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Mathematical models to guide pandemic response

Science

The ongoing coronavirus disease 2019 (COVID-19) pandemic has put mathematical models in the spotlight. As the theoretical biologist Robert May wrote: “the virtue of a mathematical model...is that it forces clarity and precision upon conjecture, thus enabling meaningful comparison between the consequences of basic assumptions and the empirical facts” ([ 1 ][1]). On page 413 of this issue, Walker et al. ([ 2 ][2]) use a mathematical model to study the impact and burden of COVID-19 across a wide range of socioeconomic and demographic settings, with a focus on low- and middle-income countries (LMICs). Their analyses show that limited health care capacity in LMICs could counterbalance the benefits of a generally younger population. Unless these countries control the spread of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), the virus causing COVID-19, high disease burdens are likely. This work adds to a growing corpus of disease modeling designed to inform and guide the pandemic response. Following the emergence of a previously unknown pathogen like SARS-CoV-2, mathematical models can be used to estimate parameters of pathogen spread, explore possible future scenarios, evaluate retrospectively the efficacy of specific interventions, and identify prospective strategies (see the figure). At every stage, communicating the scope of a model's aims and the uncertainty in its outputs is essential to ensure that models effectively inform public health policy. In early 2020, it was important to assess the global risk posed by SARS-CoV-2 ([ 3 ][3]). Models provided estimates of R (the average number of new infections caused by each infectious individual when no one in the population is immune) and the infection fatality ratio (IFR), clarifying ambiguities in the latter due to asymptomatic infections and delays between infection and death ([ 4 ][4]). Models also provided estimates of the incubation period (the time from infection to symptom onset), allowing public health agencies to decide on 14 days for quarantine of exposed individuals ([ 5 ][5]). However, this illustrates the need for careful communication: 14-day quarantine and isolation can effectively reduce disease spread on average, but some individuals may spread the disease beyond 14 days. Building on growing knowledge of mechanisms and parameter estimates allowed researchers to explore possible future scenarios, including worst-case outcomes, and determine the implications for precautionary action ([ 6 ][6], [ 7 ][7]). The results provided in Walker et al. fit within this class of analyses. Of note, the authors do not seek to provide a precise prediction of the epidemic trajectory, but rather to highlight key areas of concern for LMICs and evaluate mitigation measures. As initial pandemic responses are put in place and data accumulates, the focus shifts to retrospective estimates of the efficacy of particular strategies. Using statistical inference algorithms, researchers can fit core epidemiological models to case data and data on the timing and nature of control interventions. This allows them to account for unknown aspects of underlying transmission dynamics, to tease out the individual effects of interventions, and to quantify the degree of uncertainty in these estimates ([ 8 ][8]). The amount of mechanistic detail included in such a model depends on its aims. For example, accounting for age-dependent transmission and susceptibility is critical when building models for retrospective estimation of the impact of school closures ([ 9 ][9]), but that level of detail might be omitted when studying larger-scale interventions, such as contact tracing or bans on travel. A key challenge of retrospective statistical modeling is that interventions are often entangled with each other, and with other processes. For example, several interventions may be implemented at once, making the individual impact of each intervention on transmission difficult or impossible to estimate. One solution is to compare multiple countries, regions, or municipalities that responded differently or at different times, but this approach is complicated by inevitable differences in their social and economic characteristics. Such confounding factors can prevent mathematical models from providing “meaningful comparison between the consequences of basic assumptions and the empirical facts” ([ 1 ][1]). Robust estimates of the relative impact of different interventions could be obtained by introducing brief delays in tightening (or loosening) interventions in randomly chosen “treatment” locations. Randomization reduces the chance of systematic differences between “treatment” and “control” locations where no delay is implemented (such an approach may be ethical if the costs and benefits of interventions remain uncertain). After the delay, differences in new infections (or susceptibility) between the treatment and the control locations can be used as model inputs, and the effect of the intervention in question can be estimated ([ 10 ][10]). Such estimates could provide invaluable information: Knowing how well interventions work allows the determination of which costly restrictions can be loosened with least impact on transmission and case numbers. ![Figure][11] Modeling during an outbreak An imagined pandemic time course shows cases (blue), deaths (red), and availability of serological data (orange). Modeling can estimate associated parameters of infection and intervention efficacy. Models can also make projections about the outbreak and the effects of