Modeling Viral Spread

Mathematical modelling of reoviruses in cancer cell cultures

Oncolytic virotherapy has emerged as a potential cancer therapy, utilizing viruses to selectively target and replicate within cancer cells while preserving normal cells. In this paper, we investigate the oncolytic potential of unmodified reovirus T3wt relative to a mutated variant SV5. In animal cancer cell monolayer experiments it was found that SV5 was more oncolytic relative to T3wt. SV5 forms larger sized plaques on cancer cell monolayers and spreads to farther distances from the initial site of infection as compared to T3wt. Paradoxically, SV5 attaches to cancer cells less efficiently than T3wt, which lead us to hypothesize that there might be an optimal binding affinity with maximal oncolytic activity. To understand the relationship between the binding process and virus spread for T3wt and SV5, we employ mathematical modelling. A reaction-diffusion model is applied, which is fit to the available data and then validated on data that were not used for the fit. Analysis of our model shows that there is an optimal binding rate that leads to maximum viral infection of the cancer monolayer, and we estimate this value for T3wt and SV5. Moreover, we find that the viral burst size is an important parameter for viral spread, and that a combination of efficient binding and large burst sizes is a promising direction to further develop anti-cancer viruses.

Modelling HIV-1 control and remission

Remarkable advances are being made in developing interventions for eliciting long-term remission of HIV-1 infection. The success of these interventions will obviate the need for lifelong antiretroviral therapy, the current standard-of-care, and benefit the millions living today with HIV-1. Mathematical modelling has made significant contributions to these efforts. It has helped elucidate the possible mechanistic origins of natural and post-treatment control, deduced potential pathways of the loss of such control, quantified the effects of interventions, and developed frameworks for their rational optimization. Yet, several important questions remain, posing challenges to the translation of these promising interventions. Here, we survey the recent advances in the mathematical modelling of HIV-1 control and remission, highlight their contributions, and discuss potential avenues for future developments.

Spatial information allows inference of the prevalence of direct cell-to-cell viral infection

The role of direct cell-to-cell spread in viral infections-where virions spread between host and susceptible cells without needing to be secreted into the extracellular environment-has come to be understood as essential to the dynamics of medically significant viruses like hepatitis C and influenza. Recent work in both the experimental and mathematical modelling literature has attempted to quantify the prevalence of cell-to-cell infection compared to the conventional free virus route using a variety of methods and experimental data. However, estimates are subject to significant uncertainty and moreover rely on data collected by inhibiting one mode of infection by either chemical or physical factors, which may influence the other mode of infection to an extent which is difficult to quantify. In this work, we conduct a simulation-estimation study to probe the practical identifiability of the proportion of cell-to-cell infection, using two standard mathematical models and synthetic data that would likely be realistic to obtain in the laboratory. We show that this quantity cannot be estimated using non-spatial data alone, and that the collection of data which describes the spatial structure of the infection is necessary to infer the proportion of cell-to-cell infection. Our results provide guidance for the design of relevant experiments and mathematical tools for accurately inferring the prevalence of cell-to-cell infection in in vitro and in vivo contexts.

Self-inhibiting percolation and viral spreading in epithelial tissue

SARS-CoV-2 induces delayed type-I/III interferon production, allowing it to escape the early innate immune response. The delay has been attributed to a deficiency in the ability of cells to sense viral replication upon infection, which in turn hampers activation of the antiviral state in bystander cells. Here, we introduce a cellular automaton model to investigate the spatiotemporal spreading of viral infection as a function of virus and host-dependent parameters. The model suggests that the considerable person-to-person heterogeneity in SARS-CoV-2 infections is a consequence of high sensitivity to slight variations in biological parameters near a critical threshold. It further suggests that within-host viral proliferation can be curtailed by the presence of remarkably few cells that are primed for IFN production. Thus, the observed heterogeneity in defense readiness of cells reflects a remarkably cost-efficient strategy for protection.

Precision epidemiology at the nexus of mathematics and nanotechnology: Unraveling the dance of viral dynamics

The intersection of mathematical modeling, nanotechnology, and epidemiology marks a paradigm shift in our battle against infectious diseases, aligning with the focus of the journal on the regulation, expression, function, and evolution of genes in diverse biological contexts. This exploration navigates the intricate dance of viral transmission dynamics, highlighting mathematical models as dual tools of insight and precision instruments, a theme relevant to the diverse sections of Gene. In the context of virology, ethical considerations loom large, necessitating robust frameworks to protect individual rights, an aspect essential in infectious disease research. Global collaboration emerges as a critical pillar in our response to emerging infectious diseases, fortified by the predictive prowess of mathematical models enriched by nanotechnology. The synergy of interdisciplinary collaboration, training the next generation to bridge mathematical rigor, biology, and epidemiology, promises accelerated discoveries and robust models that account for real-world complexities, fostering innovation and exploration in the field. In this intricate review, mathematical modeling in viral transmission dynamics and epidemiology serves as a guiding beacon, illuminating the path toward precision interventions, global preparedness, and the collective endeavor to safeguard human health, resonating with the aim of advancing knowledge in gene regulation and expression.

Mathematical model explains differences in Omicron and Delta SARS-CoV-2 dynamics in Caco-2 and Calu-3 cells

Within-host infection dynamics of Omicron dramatically differs from previous variants of SARS-CoV-2. However, little is still known about which parameters of virus-cell interplay contribute to the observed attenuated replication and pathogenicity of Omicron. Mathematical models, often expressed as systems of differential equations, are frequently employed to study the infection dynamics of various viruses. Adopting such models for results of in vitro experiments can be beneficial in a number of aspects, such as model simplification (e.g., the absence of adaptive immune response and innate immunity cells), better measurement accuracy, and the possibility to measure additional data types in comparison with in vivo case. In this study, we consider a refinement of our previously developed and validated model based on a system of integro-differential equations. We fit the model to the experimental data of Omicron and Delta infections in Caco-2 (human intestinal epithelium model) and Calu-3 (lung epithelium model) cell lines. The data include known information on initial conditions, infectious virus titers, and intracellular viral RNA measurements at several time points post-infection. The model accurately explains the experimental data for both variants in both cell lines using only three variant- and cell-line-specific parameters. Namely, the cell entry rate is significantly lower for Omicron, and Omicron triggers a stronger cytokine production rate (i.e., innate immune response) in infected cells, ultimately making uninfected cells resistant to the virus. Notably, differences in only a single parameter (e.g., cell entry rate) are insufficient to obtain a reliable model fit for the experimental data.

The spatial dynamics of immune response upon virus infection through hybrid dynamical computational model

Introduction

The immune responses play important roles in the course of disease initiation and progression upon virus infection such as SARS-CoV-2. As the tissues consist of spatial structures, the spatial dynamics of immune responses upon viral infection are essential to the outcome of infection.

Methods

A hybrid computational model based on cellular automata coupled with partial differential equations is developed to simulate the spatial patterns and dynamics of the immune responses of tissue upon virus infection with several different immune movement modes.

