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Compartmental Models for COVID-19 and Control via Policy Interventions

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We demonstrate an approach to replicate and forecast the spread of the SARS-CoV-2 (COVID-19) pandemic using the toolkit of probabilistic programming languages (PPLs). Our goal is to study the impact of various modeling assumptions and motivate policy interventions enacted to limit the spread of infectious diseases. Using existing compartmental models we show how to use inference in PPLs to obtain posterior estimates for disease parameters. We improve popular existing models to reflect practical considerations such as the under-reporting of the true number of COVID-19 cases and motivate the need to model policy interventions for real-world data. We design an SEI3RD model as a reusable template and demonstrate its flexibility in comparison to other models. We also provide a greedy algorithm that selects the optimal series of policy interventions that are likely to control the infected population subject to provided constraints. We work within a simple, modular, and reproducible framework to enable immediate cross-domain access to the state-of-the-art in probabilistic inference with emphasis on policy interventions. We are not epidemiologists; the sole aim of this study is to serve as an exposition of methods, not to directly infer the real-world impact of policy-making for COVID-19.
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Compartmental Models for COVID-19 and Control via
Policy Interventions
SWAPNEEL MEHTA, New York University
NOAH KASMANOFF, New York University
We demonstrate an approach to replicate and forecast the spread of the SARS-CoV-2 (COVID-19) pandemic
using the toolkit of probabilistic programming languages (PPLs). Our goal is to study the impact of various
modeling assumptions and motivate policy interventions enacted to limit the spread of infectious diseases.
Using existing compartmental models we show how to use inference in PPLs to obtain posterior estimates
for disease parameters. We improve popular existing models to reect practical considerations such as the
under-reporting of the true number of COVID-19 cases and motivate the need to model policy interventions
for real-world data. We design an SEI3RD model as a reusable template and demonstrate its exibility in
comparison to other models. We also provide a greedy algorithm that selects the optimal series of policy
interventions that are likely to control the infected population subject to provided constraints. We work within
a simple, modular, and reproducible framework to enable immediate cross-domain access to the state-of-the-art
in probabilistic inference with emphasis on policy interventions.
We are not epidemiologists
; the sole aim
of this study is to serve as a exposition of methods, not to directly infer the real-world impact of policy-making
for COVID-19.
1 INTRODUCTION
In order to understand and control infectious diseases, it is important to build realistic models
capable of accurately replicating and projecting their transmission in a region [Atkeson 2020b;
Sameni 2020;Tang and Wang 2020]. The underlying assumption being that a model capable of
replicating disease spread has captured the true causal dynamics suciently well. Motivated by
this, there has been a spate of research in applying SIR, SEIR, and SEI3RD models to this problem
[COVID et al
.
2020;López and Rodó 2020]. These belong to a class of compartmental models that are
underpinned by Lotka-Volterra dynamics that divide a population into sections or compartments
and describe the probabilistic transitions between them through a set of partial dierential equations.
We extend this direction of work with an emphasis on modeling under-reported cases, decoupling
policy interventions from compartmental transitions, estimating the impact of policy interventions,
selecting a sequence of optimal interventions to control the spread of diseases, and using the
SEI3RD variant as an extension of existing work on SEIR models since it forms a template for many
other extensions of compartmental models [Giordano et al
.
2020;Grimm et al
.
2021;Kennedy et al
.
2020;Senapati et al
.
2020;Winters 2020;Wol 2020]. The authors of [Hong and Li 2020], much like
us, propose a new statistical tool to visualize analyses of COVID-19 data. However, our framework
makes it much simpler to inspect the intricacies of modeling assumptions, expand with a custom
set of constraints, and experiment with counterfactual simulations without the need for extensive
compute or data. We have created some demo notebooks which will be made available publicly 1.
In light of probabilistic programming languages (PPLs) [Bingham et al
.
2019;Carpenter et al
.
2017;
Salvatier et al
.
2016;van de Meent et al
.
