bibliography.bib

@article{Flaxman_et_al_2020,
	Abstract = {Following the detection of the new coronavirus1 severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) and its spread outside of China, Europe has experienced large epidemics of coronavirus disease 2019 (COVID-19). In response, many European countries have implemented non-pharmaceutical interventions, such as the closure of schools and national lockdowns. Here we study the effect of major interventions across 11 European countries for the period from the start of the COVID-19 epidemics in February 2020 until 4 May 2020, when lockdowns started to be lifted. Our model calculates backwards from observed deaths to estimate transmission that occurred several weeks previously, allowing for the time lag between infection and death. We use partial pooling of information between countries, with both individual and shared effects on the time-varying reproduction number (Rt). Pooling allows for more information to be used, helps to overcome idiosyncrasies in the data and enables more-timely estimates. Our model relies on fixed estimates of some epidemiological parameters (such as the infection fatality rate), does not include importation or subnational variation and assumes that changes in Rt are an immediate response to interventions rather than gradual changes in behaviour. Amidst the ongoing pandemic, we rely on death data that are incomplete, show systematic biases in reporting and are subject to future consolidation. We estimate that---for all of the countries we consider here---current interventions have been sufficient to drive Rt below 1 (probability Rt < 1.0 is greater than 99{\%}) and achieve control of the epidemic. We estimate that across all 11 countries combined, between 12 and 15 million individuals were infected with SARS-CoV-2 up to 4 May 2020, representing between 3.2{\%} and 4.0{\%} of the population. Our results show that major non-pharmaceutical interventions---and lockdowns in particular---have had a large effect on reducing transmission. Continued intervention should be considered to keep transmission of SARS-CoV-2 under control.},
	Author = {Flaxman, Seth and Mishra, Swapnil and Gandy, Axel and Unwin, H. Juliette T. and Mellan, Thomas A. and Coupland, Helen and Whittaker, Charles and Zhu, Harrison and Berah, Tresnia and Eaton, Jeffrey W. and Monod, M{\'e}lodie and Perez-Guzman, Pablo N. and Schmit, Nora and Cilloni, Lucia and Ainslie, Kylie E. C. and Baguelin, Marc and Boonyasiri, Adhiratha and Boyd, Olivia and Cattarino, Lorenzo and Cooper, Laura V. and Cucunub{\'a}, Zulma and Cuomo-Dannenburg, Gina and Dighe, Amy and Djaafara, Bimandra and Dorigatti, Ilaria and van Elsland, Sabine L. and FitzJohn, Richard G. and Gaythorpe, Katy A. M. and Geidelberg, Lily and Grassly, Nicholas C. and Green, William D. and Hallett, Timothy and Hamlet, Arran and Hinsley, Wes and Jeffrey, Ben and Knock, Edward and Laydon, Daniel J. and Nedjati-Gilani, Gemma and Nouvellet, Pierre and Parag, Kris V. and Siveroni, Igor and Thompson, Hayley A. and Verity, Robert and Volz, Erik and Walters, Caroline E. and Wang, Haowei and Wang, Yuanrong and Watson, Oliver J. and Winskill, Peter and Xi, Xiaoyue and Walker, Patrick G. T. and Ghani, Azra C. and Donnelly, Christl A. and Riley, Steven and Vollmer, Michaela A. C. and Ferguson, Neil M. and Okell, Lucy C. and Bhatt, Samir and Imperial College COVID-19 Response Team},
	Da = {2020/08/01},
	Date-Added = {2021-08-10 11:43:47 +0000},
	Date-Modified = {2021-08-10 11:43:47 +0000},
	Doi = {10.1038/s41586-020-2405-7},
	Id = {Flaxman2020},
	Isbn = {1476-4687},
	Journal = {Nature},
	Number = {7820},
	Pages = {257--261},
	Title = {Estimating the effects of non-pharmaceutical interventions on COVID-19 in {E}urope},
	Ty = {JOUR},
	Url = {https://doi.org/10.1038/s41586-020-2405-7},
	Volume = {584},
	Year = {2020},
	Bdsk-Url-1 = {https://doi.org/10.1038/s41586-020-2405-7}}
	
@article{Douglas_et_al_2020,
	Author = {Douglas, Margaret and Katikireddi, Srinivasa Vittal and Taulbut, Martin and McKee, Martin and McCartney, Gerry},
	Doi = {10.1136/bmj.m1557},
	Elocation-Id = {m1557},
	Eprint = {https://www.bmj.com/content/369/bmj.m1557.full.pdf},
	Journal = {BMJ},
	Publisher = {BMJ Publishing Group Ltd},
	Title = {Mitigating the wider health effects of covid-19 pandemic response},
	Url = {https://www.bmj.com/content/369/bmj.m1557},
	Volume = {369},
	Year = {2020},
	Bdsk-Url-1 = {https://www.bmj.com/content/369/bmj.m1557},
	Bdsk-Url-2 = {https://doi.org/10.1136/bmj.m1557}}
	

@book{anderson1992infectious,
  title     = {Infectious {D}iseases of {H}umans: Dynamics and {C}ontrol},
  author    = {Anderson, R.M. and May, R.M.},
  isbn      = {9780198540403},
  lccn      = {90014312},
  year      = {1992},
  publisher = {Oxford University Press}
}

@inbook{Nishiura2009,
	Abstract = {Although the basic reproduction number, R0, is useful for understanding the transmissibility of a disease and designing various intervention strategies, the classic threshold quantity theoretically assumes that the epidemic first occurs in a fully susceptible population, and hence, R0 is essentially a mathematically defined quantity. In many instances, it is of practical importance to evaluate time-dependent variations in the transmission potential of infectious diseases. Explanation of the time course of an epidemic can be partly achieved by estimating the effective reproduction number, R(t), defined as the actual average number of secondary cases per primary case at calendar time t (for t >0). R(t) shows time-dependent variation due to the decline in susceptible individuals (intrinsic factors) and the implementation of control measures (extrinsic factors). If R(t)<1, it suggests that the epidemic is in decline and may be regarded as being under control at time t (vice versa, if R(t)>1). This chapter describes the primer of mathematics and statistics of R(t) and discusses other similar markers of transmissibility as a function of time.},
	Address = {Dordrecht},
	Author = {Nishiura, Hiroshi and Chowell, Gerardo},
	Booktitle = {Mathematical and Statistical Estimation Approaches in Epidemiology},
	Doi = {10.1007/978-90-481-2313-1_5},
	Editor = {Chowell, Gerardo and Hyman, James M. and Bettencourt, Lu{\'\i}s M. A. and Castillo-Chavez, Carlos},
	Isbn = {978-90-481-2313-1},
	Pages = {103--121},
	Publisher = {Springer Netherlands},
	Title = {The {E}ffective {R}eproduction {N}umber as a {P}relude to {S}tatistical {E}stimation of {T}ime-{D}ependent {E}pidemic {T}rends},
	Url = {https://doi.org/10.1007/978-90-481-2313-1_5},
	Year = {2009},
	Bdsk-Url-1 = {https://doi.org/10.1007/978-90-481-2313-1_5}
}

@book{vynnycky2010introduction,
  title={An {I}ntroduction to {I}nfectious {D}isease {M}odelling},
  author={Vynnycky, E. and White, R.},
  isbn={9780198565765},
  lccn={2010281726},
  year={2010},
  publisher={Oxford University Press}
}

@article{Brett_et_al_2020,
	Abstract = {Despite medical advances, the emergence and re-emergence of infectious diseases continue to pose a public health threat. Low-dimensional epidemiological models predict that epidemic transitions are preceded by the phenomenon of critical slowing down (CSD). This has raised the possibility of anticipating disease (re-)emergence using CSD-based early-warning signals (EWS), which are statistical moments estimated from time series data. For EWS to be useful at detecting future (re-)emergence, CSD needs to be a generic (model-independent) feature of epidemiological dynamics irrespective of system complexity. Currently, it is unclear whether the predictions of CSD---derived from simple, low-dimensional systems---pertain to real systems, which are high-dimensional. To assess the generality of CSD, we carried out a simulation study of a hierarchy of models, with increasing structural complexity and dimensionality, for a measles-like infectious disease. Our five models included: i) a nonseasonal homogeneous Susceptible-Exposed-Infectious-Recovered (SEIR) model, ii) a homogeneous SEIR model with seasonality in transmission, iii) an age-structured SEIR model, iv) a multiplex network-based model (Mplex) and v) an agent-based simulator (FRED). All models were parameterised to have a herd-immunity immunization threshold of around 90% coverage, and underwent a linear decrease in vaccine uptake, from 92% to 70% over 15 years. We found evidence of CSD prior to disease re-emergence in all models. We also evaluated the performance of seven EWS: the autocorrelation, coefficient of variation, index of dispersion, kurtosis, mean, skewness, variance. Performance was scored using the Area Under the ROC Curve (AUC) statistic. The best performing EWS were the mean and variance, with AUC > 0.75 one year before the estimated transition time. These two, along with the autocorrelation and index of dispersion, are promising candidate EWS for detecting disease emergence.},
	Author = {Brett, Tobias AND Ajelli, Marco AND Liu, Quan-Hui AND Krauland, Mary G. AND Grefenstette, John J. AND van Panhuis, Willem G. AND Vespignani, Alessandro AND Drake, John M. AND Rohani, Pejman},
	Doi = {10.1371/journal.pcbi.1007679},
	Journal = {PLOS Computational Biology},
	Month = {03},
	Number = {3},
	Pages = {1-19},
	Publisher = {Public Library of Science},
	Title = {Detecting critical slowing down in high-dimensional epidemiological systems},
	Url = {https://doi.org/10.1371/journal.pcbi.1007679},
	Volume = {16},
	Year = {2020},
	Bdsk-Url-1 = {https://doi.org/10.1371/journal.pcbi.1007679}}

@article{Gostic_et_al_20,
	Abstract = {Estimation of the effective reproductive number Rt is important for detecting changes in disease transmission over time. During the Coronavirus Disease 2019 (COVID-19) pandemic, policy makers and public health officials are using Rt to assess the effectiveness of interventions and to inform policy. However, estimation of Rt from available data presents several challenges, with critical implications for the interpretation of the course of the pandemic. The purpose of this document is to summarize these challenges, illustrate them with examples from synthetic data, and, where possible, make recommendations. For near real-time estimation of Rt, we recommend the approach of Cori and colleagues, which uses data from before time t and empirical estimates of the distribution of time between infections. Methods that require data from after time t, such as Wallinga and Teunis, are conceptually and methodologically less suited for near real-time estimation, but may be appropriate for retrospective analyses of how individuals infected at different time points contributed to the spread. We advise caution when using methods derived from the approach of Bettencourt and Ribeiro, as the resulting Rt estimates may be biased if the underlying structural assumptions are not met. Two key challenges common to all approaches are accurate specification of the generation interval and reconstruction of the time series of new infections from observations occurring long after the moment of transmission. Naive approaches for dealing with observation delays, such as subtracting delays sampled from a distribution, can introduce bias. We provide suggestions for how to mitigate this and other technical challenges and highlight open problems in Rt estimation.},
	Author = {Gostic, Katelyn M. AND McGough, Lauren AND Baskerville, Edward B. AND Abbott, Sam AND Joshi, Keya AND Tedijanto, Christine AND Kahn, Rebecca AND Niehus, Rene AND Hay, James A. AND De Salazar, Pablo M. AND Hellewell, Joel AND Meakin, Sophie AND Munday, James D. AND Bosse, Nikos I. AND Sherrat, Katharine AND Thompson, Robin N. AND White, Laura F. AND Huisman, Jana S. AND Scire, J{\'e}r{\'e}mie AND Bonhoeffer, Sebastian AND Stadler, Tanja AND Wallinga, Jacco AND Funk, Sebastian AND Lipsitch, Marc AND Cobey, Sarah},
	Doi = {10.1371/journal.pcbi.1008409},
	Journal = {PLOS Computational Biology},
	Month = {12},
	Number = {12},
	Pages = {1-21},
	Publisher = {Public Library of Science},
	Title = {Practical considerations for measuring the effective reproductive number, {R}t},
	Url = {https://doi.org/10.1371/journal.pcbi.1008409},
	Volume = {16},
	Year = {2020},
	Bdsk-Url-1 = {https://doi.org/10.1371/journal.pcbi.1008409}}
	
