Probing the effects of the well-mixed assumption on viral infection dynamics

Choice of spatial discretisation influences the progression of viral infection within multicellular tissues

There has been an increasing recognition of the utility of models of the spatial dynamics of viral spread within tissues. Multicellular models, where cells are represented as discrete regions of space coupled to a virus density surface, are a popular approach to capture these dynamics. Conventionally, such models are simulated by discretising the viral surface and depending on the rate of viral diffusion and other considerations, a finer or coarser discretisation may be used. The impact that this choice may have on the behaviour of the system has not been studied. Here we demonstrate that under realistic parameter regimes - where viral diffusion is small enough to support the formation of familiar ring-shaped infection plaques - the choice of spatial discretisation of the viral surface can qualitatively change key model outcomes including the time scale of infection. Importantly, we show that the choice between implementing viral spread as a cell-scale process, or as a high-resolution converged PDE can generate distinct model outcomes, which raises important conceptual questions about the strength of assumptions underpinning the spatial structure of the model. We investigate the mechanisms driving these discretisation artefacts, the impacts they may have on model predictions, and provide guidance on the design and implementation of spatial and especially multicellular models of viral dynamics. We obtain our results using the simplest TIV construct for the viral dynamics, and therefore anticipate that the important effects we describe will also influence model predictions in more complex models of virus-cell-immune system interactions. This analysis will aid in the construction of models for robust and biologically realistic modelling and inference.

Modeling and Optimization of Continuous Viral Vaccine Production

A model that captures realistic viral growth dynamics has been developed based on a continuous and semi-continuous production model of an influenza A virus. This model considers viral growth parameters such as viral latency. It also captures the lag observed during the early production of viruses in a culture and explains later-phase growth dynamics. Furthermore, a sensitivity analysis was performed to investigate the effects of each input on each output. This revealed that production of defective interfering particles (DIPs) highly depends on the number of cells introduced to the viral reactor. The rationale for this is, as per the model, that a reduction in number of cells to be infected causes a reduction in DIPs formed as rate of viral infection decreases. Finally, a flowsheet model was created to optimize the continuous platform, including number of cells supplied to the viral reactor. From this, it was observed that the peak number of DIPs formed could be reduced by one-third. Finally, this model is tailorable to different viral particles using parameter estimation. Therefore, the proposed mathematical model provides a versatile, comprehensive platform that can be tailored to various viral cultures with or without a latent phase.

A multiscale multicellular spatiotemporal model of local influenza infection and immune response

Respiratory viral infections pose a serious public health concern, from mild seasonal influenza to pandemics like those of SARS-CoV-2. Spatiotemporal dynamics of viral infection impact nearly all aspects of the progression of a viral infection, like the dependence of viral replication rates on the type of cell and pathogen, the strength of the immune response and localization of infection. Mathematical modeling is often used to describe respiratory viral infections and the immune response to them using ordinary differential equation (ODE) models. However, ODE models neglect spatially-resolved biophysical mechanisms like lesion shape and the details of viral transport, and so cannot model spatial effects of a viral infection and immune response. In this work, we develop a multiscale, multicellular spatiotemporal model of influenza infection and immune response by combining non-spatial ODE modeling and spatial, cell-based modeling. We employ cellularization, a recently developed method for generating spatial, cell-based, stochastic models from non-spatial ODE models, to generate much of our model from a calibrated ODE model that describes infection, death and recovery of susceptible cells and innate and adaptive responses during influenza infection, and develop models of cell migration and other mechanisms not explicitly described by the ODE model. We determine new model parameters to generate agreement between the spatial and original ODE models under certain conditions, where simulation replicas using our model serve as microconfigurations of the ODE model, and compare results between the models to investigate the nature of viral exposure and impact of heterogeneous infection on the time-evolution of the viral infection. We found that using spatially homogeneous initial exposure conditions consistently with those employed during calibration of the ODE model generates far less severe infection, and that local exposure to virus must be multiple orders of magnitude greater than a uniformly applied exposure to all available susceptible cells. This strongly suggests a prominent role of localization of exposure in influenza A infection. We propose that the particularities of the microenvironment to which a virus is introduced plays a dominant role in disease onset and progression, and that spatially resolved models like ours may be important to better understand and more reliably predict future health states based on susceptibility of potential lesion sites using spatially resolved patient data of the state of an infection. We can readily integrate the immune response components of our model into other modeling and simulation frameworks of viral infection dynamics that do detailed modeling of other mechanisms like viral internalization and intracellular viral replication dynamics, which are not explicitly represented in the ODE model. We can also combine our model with available experimental data and modeling of exposure scenarios and spatiotemporal aspects of mechanisms like mucociliary clearance that are only implicitly described by the ODE model, which would significantly improve the ability of our model to present spatially resolved predictions about the progression of influenza infection and immune response.

