Data-Driven Methods for Present and Future Pandemics: Monitoring, Modelling and Managing
Teodoro Alamo, Daniel G. Reina, Pablo Millán Gata, Victor M. Preciado, Giulia Giordano
DData-Driven Methods for Present and Future Pandemics:Monitoring, Modelling and Managing (cid:63)
Teodoro Alamo a , Daniel G. Reina b , Pablo Mill´an Gata c , Victor M. Preciado d , Giulia Giordano e a Departamento de Ingenier´ıa de Sistemas y Autom´atica, Universidad de Sevilla, Escuela Superior de Ingenieros, Sevilla b Departamento de Ingenier´ıa Electr´onica, Universidad de Sevilla, Escuela Superior de Ingenieros, Sevilla c Departamento de Ingenier´ıa, Universidad Loyola Andaluc´ıa, Seville, Spain d Department of Electrical and Systems Engineering, University of Pennsylvania, Philadelphia, USA e Department of Industrial Engineering, University of Trento, Trento, Italy
Abstract
This survey analyses the role of data-driven methodologies for pandemic modelling and control. We provide aroadmap from the access to epidemiological data sources to the control of epidemic phenomena. We review theavailable methodologies and discuss the challenges in the development of data-driven strategies to combat the spread-ing of infectious diseases. Our aim is to bring together several di ff erent disciplines required to provide a holisticapproach to the epidemic, such as data science, epidemiology, or systems-and-control theory. A 3M-analysis is pre-sented, whose three pillars are: Monitoring, Modelling and Managing. The focus is on the potential of data-drivenschemes to address di ff erent challenges raised by a pandemic: (i) monitoring the epidemic evolution and assessing thee ff ectiveness of the adopted countermeasures; (ii) modelling and forecasting the spread of the epidemic; (iii) makingtimely decisions to manage, mitigate and suppress the contagion. For each step of this roadmap, we review consol-idated theoretical approaches (including data-driven methodologies that have been shown to be successful in othercontexts) and discuss their application to past or present epidemics, as well as their potential application to futureepidemics. Keywords:
Pandemic control, epidemiological models, machine learning, forecasting, surveillance systems,epidemic control, optimal control, model predictive control.
1. Introduction
The 2019 coronavirus pandemic (Covid-19) is one ofthe most critical public health emergencies in recent hu-man history. While facing this pandemic, governments,public institutions, health-care professionals, and re-searches of di ff erent disciplines address the problem ofe ff ectively controlling the spread of the virus while min-imizing the negative e ff ects on both the economy andsociety. The challenges raised by this pandemic re-quire a holistic approach. In this document, we ana-lyze the interplay between data science, epidemiologyand control theory, which is crucial to understand and (cid:63) The authors belong to the CONtrol COvid-19 Team, includingmore than 35 researches from universities of Spain, Italy, France, Ger-many, United Kingdom and Argentina. The main goal of the CONCO-Team is to develop data-driven methods to better understand and con-trol the Covid-19 pandemic.T. Alamo acknowledges MEyC Spain (contract PID2019-106212RB-C41). manage the spread of diseases both in human and an-imal populations. In line with current epidemiologicalneeds, this paper aims to review available methodolo-gies, while anticipating the di ffi culties and challengesencountered in the development of data-driven strate-gies to combat pandemics. We consider the Covid-19pandemic as a case study and summarise some lessonslearned from this pandemics with the hope of improvingour preparedness at handling future outbreaks.In the context of epidemics outbreaks, data-driventools are fundamental to: (i) monitor the spread of theepidemic and assess the potential impact of adoptedcountermeasures, not only from a health-care perspec-tive but also from a socioeconomic one; (ii) model andforecast the epidemic evolution; (iii) manage the epi-demic by making timely decisions to mitigate and sup-press the contagion. Optimal decision making in thecontext of a pandemic is a complex process involvinga significant amount of uncertainty; at the same time, Preprint submitted to Journal Name March 1, 2021 a r X i v : . [ ee ss . S Y ] F e b ot reacting timely and with adequate intensity, even inthe presence of overwhelming uncertainties, can lead tosevere consequences. This survey provides a holisticroadmap that encompass from the process of retriev-ing epidemiological data to the decision-making pro-cess aimed at controlling, mitigating and preventing theepidemic spread. A 3M-analysis is proposed, coveringthree main aspects: Monitoring, Modelling and Manag-ing, as shown in Figure 1. A more detailed document,focused on the Covid-19 pandemic, can be found in thepreprint [3]. Each step of this roadmap is presentedthrough a review of consolidated theoretical methodsand a discussion of their potential to help us under-stand and control pandemics. When possible, exam-ples of applications of these methodologies on past orcurrent epidemics are provided. Data-driven method-ologies that have proven successful in other biologicalcontexts, or have been identified as promising solutionsin the Covid-19 pandemic, are highlighted. This surveydoes not provide an exhaustive enumeration of method-ologies, algorithms and applications. Instead, it is con-ceived to serve as a bridge between those disciplines re-quired to develop a holistic approach to the epidemic,namely: data science, epidemiology, and control theory.Data are a fundamental pillar to understand, model,forecast, and manage many of the aspects required toprovide a comprehensive response against an epidemicoutbreak pandemic. There exists many di ff erent opendata resources and institutions providing relevant infor-mation not only in terms of specific epidemiologicalvariables but also of other auxiliary variables that fa-cilitate the assessment of the e ff ectiveness of the imple-mented interventions (see [4] for a review on open dataresources and repositories for the Covid-19 case). Sincethe available epidemiological data su ff er from severelimitations, methodologies to detect anomalies in theraw data and generate time-series with enhanced quality(like data reconciliation, data-fusion, data-clustering,signal processing, to name just a few) play a crucial role.Another important aspect of the 3M-approach is thereal-time surveillance of the epidemic, which can be im-plemented by monitoring mobility, using social mediato assess the compliance to restrictions and recommen-dations, pro-active testing, contact-tracing, etc. The de-sign and implementation of surveillance systems capa-ble of early detecting secondary epidemic waves is alsovery important.Modelling techniques are also fundamental in thefight against pandemics. Epidemiological models rangefrom coarse compartmental models to complex net-worked and agent-based models. Fundamental param-eters characterizing the dynamics of the virus can be Figure 1: 3M-Approach to data-driven control of an epidemic: Moni-toring, Modelling and Managing. identified using these models. Besides, data-driven pa-rameter estimation provides mechanisms to forecast theepidemic evolution, as well as to anticipate the e ff ec-tiveness of adopted interventions. However, fitting themodels to the available data requires specific techniquesbecause of critical issues like partial observation, non-linearities and non-identifiability. Sensitivity analysis,model selection and validation methodologies have tobe implemented [147], [39]. Apart from the forecast-ing possibilities that epidemiological models o ff er, al-ternative forecasting techniques from the field of datascience can be applied in this context. The choiceranges from simple linear parametric methods to com-plex deep-learning approaches. The methods can beparametric or non-parametric in nature. Some of thesetechniques provide probabilistic characterizations of theprovided forecasts.Several measures to mitigate the epidemic can befound in the literature, but one needs to be careful abouttheir e ff ectiveness [230]. Some measures, like an ag-gressive lockdown of an entire country, have a devas-tating e ff ect on the economy and they might be adoptedat very precise moments, preferably as early as possi-ble and for short time periods. Other measures, likepro-active testing and contact-tracing, can be very ef-2ective while having a minor impact on the economy[73]. In this direction, control theory provides a consol-idated framework to formulate and solve many relevantdecision-making problems [162], such as the optimal al-location of resources (e.g. test reagents and vaccines)and the determination of the optimal time to implementcertain interventions. The use of optimal control the-ory and (distributed) model predictive control has greatpotential in epidemic control. Mathematical tools fromthe fields of control theory and dynamic systems, suchas bifurcation theory and Lyapunov theory, have beenextensively used to characterize the di ff erent possiblequalitative behaviours of epidemics.This survey is organized as follows: Section 2 de-scribes di ff erent methodologies to monitor the currentstate of a pandemic. An overview of di ff erent tech-niques to model an epidemic is provided in Section 3.The main forecasting techniques are described in Sec-tion 4. The question of how to assess the e ff ectivenessof di ff erent non-pharmaceutical measures is analyzed inSection 5. The decision making process and its link withcontrol theory is addressed in Section 6. The review pa-per is finished with a section describing some conclu-sions and lessons learnt.
