# Adaptive Interventions for Global Health:
A Case Study of Malaria

África Periánez
benshi.ai
Barcelona, Spain
<EMAIL_ADDRESS>
&Andrew Trister
Bill and Melinda Gates Foundation
Seattle, USA
<EMAIL_ADDRESS>
&Madhav Nekkar & Ana Fernández del Río
benshi.ai
Barcelona, Spain
<EMAIL_ADDRESS>
&Pedro L. Alonso
Faculty of Medicine, Universidad de Barcelona
Barcelona. Spain
<EMAIL_ADDRESS>
###### Abstract

Malaria can be prevented, diagnosed, and treated; however, every year, there
are more than 200 million cases and 200.000 preventable deaths. Malaria
remains a pressing public health concern in low- and middle-income countries,
especially in sub-Saharan Africa. We describe how by means of mobile health
applications, machine-learning-based adaptive interventions can strengthen
malaria surveillance and treatment adherence, increase testing, measure
provider skills and quality of care, improve public health by supporting
front-line workers and patients (e.g., by capacity building and encouraging
behavioral changes, like using bed nets), reduce test stockouts in pharmacies
and clinics and informing public health for policy intervention.

##  1 Introduction

Not everyone benefits equally from the same treatment, has the same
environment, or can receive therapy at the same time. Adaptive interventions
consider human differences as treatment begins and during its course to ensure
each person gets the treatment that continuously works for them. Interventions
are adaptive in that they adapt to the individual (providing personalization)
and their evolving context and needs (i.e., adjusting dynamically).

Both clinical and patient actions are essential to high-quality, low-cost,
effective healthcare. Human behavior constitutes the primary mode for
activating health improvements, including decisions made by clinicians and
their patients. In medical practice, clinicians can be gently nudged to
improve decision-making by providing a decision architecture in which optimal
default clinical actions are suggested. Outside controlled clinical
environments (e.g., intensive care units), patient actions (e.g., lifestyle
choices and treatment adherence) primarily determine health outcomes. Mobile
health is key to promoting continuous and proactive healthcare monitoring. It
can, for example, mitigate the lack of medication adherence, which can be low
and challenging to address, as patient actions control healthcare delivery
outside medical environments.

The data generated by the users of digital applications are instrumental in
determining past and current behaviors and predicting future conduct. This
information can be used to deliver personalized mobile-mediated interventions.
Such interventions aim to provide the appropriate type or degree of support by
adapting to an individual’s changing internal and contextual states (Nahum-
Shani et al., 2018; Menictas et al., 2019; Carpenter et al., 2020;
Bidargaddi.N amd Schrader et al., 2020; Coppersmith et al., 2021).

Mobile health applications also serve as a direct channel of communication
with their users, from front-line healthcare workers to patients and the
general public. Interventions and incentives can be delivered directly to the
users through their phones. Machine learning (ML) can help generate
predictions regarding the behavior of app users, health outcomes, and their
contexts. These predictions can be integrated with real-time information
pertaining to the users’ choices and circumstances to determine the
individuals or groups needing additional support. Moreover, these predictions
can be used to personalize the timing of the interventions delivered to each
user. Similarly, reliable predictions regarding the evolution or variations in
the demand for different medical supplies can help establish reminder and
suggestion systems that can help ensure that all the supply chain actors
(e.g., pharmacists) maintain adequate stocks of essential supplies at all
times. Motivational prompts, personalized reminders, incentives, engaging
elements (such as game-designed elements), and other nudges can be used to
boost and reinforce good practices (Hrnjic & Tomczak, 2019).

Adaptive interventions in healthcare have primarily been deployed in resource-
rich countries (Nahum-Shani et al., 2018; Hardeman et al., 2019). Given the
immense disease burdens borne by low- and middle-income countries and the
increasing smartphone penetration in these regions, the use of adaptive
interventions to improve health outcomes may be highly beneficial to global
health. In this research, we explore this potential based on a case study of
malaria. A discussion of the ML methods can be found in the appendix A.

##  2 Malaria: A Case Study

Malaria, caused by the parasite _Plasmodium falciparum_ , remains a deadly
illness across much of the Global South.

Malaria in pregnancy and infancy is a major public health concern and a key
driver of maternal and newborn mortality (Tarning, 2016). In 2020, nearly
630,000 deaths worldwide were caused by malaria, a disproportionate share of
which (96%) occurred in Africa (Organization, 2022). In Sub-Saharan Africa,
malaria is responsible for 12% of all child fatalities (Roser & Ritchie,
2019), and children under five account for approximately 80% of all malaria
deaths (Organization, 2022).