2018] reaching their ’coming-of-age’ moment, COVID-19
modeling is being explored to guide and support policy decisions and decision-makers [de Witt
et al
.
2020;Wood et al
.
2020]. In this work, we provide data scientists with a concrete example
of applying probabilistic inference to understand disease spread and control. To epidemiologists,
we oer this manuscript as a guide to design compartmental models to evaluate the impact of
1https://drive.google.com/drive/folders/1Npdn4bS_qlps5EdA6vXlMvRXSljGgYCd?usp=sharing
Authors’ addresses: Swapneel Mehta, New York University, Center for Data Science,
swapneel.mehta@nyu.edu
; Noah
Kasmano, New York University, Center for Data Science, nsk367@nyu.edu.
arXiv:2203.02860v1 [stat.ML] 6 Mar 2022
2 Swapneel Mehta and Noah Kasmano
SIR
𝛽𝛾
Fig. 1. The SIR Compartmental Model
SE I R
𝛽𝜎𝛾
Fig. 2. Add an ’Exposed’ compartment to the SIR model to obtain the SEIR model
policy interventions [Giordano et al
.
2020;Mandal et al
.
2020;Wang et al
.
2020b]. We include clear
motivation and detailed real-world examples of how to manually explore the impact of such non-
pharmaceutical interventions (NPIs) using synthetically generated data and provide an algorithm
to automatically generate strategies for implementing governmental policies for disease control.
Our main contributions are as follows:
Improve existing compartmental models with the ability to deal with the under-reporting of
COVID-19 cases and decoupling NPIs from disease parameters.
Evaluate our parameter estimates through empirical comparisons with those in prevalent
literature, observed patterns in testing coverage, and highlight the relative consistency of
our predictions compared to anomalies in existing approaches.
Model xed policy interventions and describe an algorithm to select adaptive policy inter-
ventions to aid government eorts to limit the spread of disease.
Highlight the ease of using PPLs to design, extend, and t SEI3RD models using approximate
inference to obtain posterior disease parameter estimates; in addition to having an open-
source code-base and our self-contained tutorial to be released at the time of publication.
2 SUSCEPTIBLE-INFECTED-RECOVERED (SIR) MODEL DYNAMICS
The SIR model dynamics form a template for the increasingly complex compartmental models we
explore. These dynamics are encapsulated in the following dierential equations following the
Lotka-Volterra Model:
𝑑𝑆
𝑑𝑡
=𝛽𝑆 (𝑡)𝑖(𝑡),𝑑𝐼
𝑑𝑡
=𝛽𝑆 (𝑡)𝑖(𝑡) 𝛾𝐼 (𝑡),𝑑𝑅
𝑑𝑡
=𝛾𝐼 (𝑡)
3 MODELING NOISY OBSERVATIONS
The population-level disease parameters we discussed for the SIR model (transmission probability
𝛽
and recovery rate
𝛾
) are usually estimated in terms of measurable quantities from observed
data such as the mean recovery time. The response rate
𝜌
indicates the proportion of observed
infections in reality because we typically cannot expect to observe every single case of infection.
To be clear, it is often the case that people may not realize they have been infected in the absence
of extensive testing. This has been the case in most countries at least for the initial few months
of the pandemic and some strategies attempt to remedy it in dierent ways [Jagodnik et al
.
2020;
Khan et al
.
2021;Lachmann 2020]. Since the real-world data we intend to use overlaps signicantly
with this duration, it makes sense to add this variable to our model. Importantly, the same may be
exacerbated due to socioeconomic and political factors, so
𝜌
allows us to model this under-reporting
of COVID-19 cases in practice. We start by making a guess about the range for the reproduction
Compartmental Models for COVID-19 and Control via Policy Interventions 3
number
𝑅0=
𝛽
𝛾
and response rate
𝜌
. This ’guess’ is equivalent to placing empirically informed
(from past outbreaks, for instance) priors on these latent variables and estimate their posterior
distributions.
