Correction: Recommended reporting items for epidemic forecasting and prediction research: The EPIFORGE 2020 guidelines.

[This corrects the article DOI: 10.1371/journal.pmed.1003793.].

Modeling Cases and Deaths per Million using Daily-Aggregated Facebook COVID-19 Symptom Survey Data.

We develop a novel analytic approach to modeling future COVID-19 risk using COVID-19 Symptom Survey data aggregated daily by US state, joined with daily time-series data on confirmed cases and deaths. Specifically, we model N-day forward-looking estimates for per-US-state-per-day change in deaths per million (DPM) and cases per million (CPM) using a multivariate regression model to below baseline error (65% and 38% mean absolute percentage error for DPM/CPM, respectively). Additionally, we model future changes in the curvature of CPM/DPM as "increasing" or "decreasing" using a random forest classifier to above 72% accuracy. In sum, we develop and characterize models to establish a relationship between behaviors and beliefs of individuals captured via the Facebook COVID-19 Symptom Surveys and the trajectory of COVID-19 outbreaks evidenced in terms of CPM and DPM. Such information can be helpful in assessing collective risks of infection and death during a pandemic as well as in determining the effectiveness of appropriate risk mitigation strategies based on behaviors evidenced through survey responses.

Recommended reporting items for epidemic forecasting and prediction research: The EPIFORGE 2020 guidelines.

BACKGROUND: The importance of infectious disease epidemic forecasting and prediction research is underscored by decades of communicable disease outbreaks, including COVID-19. Unlike other fields of medical research, such as clinical trials and systematic reviews, no reporting guidelines exist for reporting epidemic forecasting and prediction research despite their utility. We therefore developed the EPIFORGE checklist, a guideline for standardized reporting of epidemic forecasting research. METHODS AND FINDINGS: We developed this checklist using a best-practice process for development of reporting guidelines, involving a Delphi process and broad consultation with an international panel of infectious disease modelers and model end users. The objectives of these guidelines are to improve the consistency, reproducibility, comparability, and quality of epidemic forecasting reporting. The guidelines are not designed to advise scientists on how to perform epidemic forecasting and prediction research, but rather to serve as a standard for reporting critical methodological details of such studies. CONCLUSIONS: These guidelines have been submitted to the EQUATOR network, in addition to hosting by other dedicated webpages to facilitate feedback and journal endorsement.

Modelling the aqueous transport of an infectious pathogen in regional communities: application to the cholera outbreak in Haiti.

A mathematical model is developed to describe the dynamics of the spread of a waterborne disease among communities located along a flowing waterway. The model is formulated as a system of reaction-diffusion-advection partial differential equations in this spatial setting. The compartments of the model consist of susceptible, infected, and recovered individuals in the communities along the waterway, together with a term representing the pathogen load in each community and a term representing the spatial concentration of pathogens flowing along the waterway. The model is applied to the cholera outbreak in Haiti in 2010.

A systematic review and evaluation of Zika virus forecasting and prediction research during a public health emergency of international concern.