Although global malaria incidence and mortality have substantially decreased
in the recent two decades, progress stalled since 2015 (Noor & Alonso, 2022),
and researchers worry that the COVID-19 pandemic has disrupted malaria
intervention coverage, reversing these gains (Weiss et al., 2021).

###  2.1 Surveillance: Progress Towards Elimination

Even with the World Health Organization’s (WHO) historic approval of a malaria
vaccine (Li et al., 2022), senior leaders caution that a broader approach will
be needed for malaria eradication, including improved collection and usage of
high-quality data—from health-management information systems and electronic
databases to geospatial models (Alonso, 2021)—and flexible evaluation and
implementation of interventions by local decision-makers (Alonso & O’Brien,
2022). According to a global landscape review, malaria surveillance systems in
2015–-2016 were “insufficient to support the planning and implementing of
targeted interventions and measure progress toward malaria elimination”
(Lourenço et al., 2019). Such elimination efforts require the accurate
notification of individual cases within 24 h of diagnosis to provide timely
and targeted responses, which essentially represent adaptive interventions
(WHO, 2015).

Notably, such malaria surveillance frameworks must be integrated, data-driven,
tailored, and based on mobile platforms. Mobile health for malaria
surveillance, that uses a combination of message- and application-based
reporting, can support health workers and clinicians in recording malaria case
information (Githinji et al., 2014; Baliga et al., 2019; RTI, 2020; Oo et al.,
2021; Bhowmick et al., 2021), decreasing delays in case reporting to health
officials, and improving the quality of data collection. Data fragmentation
remains a barrier to a cohesive malaria response, and previously siloed data
from diverse sources (private/public healthcare providers, government
agencies, etc.) must be integrated to create a real-time, case-based malaria
surveillance system (Rahi & Sharma, 2020). Automation is essential, as
analytics related to threat monitoring, requirement identification, and system
performance must be readily available for decision-makers (Ohrt et al., 2015).

A comprehensive malaria surveillance system can inform policy interventions,
e.g., the allocation of mosquito nets, tests, and antimalarials can be
targeted to favor individuals and communities in need. By understanding the
changes in people’s mobility and clustering, the impact of non-pharmaceutical
interventions can be evaluated, and geographic areas in which additional
actions may be helpful can be identified (Grantz et al., 2020). Several
studies revealed that the characterization of travel patterns through
geolocation data and their combination with contextual information on malaria
incidence can inform strategies to target travelers and reduce transmission
(Milusheva, 2020).

###  2.2 Prevention

For pregnant women visiting institutions for antenatal care, the WHO
recommends the administration of intermittent preventive treatment with
sulfadoxine–pyrimethamine and distribution of long-lasting insecticide-treated
mosquito nets (LLINs) (Salomão et al., 2017).

Unfortunately, many pregnant women in low- and middle-income countries do not
receive either intervention owing to stock-outs in data-fragmented health
systems (Salomão et al., 2017). Accurate demand forecasts that can integrate
real-time contextual information are needed to ensure equitable and efficient
allocation of important preventive goods. For example, by building geospatial
models and combining data from net manufacturers, national programs, and
cross-sectional household surveys, researchers can develop detailed maps of
LLIN access and usage (Bertozzi-Villa et al., 2021). These analyses can also
be automated to identify geographic areas in which additional supplies or
different actions may be required.

Patient behavior also determines the efficacy of the interventions, as many
pregnant women fail to take their pills (Mubyazi et al., 2005) or do not use
bed nets for various practical reasons (Manu et al., 2017; Gultie et al.,
2020). These aspects highlight the importance of communication to encourage
behavioral changes (Ricotta et al., 2014).

For children aged 3–-59 months, WHO recommends seasonal malaria
chemoprevention (SMC) during the months of peak malaria transmission. Although
SMC effectively controls malaria and reduces hospitalizations (Diawara et al.,
2017; Baba et al., 2020; Issiaka et al., 2020; Cairns et al., 2021), clinical
data and pharmacokinetic analyses reveal that complete adherence to treatment
is observed in fewer than 20% of children outside the study setting (Ding et
al., 2020). Targeted interventions for behavioral changes are required to
improve the real-world effectiveness of SMC.

