
X
0
1
1
1
O
X
1
For example, in line 2, we have (2)2) < K1 (x) and (1(2) < Kj (2), which means that neither of the actions that screen
disease is activated since neither K2 (x) or (x) is the largest, leading to p (x) 0. line 3, we have 62(x) < Ki (x) and K1 (x) < (x).). but (x) < (x) again meaning that p (x) = 0. But, for instance, in line 6, since we have (2( > in (I)), (x) > na(x), (x()) > na (x)), this time (x)) since (z))) is the largest.
22

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Unified Screening for Multiple Diseases
It should be observed that lines 5 and 7 inconsistent since (x) 2 4g, and (1)(x) 1 hence (x) (a).. Therefore, (x) sa (a)) implies na(x) (x) hence we cannot have K2(ar) > (1 and
(x) < (z) at the same time.
As result, we can write the relation as:
= = (2, (x),(() A p4,(xx)).
(35)
Then it is enough for the optimal referral decision boundary to satisty (2) xi (I), which will imply that Ra(r) > (z(x))
This guarantees that either K2 (z) or (x) is larger than both K1 (z) and Kz(z), because it is impossible for na (z)) and Ki (x) to be the largest, hence the activation for the first disease is guaranteed. For example, if kg(x)) > #2(2), then (1) will be
the largest. If k2(x)) > K1 (2:) is not true, then we should have both Ka(x) > (2(x)) and (a(x) > sa (z), which will make (x) the largest.
Using symmetry for the second discase, we have.
pb(x) = Ps) (p4,1 (z) A p4,(2z))
(36)
where
Pa.1(x) = (<a(()) > Ki (1 }
P1,(2) = 1f (M((x) > ki k1 }
P12(z) 1f (()(()) (2(2)) }.
B.6. Proof Proposition 4.5 Fix 2g E (0, 1). Note that
A (T,, - - (2), = To
Σ P(T > t|m))p[T S O213,
(37)
is strictly decreasing in 21, and Af, (2), - (2 = T2, (5))),, - 1) is constant in zı. Therefore, (2,1 (1, is strictly decreasing in ø1-
Fix x2 = 0. Then,
To Σ t=0
(21, 0)
P(T: 2t,0,121111
strictly increasing in 31, AC, (2 =0) is non-increasing in Tí. Therefore, ( 22 0) is strictly increasing in
T1. Also note that 4 (I1, 22 = 0) 2 42 = 0) and (f x2 = 0) - (1, x2 = 0) c2(S. - 1) > 0.
Therefore, 14,1 22 0) < (4,3(11 -2 0). Due this, and strictly increasing nature of A4,1 = 0), its zero
crossing (if any) should happen after the zero crossing of A4,2 12 0). Hence, it holds that b= (0) (0).
Fix 22 1. (f,
1) (E[# (min To, (f), 2] -1), A3, (21, 1) (21, 22 1) > by (37). Therefore, A ( 1) > (x x2 1). Since A4,((11, 32 = 1) is strictly (2
1) (L,(11,
increasing in æ its zero crossing should happen later than the zero crossing (if any) of 14, 0). Hence, it holds
that 6== (1) < (1).
Under the assumption that (=(2)) and bT (zz) have unique intercept, the continuity of these functions (which results from the continuity of A functions) imply that there exists #2T E [0, 1}. where by (x2) 2 b= (x2) for d2 2 D2T. Moreover, since (x2) < b1 (x2) for all 22, this value should be greater than or equal to T21, where by (x2) exceeds b (x²).
Based on this, we first combine these two boundaries due to the parentheses between PL.1 and (1.((1) in Lemma 4.1. Due to the "and" condition, points must lie within both regions (to satisfy both of them). Thus, we take the larger boundary. Let us introduce an intermediate boundary as:
(I2)
x2 E (0, Ø2T)
bx (12) = b- (12)
(38)
22 E [I2T, 1]
23

---

Unified Screening for Multiple Diseases
Now, we can combine this with the boundary of P2,1 (2) obtain the boundary Çi (122) for the optimal rule (z)).. Given the "or" condition between in (2,1(z) and (a)) A (4,3(x) (whose boundary we just defined bx (x22) in Lemma 4.1, select the smaller boundary value of the two, it suffices for any point to lie within at least one of the two regions. Starting from I2 = O, we initially use the lower boundary bp (22), since we have shown that by (x2) < (= for 22 < #21 in Lemma 4.3. When bu (I22) exceeds d= (x2) at 221,, the boundary switches to b== (x2), which is now smaller. We have also shown that bt (#2) & bp (x²2 for all Z2 E [0, 1] in Lemma 4.4, thus we conclude that b1 (x2) z by (22) for 22 2 221. which leads to our optimal referral decision boundary Çi (222) being obtained as:
Ci (122) = by (x2)
22 E [0, by (x2) if 22 E (#211 1]
(39)
given that 221 S 22T as we have just shown. Finally, by substituting by (x2). our optimal referral decision pi(x) = 1 1 (x2)) is obtained as:
bi (122)
if
32 E
[0,
(1x2)
b (x(2)
if
22 E (221,
(40)
(T(2)
if
22 E [
Moreover, (1)0) = (1(1) since we have shown that (0) (1) (by Lemma 4.4). Also observe that (1(0) b_ (0) S
bt (122 S by (x2) for all 22 [0, 1] (by Lemma 4.4), and also (0) = (0) = bT S bt (x2T) = b (x2T) S d= (x2)
for 22 < 22T < 1 (by Lemma 4.3 and 4.4, using that b= (22) is non-increasing). Consequently, (1 S b (x2) for
I2 E (2T). Combining all of these, we finally conclude Ç1 (0) S (12) for all xх E [0,1].
Due to the symmetry the functions by replacing the index n = 1 with n = 2 and vice-versa, and consequently due to the symmetry of the boundaries, for the second disease, we also obtain the form (1 1 (x0g > (2(x1)) where
6 (x1) if
æ E [1,0
(2(x1) = b=(x1) if
æ e (211)
(41)
(x1) if
in e [MT,1]]
B.7. Proof Proposition 4.6
Recall that, in our case for two diseases, the independent referral decisions will have the form p(X) = (pt (21), where Pn 's are independent from 2,, Vi a n. In this case, redefine action space for each disease b Pr ( - f1, 2). where 6 corresponds to the case where disease n not being screened and b = 2 corresponds to the case where disease is screened. Unlike action in joint screening, action b only depends on the risk x,, of disease
Recall that 24, = 1 1) A ( > J. In independent screening, the decision to screen for disease depends only on the variables related to disease n. Let T = T,, 1 [ + (. The independent screening problem for disease is formulated as
maximizeren.
Eimin(To,> H subject to
B[#, (min (To, T,, (7) < Ba, P(O,, = 0, (t) = 1} S on Vt e [To].
(42)
where B, is the budget reserved for screening of disease and II, represents the set of all screening and diagnosis policies that is based only on the risk and observations from disease n.
Let action b = represent no screening and b = 2 represent screening for disease n. Using posterior probability of having the disease with the calibrated threshold 7, as the diagnosis rule, under the uniform risk profile, the optimization problem for the independent referral decision of disease becomes
2 Σ
maximizega
(b) (b dx
(43)
xEX
24