@article{Cori_et_al_2013,
    author = {Cori, Anne and Ferguson, Neil M. and Fraser, Christophe and Cauchemez, Simon},
    title = "{A New Framework and Software to Estimate Time-Varying Reproduction Numbers During Epidemics}",
    journal = {American Journal of Epidemiology},
    volume = {178},
    number = {9},
    pages = {1505-1512},
    year = {2013},
    month = {09},
    abstract = "{The quantification of transmissibility during epidemics is essential to designing and adjusting public health responses. Transmissibility can be measured by the reproduction number R, the average number of secondary cases caused by an infected individual. Several methods have been proposed to estimate R over the course of an epidemic; however, they are usually difficult to implement for people without a strong background in statistical modeling. Here, we present a ready-to-use tool for estimating R from incidence time series, which is implemented in popular software including Microsoft Excel (Microsoft Corporation, Redmond, Washington). This tool produces novel, statistically robust analytical estimates of R and incorporates uncertainty in the distribution of the serial interval (the time between the onset of symptoms in a primary case and the onset of symptoms in secondary cases). We applied the method to 5 historical outbreaks; the resulting estimates of R are consistent with those presented in the literature. This tool should help epidemiologists quantify temporal changes in the transmission intensity of future epidemics by using surveillance data.}",
    issn = {0002-9262},
    doi = {10.1093/aje/kwt133},
    url = {https://doi.org/10.1093/aje/kwt133},
    eprint = {https://academic.oup.com/aje/article-pdf/178/9/1505/17341195/kwt133.pdf},
}

@article{Thompson_et_al_20,
	Abstract = {Accurate estimation of the parameters characterising infectious disease transmission is vital for optimising control interventions during epidemics. A valuable metric for assessing the current threat posed by an outbreak is the time-dependent reproduction number, i.e. the expected number of secondary cases caused by each infected individual. This quantity can be estimated using data on the numbers of observed new cases at successive times during an epidemic and the distribution of the serial interval (the time between symptomatic cases in a transmission chain). Some methods for estimating the reproduction number rely on pre-existing estimates of the serial interval distribution and assume that the entire outbreak is driven by local transmission. Here we show that accurate inference of current transmissibility, and the uncertainty associated with this estimate, requires: (i) up-to-date observations of the serial interval to be included, and; (ii) cases arising from local transmission to be distinguished from those imported from elsewhere. We demonstrate how pathogen transmissibility can be inferred appropriately using datasets from outbreaks of H1N1 influenza, Ebola virus disease and Middle-East Respiratory Syndrome. We present a tool for estimating the reproduction number in real-time during infectious disease outbreaks accurately, which is available as an R software package (EpiEstim 2.2). It is also accessible as an interactive, user-friendly online interface (EpiEstim App), permitting its use by non-specialists. Our tool is easy to apply for assessing the transmission potential, and hence informing control, during future outbreaks of a wide range of invading pathogens.},
	Author = {R.N. Thompson and J.E. Stockwin and R.D. {van Gaalen} and J.A. Polonsky and Z.N. Kamvar and P.A. Demarsh and E. Dahlqwist and S. Li and E. Miguel and T. Jombart and J. Lessler and S. Cauchemez and A. Cori},
	Doi = {https://doi.org/10.1016/j.epidem.2019.100356},
	Issn = {1755-4365},
	Journal = {Epidemics},
	Keywords = {Mathematical modelling, Infectious disease epidemiology, Parameter inference, Reproduction number, Serial interval, Disease control},
	Pages = {100356},
	Title = {Improved inference of time-varying reproduction numbers during infectious disease outbreaks},
	Url = {https://www.sciencedirect.com/science/article/pii/S1755436519300350},
	Volume = {29},
	Year = {2019},
	Bdsk-Url-1 = {https://www.sciencedirect.com/science/article/pii/S1755436519300350},
	Bdsk-Url-2 = {https://doi.org/10.1016/j.epidem.2019.100356}}
	
@article{Wallinga_Teunis_04,
    author = {Wallinga, Jacco and Teunis, Peter},
    title = "{Different Epidemic Curves for Severe Acute Respiratory Syndrome Reveal  Similar Impacts of Control Measures}",
    journal = {American Journal of Epidemiology},
    volume = {160},
    number = {6},
    pages = {509-516},
    year = {2004},
    month = {09},
    abstract = "{Severe acute respiratory syndrome (SARS) has been the first severe contagious disease to emerge in the 21st century. The available epidemic curves for SARS show marked differences between the affected regions with respect to the total number of cases and epidemic duration, even for those regions in which outbreaks started almost simultaneously and similar control measures were implemented at the same time. The authors developed a likelihood-based estimation procedure that infers the temporal pattern of effective reproduction numbers from an observed epidemic curve. Precise estimates for the effective reproduction numbers were obtained by applying this estimation procedure to available data for SARS outbreaks that occurred in Hong Kong, Vietnam, Singapore, and Canada in 2003. The effective reproduction numbers revealed that epidemics in the various affected regions were characterized by markedly similar disease transmission potentials and similar levels of effectiveness of control measures. In controlling SARS outbreaks, timely alerts have been essential: Delaying the institution of control measures by 1 week would have nearly tripled the epidemic size and would have increased the expected epidemic duration by 4 weeks.}",
    issn = {0002-9262},
    doi = {10.1093/aje/kwh255},
    url = {https://doi.org/10.1093/aje/kwh255},
    eprint = {https://academic.oup.com/aje/article-pdf/160/6/509/179728/kwh255.pdf},
}

@article{Andrade_Duggan_2020,
title = "{An evaluation of Hamiltonian Monte Carlo performance to calibrate age-structured compartmental SEIR models to incidence data}",
journal = "Epidemics",
volume = "33",
pages = "100415",
year = "2020",
issn = "1755-4365",
doi = "https://doi.org/10.1016/j.epidem.2020.100415",
url = "http://www.sciencedirect.com/science/article/pii/S1755436520300372",
author = "Jair Andrade and Jim Duggan",
keywords = "Hamiltonian Monte Carlo, Nelder–Mead optimisation, SEIR, WAIFW, Stan",
abstract = "Hamiltonian Monte Carlo (HMC) is a Markov chain Monte Carlo method to estimate unknown quantities through sample generation from a target distribution for which an analytical solution is difficult. The strength of this method lies in its geometrical foundations, which render it efficient for traversing high-dimensional spaces. First, this paper analyses the performance of HMC in calibrating five variants of inputs to an age-structured SEIR model. Four of these variants are related to restriction assumptions that modellers devise to handle high-dimensional parameter spaces. The other one corresponds to the unrestricted symmetric variant. To provide a robust analysis, we compare HMC’s performance to that of the Nelder–Mead algorithm (NMS), a common method for non-linear optimisation. Furthermore, the calibration is performed on synthetic data in order to avoid confounding effects from errors in model selection. Then, we explore the variation in the method’s performance due to changes in the scale of the problem. Finally, we fit an SEIR model to real data. In all the experiments, the results show that HMC approximates both the synthetic and real data accurately, and provides reliable estimates for the basic reproduction number and the age-dependent transmission rates. HMC’s performance is robust in the presence of underreported incidences and high-dimensional complexity. This study suggests that stringent assumptions on age-dependent transmission rates can be lifted in favour of more realistic representations. The supplementary section presents the full set of results."
}

@article{Breto_2018,
author = {Carles Bretó},
title = {{Modeling and Inference for Infectious Disease Dynamics: A Likelihood-Based Approach}},
volume = {33},
journal = {Statistical Science},
number = {1},
publisher = {Institute of Mathematical Statistics},
pages = {57 -- 69},
keywords = {compartment model, continuous-time Markov chain, environmental stochasticity, iterated filtering, Lévy-driven stochastic differential equation, maximum likelihood, particle filter},
year = {2018},
doi = {10.1214/17-STS636},
URL = {https://doi.org/10.1214/17-STS636}
}

@article{Oliva_2003,
title = "Model calibration as a testing strategy for system dynamics models",
journal = "European Journal of Operational Research",
volume = "151",
number = "3",
pages = "552 - 568",
year = "2003",
issn = "0377-2217",
doi = "https://doi.org/10.1016/S0377-2217(02)00622-7",
url = "http://www.sciencedirect.com/science/article/pii/S0377221702006227",
author = "Rogelio Oliva",
keywords = "System dynamics, Simulation, Hypotheses testing, Model calibration, Parameter estimation",
abstract = "System dynamics models are becoming increasingly common in the analysis of policy and managerial issues. The usefulness of these models is predicated on their ability to link observable patterns of behavior to micro-level structure and decision-making processes. This paper posits that model calibration––the process of estimating the model parameters (structure) to obtain a match between observed and simulated structures and behaviors––is a stringent test of a hypothesis linking structure to behavior, and proposes a framework to use calibration as a form of model testing. It tackles the issue at three levels: theoretical, methodological, and technical. First, it explores the nature of model testing, and suggests that the modeling process be recast as an experimental approach to gain confidence in the hypothesis articulated in the model. At the methodological level, it proposes heuristics to guide the testing strategy, and to take advantage of the strengths of automated calibration algorithms. Finally, it presents a set of techniques to support the hypothesis testing process. The paper concludes with an example and a summary of the argument for the proposed approach."
}