2. Estimation of the state of a pandemic
There is a plethora of indicators that can be moni-tored in order to contain a pandemic. This includes notonly estimations of the current incidence of the diseasein the population and the health-care system, but alsothe (daily) surveillance of measures that directly or in-directly a ff ect its spread, such as physical distancing andmobility, as well as testing and contact tracing. In orderto design an e ff ective response to an epidemic outbreak,it is of upmost importance to build up-to-date estima-tions of the epidemic state. This estimation process ishindered by the presence of an incubation period of theinfectious disease, which introduces a time-delay be-tween the beginning of a new infection and its potentialdetection. Another challenge in the estimation processis the presence of infectious but asymptomatic cases,which is an important transmission vector in the caseof Covid-19 [73]. These (and other) challenges moti-vate the need for specific surveillance and estimationmethodologies capable of using available informationin order to design quick and e ff ective control measures.In this section, we cover the most relevant techniquesto monitor the state of the pandemic, focusing on ap-proaches oriented towards (i) real-time monitoring ofdi ff erent aspects of the pandemic (real-time epidemi-ology); (ii) early detection of infected cases and im- mune response estimation (pro-active testing); (iii) es-timation of the current fraction of infected population,both symptomatic and asymptomatic (state estimationmethods); (iv) early detection of new waves (epidemicwave surveillance). The use of a large number of real-time data streamsto infer the status and dynamics of a population’s healthpresents enormous opportunities as well as significantscientific and technological challenges [26], [240], [62].Real-time epidemic data can vary widely in nature andorigin (e.g., mobile phone data, social media data, IoTdata and public health systems) [4],[212]. During theCovid-19 pandemic, mobile phone data, when usedproperly and carefully, have provided invaluable infor-mation for supporting public health actions across early,middle, and late-stage pandemic phases [168]. Volun-tary installation of Covid-19 apps or web-based toolshave allowed the active retrieval of data related to ex-posure and infections. The information steming fromthese sources have provided real-time epidemiologicaldata that have then been used to identify hot spots foroutbreaks [62]. Social media have also been relevant toassess the mobility of the population and its awarenesswith regard to physical distancing, as well as the stateof the economy and many other key indicators [243].Our ability to extract information regarding popula-tion mobility is essential to predict spatial transmission,identify risk areas, and decide control measures againstthe disease. Nowadays, the most e ff ective tool to accessreal-time mobility data is through Big Data technologiesand Geographic Information Systems (GIS). These sys-tems have played a relevant role when addressing pastepidemics like SARS and MERS [172], providing e ffi -cient aggregation of multi-source big data, rapid visual-ization of epidemic information, spatial tracking of con-firmed cases, surveillance of regional transmission andspatial segmentation of the epidemic risk [243], [226]. Proactive testing is key in the control of infectiousdiseases, since it allows us to identify and isolate in-fected individuals. It also provides relevant informationto identify risk areas, fraction of asymptomatic carriers,and attained levels of immunization in the population[228], [234]. There are di ff erent methodologies to ap-proach proactive testing: • Risk-based approach : In this approach, one musttest first those individuals with the highest prob-ability of being carriers of the disease (i.e. not3nly those with symptoms, but also those who havebeen heavily exposed to the disease). For example,health-care workers are at high risk and can also berelevant transmission vectors. Second, test thoseindividuals that have been exposed to a confirmedcase according to contact tracing. Finally, testthose individuals who have recently travelled to hotspots [226]. The determination of hot spots canbe done by means of government mobility surveil-lance or by personal software environments [62]. • Voucher-based system : In this system, peoplewho test positive are given an anonymous voucherthat they can share with a limited number of peoplewhom they think might have infected. The recip-ients can use this voucher to book a test and re-ceive their test results without ever revealing theiridentity. People receiving positive result are givenvouchers to further backtrack the path of infection;see [190] and [158] for the Covid-19 case. • Serology studies : Some tests (such as RT-PCRrevealing viral load) are unable to detect past in-fection. Conversely, serological tests, carried outwithin the correct time frame after disease onset,can detect both active and past infections, sincethey detect antibodies produced in response to thedisease. Serological analysis can be useful to iden-tify clusters of cases, to retrospectively delineatetransmission chains, to ascertain how long trans-mission has been ongoing, or to estimate the frac-tion of asymptomatic individuals in the population[228].
As we will see in the next section, dynamic state-space epidemiological models are fundamental to char-acterize how the virus spreads in a specific region andestimate time-varying epidemiological variables that arenot directly measurable [48],[197]. Classical state-space estimation methods, like the Kalman filter [187],are employed to estimate the fraction of currently in-fected population. The objective of the Kalman filter isto update our knowledge about the state of the systemwhenever a new observation is available [64]. Di ff erentmodifications and generalizations of the Kalman filterhave been developed and tailored to epidemic models.These methodologies are essential both to the estima-tion problem and to the inference of the parameters thatdescribe the model (see [199] and [1]). Infectious diseases often lead to recurring epidemicwaves interspersed with periods of low-level transmis-sion, as observed, for example, in the ”Spanish” flu[186], Influenza [218] and Covid-19 [83]. In this con-text, it is crucial to implement a surveillance system ableto detect, or even predict, recurring epidemic waves,so as to enable an immediate response aiming to re-duces the potential burden of the outbreak. Detectingoutbreaks requires methodologies able to process hugeamount of data stemming from various surveillance sys-tems [12], [63], [66] and determine whether the spreadof the virus has surpassed a threshold requiring mitiga-tion measures; see, e.g. [129]. A large body of literaturefocuses on epidemiological detection problems, sincemany infectious diseases undergo considerable seasonalfluctuations with peaks seriously impacting the health-care systems [205], [216]. National surveillance sys-tems are implemented world-wide to rapidly detect out-breaks of influenza-like illnesses, and assess the e ff ec-tiveness of influenza vaccines [218], [209]. Specificmethodologies to determine the baseline influenza ac-tivity and epidemic thresholds have been proposed andimplemented [219]. These methods aim at reducingfalse alarms and detection lags. Outbreak detection canbe implemented in di ff erent ways that range from sim-ple predictors based on moving average filters [71] andfusion methods [63] to complex spatial and temporalanalyses [50], [21].In the early phases of a new pandemic, such as therecent Covid-19, the detection of recurring epidemicwaves is particularly challenging because: (i) histori-cal seasonal data are lacking, (ii) determining the cur-rent fraction of infected population can be di ffi cult whenmany asymptomatic infected are present, and (iii) deter-mining baselines and thresholds requires a precise char-acterization of the regional (time-varying) reproductionnumber.