---

Unified Screening for Multiple Diseases
On (b/x))m,,b.. b=
dr S Ba,
subject to
ZE
On (b(())) 2 0, vb E [2], E X, Σ Cn (bl=)) 1, Vo X,
where bn Elmin (To, Th } (b, æ] and mn,d,z == C, [+ (b, 2/). The expected costs satisfy the following: mn,1,a 0, m1,2, = m2, 2,, = 3, 112,, + m22,,a M4,2 22,2 + m3,..
To characterize the structure of 17 (I,)). we proceed with the Lagrange dual problem associated with (43). In particular,
(xx) = 1{ [ = 2, xn] - Ac, 16 = 2, mn] > [To, H6 = 1, xn. - 1o 11|6 = 1, }.
where > 0 depends on B Basically, disease n will be screened when Ri (corresponding to screening) exceeds Ai (corresponding to not screening).
The survival functions for independent screening of disease are:
(1, b = 2) = (TT, > (pr)) + PTT < bx kn STz Then, ST₃ (1, En = 2) - SS: kn, = 1) = P(T, - 1,,0,b = 2 2 0. Note that E[min (To, Hbb x] Ejmin { (To Tx) [b = 1, -i P(T, 0,, 2, is increasing in In. Moreover, E[#n 2, Inl is
> -ppm))
non-increasing in In and [8 1, = 0. Therefore, k2(xn) ki (Ix) is increasing in In.
For nn = 0, E] [mm bb = 2, = To. 1b = 1, zn_ and B|#1 2, xn > 0. Therefore, (2 = 0) < 61 (In, = 0). For In = 1, E min (To, Tx 2m > Eimin Tx> bb = 1, _nd and 1 lb = 2, x.l = 0. Therefore,
K2(x,, = 1) > ki (2i = 1).
Since (2(xx)) Ni (x,, is increasing in 2n. there exists a unique point En (0. 1) such that k2(xn) - Kr (Ix) S 0 for In S En and (x,,) K1 (2.) 0 for F. Sn.. Assuming that the functions H2z (2)) and Kn (I,) are continuous in (0, 1), <2(Sm) Ki (5x) O.
Hence we obtain the form (x1) = 1(x1 > { and (122 = [x2 > (2), where č E (0 and Eo E (0, 1) are
constants that depend on the screening budgets reserved for each disease.
We analyze the behavior of the policies/referral decisions under constant thresholds. First, observe the following fundamental relationship under these independent referral decisions:
Higher thresholds result in fewer screenings, reducing the overall cost.
Lower thresholds result in more screenings, improving survival times.
Given these observations, the optimal constant-threshee referral decision corresponds to the threshold & that exactly satisfies the budget constraint. Thresholds below this value violate the budget as they involve higher screening costs, while thresholds above it underutilize the budget, leading to suboptimal survival times.
Consider the case when the unified budget B is split among two disease screening programs as in the optimal unified referral decision. Now, consider the extreme cases of constant thresholds:
At (0) = min &i (I22, the threshold is low, implying that this referral decision screens strictly more people compared to the overall optimal referral decision of G, As a result, (0) violates the budget due to excessive cost.
At [o.1) &i (x2). the threshold is high, resulting in strictly fewer screenings. Since this referral decision has a lower cost than (i, it under budget.
The key insight is that čL, the overall optimal referral decision, balances survival time and cost such that it satisfies the budget exactly. By continuity of the cost function with respect to the threshold, and since one extreme (G (0)) exceeds the budget while the other (maxx2ee0.1) Si (x2)) remains under budget, it follows from the Intermediate Value Theorem that ČL must lie between these two extremes. This establishes the result:
(0) = min (x & & S max (122)
2>E/0,1]
x,E[0,1]
25