@article{Dehning_et_al_20,
	Abstract = {From February to April 2020, many countries introduced variations on social distancing measures to slow the ravages of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Publicly available data show that Germany has been particularly successful in minimizing death rates. Dehning et al. quantified three governmental interventions introduced to control the outbreak. The authors predicted that the third governmental intervention{\textemdash}a strict contact ban since 22 March{\textemdash}switched incidence from growth to decay. They emphasize that relaxation of controls must be done carefully, not only because there is a 2-week lag between a measure being enacted and the effect on case reports but also because the three measures used in Germany only just kept virus spread below the growth threshold.Science, this issue p. eabb9789INTRODUCTIONWhen faced with the outbreak of a novel epidemic such as coronavirus disease 2019 (COVID-19), rapid response measures are required by individuals, as well as by society as a whole, to mitigate the spread of the virus. During this initial, time-critical period, neither the central epidemiological parameters nor the effectiveness of interventions such as cancellation of public events, school closings, or social distancing is known.RATIONALEAs one of the key epidemiological parameters, we inferred the spreading rate λ from confirmed SARS-CoV-2 infections using the example of Germany. We apply Bayesian inference based on Markov chain Monte Carlo sampling to a class of compartmental models [susceptible-infected-recovered (SIR)]. Our analysis characterizes the temporal change of the spreading rate and allows us to identify potential change points. Furthermore, it enables short-term forecast scenarios that assume various degrees of social distancing. A detailed description is provided in the accompanying paper, and the models, inference, and forecasts are available on GitHub (https://github.com/Priesemann-Group/covid19_inference_forecast). Although we apply the model to Germany, our approach can be readily adapted to other countries or regions.RESULTSIn Germany, interventions to contain the COVID-19 outbreak were implemented in three steps over 3 weeks: (i) Around 9 March 2020, large public events such as soccer matches were canceled; (ii) around 16 March 2020, schools, childcare facilities, and many stores were closed; and (iii) on 23 March 2020, a far-reaching contact ban (Kontaktsperre) was imposed by government authorities; this included the prohibition of even small public gatherings as well as the closing of restaurants and all nonessential stores.From the observed case numbers of COVID-19, we can quantify the impact of these measures on the disease spread using change point analysis. Essentially, we find that at each change point the spreading rate λ decreased by ~40\%. At the first change point, assumed around 9 March 2020, λ decreased from 0.43 to 0.25, with 95\% credible intervals (CIs) of [0.35, 0.51] and [0.20, 0.30], respectively. At the second change point, assumed around 16 March 2020, λ decreased to 0.15 (CI [0.12, 0.20]). Both changes in λ slowed the spread of the virus but still implied exponential growth (see red and orange traces in the figure).To contain the disease spread, i.e., to turn exponential growth into a decline of new cases, the spreading rate has to be smaller than the recovery rate μ = 0.13 (CI [0.09, 0.18]). This critical transition was reached with the third change point, which resulted in λ = 0.09 (CI [0.06, 0.13]; see blue trace in the figure), assumed around 23 March 2020.From the peak position of daily new cases, one could conclude that the transition from growth to decline was already reached at the end of March. However, the observed transient decline can be explained by a short-term effect that originates from a sudden change in the spreading rate (see Fig. 2C in the main text).As long as interventions and the concurrent individual behavior frequently change the spreading rate, reliable short- and long-term forecasts are very difficult. As the figure shows, the three example scenarios (representing the effects up to the first, second, and third change point) quickly diverge from each other and, consequently, span a considerable range of future case numbers.Inference and subsequent forecasts are further complicated by the delay of ~2 weeks between an intervention and the first useful estimates of the new λ (which are derived from the reported case numbers). Because of this delay, any uncertainty in the magnitude of social distancing in the previous 2 weeks can have a major impact on the case numbers in the subsequent 2 weeks. Beyond 2 weeks, the case numbers depend on our future behavior, for which we must make explicit assumptions. In sum, future interventions (such as lifting restrictions) should be implemented cautiously to respect the delayed visibility of their effects.CONCLUSIONWe developed a Bayesian framework for the spread of COVID-19 to infer central epidemiological parameters and the timing and magnitude of intervention effects. With such an approach, the effects of interventions can be assessed in a timely manner. Future interventions and lifting of restrictions can be modeled as additional change points, enabling short-term forecasts for case numbers. In general, our approach may help to infer the efficiency of measures taken in other countries and inform policy-makers about tightening, loosening, and selecting appropriate measures for containment of COVID-19.Bayesian inference of SIR model parameters from daily new cases of COVID-19 enables us to assess the impact of interventions.In Germany, three interventions (mild social distancing, strong social distancing, and contact ban) were enacted consecutively (circles). Colored lines depict the inferred models that include the impact of one, two, or three interventions (red, orange, or green, respectively, with individual data cutoff) or all available data until 21 April 2020 (blue). Forecasts (dashed lines) show how case numbers would have developed without the effects of the subsequent change points. Note the delay between intervention and first possible inference of parameters caused by the reporting delay and the necessary accumulation of evidence (gray arrows). Shaded areas indicate 50\% and 95\% Bayesian credible intervals.As coronavirus disease 2019 (COVID-19) is rapidly spreading across the globe, short-term modeling forecasts provide time-critical information for decisions on containment and mitigation strategies. A major challenge for short-term forecasts is the assessment of key epidemiological parameters and how they change when first interventions show an effect. By combining an established epidemiological model with Bayesian inference, we analyzed the time dependence of the effective growth rate of new infections. Focusing on COVID-19 spread in Germany, we detected change points in the effective growth rate that correlate well with the times of publicly announced interventions. Thereby, we could quantify the effect of interventions and incorporate the corresponding change points into forecasts of future scenarios and case numbers. Our code is freely available and can be readily adapted to any country or region.},
	Author = {Dehning, Jonas and Zierenberg, Johannes and Spitzner, F. Paul and Wibral, Michael and Neto, Joao Pinheiro and Wilczek, Michael and Priesemann, Viola},
	Doi = {10.1126/science.abb9789},
	Elocation-Id = {eabb9789},
	Eprint = {https://science.sciencemag.org/content/369/6500/eabb9789.full.pdf},
	Issn = {0036-8075},
	Journal = {Science},
	Number = {6500},
	Publisher = {American Association for the Advancement of Science},
	Title = {Inferring change points in the spread of COVID-19 reveals the effectiveness of interventions},
	Url = {https://science.sciencemag.org/content/369/6500/eabb9789},
	Volume = {369},
	Year = {2020},
	Bdsk-Url-1 = {https://science.sciencemag.org/content/369/6500/eabb9789},
	Bdsk-Url-2 = {https://doi.org/10.1126/science.abb9789}}
	

@article{Davies_2020,
	Abstract = {The COVID-19 pandemic has shown a markedly low proportion of cases among children1--4. Age disparities in observed cases could be explained by children having lower susceptibility to infection, lower propensity to show clinical symptoms or both. We evaluate these possibilities by fitting an age-structured mathematical model to epidemic data from China, Italy, Japan, Singapore, Canada and South Korea. We estimate that susceptibility to infection in individuals under 20 years of age is approximately half that of adults aged over 20 years, and that clinical symptoms manifest in 21{\%} (95{\%} credible interval: 12--31{\%}) of infections in 10- to 19-year-olds, rising to 69{\%} (57--82{\%}) of infections in people aged over 70 years. Accordingly, we find that interventions aimed at children might have a relatively small impact on reducing SARS-CoV-2 transmission, particularly if the transmissibility of subclinical infections is low. Our age-specific clinical fraction and susceptibility estimates have implications for the expected global burden of COVID-19, as a result of demographic differences across settings. In countries with younger population structures---such as many low-income countries---the expected per capita incidence of clinical cases would be lower than in countries with older population structures, although it is likely that comorbidities in low-income countries will also influence disease severity. Without effective control measures, regions with relatively older populations could see disproportionally more cases of COVID-19, particularly in the later stages of an unmitigated epidemic.},
	Author = {Davies, Nicholas G. and Klepac, Petra and Liu, Yang and Prem, Kiesha and Jit, Mark and Pearson, Carl A. B. and Quilty, Billy J. and Kucharski, Adam J. and Gibbs, Hamish and Clifford, Samuel and Gimma, Amy and van Zandvoort, Kevin and Munday, James D. and Diamond, Charlie and Edmunds, W. John and Houben, Rein M. G. J. and Hellewell, Joel and Russell, Timothy W. and Abbott, Sam and Funk, Sebastian and Bosse, Nikos I. and Sun, Yueqian Fiona and Flasche, Stefan and Rosello, Alicia and Jarvis, Christopher I. and Eggo, Rosalind M. and CMMID COVID-19 working group},
	Da = {2020/08/01},
	Date-Added = {2020-09-21 18:32:17 +0000},
	Date-Modified = {2020-09-21 18:32:17 +0000},
	Doi = {10.1038/s41591-020-0962-9},
	Id = {Davies2020},
	Isbn = {1546-170X},
	Journal = {Nature Medicine},
	Number = {8},
	Pages = {1205--1211},
	Title = {Age-dependent effects in the transmission and control of COVID-19 epidemics},
	Ty = {JOUR},
	Url = {https://doi.org/10.1038/s41591-020-0962-9},
	Volume = {26},
	Year = {2020},
	Bdsk-Url-1 = {https://doi.org/10.1038/s41591-020-0962-9}}

@article{breto_et_al_2009,
	Author = {Carles Bret{\'o} and Daihai He and Edward L. Ionides and Aaron A. King},
	Doi = {10.1214/08-AOAS201},
	Journal = {The Annals of Applied Statistics},
	Keywords = {cholera, Filtering, maximum likelihood, measles, sequential Monte Carlo, state space model},
	Number = {1},
	Pages = {319 -- 348},
	Publisher = {Institute of Mathematical Statistics},
	Title = {{Time series analysis via mechanistic models}},
	Url = {https://doi.org/10.1214/08-AOAS201},
	Volume = {3},
	Year = {2009},
	Bdsk-Url-1 = {https://doi.org/10.1214/08-AOAS201}}
	
@article{He_et_al_2010,
	Abstract = { Statistical inference for mechanistic models of partially observed dynamic systems is an active area of research. Most existing inference methods place substantial restrictions upon the form of models that can be fitted and hence upon the nature of the scientific hypotheses that can be entertained and the data that can be used to evaluate them. In contrast, the so-called plug-and-play methods require only simulations from a model and are thus free of such restrictions. We show the utility of the plug-and-play approach in the context of an investigation of measles transmission dynamics. Our novel methodology enables us to ask and answer questions that previous analyses have been unable to address. Specifically, we demonstrate that plug-and-play methods permit the development of a modelling and inference framework applicable to data from both large and small populations. We thereby obtain novel insights into the nature of heterogeneity in mixing and comment on the importance of including extra-demographic stochasticity as a means of dealing with environmental stochasticity and model misspecification. Our approach is readily applicable to many other epidemiological and ecological systems. },
	Author = {He, Daihai and Ionides, Edward L. and King, Aaron A.},
	Doi = {10.1098/rsif.2009.0151},
	Eprint = {https://royalsocietypublishing.org/doi/pdf/10.1098/rsif.2009.0151},
	Journal = {Journal of The Royal Society Interface},
	Number = {43},
	Pages = {271-283},
	Title = "{Plug-and-play inference for disease dynamics: measles in large and small populations as a case study}",
	Url = {https://royalsocietypublishing.org/doi/abs/10.1098/rsif.2009.0151},
	Volume = {7},
	Year = {2010},
	Bdsk-Url-1 = {https://royalsocietypublishing.org/doi/abs/10.1098/rsif.2009.0151},
	Bdsk-Url-2 = {https://doi.org/10.1098/rsif.2009.0151}}
	
@article{Keeling_et_al_2001,
	Abstract = {Biological phenomena offer a rich diversity of problems that can be understood using mathematical techniques. Three key features common to many biological systems are temporal forcing, stochasticity and nonlinearity. Here, using simple disease models compared to data, we examine how these three factors interact to produce a range of complicated dynamics. The study of disease dynamics has been amongst the most theoretically developed areas of mathematical biology; simple models have been highly successful in explaining the dynamics of a wide variety of diseases. Models of childhood diseases incorporate seasonal variation in contact rates due to the increased mixing during school terms compared to school holidays. This `binary' nature of the seasonal forcing results in dynamics that can be explained as switching between two nonlinear spiral sinks. Finally, we consider the stability of the attractors to understand the interaction between the deterministic dynamics and demographic and environmental stochasticity. Throughout attention is focused on the behaviour of measles, whooping cough and rubella.},
	Author = {Matt J. Keeling and Pejman Rohani and Bryan T. Grenfell},
	Doi = {https://doi.org/10.1016/S0167-2789(00)00187-1},
	Issn = {0167-2789},
	Journal = {Physica D: Nonlinear Phenomena},
	Keywords = {Childhood diseases, Stochasticity, Seasonal forcing, SIR models, Nonlinearity},
	Number = {3},
	Pages = {317-335},
	Title = {Seasonally forced disease dynamics explored as switching between attractors},
	Url = {https://www.sciencedirect.com/science/article/pii/S0167278900001871},
	Volume = {148},
	Year = {2001},
	Bdsk-Url-1 = {https://www.sciencedirect.com/science/article/pii/S0167278900001871},
	Bdsk-Url-2 = {https://doi.org/10.1016/S0167-2789(00)00187-1}}
	