Caregiver-targeted message-based interventions to increase preventive health
behaviors (such as sleeping under a net) have been noted to be successful in
decreasing the malaria prevalence in children under the age of five (Mohammed
et al., 2019). Despite the vast potential of mobile health solutions, one-
size-fits-all interventions are typically implemented in which a standard
message is sent to all participants. There exists an enormous opportunity for
the delivery of personalized interventions that can appropriately incentivize
a given user and guide targeted public health campaigns, which may include
gamification and leveraging of individual social networks (Ernst et al.,
2017). As a potential example, researchers recently tested an ML model in
conjunction with an accelerometer-based approach for measuring a range of LLIN
use behaviors: Although these technologies represent a proof of concept at
present, they can support the implementation of financial incentives based on
granular LLIN-use monitoring over longer time periods (Koudou et al., 2022).

The only way to understand which interventions work best and which incentives
drive behavioral change is by running experiments on the ground (Banerjee et
al., 2010; Bates et al., 2012; Zhou et al., 2020a; 2021).

###  2.3 Quality of Care

####  2.3.1 Diagnosis

Malaria is one of the most under-diagnosed and over-treated diseases. The
potential for severe outcomes means that any patient with a fever (especially
a child) may be administered treatment for malaria, often without having
received a diagnostic test (Amankwa et al., 2019; Ajibaye et al., 2019; Boadu
et al., 2016; Beisel et al., 2016). The administration of drugs without a
conclusive test may accelerate antimalarial resistance, an increasingly
worrying problem according to WHO malaria experts (Rasmussen et al., 2022).

Although microscopy is the gold-standard for malaria diagnosis, its
implementation remains infeasible in many resource-constrained settings
(Beisel et al., 2016). Consequently, many stakeholders have turned to rapid
diagnostic tests (RDTs). However, their access remains limited, with frequent
stock-outs (Boadu et al., 2016; Blanas et al., 2013) that may be related to
inaccurate record-keeping. Digital tools and demand forecasting algorithms can
alleviate this problem, as has been proved in India with an app that included
a supply chain management component (Rajvanshi et al., 2021).

Furthermore, many lay community health workers struggle to appropriately
perform RDTs (Boadu et al., 2016; Blanas et al., 2013; Beisel et al., 2016),
and thus, capacity-building efforts to increase their skills are urgently
required. Digital monitoring of the health workers’ performance can serve as
an effective quality control strategy and a mode of delivering feedback
(Laktabai et al., 2018). Personalized digital nudges can direct the health
workers who need additional support to specific online learning resources or
encourage them to sign up for an in-person training session.

Patient-facing steering may also be required, as sick patients who receive a
negative test may still expect treatment. Patient and provider education
around antimicrobial resistance may help, in addition to increasing awareness
and clinical decision support for the management of other febrile illnesses.

####  2.3.2 Artemisinin Combination Therapy (ACT)

ACT is the first-line malaria treatment throughout most of the malaria-endemic
world. Notable issues include stock-outs (Blanas et al., 2013; Rowe et al.,
2009; O’Connell et al., 2011) and the distribution of low-quality
antimalarials, especially in urban areas Newton et al. (2017). Most
antimalarials are distributed through the private sector, where non-
artemisinin therapies are prevalent and 5–24 times less expensive than
quality-assured ACT (O’Connell et al., 2011). In this context, it is important
to strengthen the supply chain with a data-driven approach, potentially by
using demand forecasting algorithms to send appropriately timed reminders to
pharmacists to ensure that they restock necessary supplies.

The quality of care and clinical management of malaria remains widely varying
and substandard. In Sub-Saharan Africa, less than one-third of the children
diagnosed with malaria receive both a blood test diagnosis and appropriate
antimalarial treatment (Cohen et al., 2020). Evidence for gaps in the
community health workers’ and drug dispensers’ ability to appropriately manage
malaria has been found in several countries (Rowe et al., 2009; Blanas et al.,
2013; Chowdhury et al., 2020; Buabeng, 2010; Kamuhabwa & Silumbe, 2013).
Capacity-building efforts are clearly necessary, and adaptive interventions
can facilitate the assignment of appropriate content to each worker (Katsaris
et al., 2021). AI-based user segmentation and behavioral phenotyping can yield
highly granular cohorts to be focused on. For example, we research the
capacity development of midwives by predicting their demand for specific
contents (Guitart et al., 2021a) and building recommendation systems for an
e-learning app by predicting the chance that a given user clicks on a certain
item (Guitart et al., 2021b).