---

Unified Screening for Multiple Diseases
C. Empirical Justification of Theoretical Assumptions
The assumptions that b= (22) is decreasing in ø2, and that b=(x2) and bt (x2) have a unique intercept, are difficult to prove analytically. This complexity arises from the dependence of the decision boundaries on the prior risk profiles in addition to the likelihood. To provide empirical support for these assumptions, we conduct comprehensive simulations under varying diagnosis thresholds 7, different values À model the tradeoff between survival and cost, and alternative sampling distributions for the observations y. including both Bernoulli and Gaussian noise models. Screening costs and screening periods are set to 1 for both diseases.
In Figures 6, 7, and 8, we plot b and bt for different values of under Gaussian and Bernoull likelihoods. The adverse event times Ti and T2 are assumed to be equal and deterministic. Under Gaussian likelihood, the mean is zero when no disease present and when the disease is present. The noise variance is 1. Under Bernoulli likelihood, the probability of observing 1 when a disease is present is p. while when not present, it is q. These plots demonstrate that under reasonable conditions, the assumptions hold approximately in practice, and the observed boundaries intersect uniquely.
To investigate the effects of the randomness of adverse event times on the decision boundaries, we also performed simulations under Poisson distributed and uniformly distributed adverse event times (Figures 9 and 10).
These results suggest that although analytical guarantees may be clusive, the structure of the optimal boundaries observed in practice largely adheres to the assumptions stated in Proposition 4.5. When violations do occur, they appear minimal and do not disrupt the overall interpretability of the policies.
(curved) vs. (verical)
(curved) vs. by (vertical)
walues
Jeg
08-
JeO
2w0.12
1-0
be0.00
2
2
02
02
(a) 7 0.6. 1, Tr. 10, To 20
(b) = 0.8, 1, T. 10,To To 20
Figure 6: Decision boundaries for Gaussian activations (Set 1)
(curvad) (veriica))
(curved) Aerttical
30
02
Je0
2m0
1=0 12
JeO
2e00 64
5.
[n] 7 0.9, 1, T, 15, To 20
(b) 0.95, 1, T, 10, To 20
Figure 7: Decision boundaries under Gaussian likelihood.
26

---

Unified Screening for Multiple Diseases
D. (curved) vs. D, (veriical)
D. (curved) vs. b, (vertical)
1 kausc
1mg
1m0 C2
J-g
1-0.0 cc
1-0
c0
JeO
10
06-
0.-
56
3.
(1] 0.6. p 0.7, 0.3, T, 10. To 20
(b) 7
0.6, p 0.5, 0.3. Tx 10. To 20
Figure 8: Decision boundaries under Bernoulli likelihood.
cressings metr and values
for all metrics and
200.2
02
02-
2..
(a) = 0.9. Poisson parameters An 10, To
(b) = 0.9, Poisson purameters Aa 5, To
Figure 9: Decision boundaries for Poisson distributed adverse event times and Gaussian likelihood.
Awrocraiing for all metriss and valuos
zero-4rossin for all metris and A values
240.
66
00
(61 2010
343,0e10.
68.
08
(n) 7 0.9. T, Unif[I, 20), To
(b) = 0.9, T, Uniff] Inif[10,, (0)) To
20
Figure 10: Decision boundaries for uniformly distributed adverse event times and Gaussian likelihood.
27

---

Unified Screening for Multiple Diseases
D. Supplementary Experiments
Figure 11 presents a plot of the actions/referral decisions based on solution provided by CVXPY, alongside the boundaries derived in Section for better visualization.
b (x(x)
2000
1013
(311)
- (111
(x()))
(xg)
b- (3))
(x1)
64
(x2)
003
(a)
Figure 11: (a) Optimal boundaries for the actions, as provided by CVXPY for 10, (b) Matching with the boundaries derived
Section 4.
D.1. Varying Key Parameters (B, Cu. Pn, ar, B)
series of simulations was conducted to evaluate the policy framework's performance and accuracy, with systematic variations in key parameters. Findings across different configurations include the following:
Increasing the Budget B: An increase in the budget B allowed for more frequent screening, effectively reducing the "grey region" (where specific action was taken due to cost limitations, Figure 12). As result, the regions implementing active screening policies (a 2, 3, 4) expanded, illustrating that higher budgets enable more comprehensive screening and reduce the risk of undiagnosed conditions.
(a)
(b)
(c)
(d)
Figure 12: Varying the budget (B) with other parameters fixed, (a) B 2, (b) B = (c) B (d) B 15.
Varying Mean Adverse Event Times (e,, = B(T,) for Diseases: When the mean time to adverse events was shorter for specific disease, the model showed a tendency to prioritize screening for that disease, aligning with the objective of early detection. Diseases with longer expected event times were deprioritized, as immediate screening was less urgent (see Figure 13).
28

---

Unified Screening for Multiple Diseases
(a)
(b)
(c)
(d)
Figure 13:
Varying the adverse event times (In) for the diseases (where they distributed with Gaussian distribution) with other
parameters fixed, (a) 14 37, 122 30.1, 1, (h) fr 34, 30.1, (c) μr 30.1, M42 34, (d) pa 30.1, 1,,1,,1 122
37.
Modifying Feature Distribution Parameters (a and (): As is decreased and 3 is increased, the probabilities p(x) associated with higher-risk regions requiring screening decrease. This reduction enables more extensive screening, as a greater number of points satisfy the cost constraint. Consequently, the grey regions the plot, which represent unscreened areas, shrink in size (see Figure 14).
(a)
(b)
(c)
(d)
Figure 14: Varying coefficients a and the the feature distribution with other parameters fixed, (a) 5, (b)
5, B = 3, (c) a = 3,8 3 3, (d) a = 1,8
D.2. Decision Boundaries Based 's and Comparison
After conducting simulations with the proposed optimizations, plotted ky's for each action to assess its behavior across the feature dimensions and 22. To approximate the expected value, we employed Monte Carlo integration. We then compared the values of N; across the four actions, and selected the action associated with the highest N2 at each point (#1, 22). This allowed us to delineate decision boundaries by identifying the action that maximizes Kę as a function of patient features r. In the mathematical proof, we also derived the optimal referral decisions (x(x) by determining the boundaries which one parameter K; surpasses the others. This approach based on the principle that the optimal policy corresponds to the parameter i; that achieves the maximum value.
Plots of K2 for each action (see Figure 15), with À 0.25, demonstrate patterns consistent with analytical results. Specifically: (a) without screening, decreases with both and peaking (x) = 0, 22 0): (b) when screening the first disease, K2 decreases with but increases with I1, peaking at (I) = 1, 0); (c) when screening only the second disease, K3 decreases with x, but increases with Ø2, peaking at (x: = 0, = 1); and (d) when both diseases are screened, R4 increases with both 21 and T2, peaking at (a1 = 1, x2 = 1). These observations show peak values at boundary points where 2; is either 0, indicating that high-risk values minimize cost, while zero-risk values maximize survival time.
29