@article{Liu_Stechlinski_12,
	Abstract = {Infectious disease models with time-varying parameters and general nonlinear incidence rates are analyzed. The functional form of the nonlinear incidence rate is assumed to change in time, due to, for example, environmental factors or a change in population behavior. More specifically, a new SIR model with time-varying parameters and switched nonlinear incidence rate is studied. The stability of the disease-free equilibrium is investigated, as well as disease persistence in the endemic case. A switched epidemic model with generalized compartments and time-varying parameters is also proposed and analyzed. Pulse vaccination and pulse treatment are applied to the new SIR model with seasonality and switched incidence rate. A control strategy with vaccine failure is applied to the switched epidemic model with generalized compartments. The control strategies are analyzed to determine their success in eradicating the disease. Some examples are given, with simulations, to illustrate the threshold conditions found.},
	Author = {Xinzhi Liu and Peter Stechlinski},
	Doi = {https://doi.org/10.1016/j.apm.2011.08.019},
	Issn = {0307-904X},
	Journal = {Applied Mathematical Modelling},
	Keywords = {Epidemic model, Nonlinear incidence rate, Seasonality, Pulse control, Basic reproduction number, Switched system},
	Number = {5},
	Pages = {1974-1994},
	Title = {Infectious disease models with time-varying parameters and general nonlinear incidence rate},
	Url = {https://www.sciencedirect.com/science/article/pii/S0307904X11005191},
	Volume = {36},
	Year = {2012},
	Bdsk-Url-1 = {https://www.sciencedirect.com/science/article/pii/S0307904X11005191},
	Bdsk-Url-2 = {https://doi.org/10.1016/j.apm.2011.08.019}}


@article{Liu_2010,
	Abstract = {The statistical data of tuberculosis (TB) cases show seasonal fluctuations in many countries. A TB model incorporating seasonality is developed and the basic reproduction ratio R0 is defined. It is shown that the disease-free equilibrium is globally asymptotically stable and the disease eventually disappears if R0<1, and there exists at least one positive periodic solution and the disease is uniformly persistent if R0>1. Numerical simulations indicate that there may be a unique positive periodic solution which is globally asymptotically stable if R0>1. Parameter values of the model are estimated according to demographic and epidemiological data in China. The simulation results are in good accordance with the seasonal variation of the reported cases of active TB in China.},
	Author = {Liu, Luju and Zhao, Xiao-Qiang and Zhou, Yicang},
	Da = {2010/05/01},
	Date-Added = {2021-06-23 17:56:46 +0000},
	Date-Modified = {2021-06-23 17:56:46 +0000},
	Doi = {10.1007/s11538-009-9477-8},
	Id = {Liu2010},
	Isbn = {1522-9602},
	Journal = {Bulletin of Mathematical Biology},
	Number = {4},
	Pages = {931--952},
	Title = "{A Tuberculosis Model with Seasonality}",
	Ty = {JOUR},
	Url = {https://doi.org/10.1007/s11538-009-9477-8},
	Volume = {72},
	Year = {2010},
	Bdsk-Url-1 = {https://doi.org/10.1007/s11538-009-9477-8}}
	


@article{Linka_et_al_2020,
	Abstract = {Throughout the past six months, no number has dominated the public media more persistently than the reproduction number of COVID-19. This powerful but simple concept is widely used by the public media, scientists, and political decision makers to explain and justify political strategies to control the COVID-19 pandemic. Here we explore the effectiveness of political interventions using the reproduction number of COVID-19 across Europe. We propose a dynamic SEIR epidemiology model with a time-varying reproduction number, which we identify using machine learning. During the early outbreak, the basic reproduction number was 4.22 $\pm$1.69, with maximum values of 6.33 and 5.88 in Germany and the Netherlands. By May 10, 2020, it dropped to 0.67 $\pm$0.18, with minimum values of 0.37 and 0.28 in Hungary and Slovakia. We found a strong correlation between passenger air travel, driving, walking, and transit mobility and the effective reproduction number with a time delay of 17.24 $\pm$2.00 days. Our new dynamic SEIR model provides the flexibility to simulate various outbreak control and exit strategies to inform political decision making and identify safe solutions in the benefit of global health.},
	Author = {Linka, Kevin and Peirlinck, Mathias and Kuhl, Ellen},
	Da = {2020/10/01},
	Date-Added = {2021-06-24 10:31:00 +0000},
	Date-Modified = {2021-06-24 10:31:00 +0000},
	Doi = {10.1007/s00466-020-01880-8},
	Id = {Linka2020},
	Isbn = {1432-0924},
	Journal = {Computational Mechanics},
	Number = {4},
	Pages = {1035--1050},
	Title = {The reproduction number of COVID-19 and its correlation with public health interventions},
	Ty = {JOUR},
	Url = {https://doi.org/10.1007/s00466-020-01880-8},
	Volume = {66},
	Year = {2020},
	Bdsk-Url-1 = {https://doi.org/10.1007/s00466-020-01880-8}}
	

@article{Endo_et_al_2019,
	Abstract = {The particle Markov-chain Monte Carlo (PMCMC) method is a powerful tool to efficiently explore high-dimensional parameter space using time-series data. We illustrate an overall picture of PMCMC with minimal but sufficient theoretical background to support the readers in the field of biomedical/health science to apply PMCMC to their studies. Some working examples of PMCMC applied to infectious disease dynamic models are presented with R code.},
	Author = {Akira Endo and Edwin {van Leeuwen} and Marc Baguelin},
	Doi = {https://doi.org/10.1016/j.epidem.2019.100363},
	Issn = {1755-4365},
	Journal = {Epidemics},
	Keywords = {Particle Markov-chain Monte Carlo, State-space models, Hidden Markov process, Particle filter, Sequential Monte Carlo},
	Pages = {100363},
	Title = "{Introduction to particle Markov-chain Monte Carlo for disease dynamics modellers}",
	Url = {https://www.sciencedirect.com/science/article/pii/S1755436519300301},
	Volume = {29},
	Year = {2019},
	Bdsk-Url-1 = {https://www.sciencedirect.com/science/article/pii/S1755436519300301},
	Bdsk-Url-2 = {https://doi.org/10.1016/j.epidem.2019.100363}}
	


@article{Dureau_et_al_2013,
    author = {Dureau, Joseph and Kalogeropoulos, Konstantinos and Baguelin, Marc},
    title = "{Capturing the time-varying drivers of an epidemic using stochastic dynamical systems}",
    journal = {Biostatistics},
    volume = {14},
    number = {3},
    pages = {541-555},
    year = {2013},
    month = {01},
    abstract = "{Epidemics are often modeled using non-linear dynamical systems observed through partial and noisy data. In this paper, we consider stochastic extensions in order to capture unknown influences (changing behaviors, public interventions, seasonal effects, etc.). These models assign diffusion processes to the time-varying parameters, and our inferential procedure is based on a suitably adjusted adaptive particle Markov chain Monte Carlo algorithm. The performance of the proposed computational methods is validated on simulated data and the adopted model is applied to the 2009 H1N1 pandemic in England. In addition to estimating the effective contact rate trajectories, the methodology is applied in real time to provide evidence in related public health decisions. Diffusion-driven susceptible exposed infected retired-type models with age structure are also introduced.}",
    issn = {1465-4644},
    doi = {10.1093/biostatistics/kxs052},
    url = {https://doi.org/10.1093/biostatistics/kxs052},
    eprint = {https://academic.oup.com/biostatistics/article-pdf/14/3/541/17739168/kxs052.pdf},
}




@article{Funk_et_al_2018,
	Abstract = {Real-time forecasts of infectious diseases can help public health planning, especially during outbreaks. If forecasts are generated from mechanistic models, they can be further used to target resources or to compare the impact of possible interventions. However, paremeterising such models is often difficult in real time, when information on behavioural changes, interventions and routes of transmission are not readily available. Here, we present a semi-mechanistic model of infectious disease dynamics that was used in real time during the 2013--2016 West African Ebola epidemic, and show fits to a Ebola Forecasting Challenge conducted in late 2015 with simulated data mimicking the true epidemic. We assess the performance of the model in different situations and identify strengths and shortcomings of our approach. Models such as the one presented here which combine the power of mechanistic models with the flexibility to include uncertainty about the precise outbreak dynamics may be an important tool in combating future outbreaks.},
	Author = {Sebastian Funk and Anton Camacho and Adam J. Kucharski and Rosalind M. Eggo and W. John Edmunds},
	Doi = {https://doi.org/10.1016/j.epidem.2016.11.003},
	Issn = {1755-4365},
	Journal = {Epidemics},
	Keywords = {Forecasting, Real-time modelling, Infectious disease dynamics, Outbreak},
	Note = {The RAPIDD Ebola Forecasting Challenge},
	Pages = {56-61},
	Title = {Real-time forecasting of infectious disease dynamics with a stochastic semi-mechanistic model},
	Url = {https://www.sciencedirect.com/science/article/pii/S1755436516300445},
	Volume = {22},
	Year = {2018},
	Bdsk-Url-1 = {https://www.sciencedirect.com/science/article/pii/S1755436516300445},
	Bdsk-Url-2 = {https://doi.org/10.1016/j.epidem.2016.11.003}}
	

@article{Camacho_et_al_2005,
	Abstract = {BACKGROUND: Between August and November 2014, the incidence of Ebola virus disease (EVD) rose dramatically in several districts of Sierra Leone. As a result, the number of cases exceeded the capacity of Ebola holding and treatment centres. During December, additional beds were introduced, and incidence declined in many areas. We aimed to measure patterns of transmission in different regions, and evaluate whether bed capacity is now sufficient to meet future demand. METHODS: We used a mathematical model of EVD infection to estimate how the extent of transmission in the nine worst affected districts of Sierra Leone changed between 10th August 2014 and 18th January 2015. Using the model, we forecast the number of cases that could occur until the end of March 2015, and compared bed requirements with expected future capacity. RESULTS: We found that the reproduction number, R, defined as the average number of secondary cases generated by a typical infectious individual, declined between August and December in all districts. We estimated that R was near the crucial control threshold value of 1 in December. We further estimated that bed capacity has lagged behind demand between August and December for most districts, but as a consequence of the decline in transmission, control measures caught up with the epidemic in early 2015. CONCLUSIONS: EVD incidence has exhibited substantial temporal and geographical variation in Sierra Leone, but our results suggest that the epidemic may have now peaked in Sierra Leone, and that current bed capacity appears to be sufficient to keep the epidemic under-control in most districts.},
	An = {25737806},
	Author = {Camacho, Anton and Kucharski, Adam and Aki-Sawyerr, Yvonne and White, Mark A and Flasche, Stefan and Baguelin, Marc and Pollington, Timothy and Carney, Julia R and Glover, Rebecca and Smout, Elizabeth and Tiffany, Amanda and Edmunds, W John and Funk, Sebastian},
	Date = {2015/02/10},
	Date-Added = {2021-06-24 14:40:54 +0000},
	Date-Modified = {2021-06-24 14:40:54 +0000},
	Db = {PubMed},
	Doi = {10.1371/currents.outbreaks.406ae55e83ec0b5193e30856b9235ed2},
	Isbn = {2157-3999},
	J2 = {PLoS Curr},
	Journal = {PLoS currents},
	Keywords = {ebola},
	L2 = {https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4339317/},
	La = {eng},
	Month = {02},
	Pages = {ecurrents.outbreaks.406ae55e83ec0b5193e30856b9235ed2},
	Publisher = {Public Library of Science},
	Title = "{Temporal Changes in Ebola Transmission in Sierra Leone and Implications for Control Requirements: a Real-time Modelling Study}",
	Ty = {JOUR},
	U1 = {25737806{$[$}pmid{$]$}},
	U2 = {PMC4339317{$[$}pmcid{$]$}},
	U4 = {ecurrents.outbreaks.406ae55e83ec0b5193e30856b9235ed2{$[$}PII{$]$}},
	Url = {https://pubmed.ncbi.nlm.nih.gov/25737806},
	Volume = {7},
	Year = {2015},
	Bdsk-Url-1 = {https://pubmed.ncbi.nlm.nih.gov/25737806},
	Bdsk-Url-2 = {https://doi.org/10.1371/currents.outbreaks.406ae55e83ec0b5193e30856b9235ed2}}