Results

Various patterns of the distribution of virus particles under different immune strengths and movement modes of immune cells are obtained through the computational models. The results also reveal that the directed immune cell wandering model has a better immunization effect. Several other characteristics, such as the peak level of virus density and onset time and the onset of the diseases, are also checked with different immune and physiological conditions, for example, different immune clearance strengths, and different cell-to-cell transmission rates. Furthermore, by the Lasso analysis, it is identified that the three main parameters had the most impact on the rate of onset time of disease. It is also shown that the cell-to-cell transmission rate has a significant effect and is more important for controlling the diseases than those for the cell-free virus given that the faster cell-to-cell transmission than cell-free transmission the rate of virus release is low.

Discussion

Our model simulates the process of viral and immune response interactions in the alveola repithelial tissues of infected individuals, providing insights into the viral propagation of viruses in two dimensions as well as the influence of immune response patterns and key factors on the course of infection.

Patch formation driven by stochastic effects of interaction between viruses and defective interfering particles

Defective interfering particles (DIPs) are virus-like particles that occur naturally during virus infections. These particles are defective, lacking essential genetic materials for replication, but they can interact with the wild-type virus and potentially be used as therapeutic agents. However, the effect of DIPs on infection spread is still unclear due to complicated stochastic effects and nonlinear spatial dynamics. In this work, we develop a model with a new hybrid method to study the spatial-temporal dynamics of viruses and DIPs co-infections within hosts. We present two different scenarios of virus production and compare the results from deterministic and stochastic models to demonstrate how the stochastic effect is involved in the spatial dynamics of virus transmission. We compare the spread features of the virus in simulations and experiments, including the formation and the speed of virus spread and the emergence of stochastic patchy patterns of virus distribution. Our simulations simultaneously capture observed spatial spread features in the experimental data, including the spread rate of the virus and its patchiness. The results demonstrate that DIPs can slow down the growth of virus particles and make the spread of the virus more patchy.

Making waves: Integrating wastewater surveillance with dynamic modeling to track and predict viral outbreaks

Wastewater surveillance has proved to be a valuable tool to track the COVID-19 pandemic. However, most studies using wastewater surveillance data revolve around establishing correlations and lead time relative to reported case data. In this perspective, we advocate for the integration of wastewater surveillance data with dynamic within-host and between-host models to better understand, monitor, and predict viral disease outbreaks. Dynamic models overcome emblematic difficulties of using wastewater surveillance data such as establishing the temporal viral shedding profile. Complementarily, wastewater surveillance data bypasses the issues of time lag and underreporting in clinical case report data, thus enhancing the utility and applicability of dynamic models. The integration of wastewater surveillance data with dynamic models can enhance real-time tracking and prevalence estimation, forecast viral transmission and intervention effectiveness, and most importantly, provide a mechanistic understanding of infectious disease dynamics and the driving factors. Dynamic modeling of wastewater surveillance data will advance the development of a predictive and responsive monitoring system to improve pandemic preparedness and population health.

Agent-Based and Continuum Models for Spatial Dynamics of Infection by Oncolytic Viruses

The use of oncolytic viruses as cancer treatment has received considerable attention in recent years, however the spatial dynamics of this viral infection is still poorly understood. We present here a stochastic agent-based model describing infected and uninfected cells for solid tumours, which interact with viruses in the absence of an immune response. Two kinds of movement, namely undirected random and pressure-driven movements, are considered: the continuum limit of the models is derived and a systematic comparison between the systems of partial differential equations and the individual-based model, in one and two dimensions, is carried out. In the case of undirected movement, a good agreement between agent-based simulations and the numerical and well-known analytical results for the continuum model is possible. For pressure-driven motion, instead, we observe a wide parameter range in which the infection of the agents remains confined to the center of the tumour, even though the continuum model shows traveling waves of infection; outcomes appear to be more sensitive to stochasticity and uninfected regions appear harder to invade, giving rise to irregular, unpredictable growth patterns. Our results show that the presence of spatial constraints in tumours' microenvironments limiting free expansion has a very significant impact on virotherapy. Outcomes for these tumours suggest a notable increase in variability. All these aspects can have important effects when designing individually tailored therapies where virotherapy is included.

HIV-1 cell-to-cell spread overcomes the virus entry block of non-macrophage-tropic strains in macrophages

Macrophages (MΦ) are increasingly recognized as HIV-1 target cells involved in the pathogenesis and persistence of infection. Paradoxically, in vitro infection assays suggest that virus isolates are mostly T-cell-tropic and rarely MΦ-tropic. The latter are assumed to emerge under CD4+ T-cell paucity in tissues such as the brain or at late stage when the CD4 T-cell count declines. However, assays to qualify HIV-1 tropism use cell-free viral particles and may not fully reflect the conditions of in vivo MΦ infection through cell-to-cell viral transfer. Here, we investigated the capacity of viruses expressing primary envelope glycoproteins (Envs) with CCR5 and/or CXCR4 usage from different stages of infection, including transmitted/founder Envs, to infect MΦ by a cell-free mode and through cell-to-cell transfer from infected CD4+ T cells. The results show that most viruses were unable to enter MΦ as cell-free particles, in agreement with the current view that non-M-tropic viruses inefficiently use CD4 and/or CCR5 or CXCR4 entry receptors on MΦ. In contrast, all viruses could be effectively cell-to-cell transferred to MΦ from infected CD4+ T cells. We further showed that viral transfer proceeded through Env-dependent cell-cell fusion of infected T cells with MΦ targets, leading to the formation of productively infected multinucleated giant cells. Compared to cell-free infection, infected T-cell/MΦ contacts showed enhanced interactions of R5 M- and non-M-tropic Envs with CD4 and CCR5, resulting in a reduced dependence on receptor expression levels on MΦ for viral entry. Altogether, our results show that virus cell-to-cell transfer overcomes the entry block of isolates initially defined as non-macrophage-tropic, indicating that HIV-1 has a more prevalent tropism for MΦ than initially suggested. This sheds light into the role of this route of virus cell-to-cell transfer to MΦ in CD4+ T cell rich tissues for HIV-1 transmission, dissemination and formation of tissue viral reservoirs.

Antithetic effect of interferon-α on cell-free and cell-to-cell HIV-1 infection

In HIV-1-infected individuals, transmitted/founder (TF) virus contributes to establish new infection and expands during the acute phase of infection, while chronic control (CC) virus emerges during the chronic phase of infection. TF viruses are more resistant to interferon-alpha (IFN-α)-mediated antiviral effects than CC virus, however, its virological relevance in infected individuals remains unclear. Here we perform an experimental-mathematical investigation and reveal that IFN-α strongly inhibits cell-to-cell infection by CC virus but only weakly affects that by TF virus. Surprisingly, IFN-α enhances cell-free infection of HIV-1, particularly that of CC virus, in a virus-cell density-dependent manner. We further demonstrate that LY6E, an IFN-stimulated gene, can contribute to the density-dependent enhancement of cell-free HIV-1 infection. Altogether, our findings suggest that the major difference between TF and CC viruses can be explained by their resistance to IFN-α-mediated inhibition of cell-to-cell infection and their sensitivity to IFN-α-mediated enhancement of cell-free infection.

