A method to determine the duration of the eclipse phase for in vitro infection with a highly pathogenic SHIV strain

How do viruses get around? A review of mathematical modeling of in-host viral transmission

Mathematical models of within host viral infections have provided important insights into the dynamics of viral infections. There has been much progress in adding more detailed biological processes to these models, such as incorporating the immune response, drug resistance, and viral coinfections. Unfortunately, the default assumption for the majority of these models is that virus is released from infected cells, travels through extracellular space, and deposits on another cell. This mode of transmission is known as cell-free infection. However, virus can also tunnel directly from one cell to another or cause neighboring cells to fuse, processes that also pass the infection to new cells. Additionally, most models do not explicitly include the transport of virus from one cell to another when describing cell-free transmission. In this review, we examine the current state of mathematical modeling that explicitly examines transmission beyond the cell-free assumption. While mathematical models have been developed to examine these processes, there are further improvements that can be made to better capture known viral dynamics.

Incorporating Intracellular Processes in Virus Dynamics Models

In-host models have been essential for understanding the dynamics of virus infection inside an infected individual. When used together with biological data, they provide insight into viral life cycle, intracellular and cellular virus-host interactions, and the role, efficacy, and mode of action of therapeutics. In this review, we present the standard model of virus dynamics and highlight situations where added model complexity accounting for intracellular processes is needed. We present several examples from acute and chronic viral infections where such inclusion in explicit and implicit manner has led to improvement in parameter estimates, unification of conclusions, guidance for targeted therapeutics, and crossover among model systems. We also discuss trade-offs between model realism and predictive power and highlight the need of increased data collection at finer scale of resolution to better validate complex models.

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.

Multiple cohort study of hospitalized SARS-CoV-2 in-host infection dynamics: Parameter estimates, identifiability, sensitivity and the eclipse phase profile

Within-host SARS-CoV-2 modelling studies have been published throughout the COVID-19 pandemic. These studies contain highly variable numbers of individuals and capture varying timescales of pathogen dynamics; some studies capture the time of disease onset, the peak viral load and subsequent heterogeneity in clearance dynamics across individuals, while others capture late-time post-peak dynamics. In this study, we curate multiple previously published SARS-CoV-2 viral load data sets, fit these data with a consistent modelling approach, and estimate the variability of in-host parameters including the basic reproduction number, R0, as well as the best-fit eclipse phase profile. We find that fitted dynamics can be highly variable across data sets, and highly variable within data sets, particularly when key components of the dynamic trajectories (e.g. peak viral load) are not represented in the data. Further, we investigated the role of the eclipse phase time distribution in fitting SARS-CoV-2 viral load data. By varying the shape parameter of an Erlang distribution, we demonstrate that models with either no eclipse phase, or with an exponentially-distributed eclipse phase, offer significantly worse fits to these data, whereas models with less dispersion around the mean eclipse time (shape parameter two or more) offered the best fits to the available data across all data sets used in this work. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

Effect of cellular regeneration and viral transmission mode on viral spread

Illness negatively affects all aspects of life and one major cause of illness is viral infections. Some viral infections can last for weeks; others, like influenza (the flu), can resolve quickly. During infections, uninfected cells can replicate in order to replenish the cells that have died due to the virus. Many viral models, especially those for short-lived infections like influenza, tend to ignore cellular regeneration since many think that uncomplicated influenza resolves much faster than cells regenerate. This research accounts for cellular regeneration, using an agent-based framework, and varies the regeneration rate in order to understand how cell regeneration affects viral infection dynamics under assumptions of different modes of transmission. We find that although the general trends in peak viral load, time of viral peak, and chronic viral load as regeneration rate changes are the same for cell-free or cell-to-cell transmission, the changes are more extreme for cell-to-cell transmission due to limited access of infected cells to newly generated cells.

Fractional transit compartment model for describing drug delayed response to tumors using Mittag-Leffler distribution on age-structured PKPD model

The response of a cell population is often delayed relative to drug injection, and individual cells in a population of cells have a specific age distribution. The application of transit compartment models (TCMs) is a common approach for describing this delay. In this paper, we propose a TCM in which damaged cells caused by a drug are given by a single fractional derivative equation. This model describes the delay as a single equation composed of fractional and ordinary derivatives, instead of a system of ODEs expressed in multiple compartments, applicable to the use of the PK concentration in the model. This model tunes the number of compartments in the existing model and expresses the delay in detail by estimating an appropriate fractional order. We perform model robustness, sensitivity analysis, and change of parameters based on the amount of data. Additionally, we resolve the difficulty in parameter estimation and model simulation using a semigroup property, consisting of a system with a mixture of fractional and ordinary derivatives. This model provides an alternative way to express the delays by estimating an appropriate fractional order without determining the pre-specified number of compartments.

Extended transit compartment model to describe tumor delay using Coxian distribution

The measured response of cell population is often delayed relative to drug injection, and individuals in a population have a specific age distribution. Common approaches for describing the delay are to apply transit compartment models (TCMs). This model reflects that all damaged cells caused by drugs suffer transition processes, resulting in death. In this study, we present an extended TCM using Coxian distribution, one of the phase-type distributions. The cell population attacked by a drug is described via age-structured models. The mortality rate of the damaged cells is expressed by a convolution of drug rate and age density. Then applying to Erlang and Coxian distribution, we derive Erlang TCM, representing the existing model, and Coxian TCMs, reflecting sudden death at all ages. From published data of drug and tumor, delays are compared after parameter estimations in both models. We investigate the dynamical changes according to the number of the compartments. Model robustness and equilibrium analysis are also performed for model validation. Coxian TCM is an extended model considering a realistic case and captures more diverse delays.

