Stochastic theory of early viral infection: continuous versus burst production of virions

Exploring the treatment of SARS-CoV-2 with modified vesicular stomatitis virus

SARS-CoV-2 caused a global pandemic and is now an endemic virus that will require continued antiviral and vaccine development. A possible new treatment modality was recently suggested that would use vesicular stomatitis virus (VSV) modified to express the ACE2 receptor. Since the modified VSV expresses the cell surface receptor that is used by the SARS-CoV-2 spike protein, the thought is that SARS-CoV-2 virions would bind to the modified VSV and thus be neutralized. Additionally, since SARS-CoV-2 infected cells also express the spike protein, the modified VSV could potentially infect these cells, allowing for its own replication, but also potentially interfering with replication of SARS-CoV-2. This idea has not yet been tested experimentally, but we can investigate the feasibility of this possible treatment theoretically. In this manuscript, we develop a mathematical model of this suggested treatment and explore conditions under which it might be effective. We find that treatment with modified VSV does little to change the SARS-CoV-2 time course except when the treatment is applied at the onset of the SARS-CoV-2 infection at very high doses. In this case, VSV reduces the peak SARS-CoV-2 viral load, but lengthens the duration of the SARS-CoV-2 infection. Thus, we find that modified VSV treatment is unlikely to be effective largely because it does not prevent infection of cells by SARS-CoV-2.

The kinetics of SARS-CoV-2 infection based on a human challenge study

Studying the early events that occur after viral infection in humans is difficult unless one intentionally infects volunteers in a human challenge study. Here, we use data about severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) in such a study in combination with mathematical modeling to gain insights into the relationship between the amount of virus in the upper respiratory tract and the immune response it generates. We propose a set of dynamic models of increasing complexity to dissect the roles of target cell limitation, innate immunity, and adaptive immunity in determining the observed viral kinetics. We introduce an approach for modeling the effect of humoral immunity that describes a decline in infectious virus after immune activation. We fit our models to viral load and infectious titer data from all the untreated infected participants in the study simultaneously. We found that a power-law with a power h < 1 describes the relationship between infectious virus and viral load. Viral replication at the early stage of infection is rapid, with a doubling time of ~2 h for viral RNA and ~3 h for infectious virus. We estimate that adaptive immunity is initiated ~7 to 10 d postinfection and appears to contribute to a multiphasic viral decline experienced by some participants; the viral rebound experienced by other participants is consistent with a decline in the interferon response. Altogether, we quantified the kinetics of SARS-CoV-2 infection, shedding light on the early dynamics of the virus and the potential role of innate and adaptive immunity in promoting viral decline during infection.

Modeling virus-stimulated proliferation of CD4+ T-cell, cell-to-cell transmission and viral loss in HIV infection dynamics

Human immunodeficiency virus (HIV) can persist in infected individuals despite prolonged antiretroviral therapy and it may spread through two modes: virus-to-cell and cell-to-cell transmissions. Understanding viral infection dynamics is pivotal for elucidating HIV pathogenesis. In this study, we incorporate the loss term of virions, and both virus-to-cell and cell-to-cell infection modes into a within-host HIV model, which also takes into consideration the proliferation of healthy target cells stimulated by free viruses. By constructing suitable Lyapunov function and applying geometric methods, we establish global stability results of the infection free equilibrium and the infection persistent equilibrium, respectively. Our findings highlight the crucial role of the basic reproduction number in the threshold dynamics. Moreover, we use the loss rate of virions as the bifurcation parameter to investigate stability switches of the positive equilibrium, local Hopf bifurcation, and its global continuation. Numerical simulations validate our theoretical results, revealing rich viral dynamics including backward bifurcation, saddle-node bifurcation, and bistability phenomenon in the sense that the infection free equilibrium and a limit cycle are both locally asymptotically stable. These insights contribute to a deeper understanding of HIV dynamics and inform the development of effective therapeutic strategies.

Preexisting cell state rather than stochastic noise confers high or low infection susceptibility of human lung epithelial cells to adenovirus

Viruses display large variability across all stages of their life cycle, including entry, gene expression, replication, assembly, and egress. We previously reported that the immediate early adenovirus (AdV) E1A transcripts accumulate in human lung epithelial A549 cancer cells with high variability, mostly independent of the number of incoming viral genomes, but somewhat correlated to the cell cycle state at the time of inoculation. Here, we leveraged the classical Luria-Delbrück fluctuation analysis to address whether infection variability primarily arises from the cell state or stochastic noise. The E1A expression was measured by the expression of green fluorescent protein (GFP) from the endogenous E1A promoter in AdV-C5_E1A-FS2A-GFP and found to be highly correlated with the viral plaque formation, indicating reliability of the reporter virus. As an ensemble, randomly picked clonal A549 cell isolates displayed significantly higher coefficients of variation in the E1A expression than technical noise, indicating a phenotypic variability larger than noise. The underlying cell state determining infection variability was maintained for at least 9 weeks of cell cultivation. Our results indicate that preexisting cell states tune adenovirus infection in favor of the cell or the virus. These findings have implications for antiviral strategies and gene therapy applications.IMPORTANCEViral infections are known for their variability. Underlying mechanisms are still incompletely understood but have been associated with particular cell states, for example, the eukaryotic cell division cycle in DNA virus infections. A cell state is the collective of biochemical, morphological, and contextual features owing to particular conditions or at random. It affects how intrinsic or extrinsic cues trigger a response, such as cell division or anti-viral state. Here, we provide evidence that cell states with a built-in memory confer high or low susceptibility of clonal human epithelial cells to adenovirus infection. Results are reminiscent of the Luria-Delbrück fluctuation test with bacteriophage infections back in 1943, which demonstrated that mutations, in the absence of selective pressure prior to infection, cause infection resistance rather than being a consequence of infection. Our findings of dynamic cell states conferring adenovirus infection susceptibility uncover new challenges for the prediction and treatment of viral infections.

Probability of early infection extinction depends linearly on the virus clearance rate

We provide an in silico study of stochastic viral infection extinction from a pharmacokinetical viewpoint. Our work considers a non-specific antiviral drug that increases the virus clearance rate, and we investigate the effect of this drug on early infection extinction. Infection extinction data are generated by a hybrid multiscale framework that applies both continuous and discrete mathematical approaches. The central result of our paper is the observation, analysis and explanation of a linear relationship between the virus clearance rate and the probability of early infection extinction. The derivation behind this simple relationship is given by merging different mathematical toolboxes.

Decay of HCoV-OC43 infectivity is lower in cell debris-containing media than in fresh culture media

In the quantitative description of viral dynamics within cell cultures and, more broadly, in modeling within-host viral infections, a question that commonly arises is whether the degradation of a fraction of the virus could be disregarded in comparison with the massive synthesis of new viral particles. Surprisingly, quantitative data on the synthesis and degradation rates of RNA viruses in cell cultures are scarce. In this study, we investigated the decay of the human betacoronavirus OC43 (HCoV-OC43) infectivity in cell culture lysates and in fresh media. Our findings revealed a significantly slower viral decay rate in the medium containing lysate cells compared to the fresh medium. This observation suggests that the presence of cellular debris from lysed cells may offer protection or stabilize virions, slowing down their degradation. Moreover, the growth rate of HCoV-OC43 infectivity is significantly higher than degradation as long as there are productive cells in the medium, suggesting that, as a first approximation, degradation can be neglected during early infection.