Mathematical Modeling Suggests Cooperation of Plant-Infecting Viruses

Viruses are major pathogens of agricultural crops. Viral infections often start after the virus enters the outer layer of a tissue, and many successful viruses, after local replication in the infected tissue, are able to spread systemically. Quantitative details of virus dynamics in plants, however, are poorly understood, in part, because of the lack of experimental methods which allow the accurate measurement of the degree of infection in individual plant tissues. Recently, a group of researchers followed the kinetics of infection of individual cells in leaves of Nicotiana tabacum plants using Tobacco etch virus (TEV) expressing either Venus or blue fluorescent protein (BFP). Assuming that viral spread occurs from lower to upper leaves, the authors fitted a simple mathematical model to the frequency of cellular infection by the two viral variants found using flow cytometry. While the original model could accurately describe the kinetics of viral spread locally and systemically, we found that many alternative versions of the model, for example, if viral spread starts at upper leaves and progresses to lower leaves or when virus dissemination is stopped due to an immune response, fit the data with reasonable quality, and yet with different parameter estimates. These results strongly suggest that experimental measurements of the virus infection in individual leaves may not be sufficient to identify the pathways of viral dissemination between different leaves and reasons for viral control. We propose experiments that may allow discrimination between the alternatives. By analyzing the kinetics of coinfection of individual cells by Venus and BFP strains of TEV we found a strong deviation from the random infection model, suggesting cooperation between the two strains when infecting plant cells. Importantly, we showed that many mathematical models on the kinetics of coinfection of cells with two strains could not adequately describe the data, and the best fit model needed to assume (i) different susceptibility of uninfected cells to infection by two viruses locally in the leaf vs. systemically from other leaves, and (ii) decrease in the infection rate depending on the fraction of uninfected cells which could be due to a systemic immune response. Our results thus demonstrate the difficulty in reaching definite conclusions from extensive and yet limited experimental data and provide evidence of potential cooperation between different viral variants infecting individual cells in plants.

First-wave COVID-19 daily cases obey gamma law

Modelling how a pandemic is spreading over time is a challenging issue. The new coronavirus disease called COVID-19 does not escape this rule as it has embraced over two hundred countries. As for previous pandemics, several studies have attempted to model the occurrence of cases caused by COVID-19. However, no study has succeeded in accurately modelling the impact of the infectious agent. Here we show that COVID-19 daily case distribution in humans obeys a Gamma law, which two new parameters can describe without any adjustment. Though the Gamma law has been exploited for nearly two centuries to describe the statistical distribution of spatial or temporal quantities, the goodness-of-fit rationale using two or three parameters has remained enigmatic. The new Gamma law approach we demonstrate here emerges from actual data and sheds light on the underlying mechanisms of the observed phenomenon. This finding has promising applicability in the epidemiological domain and in all disciplines involving branching systems, for which our Gamma law approach may bring a solution to hitherto unsolved problems.

Bluetongue Research at a Crossroads: Modern Genomics Tools Can Pave the Way to New Insights

Bluetongue virus (BTV) is an arthropod-borne, segmented double-stranded RNA virus that can cause severe disease in both wild and domestic ruminants. BTV evolves via several key mechanisms, including the accumulation of mutations over time and the reassortment of genome segments.Additionally, BTV must maintain fitness in two disparate hosts, the insect vector and the ruminant. The specific features of viral adaptation in each host that permit host-switching are poorly characterized. Limited field studies and experimental work have alluded to the presence of these phenomena at work, but our understanding of the factors that drive or constrain BTV's genetic diversification remains incomplete. Current research leveraging novel approaches and whole genome sequencing applications promises to improve our understanding of BTV's evolution, ultimately contributing to the development of better predictive models and management strategies to reduce future impacts of bluetongue epizootics.

The Influence of Immune Response on Spreading of Viral Infection

In this work we develop a model of viral infection in host tissues in order to study the influence of the immune response on the infection spreading speed and on the viral load characterizing, respectively, severity of symptoms and infection transmission rate. Dynamics of the interaction between viral infection and the immune response is studied with nonlocal reaction-diffusion equations for the concentrations of virus, interferon, immune cells and antibodies. Analytical results for infection spreading speed and viral load are completed by numerical simulations. At the first stage, progression of viral infection is confronted by the innate immune response mostly determined by the local interferon production. The modeling results show in this case that infection spreading speed does not depend on interferon concentration, while the total viral load decreases with the increase of its concentration. Next, we consider the influence of globally circulating interferon and show that, in contrast to local interferon diffusion, infection spreading speed decreases with increasing of global interferon level, and the total viral load also decreases. At the next stage, adaptive immune response mediated by antibodies and cytotoxic T cells (CTL) further influences infection progression. In this case, the infection propagation speed and the total viral load are decreased by the immune response. The humoral adaptive response (antibodies) increases the global interferon concentration through the viral load, while the cellular adaptive response (CTL) decreases it.

The Social Life of Viruses

Despite their simplicity, viruses exhibit certain types of social interactions. Situations in which a given virus achieves higher fitness in combination with other members of the viral population have been described at the level of transmission, replication, suppression of host immune responses, and host killing, enabling the evolution of viral cooperation. Although cellular coinfection with multiple viral particles is the typical playground for these interactions, cooperation between viruses infecting different cells is also established through cellular and viral-encoded communication systems. In general, the stability of cooperation is compromised by cheater genotypes, as best exemplified by defective interfering particles. As predicted by social evolution theory, cheater invasion can be avoided when cooperators interact preferentially with other cooperators, a situation that is promoted in spatially structured populations. Processes such as transmission bottlenecks, organ compartmentalization, localized spread of infection foci, superinfection exclusion, and even discrete intracellular replication centers promote multilevel spatial structuring in viruses. Expected final online publication date for the Annual Review of Virology, Volume 8 is September 2021. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.

HCV Spread Kinetics Reveal Varying Contributions of Transmission Modes to Infection Dynamics

The hepatitis C virus (HCV) is capable of spreading within a host by two different transmission modes: cell-free and cell-to-cell. However, the contribution of each of these transmission mechanisms to HCV spread is unknown. To dissect the contribution of these different transmission modes to HCV spread, we measured HCV lifecycle kinetics and used an in vitro spread assay to monitor HCV spread kinetics after a low multiplicity of infection in the absence and presence of a neutralizing antibody that blocks cell-free spread. By analyzing these data with a spatially explicit mathematical model that describes viral spread on a single-cell level, we quantified the contribution of cell-free, and cell-to-cell spread to the overall infection dynamics and show that both transmission modes act synergistically to enhance the spread of infection. Thus, the simultaneous occurrence of both transmission modes represents an advantage for HCV that may contribute to viral persistence. Notably, the relative contribution of each viral transmission mode appeared to vary dependent on different experimental conditions and suggests that viral spread is optimized according to the environment. Together, our analyses provide insight into the spread dynamics of HCV and reveal how different transmission modes impact each other.