interventions. Quantities that rely on, or are substantially enhanced by, serology are shown in orange. GRAPHIC: N. CARY/ SCIENCE Retrospective evaluations of specific interventions and detailed modeling efforts naturally build toward the identification of more targeted prospective strategies: Different countries, states, cities, or even workplaces vary in their capacity to respond to the pandemic and therefore require separate strategies for tightening or loosening interventions. Models can also help identify shortcomings in existing strategies, and explore opportunities for improvement or innovation. Although presymptomatic and asymptomatic transmission of COVID-19 has made contact tracing more difficult, modeling studies suggest feasibility of controlling spread through digital contact tracing, which allows for instantaneous identification of contacts ([ 11 ][12]). Many countries have started to develop and implement digital contact tracing apps, with varying degrees of success. However, digital contact tracing has been hampered by technical and ethical challenges, including the ability to accurately determine the proximity between individuals and to store such information in a privacy-preserving way. Thus, although models are valuable tools for testing new ideas, they must be accompanied by thorough consideration of practicalities. Prospective modeling not only informs strategy but also guides data collection. Registers of confirmed COVID-19 cases may miss asymptomatic infections, which can bias estimates of disease severity and IFR. In principle, serological (antibody) measurements provide a more reliable indication of whether an individual has been infected. Estimates of the IFR and other key parameters can thus be improved by combining serological surveys with data on cumulative infection-associated mortality. But if serological surveys are deployed in low incidence populations, true positive tests will be a minority of all tests, and estimates of cumulative incidence will be too uncertain to produce an informative estimate of the IFR. Epidemiological models could help reduce this uncertainty. Although valuable, mathematical models must be one facet of a diverse scientific response to a pandemic. During the early phases, models helped establish the scale of global risk and motivated action ([ 3 ][3]). But arguably the most critical evidence of severity was empirical data from Wuhan, Iran, and Northern Italy. Recently, establishing how best to loosen distancing restrictions has become a central challenge. Modeling can capture the benefits of specific interventions in terms of reduced transmission, but say nothing about their relative economic and psychological costs. Measurement of these costs is important to identifying which interventions work best ([ 10 ][10]) and which are most costly; models that encompass both epidemiological and economic aspects ([ 12 ][13]) are needed. What is next in the deployment of models? Perhaps the greatest conceptual success of infectious disease modeling was revealing that because the spread of infection depletes susceptible individuals, feedbacks are inherent to disease dynamics and can produce unexpected outcomes, including multiannual cycles of outbreaks. Yet, despite the devastating burden that COVID-19 has imposed so far, most of the world is still susceptible to SARS-CoV-2 infection. Most countries have yet to see sufficient depletion of susceptibility to meaningfully reduce infection spread, but such effects are likely to shape the future. Models projecting into this future have evaluated policy scenarios ([ 13 ][14]) and assessed the potential impact of seasonal variation in transmission ([ 14 ][15]). Notably, the immunological details on which these models rely (duration of immunity, whether immunity blocks transmission or prevents disease after infection, etc.) remain unclear. Models are one tool among many for tackling the pandemic, but they are perhaps the best framework for grappling with these possible futures. 1. [↵][16]1. R. M. May , Science 303, 790 (2004). [OpenUrl][17][Abstract/FREE Full Text][18] 2. [↵][19]1. P. G. T. Walker et al ., Science 369, 413 (2020). [OpenUrl][20][Abstract/FREE Full Text][21] 3. [↵][22]1. S. Cobey , Science 368, 713 (2020). [OpenUrl][23][Abstract/FREE Full Text][24] 4. [↵][25]1. T. W. Russell et al ., Euro Surveill. 25, 2000256 (2020). [OpenUrl][26][CrossRef][27][PubMed][28] 5. [↵][29]1. J. A. Backer et al ., Euro Surveill. 25, 2000062 (2020). [OpenUrl][30][CrossRef][31] 6. [↵][32]1. N. M. Ferguson et al ., “Report 9: Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand”; . 7. [↵][33]1. N. G. Davies et al ., Lancet Public Health 10.1016/S2468-2667(20)30133-X (2020). 8. [↵][34]1. S. Flaxman et al ., Nature 10.1038/s41586-020-24 (2020). 9. [↵][35]1. N. G. Davies et al ., Nat. Med. 10.1038/s41591-020-0962-9 (2020). 10. [↵][36]1. J. Haushofer, 2. C. J. E. Metcalf , Science 368, 1063 (2020). [OpenUrl][37][Abstract/FREE Full Text][38] 11. [↵][39]1. L. Ferretti et al ., Science 368, eabb6936 (2020). [OpenUrl][40][Abstract/FREE Full Text][41] 12. [↵][42]1. P. Klepac et al ., Proc. Natl. Acad. Sci. U.S.A. 108, 14366 (2011). [OpenUrl][43][Abstract/FREE Full Text][44] 13. [↵][45]1. S. M. Kissler et al ., Science 368, 860 (2020). [OpenUrl][46][Abstract/FREE Full Text][47] 14. [↵][48]1. R. E. Baker et al ., Science 369, 315 (2020). [OpenUrl][49][Abstract/FREE Full Text][50] Acknowledgments: Thanks to C. Buckee for the Robert May quote; and I. Werning, J. Haushofer, and B. Grenfell for helpful comments. All authors contributed equally. 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Your Wish Is My CMD