The reproduction number and its probability distribution for stochastic viral dynamics

We consider stochastic models of individual infected cells. The reproduction number, R, is understood as a random variable representing the number of new cells infected by one initial infected cell in an otherwise susceptible (target cell) population. Variability in R results partly from heterogeneity in the viral burst size (the number of viral progeny generated from an infected cell during its lifetime), which depends on the distribution of cellular lifetimes and on the mechanism of virion release. We analyse viral dynamics models with an eclipse phase: the period of time after a cell is infected but before it is capable of releasing virions. The duration of the eclipse, or the subsequent infectious, phase is non-exponential, but composed of stages. We derive the probability distribution of the reproduction number for these viral dynamics models, and show it is a negative binomial distribution in the case of constant viral release from infectious cells, and under the assumption of an excess of target cells. In a deterministic model, the ultimate in-host establishment or extinction of the viral infection depends entirely on whether the mean reproduction number is greater than, or less than, one, respectively. Here, the probability of extinction is determined by the probability distribution of R, not simply its mean value. In particular, we show that in some cases the probability of infection is not an increasing function of the mean reproduction number.

Viral expansion after transfer is a primary driver of influenza A virus transmission bottlenecks

For many viruses, narrow bottlenecks acting during transmission sharply reduce genetic diversity in a recipient host relative to the donor. Since genetic diversity represents adaptive potential, such losses of diversity are though to limit the opportunity for viral populations to undergo antigenic change and other adaptive processes. Thus, a detailed picture of evolutionary dynamics during transmission is critical to understanding the forces driving viral evolution at an epidemiologic scale. To advance this understanding, we used a novel barcoded virus library and a guinea pig model of transmission to decipher where in the transmission process diversity is lost for influenza A viruses. In inoculated guinea pigs, we show that a high level of viral genetic diversity is maintained across time. Continuity in the barcodes detected furthermore indicates that stochastic effects are not pronounced within inoculated hosts. Importantly, in both aerosol-exposed and direct contact-exposed animals, we observed many barcodes at the earliest time point(s) positive for infectious virus, indicating robust transfer of diversity through the environment. This high viral diversity is short-lived, however, with a sharp decline seen 1-2 days after initiation of infection. Although major losses of diversity at transmission are well described for influenza A virus, our data indicate that events that occur following viral transfer and during the earliest stages of natural infection have a predominant role in this process. This finding suggests that immune selection may have greater opportunity to operate during influenza A transmission than previously recognized.

The effect of random virus failure following cell entry on infection outcome and the success of antiviral therapy

A virus infection can be initiated with very few or even a single infectious virion, and as such can become extinct, i.e. stochastically fail to take hold or spread significantly. There are many ways that a fully competent infectious virion, having successfully entered a cell, can fail to cause a productive infection, i.e. one that yields infectious virus progeny. Though many stochastic models (SMs) have been developed and used to estimate a virus infection's establishment probability, these typically neglect infection failure post virus entry. The SM presented herein introduces parameter [Formula: see text] which corresponds to the probability that a virion's entry into a cell will result in a productive cell infection. We derive an expression for the likelihood of infection establishment in this new SM, and find that prophylactic therapy with an antiviral reducing [Formula: see text] is at least as good or better at decreasing the establishment probability, compared to antivirals reducing the rates of virus production or virus entry into cells, irrespective of the SM parameters. We investigate the difference in the fraction of cells consumed by so-called extinct versus established virus infections, and find that this distinction becomes biologically meaningless as the probability of establishment approaches zero. We explain why the release of virions continuously over an infectious cell's lifespan, rather than as a single burst at the end of the cell's lifespan, does not result in an increased risk of infection extinction. We show, instead, that the number of virus released, not the timing of the release, affects infection establishment and associated critical antiviral efficacy.

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.

Modeling identifies variability in SARS-CoV-2 uptake and eclipse phase by infected cells as principal drivers of extreme variability in nasal viral load in the 48 h post infection

The SARS-CoV-2 coronavirus continues to evolve with scores of mutations of the spike, membrane, envelope, and nucleocapsid structural proteins that impact pathogenesis. Infection data from nasal swabs, nasal PCR assays, upper respiratory samples, ex vivo cell cultures and nasal epithelial organoids reveal extreme variabilities in SARS-CoV-2 RNA titers within and between the variants. Some variabilities are naturally prone to clinical testing protocols and experimental controls. Here we focus on nasal viral load sensitivity arising from the timing of sample collection relative to onset of infection and from heterogeneity in the kinetics of cellular infection, uptake, replication, and shedding of viral RNA copies. The sources of between-variant variability are likely due to SARS-CoV-2 structural protein mutations, whereas within-variant population variability is likely due to heterogeneity in cellular response to that particular variant. With the physiologically faithful, agent-based mechanistic model of inhaled exposure and infection from (Chen et al., 2022), we perform statistical sensitivity analyses of the progression of nasal viral titers in the first 0-48 h post infection, focusing on three kinetic mechanisms. Model simulations reveal shorter latency times of infected cells (including cellular uptake, viral RNA replication, until the onset of viral RNA shedding) exponentially accelerate nasal viral load. Further, the rate of infectious RNA copies shed per day has a proportional influence on nasal viral load. Finally, there is a very weak, negative correlation of viral load with the probability of infection per virus-cell encounter, the model proxy for spike-receptor binding affinity.

A simple model for viral decay dynamics and the distribution of infected cell life spans in SHIV-infected infant rhesus macaques

The dynamics of HIV viral load following the initiation of antiretroviral therapy is not well-described by simple, single-phase exponential decay. Several mathematical models have been proposed to describe its more complex behavior, the most popular of which is two-phase exponential decay. The underlying assumption in two-phase exponential decay is that there are two classes of infected cells with different lifespans. However, with the exception of CD4+ T cells, there is not a consensus on all of the cell types that can become productively infected, and the fit of the two-phase exponential decay to observed data from SHIV.C.CH505 infected infant rhesus macaques was relatively poor. Therefore, we propose a new model for viral decay, inspired by the Gompertz model where the decay rate itself is a dynamic variable. We modify the Gompertz model to include a linear term that modulates the decay rate. We show that this simple model performs as well as the two-phase exponential decay model on HIV and SIV data sets, and outperforms it for the infant rhesus macaque SHIV.C.CH505 infection data set. We also show that by using a stochastic differential equation formulation, the modified Gompertz model can be interpreted as being driven by a population of infected cells with a continuous distribution of cell lifespans, and estimate this distribution for the SHIV.C.CH505-infected infant rhesus macaques. Thus, we find that the dynamics of viral decay in this model of infant HIV infection and treatment may be explained by a distribution of cell lifespans, rather than two distinct cell types.

Stochastic Modelling of HIV-1 Replication in a CD4 T Cell with an IFN Response

A mathematical model of the human immunodeficiency virus Type 1 (HIV-1) life cycle in CD4 T cells was constructed and calibrated. It describes the activation of the intracellular Type I interferon (IFN-I) response and the IFN-induced suppression of viral replication. The model includes viral replication inhibition by interferon-induced antiviral factors and their inactivation by the viral proteins Vpu and Vif. Both deterministic and stochastic model formulations are presented. The stochastic model was used to predict efficiency of IFN-I-induced suppression of viral replication in different initial conditions for autocrine and paracrine effects. The probability of virion excretion for various MOIs and various amounts of IFN-I was evaluated and the statistical properties of the heterogeneity of HIV-1 and IFN-I production characterised.