---

Unified Screening for Multiple Diseases
(a)
=
(b) 2
(c) 3
(d) 4
Figure 15: Plots of Ry for each action with À = 0.25, observe the behaviors which match mathematical results. (a) When screening made, K1 decreasing in both 21 and Iz. with the peak value occurring at (#1 0, 32 0). When the first disease screened, Ki
decreasing in 22 but increasing in æ1, with the peak value occurring at (ær 1, = 0). (c) When the second disease screened, R3
is decreasing in xı but increasing in 22, with the peak value occurring at [x] = 0,22 1). (d) When both diseases are screened, Rig is
increasing in both and with the peak value occurring at (æ1 1,72 1).
Next, we examined the values of K; across the four actions, selecting the action with the highest K; at each point. We generated decision boundaries by selecting actions with the largest values, and recreated the decision points found in mathematical proof by locating intersections where one K2 surpasses the others (see Figure 16b). A comparison of these boundaries, derived by selecting the maximum K₂ (per our theoretical framework) and those from CVXPY's linear programming solution, confirms that the boundary shapes are nearly identical. This validates our proof"s credibility and alignment with the convex optimization results (see Figure 17).
14
U8
0.1
06
0.5
(a)
(b)
Figure 16: (a) Maximum R, values among the four actions 1,2,3,4, 3,4. The maximum values of K2 is plotted by comparing the four values for cach (21, x2). The peak values occur the comners where either or 0, since having the risk minimizes the cost while having no risk maximizes the survival time. (b) decision boundary obtained by choosing the action with largest value. This is essentially what we did the proof, we formed the boundaries by finding the intersection points which one of K; 's exceeds the other.
D.3. Supplementary Material for Comparison with Independent Screening
We present the plots of chosen actions for independent screening and unified screening in Figure 18 for better visualization. We determine the survival times (E [mii T" Ha" rr) for the first and second diseases separately, and by choosing the minimum of these two survival times (as the adverse event that occurs first is considered, like in our joint model), we obtain the plot of survival times for the independent model (see Figure 19a). We also created a survival time plot for our joint screening model (see Figure 19b).
30

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Unified Screening for Multiple Diseases
1221
2.6
0.5
0. 146 08
01 0.4 1.5
(a)
(b)
Figure 17: Comparison of boundaries (a) obtained by choosing the action/referral with the largest value of Ri (as did in the mathematical proof) and (b) by solving linear programming by CVXPY. Observe that the boundary shapes almost identical, verifying credibility of our proof and aligns with convex optimization solution provided by CVXPY.
- 5-03
0.3
0.1
0.2
0.2
(h)
Figure 18: Comparison of chosen actions (a) for independent screening (B) = -B = 5) (b) for unified screening (B = 10).
58
38
2.5
0.4
04-
0.2
0.
20
62
3.4
06
0.6
06
62
0.4
06
0.6
(a)
(b)
Figure 19: Survival times (E (min/To, Ma, 2)) in (a) independent screening (b) unified screening.
E. Additional Experimental Details
To assess the robustness of our results, conduct Monte Carlo simulation across 50 independent outer iterations. For each iteration, we store the resulting gain and cost estimates and solve separate instance of the linear program described in Equation (5) using CVXPY to obtain an optimal screening policy. The resulting policies yield average survival times whose standard deviations are 0.0089 for the unified approach and 0.0117 for the independent approach. Given that the mean difference in survival time between the two methods approximately 0.2, the 95% confidence intervals (mean + std) for each method do not overlap. This provides strong empirical evidence that the observed advantage of unified screening is statistically significant and not due random variation.
31

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Unified Screening for Multiple Diseases
F. Table of Notations
Table 2: Table of Notations
Symbol
Description
N
Number of diseases
Latent disease state for the nth disease, On E {0, 1}
T,
Time of adverse event for the nth disease, Tn, E N+
X
Patient risk vector, X E [0, 11 X = (,, XN]
X;
Prior probability of having disease i, X; E [0,
(t)))
Value of the nith screening target at time t, (t()) E R
(t)
Screening action for the nth target at time t, 6,(() E {0, (0,1}
(1)
Diagnostic action for nth disease at time t, (t) E {0,1}
Diagnosis time for the wihh disease
On
The event that diagnosis on time of the nath disease, such that O, {O,, Att < <T,).
7/
Diagnosis threshold for the rih disease
On.
Cost per sample from the noth screening target
B
Budget for screening costs
On
Maximum allowed false positive rate for the nth disease
Is(T')
Information available to the screening policy at time T
IT(T)
Information available to the diagnostic policy at time T
T
Time of the first adverse event that is not diagnosed on time
To
Survival time limit
P(X)
Referral decision vector based on risk vector X, p(X) ( (N())
p(X)
Independent referral decision vector such that ((X) = (21(21) (.. (N,)))
Action index representing referral decisions, a E [I,....,2N)
((alar)
Probability of selecting action a given risk vector x
Ta.x
Expected survival time given action a and risk æ (Efmin T T* (a,x)).
(Tnl...,, Tn
The sampling schedule for the nith disease, where Tui the time point that the ith sample taken.
E [H,
Expected total screening cost for the nthh disease.
Ma,x
The total screening cost given action and (2n c, E,
A
Lagrange multiplier for cost constraint.
(x(x)
Parameter defined as k₂(2)) := E min (To, = i, i XE and given the vector sc.
i, for the ith action
(7)(x)
Optimal referral decision for disease n based on risk vector æ.
(2(x)
Optimal referral decision based on risk vector æ, p² (x) = (X))
32