3. Epidemiological models
Mathematical epidemiology is a well-establishedfield aiming to model the spread of diseases both in hu-man and animal populations [191], [147], [210]. Giventhe high complexity of these phenomena, models arekey to understand epidemiological patterns and supportdecision making processes [101]. There are in-hostmodels that take into account the complexity of virus-host dynamics at the microscopic scale, describing howthe pathogen interacts with cellular biomolecular pro-cesses and with the immune system, and between-host4odels that describe how the epidemic spreads withina population at the macroscopic scale, by consideringthe contagion either at an aggregate level (compartmen-tal models) or through agent-based networked modelsof the population. Approaches for epidemic multi-scalemodelling, which include the interplay between im-munological and epidemiological phenomena, are veryrecent and mostly rely on partial di ff erential equations,sometimes reduced to small-size ordinary di ff erential-equation systems, see e.g. [11], [10], [20], [41], [72],[79], [89], [95]. Multi-scale epidemic modelling withan interdisciplinary approach integrating epidemiology,immunology, economy and mathematics is advocated in[23]. In-host infection dynamics capture the interplay be-tween virus and host. Models describing the dy-namics of the immune response [46] in the presenceof an infectious disease have been proposed for in-fluenza [92],[135],[232],[239] and generic viral infec-tions [154]. Very recently an immunological descriptionfor Covid-19 has been provided [148] and has enabledthe characterization of virus-host dynamics for SARS-CoV-2 [2], [105].The evolution of a disease and its infectiousness overtime can be characterized through some key epidemi-ological parameters (see e.g. [102], [224], [221], and[104]): • Latency time : Time during which an individualis infected but not yet infectious. For Covid-19,initial estimates are of 3-4 days [137]. • Incubation time : Time between infection and on-set of symptoms. For Covid-19, the median in-cubation period is estimated to be 5.1 days, and97.5% of those who develop symptoms will do sowithin 11.5 days of infection [128]; the mediantime from the onset of symptoms to death is closeto 3 weeks [244]. • Serial interval : Time between symptom onsetsof successive cases in a transmission chain [221].For Covid-19, initial estimates of the median se-rial interval yield a value of around 4 days, whichis shorter than its median incubation period [161];this implies that a substantial proportion of sec-ondary transmission may occur prior to illness on-set [100]. • Infectiousness profile : It characterizes the infec-tiousness of an infected individual over time. For Covid-19, the median duration of viral sheddingestimation was 20 days in survivors, while the mostprolonged observed duration of viral shedding insurvivors was 37 days [244]. • Basic reproduction number R : It represents theaverage number of new infections generated by aninfectious person at the early stages of the out-break, when everyone is susceptible, and no coun-termeasures have been taken [102],[224],[140].For Covid-19, first estimations range from 2.24 to3.58 [241]; the e ff ect of temperature and humidityin this parameter is addressed in di ff erent studies,see e.g. [151].The basic reproduction number, along with the serialinterval, can be used to estimate the number of infec-tions that are caused by a single case in a given time pe-riod. Without any control measure, at the early stages ofthe outbreak, more than 400 people can be infected bya single Covid-19 case in one month [160]. Estimatesof the basic reproductive number are of interest duringan outbreak because they provide information about thelevel of intervention required to interrupt transmissionand about the potential final size of the outbreak [102].The aforementioned parameters are often inferredfrom epidemiological models, once they have been fit-ted to available data on the number of confirmed infec-tion cases and deaths [191], [224]. Compartmental models partition a population intodi ff erent groups, called compartments , associated withmutually exclusive stages of the disease. Each compart-ment is associated with a variable that counts the indi-viduals who are in that stage of the infection [34].The simplest compartmental model is the SIR model,introduced by Kermack and McKendrick at the begin-ning of the 20th century [115]. The SIR model in-cludes three compartments: Susceptible ( S ), represent-ing healthy individuals susceptible of getting infected, Infected ( I ), and Recovered / Removed ( R ). For possiblyfatal diseases, this last compartment can take into ac-count both recovered (with permanent immunity) anddeceased individuals; however, for low mortality ratediseases, including only recovered individuals can be agood approximation.The SIR model describes the dynamics of an epi-demic according to the following set of nonlinear dif-5erential equations: dS ( t ) dt = − β S ( t ) I ( t ) , (1) dI ( t ) dt = β S ( t ) I ( t ) − µ I ( t ) , (2) dR ( t ) dt = µ I ( t ) , (3)where β is the infection rate, while µ is the recoveryrate; the variables S , I and R represent the fraction ofsusceptible, infected and recovered (or removed) indi-viduals within the population, and S ( t ) + I ( t ) + R ( t ) = t . At the onset of a new epidemic, S equalsapproximately the entire population, and thus from (2)it holds that I ( t ) = I e ( β − µ ) t = I e µ ( R − t , where I rep-resents the initial number of infected I = I (0) and R = β/µ is the basic reproduction number , i.e. theaverage number of secondary cases produced by an in-fectious individual when S ≈
1. Clearly, when R isgreater than 1, there is an exponential increase in thenumber of infected individuals during the early days ofthe epidemic. The same equation can also be used to es-timate the point at which the number of newly infectedindividuals begins to decrease, S ( t ) = / R . At thispoint, the given population has reached what is knownas herd immunity [76].To account for the latency time, an extended versionof the SIR model, called the SEIR model, includes anextra compartment for Exposed (E) individuals, whohave been infected but are not yet infectious, and aretransitioning into the Infectious compartment at a fixedrate.
To model the specific dynamics of a given infec-tious disease, extended compartmental models includ-ing additional compartments and transitions are oftenproposed. In particular, it is possible to consider symp-tomatic and asymptomatic compartments, vaccinatedand unvaccinated, the possibility of reinfection after re-covery, quarantined individuals, hospitalized, etc. Com-prehensive books, surveys and works on compartmentalmodels and their extensions are [14],[35],[37],[44],[60],[90],[106].The number of compartments required to model a dis-ease depends on a variety of factors. For example, whenmodeling the dynamics of a new disease, for which novaccine is available, it makes no sense to consider thevaccinated group. However, in other cases, as whenmodelling seasonal influenza, it is relevant to distin-guish between vaccinated and unvaccinated populations[34]. Many diseases, like malaria, West Nile virus, etc.,
Figure 2: Illustration of an extended compartmental epidemic modelwith seven compartments used in [188] to model SARS : Susceptible(S), Latent (L), Asymptomatic and potientially infectious (I), Symp-tomatic Diagnosed (Y), Hospitalized that die ( H D ), Hospitalized thatrecover ( H R ) and Recovered R . are transmitted not directly from human to human butby infected animals (usually insects) [208]. For thesecases, the corresponding animal compartments are in-cluded in the model. Another relevant factor influencingwhat compartments to include in a model is the quantityand quality of available data. Complex models requiremore data to fit the parameters, so in the early states of anew disease outbreak simple compartmental models areoften employed.Many applications of extended compartmental mod-els can be found in the literature. For example, in [188],the authors use a dynamical compartmental model toanalyze the e ff ective transmission rate of the SARS epi-demic in Hong Kong. The model consists of 7 com-partments: Susceptible individuals (S) become infectedand enter a latent state (L). They then progress to ashort asymptomatic and potentially infectious stage (I)followed by a symptomatic state that leads to diagnose(Y) and hospitalization. It is assumed that every symp-tomatic case is eventually hospitalized and either dies( H D ) or, after treatment in the hospital ( H R ), recovers(R) (see Figure 2). In [51], a stochastic SEIR modelis used to estimate the basic reproduction number of6ERS-CoV in the Arabian Peninsula, distinguishingbetween cases transmitted by animals and secondarycases.In the spread of the Covid-19 pandemic, asymp-tomatic infected individuals play a crucial role (see [73]and [82]); the large prevalence of asymptomatic in-fections makes it harder to detect all cases and, thus,timely break the contagion chain. In [82], a SIDARTHEmodel with eight compartments is proposed. Thismodel distinguishes between asymptomatic and symp-tomatic infected, as well as detected and undetected in-fection cases, and partitions the population into Sus-ceptible, Infected (asymptomatic infected, undetected),Diagnosed (asymptomatic infected, detected), Ailing(symptomatic infected, undetected), Recognised (symp-tomatic infected, detected), Threatened (infected withlife-threatening symptoms, detected), Healed (recov-ered) and Extinct (dead) individuals. In [73], theepidemic model includes a transmission rate β thattakes into account the contributions of asymptomatic,presymptomatic and symptomatic transmissions, aswell as environmental transmission. In both works, theresults indicate that the contribution of asymptomaticinfected to R is higher than that of symptomatic in-fected and other transmission modalities. In fact, symp-tomatic infected are often rapidly detected and isolated. Age-structured epidemic models incorporate hetero-geneous, age-dependent contact rates between individu-als [58]. In [231] and [193], stability results for di ff erentage-structured SEIR models are given. For Covid-19,an age-structured model, aiming at estimating the e ff ectof physical distancing measures in Wuhan, is presentedin [179]. In [195], a stratified approach is used to modelthe epidemic in France. Some works have studied the influence of tempera-ture and humidity on the spread of viruses [86], [223].In the case of Covid-19 , it has been reported that bothvariables have an e ff ect on the basic reproduction num-ber R [151], [194]. This influence might be included inthe epidemic models to capture the seasonal behaviourof Covid-19; for instance, by considering the parame-ters β and µ as functions of both temperature and rela-tive humidity. Yet, it remains unclear under which cir-cumstances seasonal and geographic variations in cli-mate can substantially alter the dynamics of a given pan-demic, specially in the case of high susceptibility [17]. Compartmental models are well-suited to describethe evolution of epidemics in a single, well-mixed popu-lation where each individual is assumed to interact withevery other at a common rate (homogeneous contacts).While this can be a reasonable approximation in somecontexts, it is not appropriate to study the global spreadof a pandemic spreading over a large, geographicallydispersed population. In the last decades, compartmen-tal models have been successfully extended to spatialepidemiological models in order to analyze spreadingphenomena where spatial patterns need to be more accu-rately described. Graphs and networks have often beenused to achieve this, see for instance [107], [114], [118],[152], [162], [171], [134], [166], [165], [170], [169],[245]. Two widely used classes of models are describedin the following sub-subsections.