@article{Davies_et_al_2021,
	Abstract = {Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) has the capacity to generate variants with major genomic changes. The UK variant B.1.1.7 (also known as VOC 202012/01) has many mutations that alter virus attachment and entry into human cells. Using a variety of statistical and dynamic modeling approaches, Davies et al. characterized the spread of the B.1.1.7 variant in the United Kingdom. The authors found that the variant is 43 to 90\% more transmissible than the predecessor lineage but saw no clear evidence for a change in disease severity, although enhanced transmission will lead to higher incidence and more hospital admissions. Large resurgences of the virus are likely to occur after the easing of control measures, and it may be necessary to greatly accelerate vaccine roll-out to control the epidemic.Science, this issue p. eabg3055INTRODUCTIONSeveral novel variants of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), the virus that causes COVID-19, emerged in late 2020. One of these, Variant of Concern (VOC) 202012/01 (lineage B.1.1.7), was first detected in southeast England in September 2020 and spread to become the dominant lineage in the United Kingdom in just a few months. B.1.1.7 has since spread to at least 114 countries worldwide.RATIONALEThe rapid spread of VOC 202012/01 suggests that it transmits more efficiently from person to person than preexisting variants of SARS-CoV-2. This could lead to global surges in COVID-19 hospitalizations and deaths, so there is an urgent need to estimate how much more quickly VOC 202012/01 spreads, whether it is associated with greater or lesser severity of disease, and what control measures might be effective in mitigating its impact. We used social contact and mobility data, as well as demographic indicators linked to SARS-CoV-2 community testing data in England, to assess whether the spread of the new variant may be an artifact of higher baseline transmission rates in certain geographical areas or among specific demographic subpopulations. We then used a series of complementary statistical analyses and mathematical models to estimate the transmissibility of VOC 202012/01 across multiple datasets from the UK, Denmark, Switzerland, and the United States. Finally, we extended a mathematical model that has been extensively used to forecast COVID-19 dynamics in the UK to consider two competing SARS-CoV-2 lineages: VOC 202012/01 and preexisting variants. By fitting this model to a variety of data sources on infections, hospitalizations, and deaths across seven regions of England, we assessed different hypotheses for why the new variant appears to be spreading more quickly, estimated the severity of disease associated with the new variant, and evaluated control measures including vaccination and nonpharmaceutical interventions. Combining multiple lines of evidence allowed us to draw robust inferences.RESULTSThe rapid spread of VOC 202012/01 is not an artifact of geographical differences in contact behavior and does not substantially differ by age, sex, or socioeconomic stratum. We estimate that the new variant has a 43 to 90\% higher reproduction number (range of 95\% credible intervals, 38 to 130\%) than preexisting variants. Similar increases are observed in Denmark, Switzerland, and the United States. The most parsimonious explanation for this increase in the reproduction number is that people infected with VOC 202012/01 are more infectious than people infected with a preexisting variant, although there is also reasonable support for a longer infectious period and multiple mechanisms may be operating. Our estimates of severity are uncertain and are consistent with anything from a moderate decrease to a moderate increase in severity (e.g., 32\% lower to 20\% higher odds of death given infection). Nonetheless, our mathematical model, fitted to data up to 24 December 2020, predicted a large surge in COVID-19 cases and deaths in 2021, which has been borne out so far by the observed burden in England up to the end of March 2021. In the absence of stringent nonpharmaceutical interventions and an accelerated vaccine rollout, COVID-19 deaths in the first 6 months of 2021 were projected to exceed those in 2020 in England.CONCLUSIONMore than 98\% of positive SARS-CoV-2 infections in England are now due to VOC 202012/01, and the spread of this new variant has led to a surge in COVID-19 cases and deaths. Other countries should prepare for potentially similar outcomes.Impact of SARS-CoV-2 Variant of Concern 202012/01.(A) Spread of VOC 202012/01 (lineage B.1.1.7) in England. (B) The estimated relative transmissibility of VOC 202012/01 (mean and 95\% confidence interval) is similar across the United Kingdom as a whole, England, Denmark, Switzerland, and the United States. (C) Projected COVID-19 deaths (median and 95\% confidence interval) in England, 15 December 2020 to 30 June 2021. Vaccine rollout and control measures help to mitigate the burden of VOC 202012/01.A severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) variant, VOC 202012/01 (lineage B.1.1.7), emerged in southeast England in September 2020 and is rapidly spreading toward fixation. Using a variety of statistical and dynamic modeling approaches, we estimate that this variant has a 43 to 90\% (range of 95\% credible intervals, 38 to 130\%) higher reproduction number than preexisting variants. A fitted two-strain dynamic transmission model shows that VOC 202012/01 will lead to large resurgences of COVID-19 cases. Without stringent control measures, including limited closure of educational institutions and a greatly accelerated vaccine rollout, COVID-19 hospitalizations and deaths across England in the first 6 months of 2021 were projected to exceed those in 2020. VOC 202012/01 has spread globally and exhibits a similar transmission increase (59 to 74\%) in Denmark, Switzerland, and the United States.},
	Author = {Davies, Nicholas G. and Abbott, Sam and Barnard, Rosanna C. and Jarvis, Christopher I. and Kucharski, Adam J. and Munday, James D. and Pearson, Carl A. B. and Russell, Timothy W. and Tully, Damien C. and Washburne, Alex D. and Wenseleers, Tom and Gimma, Amy and Waites, William and Wong, Kerry L. M. and van Zandvoort, Kevin and Silverman, Justin D. and , and , and Diaz-Ordaz, Karla and Keogh, Ruth and Eggo, Rosalind M. and Funk, Sebastian and Jit, Mark and Atkins, Katherine E. and Edmunds, W. John},
	Doi = {10.1126/science.abg3055},
	Elocation-Id = {eabg3055},
	Eprint = {https://science.sciencemag.org/content/372/6538/eabg3055.full.pdf},
	Issn = {0036-8075},
	Journal = {Science},
	Number = {6538},
	Publisher = {American Association for the Advancement of Science},
	Title = "{Estimated transmissibility and impact of SARS-CoV-2 lineage B.1.1.7 in England}",
	Url = {https://science.sciencemag.org/content/372/6538/eabg3055},
	Volume = {372},
	Year = {2021},
	Bdsk-Url-1 = {https://science.sciencemag.org/content/372/6538/eabg3055},
	Bdsk-Url-2 = {https://doi.org/10.1126/science.abg3055}}
	

@article{King_et_al_2016,
   author = {Aaron A. King and Dao Nguyen and Edward L. Ionides},
   title = "{Statistical Inference for Partially Observed Markov Processes via the R Package pomp}",
   journal = {Journal of Statistical Software, Articles},
   volume = {69},
   number = {12},
   year = {2016},
   keywords = {Markov processes; hidden Markov model; state space model; stochastic dynamical system; maximum likelihood; plug-and-play; time series; mechanistic model; sequential Monte Carlo; R},
   abstract = {Partially observed Markov process (POMP) models, also known as hidden Markov models or state space models, are ubiquitous tools for time series analysis. The R package pomp provides a very flexible framework for Monte Carlo statistical investigations using nonlinear, non-Gaussian POMP models. A range of modern statistical methods for POMP models have been implemented in this framework including sequential Monte Carlo, iterated filtering, particle Markov chain Monte Carlo, approximate Bayesian computation, maximum synthetic likelihood estimation, nonlinear forecasting, and trajectory matching. In this paper, we demonstrate the application of these methodologies using some simple toy problems. We also illustrate the specification of more complex POMP models, using a nonlinear epidemiological model with a discrete population, seasonality, and extra-demographic stochasticity. We discuss the specification of user-defined models and the development of additional methods within the programming environment provided by pomp.},
   issn = {1548-7660},
   pages = {1--43},
   doi = {10.18637/jss.v069.i12},
   url = {https://www.jstatsoft.org/v069/i12}
}
	


@article{carpenter2017stan,
title={Stan: A probabilistic programming language},
author={Carpenter, Bob and Gelman, Andrew and Hoffman, Matthew D and Lee, Daniel and Goodrich, Ben and Betancourt, Michael and Brubaker, Marcus and Guo, Jiqiang and Li, Peter and Riddell, Allen},
journal={Journal of statistical software},
volume={76},
number={1},
year={2017},
publisher={Columbia Univ., New York, NY (United States); Harvard Univ., Cambridge, MA (United States)}
}

@article{owidcoronavirus,
    author = {Hannah Ritchie, Esteban Ortiz-Ospina, Diana Beltekian, Edouard Mathieu, Joe Hasell, Bobbie Macdonald, Charlie Giattino, Cameron Appel, Lucas Rodés-Guirao and Max Roser},
    title = "{Coronavirus Pandemic (COVID-19)}",
    journal = {Our World in Data},
    year = {2020},
    note = {https://ourworldindata.org/coronavirus}
}

@book{Chopin_Papaspiliopoulos_2020,
  title = "{An Introduction to Sequential Monte Carlo}",
  author={Chopin, N. and Papaspiliopoulos, O.},
  isbn={9783030478452},
  series={Springer Series in Statistics},
  year={2020},
  publisher={Springer International Publishing}
}

@article{Ionides_et_al_2006,
	Abstract = {Nonlinear stochastic dynamical systems are widely used to model systems across the sciences and engineering. Such models are natural to formulate and can be analyzed mathematically and numerically. However, difficulties associated with inference from time-series data about unknown parameters in these models have been a constraint on their application. We present a new method that makes maximum likelihood estimation feasible for partially-observed nonlinear stochastic dynamical systems (also known as state-space models) where this was not previously the case. The method is based on a sequence of filtering operations which are shown to converge to a maximum likelihood parameter estimate. We make use of recent advances in nonlinear filtering in the implementation of the algorithm. We apply the method to the study of cholera in Bangladesh. We construct confidence intervals, perform residual analysis, and apply other diagnostics. Our analysis, based upon a model capturing the intrinsic nonlinear dynamics of the system, reveals some effects overlooked by previous studies.},
	Author = {Ionides, E. L. and Bret{\'o}, C. and King, A. A.},
	Doi = {10.1073/pnas.0603181103},
	Eprint = {https://www.pnas.org/content/103/49/18438.full.pdf},
	Issn = {0027-8424},
	Journal = {Proceedings of the National Academy of Sciences},
	Number = {49},
	Pages = {18438--18443},
	Publisher = {National Academy of Sciences},
	Title = {Inference for nonlinear dynamical systems},
	Url = {https://www.pnas.org/content/103/49/18438},
	Volume = {103},
	Year = {2006},
	Bdsk-Url-1 = {https://www.pnas.org/content/103/49/18438},
	Bdsk-Url-2 = {https://doi.org/10.1073/pnas.0603181103}}
	