INTRODUCTION: Epidemic forecasting and prediction tools have the potential to provide actionable information in the midst of emerging epidemics. While numerous predictive studies were published during the 2016-2017 Zika Virus (ZIKV) pandemic, it remains unknown how timely, reproducible, and actionable the information produced by these studies was. METHODS: To improve the functional use of mathematical modeling in support of future infectious disease outbreaks, we conducted a systematic review of all ZIKV prediction studies published during the recent ZIKV pandemic using the PRISMA guidelines. Using MEDLINE, EMBASE, and grey literature review, we identified studies that forecasted, predicted, or simulated ecological or epidemiological phenomena related to the Zika pandemic that were published as of March 01, 2017. Eligible studies underwent evaluation of objectives, data sources, methods, timeliness, reproducibility, accessibility, and clarity by independent reviewers. RESULTS: 2034 studies were identified, of which n = 73 met the eligibility criteria. Spatial spread, R0 (basic reproductive number), and epidemic dynamics were most commonly predicted, with few studies predicting Guillain-Barré Syndrome burden (4%), sexual transmission risk (4%), and intervention impact (4%). Most studies specifically examined populations in the Americas (52%), with few African-specific studies (4%). Case count (67%), vector (41%), and demographic data (37%) were the most common data sources. Real-time internet data and pathogen genomic information were used in 7% and 0% of studies, respectively, and social science and behavioral data were typically absent in modeling efforts. Deterministic models were favored over stochastic approaches. Forty percent of studies made model data entirely available, 29% provided all relevant model code, 43% presented uncertainty in all predictions, and 54% provided sufficient methodological detail to allow complete reproducibility. Fifty-one percent of predictions were published after the epidemic peak in the Americas. While the use of preprints improved the accessibility of ZIKV predictions by a median of 119 days sooner than journal publication dates, they were used in only 30% of studies. CONCLUSIONS: Many ZIKV predictions were published during the 2016-2017 pandemic. The accessibility, reproducibility, timeliness, and incorporation of uncertainty in these published predictions varied and indicates there is substantial room for improvement. To enhance the utility of analytical tools for outbreak response it is essential to improve the sharing of model data, code, and preprints for future outbreaks, epidemics, and pandemics.

Mathematical model for positioning the FtsZ contractile ring in Escherichia coli.

During bacterial cytokinesis, a proteinaceous contractile ring assembles in the cell middle. The Z ring tethers to the membrane and contracts, when triggered, to form two identical daughter cells. One mechanism for positioning the ring involves the MinC, MinD and MinE proteins, which oscillate between cell poles to inhibit ring assembly. Averaged over time, the concentration of the inhibitor MinC is lowest at midcell, restricting ring assembly to this region. A second positioning mechanism, called Nucleoid Occlusion, acts through protein SlmA to inhibit ring polymerization in the location of the nucleoid. Here, a mathematical model was developed to explore the interactions between Min oscillations, nucleoid occlusion, Z ring assembly and positioning. One-dimensional advection-reaction-diffusion equations were built to simulate the spatio-temporal concentrations of Min proteins and their effect on various forms of FtsZ. The resulting partial differential equations were numerically solved using a finite volume method. The reduced chemical model assumed that the ring is composed of overlapping FtsZ filaments and that MinC disrupts lateral interactions between filaments. SlmA was presumed to break long FtsZ filaments into shorter units. A term was developed to account for the movement of FtsZ subunits in membrane-bound filaments as they touch and align with other filaments. This alignment was critical in forming sharp stable rings. Simulations qualitatively reproduced experimental results showing the incorrect positioning of rings when Min proteins were not expressed, and the formation of multiple rings when FtsZ was overexpressed.

Mathematical model of a cell size checkpoint.

How cells regulate their size from one generation to the next has remained an enigma for decades. Recently, a molecular mechanism that links cell size and cell cycle was proposed in fission yeast. This mechanism involves changes in the spatial cellular distribution of two proteins, Pom1 and Cdr2, as the cell grows. Pom1 inhibits Cdr2 while Cdr2 promotes the G2 → M transition. Cdr2 is localized in the middle cell region (midcell) whereas the concentration of Pom1 is highest at the cell tips and declines towards the midcell. In short cells, Pom1 efficiently inhibits Cdr2. However, as cells grow, the Pom1 concentration at midcell decreases such that Cdr2 becomes activated at some critical size. In this study, the chemistry of Pom1 and Cdr2 was modeled using a deterministic reaction-diffusion-convection system interacting with a deterministic model describing microtubule dynamics. Simulations mimicked experimental data from wild-type (WT) fission yeast growing at normal and reduced rates; they also mimicked the behavior of a Pom1 overexpression mutant and WT yeast exposed to a microtubule depolymerizing drug. A mechanism linking cell size and cell cycle, involving the downstream action of Cdr2 on Wee1 phosphorylation, is proposed.

Mathematical modeling of a minimal protocell with coordinated growth and division.