A hybrid discrete-continuum model of immune responses to SARS-CoV-2 infection in the lung alveolar region, with a focus on interferon induced innate response

We develop a lattice-based, hybrid discrete-continuum modeling framework for SARS-CoV-2 exposure and infection in the human lung alveolar region, or parenchyma, the massive surface area for gas exchange. COVID-19 pneumonia is alveolar infection by the SARS-CoV-2 virus significant enough to compromise gas exchange. The modeling framework orchestrates the onset and progression of alveolar infection, spatially and temporally, beginning with a pre-immunity baseline, upon which we superimpose multiple mechanisms of immune protection conveyed by interferons and antibodies. The modeling framework is tunable to individual profiles, focusing here on degrees of innate immunity, and to the evolving infection-replication properties of SARS-CoV-2 variant strains. The model employs partial differential equations for virion, interferon, and antibody concentrations governed by diffusion in the thin fluid coating of alveolar cells, species and lattice interactions corresponding to sources and sinks for each species, and multiple immune protections signaled by interferons. The spatial domain is a two-dimensional, rectangular lattice of alveolar type I (non-infectable) and type II (infectable) cells with a stochastic, species-concentration-governed, switching dynamics of type II lattice sites from healthy to infected. Once infected, type II cells evolve through three phases: an eclipse phase during which RNA copies (virions) are assembled; a shedding phase during which virions and interferons are released; and then cell death. Model simulations yield the dynamic spread of, and immune protection against, alveolar infection and viral load from initial sites of exposure. We focus in this paper on model illustrations of the diversity of outcomes possible from alveolar infection, first absent of immune protection, and then with varying degrees of four known mechanisms of interferon-induced innate immune protection. We defer model illustrations of antibody protection to future studies. Results presented reinforce previous recognition that interferons produced solely by infected cells are insufficient to maintain a high efficacy level of immune protection, compelling additional mechanisms to clear alveolar infection, such as interferon production by immune cells and adaptive immunity (e.g., T cells). This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Viro-fluidics: Real-time analysis of virus production kinetics at the single-cell level

Real-time visualization and quantification of viruses released by a cell are crucial to further decipher infection processes. Kinetics studies at the single-cell level will circumvent the limitations of bulk assays with asynchronous virus replication. We have implemented a "viro-fluidic" method, which combines microfluidics and virology at single-cell and single-virus resolutions. As an experimental model, we used standard cell lines producing fluorescent HIV-like particles (VLPs). First, to scale the strategy to the single-cell level, we validated a sensitive flow virometry system to detect VLPs in low concentration samples (≥10⁴ VLPs/mL). Then, this system was coupled to a single-cell trapping device to monitor in real-time the VLPs released, one at a time, from single cells under cell culture conditions. Our results revealed an average production rate of 50 VLPs/h/cell similar to the rate estimated for the same cells grown in population. Thus, the virus-producing capacities of the trapped cells were preserved and its real-time monitoring was accurate. Moreover, single-cell analysis revealed a release of VLPs with stochastic bursts with typical time intervals of few minutes, revealing the existence of limiting step(s) in the virus biogenesis process. Our tools can be applied to other pathogens or to extracellular vesicles to elucidate the dissemination mechanisms of these biological nanoparticles.

Modeling recapitulates the heterogeneous outcomes of SARS-CoV-2 infection and quantifies the differences in the innate immune and CD8 T-cell responses between patients experiencing mild and severe symptoms

SARS-CoV-2 infection results in highly heterogeneous outcomes, from cure without symptoms to acute respiratory distress and death. Empirical evidence points to the prominent roles of innate immune and CD8 T-cell responses in determining the outcomes. However, how these immune arms act in concert to elicit the outcomes remains unclear. Here, we developed a mathematical model of within-host SARS-CoV-2 infection that incorporates the essential features of the innate immune and CD8 T-cell responses. Remarkably, by varying the strengths and timings of the two immune arms, the model recapitulated the entire spectrum of outcomes realized. Furthermore, model predictions offered plausible explanations of several confounding clinical observations, including the occurrence of multiple peaks in viral load, viral recrudescence after symptom loss, and prolonged viral positivity. We applied the model to analyze published datasets of longitudinal viral load measurements from patients exhibiting diverse outcomes. The model provided excellent fits to the data. The best-fit parameter estimates indicated a nearly 80-fold stronger innate immune response and an over 200-fold more sensitive CD8 T-cell response in patients with mild compared to severe infection. These estimates provide quantitative insights into the likely origins of the dramatic inter-patient variability in the outcomes of SARS-CoV-2 infection. The insights have implications for interventions aimed at preventing severe disease and for understanding the differences between viral variants.

Viral Aggregation: The Knowns and Unknowns

Viral aggregation is a complex and pervasive phenomenon affecting many viral families. An increasing number of studies have indicated that it can modulate critical parameters surrounding viral infections, and yet its role in viral infectivity, pathogenesis, and evolution is just beginning to be appreciated. Aggregation likely promotes viral infection by increasing the cellular multiplicity of infection (MOI), which can help overcome stochastic failures of viral infection and genetic defects and subsequently modulate their fitness, virulence, and host responses. Conversely, aggregation can limit the dispersal of viral particles and hinder the early stages of establishing a successful infection. The cost–benefit of viral aggregation seems to vary not only depending on the viral species and aggregating factors but also on the spatiotemporal context of the viral life cycle. Here, we review the knowns of viral aggregation by focusing on studies with direct observations of viral aggregation and mechanistic studies of the aggregation process. Next, we chart the unknowns and discuss the biological implications of viral aggregation in their infection cycle. We conclude with a perspective on harnessing the therapeutic potential of this phenomenon and highlight several challenging questions that warrant further research for this field to advance.

Sensitivity of SARS-CoV-2 Life Cycle to IFN Effects and ACE2 Binding Unveiled with a Stochastic Model

Mathematical modelling of infection processes in cells is of fundamental interest. It helps to understand the SARS-CoV-2 dynamics in detail and can be useful to define the vulnerability steps targeted by antiviral treatments. We previously developed a deterministic mathematical model of the SARS-CoV-2 life cycle in a single cell. Despite answering many questions, it certainly cannot accurately account for the stochastic nature of an infection process caused by natural fluctuation in reaction kinetics and the small abundance of participating components in a single cell. In the present work, this deterministic model is transformed into a stochastic one based on a Markov Chain Monte Carlo (MCMC) method. This model is employed to compute statistical characteristics of the SARS-CoV-2 life cycle including the probability for a non-degenerate infection process. Varying parameters of the model enables us to unveil the inhibitory effects of IFN and the effects of the ACE2 binding affinity. The simulation results show that the type I IFN response has a very strong effect on inhibition of the total viral progeny whereas the effect of a 10-fold variation of the binding rate to ACE2 turns out to be negligible for the probability of infection and viral production.

Stochastic investigation of HIV infection and the emergence of drug resistance

Drug-resistant HIV-1 has caused a growing concern in clinic and public health. Although combination antiretroviral therapy can contribute massively to the suppression of viral loads in patients with HIV-1, it cannot lead to viral eradication. Continuing viral replication during sub-optimal therapy (due to poor adherence or other reasons) may lead to the accumulation of drug resistance mutations, resulting in an increased risk of disease progression. Many studies also suggest that events occurring during the early stage of HIV-1 infection (i.e., the first few hours to days following HIV exposure) may determine whether the infection can be successfully established. However, the numbers of infected cells and viruses during the early stage are extremely low and stochasticity may play a critical role in dictating the fate of infection. In this paper, we use stochastic models to investigate viral infection and the emergence of drug resistance of HIV-1. The stochastic model is formulated by a continuous-time Markov chain (CTMC), which is derived based on an ordinary differential equation model proposed by Kitayimbwa et al. that includes both forward and backward mutations. An analytic estimate of the probability of the clearance of HIV infection of the CTMC model near the infection-free equilibrium is obtained by a multitype branching process approximation. The analytical predictions are validated by numerical simulations. Unlike the deterministic dynamics where the basic reproduction number R₀ serves as a sharp threshold parameter (i.e., the disease dies out if R₀<1 and persists if R₀>1), the stochastic models indicate that there is always a positive probability for HIV infection to be eradicated in patients. In the presence of antiretroviral therapy, our results show that the chance of clearance of the infection tends to increase although drug resistance is likely to emerge.

Qualitative behaviour of a stochastic hepatitis C epidemic model in cellular level

In this paper, a mathematical model describing the dynamical of the spread of hepatitis C virus (HCV) at a cellular level with a stochastic noise in the transmission rate is developed from the deterministic model. The unique time-global solution for any positive initial value is served. The Ito's Formula, the suitable Lyapunov function, and other stochastic analysis techniques are used to analyze the model dynamics. The numerical simulations are carried out to describe the analytical results. These results highlight the impact of the noise intensity accelerating the extinction of the disease.