Iterative community-driven development of a SARS-CoV-2 tissue simulator

The 2019 novel coronavirus, SARS-CoV-2, is a pathogen of critical significance to international public health. Knowledge of the interplay between molecular-scale virus-receptor interactions, single-cell viral replication, intracellular-scale viral transport, and emergent tissue-scale viral propagation is limited. Moreover, little is known about immune system-virus-tissue interactions and how these can result in low-level (asymptomatic) infections in some cases and acute respiratory distress syndrome (ARDS) in others, particularly with respect to presentation in different age groups or pre-existing inflammatory risk factors. Given the nonlinear interactions within and among each of these processes, multiscale simulation models can shed light on the emergent dynamics that lead to divergent outcomes, identify actionable "choke points" for pharmacologic interventions, screen potential therapies, and identify potential biomarkers that differentiate patient outcomes. Given the complexity of the problem and the acute need for an actionable model to guide therapy discovery and optimization, we introduce and iteratively refine a prototype of a multiscale model of SARS-CoV-2 dynamics in lung tissue. The first prototype model was built and shared internationally as open source code and an online interactive model in under 12 hours, and community domain expertise is driving regular refinements. In a sustained community effort, this consortium is integrating data and expertise across virology, immunology, mathematical biology, quantitative systems physiology, cloud and high performance computing, and other domains to accelerate our response to this critical threat to international health. More broadly, this effort is creating a reusable, modular framework for studying viral replication and immune response in tissues, which can also potentially be adapted to related problems in immunology and immunotherapy.

Generation of multicellular spatiotemporal models of population dynamics from ordinary differential equations, with applications in viral infection

BACKGROUND

The biophysics of an organism span multiple scales from subcellular to organismal and include processes characterized by spatial properties, such as the diffusion of molecules, cell migration, and flow of intravenous fluids. Mathematical biology seeks to explain biophysical processes in mathematical terms at, and across, all relevant spatial and temporal scales, through the generation of representative models. While non-spatial, ordinary differential equation (ODE) models are often used and readily calibrated to experimental data, they do not explicitly represent the spatial and stochastic features of a biological system, limiting their insights and applications. However, spatial models describing biological systems with spatial information are mathematically complex and computationally expensive, which limits the ability to calibrate and deploy them and highlights the need for simpler methods able to model the spatial features of biological systems.

RESULTS

In this work, we develop a formal method for deriving cell-based, spatial, multicellular models from ODE models of population dynamics in biological systems, and vice versa. We provide examples of generating spatiotemporal, multicellular models from ODE models of viral infection and immune response. In these models, the determinants of agreement of spatial and non-spatial models are the degree of spatial heterogeneity in viral production and rates of extracellular viral diffusion and decay. We show how ODE model parameters can implicitly represent spatial parameters, and cell-based spatial models can generate uncertain predictions through sensitivity to stochastic cellular events, which is not a feature of ODE models. Using our method, we can test ODE models in a multicellular, spatial context and translate information to and from non-spatial and spatial models, which help to employ spatiotemporal multicellular models using calibrated ODE model parameters. We additionally investigate objects and processes implicitly represented by ODE model terms and parameters and improve the reproducibility of spatial, stochastic models.

CONCLUSION

We developed and demonstrate a method for generating spatiotemporal, multicellular models from non-spatial population dynamics models of multicellular systems. We envision employing our method to generate new ODE model terms from spatiotemporal and multicellular models, recast popular ODE models on a cellular basis, and generate better models for critical applications where spatial and stochastic features affect outcomes.

A modular framework for multiscale, multicellular, spatiotemporal modeling of acute primary viral infection and immune response in epithelial tissues and its application to drug therapy timing and effectiveness

Simulations of tissue-specific effects of primary acute viral infections like COVID-19 are essential for understanding disease outcomes and optimizing therapies. Such simulations need to support continuous updating in response to rapid advances in understanding of infection mechanisms, and parallel development of components by multiple groups. We present an open-source platform for multiscale spatiotemporal simulation of an epithelial tissue, viral infection, cellular immune response and tissue damage, specifically designed to be modular and extensible to support continuous updating and parallel development. The base simulation of a simplified patch of epithelial tissue and immune response exhibits distinct patterns of infection dynamics from widespread infection, to recurrence, to clearance. Slower viral internalization and faster immune-cell recruitment slow infection and promote containment. Because antiviral drugs can have side effects and show reduced clinical effectiveness when given later during infection, we studied the effects on progression of treatment potency and time-of-first treatment after infection. In simulations, even a low potency therapy with a drug which reduces the replication rate of viral RNA greatly decreases the total tissue damage and virus burden when given near the beginning of infection. Many combinations of dosage and treatment time lead to stochastic outcomes, with some simulation replicas showing clearance or control (treatment success), while others show rapid infection of all epithelial cells (treatment failure). Thus, while a high potency therapy usually is less effective when given later, treatments at late times are occasionally effective. We illustrate how to extend the platform to model specific virus types (e.g., hepatitis C) and add additional cellular mechanisms (tissue recovery and variable cell susceptibility to infection), using our software modules and publicly-available software repository.