Meta-population models integrate two types of dy-namics: one related to the disease, typically driven bya compartmental model, and the other to the mobil-ity of individuals (agent-based model) across the sub-populations that build the meta-population under anal-ysis [87], [18]. As a representative example, in [38]the authors introduce the notion of e ff ective distanceto capture the spatio-temporal dynamics of epidemics,combining the SIR model of n = , , . . . , p popula-tions with mobility among them. The resulting modelfor each population is dS n ( t ) dt = − β S n ( t ) I n ( t ) + (cid:88) m (cid:44) n ( w nm S m − w mn S n ) dI n ( t ) dt = β S n ( t ) I n ( t ) − µ I n ( t ) + (cid:88) m (cid:44) n ( w nm I m − w mn I n ) , dR n ( t ) dt = µ I n ( t ) + (cid:88) m (cid:44) n ( w nm R m − w mn R n ) , where w nm is the per capita tra ffi c flux from popu-lation m to population n . In [7], the authors use aSEIR compartmental model together with stochasticdata-driven simulations to capture the mobility in allSpanish provinces. The work focuses on evaluating thee ff ectiveness of contention measurements in Spain onFebruary 28th, when a few dozen cases of Covid-19 hadbeen detected. Meta-population models to capture thespatio-temporal dynamics of the Covid-19 epidemics inItaly have been proposed in [25], [80] and [59]. By cap-turing both temporal and spatial evolution of epidemics,7eta-population models are also capable of forecastingthe e ff ectiveness of mobility restrictions. Social network models consider that transmission canonly occur along linked or connected individuals [65],which allows to explicitly model heterogeneity in con-tact patterns. Small-world networks have been used incombination with compartmental models to model dis-ease transmission of SARS [204] and Covid-19 [211],and also to assess the e ffi cacy of contact tracing [117].In general, network models produce a more accurateprediction of the disease spread [170]. In particular, theuse of homogeneous compartmental models in popula-tion with heterogeneous contacts tends to underestimatedisease burden early in the outbreak and overestimate ittowards the end, although for certain kinds of networkscompartmental models can be modified to prevent thisproblem [19]. Another interesting aspect of studyingepidemic spreads with network models is the observa-tion of a percolation phase transition [107] [171], i.e., anabrupt change in the global dynamics of the epidemics.Percolation theory has been widely studied in randomnetworks [5]. In the context of epidemic modelling,the transition phase occurs where isolated clusters of in-fected people join to form a giant component that is ableto infect many people [94]. Computer-based simulation methods to predict thespread of epidemics can take into account numerousfactors, such as heterogeneous behavioural patterns,mobility patterns, both at long and short scales, de-mographics, epidemiological data, or disease-specificmechanisms [146], [103]. The real-world accuracy ofmathematical and computational models used in epi-demiology has been considerably improved by the inte-gration of large-scale data sets and explicit simulationsof entire populations down to the scale of single individ-uals. These computational tools have recently gainedimportance in the field of infectious disease epidemiol-ogy, by providing rationales and quantitative analysis tosupport decision-making and policy-making processes[214]. As a representative example, the Global Epi-demic and Mobility simulation framework (GLEAM)allows performing stochastic simulations of a globalepidemic with di ff erent global-local mobility patterns,as well as data regarding demographics or hospitaliza-tion [217].However, detailed simulation-based methods dependon a significant number of parameters, which need to be chosen and fixed for a specific simulation. This is espe-cially di ffi cult in the early days of an epidemic outbreak.Furthermore, these approaches might not reveal whichfactors are actually relevant in the spread of epidemics.Simpler data-driven tools have also been developed toovercome these di ffi culties [146]. ff ect of containment measures Controlling an emerging infectious disease requiresboth the prompt implementation of countermeasuresand the rapid assessment of their e ffi cacy [47], [90],[52], [36], [97], [88]. In what follows, we enumerate themost relevant non-pharmaceutical interventions, focus-ing on di ff erent research works that assess their e ffi cacy. • Quarantine : Quarantine of diagnosed cases, orprobably infected, is crucial in every epidemic out-break. In order to model the e ff ect of quarantine,specific compartments are included in the epidemicmodels for SARs [90], [52]. If a significant frac-tion of the infected population is not diagnosed (ordiagnosed with a significant delay), then the mod-elling is harder and non-diagnosed groups are in-cluded in the models [149], [82], [15].Quarantine of a whole population (i.e., lockdown)is the most extreme measure in the scope of phys-ical distancing / mobility restrictions. The extremeimpact of Covid-19 yield to the quarantine of theepicentre of the pandemic (Wuhan) on January24th, 2020, and the same measures were subse-quently adopted in di ff erent countries of Europeand America [80]. In this case, the e ff ect of a lock-down can be modelled by means of time-varyingepidemic models, see e.g. [42]. • Physical distancing : Physical (or social) distanc-ing is another measure promoted by governments,public and private institutions in an attempt toreduce disease transmission [179], [156], [176].Population-wide wearing of masks, capacity re-duction on public transport, reducing or stoppingthe activity in educational institutions or factoriesare examples of this. In [142], the authors con-duct a simulation-based analysis to determine thee ff ects of physical distancing both in public healthand in the economy. Two social network mod-els (regular and small-world networks) are com-bined with a compartmental SIR model, and theeconomic impact takes into account the costs ofindividuals falling ill and the cost of a reductionin social contacts.8 Mobility restrictions : Governments often in-troduce long-range or local mobility restrictionsaimed at reducing disease transmission. Spatialepidemiology is particularly useful to model the ef-fects of such measures. For instance, in [211], theauthors show, by means of a small-world networkmodel, that the onset of mobility restrictions influ-ences the final size of the outbreak, which is wellbelow the levels of herd immunity. • Proactive testing : Proactive testing of asymp-tomatic individuals is very relevant for the moni-toring and control of the Covid-19 pandemic [229],since it allows to isolate infectious individuals andimplement contact tracing strategies, which havebeen shown to be crucial to an e ff ective control thepandemic [82]. • Contact tracing : Contact tracing is a widely usedepidemic control measure that aims to identify andisolate infected individuals by following the socialcontacts of individuals that are known to be in-fectious. A review of contact-tracing based epi-demic models for SARS and MERS can be foundin [124]. In [117], a small-world, free-scale net-work model is combined with a compartmentalmodel to assess the e ffi cacy of contact tracing. Dynamic epidemiological models rely on a set of pa-rameters that have to be tuned in order to provide real-istic predictions and / or infer essential features, such asthe (time-varying) e ff ective reproduction number [57],or the latent period. Fitting epidemic models to data isa fundamental problem in epidemiology that can be ap-proached in di ff erent ways. We can distinguish betweenclassical methods, in which the parameters of the modelare unknown but fixed, and Bayesian methods, in whichthey are assumed to be random variables [125]. Anotherclassification follows from the accessibility to the pop-ulations considered in the compartments of the model: • Full access to the evolution of the number of casesin each compartment: In most models, the parame-ters that determine the dynamics multiply linear orbi-linear terms, depending on the current numberof cases in each compartment. This means that a(vector) equality constraint, that depends (bi-) lin-early on the parameters to fit, can be obtained ateach sample time. In the case of linear constraints,standard linear identification techniques, such asleast-square methods, can be applied to estimate the parameters that best fit the model to the data.See, for example, [147, Chapter 6] and [9]. • Partial access to the number of cases in each com-partment: In many situations, there are no avail-able time series for one or more of the groups con-sidered in the model. This complicates the data-fitting process considerably because it is no longerpossible to obtain, in a simple way, the equalityconstraints described in the full access case. Thestandard approach in this case is to resort to non-linear identification techniques (see [199] and [1]).In this context, Monte Carlo based methods (e.g.Markov Chain Monte Carlo and Sequential MonteCarlo algorithms) play a crucial role in addressingthe challenges that lie in reconciling predictionsand observations [150].