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Unified Screening for Multiple Diseases
Table Table Notations (cont.d)
Symbol
Description
(z.)(z)
Referral sub-rule to determine (z) for = 2.
Sy- (t, x.
Survival function based on 7 such that ST-(t, a,) = >> x}
Fr. (4)
The CDF of the adverse event times Ta.
ry (222
Boundary function for the first disease's s screening sub-rule (2,1 such that p2,1 1 >
bi
(x2))
b= (x22
Boundary function for the first disease's screening sub-rule (4,1 such that Pa,1 (x) 1
b (x2)}
br (x2)
Boundary function for the first disease's screening sub-rule (4,((1) such that (4,3(1) 1 ( >
bt (22))
(x22)
Optimal referral decision boundary for the first disease dependent on 22, such that pi = 1 { > (I22)).
(2(x))
Optimal referral decision boundary for the second disease dependent on æ1, such that p5(x))
1 {x2 > (2(x1))
(xx)
Optimal independent referral decision for the nth disease that is dependent on Ze, Vi n, when the referral decisions restricted to the set of the independent referral decisions pr. (x..).
C1
Optimal independent referral decision boundary for the first disease that is not dependent 22, such that pi (21) = 1 (x1 > (
E2
Optimal independent referral decision boundary for the second disease that is not dependent on such that 35 (#2) 1 (.)
33