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.

Ascension of Chlamydia is moderated by uterine peristalsis and the neutrophil response to infection

Chlamydia trachomatis is a common sexually transmitted infection that is associated with a range of serious reproductive tract sequelae including in women Pelvic Inflammatory Disease (PID), tubal factor infertility, and ectopic pregnancy. Ascension of the pathogen beyond the cervix and into the upper reproductive tract is thought to be necessary for these pathologies. However, Chlamydia trachomatis does not encode a mechanism for movement on its genome, and so the processes that facilitate ascension have not been elucidated. Here, we evaluate the factors that may influence chlamydial ascension in women. We constructed a mathematical model based on a set of stochastic dynamics to elucidate the moderating factors that might influence ascension of infections in the first month of an infection. In the simulations conducted from the stochastic model, 36% of infections ascended, but only 9% had more than 1000 bacteria ascend. The results of the simulations indicated that infectious load and the peristaltic contractions moderate ascension and are inter-related in impact. Smaller initial loads were much more likely to ascend. Ascension was found to be dependent on the neutrophil response. Overall, our results indicate that infectious load, menstrual cycle timing, and the neutrophil response are critical factors in chlamydial ascension in women.

Targeted antiviral treatment using non-ionizing radiation therapy for SARS-CoV-2 and viral pandemics preparedness: Technique, methods and practical notes for clinical application

OBJECTIVE

Pandemic outbreaks necessitate effective responses to rapidly mitigate and control the spread of disease and eliminate the causative organism(s). While conventional chemical and biological solutions to these challenges are characteristically slow to develop and reach public availability; recent advances in device components operating at Super High Frequency (SHF) bands (3-30 GHz) of the electromagnetic spectrum enable novel approaches to such problems.

METHODS

Based on experimentally documented evidence, a clinically relevant in situ radiation procedure to reduce viral loads in patients is devised and presented. Adapted to the currently available medical device technology to cause viral membrane fracture, this procedure selectively inactivates virus particles by forced oscillations arising from Structure Resonant Energy Transfer (SRET) thereby reducing infectivity and disease progression.

RESULTS

Effective resonant frequencies for pleiomorphic Coronavirus SARS-CoV-2 is calculated to be in the 10-17 GHz range. Using the relation y = -3.308x + 42.9 with x and y representing log10 number of virus particles and the clinical throat swab Ct value respectively; in situ patient-specific exposure duration of ~15x minutes can be utilized to inactivate up to 100% of virus particles in the throat-lung lining, using an irradiation dose of 14.5 ± 1 W/m2; which is within the 200 W/m2 safety standard stipulated by the International Commission on Non-Ionizing Radiation Protection (ICNIRP).

CONCLUSIONS

The treatment is designed to make patients less contagious enhancing faster recoveries and enabling timely control of a spreading pandemic.

ADVANCES IN KNOWLEDGE

The article provides practically applicable parameters for effective clinical adaptation of this technique to the current pandemic at different levels of healthcare infrastructure and disease prevention besides enabling rapid future viral pandemics response.

Mutational pressure by host APOBEC3s more strongly affects genes expressed early in the lytic phase of herpes simplex virus-1 (HSV-1) and human polyomavirus (HPyV) infection

Herpes-Simplex Virus 1 (HSV-1) infects most humans when they are young, sometimes with fatal consequences. Gene expression occurs in a temporal order upon lytic HSV-1 infection: immediate early (IE) genes are expressed, then early (E) genes, followed by late (L) genes. During this infection cycle, the HSV-1 genome has the potential for exposure to APOBEC3 (A3) proteins, a family of cytidine deaminases that cause C>U mutations on single-stranded DNA (ssDNA), often resulting in a C>T transition. We developed a computational model for the mutational pressure of A3 on the lytic cycle of HSV-1 to determine which viral kinetic gene class is most vulnerable to A3 mutations. Using in silico stochastic methods, we simulated the infectious cycle under varying intensities of A3 mutational pressure. We found that the IE and E genes are more vulnerable to A3 than L genes. We validated this model by analyzing the A3 evolutionary footprints in 25 HSV-1 isolates. We find that IE and E genes have evolved to underrepresent A3 hotspot motifs more so than L genes, consistent with greater selection pressure on IE and E genes. We extend this model to two-step infections, such as those of polyomavirus, and find that the same pattern holds for over 25 human Polyomavirus (HPyVs) genomes. Genes expressed earlier during infection are more vulnerable to mutations than those expressed later.

Mechanistic Modeling of SARS-CoV-2 and Other Infectious Diseases and the Effects of Therapeutics

Modern viral kinetic modeling and its application to therapeutics is a field that attracted the attention of the medical, pharmaceutical, and modeling communities during the early days of the AIDS epidemic. Its successes led to applications of modeling methods not only to HIV but a plethora of other viruses, such as hepatitis C virus (HCV), hepatitis B virus and cytomegalovirus, which along with HIV cause chronic diseases, and viruses such as influenza, respiratory syncytial virus, West Nile virus, Zika virus, and severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), which generally cause acute infections. Here we first review the historical development of mathematical models to understand HIV and HCV infections and the effects of treatment by fitting the models to clinical data. We then focus on recent efforts and contributions of applying these models towards understanding SARS-CoV-2 infection and highlight outstanding questions where modeling can provide crucial insights and help to optimize nonpharmaceutical and pharmaceutical interventions of the coronavirus disease 2019 (COVID-19) pandemic. The review is written from our personal perspective emphasizing the power of simple target cell limited models that provided important insights and then their evolution into more complex models that captured more of the virology and immunology. To quote Albert Einstein, "Everything should be made as simple as possible, but not simpler," and this idea underlies the modeling we describe below.

Success of prophylactic antiviral therapy for SARS-CoV-2: Predicted critical efficacies and impact of different drug-specific mechanisms of action

Repurposed drugs that are safe and immediately available constitute a first line of defense against new viral infections. Despite limited antiviral activity against SARS-CoV-2, several drugs are being tested as medication or as prophylaxis to prevent infection. Using a stochastic model of early phase infection, we evaluate the success of prophylactic treatment with different drug types to prevent viral infection. We find that there exists a critical efficacy that a treatment must reach in order to block viral establishment. Treatment by a combination of drugs reduces the critical efficacy, most effectively by the combination of a drug blocking viral entry into cells and a drug increasing viral clearance. Below the critical efficacy, the risk of infection can nonetheless be reduced. Drugs blocking viral entry into cells or enhancing viral clearance reduce the risk of infection more than drugs that reduce viral production in infected cells. The larger the initial inoculum of infectious virus, the less likely is prevention of an infection. In our model, we find that as long as the viral inoculum is smaller than 10 infectious virus particles, viral infection can be prevented almost certainly with drugs of 90% efficacy (or more). Even when a viral infection cannot be prevented, antivirals delay the time to detectable viral loads. The largest delay of viral infection is achieved by drugs reducing viral production in infected cells. A delay of virus infection flattens the within-host viral dynamic curve, possibly reducing transmission and symptom severity. Thus, antiviral prophylaxis, even with reduced efficacy, could be efficiently used to prevent or alleviate infection in people at high risk.

A delayed HIV-1 model with cell-to-cell spread and virus waning

In this paper, we propose and analyse a delayed HIV-1 model with both viral and cellular transmissions and virus waning. We obtain the threshold dynamics of the proposed model, characterized by the basic reproduction number R0 . If R0<1 , the infection-free steady state is globally asymptotically stable; whereas if R0>1 , the system is uniformly persistent. When the delays are positive, we show that the intracellular delays in both viral and cellular infections may lead to stability switches of the infected steady state. Both analytical and numerical results indicate that if the effect of cell-to-cell transmission is ignored, then the risk of HIV-1 infection will be underestimated. Moreover, the viral load of model without virus waning is higher than the one of model with virus waning. These results highlight the important role of two ways of viral transmission and virus waning on HIV-1 infection.