A modular framework for multiscale, multicellular, spatiotemporal modeling of acute primary viral infection and immune response in epithelial tissues and its application to drug therapy timing and effectiveness: A multiscale model of viral infection in epithelial tissues

Simulations of tissue-specific effects of primary acute viral infections like COVID-19 are essential for understanding disease outcomes and optimizing therapies. Such simulations need to support continuous updating in response to rapid advances in understanding of infection mechanisms, and parallel development of components by multiple groups. We present an open-source platform for multiscale spatiotemporal simulation of an epithelial tissue, viral infection, cellular immune response and tissue damage, specifically designed to be modular and extensible to support continuous updating and parallel development. The base simulation of a simplified patch of epithelial tissue and immune response exhibits distinct patterns of infection dynamics from widespread infection, to recurrence, to clearance. Slower viral internalization and faster immune-cell recruitment slow infection and promote containment. Because antiviral drugs can have side effects and show reduced clinical effectiveness when given later during infection, we studied the effects on progression of treatment potency and time-of-first treatment after infection. In simulations, even a low potency therapy with a drug which reduces the replication rate of viral RNA greatly decreases the total tissue damage and virus burden when given near the beginning of infection. Many combinations of dosage and treatment time lead to stochastic outcomes, with some simulation replicas showing clearance or control (treatment success), while others show rapid infection of all epithelial cells (treatment failure). Thus, while a high potency therapy usually is less effective when given later, treatments at late times are occasionally effective. We illustrate how to extend the platform to model specific virus types (e.g., hepatitis C) and add additional cellular mechanisms (tissue recovery and variable cell susceptibility to infection), using our software modules and publicly-available software repository.

A mathematical model describing the localization and spread of influenza A virus infection within the human respiratory tract

Within the human respiratory tract (HRT), virus diffuses through the periciliary fluid (PCF) bathing the epithelium. But virus also undergoes advection: as the mucus layer sitting atop the PCF is pushed along by the ciliated cell's beating cilia, the PCF and its virus content are also pushed along, upwards towards the nose and mouth. While many mathematical models (MMs) have described the course of influenza A virus (IAV) infections in vivo, none have considered the impact of both diffusion and advection on the kinetics and localization of the infection. The MM herein represents the HRT as a one-dimensional track extending from the nose down towards the lower HRT, wherein stationary cells interact with IAV which moves within (diffusion) and along with (advection) the PCF. Diffusion was found to be negligible in the presence of advection which effectively sweeps away IAV, preventing infection from disseminating below the depth at which virus first deposits. Higher virus production rates (10-fold) are required at higher advection speeds (40 μm/s) to maintain equivalent infection severity and timing. Because virus is entrained upwards, upper parts of the HRT see more virus than lower parts. As such, infection peaks and resolves faster in the upper than in the lower HRT, making it appear as though infection progresses from the upper towards the lower HRT, as reported in mice. When the spatial MM is expanded to include cellular regeneration and an immune response, it reproduces tissue damage levels reported in patients. It also captures the kinetics of seasonal and avian IAV infections, via parameter changes consistent with reported differences between these strains, enabling comparison of their treatment with antivirals. This new MM offers a convenient and unique platform from which to study the localization and spread of respiratory viral infections within the HRT.

Varying Inoculum Dose to Assess the Roles of the Immune Response and Target Cell Depletion by the Pathogen in Control of Acute Viral Infections

It is difficult to determine whether an immune response or target cell depletion by the infectious agent is most responsible for the control of acute primary infection. Both mechanisms can explain the basic dynamics of an acute infection-exponential growth of the pathogen followed by control and clearance-and can also be represented by many different differential equation models. Consequently, traditional model comparison techniques using time series data can be ambiguous or inconclusive. We propose that varying the inoculum dose and measuring the subsequent infectious load can rule out target cell depletion by the pathogen as the main control mechanism. Infectious load can be any measure that is proportional to the number of infected cells, such as viraemia. We show that a twofold or greater change in infectious load is unlikely when target cell depletion controls infection, regardless of the model details. Analyzing previously published data from mice infected with influenza, we find the proportion of lung epithelial cells infected was 21-fold greater (95% confidence interval 14-32) in the highest dose group than in the lowest. This provides evidence in favor of an alternative to target cell depletion, such as innate immunity, in controlling influenza infections in this experimental system. Data from other experimental animal models of acute primary infection have a similar pattern.

Improving risk assessment of the emergence of novel influenza A viruses by incorporating environmental surveillance

Reassortment is an evolutionary mechanism by which influenza A viruses (IAV) generate genetic novelty. Reassortment is an important driver of host jumps and is widespread according to retrospective surveillance studies. However, predicting the epidemiological risk of reassortant emergence in novel hosts from surveillance data remains challenging. IAV strains persist and co-occur in the environment, promoting co-infection during environmental transmission. These conditions offer opportunity to understand reassortant emergence in reservoir and spillover hosts. Specifically, environmental RNA could provide rich information for understanding the evolutionary ecology of segmented viruses, and transform our ability to quantify epidemiological risk to spillover hosts. However, significant challenges with recovering and interpreting genomic RNA from the environment have impeded progress towards predicting reassortant emergence from environmental surveillance data. We discuss how the fields of genomics, experimental ecology and epidemiological modelling are well positioned to address these challenges. Coupling quantitative disease models and natural transmission studies with new molecular technologies, such as deep-mutational scanning and single-virus sequencing of environmental samples, should dramatically improve our understanding of viral co-occurrence and reassortment. We define observable risk metrics for emerging molecular technologies and propose a conceptual research framework for improving accuracy and efficiency of risk prediction. This article is part of the theme issue 'Dynamic and integrative approaches to understanding pathogen spillover'.