@book{blitzstein_2019,
  title= "{Introduction to Probability, Second Edition}",
  author={Blitzstein, J.K. and Hwang, J.},
  isbn={9780429766732},
  series={Chapman \& Hall/CRC Texts in Statistical Science},
  year={2019},
  publisher={CRC Press}
}

	


@ARTICLE{Arlampalam_et_al_2002,

  author={Arulampalam, M.S. and Maskell, S. and Gordon, N. and Clapp, T.},

  journal={IEEE Transactions on Signal Processing}, 

  title= "{A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking}", 

  year={2002},

  volume={50},

  number={2},

  pages={174-188},

  doi={10.1109/78.978374}}
  

@book{Keeling_Rohani_2011,
  title="{Modeling Infectious Diseases in Humans and Animals}",
  author={Keeling, M.J. and Rohani, P.},
  isbn={9781400841035},
  lccn={2006939548},
  year={2011},
  publisher={Princeton University Press}
}

@article{Gleeson_et_al_2021,
author = {Gleeson, James P.  and Brendan Murphy, Thomas  and O’Brien, Joseph D.  and Friel, Nial  and Bargary, Norma  and O'Sullivan, David J. P. },
title = {Calibrating COVID-19 susceptible-exposed-infected-removed models with time-varying effective contact rates},
journal = {Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences},
volume = {380},
number = {2214},
pages = {20210120},
year = {2022},
doi = {10.1098/rsta.2021.0120},

URL = {https://royalsocietypublishing.org/doi/abs/10.1098/rsta.2021.0120},
eprint = {https://royalsocietypublishing.org/doi/pdf/10.1098/rsta.2021.0120}
,
    abstract = { We describe the population-based susceptible-exposed-infected-removed (SEIR) model developed by the Irish Epidemiological Modelling Advisory Group (IEMAG), which advises the Irish government on COVID-19 responses. The model assumes a time-varying effective contact rate (equivalently, a time-varying reproduction number) to model the effect of non-pharmaceutical interventions. A crucial technical challenge in applying such models is their accurate calibration to observed data, e.g. to the daily number of confirmed new cases, as the history of the disease strongly affects predictions of future scenarios. We demonstrate an approach based on inversion of the SEIR equations in conjunction with statistical modelling and spline-fitting of the data to produce a robust methodology for calibration of a wide class of models of this type. This article is part of the theme issue ‘Data science approaches to infectious disease surveillance’. }
}


@article{Rasmussen_et_al_2011,
	Abstract = {Phylodynamics - the field aiming to quantitatively integrate the ecological and evolutionary dynamics of rapidly evolving populations like those of RNA viruses -- increasingly relies upon coalescent approaches to infer past population dynamics from reconstructed genealogies. As sequence data have become more abundant, these approaches are beginning to be used on populations undergoing rapid and rather complex dynamics. In such cases, the simple demographic models that current phylodynamic methods employ can be limiting. First, these models are not ideal for yielding biological insight into the processes that drive the dynamics of the populations of interest. Second, these models differ in form from mechanistic and often stochastic population dynamic models that are currently widely used when fitting models to time series data. As such, their use does not allow for both genealogical data and time series data to be considered in tandem when conducting inference. Here, we present a flexible statistical framework for phylodynamic inference that goes beyond these current limitations. The framework we present employs a recently developed method known as particle MCMC to fit stochastic, nonlinear mechanistic models for complex population dynamics to gene genealogies and time series data in a Bayesian framework. We demonstrate our approach using a nonlinear Susceptible-Infected-Recovered (SIR) model for the transmission dynamics of an infectious disease and show through simulations that it provides accurate estimates of past disease dynamics and key epidemiological parameters from genealogies with or without accompanying time series data.},
	Author = {Rasmussen, David A. AND Ratmann, Oliver AND Koelle, Katia},
	Doi = {10.1371/journal.pcbi.1002136},
	Journal = {PLOS Computational Biology},
	Month = {08},
	Number = {8},
	Pages = {1-11},
	Publisher = {Public Library of Science},
	Title = "{Inference for Nonlinear Epidemiological Models Using Genealogies and Time Series}",
	Url = {https://doi.org/10.1371/journal.pcbi.1002136},
	Volume = {7},
	Year = {2011},
	Bdsk-Url-1 = {https://doi.org/10.1371/journal.pcbi.1002136}}
	
@book{Wiersema_2008,
  title= "{Brownian Motion Calculus}",
  author={Wiersema, U.F.},
  isbn={9780470021712},
  series={Wiley Finance},
  year={2008},
  publisher={Wiley}
}


@article{Ionides_et_al_2011,
	Author = {Edward L. Ionides and Anindya Bhadra and Yves Atchad{\'e} and Aaron King},
	Doi = {10.1214/11-AOS886},
	Journal = {The Annals of Statistics},
	Keywords = {Dynamic systems, Filtering, importance sampling, partially observed Markov process, sequential Monte Carlo, state space model},
	Number = {3},
	Pages = {1776 -- 1802},
	Publisher = {Institute of Mathematical Statistics},
	Title = {{Iterated filtering}},
	Url = {https://doi.org/10.1214/11-AOS886},
	Volume = {39},
	Year = {2011},
	Bdsk-Url-1 = {https://doi.org/10.1214/11-AOS886}}
	
	
@article{King_et_al_2015,
	Abstract = { As an emergent infectious disease outbreak unfolds, public health response is reliant on information on key epidemiological quantities, such as transmission potential and serial interval. Increasingly, transmission models fit to incidence data are used to estimate these parameters and guide policy. Some widely used modelling practices lead to potentially large errors in parameter estimates and, consequently, errors in model-based forecasts. Even more worryingly, in such situations, confidence in parameter estimates and forecasts can itself be far overestimated, leading to the potential for large errors that mask their own presence. Fortunately, straightforward and computationally inexpensive alternatives exist that avoid these problems. Here, we first use a simulation study to demonstrate potential pitfalls of the standard practice of fitting deterministic models to cumulative incidence data. Next, we demonstrate an alternative based on stochastic models fit to raw data from an early phase of 2014 West Africa Ebola virus disease outbreak. We show not only that bias is thereby reduced, but that uncertainty in estimates and forecasts is better quantified and that, critically, lack of model fit is more readily diagnosed. We conclude with a short list of principles to guide the modelling response to future infectious disease outbreaks. },
	Author = {King, Aaron A. and Domenech de Cell{\`e}s, Matthieu and Magpantay, Felicia M. G. and Rohani, Pejman},
	Doi = {10.1098/rspb.2015.0347},
	Eprint = {https://royalsocietypublishing.org/doi/pdf/10.1098/rspb.2015.0347},
	Journal = {Proceedings of the Royal Society B: Biological Sciences},
	Number = {1806},
	Pages = {20150347},
	Title = "{Avoidable errors in the modelling of outbreaks of emerging pathogens, with special reference to Ebola}",
	Url = {https://royalsocietypublishing.org/doi/abs/10.1098/rspb.2015.0347},
	Volume = {282},
	Year = {2015},
	Bdsk-Url-1 = {https://royalsocietypublishing.org/doi/abs/10.1098/rspb.2015.0347},
	Bdsk-Url-2 = {https://doi.org/10.1098/rspb.2015.0347}}
	
@article{Wale_et_al_2019,
	Abstract = {What makes an infected host sick{\textemdash}the pathogen or the host response to it? How should pathogens defend themselves? Answering these questions requires an understanding of how, and how much, the immune response changes populations of host and pathogen cells. We break down the immune response into components with distinct effects on cell birth and death, and quantify the impact of each on disease and pathogens. We find that hosts control infections not only by killing pathogens, but by starving parasites and shortening the lifespans of cells on which they depend. This work reveals that some immune responses{\textemdash}often seen as harmful to the host{\textemdash}may in fact be helpful and suggests simple rules that govern the immune response{\textquoteright}s deployment.Hosts defend themselves against pathogens by mounting an immune response. Fully understanding the immune response as a driver of host disease and pathogen evolution requires a quantitative account of its impact on parasite population dynamics. Here, we use a data-driven modeling approach to quantify the birth and death processes underlying the dynamics of infections of the rodent malaria parasite, Plasmodium chabaudi, and the red blood cells (RBCs) it targets. We decompose the immune response into 3 components, each with a distinct effect on parasite and RBC vital rates, and quantify the relative contribution of each component to host disease and parasite density. Our analysis suggests that these components are deployed in a coordinated fashion to realize distinct resource-directed defense strategies that complement the killing of parasitized cells. Early in the infection, the host deploys a strategy reminiscent of siege and scorched-earth tactics, in which it both destroys RBCs and restricts their supply. Late in the infection, a {\textquotedblleft}juvenilization{\textquotedblright} strategy, in which turnover of RBCs is accelerated, allows the host to recover from anemia while holding parasite proliferation at bay. By quantifying the impact of immunity on both parasite fitness and host disease, we reveal that phenomena often interpreted as immunopathology may in fact be beneficial to the host. Finally, we show that, across mice, the components of the host response are consistently related to each other, even when infections take qualitatively different trajectories. This suggests the existence of simple rules that govern the immune system{\textquoteright}s deployment.},
	Author = {Wale, Nina and Jones, Matthew J. and Sim, Derek G. and Read, Andrew F. and King, Aaron A.},
	Doi = {10.1073/pnas.1908147116},
	Eprint = {https://www.pnas.org/content/116/44/22386.full.pdf},
	Issn = {0027-8424},
	Journal = {Proceedings of the National Academy of Sciences},
	Number = {44},
	Pages = {22386--22392},
	Publisher = {National Academy of Sciences},
	Title = "{The contribution of host cell-directed vs. parasite-directed immunity to the disease and dynamics of malaria infections}",
	Url = {https://www.pnas.org/content/116/44/22386},
	Volume = {116},
	Year = {2019},
	Bdsk-Url-1 = {https://www.pnas.org/content/116/44/22386},
	Bdsk-Url-2 = {https://doi.org/10.1073/pnas.1908147116}}
	
@book{Pawitan_2013,
  title= "{In All Likelihood: Statistical Modelling and Inference Using Likelihood}",
  author={Pawitan, Y.},
  isbn={9780199671229},
  lccn={2013474192},
  series={In All Likelihood: Statistical Modelling and Inference Using Likelihood},
  year={2013},
  publisher={OUP Oxford}
}

@article{Ionides_et_al_2017,
	Abstract = { Monte Carlo methods to evaluate and maximize the likelihood function enable the construction of confidence intervals and hypothesis tests, facilitating scientific investigation using models for which the likelihood function is intractable. When Monte Carlo error can be made small, by sufficiently exhaustive computation, then the standard theory and practice of likelihood-based inference applies. As datasets become larger, and models more complex, situations arise where no reasonable amount of computation can render Monte Carlo error negligible. We develop profile likelihood methodology to provide frequentist inferences that take into account Monte Carlo uncertainty. We investigate the role of this methodology in facilitating inference for computationally challenging dynamic latent variable models. We present examples arising in the study of infectious disease transmission, demonstrating our methodology for inference on nonlinear dynamic models using genetic sequence data and panel time-series data. We also discuss applicability to nonlinear time-series and spatio-temporal data. },
	Author = {Ionides, E. L. and Breto, C. and Park, J. and Smith, R. A. and King, A. A.},
	Doi = {10.1098/rsif.2017.0126},
	Eprint = {https://royalsocietypublishing.org/doi/pdf/10.1098/rsif.2017.0126},
	Journal = {Journal of The Royal Society Interface},
	Number = {132},
	Pages = {20170126},
	Title = "{Monte Carlo profile confidence intervals for dynamic systems}",
	Url = {https://royalsocietypublishing.org/doi/abs/10.1098/rsif.2017.0126},
	Volume = {14},
	Year = {2017},
	Bdsk-Url-1 = {https://royalsocietypublishing.org/doi/abs/10.1098/rsif.2017.0126},
	Bdsk-Url-2 = {https://doi.org/10.1098/rsif.2017.0126}}