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.

Models of SIV rebound after treatment interruption that involve multiple reactivation events

In order to assess the efficacy of novel HIV-1 treatments leading to a functional cure, the time to viral rebound is frequently used as a surrogate endpoint. The longer the time to viral rebound, the more efficacious the therapy. In support of such an approach, mathematical models serve as a connection between the size of the latent reservoir and the time to HIV-1 rebound after treatment interruption. The simplest of such models assumes that a single successful latent cell reactivation event leads to observable viremia after a period of exponential viral growth. Here we consider a generalization developed by Pinkevych et al. and Hill et al. of this simple model in which multiple reactivation events can occur, each contributing to the exponential growth of the viral load. We formalize and improve the previous derivation of the dynamics predicted by this model, and use the model to estimate relevant biological parameters from SIV rebound data. We confirm a previously described effect of very early antiretroviral therapy (ART) initiation on the rate of recrudescence and the viral load growth rate after treatment interruption. We find that every day ART initiation is delayed results in a 39% increase in the recrudescence rate (95% credible interval: [18%, 62%]), and a 11% decrease of the viral growth rate (95% credible interval: [4%, 20%]). We show that when viral rebound occurs early relative to the viral load doubling time, a model with multiple successful reactivation events fits the data better than a model with only a single successful reactivation event.

Detailed Analyses of Zika Virus Tropism in Culex quinquefasciatus Reveal Systemic Refractoriness

The role of Culex quinquefasciatus in Zika virus transmission has been debated since the epidemic of Zika occurred in the Americas in 2015 to 2016. The majority of studies have found no evidence that C. quinquefasciatus or other Culex species are competent vectors of Zika virus, and the few studies that have proposed Zika vector status for C. quinquefasciatus have relied predominantly on quantitative real-time PCR (qRT-PCR) for viral detection. We assessed the infectious range of pre- and post-epidemic Zika virus isolates in order to classify mosquito samples based on titer infectiousness and demonstrated that two strains of C. quinquefasciatus, including one previously found to be competent, are highly resistant to infection with these Zika isolates compared to Aedes aegypti and are not competent for virus transmission. Further dissection of the dynamics of Zika exposure in both A. aegypti and C. quinquefasciatus revealed that while virus transmission by C. quinquefasciatus is blocked at the levels of the midgut and salivary glands, viral RNA persists in these tissues for prolonged periods post-exposure. We assessed Zika entry dynamics in both Aedes and Culex cells, and our results suggest that Zika virus infection in Culex cells may be blocked downstream of cell entry. These findings strongly suggest that C. quinquefasciatus is not a vector of Zika virus and additionally inform the use of qRT-PCR in vector competence assays as well as our understanding of barriers to arbovirus infection in non-susceptible mosquito species.IMPORTANCE Understanding which mosquito species transmit an emerging arbovirus is critical to effective vector control. During the Zika virus epidemic in 2015 to 2016, Aedes mosquitoes were confirmed as vectors. However, studies addressing the vector status of Culex quinquefasciatus mosquitoes presented conflicting evidence and remain an outstanding source of confusion in the field. Here, we established a robust cell-based assay to identify infectious titers of Zika virus and assessed the virus titers in C. quinquefasciatus by quantitative real-time PCR (qRT-PCR). We found that while low levels of virus were detected in C. quinquefasciatus, these titers did not correspond to infectious virus, and these mosquitoes did not transmit virus in the saliva. We also present evidence that the virus may enter Culex cells before infection is disrupted. Our findings are important for future studies incriminating vector species using qRT-PCR for virus detection and offer new information on how virus transmission is blocked by mosquitoes.

Viral Infection Dynamics Model Based on a Markov Process with Time Delay between Cell Infection and Progeny Production

Many human virus infections including those with the human immunodeficiency virus type 1 (HIV) are initiated by low numbers of founder viruses. Therefore, random effects have a strong influence on the initial infection dynamics, e.g., extinction versus spread. In this study, we considered the simplest (so-called, ‘consensus’) virus dynamics model and incorporated a delay between infection of a cell and virus progeny release from the infected cell. We then developed an equivalent stochastic virus dynamics model that accounts for this delay in the description of the random interactions between the model components. The new model is used to study the statistical characteristics of virus and target cell populations. It predicts the probability of infection spread as a function of the number of transmitted viruses. A hybrid algorithm is suggested to compute efficiently the system dynamics in state space domain characterized by the mix of small and large species densities.

Integrating mathematical models with experimental data to investigate the within-host dynamics of bacterial infections

Bacterial infections still constitute a major cause of mortality and morbidity worldwide. The unavailability of therapeutics, antimicrobial resistance and the chronicity of infections due to incomplete clearance contribute to this phenomenon. Despite the progress in antimicrobial and vaccine development, knowledge about the effect that therapeutics have on the host–bacteria interactions remains incomplete. Insights into the characteristics of bacterial colonization and migration between tissues and the relationship between replication and host- or therapeutically induced killing can enable efficient design of treatment approaches. Recently, innovative experimental techniques have generated data enabling the qualitative characterization of aspects of bacterial dynamics. Here, we argue that mathematical modeling as an adjunct to experimental data can enrich the biological insight that these data provide. However, due to limited interdisciplinary training, efforts to combine the two remain limited. To promote this dialogue, we provide a categorization of modeling approaches highlighting their relationship to data generated by a range of experimental techniques in the area of in vivo bacterial dynamics. We outline common biological themes explored using mathematical models with case studies across all pathogen classes. Finally, this review advocates multidisciplinary integration to improve our mechanistic understanding of bacterial infections and guide the use of existing or new therapies.

Stochastic dynamics of Francisella tularensis infection and replication

We study the pathogenesis of Francisella tularensis infection with an experimental mouse model, agent-based computation and mathematical analysis. Following inhalational exposure to Francisella tularensis SCHU S4, a small initial number of bacteria enter lung host cells and proliferate inside them, eventually destroying the host cell and releasing numerous copies that infect other cells. Our analysis of disease progression is based on a stochastic model of a population of infectious agents inside one host cell, extending the birth-and-death process by the occurrence of catastrophes: cell rupture events that affect all bacteria in a cell simultaneously. Closed expressions are obtained for the survival function of an infected cell, the number of bacteria released as a function of time after infection, and the total bacterial load. We compare our mathematical analysis with the results of agent-based computation and, making use of approximate Bayesian statistical inference, with experimental measurements carried out after murine aerosol infection with the virulent SCHU S4 strain of the bacterium Francisella tularensis, that infects alveolar macrophages. The posterior distribution of the rate of replication of intracellular bacteria is consistent with the estimate that the time between rounds of bacterial division is less than 6 hours in vivo.

Infecting a host cell is required for the replication of many types of bacteria and viruses. In some cases, infected cells release new infectious agents continuously over their lifetime. In others, such as the Francisella tularensis bacterium studied here, they are released in a single burst that coincides with the cell’s death. We show how a stochastic model, the birth-and-death process with catastrophe, can be used to characterise infection in a single cell, thereby allowing us to account for burst events and quantify the kinetics of pathogenesis in the lung, the initial site of infection, as well as in other organs that the infection spreads to. We learn about the parameters of the mathematical model of Francisella tularensis infection making use of the experimental measurements of bacterial loads, together with approximate Bayesian statistical inference methods. The most important parameter describing the pathogenesis is the rate of replication of intracellular bacteria.