Causes and Consequences of Spatial Within-Host Viral Spread

The spread of viral pathogens both between and within hosts is inherently a spatial process. While the spatial aspects of viral spread at the epidemiological level have been increasingly well characterized, the spatial aspects of viral spread within infected hosts are still understudied. Here, with a focus on influenza A viruses (IAVs), we first review experimental studies that have shed light on the mechanisms and spatial dynamics of viral spread within hosts. These studies provide strong empirical evidence for highly localized IAV spread within hosts. Since mathematical and computational within-host models have been increasingly used to gain a quantitative understanding of observed viral dynamic patterns, we then review the (relatively few) computational modeling studies that have shed light on possible factors that structure the dynamics of spatial within-host IAV spread. These factors include the dispersal distance of virions, the localization of the immune response, and heterogeneity in host cell phenotypes across the respiratory tract. While informative, we find in these studies a striking absence of theoretical expectations of how spatial dynamics may impact the dynamics of viral populations. To mitigate this, we turn to the extensive ecological and evolutionary literature on range expansions to provide informed theoretical expectations. We find that factors such as the type of density dependence, the frequency of long-distance dispersal, specific life history characteristics, and the extent of spatial heterogeneity are critical factors affecting the speed of population spread and the genetic composition of spatially expanding populations. For each factor that we identified in the theoretical literature, we draw parallels to its analog in viral populations. We end by discussing current knowledge gaps related to the spatial component of within-host IAV spread and the potential for within-host spatial considerations to inform the development of disease control strategies.

Accounting for Space—Quantification of Cell-To-Cell Transmission Kinetics Using Virus Dynamics Models

Mathematical models based on ordinary differential equations (ODE) that describe the population dynamics of viruses and infected cells have been an essential tool to characterize and quantify viral infection dynamics. Although an important aspect of viral infection is the dynamics of viral spread, which includes transmission by cell-free virions and direct cell-to-cell transmission, models used so far ignored cell-to-cell transmission completely, or accounted for this process by simple mass-action kinetics between infected and uninfected cells. In this study, we show that the simple mass-action approach falls short when describing viral spread in a spatially-defined environment. Using simulated data, we present a model extension that allows correct quantification of cell-to-cell transmission dynamics within a monolayer of cells. By considering the decreasing proportion of cells that can contribute to cell-to-cell spread with progressing infection, our extension accounts for the transmission dynamics on a single cell level while still remaining applicable to standard population-based experimental measurements. While the ability to infer the proportion of cells infected by either of the transmission modes depends on the viral diffusion rate, the improved estimates obtained using our novel approach emphasize the need to correctly account for spatial aspects when analyzing viral spread.

The impact of cell regeneration on the dynamics of viral coinfection

Many mathematical models of respiratory viral infections do not include regeneration of cells within the respiratory tract, arguing that the infection is resolved before there is significant cellular regeneration. However, recent studies have found that ∼40% of patients hospitalized with influenza-like illness are infected with at least two different viruses, which could potentially lead to longer-lasting infections. In these longer infections, cell regeneration might affect the infection dynamics, in particular, allowing for the possibility of chronic coinfections. Several mathematical models have been used to describe cell regeneration in infection models, though the effect of model choice on the predicted time course of viral coinfections is not clear. We investigate four mathematical models incorporating different mechanisms of cell regeneration during respiratory viral coinfection to determine the effect of cell regeneration on infection dynamics. We perform linear stability analysis for each of the models and find the steady states analytically. The analysis suggests that chronic illness is possible but only with one viral species; chronic coexistence of two different viral species is not possible with the regeneration models considered here.

Spatiotemporal Dynamics of Virus Infection Spreading in Tissues

Virus spreading in tissues is determined by virus transport, virus multiplication in host cells and the virus-induced immune response. Cytotoxic T cells remove infected cells with a rate determined by the infection level. The intensity of the immune response has a bell-shaped dependence on the concentration of virus, i.e., it increases at low and decays at high infection levels. A combination of these effects and a time delay in the immune response determine the development of virus infection in tissues like spleen or lymph nodes. The mathematical model described in this work consists of reaction-diffusion equations with a delay. It shows that the different regimes of infection spreading like the establishment of a low level infection, a high level infection or a transition between both are determined by the initial virus load and by the intensity of the immune response. The dynamics of the model solutions include simple and composed waves, and periodic and aperiodic oscillations. The results of analytical and numerical studies of the model provide a systematic basis for a quantitative understanding and interpretation of the determinants of the infection process in target organs and tissues from the image-derived data as well as of the spatiotemporal mechanisms of viral disease pathogenesis, and have direct implications for a biopsy-based medical testing of the chronic infection processes caused by viruses, e.g. HIV, HCV and HBV.

Predicting pathogen-specific CD8 T cell immune responses from a modeling approach

The primary CD8 T cell immune response constitutes a major mechanism to fight an infection by intra-cellular pathogens. We aim at assessing whether pathogen-specific dynamical parameters of the CD8 T cell response can be identified, based on measurements of CD8 T cell counts, using a modeling approach. We generated experimental data consisting in CD8 T cell counts kinetics during the response to three different live intra-cellular pathogens: two viruses (influenza, vaccinia) injected intranasally, and one bacteria (Listeria monocytogenes) injected intravenously. All pathogens harbor the same antigen (NP68), but differ in their interaction with the host. In parallel, we developed a mathematical model describing the evolution of CD8 T cell counts and pathogen amount during an immune response. This model is characterized by 9 parameters and includes relevant feedback controls. The model outputs were compared with the three data series and an exhaustive estimation of the parameter values was performed. By focusing on the ability of the model to fit experimental data and to produce a CD8 T cell population mainly composed of memory cells at the end of the response, critical parameters were identified. We show that a small number of parameters (2-4) define the main features of the CD8 T cell immune response and are characteristic of a given pathogen. Among these parameters, two are related to the effector CD8 T cell mediated control of cell and pathogen death. The parameter associated with memory cell death is shown to play no relevant role during the main phases of the CD8 T cell response, yet it becomes essential when looking at the predictions of the model several months after the infection.