Lastly, patient adherence to antimalarial regimes is a notable issue, with
estimates suggesting that it ranges from 40% to 65% (Amponsah et al., 2015;
Mace et al., 2011; Onyango et al., 2012). Trials of message-based-reminders to
healthcare workers (Zurovac et al., 2011; Kaunda-Khangamwa et al., 2018) and
patients (Macías Saint-Gerons et al., 2022) have shown promise while
highlighting the necessity of personalized interventions involving relevant
and actionable messages (Buabeng, 2010). In the case of tuberculosis, where
adherence is also a problem, researchers have noted that an ML-driven approach
to predict TB adherence risk in conjunction with adaptive interventions
dynamically increases the odds ratio of next-day treatment adherence
verification by 35% (Boutilier et al., 2021).

####  2.3.3 Pre-Referral Treatment

Another crucial area where malaria care delivery can be strengthened is pre-
referral treatment, usually rectal artesunate (RAS), recommended by WHO for
young children with suspected severe malaria when injections are not available
(WHO, 2022). Although the effectiveness of RAS has been demonstrated in
placebo-controlled trials (Gomes et al., 2009), more recent studies have
indicated that its introduction in real-world conditions may increase the
malaria case-fatality rates (WHO, 2022). This phenomenon is attributable to
several reasons, such as low patient adherence to referral guidelines (Simba
et al., 2010) and inadequate ability of health workers to deliver pre-referral
care (Amboko et al., 2022) appropriately. Patient- and provider-facing
steering and capacity building are needed to strengthen health systems and
ensure that guidelines are followed.

##  3 Conclusions

Combating the devastating toll of malaria requires multifaceted and innovative
strategies. Technology alone is not a panacea, and no simple solution exists.
However, ensuring high-quality data collection from diverse sources, such as
mobile phones, supply chains, public surveys, and electronic health records,
can play a crucial step. These data can be integrated to obtain intrinsic and
contextual information that can drive personalized and adaptive interventions.

The use cases are vast: Reminding antenatal care clinics to stock up on LLINs;
providing capacity-building resources to struggling community health workers;
supporting a district health officer in responding to a new malaria outbreak;
incentivizing pharmacists to administer diagnostic tests before prescribing
treatments; or encouraging patients to adhere to a treatment regime.

### Acknowledgments

This work was supported, in whole or in part, by the Bill & Melinda Gates
Foundation INV-022480. Under the grant conditions of the Foundation, a
Creative Commons Attribution 4.0 Generic License has been assigned to the
Author Accepted Manuscript version that might arise from this submission.

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##  Appendix A Appendix: Tools and Methods

This appendix describes the proposed conceptual and methodological framework:
a data-centric behavioral ML platform (Tang et al., 2021) that leverages logs
from different types of mobile health solutions—together with contextual
information—to deliver adaptive interventions to healthcare workers and their
patients directly through their phones.

###  A.1 Data-Centric Platform

Our platform is data-centric in that data tracking, and labeling lies at its
core. Integration with the different apps is achieved through our _Software
Developer Kit_ (HealthKit SDK), which provides specifications on what
information should be logged (and how) and the messaging service that delivers
the interventions. Integration through this SDK ensures that various user
metrics and traits characterizing engagement and behavior are readily
available, which can help clarify the individuals to be targeted by specific
interventions and the time for delivering the interventions.

###  A.2 Machine Learning

The application of data science and ML methodologies to extract and predict
valuable information to inform intervention design lies at the core of the
software we build. As is generally the case, no single model is best across
all datasets and use cases. Use-case-specific data pipelines transform
incoming information through the HealthKit SDK into metrics ready to be
consumed by the statistical and ML models, which in turn produce additional
metrics that can be used in the intervention definition.

####  A.2.1 Reinforcement Learning

Reinforcement learning (RL) is the ideal paradigm for sequential decision-
making in dynamically evolving contexts that respond to those decisions. It
allows us to continually improve how we make choices for a patient at any
given moment, maximizing the potential for positive outcomes while minimizing
undesired side effects. A competition on policy learning for malaria control
using RL was for example included as part of the KDD Cup Challenge 2019 (Zhou
et al., 2020b; Nguyen et al., 2019; Zou, 2021).