3.1 Using PPLs for Modeling and Inference
PPLs are a natural candidate among the tools we considered for this problem. We would like
to dene a time-series probabilistic model with some underlying dynamics that encodes our
assumptions about the data-generating process; then we want to t this model to the data through
stochastic variational inference [Homan et al
.
2013;Ranganath et al
.
2014], and use the posterior
distributions over its parameters to make predictions about future time-steps. As a sanity check, we
also want to test how well it replicates the historical trajectory of the disease through simulations.
Pyro
[Bingham et al
.
2019], a deep universal probabilistic programming language, is an excellent
candidate for this because of the following reasons:
It comprises of thin wrappers around
PyTorch
[Paszke et al
.
2019] distributions allowing us
to write complex generative models interweaving stochastic and deterministic control ow
while still within a familiar and popular machine learning (ML) framework.
It oers general-purpose inference algorithms out of the box that allows us to shift the focus
from designing custom inference algorithms to building expressive models.
It recently extended support in terms of an API for epidemiological models. We perform our
experiments in
Pyro
since it is a much more accessible framework due to the community
around it in comparison to modern alternatives like
Pyprob
[Baydin et al
.
2019]. We can
conrm this empirically since we spent some time reproducing work along similar lines in
[Wood et al
.
2020] and found much less time required to do the same in
Pyro
. We do still
appreciate the relative ease of reproducing it compared to other machine learning research
for healthcare.
We consider a partially observed population and want to understand the putative controls that
could achieve a goal dened as "control the spread of the disease", "reduce the death rate", or other
such equivalent outcomes. While some of the parameters that dene disease spread are controllable,
there are certain non-controllable parameters including properties of the disease that we will infer.
From a computational standpoint, we demonstrate the capability of the universal probabilistic
programming language,
Pyro
, to combine a system of equations that dene a simulator and perform
inference over the latent variables within the simulation to obtain a posterior distribution over the
model parameters. In addition to an inference engine,
Pyro
oers the capability to intervene on the
variables within this simulation in order to obtain potential outcomes of policy changes that can be
expressed within the language as xing the values of certain stochastic parameters.
(a) Posterior distribution for 𝑅0(b) Posterior distribution for 𝜌
4 Swapneel Mehta and Noah Kasmano
Fig. 4. Predicting Future Infections for a Simulated Dataset in a Highly Infectious Seing
4 EXPERIMENTAL EVALUATION OF COMPARTMENTAL MODELS IN REAL-WORLD
SETTINGS
For our simulation-based study, we used models
𝑀=𝑆𝐼 𝑅, 𝑆 𝐸𝐼 𝑅, 𝑆𝐸𝐼 3𝑅𝐷
and rened versions of
each,
𝑀(𝑖)
, wherein we start with an infected population of 0.01%. We conduct some simulation
studies for instance by setting the
𝑅0=
3
.
0and
𝜌=
0
.
85. We perform variational inference [Homan
et al
.
2013;Ranganath et al
.
2014] to obtain posterior parameter estimates close to the true values.
These are indicated by the vertical black line in 3a and 3b. Since the posterior point-estimates are
close to the true values for the simulation, we can see why we are able to accurately replicate
the disease spread shown in 4. We plot the daily infections versus time and the plot resembles a
single wave of infected patients. The shaded portion indicates a 90% condence interval for the
model’s predictions which closely tracks the true values of disease spread. While it is no surprise
that SIR, SEIR, and SEI3RD models perform well on simulated data, there are extensive studies
in applying the SEIR model in practice. However, we believe that the SEIR model also has its
failures (see Qualitative Analysis of Successes and Failures) that often go undetected due to a lack
of comprehensive evaluation. We show this through an extensive set of experiments on real-world
data across multiple geographies. The tabular comparison of the reproduction number or
𝑅
0and
the
𝜌
estimated via our models are shown in Table 1for dierent time periods which oers the
following insights:
The reported cases for the initial period of January - July 2020 indicate far less testing than
the full periods (upto May 2021 for the USA and January 2021 for the rest of the world, in
our dataset).