The inflammatory response to influenza A virus (H1N1): An experimental and mathematical study

Mortality from influenza infections continues as a global public health issue, with the host inflammatory response contributing to fatalities related to the primary infection. Based on Ordinary Differential Equation (ODE) formalism, a computational model was developed for the in-host response to influenza A virus, merging inflammatory, innate, adaptive and humoral responses to virus and linking severity of infection, the inflammatory response, and mortality. The model was calibrated using dense cytokine and cell data from adult BALB/c mice infected with the H1N1 influenza strain A/PR/8/34 in sublethal and lethal doses. Uncertainty in model parameters and disease mechanisms was quantified using Bayesian inference and ensemble model methodology that generates probabilistic predictions of survival, defined as viral clearance and recovery of the respiratory epithelium. The ensemble recovers the expected relationship between magnitude of viral exposure and the duration of survival, and suggests mechanisms primarily responsible for survival, which could guide the development of immuno-modulatory interventions as adjuncts to current anti-viral treatments. The model is employed to extrapolate from available data survival curves for the population and their dependence on initial viral aliquot. In addition, the model allows us to illustrate the positive effect of controlled inflammation on influenza survival.

Modeling HIV-1 dynamics and fitness in cell culture across scales

A common approach to understand and analyze complex biological systems is to describe the dynamics in terms of a system of ordinary differential equations (ODE) depending on numerous biologically meaningful and descriptive parameters that are estimated using observed data. The ODE models are often based on the implicit assumption of well-mixed dynamics, i.e., the delay of interaction due to spatial distribution is not included in the model. In this article, we address the question how the heterogeneity of the underlying system affects the estimated parameter values of the ODE model, and on the other hand, what information about the microscopic system can be drawn from these values. The system we are considering is a pairwise growth competition assay used to quantify ex vivo replicative fitness of different HIV-1 isolates. To overcome the lack of ground truth, we generate data using a detailed microscopic spatially distributed hybrid stochastic-deterministic (HSD) infection model in which the dynamics is controlled by parameters directly related to cell level infection, virus production processes, and diffusion of virus particles. The synthetic data sets are then analyzed using an ODE based well-mixed model, in which the corresponding macroscopic parameter distributions are estimated using Markov chain Monte Carlo (MCMC) methods. This approach provides a comprehensive picture of the statistical dependencies of the model parameter across different scales.

How sticky should a virus be? The impact of virus binding and release on transmission fitness using influenza as an example

Budding viruses face a trade-off: virions need to efficiently attach to and enter uninfected cells while newly generated virions need to efficiently detach from infected cells. The right balance between attachment and detachment-the right amount of stickiness-is needed for maximum fitness. Here, we design and analyse a mathematical model to study in detail the impact of attachment and detachment rates on virus fitness. We apply our model to influenza, where stickiness is determined by a balance of the haemagglutinin (HA) and neuraminidase (NA) proteins. We investigate how drugs, the adaptive immune response and vaccines impact influenza stickiness and fitness. Our model suggests that the location in the 'stickiness landscape' of the virus determines how well interventions such as drugs or vaccines are expected to work. We discuss why hypothetical NA enhancer drugs might occasionally perform better than the currently available NA inhibitors in reducing virus fitness. We show that an increased antibody or T-cell-mediated immune response leads to maximum fitness at higher stickiness. We further show that antibody-based vaccines targeting mainly HA or NA, which leads to a shift in stickiness, might reduce virus fitness above what can be achieved by the direct immunological action of the vaccine. Overall, our findings provide potentially useful conceptual insights for future vaccine and drug development and can be applied to other budding viruses beyond influenza.

A Simulation Framework to Investigate in vitro Viral Infection Dynamics

Virus infection is a complex biological phenomenon for which in vitro experiments provide a uniquely concise view where data is often obtained from a single population of cells, under controlled environmental conditions. Nonetheless, data interpretation and real understanding of viral dynamics is still hampered by the sheer complexity of the various intertwined spatio-temporal processes. In this paper we present a tool to address these issues: a cellular automata model describing critical aspects of in vitro viral infections taking into account spatial characteristics of virus spreading within a culture well. The aim of the model is to understand the key mechanisms of SARS-CoV infection dynamics during the first 24 hours post infection. Using a simulated annealing algorithm we tune free parameters with data from SARS-CoV infection of cultured lung epithelial cells. We also interrogate the model using a Latin Hypercube sensitivity analysis to identify which mechanisms are critical to the observed infection of host cells and the release of measured virus particles.