Communications of the ACM

As artificial intelligence (AI) techniques advance, they are beginning to automate tasks that, until recently, only humans could perform--tasks such as translating text from one language to another or making medical diagnoses. It seems only logical to turn that computer power on computers themselves and use AI to automate programming. In fact, computer scientists are working on just that idea, using various AI techniques to develop new methods of automating the writing of code. "The ultimate goal of this is that you would have professional software engineers not actually write code anymore," says Chris Jermaine, a professor of computer science at Rice University in Houston, TX. Instead, the engineer would tell a computer what a piece of software should do, and the AI system would write the code, perhaps stopping along the way to pose questions to the engineer.


Applying Inductive Logic Programming to Predicting Gene Function

AI Magazine

One of the fastest advancing areas of modern science is functional genomics. This science seeks to understand how the complete complement of molecular components of living organisms (nucleic acid, protein, small molecules, and so on) interact together to form living organisms. Functional genomics is of interest to AI because the relationship between machines and living organisms is central to AI and because the field is an instructive and fun domain to apply and sharpen AI tools and ideas, requiring complex knowledge representation, reasoning, learning, and so on. This article describes two machine learning (inductive logic programming [ILP])-based approaches to the bioinformatic problem of predicting protein function from amino acid sequence. The first approach is based on using ILP as a way of bootstrapping from conventional sequence-based homology methods.


The Voice of the Turtle: Whatever Happened to AI?

AI Magazine

Doug Lenat has worked in diverse parts of AI – natural language understanding and generation, automatic program synthesis, expert systems, machine learning, etc. – for going on 40 years now, just long enough to dare to write this article. His 1976 Stanford PhD thesis, AM, demonstrated that creative discoveries in mathematics could be produced by a computer program (a theorem proposer, rather than a theorem prover) guided by a corpus of hundreds of heuristic rules for deciding which experiments to perform and judging "interestingness" of their outcomes. That work earned him the IJCAI Computers and Thought Award, and sparked a renaissance in machine learning research. Dr. Lenat was on the CS faculty at CMU and Stanford, was one of the founders of Teknowledge, and was in the first batch of AAAI Fellows. He worked with Bill Gates and Nathan Myhrvold to launch Microsoft Research Labs, and to this day he remains the only person to have served on the technical advisory boards of both Apple and Microsoft.