The estimated
𝑅
0seemed high for most regions upto April 2020, and even though longer-term
estimates denoted an
𝑅
0of a little over 1 for most regions, the spate of cases underscores the
need to reduce the spread of the infection. At the same time, the reproduction number is not
Compartmental Models for COVID-19 and Control via Policy Interventions 5
all that matters and a careful study of mortality rates is warranted as testing increases and fa-
talities decrease in order to draw concrete scientic conclusions and policy recommendations
given this evidence.
We show qualitatively why it is necessary to build more granular, exible compartmental
models and underscore the need for separately modeling policy interventions as human
inuence on these processes.
The references for Comparing 𝑅0estimates with literature are drawn from prevalent literature
and expert-curated resources
2 3
on the subject [Gunzler and Sehgal 2020;Kamalich Muniz-Rodriguez
et al
.
[n. d.];Lau et al
.
2020;Prodanov 2021]. A certain George E. P. Box would be wont to say "All
𝑅
0estimates are wrong but some may be useful", and we illustrate this via an empirical and rather
qualitative route. Firstly, in some cases the reference method itself presents anomalous forecasts
like the
𝑅
0for Germany throughout the year being predicted as an unusual 22.032, and that of the
Netherlands, 9.103 from [Prodanov 2021]. In comparison, we observe relative consistency at least at
the model level, which might be a strong signal to consider more realistic modeling choices when
using real, noisy data for estimation. Where we lack consistency, we have a signal in the form of a
set of forecasts as well as an additional estimate of
𝜌
. An oddly low
𝜌
, for instance, might convince
us of an estimation error despite having high-condence and a good t to the observed data. We
observe that the SEI3RD models perform better than the SEIR and SIR models which might lack
the capacity to eectively model transitions. We also study a rened set of compartmental models
which consider an initial, partially infected (0.01%) population instead of starting from a single
infection (’patient zero’).
Once we successfully model disease progression and forecast for a set of future time-steps to
recover potential new infections. We can examine counterfactual questions of the nature ’What
would have happened to the number of cases if the government enacted
𝑋
steps at
𝑌
time?’
by introducing policy interventions
𝑢
at this stage and gure out the global minimally invasive
intervention to perform that will remain within the desired thresholds for the infected populace.
An adaptive algorithm to determine the optimal policy interventions is developed in this paper
as we continue to explore breadth-wise analyses pertaining to COVID-19 control, and compare
our models to the impact of real-world interventions. This discussion, albeit a core contribution, is
relegated to the appendix.
5 THE REPRODUCTION NUMBER
As responsible data scientists attempting to provide a tool for epidemiological analysis, it is impor-
tant not to overstate the relevance of estimating the correct 𝑅0from the data since that is not the
only parameter of interest. Many numbers have been bandied about in the news under the assump-
tion that controlling
𝑅
0implies controlling the pandemic. To a certain degree this is true, however
there are important caveats to this notion that some have expanded upon [Hébert-Dufresne et al
.
2020;Maruotti et al
.
2021]. The summary of the discussion is that
𝑅
0must be used as a tool to
paint a partial picture of a disease spread, in conjunction with multiple factors. In this regard,
our consideration of
𝜌
, modeling of infected fractions of populations, and consideration of policy
interventions are signicant steps taken to provide a comprehensive idea of the state of a pandemic.
ACKNOWLEDGMENTS
We would like to acknowledge guidance and support from Kyle Cranmer and Rajesh Ranganath
that laid the foundations for this research project.