Sensitivity analysis (SA) is the study of how the un-certainty in the output of a model (numerical or other-wise) can be apportioned to di ff erent sources of uncer-tainty in the model input [196]. See the review paper[182] on the use of this technique in the context of bio-logical sciences. A monovariate and multivariate sensi-tivity analysis for a data-fitted SARS model is given in[13]. The use of SA is common in many research paperson modelling Covid-19 (see e.g. [70] and [195]). The ultimate test of the validity of any model is thatits behaviour is in accord with real data. Because of thesimplifications introduced in any mathematical modelof a biological system, we must expect some divergencebetween the results of a model and reality, even for themost carefully collected data and most detailed model.Di ff erent questions arise in this context: (i) How can wedetermine if a model describes data well? (ii) How canwe determine the parameter values in a model that areappropriate for describing real data? These questionsare too broad to have a single answer [9], [222].Epidemic models depend on their data calibration.However, many possible models are potentially suitedto analyze the spread of a pandemic in a given moment.The models are inherently linked to the goal for whichthey were envisaged. For a given goal (for example sec-ond outbreak detection), di ff erent models can be consid-ered. Model selection techniques are used on a regularbasis in epidemiology [174]. They address the prob-lem of choosing, among a set of candidate models, themost suitable for a given purpose [39]. The selection isbased on di ff erent aspects: (i) How the calibrated model9s able to reconcile and match observations and (ii) thecomplexity of the model. Under similar adjustment toobservations, simpler models are preferred since theyare more robust from an information-theoretic point ofview [108].There are often di ff erent sets of parameters yieldinga similar fit to data, but providing significantly di ff er-ent estimations of the main characteristics of the spreadof the epidemic (like peak size, reproduction number,etc.). This issue is known as non-identifiability [189],[91]. Identifiability issues may lead to inferences thatare driven more by prior assumptions than by the datathemselves [139]. There are some approaches to ad-dress this di ffi culty. The first one is to resort to sim-plified models (SIR and SEIR models, for example) inwhich the number of parameters to adjust is small, see[189] and [175]. The second one is to use data fromdi ff erent regions in a not aggregated way, which re-duces the probability of parametric over-fitting [74]. Inthis context, model selection theory provides systematicmethodologies to determine which model structure bestsuit the purposes of the model [39], [174].
4. Forecasting
The task of forecasting a time series can be stated as asupervised learning problem in which a number of tem-poral variables (also called predictors or features in themachine learning literature) are used to learn a modelable to predict the future value of an output variable ofinterest [29]. In our context, we focus on forecastingmethods aiming to predict the future evolution of epi-demiological variables [206], [53]. We find in the lit-erature numerous approaches to forecast temporal vari-ables describing the evolution of Covid-19 [207], [173],[42], from black-box approaches to estimates based onlearning the internal parameters of compartmental epi-demic models. Forecasting in the context of globalpandemics faces many di ffi culties [109] and requiresthe implementation of validation and sensitivity anal-ysis [39]. We now introduce some considerations thatshould be taken into account in order to select and traina suitable forecasting model.First, we start with some statistical considerations: • Frequentist versus Bayesian statistical methods: Inthe former, probabilities are assigned according toexperiment repetition and occurrence. In the latter,the parameters of a model are learned using Bayes’theorem and prior knowledge about the probabilitydistributions of unknown variables [32]. • Parametric versus non-parametric approaches: Inthe former, we assume a parametric function map-ping past variables input into future predictions.This function contains several unknown parame-ters that are learned using historic time series. Inthe non-parametric approach, we do not assumesuch a parametric function [143]; for example, onecan make future predictions for a given time seriesby analyzing the behavior of historic past behav-iors resembling the behavior of the time series un-der consideration.Other considerations to keep in mind are: • The model should be trained with reliable data. Ifthe available data is poor, the forecasts producedwill be unreliable. In this direction, data-cleaningtechniques such as data reconciliation, standard-ization, filtering, and outlier detection should beutilized to improve the quality of the input data col-lected [6]. • The amount of data collected should be appropri-ate for the forecasting technique under consider-ation. For instance, black-box models, such asdeep learning, require vast amounts of data com-pared with physics-informed models, such as com-partmental models [215],[233]; therefore, whiledealing with relatively short time series, makingpredictions using compartmental models is moreappropriate than using deep learning (and otherblack-box techniques). • Learning procedures should include training, val-idation, and test phases executed separately. Inother words, available data set should be dividedinto three parts, each one used for a di ff erent pur-pose. In the training stage, model parameters arelearned using training data. In the validation step,one adjusts model hyper-parameters and performscomparisons with other competing approaches. Fi-nally, the final test of a model should be carried outwith data that has not been used during the trainingor validation phases [39]. • Interpretability of the model. While deep learn-ing (and other black-box techniques) may pro-duce high-quality predictions, the obtained modelis hard to interpret; in other words, we typicallydo not have an intuitive understanding of why themodel is making a prediction [16]. However, whenpolicy-makers make critical decisions based on theforecast of a model, it is important for them to un-derstand why the model is behaving in a certain10ay. Therefore, it is sometimes reasonable to usedmore interpretable models, with parameters hav-ing a clear physical / biological interpretation, evenat the expense of having a lower performance thanblack-box approaches.
5. Impact Assessment Tools
In order to design e ff ective control strategies, it is im-portant to define the control goals first. In the contextof the current pandemic, the ultimate goal is to main-tain the spread of the virus within an adequate threshold(e.g., a low level of infectious [180]), while minimiz-ing the economic and social impacts of the interven-tions. Once this goal is quantified in terms of a costfunction, we should then consider the types of interven-tions that can be taken to achieve our goals, as well astheir associated costs. For example, there are severalnon-pharmaceutical interventions that can be used be-fore a vaccine is widely available, such as physical dis-tancing, border closures, school closures, isolation ofsymptomatic individuals, among others (see subsection3.7). Each one of these interventions have an associ-ated economic and social cost that should be consideredwhile making a decision.In order to use disciplined decision-making tech-niques, like the ones described below, one needs toclearly state the control objectives in a precise, quan-titative form. Furthermore, it is necessary to quantifythe impact and costs of all possible interventions, aswell as their actuation limits [47], [36]. In this direc-tion, we can quantify the impact of our actions by usingsuitable indexes such as the mean reproductive number,the mortality index, or the unemployment rate or publicdebt, to name just a few. Once the decision-maker hasdecided how to use these indexes to measure the im-pact and cost of potential actions, the decision-makingprocess can be stated as a formal optimization problemwith constraints. For example, the goal could the min-imization of a weighted index measuring the economicand social impact of our actions while keeping the re-productive number smaller than one.We would like to remark that the numerical estima-tion of certain indexes is not an easy task because theyrequire the design of data-driven strategies to assess thee ff ect of each potential decision on di ff erent indexes.This could be done by means of