Self-replication is an essential attribute of life but the molecular-level mechanisms involved are not well understood. Cellular self-replication requires not only duplicating all cellular components and doubling volume and membrane area, but also replicating cellular geometry. A whole-cell modeling framework is presented in which an assumed reaction network determines both concentration changes of cellular components and cell geometry. Cell shape is calculated by minimizing membrane-bending energy. Using this framework, simultaneous doubling of volume, surface area, and all components was found to be insufficient to provide mid-cell "pinching" of the parental cell to form two daughter cells. This prompted the design of a minimal protocell that includes a growing shell, a cell-cycle engine, and a contractile ring to enforce cytokinesis. Kinetic parameters were found such that the system exhibited periodic behavior with fundamental aspects of self-replication. This involved simultaneous doubling of all cellular components during a cell cycle, doubling cell volume and membrane area, achieving periodic changes in surface/volume ratio, and forming daughter cells that were geometrically equivalent to each other and to the "newborn" parental cell. The results presented here impact the design of laboratory protocells and the development of a modular strategy for constructing a comprehensive in silico whole-cell model.

Kinetic modeling of the assembly, dynamic steady state, and contraction of the FtsZ ring in prokaryotic cytokinesis.

Cytokinesis in prokaryotes involves the assembly of a polymeric ring composed of FtsZ protein monomeric units. The Z ring forms at the division plane and is attached to the membrane. After assembly, it maintains a stable yet dynamic steady state. Once induced, the ring contracts and the membrane constricts. In this work, we present a computational deterministic biochemical model exhibiting this behavior. The model is based on biochemical features of FtsZ known from in vitro studies, and it quantitatively reproduces relevant in vitro data. An essential part of the model is a consideration of interfacial reactions involving the cytosol volume, where monomeric FtsZ is dispersed, and the membrane surface in the cell's mid-zone where the ring is assembled. This approach allows the same chemical model to simulate either in vitro or in vivo conditions by adjusting only two geometrical parameters. The model includes minimal reactions, components, and assumptions, yet is able to reproduce sought-after in vivo behavior, including the rapid assembly of the ring via FtsZ-polymerization, the formation of a dynamic steady state in which GTP hydrolysis leads to the exchange of monomeric subunits between cytoplasm and the ring, and finally the induced contraction of the ring. The model gives a quantitative estimate for coupling between the rate of GTP hydrolysis and of FtsZ subunit turnover between the assembled ring and the cytoplasmic pool as observed. Membrane constriction is chemically driven by the strong tendency of GTP-bound FtsZ to self-assembly. The model suggests a possible mechanism of membrane contraction without a motor protein. The portion of the free energy of GTP hydrolysis released in cyclization is indirectly used in this energetically unfavorable process. The model provides a limit to the mechanistic complexity required to mimic ring behavior, and it highlights the importance of parallel in vitro and in vivo modeling.

Whole-cell modeling framework in which biochemical dynamics impact aspects of cellular geometry.

A mathematical framework for modeling biological cells from a physicochemical perspective is described. Cells modeled within this framework consist of at least two regions, including a cytosolic volume encapsulated by a membrane surface. The cytosol is viewed as a well-stirred chemical reactor capable of changing volume while the membrane is assumed to be an oriented 2-D surface capable of changing surface area. Two physical properties of the cell, namely volume and surface area, are determined by (and determine) the reaction dynamics generated from a set of chemical reactions designed to be occurring in the cell. This framework allows the modeling of complex cellular behaviors, including self-replication. This capability is illustrated by constructing two self-replicating prototypical whole-cell models. One protocell was designed to be of minimal complexity; the other to incorporate a previously reported well-known mechanism of the eukaryotic cell cycle. In both cases, self-replicative behavior was achieved by seeking stable physically possible oscillations in concentrations and surface-to-volume ratio, and by synchronizing the period of such oscillations to the doubling of cytosolic volume and membrane surface area. Rather than being enforced externally or artificially, growth and division occur naturally as a consequence of the assumed chemical mechanism operating within the framework.

A framework for whole-cell mathematical modeling.