2https://epiforecasts.io/covid/posts/national/united-states/
3https://covidestim.org/us/GA
6 Swapneel Mehta and Noah Kasmano
Table 1. Comparing 𝑅0estimates with literature
Region Model Jan - April, ’20 Jan - Dec, ’20
𝑅0𝜌Ref. 𝑅0𝑅0𝜌Ref. 𝑅0
Italy
SIR 1.48 𝜎=.01 0.187 2.676 1.34 0.125 1.8024
SIR(i) 1.712 0.76 𝜎=.03 - 0.608 0.506 -
SEIR 2.27 𝜎=.02 0.11 - 1.49 0.14 -
SEIR(i) 0.403 0.56 - 1.96 0.178 -
SEI3RD 3.03 𝜎=.07 0.557 - 5.78 0.001 -
SEI3RD(i) 1.247 0.532 - 0.215 0.505 -
Netherlands
SIR 1.69 0.36 𝜎=.01 1.9962 1.724 𝜎=0.84 0.71 9.103
SIR(i) 1.699 𝜎=.01 0.54 𝜎=.015 - 0.490 0.5079 -
SEIR 4.51 𝜎=.011 3.24 𝜎=.01 - 1.76 0.45 -
SEIR(i) 10.048 0.71 - 0.236 0.50 -
SEI3RD 4.23 𝜎=.04 0.735 - 2.78 0.154 -
SEI3RD(i) 3.763 𝜎=.03 0.70 - 3.51 0.108 -
New York
SIR 1.432 0.54 𝜎=.016 1.21 1.475 0.027 0.81
SIR(i) 1.643 0.594 - 2.078 0.619 -
SEIR 0.88 0.50 - 0.81 0.50 -
SEIR(i) 4.09 𝜎=.01 0.04 - 4.98 0.14 -
SEI3RD 1.53 0.504 - 1.67 0.503 -
SEI3RD(i) 10.79 𝜎=.006 0.0795 - 12.295 0.253 -
Germany
SIR 0.108 0.5186 1.637 1.610 0.120 22.032
SIR(i) 1.643 0.594 - 2.078 0.619 -
SEIR 2.51 𝜎=.013 0.76 𝜎=.01 - 2.93 𝜎=.10.154 -
SEIR(i) 1.973 0.213 - 5.02 𝜎=.47 0.57 -
SEI3RD 0.572 0.55 - 5.56 𝜎=.08 0.01 -
SEI3RD(i) 1.37 0.52 - 4.7 0.124 -
Georgia
SIR 3.74 0.303 1.45 3.296 0.343 0.84
SIR(i) 4.075 0.302 - 3.661 0.039 -
SEIR 5.3 𝜎=.17 0.015 - 3.47 0.56 -
SEIR(i) 4.97 𝜎=.078 0.014 - 5.36 0.096 -
SEI3RD 4.09 𝜎=.17 0.5483 - 1.32 0.501 -
SEI3RD(i) 6.8 0.049 - 5.62 0.205 -
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10 Swapneel Mehta and Noah Kasmano
A APPENDIX
B QUALITATIVE ANALYSIS OF SUCCESSES AND FAILURES
We have attempted to t many dierent kinds of models and obtained certain parameter estimates
which we compare to other, equally noisy parameter estimates from the literature. The fact is, each
country has dealt with COVID-19 in dierent ways. While China seemed to be able to take stringent
measures and limit the spread, others like Italy were not able to follow the same approach. There
was a strong focus on the fast-rising numbers of infected people in the United States (September,
2020), quickly overshadowed by a worse state of aairs in India (May, 2021). It is crucial to analyse
how well each model is able to not only t the data but also forecast the spread of disease accurately.
Since we have trained some models on partial amounts of data, let us take a look at which of the
models can identify upcoming ’waves’ in the progress of the disease.