Reviews:
[{"review_id": "iOiJXfIiCa", "reviewer": "Reviewer_yUUd", "summary": "This article offers a novel optimization framework for the complex task of engaging in unified screening for multiple diseases. They offer a novel optimization framework, attempting to balance multiple factors including disease risk, budget, and diagnostic test characteristics.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the thorough review of our paper and the constructive comments. \n\n**Why screening of one disease depends on the screening of another:** In the example of pulmonary hypertension and cardiac disease, risks are clearly contingent. Our approach is not limited to such examples, as we offer a non-trivial solution even when disease risks are independent. Due to the nature of our objective function (related to survival time) and our constraint function (screening cost budget), the optimal threshold for activating the screening of one disease depends on the risk of the other disease. This coupling distinguishes our model and solution from independent screening, which yields a suboptimal policy for our referral problem. We will further clarify this in the revised paper. \n\n**Regarding the benefit of screening:** Thanks for pointing out this important review paper. We agree that screening does not always offer benefits and sometimes can even be harmful, as in the case of overdiagnosis. We will mention this review paper in the revised version and explicitly discuss the limitations of screening in line with this work. In our work, we focus on the case when screening does not hurt. In our case, the only detrimental effect of screening is a false positive rate, which is expressed as a constraint in the optimization problem. We cannot screen everyone for every disease since screening is costly and screening resources are limited. We characterize exactly how limited screening resources should be distributed over the population so that the expected benefit is maximized.\n\n**Clarification of the screening example:** Consider a patient who (potentially) has both heart disease and cancer but is only screened for cancer, where the adverse event from cancer is expected to occur after the adverse event from heart disease. If we only screen for cancer, but the patient unexpectedly dies from heart disease, then the cancer screening offers no benefit to the patient in terms of lifetime gain. If the patient is screened for both diseases, heart disease will be identified earlier, the adverse event due to heart disease will be prevented, and cancer screening will result in lifetime gain by detecting cancer before the adverse event associated with it happens. \n\nWe hope that our response has addressed your concerns. Please let us know if you have any other concerns.", "claims_and_evidence": "The authors begin with a reasonable summary of the challenges of disease screening and the optimal use of resources in this context, although this could be clarified with respect to why we might not just stack these individual screening programs in all patients (which is implicit but may benefit from being made explicit). It is not entirely clear to me why the screening for one disease depends on the screening for another, outside of contexts where the risks are clearly contingent (e.g. pulmonary hypertension and cardiac disease). Is this approach meant to be limited to such contexts?\n\nThey further offer an in silico evaluation, attempting to assess the impact of their various policies in a simulated environment. While the in silico analysis is reasonably performed within these boundaries, I worry that the overall approach overstates the benefit of many screening programs implicitly. While the benefit of screening is intuitive, it often does not play out in practice (this excellent recent review should be discussed, as most screening programs have not actually been shown to offer any significant mortality benefit https://pmc.ncbi.nlm.nih.gov/articles/PMC10463170/). This should be softened. \n\nThe authors claim \"For example, treating one condition, such as heart disease, can enhance the effectiveness of screening for another, such as lung cancer\" without citation. What is meant by this? Is this referring to improving the accuracy of e.g. a chest x-ray because  there is less pulmonary edema? \n\nOverall, however, despite some of my concerns about the broader direction of the field (and my deeper skepticism of screening as naively applied), I feel that this paper makes a reasonable contribution to its literature field. \n\nI do, however, believe that these results must be discussed more cautiously, with the recognition that these toy formulations of screening may not fully align with the complex realities of clinical medicine in this context. Similarly, I feel the authors should provide some further justification of the value of this multi-disease screening approach and the specifics of their contentions regarding the relatedness of these diseases.", "methods_and_evaluation_criteria": "The authors offer a detailed mathematical optimization formula to combine multiple different policies. It is commendable that their methods attempt to incorporate multiple different areas of analysis. \n\nI am not able to comment in detail on the accuracy of their formulae, as my background is more in clinical medicine and practical applications of machine learning in clinical contexts.", "theoretical_claims": "As outlined above, I worry that this work (as with much of the work in this field) overstates the effectiveness of screening in general.", "experimental_designs_or_analyses": "I am not able to review the mathematical proofs in detail given my background, however their in silico analyses are reasonable overall.", "relation_to_broader_scientific_literature": "The authors offer an excellent engagement with the relevant literature, with appropriate commentary in the related works section. I also appreciate the excellent Table 1 approach to clearly situating this project within the broader literature. My concerns as outlined above are with a lack of engagement with literature skeptical of screening overall.", "essential_references_not_discussed": "As discussed above, further engagement with some of the medical literature skeptical of the broader benefits of screening may be worthwhile here.", "other_strengths_and_weaknesses": "Discussed elsewhere.", "other_comments_or_suggestions": "One quibble with the introduction is with the statement \"For example, screening a patient with high risks for both lung cancer and cardiovascular disease for only one condition might fail to improve their overall health outcomes, whereas a unified screening approach could yield better results\". This does not make it clear why one would not just screen for both in this patient. I believe it is substantiated elsewhere, bus this should be more clearly explained in the introduction.", "questions_for_authors": "See above - I have several questions regarding the underlying theoretical assertion of the connection between the multiple diseases being screened.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "33VEPERWnQ", "reviewer": "Reviewer_RcdJ", "summary": "This paper proposes a framework for unified screening of multiple diseases under budget constraints and competing risks. The authors formulate this as a referral problem where they choose which screening policies to activate based on patient risk profiles. They characterize optimal decision boundaries for the two-disease case and conduct in-silico experiments to compare against independent screening approaches.\n\n## update after rebuttal\n\nAfter carefully considering the authors' rebuttal, my assessment remains unchanged. While the authors addressed some technical concerns, the fundamental issue of limited practical applicability persists. The proposed method still lacks validation with real-world medical data, which is crucial for demonstrating clinical relevance and potential impact. For a method targeting medical applications, empirical validation with representative clinical data is essential to establish both reliability and utility.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "Thank you for the thorough review of our paper and constructive comments. \n\n**Figure 1** 1(a) shows the decision boundaries for independent screening (current standard). 2(a) shows the boundaries that characterize the optimal policy for our referral problem (2). Our main contribution is to mathematically characterize these boundaries (Lemma 4.1 to Lemma 4.4, Proposition 4.5). Main message: to act optimally, screening for disease 1 should also depend on the risk of disease 2. Boundary shapes also depend on the screening budget, screening costs, adverse event times, and risk distribution (see Appendix C). Practical interpretation: Clinics can use 1(b) to decide for what a patient with risk vector $(x_1,x_2)$ should be screened. \n\n**Scaling to more diseases:** $>2$ diseases does not put significant computational burden. Our referral policy optimization problem is a linear program (LP); hence, SoTA complexity bounds for LP apply to our setting.