The default framework for modeling biochemical processes is that of a constant-volume reactor operating under steady-state conditions. This is satisfactory for many applications, but not for modeling growth and division of cells. In this study, a whole-cell modeling framework is developed that assumes expanding volumes and a cell-division cycle. A spherical newborn cell is designed to grow in volume during the growth phase of the cycle. After 80% of the cycle period, the cell begins to divide by constricting about its equator, ultimately affording two spherical cells with total volume equal to twice that of the original. The cell is partitioned into two regions or volumes, namely the cytoplasm (Vcyt) and membrane (Vmem), with molecular components present in each. Both volumes change during the cell cycle; Vcyt changes in response to osmotic pressure changes as nutrients enter the cell from the environment, while Vmem changes in response to this osmotic pressure effect such that membrane thickness remains invariant. The two volumes change at different rates; in most cases, this imposes periodic or oscillatory behavior on all components within the cell. Since the framework itself rather than a particular set of reactions and components is responsible for this behavior, it should be possible to model various biochemical processes within it, affording stable periodic solutions without requiring that the biochemical process itself generates oscillations as an inherent feature. Given that these processes naturally occur in growing and dividing cells, it is reasonable to conclude that the dynamics of component concentrations will be more realistic than when modeled within constant-volume and/or steady-state frameworks. This approach is illustrated using a symbolic whole cell model.

Dynamic responses of protein homeostatic regulatory mechanisms to perturbations from steady state.

Nineteen hypothetical protein homeostatic regulatory mechanisms were constructed and analysed in terms of the rate at which they recovered from a perturbation in the steady-state concentration of any component. Systems were constructed to symbolize transcription/translation processes of the average protein from Escherichia coli (1000 copies of protein P along with 1 gene G per cell). In some model systems, G catalysed the synthesis of P directly, while in others G catalysed the synthesis of mRNA (called M), and M catalysed the synthesis of P in a subsequent step. Recovery rates for each regulatory mechanism were obtained by generating the corresponding system of differential equations, linearizing the system about the steady state, and determining eigenvalues of the associated coefficient matrix. The optimal rate of recovery for a given mechanism, R(D), was determined by combining random and gradient search approaches to find rate constants for which the system recovered fastest. Regulatory elements that improved dynamic regulation were identified. These consisted of negative feedback relationships that involved P binding to either G (to shut off the synthesis of P) or M (to stimulate its degradation). Regulation improved as increasing numbers of P's bound to either G or M; however, the binding to M was more effective. In other mechanisms PP dimers bound G. Dimer-binding mechanisms were roughly twice as effective in terms of regulation as those that bound P monomers. The effect of linking two regulatory "modules" was also investigated. Linking had no effect on R(D), but optimal rate constants for the linked system were similar to those of the unlinked modules, suggesting that it may be feasible to construct regulatory networks by linking individual modules of this type.

Analysis of protein homeostatic regulatory mechanisms in perturbed environments at steady state.

Nine different protein homeostatic regulatory mechanisms were analysed for their ability to maintain a generic protein P within a specified range of a set-point steady-state concentration while perturbed by external processes that altered the rates at which P was produced and/or consumed. Steady state regulatory effectiveness was defined by the area within a rectangular region of "perturbation space", where axes correspond to rates of positive and negative perturbations. The size of this region differed in accordance with the regulatory elements composing the homeostatic mechanism. Such elements included basic negative feedback control of transcription (in which P, at some high concentration relative to its set-point value, binds to the gene G that encodes it, thereby inhibiting transcription), multiple sequential binding of a feedback effector (two P's bind sequentially to G), and dimerization of a feedback effector (a P(2) dimer binds to G). Two homeostatic mechanisms included a cascade structure, one with and one without translational feedback control. Another mechanism included feedback control of P degradation. Finally, two mechanisms illustrated the limits of regulatory systems. One lacked all regulatory elements (and included only an invariant rate of P synthesis and degradation) while the other assumed perfect (Boolean) regulation, in which transcription is completely inhibited at [P]>[P](sp) and is fully active at [P]<[P](sp). All of the systems evaluated are known, but the analytical expressions developed here allow quantitative comparisons between them. These expressions were evaluated at values typical of the average protein in Escherichia coli. A method for building regulatory networks by linking semi-independent regulatory modules is discussed.