For instance, in the gures 5(vertical bars mark the beginning of forecasts), the SEIR and the
SEI3RD models are able to capture the second wave, with the latter being able to, impressively
enough, predict the second one having only observed the rst wave. Similarly, in 6we can see
that both the SIR and SEIR models seem to perform poorly at data tting and forecasting. Note
that in both cases, the notion of the ’best model’ as indicated by comparing estimated
𝑅
0does
not seem to hold, emphasising the need to consider the least restrictive models when conducting
such studies. Of course, the SEI3RD model is not a panacea. We oer two additional examples
of its forecasts 8that convince us we need to model an external factor that allows us to modify
the second, larger wave (peak) in a manner similar to how government interventions delayed
the infections from successively peaking as per the predictions of the SEI3RD model. Also in 8
is the availability of condence intervals for the forecast. In general, while the SEI3RD model
performs well at capturing multiple waves of COVID-19 infections, we are making a fundamentally
incorrect modeling assumption by ignoring the fact that disease evolution was hindered by human
intervention, in particular via government policies designed to limit its spread. In fact, we can
utilise these ’vanilla’ SEI3RD models that are reasonably good at predicting the potential spread of
a disease to help us gure out how best to limit it! This motivates us to study Policy Interventions.
C COMPARTMENTAL MODEL EXTENSIONS IN PYRO
There is a vast body of literature on compartmental models for epidemiology. In particular, the
SEIR model seems the most popular and widely applied for real-world correspondence possibly
owing to a trade-o between simplicity and eectiveness [Buckman et al
.
2020]. In [COVID et al
.
2020] the team ts an SEIR model to mortality data in an eort to examine possible trajectories of
COVID-19 infections at the state level. In particular, they conduct a review of COVID-19 model on a
state-wise basis with particular emphasis on charting out potential scenarios in terms of the number
of fatalities in the presence of various non-pharmaceutical interventions. They forecast the best and
worst case outcomes in terms of these numbers and make recommendations to ensure the safety
of the US population in case of epidemic resurgences in many states. The authors of [López and
Rodó 2020] use an SEIR model to conduct a study of the recurrence of the COVID-19 pandemic via
dierent post-connement scenarios. Their work highlights the importance of non-pharmaceutical
interventions due to the re-emergence risk from the time-decay of acquired immunity and lack of
eective pharmaceutical interventions.
There is however signicant work that has gone into building more expressive, realistic, informed
models some of which are SIDARTHE [Giordano et al
.
2020], SEI3HR in India [Senapati et al
.
2020], SUEIHCDR in Brazil [Kennedy et al
.
2020], SEI3RSD [Wol 2020], SEI3Q3RD [Grimm
et al
.
2021], SEI3R2S [Winters 2020], and others [Ndaïrou et al
.
2020]. SEIR models for the spread
of disease [Buckman et al
.
2020,?;COVID et al
.
2020] are prevalent in the existing literature
Compartmental Models for COVID-19 and Control via Policy Interventions 11
Fig. 5. Fiing SIR (top le), SEI3RD (boom right) and SEIR models to data from Italy
Fig. 6. Fiing SIR (top le), SEIR (top right) and SEI3RD models to data from the Netherlands
with multitudinous global, national, and regional studies conducted through studying the disease
transmission dynamics modeled by its compartmental transitions. We conjecture that the SEIR
model can be improved by reducing the modeling assumptions implicit in its denition such as
the explicit separation of the recovered individuals from the fatalities out of all those in the nal
’removed’ compartment. In the framework oered by
Pyro
, it is straightforward to modify the
12 Swapneel Mehta and Noah Kasmano
Fig. 7. Fiing SIR (top le), SEIR (top right) and SEI3RD models to data from New York
Fig. 8. Fiing SEI3RD models to data from Germany and Georgia (right)
compartmental model structure which makes it extremely useful as a toolkit for practitioners
interested in an experimental perspective on epidemiological modeling.
As a template, we consider the idea of three-layered infection states corresponding to increasingly
infectious ’spreaders’ motivated by the need to tie epidemiology with medical physiology through
modeling causal dynamic processes in time [Winters 2020] for dening an SEI3RD model as shown
in 9.
For researchers, an excellent approach to reproducibly expand upon the existing literature could
be following the approach of [Ndaïrou et al
.
2020]. They examine an eight-compartment model to
obtain disease parameter estimates of a COVID-19 variant in Wuhan, China. They then conduct
a sensitivity analysis of their model to examine the variance with respect to each parameter and
compare their results with a numerical simulation to examine its suitability. Our motivation with
the SEI3RD model is, in a similar vein, to expand upon the line of work leading to the SEIR model but
with the dierence of oering an open-source template to introduce new variants of compartmental
models that can dier regionally.