predictive models andforecasting schemes analyzed in the previous sections.In some cases, quantifying the e ff ect of one interventionover the spread of an epidemic is a non-trivial task, sincemultiple interventions are typically present at the sametime [98]. In these scenarios, correlation analyses, like Pearson Correlation Coe ffi cient (PCC), can be a naiveway to assess causalities. Wheneven possible, a reli-able approach to establish causalities is to perform Ran-domized Control Trials (RTC) [61], [98]. In an RTC, asubset of randomly chosen individuals receives an inter-vention, while the rest of individuals receives no inter-vention. A standard statistical analysis of the observedresults can be used to reliable evaluate the impact ofthis intervention. In the following subsections, we dis-cuss a collection of indexes that could be included in thedecision-making process of managing a pandemic. It is natural to express the e ff ectiveness of controlstrategies in terms of the e ff ective reproductive number R ( t ). As introduced in Section 3, the basic reproduc-tion number R determines the potential of an epidemicto spread exponentially at its early stage by measuringthe number of secondary infections induced by a typicalinfectious individual in a population when everyone issusceptible. In contrast, when an epidemic is on course,the e ff ective reproduction number, denoted by R ( t ), isused to quantify the average number of secondary in-fectious per infectious case in a population made up ofboth susceptible and non-susceptible hosts. The e ff ec-tive reproduction number can be used to assess the abil-ity of available control measures to contain the spreadof an epidemic. By implementing interventions able tomaintain R ( t ) below 1, the incidence of new infectionsdecreases and the spread of epidemics fades with time.In [57], the authors presented a software tool that wasvalidated with 5 di ff erent epidemics, including SARSand influenza. This tool can be used to estimate thedaily reproductive number R ( t ) and its variation in thepresence of vaccination and super spreading events.For Covid-19, a numerical analysis of the e ff ectivereproductive ratio can be found in [70], where usingreal data and a SEIR model, the authors estimate R ( t )in Wuhan and quantify the e ff ectiveness of governmentmeasures. Based on the number of deaths, in [77],the Imperial College Covid-19 Response Team used asemi-mechanistic Bayesian model to estimate the evolu-tion of R ( t ) when non-pharmaceutical measures, such asphysical distancing, self-isolation, school closure, pub-lic events banned, and complete lock-down, were rec-ommended / enforced.Limitations in the use of R ( t ) as an assessment toolstem from the unreliability of available data sources.As a result, determining the real value of R ( t ) is di ffi -cult. Other indirect measures, like the number of deaths,ICU cases, saturation of health-care systems can also be11mployed to assess the current epidemic burden, as de-scribed in the next subsection. The capacity of a country to prevent, detect, and re-spond to epidemic outbreaks vary widely across coun-tries. The preparedness and resilience of a health-caresystem is a particularly relevant factor to analyze thefuture impact of an infectious outbreak in the popula-tion [113]. The capacity of a health system to continueto deliver the same level (quantity, quality and equity)of basic healthcare services and protection to popula-tions can severely degrade during an epidemic outbreak[30],[67]. At the early stages of the Covid-19 outbreak,its virulence and high contagiousness quickly saturatedthe health-care system of many cities around the world,resulting in higher mortality rates [153],[126]. Fur-thermore, in countries with low capacity, like Africanand South American countries, saturation levels arereached even with a significantly smaller number ofcases [220, 156].To limit the saturation of health-care systems andplan resource distribution e ff ectively, tools that assessthe e ff ect of di ff erent interventions on the magnitudeand timing of the epidemic peak during first and sec-ondary outbreaks (see Sections 3 and 4) are fundamen-tal. However, precise tools to forecast these peaks arechallenging to obtain, due to the limitations of the avail-able data and the time-varying nature of the mitigatione ff orts and potential seasonal behaviour of a pandemic.Another issue is the uncertain adherence of the pop-ulation to the interventions (see next subsection). Inorder to partially circumvent these issues, forecasts ofcumulative disease burden are often looked for. Whilemissing the intensity and timing of the peaks, these pro-jections can at least allow to identify areas with heavypresent and / or future pandemic incidence. Analyses of the relationship between risk percep-tion and preventive behaviours can be found in the so-cial epidemiology literature [24],[133]. Moreover, thelevel of belief in the e ff ectiveness of recommended be-haviours and trust in authorities are important predic-tors of adherence to preventive behaviour (see surveypaper [28]), which is fundamental to deploy e ff ectivecontainment strategies [155]. Here, we review some ofthe methodologies that could be helpful to design in-dexes aiming to monitor the adherence to interventionsof the population and the social burden of the pandemic. • Social network analysis : Online social networks,such as Facebook and Twitter, can be used to assessthe impact of an infectious disease in society. Peo-ple post in these social networks their feelings andworries. In [202], 530,206 tweets in the USA wereanalyzed to measure the social impact of Covid-19. The hashtags were classified into six cate-gories, including general covid, quarantine, schoolclosures, panic buying, lockdowns, frustration andhope. Thus, the number of tweets in each categorycan be used as a metric of social impact and overallsentiment. Similarly, Weibo microblogging socialnetwork was used in [136] to study the propaga-tion of situational information related to Covid-19in China. In [110], the political polarization withregards to Covid-19 in the United States was ana-lyzed using a large Twitter dataset. • Search engines : Online searches made by citizensin search engines, such as Google, Bing, or Baidu,can be used to measure the social impact of theepidemic in di ff erent locations. Normally, peopletry to find information of unknown diseases, drugs,vaccines, and treatments on the Internet. Alongthis line, the authors of [81] found a correlationbetween the relative frequency of certain queriesin Google and the percentage of physician visitsin which a patient presents influenza-like symp-toms. Furthermore, other works have performedsimilar studies for other epidemics like InfluenzaVirus A (H1N1) [56]. Regarding Covid-19, in[183], the Baidu engine is used to estimate thenumber of new cases of Covid-19 in China by thenumber of searches of five keywords, such as drycough, fever, chest distress, coronavirus, and pneu-monia. These five keywords showed a high corre-lation with the number of new cases. • News : The number and the content of posts inonline newspapers can also be used to assess thespread of the virus. Along this line, in [242], Nat-ural Language Processing (NLP) is used to extractthe relevant features of news media in China. • Online questionnaires : Another tool for mea-suring the social impact of a sanitary emergenceis through online questionnaires such as [167](Spain, 146,728 participants) [184] (China, 52,730participants) and [120] (UK, 2,500 participants),which were implemented for the Covid-19 pan-demic. These questionnaires allow to rapidly askcitizens multiple questions related to adherence tointerventions, as well as psychological, social and12conomic impact, among other aspects. The maindi ffi culty is to spread the questionnaires through-out the population, although social networks andweb-based tools help to reach a large amount ofpopulation. • Mobility : One of the most relevant indexes tounderstand the spread of a pandemic is mobility[213]. See Subsection 8.4 in [4] for a relation ofmobility data sets in the context of Covid-19. Thereduction of mobility is not only due to the im-posed quarantines and lockdowns by governmentsbut also by the increasing population’s fear of get-ting infected. In [68], a perceived risk index ofcontracting Covid-19 is defined. This metric mea-sures the individuals’ perception of risk, and it isdetermined by several variables, such as preva-lence in both local and neighbouring locations, aswell as population demographics. The results in[68] indicate that a rise of local infection rate from0% to 0.003% reduces mobility by 2.31%.