@article{Gordon_et_al_1993,
	Abstract = {An algorithm, the bootstrap filter, is proposed for implementing recursive Bayesian filters. The required density of the state vector is represented as a set of random samples, which are updated and propagated by the algorithm. The method is not restricted by assumptions of linearity or Gaussian noise: it may be applied to any state transition or measurement model. A simulation example of the bearings only tracking problem is presented. This simulation includes schemes for improving the efficiency of the basic algorithm. For this example, the performance of the bootstrap filter is greatly superior to the standard extended Kalman filter.},
	Author = {Gordon, N.J. and Smith, A.F.M. and Salmond, D.J.},
	Copyright = {{\copyright} The Institution of Electrical Engineers},
	Issn = {0956-375X},
	Issue = {2},
	Journal = {IEE Proceedings F (Radar and Signal Processing)},
	Keywords = {state transition model;algorithm;bootstrap filter;recursive Bayesian filters;extended Kalman filter;simulation;measurement model;Gaussian noise;nonGaussian Bayesian state estimation;bearings only tracking problem;nonlinear Bayesian state estimation;state vector density;random samples;},
	Language = {English},
	Month = {April},
	Pages = {107-113(6)},
	Doi = {10.1049/ip-f-2.1993.0015},
	Title = "{Novel approach to nonlinear/non-Gaussian Bayesian state estimation}",
	Volume = {140},
	Year = {1993},
	Bdsk-Url-1 = {https://digital-library.theiet.org/content/journals/10.1049/ip-f-2.1993.0015}}
	
@online{HPSC_2020,
  author = {HPSC},
  title = "{Preliminary report of the results of the Study to Investigate COVID-19 Infection in People Living in Ireland (SCOPI):  A national seroprevalence study, June-July 2020}",
  year = 2020,
  url = {https://www.hpsc.ie/a-z/respiratory/coronavirus/novelcoronavirus/scopi/},
  urldate = {2020-08-20}
}

@inbook{Barlas_Yasarcan_2008,
	Address = {Berlin, Heidelberg},
	Author = {Barlas, Yaman and Yasarcan, Hakan},
	Booktitle = {Complex Decision Making: Theory and Practice},
	Doi = {10.1007/978-3-540-73665-3_15},
	Editor = {Qudrat-Ullah, H. and Spector, J.M. and Davidsen, P.I.},
	Isbn = {978-3-540-73665-3},
	Pages = {295--320},
	Publisher = {Springer Berlin Heidelberg},
	Title = "{A Comprehensive Model of Goal Dynamics in Organizations: Setting, Evaluation and Revision}",
	Url = {https://doi.org/10.1007/978-3-540-73665-3_15},
	Year = {2008},
	Bdsk-Url-1 = {https://doi.org/10.1007/978-3-540-73665-3_15}}

	
@article{Cox_et_al_1985,
 ISSN = {00129682, 14680262},
 Doi = {https://doi.org/10.2307/1911242},
 abstract = {This paper uses an intertemporal general equilibrium asset pricing model to study the term structure of interest rates. In this model, anticipations, risk aversion, investment alternatives, and preferences about the timing of consumption all play a role in determining bond prices. Many of the factors traditionally mentioned as influencing the term structure are thus included in a way which is fully consistent with maximizing behavior and rational expectations. The model leads to specific formulas for bond prices which are well suited for empirical testing.},
 author = {John C. Cox and Jonathan E. Ingersoll and Stephen A. Ross},
 journal = {Econometrica},
 number = {2},
 pages = {385--407},
 publisher = {Wiley},
 title = "{A Theory of the Term Structure of Interest Rates}",
 volume = {53},
 year = {1985}
}




@article{Sklar_1996,
 ISSN = {07492170},
 URL = {http://www.jstor.org/stable/4355880},
 abstract = {The author recalls his initial involvement with the basic notions of probability theory, which began in the late forties in the context of number theory, continued through his work with B. Schweizer on probabilistic metric spaces, and culminated in a correspondence with Fréchet that led to the identification and naming of copulas. The author speculates about possible future applications of the theory of distribution functions with given margins: In particular, there is the prospect of productive treatment of situations where, say, no common probability space can be found for a given set of "random variables," but such common probability spaces exist for arbitrary proper subsets of the given set.},
 author = {A. Sklar},
 journal = {Lecture Notes-Monograph Series},
 pages = {1--14},
 publisher = {Institute of Mathematical Statistics},
 title = "{Random Variables, Distribution Functions, and Copulas: A Personal Look Backward and Forward}",
 volume = {28},
 year = {1996}
}

@book{Sterman_2000,
  title="{Business Dynamics: Systems Thinking and Modeling for a Complex World}",
  author={Sterman, J.},
  isbn={9780072311358},
  lccn={99056030},
  series={McGraw-Hill Higher Education},
  year={2000},
  publisher={Irwin/McGraw-Hill}
}

@book{Fisher_2006,
  title="{The Purchasing Power of Money: Its' Determination And Relation to Credit Interest And Crises}",
  author={Fisher, I.},
  isbn={9781596056138},
  series={Cosimo classics economics},
  year={2006},
  publisher={Cosimo Classics}
}

@misc{Betancourt_2018,
      title="{A Conceptual Introduction to Hamiltonian Monte Carlo}", 
      author={Michael Betancourt},
      year={2018},
      eprint={1701.02434},
      archivePrefix={arXiv},
      primaryClass={stat.ME}
}

@article{Hyndman_2006,
title = "Another look at measures of forecast accuracy",
journal = "International Journal of Forecasting",
volume = "22",
number = "4",
pages = "679 - 688",
year = "2006",
issn = "0169-2070",
doi = "https://doi.org/10.1016/j.ijforecast.2006.03.001",
url = "http://www.sciencedirect.com/science/article/pii/S0169207006000239",
author = "Rob J. Hyndman and Anne B. Koehler",
keywords = "Forecast accuracy, Forecast evaluation, Forecast error measures, M-competition, Mean absolute scaled error",
abstract = "We discuss and compare measures of accuracy of univariate time series forecasts. The methods used in the M-competition as well as the M3-competition, and many of the measures recommended by previous authors on this topic, are found to be degenerate in commonly occurring situations. Instead, we propose that the mean absolute scaled error become the standard measure for comparing forecast accuracy across multiple time series."
}


@article{Funk_King_2020,
	Abstract = {Inference using mathematical models of infectious disease dynamics can be an invaluable tool for the interpretation and analysis of epidemiological data. However, researchers wishing to use this tool are faced with a choice of models and model types, simulation methods, inference methods and software packages. Given the multitude of options, it can be challenging to decide on the best approach. Here, we delineate the choices and trade-offs involved in deciding on an approach for inference, and discuss aspects that might inform this decision. We provide examples of inference with a dataset of influenza cases using the R packages pomp and rbi.},
	Author = {Sebastian Funk and Aaron A. King},
	Doi = {https://doi.org/10.1016/j.epidem.2019.100383},
	Issn = {1755-4365},
	Journal = {Epidemics},
	Keywords = {Inference, Infectious disease model, Bayesian, Frequentist, Model fitting},
	Pages = {100383},
	Title = {Choices and trade-offs in inference with infectious disease models},
	Url = {https://www.sciencedirect.com/science/article/pii/S1755436519300441},
	Volume = {30},
	Year = {2020},
	Bdsk-Url-1 = {https://www.sciencedirect.com/science/article/pii/S1755436519300441},
	Bdsk-Url-2 = {https://doi.org/10.1016/j.epidem.2019.100383}}


@article{Delamater_et_al_2019,
	Abstract = {The basic reproduction number (R0), also called the basic reproduction ratio or rate or the basic reproductive rate, is an epidemiologic metric used to describe the contagiousness or transmissibility of infectious agents. R0 is affected by numerous biological, sociobehavioral, and environmental factors that govern pathogen transmission and, therefore, is usually estimated with various types of complex mathematical models, which make R0 easily misrepresented, misinterpreted, and misapplied. R0 is not a biological constant for a pathogen, a rate over time, or a measure of disease severity, and R0 cannot be modified through vaccination campaigns. R0 is rarely measured directly, and modeled R0 values are dependent on model structures and assumptions. Some R0 values reported in the scientific literature are likely obsolete. R0 must be estimated, reported, and applied with great caution because this basic metric is far from simple.},
	Author = {Delamater, Paul and Street, Erica and Leslie, Timothy and Yang, Y. Tony and Jacobsen, Kathryn},
	Booktitle = {Emerging Infectious Disease journal},
	Date-Added = {2021-08-05 11:43:17 +0000},
	Date-Modified = {2021-08-05 11:43:17 +0000},
	Doi = {10.3201/eid2501.171901},
	Isbn = {1080-6059},
	Keywords = {basic reproduction number; theoretical models; disease outbreaks; infectious disease transmission; R0; mathematical modeling; outbreaks; basic reproductive ratio; basic reproductive rate; basic reproduction ratio; basic reproduction rate},
	Number = {1},
	Pages = {1},
	Title = "{Complexity of the Basic Reproduction Number (R$_{0}$)}",
	Ty = {JOUR},
	Url = {https://wwwnc.cdc.gov/eid/article/25/1/17-1901_article},
	Volume = {25},
	Year = {2019},
	Bdsk-Url-1 = {https://wwwnc.cdc.gov/eid/article/25/1/17-1901_article},
	Bdsk-Url-2 = {https://doi.org/10.3201/eid2501.171901}}
	

@article{Li_et_al_2020,
	Author = {Li, Qun and Guan, Xuhua and Wu, Peng and Wang, Xiaoye and Zhou, Lei and Tong, Yeqing and Ren, Ruiqi and Leung, Kathy S.M. and Lau, Eric H.Y. and Wong, Jessica Y. and Xing, Xuesen and Xiang, Nijuan and Wu, Yang and Li, Chao and Chen, Qi and Li, Dan and Liu, Tian and Zhao, Jing and Liu, Man and Tu, Wenxiao and Chen, Chuding and Jin, Lianmei and Yang, Rui and Wang, Qi and Zhou, Suhua and Wang, Rui and Liu, Hui and Luo, Yinbo and Liu, Yuan and Shao, Ge and Li, Huan and Tao, Zhongfa and Yang, Yang and Deng, Zhiqiang and Liu, Boxi and Ma, Zhitao and Zhang, Yanping and Shi, Guoqing and Lam, Tommy T.Y. and Wu, Joseph T. and Gao, George F. and Cowling, Benjamin J. and Yang, Bo and Leung, Gabriel M. and Feng, Zijian},
	Doi = {10.1056/NEJMoa2001316},
	Eprint = {https://doi.org/10.1056/NEJMoa2001316},
	Journal = {New England Journal of Medicine},
	Note = {PMID: 31995857},
	Number = {13},
	Pages = {1199-1207},
	Title = "{Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus--Infected Pneumonia}",
	Url = {https://doi.org/10.1056/NEJMoa2001316},
	Volume = {382},
	Year = {2020},
	Bdsk-Url-1 = {https://doi.org/10.1056/NEJMoa2001316}}
	