Towards multiscale modeling of influenza infection

Aided by recent advances in computational power, algorithms, and higher fidelity data, increasingly detailed theoretical models of infection with influenza A virus are being developed. We review single scale models as they describe influenza infection from intracellular to global scales, and, in particular, we consider those models that capture details specific to influenza and can be used to link different scales. We discuss the few multiscale models of influenza infection that have been developed in this emerging field. In addition to discussing modeling approaches, we also survey biological data on influenza infection and transmission that is relevant for constructing influenza infection models. We envision that, in the future, multiscale models that capitalize on technical advances in experimental biology and high performance computing could be used to describe the large spatial scale epidemiology of influenza infection, evolution of the virus, and transmission between hosts more accurately.

A hybrid stochastic-deterministic computational model accurately describes spatial dynamics and virus diffusion in HIV-1 growth competition assay

We present a new hybrid stochastic-deterministic, spatially distributed computational model to simulate growth competition assays on a relatively immobile monolayer of peripheral blood mononuclear cells (PBMCs), commonly used for determining ex vivo fitness of human immunodeficiency virus type-1 (HIV-1). The novel features of our approach include incorporation of viral diffusion through a deterministic diffusion model while simulating cellular dynamics via a stochastic Markov chain model. The model accounts for multiple infections of target cells, CD4-downregulation, and the delay between the infection of a cell and the production of new virus particles. The minimum threshold level of infection induced by a virus inoculum is determined via a series of dilution experiments, and is used to determine the probability of infection of a susceptible cell as a function of local virus density. We illustrate how this model can be used for estimating the distribution of cells infected by either a single virus type or two competing viruses. Our model captures experimentally observed variation in the fitness difference between two virus strains, and suggests a way to minimize variation and dual infection in experiments.

Immature oxidative stress management as a unifying principle in the pathogenesis of necrotizing enterocolitis: insights from an agent-based model

BACKGROUND

Necrotizing enterocolitis (NEC) is a complex disease involving prematurity, enteral feeding, and bacterial effects. We propose that the underlying initial condition in its pathogenesis is reduced ability of the neonatal gut epithelial cells (NGECs) to clear oxidative stress (OS), and that when such a NGEC population is exposed to enteral feeding, the increased metabolic OS tips the population toward apoptosis, inflammation, bacterial activation, and eventual necrosis. The multi-factorial complexity of NEC requires characterization with computational modeling, and herein, we used an agent-based model (ABM) to instantiate and examine our unifying hypothesis of the pathogenesis of NEC.

METHODS

An ABM of the neonatal gut was created with NGEC computational agents incorporating rules for pathways for OS, p53, tight junctions, Toll-like receptor (TLR)-4, nitric oxide, and nuclear factor-kappa beta (NF-κB). The modeled bacteria activated TLR-4 on contact with NGECs. Simulations included parameter sweeps of OS response, response to feeding, addition of bacteria, and alterations in gut mucus production.

RESULTS

The ABM reproduced baseline cellular respiration and clearance of OS. Reduction in OS clearance consistent with clinical NEC led to senescence, apoptosis, or inflammation, with disruption of tight junctions, but rarely to NGEC necrosis. An additional "hit" of bacteria activating TLR-4 potentiated a shift to NGEC necrosis across the entire population. The mucus layer was modeled to limit bacterial-NGEC interactions and reduce this effect, but concomitant apoptosis in the goblet cell population reduced the efficacy of the mucus layer and limited its protective effect in simulated experiments. This finding suggests a means by which increased apoptosis at the cellular population level can lead to a transition to the necrosis outcome.

CONCLUSIONS

Our ABM incorporates known components of NEC and demonstrates that impaired OS management can lead to apoptosis and inflammation of NGECs, rendering the system susceptible to an additional insult involving regionalized mucus barrier failure and TLR-4 activation, which potentiates the necrosis outcome. This type of integrative dynamic knowledge representation can be a useful adjunct to help guide and contextualize research.

Agent-based dynamic knowledge representation of Pseudomonas aeruginosa virulence activation in the stressed gut: Towards characterizing host-pathogen interactions in gut-derived sepsis

Background

There is a growing realization that alterations in host-pathogen interactions (HPI) can generate disease phenotypes without pathogen invasion. The gut represents a prime region where such HPI can arise and manifest. Under normal conditions intestinal microbial communities maintain a stable, mutually beneficial ecosystem. However, host stress can lead to changes in environmental conditions that shift the nature of the host-microbe dialogue, resulting in escalation of virulence expression, immune activation and ultimately systemic disease. Effective modulation of these dynamics requires the ability to characterize the complexity of the HPI, and dynamic computational modeling can aid in this task. Agent-based modeling is a computational method that is suited to representing spatially diverse, dynamical systems. We propose that dynamic knowledge representation of gut HPI with agent-based modeling will aid in the investigation of the pathogenesis of gut-derived sepsis.

Methodology/Principal Findings

An agent-based model (ABM) of virulence regulation in Pseudomonas aeruginosa was developed by translating bacterial and host cell sense-and-response mechanisms into behavioral rules for computational agents and integrated into a virtual environment representing the host-microbe interface in the gut. The resulting gut milieu ABM (GMABM) was used to: 1) investigate a potential clinically relevant laboratory experimental condition not yet developed - i.e. non-lethal transient segmental intestinal ischemia, 2) examine the sufficiency of existing hypotheses to explain experimental data - i.e. lethality in a model of major surgical insult and stress, and 3) produce behavior to potentially guide future experimental design - i.e. suggested sample points for a potential laboratory model of non-lethal transient intestinal ischemia. Furthermore, hypotheses were generated to explain certain discrepancies between the behaviors of the GMABM and biological experiments, and new investigatory avenues proposed to test those hypotheses.