Compartmental Models for COVID-19 and Control via Policy Interventions 13
Fig. 9. The SEI3RD Model from [Wood et al. 2020] along with our policy intervention parameter 𝑢
C.1 Policy Interventions
The use of surgical face masks and face shields by healthcare and non-healthcare workers alike
has been shown to signicantly reduce or prevent the transmission of human coronaviruses and
inuenza viruses through respiratory droplets from symptomatic individuals in conned spaces
[Chu et al
.
2020;Leung et al
.
2020;Liang et al
.
2020]. This is an example of a type of intervention that
falls into the class of non-pharmaceutical interventions (NPIs) which are important to model in light
of the time it requires to implement substantial pharmaceutical interventions (PIs) such as vaccines.
Our simulation, therefore, focuses on the modeling of NPIs, or what we term policy interventions,
to allow us to build world-models that are reective of disease spread in the absence of eective PIs.
While vaccines may be the most eective long-term solution, there is signicant impact of short
term NPIs such as isolation and contact tracing [Bertozzi et al
.
2020;Grimm et al
.
2021]. For this
reason, our work focuses extensively on a framework to explore intervention strategies without
the need for human-supervised search.
When governments deal with diseases, they may take certain measures that result in limiting
the exposure of the population to the disease. These measures may range from mild as in washing
hands to stringent as in enforcing a complete lockdown [International Monetary Fund [n. d.];
United States Government [n. d.]]. While some lines of work focus on how to monitor their impact
[Giudici and Ranetti 2020;Vasconcelos et al
.
2020], we propose an algorithm to explore new
strategies for control. The goal of epidemiological modeling, particularly the spread of pandemics
is to be able to infer the actions necessary to limit their spread. Every policy to address this is
designed to intervene on the rate of spread through natural or articial means, for a short or
long term duration. However, most modeling approaches treat it as aecting the transmission in a
deterministic manner whereas in reality, the impact of government policies varies with time. For
this reason, we decouple modeling of the policy intervention
𝑢
from the transmission probability
𝛽
.
This allows us to introduce our greedy search algorithm described in (also see 12).
Modelling policy interventions for COVID-19 has been the focus of a large body of work ([Chen
et al
.
2020;Giordano et al
.
2020;Giudici and Ranetti 2020;Vasconcelos et al
.
2020]). In our work,
we start with the similar implementation of a policy intervention dened by the parameter
𝑢
which
modies the SIR compartmental model as per the equation below:
𝛽1=(1𝑢)𝛽
𝛽=𝛽1
Recall that
𝛽
was the transmission probability of an individual from the pool of susceptible
individuals to the next compartment which is model-dependent. Changing the values of this policy
14 Swapneel Mehta and Noah Kasmano
Fig. 10. Visualizing the disease progression with time (along x-axis) and infected population fraction (along
the y-axis) under varying levels of the policy intervention parameter 𝑢for SIR.
intervention parameter manually, as in
𝑢
taking on a deterministic set of values, and simulating
the resulting trajectories results in the plot shown in 10 for the SIR model and 11 for the SEIR
model. The horizontal dashed line indicates the 10% threshold of infections. This approach allows
us to monitor what could have been the impact of governmental policies such as a lockdown
implemented over a long time period. However,
𝑢
may not be the same at all points in time. Thus,
we introduce the algorithm C.1 to consider a sequence of policy interventions that is analogous to
periodic policy updates or a variance in the impact of the same policies. A visual analogy of the
algorithm is oered by illustrating the best and worst case outcomes of selecting dierent policy
interventions at two time-steps in 12 with a description below.
Compartmental Models for COVID-19 and Control via Policy Interventions 15
Fig. 11. Visualizing the disease progression with time (along x-axis) and infected population fraction (along
the y-axis) under varying levels of the policy intervention parameter 𝑢for the SEIR model.