Controlling of pandemic COVID-19 using optimal control theory

In 2019, a new infectious disease called pandemic COVID-19 began to spread from Wuhan, China. In spite of the efforts to stop the disease, being out of the control of the governments it spread rapidly all over the world. From then on, much research has been done in the world with the aim of controlling this contagious disease. A mathematical model for modeling the spread of COVID-19 and also controlling the spread of the disease has been presented in this paper. We find the disease-free equilibrium points as trivial equilibrium (TE), virus absenteeism equilibrium (VAE) and virus incidence equilibrium (VIE) for the proposed model; and at the trivial equilibrium point for the presented dynamic system we obtain the Jacobian matrix so as to be used in finding the largest eigenvalue. Radius spectral method has been used for finding the reproductive number. In the following, by adding a controller to the model and also using the theory of optimal control, we can improve the performance of the model. We must have a correct understanding of the system i.e. how it works, the various variables affecting the system, and the interaction of the variables on each other. To search for the optimal values, we need to use an appropriate optimization method. Given the limitations and needs of the problem, the aim of the optimization is to find the best solutions, to find conditions that result in the maximum of susceptiblity, the minimum of infection, and optimal quarantination.

Dynamical characterization of antiviral effects in COVID-19

Mathematical models describing SARS-CoV-2 dynamics and the corresponding immune responses in patients with COVID-19 can be critical to evaluate possible clinical outcomes of antiviral treatments. In this work, based on the concept of virus spreadability in the host, antiviral effectiveness thresholds are determined to establish whether or not a treatment will be able to clear the infection. In addition, the virus dynamic in the host - including the time-to-peak and the final monotonically decreasing behavior - is characterized as a function of the time to treatment initiation. Simulation results, based on nine patient data, show the potential clinical benefits of a treatment classification according to patient critical parameters. This study is aimed at paving the way for the different antivirals being developed to tackle SARS-CoV-2.

Cooperative nature of viral replication

The ability of viruses to infect their hosts depends on rapid dissemination following transmission. The notion that viral particles function as independent propagules has been challenged by recent observations suggesting that viral aggregates show enhanced infectivity and faster spread. However, these observations remain poorly understood. Here, we show that viral replication is a cooperative process, such that entry of multiple viral genome copies into the same cell disproportionately increases short-term viral progeny production. This cooperativity arises from the positive feedback established between replication templates and virus-encoded products involved in replication and should be a general feature of viruses. We develop a simple model that captures this effect, verify that cooperativity also emerges in more complex models for specific human viruses, validate our predictions experimentally using different mammalian viruses, and discuss the implications of cooperative replication for viral fitness.

Mathematical modeling of hepatitis C RNA replication, exosome secretion and virus release

Hepatitis C virus (HCV) causes acute hepatitis C and can lead to life-threatening complications if it becomes chronic. The HCV genome is a single plus strand of RNA. Its intracellular replication is a spatiotemporally coordinated process of RNA translation upon cell infection, RNA synthesis within a replication compartment, and virus particle production. While HCV is mainly transmitted via mature infectious virus particles, it has also been suggested that HCV-infected cells can secrete HCV RNA carrying exosomes that can infect cells in a receptor independent manner. In order to gain insight into these two routes of transmission, we developed a series of intracellular HCV replication models that include HCV RNA secretion and/or virus assembly and release. Fitting our models to in vitro data, in which cells were infected with HCV, suggests that initially most secreted HCV RNA derives from intracellular cytosolic plus-strand RNA, but subsequently secreted HCV RNA derives equally from the cytoplasm and the replication compartments. Furthermore, our model fits to the data suggest that the rate of virus assembly and release is limited by host cell resources. Including the effects of direct acting antivirals in our models, we found that in spite of decreasing intracellular HCV RNA and extracellular virus concentration, low level HCV RNA secretion may continue as long as intracellular RNA is available. This may possibly explain the presence of detectable levels of plasma HCV RNA at the end of treatment even in patients that ultimately attain a sustained virologic response.

Approximately 70 million people are chronically infected with hepatitis C virus (HCV), which if left untreated may lead to cirrhosis and liver cancer. However, modern drug therapy is highly effective and hepatitis C is the first chronic virus infection that can be cured with short-term therapy in almost all infected individuals. The within-host transmission of HCV occurs mainly via infectious virus particles, but experimental studies suggest that there may be additional receptor-independent cell-to-cell transmission by exosomes that carry the HCV genome. In order to understand the intracellular HCV lifecycle and HCV RNA spread, we developed a series of mathematical models that take both exosomal secretion and viral secretion into account. By fitting these models to in vitro data, we found that secretion of both HCV RNA as well as virus probably occurs and that the rate of virus assembly is likely limited by cellular co-factors on which the virus strongly depends for its own replication. Furthermore, our modeling predicted that the parameters governing the processes in the viral lifecycle that are targeted by direct acting antivirals are the most sensitive to perturbations, which may help explain their ability to cure this infection.

Virus spread in the liver: mechanisms, commonalities, and unanswered questions

The liver is home to five known human hepatitis viruses (hepatitis A virus-hepatitis E virus). Despite being phylogenetically unrelated, these viruses replicate and spread in the liver without causing apparent cytopathic effects, and all have evolved strategies to counteract antibody-mediated inhibition of virus spread. In this review, we discuss the current understanding regarding the spread mechanisms for these viruses with an attempt to extract common principles and identify key questions for future studies.

Tracking Mechanisms of Viral Dissemination In Vivo

Dissemination and replication of viruses into hosts is a multistep process where viral particles infect, navigate, and indoctrinate various cell types. Viruses can reach tissues that are distant from their infection site by subverting subcellular mechanisms in ways that are, sometimes, disruptive. Modeling these steps, at appropriate resolution and within animal models, is cumbersome. Yet, mimicking these strategies in vitro fails to recapitulate the complexity of the cellular ecosystem. Here, we will discuss relevant in vivo platforms to dissect the cellular and molecular programs governing viral dissemination and briefly discuss organoid and ex vivo alternatives. We will focus on the zebrafish model and will describe how it provides a transparent window to unravel new cellular mechanisms of viral dissemination in vivo.

The zebrafish model allows in vivo investigations of virus-induced molecular processes at subcellular resolution.

Viruses have evolved multiple strategies for disseminating over long distance, including by indoctrinating host cell types with high migration potential.

Organoids derived from stem cells emerge as powerful alternatives to unravel new molecular mechanisms of viral dissemination.