Modeling Favipiravir Antiviral Efficacy Against Emerging Viruses: From Animal Studies to Clinical Trials

In 2014, our research network was involved in the evaluation of favipiravir, an anti-influenza polymerase inhibitor, against Ebola virus. In this review, we discuss how mathematical modeling was used, first to propose a relevant dosing regimen in humans, and then to optimize its antiviral efficacy in a nonhuman primate (NHP) model. The data collected in NHPs were finally used to develop a model of Ebola pathogenesis integrating the interactions among the virus, the innate and adaptive immune response, and the action of favipiravir. We conclude the review of this work by discussing how these results are of relevance for future human studies in the context of Ebola virus, but also for other emerging viral diseases for which no therapeutics are available.

Mathematical modeling to reveal breakthrough mechanisms in the HIV Antibody Mediated Prevention (AMP) trials

The ongoing Antibody Mediated Prevention (AMP) trials will uncover whether passive infusion of the broadly neutralizing antibody (bNAb) VRC01 can protect against HIV acquisition. Previous statistical simulations indicate these trials may be partially protective. In that case, it will be crucial to identify the mechanism of breakthrough infections. To that end, we developed a mathematical modeling framework to simulate the AMP trials and infer the breakthrough mechanisms using measurable trial outcomes. This framework combines viral dynamics with antibody pharmacokinetics and pharmacodynamics, and will be generally applicable to forthcoming bNAb prevention trials. We fit our model to human viral load data (RV217). Then, we incorporated VRC01 neutralization using serum pharmacokinetics (HVTN 104) and in vitro pharmacodynamics (LANL CATNAP database). We systematically explored trial outcomes by reducing in vivo potency and varying the distribution of sensitivity to VRC01 in circulating strains. We found trial outcomes could be used in a clinical trial regression model (CTRM) to reveal whether partially protective trials were caused by large fractions of VRC01-resistant (IC50>50 μg/mL) circulating strains or rather a global reduction in VRC01 potency against all strains. The former mechanism suggests the need to enhance neutralizing antibody breadth; the latter suggests the need to enhance VRC01 delivery and/or in vivo binding. We will apply the clinical trial regression model to data from the completed trials to help optimize future approaches for passive delivery of anti-HIV neutralizing antibodies.

Infusions of broadly neutralizing antibodies are currently being tested as a novel HIV prevention modality. To help interpret the results of these antibody mediated prevention (AMP) studies we developed a mathematical modeling framework. The approach combines antibody potency and drug levels with models of HIV viral dynamics, which will be generally applicable to future studies. Through simulating these clinical trials, we found trial outcomes can be used in combination to infer whether breakthrough infections are caused by large fractions of antibody-resistant circulating strains or some reduction in potency against all strains. This distinction helps to focus future trials on enhancing neutralizing antibody breadth or antibody delivery and/or in vivo binding.

Principles Governing Establishment versus Collapse of HIV-1 Cellular Spread

A population at low census might go extinct or instead transition into exponential growth to become firmly established. Whether this pivotal event occurs for a within-host pathogen can be the difference between health and illness. Here, we define the principles governing whether HIV-1 spread among cells fails or becomes established by coupling stochastic modeling with laboratory experiments. Following ex vivo activation of latently infected CD4 T cells without de novo infection, stochastic cell division and death contributes to high variability in the magnitude of initial virus release. Transition to exponential HIV-1 spread often fails due to release of an insufficient amount of replication-competent virus. Establishment of exponential growth occurs when virus produced from multiple infected cells exceeds a critical population size. We quantitatively define the crucial transition to exponential viral spread. Thwarting this process would prevent HIV transmission or rebound from the latent reservoir.

A personalized computational model of edema formation in myocarditis based on long-axis biventricular MRI images

Background

Myocarditis is defined as the inflammation of the myocardium, i.e. the cardiac muscle. Among the reasons that lead to this disease, we may include infections caused by a virus, bacteria, protozoa, fungus, and others. One of the signs of the inflammation is the formation of edema, which may be a consequence of the interaction between interstitial fluid dynamics and immune response. This complex physiological process was mathematically modeled using a nonlinear system of partial differential equations (PDE) based on porous media approach. By combing a model based on Biot’s poroelasticity theory with a model for the immune response we developed a new hydro-mechanical model for inflammatory edema. To verify this new computational model, T2 parametric mapping obtained by Magnetic Resonance (MR) imaging was used to identify the region of edema in a patient diagnosed with unspecific myocarditis.

Results

A patient-specific geometrical model was created using MRI images from the patient with myocarditis. With this model, edema formation was simulated using the proposed hydro-mechanical mathematical model in a two-dimensional domain. The computer simulations allowed us to correlate spatiotemporal dynamics of representative cells of the immune systems, such as leucocytes and the pathogen, with fluid accumulation and cardiac tissue deformation.

Conclusions

This study demonstrates that the proposed mathematical model is a very promising tool to better understand edema formation in myocarditis. Simulations obtained from a patient-specific model reproduced important aspects related to the formation of cardiac edema, its area, position, and shape, and how these features are related to immune response.

Within-host modeling of blood-stage malaria

Malaria infection continues to be a major health problem worldwide and drug resistance in the major human parasite species, Plasmodium falciparum, is increasing in South East Asia. Control measures including novel drugs and vaccines are in development, and contributions to the rational design and optimal usage of these interventions are urgently needed. Infection involves the complex interaction of parasite dynamics, host immunity, and drug effects. The long life cycle (48 hours in the common human species) and synchronized replication cycle of the parasite population present significant challenges to modeling the dynamics of Plasmodium infection. Coupled with these, variation in immune recognition and drug action at different life cycle stages leads to further complexity. We review the development and progress of "within-host" models of Plasmodium infection, and how these have been applied to understanding and interpreting human infection and animal models of infection.

Modeling HIV Dynamics Under Combination Therapy with Inducers and Antibodies

A mathematical model is proposed to simulate the "shock-kill" strategy where broadly neutralizing antibodies (bNAbs) are injected with a combination of HIV latency activators to reduce persistent HIV reservoirs. The basic reproductive ratio of virus is computed to extrapolate how the combinational therapy of inducers and antibodies affects the persistence of HIV infection. Numerical simulations demonstrate that a proper combination of inducers and bNAbs can drive the basic reproductive ratio below unity. Interestingly, it is found that a longer dosage interval leads to the higher HIV survival opportunity and a smaller dosage interval is preferred, which is fundamental to design an optimal therapeutic scheme. Further simulations reveal the conditions under which the joint therapy of inducer and antibodies induces a large extension of viral rebound time, which highlights the mechanism of delayed viral rebound from the experiment (Halper-Stromberg et al. in Cell 158:989-999, 2014). Optimal time for cessation of treatment is also analyzed to aid practical applications.

Effect of stochasticity on coinfection dynamics of respiratory viruses

Background

Respiratory viral infections are a leading cause of mortality worldwide. As many as 40% of patients hospitalized with influenza-like illness are reported to be infected with more than one type of virus. However, it is not clear whether these infections are more severe than single viral infections. Mathematical models can be used to help us understand the dynamics of respiratory viral coinfections and their impact on the severity of the illness. Most models of viral infections use ordinary differential equations (ODE) that reproduce the average behavior of the infection, however, they might be inaccurate in predicting certain events because of the stochastic nature of viral replication cycle. Stochastic simulations of single virus infections have shown that there is an extinction probability that depends on the size of the initial viral inoculum and parameters that describe virus-cell interactions. Thus the coinfection dynamics predicted by the ODE might be difficult to observe in reality.

Results

In this work, a continuous-time Markov chain (CTMC) model is formulated to investigate probabilistic outcomes of coinfections. This CTMC model is based on our previous coinfection model, expressed in terms of a system of ordinary differential equations. Using the Gillespie method for stochastic simulation, we examine whether stochastic effects early in the infection can alter which virus dominates the infection.

Conclusions

We derive extinction probabilities for each virus individually as well as for the infection as a whole. We find that unlike the prediction of the ODE model, for similar initial growth rates stochasticity allows for a slower growing virus to out-compete a faster growing virus.