6. Managing and Decision Making
Deciding which ones of the far-reaching social andeconomic restrictions are most e ff ective to contain thespread of a disease, as well as the conditions underwhich they can be safely lifted is one of the maingoals of data-driven decision approaches to combatpandemics. Unlike an unmitigated pandemic, whichspreads through the susceptible population out of con-trol and eventually fades out, a mitigated pandemicpresents waves. For example, a first wave grows whena very transmissible disease appears and decreases dueto, for example, social distancing measures. However,as soon as social distancing measures are relaxed, a newwave can appear as long as we have a large number ofindividuals susceptible to the infection. To avoid recur-sive waves, it is important to put in place surveillancesystems and reactive mechanisms to reduce the poten-tial burden of secondary epidemic waves. The decision-making process in this context is complex for many rea-sons: • The presence of uncertainty in some crucial param-eters characterizing the spread, such as seasonal-ity, extent and duration of immunity of a new pan-demic outbreak [55], [119]. • The di ffi culties in assessing the quantitative e ff ectof a specific set of mitigation interventions on thee ff ective reproduction number [98]. • The possibility of significant non-symptomatictransmission (as in the case of Covid-19), whichrenders some interventions less e ff ective [161],[176] . • The di ff erent regional incidence and adherence tointerventions, which motivates spatially distributeddecisions [200, 59]. • The limited capacity of health-care systems andthe logistic challenges to address mass testing andmass vaccination. • The necessity to mitigate the spread of the epi-demic and, at the same time, reducing the socioe-conomic impact. • The time-delay induced by the incubation periodof the disease, as well as the testing system, whichdoes not allow for a prompt evaluation of the e ff ectof the implemented actions. • The di ffi culties of assessing in a quantitative waythe disruptive e ff ects of the undertaken measuresin relevant macroeconomic variables.In what follows, we analyze under which circum-stances the epidemic can be mitigated (controllabilityof the pandemic). After that, we also discuss somemethodologies that have been applied to combat infec-tious diseases, including the Covid-19 pandemic, andthat could potentially be applied in the context of futurepandemics. See also the review papers [162], [40] forthe use of control theory in the context of disease con-trol, or [178], [170], [15] for the stability analysis of anepidemic. In this subsection, we review the most important fac-tors determining the controllability of a pandemic: theaspects that have a relevant impact on the e ff ective re-production number. We link them with standard epi-demic threshold theorems (e.g. [22], [227], [115]).The epidemic threshold theorem of Kermack andMcKendrick [115], stated in 1927, and in particular itsstochastic form as given by Whittle [227] are funda-mental to predict the size and nature of an infectiousdisease outbreak. The theorem indicates that, in homo-geneously mixed communities, major epidemics can beprevented by keeping the product of the size of the sus-ceptible population, the infection rate, and the mean du-ration of the infectious period, su ffi ciently small [22].We now discuss how to have an impact on each of thesefactors by means of control actions.13 Size of the susceptible population:
The most ef-fective way to reduce the susceptible population isby means of vaccines: vaccination campaigns in-crease herd immunity to a level that prevents fur-ther spread of the disease [198]. Protection againstan infectious disease can either be achieved bywidespread vaccination or by repeated waves of in-fection over the years, until a large enough fractionof the population is immunized [85]. However,an issue is the duration of the acquired immunity[119], which in some infectious diseases, like theseasonal influenza, is not long enough to preventrecurring seasonal peaks [55]. • Infection rate:
This factor can be reduced bymeans of di ff erent control actions like physical dis-tancing, mobility constraints or prohibition of cer-tain activities [123], [159]. Depending on the sea-sonality and the specific demographic characteris-tics of a given population, the implemented mea-sures can exhibit a time-varying e ff ect on the in-fection rate [57], [70]. This might cause flowsfrom tropical to temperate regions and back in eachhemisphere’s respective winter, limiting opportu-nities for global population declines [55] and im-plying that surveillance methods to detect a sea-sonal peak should be put in place. • Mean duration of the infectious period:
An ef-fective way to reduce the infectious period con-sists in detecting infected cases and setting theminto quarantine [52]. Challenges are posed by rel-atively short latent periods and by the presence ofmany asymptomatic cases, as in the Covid-19 pan-demic; then, the impact of quarantine measures de-pends very much on how fast the detection is tak-ing place. It has been shown that the probability ofcontrol decreases with long delays from symptomonset to isolation [104], [73]. A large prevalenceof asymptomatic cases is indeed an issue due tothe significant probability that transmission occursbefore the onset of symptoms (median latent delayis slightly smaller than median incubation time),hence before the infection can be detected [73],[82].
During a major health crisis, policy makers facethe problem of optimally allocating limited resources,such as intensive care beds, ventilators, tests, high-filtration masks and Individual Protection Equipment(IPE), medicines, vaccines, etc. [33], [238]. This fact has led to the problem of how to ethically and con-sistently allocate resources [67]. In this context, theterm “resource allocation problem” extends to issues aswhere and when to allocate available resources.A rigorous and precise allocation method should leadto the formulation of an optimization problem, com-posed of a mathematical formulation and e ffi cient al-gorithms to obtain its numerical solution [93]. In themathematical model, resource allocations are the deci-sion variables while the objectives are encoded in costfunctions and equality or inequality constraints. For ex-ample, in [238] and [33], budget allocation models formultiple populations are provided. In [177], a networkmodel is used to optimally allocate vaccines to eradicatean initial epidemic outbreak using linear matrix inequal-ities. An extension to this work to the case of directedand weighted networks can be found in [178] and [164],where geometric programming was proposed to find anoptimal solution. The same authors extend this last re-sult to more general compartmental models in [163].See also [99] for an application of geometric program-ming and multi-task learning in the context of Covid-19.In [127] an optimization problem is formulated tofind the number of tests to be performed in the di ff er-ent Italian regions in order to maximize the overall de-tection capabilities. The problem is a quadratic, convexoptimization program. In [84], a group testing [225]approach is considered, and it is shown how the opti-mization of the group size can save between 85% and95% of tests with respect to individual testing. See also[234] for an strategy that optimizes testing resources inthe context of the Covid-19 pandemic.Estimation, forecasting, and impact assessment tech-niques are often used to allocate resources, as theyenable decision-makers to predict imbalances betweensupply and demand and to evaluate the overall e ffi ciencyof di ff erent alternatives of allocation. In [67], the au-thors propose fair resource allocation guidelines in thetime of Covid-19, which can be a reference for futurepandemics. These guidelines come from four funda-mental values: (i) maximizing the benefits, (ii) treatingpeople equally, (iii) promoting instrumental value, and(iv) giving priority to the worst o ff . As a result, theseguidelines are condensed in some recommendations:1. To maximize the number of saved lives and live-years, with the latter metric subordinated to the for-mer.2. To prioritize critical interventions for health careworkers and others who take care of sick patientsbecause of their instrumental value.3. For patients with similar prognoses, equality14hould be invoked and operationalized throughrandom allocation.4. To distinguish priorities depending on the inter-ventions and the scientific evidence (e.g. vaccinescould be prioritized for older persons while alloca-tion ICU resources depending on prognosis mightmean giving priority to younger patients).5. People who participate in research to prove thesafety and e ff ectiveness of vaccines and therapeu-tics should receive some priority for interventions. A strategy to modulate the intensity of non-pharmaceutical interventions consists in implementinga trigger mechanism to maintain the e ff ective reproduc-tion number close to one, avoiding the saturation ofthe health care system while reducing, when possible,the economic and social burden of the pandemic [47],[178], [59], [27]. The on-line surveillance of the pan-demic permits to estimate the time-varying value of thee ff ective reproduction number. Three cases are possi-ble: The e ff ective reproduction number is largely under1: in this case, one could consider lifting one, or morenon-pharmaceutical measures. However, other criteriashould be met in order to implement a reduction on theconfinements measures in a safe way [180]. The threecriteria highlighted by the European Commission to de-cide on the lifting of confinement measures for Covid-19 [69] are:1. Epidemiological criteria showing that the spreadof the disease has significantly decreased and sta-bilised for a sustained period of time. This can, forexample, be indicated by a sustained reduction inthe number of new infections, hospitalisations andpatients in intensive care.2. Su ffi cient health system capacity, in terms of,for instance, occupancy rate for Intensive CareUnits; adequate number of hospital beds; access topharmaceutical products required in intensive careunits; reconstitution of stocks of equipment; accessto care, in particular for vulnerable groups; avail-ability of primary care structures, as well as suf-ficient sta ff with appropriate skills to care for pa-tients discharged from hospitals or maintained athome and to engage in measures to lift confinement(testing for example). This criterion is essential asit indicates that the di ff erent national health caresystems can cope with future increases in cases af-ter lifting of the measures. At the same time, hos-pitals are likely to face a backlog of elective inter- ventions that had been temporarily postponed dur-ing the pandemic’s peak. Therefore, states’ healthsystems should have recovered su ffi