Reopening schools is 'unlikely' to trigger a second wave of COVID-19 infection, say scientists

Daily Mail - Science & tech

Scientists say the reopening of schools in England, both primary and secondary, is'unlikely' to lead to a second wave of coronavirus infections. The gradual reopening of schools, starting with primary schools, wouldn't drive the average coronavirus transmission rate above one, infectious disease experts claim. Their mathematical modelling study shows that the impact of less social distancing on the part of adults would in fact be more likely to cause a'second wave'. As part of a phased return to schools in effect since Monday, the government is allowing pupils in reception, year one and year six to return to classes. While this policy could slightly raise the'R' number – the average amount of people that one infection person would pass the virus on to – it's unlikely to push it above one, the research team claims.


Bridging the Gap Between Probabilistic Model Checking and Probabilistic Planning: Survey, Compilations, and Empirical Comparison

Journal of Artificial Intelligence Research

Markov decision processes are of major interest in the planning community as well as in the model checking community. But in spite of the similarity in the considered formal models, the development of new techniques and methods happened largely independently in both communities. This work is intended as a beginning to unite the two research branches. We consider goal-reachability analysis as a common basis between both communities. The core of this paper is the translation from Jani, an overarching input language for quantitative model checkers, into the probabilistic planning domain definition language (PPDDL), and vice versa from PPDDL into Jani. These translations allow the creation of an overarching benchmark collection, including existing case studies from the model checking community, as well as benchmarks from the international probabilistic planning competitions (IPPC). We use this benchmark set as a basis for an extensive empirical comparison of various approaches from the model checking community, variants of value iteration, and MDP heuristic search algorithms developed by the AI planning community. On a per benchmark domain basis, techniques from one community can achieve state-ofthe-art performance in benchmarks of the other community. Across all benchmark domains of one community, the performance comparison is however in favor of the solvers and algorithms of that particular community. Reasons are the design of the benchmarks, as well as tool-related limitations. Our translation methods and benchmark collection foster crossfertilization between both communities, pointing out specific opportunities for widening the scope of solvers to different kinds of models, as well as for exchanging and adopting algorithms across communities.


Reasoning about Typicality and Probabilities in Preferential Description Logics

arXiv.org Artificial Intelligence

In this work we describe preferential Description Logics of typicality, a nonmonotonic extension of standard Description Logics by means of a typicality operator T allowing to extend a knowledge base with inclusions of the form T(C) D, whose intuitive meaning is that "normally/typically Cs are also Ds". This extension is based on a minimal model semantics corresponding to a notion of rational closure, built upon preferential models. We recall the basic concepts underlying preferential Description Logics. We also present two extensions of the preferential semantics: on the one hand, we consider probabilistic extensions, based on a distributed semantics that is suitable for tackling the problem of commonsense concept combination, on the other hand, we consider other strengthening of the rational closure semantics and construction to avoid the so called "blocking of property inheritance problem".


Human-Machine Collaboration for Democratizing Data Science

arXiv.org Artificial Intelligence

Data science is a cornerstone of current business practices. A major obstacle to its adoption is that most data analysis techniques are beyond the reach of typical end-users. Spreadsheets are a prime example of this phenomenon: despite being central in all sorts of data processing pipelines, the functionality necessary for processing and analyzing spreadsheets is hidden behind the high wall of spreadsheet formulas, which most end-users can neither write nor understand [Chambers and Scaffidi, 2010]. As a result, spreadsheets are often manipulated and analyzed manually. This increases the chance of making mistakes and prevents scaling beyond small data sets. Lowering the barrier to entry for specifying and solving data science tasks would help ameliorating these issues. Making data science tools more accessible would lower the cost of designing data processing pipelines and taking datadriven decisions. At the same time, accessible data science tools can prevent non-experts from relying on fragile heuristics and improvised solutions. The question we ask is then: is it possible to enable nontechnical end-users to specify and solve data science tasks that match their needs?


Indistinguishability

Communications of the ACM

I know thou wilt say "ay," And I will take thy word. Yet if thou swear'st Thou mayst prove false. At lovers' perjuries, They say, Jove laughs.


Three Modern Roles for Logic in AI

arXiv.org Artificial Intelligence

We consider three modern roles for logic in artificial intelligence, which are based on the theory of tractable Boolean circuits: (1) logic as a basis for computation, (2) logic for learning from a combination of data and knowledge, and (3) logic for reasoning about the behavior of machine learning systems.