Conclusions/Significance

Agent-based modeling can account for the spatio-temporal dynamics of an HPI, and, even when carried out with a relatively high degree of abstraction, can be useful in the investigation of system-level consequences of putative mechanisms operating at the individual agent level. We suggest that an integrated and iterative heuristic relationship between computational modeling and more traditional laboratory and clinical investigations, with a focus on identifying useful and sufficient degrees of abstraction, will enhance the efficiency and translational productivity of biomedical research.

Impact of the topology of metapopulations on the resurgence of epidemics rendered by a new multiscale hybrid modeling approach

Simulating epidemics in metapopulations is a challenging issue due to the large demographic and geographic scales to incorporate. Traditional epidemiologic models choose to simplify reality by ignoring both the spatial distribution of populations and possible intrapopulation heterogeneities, whereas more recent solutions based on Individual-Based Modeling (IBM) can achieve high precision but are costly to compute and analyze. We introduce here an original alternative to these two approaches, which relies on a novel hybrid modeling framework and incarnates a multiscale view of epidemics. The model relies on a technical fusion of two modeling paradigms: System Dynamics (SD) and Individual-Based Modeling. It features an aggregated representation of local outbreaks rendered in SD, and at the same time a spatially-explicit simulation of the spread between populations simulated in IBM. We first present the design of this deterministic model, show that it can reproduce the dynamics of real resurgent epidemics, and infer from the sensitivity of several spatial factors absent in compartmental models the importance of having large-scale epidemiological processes represented inside of an explicitly disaggregated metapopulation. After discussing the implications of results obtained from simulation runs and the applicability of this model, we conclude that SD–IB hybrid modeling can be an interesting choice to represent epidemics in a spatially-explicit way without necessarily taking into account individual heterogeneities, and therefore it can be considered as a valuable alternative to simple compartmental models suffering from detrimental effects of the well-mixed assumption.

A review of mathematical models of influenza A infections within a host or cell culture: lessons learned and challenges ahead

Most mathematical models used to study the dynamics of influenza A have thus far focused on the between-host population level, with the aim to inform public health decisions regarding issues such as drug and social distancing intervention strategies, antiviral stockpiling or vaccine distribution. Here, we investigate mathematical modeling of influenza infection spread at a different scale; namely that occurring within an individual host or a cell culture. We review the models that have been developed in the last decades and discuss their contributions to our understanding of the dynamics of influenza infections. We review kinetic parameters (e.g., viral clearance rate, lifespan of infected cells) and values obtained through fitting mathematical models, and contrast them with values obtained directly from experiments. We explore the symbiotic role of mathematical models and experimental assays in improving our quantitative understanding of influenza infection dynamics. We also discuss the challenges in developing better, more comprehensive models for the course of influenza infections within a host or cell culture. Finally, we explain the contributions of such modeling efforts to important public health issues, and suggest future modeling studies that can help to address additional questions relevant to public health.

Design considerations in building in silico equivalents of common experimental influenza virus assays

Experimentation in vitro is a vital part of the process by which the clinical and epidemiological characteristics of a particular influenza virus strain are determined. We detail the considerations which must be made in designing appropriate theoretical/mathematical models of these experiments and show how modeling can increase the information output of such experiments. Starting from a traditional system of ordinary differential equations, common to infectious disease modeling, we broaden the approach by using an agent-based model, applicable to more general experimental geometries and assumptions about the biological properties of viruses, cell and their interaction. Within this framework, we explore the limits of the assumptions made by more traditional models and the conditions under which these assumptions begin to break down, requiring the use of more sophisticated models. We apply the agent-based model to experimental plaque growth of two influenza strains, one resistant to the antiviral oseltamivir, and extract the values of key infection parameters specific to each strain.

A Simulation Framework to Investigate in vitro Viral Infection Dynamics

Virus infection is a complex biological phenomenon for which in vitro experiments provide a uniquely concise view where data is often obtained from a single population of cells, under controlled environmental conditions. Nonetheless, data interpretation and real understanding of viral dynamics is still hampered by the sheer complexity of the various intertwined spatio-temporal processes. In this paper we present a tool to address these issues: a cellular automata model describing critical aspects of in vitro viral infections taking into account spatial characteristics of virus spreading within a culture well. The aim of the model is to understand the key mechanisms of SARS-CoV infection dynamics during the first 24 hours post infection. We interrogate the model using a Latin Hypercube sensitivity analysis to identify which mechanisms are critical to the observed infection of host cells and the release of measured virus particles.