Contextual are a formulation of the RL problem with limited state
representation, and dynamics (Burtini et al., 2015; Yao et al., 2021; Dwivedi
et al., 2022; Zhang et al., 2022). Similarly, restless bandits can be used for
resource allocation (Mate et al., 2022). They are significantly less data-
intensive than other RL approaches while being robust and tractable. This
framework is instrumental in the context of adaptive intervention delivery. At
their core lies the exploration-exploitation trade-off, i.e., the compromise
between clinical research (to gather knowledge about treatments) and clinical
practice (to benefit the patients) by assigning the best intervention possible
based on all available information at that point.

####  A.2.2 Time-varying and Dynamic Prediction Modeling

Time series datasets such as clinical records represent valuable sources of
information sometimes spanning a patient’s entire lifetime of care. The
approach to decision-making described in this paper is dynamic (i.e.,
sequential and adaptive). The analytic and predictive modeling to support this
needs to be similarly dynamic, with the time-dependent evolution of the
systems of interest and their characteristics at its core. Supervised and
unsupervised ML with time-varying data (such as the survival analysis
described in the next section), time series modeling, and longitudinal data
processing should be part of the toolbox. We can use them to understand and
predict how the systems of interest behave and evolve to inform our decision-
making.

For behavioral nudging, besides the individual predictions generated using
survival analysis described below, multivariate time series forecasting can be
critical to optimizing specific interventions, such as reminders to prevent
stockouts of medical supplies based on demand prediction. Furthermore,
considering the time-varying nature of data and interventions within an RL
framework allows us to understand and optimize a patient’s treatment as
different sequential interventions in time instead of as a single-point
decision.

Predictive modeling can help define the target individuals for the
intervention and appropriate delivery time. As discussed, accurate demand
prediction is key to optimizing the supply chain and inventory. Forecasting
methods that can learn simultaneously from multiple time series, often
combining deep and state space modeling elements, have been noted to be
effective (Seeger et al., 2017; Salinas et al., 2019; Lim et al., 2019;
Salinas et al., 2020; Benidis et al., 2020).

####  A.2.3 Deep and Ensemble Survival Analysis

With time dependence at its core, survival analysis refers to a collection of
algorithms used to predict time to an event of interest (which was
traditionally death or organ failure) (Wright et al., 2017; Fu & Simonoff,
2016; Lee et al., 2020). These methodologies can be used to predict behavior
and health outcomes at the individual level, as they are well suited to
extract information from censored data (i.e., models that can learn from
subjects who have yet to experience the event of interest besides those that
have). Their output is a survival curve for each subject, indicating the
probability of not having experienced the event of interest depending on the
time. By carefully selecting the events we predict and how we measure time, we
can profile individuals using multiple predictions and risk scores (of
clinical complications or of lack of adherence to treatment, for example).

####  A.2.4 Recommendation Systems

Recommendation algorithms can be used to decide the techniques to target
health workers, patients, and the general public. Different methodologies are
expected to be suitable for different apps and use cases, from deep learning-
based click-through-rate predictions (Qu et al., 2016; Guo et al., 2017; Lian
et al., 2018; Lu et al., 2020; Guitart et al., 2021b) to collaborative,
interactive recommendation systems Ie et al. (2019); Liu et al. (2019);
Lillicrap et al. (2019); Chen et al. (2020). We foresee a growing shift toward
reinforcement-learning-based methods beyond the (relatively widespread) use of
contextual bandits (Barraza-Urbina & Glowacka, 2020). This approach is
particularly useful for optimizing a _sequence of interventions_ while
reconciling short- and long-term goals in the desired outcome.


$C_{p,d,S}^{\operatorname{pr}}>0$ such that for every $d$-linear form
$m:(\mathbb{F}_{p}^{n})^{d}\rightarrow\mathbb{F}_{p}$, if
$\max_{t\in\mathbb{F}_{p}^{*}}|\operatorname{bias}_{t,S}m|\geq\epsilon$ then
$\operatorname{pr}m\leq C_{p,d,S}^{\operatorname{pr}}\log\epsilon^{-1}$.

Conjecture 7.2 specialises in the case $S=\mathbb{F}_{p}$ to the well-known
conjecture that partition and analytic rank are equal up to a constant, and
which was recently established in the large fields case by Cohen and
Moshkovitz [2]. As we explained (following Janzer and Milićević) in the
introduction, in the case $S=\mathbb{F}_{p}$ a bound $\operatorname{pr}\leq
A_{d,p}(\operatorname{ar})$ translates into a bound of the type
$O_{p,d}(1)A_{d,p}(\operatorname{ar})$ in Theorem 1.4. Although the proof of
this implication no longer holds for an arbitrary non-empty subset $S$ of
$\mathbb{F}_{p}$, it seems at least plausible to us that if Conjecture 7.2 is
true then we can take linear bounds in $\log\epsilon^{-1}$ in Theorem 1.4.