In-host Mathematical Modelling of COVID-19 in Humans

COVID-19 pandemic has underlined the impact of emergent pathogens as a major threat to human health. The development of quantitative approaches to advance comprehension of the current outbreak is urgently needed to tackle this severe disease. Considering different starting times of infection, mathematical models are proposed to represent SARS-CoV-2 dynamics in infected patients. Based on the target cell limited model, the within-host reproductive number for SARS-CoV-2 is consistent with the broad values of human influenza infection. The best model to fit the data was including immune cell response, which suggests a slow immune response peaking between 5 to 10 days post-onset of symptoms. The model with the eclipse phase, time in a latent phase before becoming productively infected cells, was not supported. Interestingly, model simulations predict that SARS-CoV-2 may replicate very slowly in the first days after infection, and viral load could be below detection levels during the first 4 days post infection. A quantitative comprehension of SARS-CoV-2 dynamics and the estimation of standard parameters of viral infections is the key contribution of this pioneering work. These models can serve for future evaluation of control theoretical approaches to tailor new drugs against COVID-19.

Viral cell-to-cell spread: Conventional and non-conventional ways

A critical step in the life cycle of a virus is spread to a new target cell, which generally involves the release of new viral particles from the infected cell which can then initiate infection in the next target cell. While cell-free viral particles released into the extracellular environment are necessary for long distance spread, there are disadvantages to this mechanism. These include the presence of immune system components, the low success rate of infection by single particles, and the relative fragility of viral particles in the environment. Several mechanisms of direct cell-to-cell spread have been reported for animal viruses which would avoid the issues associated with cell-free particles. A number of viruses can utilize several different mechanisms of direct cell-to-cell spread, but our understanding of the differential usage by these pathogens is modest. Although the mechanisms of cell-to-cell spread differ among viruses, there is a common exploitation of key pathways and components of the cellular cytoskeleton. Remarkably, some of the viral mechanisms of cell-to-cell spread are surprisingly similar to those used by bacteria. Here we summarize the current knowledge of the conventional and non-conventional mechanisms of viral spread, the common methods used to detect viral spread, and the impact that these mechanisms can have on viral pathogenesis.

Modeling HIV multiple infection

Multiple infection of target cells by human immunodeficiency virus (HIV) may lead to viral escape from host immune responses and drug resistance to antiretroviral therapy, bringing more challenges to the control of infection. The mechanisms underlying HIV multiple infection and their relative contributions are not fully understood. In this paper, we develop and analyze a mathematical model that includes sequential cell-free virus infection (i.e.one virus is transmitted each time in a sequential infection of target cells by virus) and cell-to-cell transmission (i.e.multiple viral genomes are transmitted simultaneously from infected to uninfected cells). By comparing model prediction with the distribution data of proviral genomes in HIV-infected spleen cells, we find that multiple infection can be well explained when the two modes of viral transmission are both included. Numerical simulation using the parameter estimates from data fitting shows that the majority of T cell infections are attributed to cell-to-cell transmission and this transmission mode also accounts for more than half of cell's multiple infections. These results suggest that cell-to-cell transmission plays a critical role in forming HIV multiple infection and thus has important implications for HIV evolution and pathogenesis.

Extracellular Vesicles in Viral Spread and Antiviral Response

Viral spread by both enveloped and non-enveloped viruses may be mediated by extracellular vesicles (EVs), including microvesicles (MVs) and exosomes. These secreted vesicles have been demonstrated to be an efficient mechanism that viruses can use to enter host cells, enhance spread or evade the host immune response. However, the complex interplay between viruses and EVs gives rise to antagonistic biological tasks-to benefit the viruses, enhancing infection and interfering with the immune system or to benefit the host, by mediating anti-viral responses. Exosomes from cells infected with herpes simplex type 1 (HSV-1) may transport viral and host transcripts, proteins and innate immune components. This virus may also use MVs to expand its tropism and evade the host immune response. This review aims to describe the current knowledge about EVs and their participation in viral infection, with a specific focus on the role of exosomes and MVs in herpesvirus infections, particularly that of HSV-1.

Environmental Restrictions: A New Concept Governing HIV-1 Spread Emerging from Integrated Experimental-Computational Analysis of Tissue-Like 3D Cultures

HIV-1 can use cell-free and cell-associated transmission modes to infect new target cells, but how the virus spreads in the infected host remains to be determined. We recently established 3D collagen cultures to study HIV-1 spread in tissue-like environments and applied iterative cycles of experimentation and computation to develop a first in silico model to describe the dynamics of HIV-1 spread in complex tissue. These analyses (i) revealed that 3D collagen environments restrict cell-free HIV-1 infection but promote cell-associated virus transmission and (ii) defined that cell densities in tissue dictate the efficacy of these transmission modes for virus spread. In this review, we discuss, in the context of the current literature, the implications of this study for our understanding of HIV-1 spread in vivo, which aspects of in vivo physiology this integrated experimental-computational analysis takes into account, and how it can be further improved experimentally and in silico.

Immunological Paradigms, Mechanisms, and Models: Conceptual Understanding Is a Prerequisite to Effective Modeling

Most mathematical models that describe the individual or collective actions of cells aim at creating faithful representations of limited sets of data in a self-consistent manner. Consistency with relevant physiological rules pertaining to the greater picture is rarely imposed. By themselves, such models have limited predictive or even explanatory value, contrary to standard claims. Here I try to show that a more critical examination of currently held paradigms is necessary and could potentially lead to models that pass the test of time. In considering the evolution of paradigms over the past decades I focus on the “smart surveillance” theory of how T cells can respond differentially, individually and collectively, to both self- and foreign antigens depending on various “contextual” parameters. The overall perspective is that physiological messages to cells are encoded not only in the biochemical connections of signaling molecules to the cellular machinery but also in the magnitude, kinetics, and in the time- and space-contingencies, of sets of stimuli. By rationalizing the feasibility of subthreshold interactions, the “dynamic tuning hypothesis,” a central component of the theory, set the ground for further theoretical and experimental explorations of dynamically regulated immune tolerance, homeostasis and diversity, and of the notion that lymphocytes participate in nonclassical physiological functions. Some of these efforts are reviewed. Another focus of this review is the concomitant regulation of immune activation and homeostasis through the operation of a feedback mechanism controlling the balance between renewal and differentiation of activated cells. Different perspectives on the nature and regulation of chronic immune activation in HIV infection have led to conflicting models of HIV pathogenesis—a major area of research for theoretical immunologists over almost three decades—and can have profound impact on ongoing HIV cure strategies. Altogether, this critical review is intended to constructively influence the outlook of prospective model builders and of interested immunologists on the state of the art and to encourage conceptual work.

Proteomics Analysis Identifies IRSp53 and Fascin as Critical for PRV Egress and Direct Cell-Cell Transmission

Pseudorabies virus (PRV) has been widely used as a live trans-synaptic tracer for mapping neuronal circuits. Systematically identifying mature PRV virion proteomes and defining co-purified host proteins are necessary to fully understand the detailed mechanism underlying PRV transmission processes. Here, a PRV virion purification strategy based on sorting with flow cytometry is developed and the mature extracellular and intracellular PRV virion proteomes using LC coupled with MS/MS are characterized. In addition to viral proteins, a large number of host proteins are also identified, including proteins related to actin cytoskeletal dynamics and membrane protrusion. How many of these host proteins are true virion components are unknown and the majority of these may not be. Through functional analysis, it is found that IRSp53 and fascin are critical for the egress process and play a role in direct cell-cell transmission. Moreover, it is shown that CDC42 and Rac1 are also involved in the production of mature extracellular virions. The results suggest that the formation of the filopodia-like cytoskeleton and the rearrangement of the membrane, which are both associated with IRSp53 and fascin, may be important for the transmission of viruses used in neuronal tracing.