cient capacityin general, and not only related to the managementof Covid-19.3. Appropriate monitoring capacity, including large-scale testing capacity to detect and monitor thespread of the virus combined with contact tracingand possibilities to isolate people in case of reap-pearance and further spread of infections. Anti-body detection capacities, when confirmed specif-ically for Covid-19, will provide complementarydata on the share of the population that has suc-cessfully overcome the disease and eventuallymeasure the acquired immunity. The e ff ective reproduction number has increased to alevel clearly above 1: this would demand, in most cases,extremely prompt strengthening of the mitigation inter-ventions. The stringency of the new measures shouldguarantee that the healthcare system is not overwhelmedby a new epidemic wave. This requires the implementa-tion of forecasting tools that help to decision-makers todetermine the most suitable set of mitigating measures. The e ff ective reproductive factor is close to 1: inthis case, a deeper analysis is required. The decisionon whether to keep the same set of current mitigationmeasures or not will depend on the current fractionof infected population, the healthcare system capacity,and the potentiality of implementing in a short periodof time a mitigating intervention, which is capable ofbringing the e ff ective reproductive number to admissi-ble values. That is, the decision could be determined bythe worst-case cost of delaying in one week the imple-mentation of new measures. It is worth stressing thatpreemptive actions are always preferable: the earlier acountermeasure is adopted, the better in terms of its ef-ficacy and potential to save lives.In order to develop a timely and appropriate response,di ff erent methodologies from the field of control theoryare available (see the review paper [162]). Relying onPontryagin’s maximum principle, optimal control ap-proaches have been proposed to design optimal treat-ment plans, or vaccination plans, that minimize the costof the epidemics, including both the cost of infectionand the cost of treatment or vaccination [31] [78] [93][157]. Robust control approaches have also been pro-posed to control the spreading of infectious diseases,seen as uncertain dynamical systems [130] [131]. Weprovide more details on optimal control approaches inthe following subsections.15 .4. Optimal Control Theory Optimal control theory [138] can be applied to reducein an e ff ective way the burden of an epidemic [132],[147, Chapter 9]. The dynamic optimization techniquesof the calculus of variations and of optimal control the-ory provide methods for solving planning problems incontinuous time. The solution is a continuous function(or a set of functions) indicating the optimal path to befollowed by the variables through time or space [112].We present here a common formulation of a continuousdynamical optimization problem [96, Section 2]:min x ( · ) , u ( · ) S ( x ( T ) , T ) + T (cid:90) F ( x ( T ) , u ( t ) , t ) dts . t . x (0) = x , ˙ x = f ( x ( t ) , u ( t ) , t ) , g ( x ( t ) , u ( t ) , t ) ≥ , (4) h ( x ( t ) , t ) ≥ , (5) a ( x ( T ) , T ) ≥ , (6) b ( x ( T ) , T ) = . (7)In an epidemic control problem x ( t ) represents the stateof the pandemic at time t (for example, in terms of thepopulations of the di ff erent compartments), u ( t ) is thecontrol action which can be stated in a direct way (in-tensity of the interventions, number of vaccines, treat-ments), or in an indirect way (infection rate, immuno-logic protection, recovery rate). The di ff erential equa-tion ˙ x ( · ) = f ( · , · , · ) represents the epidemic model, in-equality (4) allows us to incorporate (mixed) constraintson x ( · ) and u ( · ) whereas the (pure) constraint (5) can beused to impose limits on the size of the components of x ( · ). Finally, (6) and (7) are terminal constraints. Thequestion of existence of optimal pairs ( x ∗ ( · ) , u ∗ ( · )) wasstudied in [49] and [75]. See also [96, Section 3] andreferences therein.Pontryagin’s maximum principle provides necessaryconditions that characterize the optimal solutions in thepresence of inequality constraints [138], [116]. Thesenecessary conditions become su ffi cient under certainconvexity conditions on the objective and constraintfunctions [145], [111]. In general, the solution of theoptimal problem in the presence of nonlinear dynam-ics and constraints requires iterative numerical meth-ods to solve the so-called Hamiltonian system, which isa two-point boundary value problem, plus a maximum(minimum) condition of the Hamiltonian (see e.g. [116,Chapter 6]).We now describe some examples of the use of theoptimal control theory in epidemic control. In [237], the dynamic optimal vaccination strategy for a SIR epi-demic model is described. The optimal solution is ob-tained using a forward-backward iterative method witha Runge-Kutta fourth-order solver. An example of howto deploy scarce resources for disease control when epi-demics occur in di ff erent but interconnected regions ispresented in [192]. The authors solve the optimal con-trol problem of minimizing the total level of infectionwhen the control actions are bounded.In [236] the authors apply Pontryagin’s Theorem toobtain an optimal Bang-Bang strategy to minimize thetotal number of infection cases during the spread of SIRepidemics in contact networks. Optimal control theoryis employed to design the best policies to control thespread of seasonal and novel A-H1N1 strains in [181].An example of the use of optimal control theory to con-trol the present Covid-19 pandemic is presented in [144]and [99], where the authors design an optimal strategy,for a five compartmental model, in order to minimizethe number of infected cases while minimizing the costof the non-pharmaceutical interventions. Model predictive control (MPC) provides optimal so-lutions to a control decision problem subject to con-straints [43], [185]. MPC is a receding horizon method-ology that involves repeatedly solving a constrained op-timization problem, using predictions of future costs,disturbances, and constraints over a moving time hori-zon. In epidemic control, the aforementioned optimiza-tion problem is solved daily, or weekly, in order to de-cide the optimal control action (for example, the inten-sity of mitigation interventions, or the optimal alloca-tion of resources). The output of the model predictivecontroller is adaptive in the sense that it takes into con-sideration the latest available information on the stateof the pandemic [201], [40]. See, for example, [156],[122], [8] for MPC formulations that address the controlof the Covid-19 pandemic. See [45] for a review paperon the application of MPC in the context of Covid-19pandemic.Because of the spatial clustered distribution of an epi-demic, it is possible to apply specific control techniquesfrom the field of distributed model predictive control[141], [54]. For example, non-linear model predictivecontrol can be used to control the epidemics by solelyacting upon the individuals’ contact pattern or network[200]. Another example of distributed MPC in the con-trol of epidemics is given in [121], where the problem ofdynamically allocating limited resources (vaccines andantidotes) to control an epidemic spreading process overa network is addressed.16 .6. Multi-objective control
Pareto optimality is used in multi-objective controlproblems with counter-balanced objectives. For in-stance, in a counter-balanced bi-objective problem, im-proving one objective implies to worsen the other one.Pareto optimality is based on the Pareto dominance,which defines that one solution dominates another oneif it is strictly superior in all the objectives. Thus, thegoal of the optimization algorithm is to find the Paretofront, which includes all dominant solutions of the con-trol problem. Therefore, there is a set of optimal solu-tions instead of one optimal solution. The Pareto front isa useful tool for decision-makers that allows to visual-ize all the possible optimal solutions (for two objectivesis a curve, for three objectives a plane, and so forth) andto evaluate the trade-o ff between di ff erent strategies. Inthe context of epidemic control [203], Pareto optimalityhas been used in [235] in a bi-objective control prob-lem, the goals are related to epidemic measures like thenumber of cases and economic costs.
7. Conclusions
This document has presented a roadmap for control-ling present and future pandemics from a data-drivenperspective, based on three pillars: Monitoring, Mod-elling, and Managing. We have highlighted the inter-play between data science, epidemiology, and controltheory to address the di ff erent challenges raised by apandemic.Methodologies and approaches proposed for previ-ous epidemics and the present Covid-19 pandemic havebeen reviewed, without claiming exhaustiveness, giventhe huge and continuously growing literature on thissubject. Although the relevant body of literature is ex-tremely large and many approaches have been studiedin the past, further research is still needed. Implement-ing e ff ective control strategies to mitigate a pandemicis di ffi cult because of various reasons: (i) the unavoid-able uncertainty on some crucial parameters character-izing the spread, including compliance issues due to theunpredictable human behaviour, (ii) the di ffi culties inassessing the quantitative e ff ect of mitigating interven-tions, (iii) the impossibility of obtaining a prompt evalu-ation of the e ff ect of the implemented interventions, dueto the intrinsic time-delay, and at the same time the crit-ical importance of acting quickly, due to the exponentialnature of the spreading phenomenon: even a small delayin interventions can lead to a much heavier healthcareburden and a much larger death toll.The first step for modelling di ff erent aspects of thepandemic is the processing of the available raw data to obtain consolidated time-series. In order to obtainpredictive models, which are crucial for the decision-making process, we have discussed several techniquesfrom epidemiology and machine learning. We have de-scribed the most relevant modelling and forecasting ap-proaches, focusing on the adjustment of the predictionmodels to the available data, model selection and vali-dation processes.Di ff erent surveillance systems able to detect, or an-ticipate, possible recurring epidemic waves have beensurveyed. These systems enable an immediate responsethat reduces the potential burden of the outbreak. Dif-ferent methods from control theory can be applied toprovide an optimal, robust and adaptive response to thetime-varying incidence of an epidemic. These methodscan be applied to the optimal allocation of resources,useful for testing campaigns and vaccination plans, andto determine trigger control schemes that modulate thestringency of the adopted interventions. 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