Electronic supplementary material

The online version of this article (10.1186/s12859-019-2793-6) contains supplementary material, which is available to authorized users.

Competing evolutionary paths in growing populations with applications to multidrug resistance

Investigating the emergence of a particular cell type is a recurring theme in models of growing cellular populations. The evolution of resistance to therapy is a classic example. Common questions are: when does the cell type first occur, and via which sequence of steps is it most likely to emerge? For growing populations, these questions can be formulated in a general framework of branching processes spreading through a graph from a root to a target vertex. Cells have a particular fitness value on each vertex and can transition along edges at specific rates. Vertices represent cell states, say genotypes or physical locations, while possible transitions are acquiring a mutation or cell migration. We focus on the setting where cells at the root vertex have the highest fitness and transition rates are small. Simple formulas are derived for the time to reach the target vertex and for the probability that it is reached along a given path in the graph. We demonstrate our results on several scenarios relevant to the emergence of drug resistance, including: the orderings of resistance-conferring mutations in bacteria and the impact of imperfect drug penetration in cancer.

How long does it take for a treatment naive, growing bacterial colony to be able to survive exposure to a cocktail of antibiotics? En route to multidrug resistance, what order did the drugs become impotent in? Questions such as these that pertain to the emergence of a significant cell type in a growing population arise frequently. They are often investigated via mathematical modelling but biologically insightful results are challenging to obtain. Here we outline a general framework of a stochastically growing population spreading through a graph to study such questions and provide simple formulas as answers. The significant cell type appears upon the population reaching a target vertex. Due to their simplicity, the derived formulas are widely accessible and can be used to guide and develop intuition on a range of biological scenarios. We demonstrate this on several settings including: how a region where drugs cannot penetrate affects the emergence of resistance, and, the ordering of mutations that leads to drugs being ineffective.

The Utility of Efavirenz-based Prophylaxis Against HIV Infection. A Systems Pharmacological Analysis

Pre-exposure prophylaxis (PrEP) is considered one of the five “pillars” by UNAIDS to reduce HIV transmission. Moreover, it is a tool for female self-protection against HIV, making it highly relevant to sub-Saharan regions, where women have the highest infection burden. To date, Truvada is the only medication for PrEP. However, the cost of Truvada limits its uptake in resource-constrained countries. Similarly, several currently investigated, patent-protected compounds may be unaffordable in these regions. We set out to explore the potential of the patent-expired antiviral efavirenz (EFV) as a cost-efficient PrEP alternative. A population pharmacokinetic model utilizing data from the ENCORE1 study was developed. The model was refined for metabolic autoinduction. We then explored EFV cellular uptake mechanisms, finding that it is largely determined by plasma protein binding. Next, we predicted the prophylactic efficacy of various EFV dosing schemes after exposure to HIV using a stochastic simulation framework. We predicted that plasma concentrations of 11, 36, 1287 and 1486ng/mL prevent 90% sexual transmissions with wild type and Y181C, K103N and G190S mutants, respectively. Trough concentrations achieved after 600 mg once daily dosing (median: 2017 ng/mL, 95% CI:445–9830) and after reduced dose (400 mg) efavirenz (median: 1349ng/mL, 95% CI: 297–6553) provided complete protection against wild-type virus and the Y181C mutant, and median trough concentrations provided about 90% protection against the K103N and G190S mutants. As reduced dose EFV has a lower toxicity profile, we predicted the reduction in HIV infection when 400 mg EFV-PrEP was poorly adhered to, when it was taken “on demand” and as post-exposure prophylaxis (PEP). Once daily EFV-PrEP provided 99% protection against wild-type virus, if ≥50% of doses were taken. PrEP “on demand” provided complete protection against wild-type virus and prevented ≥81% infections in the mutants. PEP could prevent >98% infection with susceptible virus when initiated within 24 h after virus exposure and continued for at least 9 days. We predict that 400 mg oral EFV may provide superior protection against wild-type HIV. However, further studies are warranted to evaluate EFV as a cost-efficient alternative to Truvada. Predicted prophylactic concentrations may guide release kinetics of EFV long-acting formulations for clinical trial design.

Mechanistic framework predicts drug-class specific utility of antiretrovirals for HIV prophylaxis

Currently, there is no effective vaccine to halt HIV transmission. However, pre-exposure prophylaxis (PrEP) with the drug combination Truvada can substantially decrease HIV transmission in individuals at risk. Despite its benefits, Truvada-based PrEP is expensive and needs to be taken once-daily, which often leads to inadequate adherence and incomplete protection. These deficits may be overcome by next-generation PrEP regimen, including currently investigated long-acting formulations, or patent-expired drugs. However, poor translatability of animal- and ex vivo/in vitro experiments, and the necessity to conduct long-term (several years) human trials involving considerable sample sizes (N>1000 individuals) are major obstacles to rationalize drug-candidate selection. We developed a prophylaxis modelling tool that mechanistically considers the mode-of-action of all available drugs. We used the tool to screen antivirals for their prophylactic utility and identify lower bound effective concentrations that can guide dose selection in PrEP trials. While in vitro measurable drug potency usually guides PrEP trial design, we found that it may over-predict PrEP potency for all drug classes except reverse transcriptase inhibitors. While most drugs displayed graded concentration-prophylaxis profiles, protease inhibitors tended to switch between none- and complete protection. While several treatment-approved drugs could be ruled out as PrEP candidates based on lack-of-prophylactic efficacy, darunavir, efavirenz, nevirapine, etravirine and rilpivirine could more potently prevent infection than existing PrEP regimen (Truvada). Notably, some drugs from this candidate set are patent-expired and currently neglected for PrEP repurposing. A next step is to further trim this candidate set by ruling out compounds with ominous safety profiles, to assess different administration schemes in silico and to test the remaining candidates in human trials.

Pre-exposure prophylaxis (PrEP) is a novel, promising strategy to halt HIV transmission. PrEP with Truvada can substantially decrease the risk of infection. However, individuals often inadequately adhere to the once-daily regimen and the drug is expensive. These shortcomings may be overcome by next-generation PrEP compounds, including long-acting formulations. However, poor translatability of animal- and ex vivo/in vitro experiments, and difficulties in conducting long-term trials involving considerable sample sizes (N > 1000 individuals) make drug-candidate selection and optimization of administration schemes costly and often infeasible. We developed a simulation tool that mechanistically considers the mode-of-action of all antivirals. We used the tool to screen all available antivirals for their prophylactic utility and identified lower bound effective concentrations for designing PrEP dosing regimen in clinical trials. We found that in vitro measured drug potency may over-predict PrEP potency, for all antiviral classes except reverse transcriptase inhibitors. We could rule out a number of antivirals for PrEP repurposing and predicted that darunavir, efavirenz, nevirapine, etravirine and rilpivirine provide complete protection at clinically relevant concentrations. Further trimming of this candidate set by compound-safety and by assessing different implementation schemes is envisaged.

The effect of delay in viral production in within-host models during early infection

Delay in viral production may have a significant impact on the early stages of infection. During the eclipse phase, the time from viral entry until active production of viral particles, no viruses are produced. This delay affects the probability that a viral infection becomes established and timing of the peak viral load. Deterministic and stochastic models are formulated with either multiple latent stages or a fixed delay for the eclipse phase. The deterministic model with multiple latent stages approaches in the limit the model with a fixed delay as the number of stages approaches infinity. The deterministic model framework is used to formulate continuous-time Markov chain and stochastic differential equation models. The probability of a minor infection with rapid viral clearance as opposed to a major full-blown infection with a high viral load is estimated from a branching process approximation of the Markov chain model and the results are confirmed through numerical simulations. In addition, parameter values for influenza A are used to numerically estimate the time to peak viral infection and peak viral load for the deterministic and stochastic models. Although the average length of the eclipse phase is the same in each of the models, as the number of latent stages increases, the numerical results show that the time to viral peak and the peak viral load increase.