Spatiotemporal immunofluorescent evaluation of porcine reproductive and respiratory syndrome virus transmission across the maternal-fetal interface

Porcine reproductive and respiratory syndrome virus (PRRSV) infection causes severe reproductive failure characterized by high fetal morbidity and mortality leading to substantial economic losses to the swine industry. Evaluation of spatiotemporal transmission of PRRSV at the maternal-fetal interface (MFI) is critical for understanding fetal infection. Localization of PRRSV-2 strain NVSL 97-7895 at different regions of the MFI in 20 pregnant gilts at 2, 5, 8, 12 and 14 days post-inoculation (dpi) were analyzed by immunofluorescence (IF). Samples of MFI were collected from 15 inoculated and 5 control gilts and transplacental PRRSV transmission assessed in randomly selected fetuses from each litter. Localization of NVSL 97-7895 antigen immunoreactivity in the MFI was focused in three major areas: endometrial connective tissues (ENDO), the feto-maternal junction (FMJ) and fetal placenta (PLC). NVSL 97-7895 was detected at the FMJ by 2 dpi. At 2, 5 and 8 dpi, NVSL 97-7895 was localized within the ENDO and FMJ, whereas at 12 and 14 dpi, it was mainly localized in the PLC. Using a novel IF strategy for counting and size sorting NVSL 97-7895 viral antigen in situ, results of this study indicate that non-cell-associated mechanisms are involved in PRRSV transmission across the MFI.

Global Properties of a Delay-Distributed HIV Dynamics Model Including Impairment of B-Cell Functions

In this paper, we construct an Human immunodeficiency virus (HIV) dynamics model with impairment of B-cell functions and the general incidence rate. We incorporate three types of infected cells, (i) latently-infected cells, which contain the virus, but do not generate HIV particles, (ii) short-lived productively-infected cells, which live for a short time and generate large numbers of HIV particles, and (iii) long-lived productively-infected cells, which live for a long time and generate small numbers of HIV particles. The model considers five distributed time delays to characterize the time between the HIV contact of an uninfected CD4 + T-cell and the creation of mature HIV. The nonnegativity and boundedness of the solutions are proven. The model admits two equilibria, infection-free equilibrium E P 0 and endemic equilibrium E P 1 . We derive the basic reproduction number R 0 , which determines the existence and stability of the two equilibria. The global stability of each equilibrium is proven by utilizing the Lyapunov function and LaSalle’s invariance principle. We prove that if R 0 < 1 , then E P 0 is globally asymptotically stable, and if R 0 > 1 , then E P 1 is globally asymptotically stable. These theoretical results are illustrated by numerical simulations. The effect of impairment of B-cell functions, time delays, and antiviral treatment on the HIV dynamics are studied. We show that if the functions of B-cells are impaired, then the concentration of HIV is increased in the plasma. Moreover, we observe that the time delay has a similar effect to drug efficacy. This gives some impression for developing a new class of treatments to increase the delay period and then suppress the HIV replication.

Herpes simplex virus-2 dynamics as a probe to measure the extremely rapid and spatially localized tissue-resident T-cell response

Herpes simplex virus-2 infection is characterized by frequent episodic shedding in the genital tract. Expansion in HSV-2 viral load early during episodes is extremely rapid. However, the virus invariably peaks within 18 hours and is eliminated nearly as quickly. A critical feature of HSV-2 shedding episodes is their heterogeneity. Some episodes peak at 10⁸ HSV DNA copies, last for weeks due to frequent viral re-expansion, and lead to painful ulcers, while others only reach 10³ HSV DNA copies and are eliminated within hours and without symptoms. Within single micro-environments of infection, tissue-resident CD8+ T cells (T_RM ) appear to contain infection within a few days. Here, we review components of T_RM biology relevant to immune surveillance between HSV-2 shedding episodes and containment of infection upon detection of HSV-2 cognate antigen. We then describe the use of mathematical models to correlate large spatial gradients in T_RM density with the heterogeneity of observed shedding within a single person. We describe how models have been leveraged for clinical trial simulation, as well as future plans to model the interactions of multiple cellular subtypes within mucosa, predict the mechanism of action of therapeutic vaccines, and describe the dynamics of 3-dimensional infection environment during the natural evolution of an HSV-2 lesion.

Antiviral interferon response at single-cell resolution

Experimental studies of the innate immune response of mammalian cells to viruses reveal pervasive heterogeneity at the level of single cells. Interferons are induced only in a fraction of virus-infected cells; subsequently a fraction of cells exposed to interferons upregulate interferon-stimulated genes. Nevertheless, quantitative experiments and linked mathematical models show that the interferon response can be effective in curbing viral spread through two distinct mechanisms. First, paracrine interferon signals from scattered source cells can protect many uninfected cells, and the self-amplification of interferon production might serve to calibrate response amplitude to strength of viral infection. Second, models of the tug-of-war between viral replication and the innate interferon response imply a pivotal role of interferon action on already infected cells in curbing viral spread, through effectively lowering virus replication rate. This finding is in line with the observation that several pathogenic viruses selectively abrogate interferon action on infected cells. Thus, interferons may delay viral spread in acute infections by acting as sentinels, warning uninfected cells of imminent danger, or as negative feedback regulators of virus replication in infected cells. The timing of the interferon response relative to the onset of viral replication is critical for its effectiveness in curbing viral spread.

Virus dynamics and phyloanatomy: Merging population dynamic and phylogenetic approaches

In evolutionary biology and epidemiology, phylodynamic methods are widely used to infer population biological characteristics, such as the rates of replication, death, migration, or, in the epidemiological context, pathogen spread. More recently, these methods have been used to elucidate the dynamics of viruses within their hosts. Especially the application of phylogeographic approaches has the potential to shed light on anatomical colonization pathways and the exchange of viruses between distinct anatomical compartments. We and others have termed this phyloanatomy. Here, we review the promise and challenges of phyloanatomy, and compare them to more classical virus dynamics and population genetic approaches. We argue that the extremely strong selection pressures that exist within the host may represent the main obstacle to reliable phyloanatomic analysis.