Higher level of replication efficiency of 2009 (H1N1) pandemic influenza virus than those of seasonal and avian strains: kinetics from epithelial cell culture and computational modeling

The pathogenicity and transmission of influenza A viruses are likely determined in part by replication efficiency in human cells, which is the net effect of complex virus-host interactions. H5N1 avian, H1N1 seasonal, and H1N1 2009 pandemic influenza virus strains were compared by infecting human differentiated bronchial epithelial cells in air-liquid interface cultures at relatively low virus particle/cell ratios. Differential equation and computational models were used to characterize the in vitro kinetic behaviors of the three strains. The models were calibrated by fitting experimental data in order to estimate difficult-to-measure parameters. Both models found marked differences in the relative values of p, the virion production rate per cell, and R(0), an index of the spread of infection through the monolayer, with the values for the strains in the following rank order (from greatest to least): pandemic strain, followed by seasonal strain, followed by avian strain, as expected. In the differential equation model, which treats virus and cell populations as well mixed, R(0) and p varied proportionately for all 3 strains, consistent with a primary role for productivity. In the spatially explicit computational model, R(0) and p also varied proportionately except that R(0) derived for the pandemic strain was reduced, consistent with constrained viral spread imposed by multiple host defenses, including mucus and paracrine antiviral effects. This synergistic experimental-computational strategy provides relevant parameters for identifying and phenotyping potential pandemic strains.

Parameter estimation and sensitivity analysis in an agent-based model of Leishmania major infection

Computer models of disease take a systems biology approach toward understanding host-pathogen interactions. In particular, data driven computer model calibration is the basis for inference of immunological and pathogen parameters, assessment of model validity, and comparison between alternative models of immune or pathogen behavior. In this paper we describe the calibration and analysis of an agent-based model of Leishmania major infection. A model of macrophage loss following uptake of necrotic tissue is proposed to explain macrophage depletion following peak infection. Using Gaussian processes to approximate the computer code, we perform a sensitivity analysis to identify important parameters and to characterize their influence on the simulated infection. The analysis indicates that increasing growth rate can favor or suppress pathogen loads, depending on the infection stage and the pathogen's ability to avoid detection. Subsequent calibration of the model against previously published biological observations suggests that L. major has a relatively slow growth rate and can replicate for an extended period of time before damaging the host cell.

Agent-based modeling of host-pathogen systems: The successes and challenges

Agent-based models have been employed to describe numerous processes in immunology. Simulations based on these types of models have been used to enhance our understanding of immunology and disease pathology. We review various agent-based models relevant to host–pathogen systems and discuss their contributions to our understanding of biological processes. We then point out some limitations and challenges of agent-based models and encourage efforts towards reproducibility and model validation.

Sharing the burden: antigen transport and firebreaks in immune responses

Communication between cells is crucial for immune responses. An important means of communication during viral infections is the presentation of viral antigen on the surface of an infected cell. Recently, it has been shown that antigen can be shared between infected and uninfected cells through gap junctions, connexin-based channels, that allow the transport of small molecules. The uninfected cell receiving antigen can present it on its surface. Cells presenting viral antigen are detected and killed by cytotoxic T lymphocytes. The killing of uninfected cells can lead to increased immunopathology. However, the immune response might also profit from killing those uninfected bystander cells. One benefit might be the removal of future 'virus factories'. Another benefit might be through the creation of 'firebreaks', areas void of target cells, which increase the diffusion time of free virions, making their clearance more likely. Here, we use theoretical models and simulations to explore how the mechanism of gap junction-mediated antigen transport (GMAT) affects the dynamics of the virus and immune response. We show that under the assumption of a well-mixed system, GMAT leads to increased immunopathology, which always outweighs the benefit of reduced virus production due to the removal of future virus factories. By contrast, a spatially explicit model leads to quite different results. Here we find that the firebreak mechanism reduces both viral load and immunopathology. Our study thus shows the potential benefits of GMAT and illustrates how spatial effects may be crucial for the quantitative understanding of infection dynamics and immune responses.

A virtual look at Epstein-Barr virus infection: simulation mechanism

Epstein-Barr virus (EBV) is an important human pathogen that establishes a life-long persistent infection and for which no precise animal model exists. In this paper, we describe in detail an agent-based model and computer simulation of EBV infection. Agents representing EBV and sets of B and T lymphocytes move and interact on a three-dimensional grid approximating Waldeyer's ring, together with abstract compartments for lymph and blood. The simulation allows us to explore the development and resolution of virtual infections in a manner not possible in actual human experiments. Specifically, we identify parameters capable of inducing clearance, persistent infection, or death.

Characterizing T cell movement within lymph nodes in the absence of antigen

The recent application of two-photon microscopy to the visualization of T cell movement has presented trajectories of individual T cells within lymphoid organs both in the presence and in the absence of Ag-loaded dendritic cells. Remarkably, even though T cells largely move along conduits of the fibroblastic reticular cell network, they appear to execute random walks in lymphoid organs rather than chemotaxis. In this study, we analyze experimental trajectories of T cells using computer simulations of idealized random walks. Comparisons of simulations with experimental data provide estimates of key parameters that characterize T cell motion in vivo. For example, we find that the distance moved before turning is about twice the distance between intersections in the fibroblastic reticular cell network, suggesting that at an intersection a T cell will turn onto a new fiber approximately 50% of the time. Although the calibrated model appears to offer an accurate representation of T cell movement, it has also uncovered inconsistencies across different experimental data sets.

Computer immunology

This review describes a body of work on computational immune systems that behave analogously to the natural immune system. These artificial immune systems (AIS) simulate the behavior of the natural immune system and in some cases have been used to solve practical engineering problems such as computer security. AIS have several strengths that can complement wet lab immunology. It is easier to conduct simulation experiments and to vary experimental conditions, for example, to rule out hypotheses; it is easier to isolate a single mechanism to test hypotheses about how it functions; agent-based models of the immune system can integrate data from several different experiments into a single in silico experimental system.