\n\nCharacterization of the optimal referral policy will be much more complex and far less intuitive for the case of $>2$ diseases. One would expect the number of boundaries of interest (e.g., $5$ as shown in Figure 1(b) for $N=2$) to grow combinatorially with $N$ as there can be boundaries separating screening of one subset of diseases from another. \n\n**Clinical validation:** Any simulation study will not fully capture the real-world benefits of joint screening. A definitive evaluation of our joint screening protocol versus independent screening (current standard) would require a large-scale randomized controlled trial with two arms—an undertaking that is beyond the scope of our current work. Nonetheless, we hope that our theoretical results and in-silico experiments, which demonstrate the potential benefits of joint screening in a simplified yet intuitive and representative setting, will inspire further empirical research in this area.\n\n**Uncertain disease risks and variable screening costs:** One can use off-the-shelf risk prediction models to obtain disease risks for particular diseases. For instance, Gail model (for breast cancer), QRISK3 (for cardiovascular disease), (normalized) polygenic risk scores, and even AI-based models can be used to provide risk scores. In our formulation, screening costs for different diseases can be different. In Figure 4, we also show the effects of varying screening costs. \n\n**Statistical significance:** Since our experiments are on simulated data, we report exact gains obtained by solving (2). Significance tests are not necessary since we are not making claims based on limited data. \n\n**Parameter selection and sensitivity of our results:** The choice of $T_0$ and $\\mu_n$ is done by examining clinical literature (first paragraph of Section 5.1). While our choices of screening budget and costs are not borrowed from a real-world study, in Appendix C, we discuss in detail how the behavior and the performance of the optimal referral policy changes as we vary each of these parameters. We hope that our detailed appendix resolves the reviewer's concerns about parameters. \n\n**Limited practical applicability:** There are many other examples where this methodology could be used to guide policymakers. One example can be found in sexual health services. For many settings, human immunodeficiency virus (HIV) and human papillomavirus (HPV) screening protocols are conducted at different health services and hence are independently considered (e.g., \nhttps://www.who.int/publications/i/item/9789240024168, https://pubmed.ncbi.nlm.nih.gov/38297406).\n\nMost of these screening protocols are guided by a certain level of risk (e.g., sexual behaviors). These screening measures carry varying costs (HIV antibody is not costly, whereas HPV requires more costs associated with invasive procedures carried out by a gynecologist (for cervical neoplasia) or proctologist (for anal neoplasia)). Finally, delayed screening has implications on the time to present with severe HIV disease (for HIV) and time to develop CIN2/3 or AIN2/3, that is, cervical or anal pre-cancer (for HPV). This methodology could help define which risk levels would warrant joint screening protocols (to plead for streamlined services) or if screening protocols can remain part to minimize these times to undesirable outcomes.\n\n**Minor improvements:** Compute complexity is not an issue, as the problem is LP. For $N=2$, our characterization of the decision boundaries and visualizations (e.g., Fig 1(b)) provide interpretable explanations of who should be screened. The same screening resources can be used more efficiently (no increase in screening costs for the population). We improve over independent screening in all cases (even when disease risks are independent). Reported numbers may vary based on the experimental setup, but the main message remains the same.\n\nWe hope that our response has addressed your concerns. Please let us know if you have any other concerns.", "claims_and_evidence": "The core claims are supported by clear and convincing evidence through a combination of rigorous theoretical analysis and comprehensive experimental validation. The limitations are appropriately acknowledged, and the evidence presented aligns well with the scope of claims made. The paper maintains high standards of scientific rigor in both its theoretical and empirical components.", "methods_and_evaluation_criteria": "While the theoretical method is sound, the evaluation would be much stronger with real medical datasets and more realistic screening scenarios that reflect actual clinical practice. The current evaluation framework demonstrates mathematical correctness but not practical applicability.", "theoretical_claims": "I reviewed some of the theoretical proofs, focusing primarily on Section 4 which contains the key theoretical claims about optimal screening policies. Other results seem to be correct but I have not checked the detailed proofs in the appendix.  Additional verification by other reviewers would be valuable.", "experimental_designs_or_analyses": "\bThe experimental design lacks clinical realism and proper statistical rigor is needed to support the practical claims. For example, parameter selection lacks justification (arbitrary choices for budget and screening costs, no explanation for choosing $T_0$ and $\\mu_n$ for survival times). For methodology comparison, independent screening baseline uses equal budget split without exploring other allocations. Only minor improvement in survival time (37.70 vs 37.47 years) is observed, which may not be statistically significant, but no statistical analysis provided. Provided that this is on a simulated data, the practical impact of the paper's results appears to be questionable.  The paper would be significantly strengthened by additional comparison to any existing clinical screening protocols.", "relation_to_broader_scientific_literature": "While prior work like Wright et al. (2015) and Peng & Xiang (2021) studied competing risks in single-disease contexts, this paper extends the framework to optimize screening decisions across multiple diseases simultaneously. However, there are some limitations in connecting to literature.\n\n- Does not fully explain how their optimal policy compares to existing clinical guidelines.\n- Limited discussion of how their theoretical results relate to real-world screening scenarios.\n- No comparison to other optimization approaches in healthcare resource allocation.\n\nIn summary, while the paper builds on existing ideas in disease screening and risk modeling, it could better contextualize its theoretical advances against practical clinical approaches in the field.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "Strengths:\n- The mathematical characterization of optimal policies is rigorous and well-developed.\n\nWeaknesses:\n1. Limited practical applicability:\n- Only handles 2 diseases\n- Relies on simplified assumptions about screening schedules\n- No validation with real-world medical data\n- Doesn't address clinical implementation challenges\n\n2. Minor improvements:\n- Small gain in survival times (37.70 vs 37.47 years) may not justify the increased complexity\n- No cost-benefit analysis of implementation effort versus expected gains\n\n3. Missing context:\n- Limited discussion of how this would integrate with existing clinical protocols\n- No consideration of practical constraints like patient preferences or healthcare system limitations\n- Doesn't address how to handle more realistic scenarios with uncertain disease risks\n\nThe paper makes solid theoretical contributions but needs stronger connections to clinical practice to demonstrate real-world significance.", "other_comments_or_suggestions": "NA", "questions_for_authors": "1. Figure 1 interpretation:\n- Could you explain the meaning of the curved boundaries in Figure 1(b)? What determines their shape and why do they differ from the straight-line boundaries in 1(a)?\n- What is the practical interpretation of the different colored regions in terms of clinical decision-making?\nA clear explanation would help evaluate the practical relevance of your theoretical results.\n\n2. Scaling to more diseases:\n- What are the theoretical and computational challenges in extending your approach to 3+ diseases?\n- Do you expect qualitatively different behavior in the decision boundaries with more diseases?\n- How would the computational complexity scale?\n\n3. Clinical validation:\n- Have you validated any aspects of your model against real clinical screening data or protocols?\n- What modifications would be needed to handle real-world factors like uncertain disease risks and variable screening costs?\nThis would help evaluate whether the theoretical gains would translate to practice.\n\n4. Statistical significance:\n- Did you perform statistical analysis on the survival time improvement (37.70 vs 37.47 years)?\n- How sensitive are these results to your parameter choices and assumptions?\nUnderstanding the robustness of the improvements would affect assessment of the paper's impact.", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Lg3B2N43um", "reviewer": "Reviewer_kr4k", "summary": "The problem of screening for multiple diseases is formalized as an optimization problem, specifically for the case where policies for each disease are predefined and the task is to decide which policies to activate given a vector of prior risks. Under a fixed budget and a few simplifying assumptions like sequential test independence, the optimal decision boundaries are studied analytically and through simulations.