Why viruses sometimes disperse in groups?†

Many organisms disperse in groups, yet this process is understudied in viruses. Recent work, however, has uncovered different types of collective infectious units, all of which lead to the joint delivery of multiple viral genome copies to target cells, favoring co-infections. Collective spread of viruses can occur through widely different mechanisms, including virion aggregation driven by specific extracellular components, cloaking inside lipid vesicles, encasement in protein matrices, or binding to cell surfaces. Cell-to-cell viral spread, which allows the transmission of individual virions in a confined environment, is yet another mode of clustered virus dissemination. Nevertheless, the selective advantages of dispersing in groups remain poorly understood in most cases. Collective dispersal might have emerged as a means of sharing efficacious viral transmission vehicles. Alternatively, increasing the cellular multiplicity of infection may confer certain short-term benefits to viruses, such as overwhelming antiviral responses, avoiding early stochastic loss of viral components required for initiating infection, or complementing genetic defects present in different viral genomes. However, increasing infection multiplicity may also entail long-term costs, such as mutation accumulation and the evolution of defective particles or other types of cheater viruses. These costs and benefits, in turn, should depend on the genetic relatedness among collective infectious unit members. Establishing the genetic basis of collective viral dispersal and performing controlled experiments to pinpoint fitness effects at different spatial and temporal scales should help us clarify the implications of these spread modes for viral fitness, pathogenicity, and evolution.

The effect of genetic complementation on the fitness and diversity of viruses spreading as collective infectious units

Viruses can spread collectively using different types of structures such as extracellular vesicles, virion aggregates, polyploid capsids, occlusion bodies, and even cells that accumulate virions at their surface, such as bacteria and dendritic cells. Despite the mounting evidence for collective spread, its implications for viral fitness and diversity remain poorly understood. It has been postulated that, by increasing the cellular multiplicity of infection, collective spread could enable mutually beneficial interactions among different viral genetic variants. One such interaction is genetic complementation, whereby deleterious mutations carried by different genomes are compensated. Here, we used simulations to evaluate whether complementation is likely to increase the fitness of viruses spreading collectively. We show that complementation among co-spreading viruses initially buffers the deleterious effects of mutations, but has no positive effect on mean population fitness over the long term, and even promotes error catastrophe at high mutation rates. Additionally, we found that collective spread increases the risk of invasion by social cheaters such as defective interfering particles. We also show that mutation accumulation depends on the type of collective infectious units considered. Co-spreading viral genomes produced in the same cell (e.g. extracellular vesicles, polyploid capsids, occlusion bodies) should exhibit higher genetic relatedness than groups formed extracellularly by viruses released from different cells (aggregates, binding to bacterial or dendritic cell surfaces), and we found that increased relatedness limits the adverse effects of complementation as well cheater invasion risk. Finally, we found that the costs of complementation can be offset by recombination. Based on our results, we suggest that alternative factors promoting collective spread should be considered.

The evolution of collective infectious units in viruses

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Many viruses disperse in groups, as part of collective infectious units (CIUs).

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We modelled different factors that could influence the evolution of CIUs.

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Group infectivity benefits favoured CIUs, especially if CIUs were more efficient.

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Defective genomes did not favour or disfavour CIUs.

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Defective interfering genomes disfavoured CIUs.

Viruses frequently spread among cells or hosts in groups, with multiple viral genomes inside the same infectious unit. These collective infectious units can consist of multiple viral genomes inside the same virion, or multiple virions inside a larger structure such as a vesicle. Collective infectious units deliver multiple viral genomes to the same cell simultaneously, which can have important implications for viral pathogenesis, antiviral resistance, and social evolution. However, little is known about why some viruses transmit in collective infectious units, whereas others do not. We used a simple evolutionary approach to model the potential costs and benefits of transmitting in a collective infectious unit. We found that collective infectious units could be favoured if cells infected by multiple viral genomes were significantly more productive than cells infected by just one viral genome, and especially if there were also efficiency benefits to packaging multiple viral genomes inside the same infectious unit. We also found that if some viral sequences are defective, then collective infectious units could evolve to become very large, but that if these defective sequences interfered with wild-type virus replication, then collective infectious units were disfavoured.

Building a mechanistic mathematical model of hepatitis C virus entry

The mechanism by which hepatitis C virus (HCV) gains entry into cells is a complex one, involving a broad range of host proteins. Entry is a critical phase of the viral lifecycle, and a potential target for therapeutic or vaccine-mediated intervention. However, the mechanics of HCV entry remain poorly understood. Here we describe a novel computational model of viral entry, encompassing the relationship between HCV and the key host receptors CD81 and SR-B1. We conduct experiments to thoroughly quantify the influence of an increase or decrease in receptor availability upon the extent of viral entry. We use these data to build and parameterise a mathematical model, which we then validate by further experiments. Our results are consistent with sequential HCV-receptor interactions, whereby initial interaction between the HCV E2 glycoprotein and SR-B1 facilitates the accumulation CD81 receptors, leading to viral entry. However, we also demonstrate that a small minority of viruses can achieve entry in the absence of SR-B1. Our model estimates the impact of the different obstacles that viruses must surmount to achieve entry; among virus particles attaching to the cell surface, around one third of viruses accumulate sufficient CD81 receptors, of which 4-8% then complete the subsequent steps to achieve productive infection. Furthermore, we make estimates of receptor stoichiometry; in excess of 10 receptors are likely to be required to achieve viral entry. Our model provides a tool to investigate the entry characteristics of HCV variants and outlines a framework for future quantitative studies of the multi-receptor dynamics of HCV entry.

Epithelial stratification shapes infection dynamics

Infections of stratified epithelia contribute to a large group of common diseases, such as dermatological conditions and sexually transmitted diseases. To investigate how epithelial structure affects infection dynamics, we develop a general ecology-inspired model for stratified epithelia. Our model allows us to simulate infections, explore new hypotheses and estimate parameters that are difficult to measure with tissue cell cultures. We focus on two contrasting pathogens: Chlamydia trachomatis and Human papillomaviruses (HPV). Using cervicovaginal parameter estimates, we find that key infection symptoms can be explained by differential interactions with the layers, while clearance and pathogen burden appear to be bottom-up processes. Cell protective responses to infections (e.g. mucus trapping) generally lowered pathogen load but there were specific effects based on infection strategies. Our modeling approach opens new perspectives for 3D tissue culture experimental systems of infections and, more generally, for developing and testing hypotheses related to infections of stratified epithelia.

Validated Models of Immune Response to Virus Infection

Viruses are a main cause of disease worldwide and many are without effective therapeutics or vaccines. A lack of understanding about how host responses work to control viral spread is one factor limiting effective management. How different immune components regulate infection dynamics is beginning to be better understood with the help of mathematical models. These models have been key in discriminating between hypotheses and in identifying rates of virus growth and clearance, dynamical control by different host factors and antivirals, and synergistic interactions during multi-pathogen infections. A recent focus in evaluating model predictions in the laboratory and clinic has illuminate the accuracy of models for a variety of viruses and highlighted the critical nature of theoretical approaches in virology. Here, I discuss recent model-driven exploration of host-pathogen interactions that have illustrated the importance of model validation in establishing the model's predictive capability and in defining new biology.