@article{Petersen_et_2020,
	Abstract = {Summary
The objective of this Personal View is to compare transmissibility, hospitalisation, and mortality rates for severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) with those of other epidemic coronaviruses, such as severe acute respiratory syndrome coronavirus (SARS-CoV) and Middle East respiratory syndrome coronavirus (MERS-CoV), and pandemic influenza viruses. The basic reproductive rate (R0) for SARS-CoV-2 is estimated to be 2·5 (range 1·8--3·6) compared with 2·0--3·0 for SARS-CoV and the 1918 influenza pandemic, 0·9 for MERS-CoV, and 1·5 for the 2009 influenza pandemic. SARS-CoV-2 causes mild or asymptomatic disease in most cases; however, severe to critical illness occurs in a small proportion of infected individuals, with the highest rate seen in people older than 70 years. The measured case fatality rate varies between countries, probably because of differences in testing strategies. Population-based mortality estimates vary widely across Europe, ranging from zero to high. Numbers from the first affected region in Italy, Lombardy, show an all age mortality rate of 154 per 100 000 population. Differences are most likely due to varying demographic structures, among other factors. However, this new virus has a focal dissemination; therefore, some areas have a higher disease burden and are affected more than others for reasons that are still not understood. Nevertheless, early introduction of strict physical distancing and hygiene measures have proven effective in sharply reducing R0 and associated mortality and could in part explain the geographical differences.},
	Author = {Eskild Petersen and Marion Koopmans and Unyeong Go and Davidson H Hamer and Nicola Petrosillo and Francesco Castelli and Merete Storgaard and Sulien {Al Khalili} and Lone Simonsen},
	Doi = {https://doi.org/10.1016/S1473-3099(20)30484-9},
	Issn = {1473-3099},
	Journal = {The Lancet Infectious Diseases},
	Number = {9},
	Pages = {e238-e244},
	Title = {Comparing SARS-CoV-2 with SARS-CoV and influenza pandemics},
	Url = {https://www.sciencedirect.com/science/article/pii/S1473309920304849},
	Volume = {20},
	Year = {2020},
	Bdsk-Url-1 = {https://www.sciencedirect.com/science/article/pii/S1473309920304849},
	Bdsk-Url-2 = {https://doi.org/10.1016/S1473-3099(20)30484-9}}
	

@article{Katul_et_al_2020,
	Abstract = {The SIR (`susceptible-infectious-recovered') formulation is used to uncover the generic spread mechanisms observed by COVID-19 dynamics globally, especially in the early phases of infectious spread. During this early period, potential controls were not effectively put in place or enforced in many countries. Hence, the early phases of COVID-19 spread in countries where controls were weak offer a unique perspective on the ensemble-behavior of COVID-19 basic reproduction number Ro inferred from SIR formulation. The work here shows that there is global convergence (i.e., across many nations) to an uncontrolled Ro = 4.5 that describes the early time spread of COVID-19. This value is in agreement with independent estimates from other sources reviewed here and adds to the growing consensus that the early estimate of Ro = 2.2 adopted by the World Health Organization is low. A reconciliation between power-law and exponential growth predictions is also featured within the confines of the SIR formulation. The effects of testing ramp-up and the role of `super-spreaders' on the inference of Ro are analyzed using idealized scenarios. Implications for evaluating potential control strategies from this uncontrolled Ro are briefly discussed in the context of the maximum possible infected fraction of the population (needed to assess health care capacity) and mortality (especially in the USA given diverging projections). Model results indicate that if intervention measures still result in Ro > 2.7 within 44 days after first infection, intervention is unlikely to be effective in general for COVID-19.},
	Author = {Katul, Gabriel G. AND Mrad, Assaad AND Bonetti, Sara AND Manoli, Gabriele AND Parolari, Anthony J.},
	Doi = {10.1371/journal.pone.0239800},
	Journal = {PLOS ONE},
	Month = {09},
	Number = {9},
	Pages = {1-22},
	Publisher = {Public Library of Science},
	Title = {Global convergence of COVID-19 basic reproduction number and estimation from early-time SIR dynamics},
	Url = {https://doi.org/10.1371/journal.pone.0239800},
	Volume = {15},
	Year = {2020},
	Bdsk-Url-1 = {https://doi.org/10.1371/journal.pone.0239800}}
	

@article{Sanche_et_al_2020,
	Abstract = {Severe acute respiratory syndrome coronavirus 2 is the causative agent of the ongoing coronavirus disease pandemic. Initial estimates of the early dynamics of the outbreak in Wuhan, China, suggested a doubling time of the number of infected persons of 6--7 days and a basic reproductive number (R0) of 2.2--2.7. We collected extensive individual case reports across China and estimated key epidemiologic parameters, including the incubation period (4.2 days). We then designed 2 mathematical modeling approaches to infer the outbreak dynamics in Wuhan by using high-resolution domestic travel and infection data. Results show that the doubling time early in the epidemic in Wuhan was 2.3--3.3 days. Assuming a serial interval of 6--9 days, we calculated a median R0 value of 5.7 (95{\%} CI 3.8--8.9). We further show that active surveillance, contact tracing, quarantine, and early strong social distancing efforts are needed to stop transmission of the virus.},
	Author = {Sanche, Steven and Lin, Yen Ting and Xu, Chonggang and Romero-Severson, Ethan and Hengartner, Nick and Ke, Ruian},
	Booktitle = {Emerging Infectious Disease journal},
	Date-Added = {2021-08-05 12:26:10 +0000},
	Date-Modified = {2021-08-05 12:26:10 +0000},
	Doi = {10.3201/eid2607.200282},
	Isbn = {1080-6059},
	Keywords = {COVID-19; 2019 novel coronavirus disease; SARS-CoV-2; severe acute respiratory syndrome coronavirus 2; viruses; respiratory infections; zoonoses; Wuhan; China; transmission potential; modeling},
	Number = {7},
	Pages = {1470},
	Title = "{High Contagiousness and Rapid Spread of Severe Acute Respiratory Syndrome Coronavirus 2}",
	Ty = {JOUR},
	Url = {https://wwwnc.cdc.gov/eid/article/26/7/20-0282_article},
	Volume = {26},
	Year = {2020},
	Bdsk-Url-1 = {https://wwwnc.cdc.gov/eid/article/26/7/20-0282_article},
	Bdsk-Url-2 = {https://doi.org/10.3201/eid2607.200282}}
	


	
@article{Diekmann_et_al_1990,
	Abstract = {The expected number of secondary cases produced by a typical infected individual during its entire period of infectiousness in a completely susceptible population is mathematically defined as the dominant eigenvalue of a positive linear operator. It is shown that in certain special cases one can easily compute or estimate this eigenvalue. Several examples involving various structuring variables like age, sexual disposition and activity are presented.},
	Author = {Diekmann, O. and Heesterbeek, J. A. P. and Metz, J. A. J.},
	Da = {1990/06/01},
	Date-Added = {2021-08-10 17:02:13 +0000},
	Date-Modified = {2021-08-10 17:02:13 +0000},
	Doi = {10.1007/BF00178324},
	Id = {Diekmann1990},
	Isbn = {1432-1416},
	Journal = {Journal of Mathematical Biology},
	Number = {4},
	Pages = {365--382},
	Title = {On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations},
	Ty = {JOUR},
	Url = {https://doi.org/10.1007/BF00178324},
	Volume = {28},
	Year = {1990},
	Bdsk-Url-1 = {https://doi.org/10.1007/BF00178324}}
	

@article{van_den_Driessche_2017,
	Abstract = {This primer article focuses on the basic reproduction number, ℛ0, for infectious diseases, and other reproduction numbers related to ℛ0 that are useful in guiding control strategies. Beginning with a simple population model, the concept is developed for a threshold value of ℛ0 determining whether or not the disease dies out. The next generation matrix method of calculating ℛ0 in a compartmental model is described and illustrated. To address control strategies, type and target reproduction numbers are defined, as well as sensitivity and elasticity indices. These theoretical ideas are then applied to models that are formulated for West Nile virus in birds (a vector-borne disease), cholera in humans (a disease with two transmission pathways), anthrax in animals (a disease that can be spread by dead carcasses and spores), and Zika in humans (spread by mosquitoes and sexual contacts). Some parameter values from literature data are used to illustrate the results. Finally, references for other ways to calculate ℛ0 are given. These are useful for more complicated models that, for example, take account of variations in environmental fluctuation or stochasticity.},
	Author = {Pauline {van den Driessche}},
	Doi = {https://doi.org/10.1016/j.idm.2017.06.002},
	Issn = {2468-0427},
	Journal = {Infectious Disease Modelling},
	Keywords = {Basic reproduction number, Disease control, West Nile virus, Cholera, Anthrax, Zika virus},
	Number = {3},
	Pages = {288-303},
	Title = {Reproduction numbers of infectious disease models},
	Url = {https://www.sciencedirect.com/science/article/pii/S2468042717300209},
	Volume = {2},
	Year = {2017},
	Bdsk-Url-1 = {https://www.sciencedirect.com/science/article/pii/S2468042717300209},
	Bdsk-Url-2 = {https://doi.org/10.1016/j.idm.2017.06.002}}
	
@article{Priestley_1980,
author = {Priestley, M. B.},
title = {STATE-DEPENDENT MODELS: A GENERAL APPROACH TO NON-LINEAR TIME SERIES ANALYSIS},
journal = {Journal of Time Series Analysis},
volume = {1},
number = {1},
pages = {47-71},
keywords = {Non-linear time series models, state dependent models, bilinear, threshold autoregressive, exponential autoregressive, linear analytic systems},
doi = {https://doi.org/10.1111/j.1467-9892.1980.tb00300.x},
url = {https://onlinelibrary.wiley.com/doi/abs/10.1111/j.1467-9892.1980.tb00300.x},
eprint = {https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1467-9892.1980.tb00300.x},
abstract = {Abstract. We construct a general class of non-linear models, called ‘state-dependent models’, which have a very flexible non-linear structure and which contain, as special cases, bilinear, threshold autoregressive, and exponential autoregressive models. We describe a sequential type of recursive algorithm for identifying state-dependent models, and show how such models may be used for forecasting and for indicating specific types of non-linear behaviour.},
year = {1980}
}

@article{Kimeldorf_Wahba_1970,
author = {George S. Kimeldorf and Grace Wahba},
title = {{A Correspondence Between Bayesian Estimation on Stochastic Processes and Smoothing by Splines}},
volume = {41},
journal = {The Annals of Mathematical Statistics},
number = {2},
publisher = {Institute of Mathematical Statistics},
pages = {495 -- 502},
year = {1970},
doi = {10.1214/aoms/1177697089},
URL = {https://doi.org/10.1214/aoms/1177697089}
}

@article{Hastie_Tibshirani_1993,
 ISSN = {00359246},
 URL = {http://www.jstor.org/stable/2345993},
 abstract = {We explore a class of regression and generalized regression models in which the coefficients are allowed to vary as smooth functions of other variables. General algorithms are presented for estimating the models flexibly and some examples are given. This class of models ties together generalized additive models and dynamic generalized linear models into one common framework. When applied to the proportional hazards model for survival data, this approach provides a new way of modelling departures from the proportional hazards assumption.},
 author = {Trevor Hastie and Robert Tibshirani},
 journal = {Journal of the Royal Statistical Society. Series B (Methodological)},
 number = {4},
 pages = {757--796},
 publisher = {[Royal Statistical Society, Wiley]},
 title = "{Varying-Coefficient Models}",
 volume = {55},
 year = {1993}
}