######  Conjecture 7.3.

Let $p$ be a prime, let $d\geq 2$ be a positive integer, and let $S$ be a non-
empty subset of $\mathbb{F}_{p}$. Then there exists a constant
$C_{p,d,S}^{\operatorname{rk}}>0$ such that for every polynomial
$P:\mathbb{F}_{p}^{n}\rightarrow\mathbb{F}_{p}$ with $\deg P=d$, if
$\max_{t\in\mathbb{F}_{p}^{*}}|\operatorname{bias}_{t,S}P|\geq\epsilon$ then
$\operatorname{rk}P\leq C_{p,d,S}^{\operatorname{rk}}\log\epsilon^{-1}$.

Even if Conjecture 7.2 is true, it would still not guarantee linear bounds in
$\log\epsilon^{-1}$ in Theorem 1.4. It is conjectured in [9] that the disjoint
partition rank and essential partition rank are also equal up to a constant.
If this were proved then the bounds of Theorem 1.4 would significantly
improve, but owing to the inductive structure of our proof of Theorem 1.4 this
would still not yield linear bounds in $\log\epsilon^{-1}$. We therefore
expect that arguments significantly different from those that we have used in
our inductive proof would be required to prove Conjecture 7.3, if it is true.

In another direction we can ask for qualitative strengthenings of Theorem 1.4:
even in the case $S=\mathbb{F}_{p}$, having high rank is merely a sufficient,
and not a necessary condition for a polynomial to be approximately uniformly
distributed. For instance, if $I,J$ are two disjoint subsets of $[n]$, and
$P_{1},P_{2}$ are two polynomials both with degree at least $2$ such that
$P_{1}$ is a polynomial in the variables $x_{i}$, $i\in I$ and $P_{2}$ is a
polynomial in the variables $x_{j}$, $j\in J$, then it suffices that the rank
of either of the individual polynomials $P_{1},P_{2}$ is large for their sum
$P_{1}+P_{2}$ to be approximately uniformly distributed, but if $\deg
P_{1}<\deg P_{2}$ and $\operatorname{rk}P_{2}=1$, then
$\operatorname{rk}P_{1}+P_{2}=2$.

######  Question 7.4.

Let $p$ be a prime, let $d\geq 2$ be a positive integer, let $S$ be a non-
empty subset of $\mathbb{F}_{p}$, and let $\epsilon>0$. Let
$P:\mathbb{F}_{p}^{n}\rightarrow\mathbb{F}_{p}$ be a polynomial such that
$\max_{t\in\mathbb{F}_{p}^{*}}|\operatorname{bias}_{t,S}P|\geq\epsilon$. Can
we describe the structure of $P$ over and above the fact that there exists a
polynomial $P_{0}:\mathbb{F}_{p}^{n}\rightarrow\mathbb{F}_{p}$ such that
$P_{0}(S^{n})=\\{0\\}$ and $P-P_{0}$ has bounded rank ?

In fact, to our knowledge not much is known about Question 7.4 even in the
case $S=\mathbb{F}_{p}$. Theorem 1.4 is an analogue of Proposition 1.8 for
polynomials, and we can also ask whether we can obtain an analogue of
Proposition 6.3 for polynomials.

######  Conjecture 7.5.

Let $p$ be a prime, let $d\geq 2$ be a positive integer, and let $S$ be a non-
empty subset of $\mathbb{F}_{p}$. Then there exists a positive integer
$R(p,d)$ such that if $P:\mathbb{F}_{p}^{n}\rightarrow\mathbb{F}_{p}$ is a
polynomial and the restriction of $P$ to $S^{n}$ is not surjective, then there
exists a polynomial $P_{0}:\mathbb{F}_{p}^{n}\rightarrow\mathbb{F}_{p}$ such
that $P_{0}(S^{n})=\\{0\\}$ and we can write

$P-P_{0}=\sum_{i=1}^{R(p,d)}l_{i,1}\dots l_{i,d}$

for some linear forms $l_{i,j}:\mathbb{F}_{p}^{n}\rightarrow\mathbb{F}_{p}$.

Proposition 3.7 no longer seems to be of any help for the task of proving
Conjecture 7.5, but it is conceivable that some similar decoupling strategy
could work to reduce Conjecture 7.5 to Proposition 6.3.

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