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "We thank the reviewer for the thorough review of our paper and constructive comments. \n\n**Limited results:** In (1), we formalize the joint screening and diagnosis problem. This is distinct from (2), which focuses on the referral problem. We choose to solve (2) rather than (1) because our aim is not to propose entirely new screening and diagnosis policies—which can be difficult to justify and implement in clinical practice. Instead, we build upon existing screening policies already endorsed in clinical guidelines.\n\nCharacterizing the optimal referral policy becomes significantly more complex and far less intuitive when considering more than two diseases. One would expect the number of boundaries of interest (e.g., $5$ as shown in Figure 1(b) for $N=2$) to grow combinatorial with $N$, as one must account for boundaries that separate screening decisions across different subsets of diseases.\n\nWe adopt periodic screening policies with deterministic schedules because such approaches are widely used in clinical practice, as recommended by numerous guidelines—particularly for the early detection of chronic or progressive diseases. Common examples include annual mammograms for breast cancer screening, biennial colonoscopies for colorectal cancer, and regular HbA1c tests for diabetes monitoring. \n\n**Screening target $Y_n$:** May represent a biomarker value, clinical metric, or similar quantity. Conditional independence over time is utilized in likelihood ratio and in proofs characterizing the decision boundaries. While the underlying disease state may evolve over time, this evolution can be modeled as a drift in the expectation of $Y_n$. The randomness in $Y_n$ is then primarily attributed to measurement noise or natural biological fluctuations, which we assume to be independent across time.\n\n**Why $x$ is entirely observable?** One can use off-the-shelf risk prediction models to obtain disease risks for particular diseases. For instance, Gail model (for breast cancer), QRISK3 (for cardiovascular disease), (normalized) polygenic risk scores, and even AI-based models can be used to provide risk scores.\n\n**Why objective make sense:** Our objective can be seen as a simplification of the standard Quality-Adjusted Life Years (QALYs) framework. Each year of life is either $1$ (acceptable quality, e.g., healthy) or $0$ (unacceptable quality, e.g., impaired, suffering). If the disease is detected before the adverse event happens, we assume issues related to the disease can be resolved or managed at a level such that the disease will not cause an unacceptable quality of life. However, after the adverse event happens, the disease causes unacceptable quality of life. \n\n**Relation to broader literature:** To the best of our knowledge, there is no existing decision-making problem in the literature that yields the same structure of decision boundaries as those derived in our work. While our screening cost budget constraint bears some resemblance to the constraint in knapsack problems, the underlying objective in our setting is fundamentally different. \n\n**About empirical results:** We agree that any simulation experiment may not fully capture the real-world benefits of joint screening. A definitive evaluation of our joint screening protocol versus independent screening (current standard practice as indicated by many guidelines) would require a large-scale randomized controlled trial with two arms—an undertaking that is beyond the scope of our current work. Nonetheless, we hope that our theoretical results and in-silico experiments, which demonstrate the potential benefits of joint screening in a simplified yet intuitive and representative setting, will inspire further empirical research in this area.\n\n**About an actual algorithm:** Our referral problem (2) is a linear program (LP) and can be solved by utilizing any LP solver. Some coefficients in this LP are computed by Monte Carlo sampling. \n\n**Relation to policy learning:** If the goal were to learn the optimal referral rule directly from a dataset of patient screening trajectories, the problem could indeed be framed as a policy learning task with an appropriately defined cost function. However, our work takes a complementary, orthogonal approach.\n\nWe begin with a well-defined optimization problem (2) and focus on analytically characterizing the structure of its optimal solution (the decision boundaries described in Figure 1(b) and formalized in Proposition 4.5). Rather than learning from data, our emphasis is on understanding the geometry and properties of the optimal referral rule under a known probabilistic model.\n\nOur results suggest a follow-up question: can the characterized decision boundaries be leveraged to develop sample-efficient algorithms for learning optimal referral rules from limited data? We view this as an important direction for future work.\n\nWe hope that our response has addressed your concerns. Please let us know if you have any other concerns.", "claims_and_evidence": "The main claims are largely supported. However, the results in this paper are more limited than what is stated in the introduction, and not all of the simplifying assumptions are clearly stated throughout. The referral problem that is ultimately considered has just two diseases with competing risks, policies with deterministic screening schedules, and independent screening tests. \n\nI believe that the language used to describe the method is overly complex, obscuring key design choices while taking up unnecessary space. For instance, the main optimization problem presented in Equation 2 relies on an integer mapping for sets of binary decisions that seems to be a minor implementation detail, and greatly increases the complexity of the notation. It was a challenge to parse through details like these and figure out what is really going on in the core problem presented by this paper.", "methods_and_evaluation_criteria": "Unfortunately, the verbosity mentioned above appears to have missed the need to justify key parts of the problem setting and the methodology. Starting from the beginning: \n * Please clarify the role of the screening target $Y_n$ when it is introduced. \n * Why can you assume that screening samples taken at different times are independent? Wouldn't nearby samples be more correlated? Or are these errors entirely due to measurement noise rather than an evolving disease?\n * Why can we assume that we observe the whole $x$ vector? It would be helpful to relate this to the medical examples that are mentioned earlier in the paper.\n * The objective of maximizing the time of the first adverse event that goes undiagnosed needs to be supported. Intuitively, it makes sense, but one could imagine many other objectives that accomplish similar things using expectations rather than minima.", "theoretical_claims": "The proofs for the lemmas appear to be correct, although I did not verify every detail.", "experimental_designs_or_analyses": "The experimental settings are fully synthetic and rather simple.", "relation_to_broader_scientific_literature": "My main concern relating to the broader literature is that the analysis of the problem might be artificially complicated by all the details that were introduced, and that the decision boundaries possibly correspond to much simpler results from decision theory, especially after you take into account all of the simplifying assumptions that were made along the way.", "essential_references_not_discussed": "Not to my knowledge.", "other_strengths_and_weaknesses": "The problem under consideration is highly significant. It would have strengthened this paper to present an actual algorithm for computing decision rules in one of the settings considered. All of the empirical results come from simple synthetic experiments.", "other_comments_or_suggestions": "Line 246: the equation has a typo. The variable $n$ appears to be used for two different purposes at once, $\\delta_n$ and also $\\exists n$ within the definition.", "questions_for_authors": "Could you give some intuition on how the referral optimization task is not equivalent to a simpler policy learning task, with a cost function that takes on a more specific structure?", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: This paper describes a novel optimization framework for unified screening of multiple diseases. All three reviewers gave weakly positive ratings, but they raised several important concerns. While the problem framework is novel, the resulting claims are sound, and the in-silico experiments are well-reasoned, the paper itself lacks grounding in actual clinical motivation. The lack of real-world data and relationship to existing clinical guidelines (raised by RcdJ) raise concerns about the overall contribution of this work for clinical science, which it claims as the main motivation for the paper. The paper only focuses on 2 diseases for screening, and there were concerns that the work appears to not consider the potentially detrimental effects of screening (yUUd), it does not explain how the optimal policy compares to existing clinical guidelines (RcdJ), and the presentation of material is overly unclear (kr4k). 

With the acknowledgement of the limitations, I still find the framing of the paper interesting, and the theoretical and in-silico claims are sound. For this reason, I'm recommending a weak accept.
Decision: Accept (poster)