Algorithm 1: Adaptive Interventions through Greedy Search
Result: Sequence of policy interventions
initialize sequence;
while 𝑡𝑇do
simulate compartmental transitions;
for 𝑢=𝑢𝑖, increasing from 0to 1do
simulate a trajectory with 𝑅0,𝜌;
if infections threshold then
add 𝑢𝑖to sequence;
break;
end
end
end
The current fraction of infected population is represented in bright red as a solid line. At
each timestep, we perform a simulation of possible trajectories for dierent values of the policy
16 Swapneel Mehta and Noah Kasmano
Fig. 12. Adaptively Changing the Policy Intervention
intervention
𝑢
. For example, at a time
𝑡=
50 days we simulate the possible trajectories (dashed
lines). The plots show the best and worst-case options for the progress of the disease depending on
the magnitude of the policy intervention parameter
𝑢
. This is repeated at each time step
𝑡
, with the
illustration of 𝑡=100 oered in a separate plot.
A greedy solution, as demonstrated in the algorithm C.1, is to pick the
𝑢
in a short-term optimal
manner such that it keeps the infected fraction just within the threshold. However, this might still
result in a sequence of large interventions for a highly infectious disease (long-term lockdowns).
There are better strategies that enact more stringent measures early on so that the overall impact
of interventions across the time series is not as high. Alternatively, we might weight the outcomes
of infected individuals exceeding a threshold by the amount of time it would take to breach
the threshold and thereby determine a reasonable intervention to perform. There are dierent
ways of picking the optimal policy intervention depending on the choice of utility function and
optimisation technique, including neural networks. We believe that the choice should vary subject
to demographic factors, eectiveness of government policies, durations for which governments
can implement stringent measures and acceptable thresholds of infected individuals.
C.2 Limitations
We study the spread of infectious diseases in terms of the COVID-19 pandemic in an attempt to
unify the dierent models to highlight the utility of PPLs. For this reason, while we do study the
predictions of our models by evaluating the parameter estimates against those described in the
existing literature [Hoseinpour Dehkordi et al
.
2020;Kucharski et al
.
2020;Li et al
.
2020;Verity
et al
.
2020;Wang et al
.
2020a] in some experiments, it is more interesting for us to focus, in this
work, on our technique and its extensions than the fact that we achieve similar results to those in
related literature. We provide a complete comparison of some region-wise parameter estimates with
real-world data in 1but emphasize that the novelty is in the ease of application of this technique
across the table, its adaptability to incorporate modeling assumptions, and exibility of inference
across a spectrum of compartmental models.
It is challenging to model a disease with convenient mathematical assumptions implicit in
compartmental models, that largely ignore individual-level dierences. While there are some
inferences that are plausible to make, it is clear that we must not jump to immediate conclusions
Compartmental Models for COVID-19 and Control via Policy Interventions 17
in disease modeling due to the fact that there is almost always non-trivial uncertainty associated
with parameter estimates in that multiple models might reect similar initial trajectories [Atkeson
2020a]. Furthermore, certain types of interventions might not be as eective as others [Manchein
et al
.
2020]. However, modeling interventions is one way of translating theory into practical
advice for making policy decisions [Thompson 2020] and it is promising that SEIR models perform
well over time in multiple demographies with regards to predicting the spread of the disease
[Atkeson 2021]. This strengthens the argument that we would like to further reduce our modeling
assumptions and therefore, uncertainty in parameter estimates, in order to make more condent
policy recommendations eventually. This motivates the use of the SEI3RD model (among other
variants in Compartmental Model Extensions in Pyro) which is more powerful than the class of
SEIR models since it is a derivative that allows for more granular transmission dynamics between
compartments. However, even with the SEI3RD model, we need to add in external interventions,
consider jointly conditioning on observed fatalities, and inductive biases on transition probabilities
based on estimated incubation period, infection severity, stratication by demographic factors, and
consider utility
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