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.

Correlation Between Anti-gp41 Antibodies and Virus Infectivity Decay During Primary HIV-1 Infection

Recent experiments have suggested that the infectivity of simian immunodeficiency virus (SIV) and human immunodeficiency virus type-1 (HIV-1) in plasma decreases over time during primary infection. Because anti-gp41 antibodies are produced early during HIV-1 infection and form antibody-virion complexes, we studied if such early HIV-1 specific antibodies are correlated with the decay in HIV-1 infectivity. Using a viral dynamic model that allows viral infectivity to decay and frequent early viral load data obtained from 6 plasma donors we estimate that HIV-1 infectivity begins to decay after about 2 weeks of infection. The length of this delay is consistent with the time before antibody-virion complexes were detected in the plasma of these donors and is correlated (p = 0.023, r = 0.87) with the time for antibodies to be first detected in plasma. Importantly, we identify that the rate of infectivity decay is significantly correlated with the rate of increase in plasma anti-gp41 IgG concentration (p = 0.046, r = 0.82) and the increase in IgM+IgG anti-gp41 concentration (p = 8.37 × 10⁻⁴, r = 0.98). Furthermore, we found that the viral load decay after the peak did not have any significant correlation with the rate of anti-gp41 IgM or IgG increase. These results indicate that early anti-gp41 antibodies may cause viral infectivity decay, but may not contribute significantly to controlling post-peak viral load, likely due to insufficient quantity or affinity. Our findings may be helpful to devise strategies, including antibody-based vaccines, to control acute HIV-1 infection.

A Bistable Switch in Virus Dynamics Can Explain the Differences in Disease Outcome Following SIV Infections in Rhesus Macaques

Experimental studies have shown that the size and infectious-stage of viral inoculum influence disease outcomes in rhesus macaques infected with simian immunodeficiency virus. The possible contribution to disease outcome of antibody developed after transmission and/or present in the inoculum in free or bound form is not understood. In this study, we develop a mathematical model of virus-antibody immune complex formation and use it to predict their role in transmission and protection. The model exhibits a bistable switch between clearance and persistence states. We fitted it to temporal virus data and estimated the parameter values for free virus infectivity rate and antibody carrying capacity for which the model transitions between virus clearance and persistence when the initial conditions (in particular the ratio of immune complexes to free virus) vary. We used these results to quantify the minimum virus amount in the inoculum needed to establish persistent infections in the presence and absence of protective antibodies.

Hybrid stochastic framework predicts efficacy of prophylaxis against HIV: An example with different dolutegravir prophylaxis schemes

To achieve the 90-90-90 goals set by UNAIDS, the number of new HIV infections needs to decrease to approximately 500,000 by 2020. One of the 'five pillars' to achieve this goal is pre-exposure prophylaxis (PrEP). Truvada (emtricitabine-tenofovir) is currently the only medication approved for PrEP. Despite its advantages, Truvada is costly and requires individuals to adhere to the once-daily regimen. To improve PrEP, many next-generation regimen, including long-acting formulations, are currently investigated. However, pre-clinical testing may not guide candidate selection, since it often fails to translate into clinical efficacy. On the other hand, quantifying prophylactic efficacy in the clinic is ethically problematic and requires to conduct long (years) and large (N>1000 individuals) trials, precluding systematic evaluation of candidates and deployment strategies. To prioritize- and help design PrEP regimen, tools are urgently needed that integrate pharmacological-, viral- and host factors determining prophylactic efficacy. Integrating the aforementioned factors, we developed an efficient and exact stochastic simulation approach to predict prophylactic efficacy, as an example for dolutegravir (DTG). Combining the population pharmacokinetics of DTG with the stochastic framework, we predicted that plasma concentrations of 145.18 and 722.23nM prevent 50- and 90% sexual transmissions respectively. We then predicted the reduction in HIV infection when DTG was used in PrEP, PrEP 'on demand' and post-exposure prophylaxis (PEP) before/after virus exposure. Once daily PrEP with 50mg oral DTG prevented 99-100% infections, and 85% of infections when 50% of dosing events were missed. PrEP 'on demand' prevented 79-84% infections and PEP >80% when initiated within 6 hours after virus exposure and continued for as long as possible. While the simulation framework can easily be adapted to other PrEP candidates, our simulations indicated that oral 50mg DTG is non-inferior to Truvada. Moreover, the predicted 90% preventive concentrations can guide release kinetics of currently developed DTG nano-formulations.

Population extinction under bursty reproduction in a time-modulated environment

In recent years nondemographic variability has been shown to greatly affect dynamics of stochastic populations. For example, nondemographic noise in the form of a bursty reproduction process with an a priori unknown burst size, or environmental variability in the form of time-varying reaction rates, have been separately found to dramatically impact the extinction risk of isolated populations. In this work we investigate the extinction risk of an isolated population under the combined influence of these two types of nondemographic variation. Using the so-called momentum-space Wentzel-Kramers-Brillouin (WKB) approach and accounting for the explicit time dependence in the reaction rates, we arrive at a set of time-dependent Hamilton equations. To this end, we evaluate the population's extinction risk by finding the instanton of the time-perturbed Hamiltonian numerically, whereas analytical expressions are presented in particular limits using various perturbation techniques. We focus on two classes of time-varying environments: periodically varying rates corresponding to seasonal effects and a sudden decrease in the birth rate corresponding to a catastrophe. All our theoretical results are tested against numerical Monte Carlo simulations with time-dependent rates and also against a numerical solution of the corresponding time-dependent Hamilton equations.

Effects of spatiotemporal HSV-2 lesion dynamics and antiviral treatment on the risk of HIV-1 acquisition

Patients with Herpes Simplex Virus-2 (HSV-2) infection face a significantly higher risk of contracting HIV-1. This is thought to be due to herpetic lesions serving as entry points for HIV-1 and tissue-resident CD4+ T cell counts increasing during HSV-2 lesional events. We have created a stochastic and spatial mathematical model describing the dynamics of HSV-2 infection and immune response in the genital mucosa. Using our model, we first study the dynamics of a developing HSV-2 lesion. We then use our model to quantify the risk of infection with HIV-1 following sexual exposure in HSV-2 positive women. Untreated, we find that HSV-2 infected women are up to 8.6 times more likely to acquire HIV-1 than healthy patients. However, when including the effects of the HSV-2 antiviral drug, pritelivir, the risk of HIV-1 infection is predicted to decrease by up to 35%, depending on drug dosage. We estimate the relative importance of decreased tissue damage versus decreased CD4+ cell presence in determining the effectiveness of pritelivir in reducing HIV-1 infection. Our results suggest that clinical trials should be performed to evaluate the effectiveness of pritelivir or similar agents in preventing HIV-1 infection in HSV-2 positive women.

The risk of contracting HIV-1 is significantly higher in people who have genital HSV-2 infections. Here, we put forward a new mathematical model to describe HSV-2 infection and the process of HIV-1 infection in the genital mucosa surrounding HSV-2 lesions. We determine how the characteristics of HSV-2 infection affect the risk of HIV-1 infection, and determine whether reducing the severity of HSV-2 symptoms with antiviral drugs can be expected to decrease the risk of HIV-1 infection. We find that the risk of HIV-1 infection is dependent on three factors: the amount of HIV-1 the patient is exposed to, the severity of HSV-2 lesions, and the number of CD4+ T immune cells in the genital mucosa. Our model predicts that antiviral drugs targeting HSV-2 can cause a therapeutic decrease in lesion severity and CD4+ T cell count in the genital mucosa. This furthermore causes a significant decrease in the risk of HIV-1 infection but the dose of HSV-2 antiviral drug must be sufficiently high. Our results support further development and testing of new HSV-2 antiviral drugs to help decrease the world-wide burden of HIV-1.