Interplay between reaction and diffusion processes in governing the dynamics of virus infections

Spreading of viral infection in the tissues such as lymph nodes or spleen depends on virus multiplication in the host cells, their transport and on the immune response. Reaction-diffusion systems of equations with delays in cell proliferation and death by apoptosis represent an appropriate model to study this process. The properties of the cells of the immune system and the initial viral load determine the spatiotemporal regimes of infection spreading. Infection can be completely eliminated or it can persist at some level together with a certain chronic immune response in a spatially uniform or oscillatory mode. Finally, the immune cells can be completely exhausted leading to a high viral load persistence in the tissue. It has been found experimentally, that virus proteins can affect the immune cell migration. Our study shows that both the motility of immune cells and the virus infection spreading represented by the diffusion rate coefficients are relevant control parameters determining the fate of virus-host interaction.

Superinfection and cure of infected cells as mechanisms for hepatitis C virus adaptation and persistence

RNA viruses exist as a genetically diverse quasispecies with extraordinary ability to adapt to abrupt changes in the host environment. However, the molecular mechanisms that contribute to their rapid adaptation and persistence in vivo are not well studied. Here, we probe hepatitis C virus (HCV) persistence by analyzing clinical samples taken from subjects who were treated with a second-generation HCV protease inhibitor. Frequent longitudinal viral load determinations and large-scale single-genome sequence analyses revealed rapid antiviral resistance development, and surprisingly, dynamic turnover of dominant drug-resistant mutant populations long after treatment cessation. We fitted mathematical models to both the viral load and the viral sequencing data, and the results provided strong support for the critical roles that superinfection and cure of infected cells play in facilitating the rapid turnover and persistence of viral populations. More broadly, our results highlight the importance of considering viral dynamics and competition at the intracellular level in understanding rapid viral adaptation. Thus, we propose a theoretical framework integrating viral and molecular mechanisms to explain rapid viral evolution, resistance, and persistence despite antiviral treatment and host immune responses.

Mathematical Analysis of Viral Replication Dynamics and Antiviral Treatment Strategies: From Basic Models to Age-Based Multi-Scale Modeling

Viral infectious diseases are a global health concern, as is evident by recent outbreaks of the middle east respiratory syndrome, Ebola virus disease, and re-emerging zika, dengue, and chikungunya fevers. Viral epidemics are a socio-economic burden that causes short- and long-term costs for disease diagnosis and treatment as well as a loss in productivity by absenteeism. These outbreaks and their socio-economic costs underline the necessity for a precise analysis of virus-host interactions, which would help to understand disease mechanisms and to develop therapeutic interventions. The combination of quantitative measurements and dynamic mathematical modeling has increased our understanding of the within-host infection dynamics and has led to important insights into viral pathogenesis, transmission, and disease progression. Furthermore, virus-host models helped to identify drug targets, to predict the treatment duration to achieve cure, and to reduce treatment costs. In this article, we review important achievements made by mathematical modeling of viral kinetics on the extracellular, intracellular, and multi-scale level for Human Immunodeficiency Virus, Hepatitis C Virus, Influenza A Virus, Ebola Virus, Dengue Virus, and Zika Virus. Herein, we focus on basic mathematical models on the population scale (so-called target cell-limited models), detailed models regarding the most important steps in the viral life cycle, and the combination of both. For this purpose, we review how mathematical modeling of viral dynamics helped to understand the virus-host interactions and disease progression or clearance. Additionally, we review different types and effects of therapeutic strategies and how mathematical modeling has been used to predict new treatment regimens.

Early HIV infection predictions: role of viral replication errors

In order to prevent and/or control infections it is necessary to understand their early-time dynamics. However this is precisely the phase of HIV about which the least is known. To investigate the initial stages of HIV infection within a host we have developed a multi-type, continuous-time branching process model. This model is a stochastic extension of the standard viral dynamics model, under the assumption that the number of cell targets for viral infection is constant, biologically reasonable since, during the earliest stages of HIV infection, very few cells are infected relative to their total population size. We use our model to investigate three important clinical characteristics of early HIV infection following intravenous challenge: risk of infection, time to infection clearance (assuming failed infection), and time to infection detection. Our focus is on the impact of errors in viral replication that result in non-infectious virus production on these characteristics. Only a small fraction of circulating virus in any chronically infected individual is capable of infecting susceptible cells: estimates range from 1/10⁴ - 1/10³. Characterization and quantification of the processes by which virus becomes defective remains incomplete. We consider two mechanisms that result in defective virus: (1) Copying errors, i.e., lethal errors in reverse transcription, which introduce mutations into the HIV-1 proviral genome, some of which may cripple the viral genome produced, and (2) Packaging errors, i.e., errors during viral packaging, at the end of the viral replication cycle, which cause defective virus by packaging new virions without, for example, viral RNA or key proteins required for infectivity. We show that assumptions on mechanisms of defective virus production can significantly impact early HIV infection model predictions. For example, the risk of infection is orders of magnitude higher if all defective virus is associated with packaging errors, but infection is predicted to be detectable sooner following HIV exposure if all defective virus is associated with copying errors. Thus, in order to make reliable predictions of risk, clearance time, and detection time, better characterization of viral replication is required.

The dynamics of integration, viral suppression and cell-cell transmission in the development of occult Hepatitis B virus infection

BACKGROUND

Out of several phases of HBV infection, the least understood phase is occult hepatitis B virus infection. The paucity of data due to non-availability of biological tissues and the prerequisite of ultra-sensitive assays for the detection of occult hepatitis B virus infection prompted us to utilize mathematical modeling in determining mechanisms that lead to occult hepatitis B virus infection and characteristics of HBV infection during occult hepatitis B virus infection.

METHODS

We proposed two mathematical models (M1 and M2), considering two different phenomenon for episomal maintenance and accumulation of covalently closed circular DNA (cccDNA) in infected hepatocytes: (i) M1 - recirculation of the relaxed circular DNA/double-stranded linear DNA from cytoplasm to the nucleus, and (ii) M2 - reinfection of infected hepatocytes with virions. We further incorporated the dynamics of integrated Hepatitis B virus DNA (iHBV) to investigate its role in the development of occult hepatitis B virus infection.

RESULTS

The analysis showed that the main mechanism for the spread of infection during occult hepatitis B virus infection is cell-to-cell transmission and not cell-free virus transmission. A significant viral suppression (of at least 99% from its peak production values) was essential but not sufficient in the development of occult hepatitis B virus infection under M1; however under M2, the viral suppression was neither sufficient nor essential as the inhibition of the production of HBsAg without viral suppression can also explain the development of occult hepatitis B virus infection. Our analysis also revealed that occult hepatitis B virus infection seropositive cases are more likely to progress into liver cirrhosis compared to occult hepatitis B virus infection seronegative cases. The iHBV was found to be mostly silent (by either being absent or non-productive for HBsAg) during occult hepatitis B virus infection.

CONCLUSION

The viral suppression is neither essential nor sufficient to explain the development of occult hepatitis B virus infection on its own. Not only the viral suppression but the inhibition -of the production and the export of HBsAg from cccDNA and iHBV also plays an important role in the development of occult hepatitis B virus infection. This is the first study, which incorporates the dynamics of iHBV and shows that HBV primarily spreads via cell-cell transmission during occult hepatitis B virus infection.