SARS-coronavirus-2 infections: biological instabilities characterized by order parameters

A four-variable virus dynamics TIIV model was considered that involves infected cells in an eclipse phase. The state space description of the model was transferred into an amplitude space description which is the appropriate general, nonlinear physics framework to describe instabilities. In this context, the unstable eigenvector or order parameter of the model was determined. Subsequently, a model-based analysis of viral load data from eight symptomatic COVID-19 patients was conducted. For all patients, it was found that the initial SARS-CoV-2 infection evolved along the respective patient-specific order parameter, as expected by theoretical considerations. The order parameter amplitude that described the initial virus multiplication showed doubling times between 30 minutes and 3 hours. Peak viral loads of patients were linearly related to the amplitudes of the patient order parameters. Finally, it was found that the patient order parameters determined qualitatively and quantitatively the relationships between the increases in virus-producing infected cells and infected cells in the eclipse phase. Overall, the study echoes the 40 years old suggestion by Mackey and Glass to consider diseases as instabilities.

Mathematical modelling of SARS-CoV-2 infection of human and animal host cells reveals differences in the infection rates and delays in viral particle production by infected cells

Severe acute respiratory syndrome coronavirus 2 (SARS-CoV -2), a causative agent of COVID-19 disease, poses a significant threat to public health. Since its outbreak in December 2019, Wuhan, China, extensive collection of diverse data from cell culture and animal infections as well as population level data from an ongoing pandemic, has been vital in assessing strategies to battle its spread. Mathematical modelling plays a key role in quantifying determinants that drive virus infection dynamics, especially those relevant for epidemiological investigations and predictions as well as for proposing efficient mitigation strategies. We utilized a simple mathematical model to describe and explain experimental results on viral replication cycle kinetics during SARS-CoV-2 infection of animal and human derived cell lines, green monkey kidney cells, Vero-E6, and human lung epithelium cells, A549-ACE2, respectively. We conducted cell infections using two distinct initial viral concentrations and quantified viral loads over time. We then fitted the model to our experimental data and quantified the viral parameters. We showed that such cellular tropism generates significant differences in the infection rates and incubation times of SARS-CoV-2, that is, the times to the first release of newly synthesised viral progeny by SARS-CoV-2-infected cells. Specifically, the rate at which A549-ACE2 cells were infected by SARS-CoV-2 was 15 times lower than that in the case of Vero-E6 cell infection and the duration of latent phase of A549-ACE2 cells was 1.6 times longer than that of Vero-E6 cells. On the other hand, we found no statistically significant differences in other viral parameters, such as viral production rate or infected cell death rate. Since in vitro infection assays represent the first stage in the development of antiviral treatments against SARS-CoV-2, discrepancies in the viral parameter values across different cell hosts have to be identified and quantified to better target vaccine and antiviral research.

A delay model for persistent viral infections in replicating cells

Persistently infecting viruses remain within infected cells for a prolonged period of time without killing the cells and can reproduce via budding virus particles or passing on to daughter cells after division. The ability for populations of infected cells to be long-lived and replicate viral progeny through cell division may be critical for virus survival in examples such as HIV latent reservoirs, tumor oncolytic virotherapy, and non-virulent phages in microbial hosts. We consider a model for persistent viral infection within a replicating cell population with time delay in the eclipse stage prior to infected cell replicative form. We obtain reproduction numbers that provide criteria for the existence and stability of the equilibria of the system and provide bifurcation diagrams illustrating transcritical (backward and forward), saddle-node, and Hopf bifurcations, and provide evidence of homoclinic bifurcations and a Bogdanov-Takens bifurcation. We investigate the possibility of long term survival of the infection (represented by chronically infected cells and free virus) in the cell population by using the mathematical concept of robust uniform persistence. Using numerical continuation software with parameter values estimated from phage-microbe systems, we obtain two parameter bifurcation diagrams that divide parameter space into regions with different dynamical outcomes. We thus investigate how varying different parameters, including how the time spent in the eclipse phase, can influence whether or not the virus survives.

Competitive exclusion during co-infection as a strategy to prevent the spread of a virus: A computational perspective

Inspired by the competition exclusion principle, this work aims at providing a computational framework to explore the theoretical feasibility of viral co-infection as a possible strategy to reduce the spread of a fatal strain in a population. We propose a stochastic-based model—called Co-Wish—to understand how competition between two viruses over a shared niche can affect the spread of each virus in infected tissue. To demonstrate the co-infection of two viruses, we first simulate the characteristics of two virus growth processes separately. Then, we examine their interactions until one can dominate the other. We use Co-Wish to explore how the model varies as the parameters of each virus growth process change when two viruses infect the host simultaneously. We will also investigate the effect of the delayed initiation of each infection. Moreover, Co-Wish not only examines the co-infection at the cell level but also includes the innate immune response during viral infection. The results highlight that the waiting times in the five stages of the viral infection of a cell in the model—namely attachment, penetration, eclipse, replication, and release—play an essential role in the competition between the two viruses. While it could prove challenging to fully understand the therapeutic potentials of viral co-infection, we discuss that our theoretical framework hints at an intriguing research direction in applying co-infection dynamics in controlling any viral outbreak’s speed.

Initial Inoculum and the Severity of COVID-19: A Mathematical Modeling Study of the Dose-Response of SARS-CoV-2 Infections

SARS-CoV-2 (Severe acute respiratory syndrome coronavirus 2) causes a variety of responses in those who contract the virus, ranging from asymptomatic infections to acute respiratory failure and death. While there are likely multiple mechanisms triggering severe disease, one potential cause of severe disease is the size of the initial inoculum. For other respiratory diseases, larger initial doses lead to more severe outcomes. We investigate whether there is a similar link for SARS-CoV-2 infections using the combination of an agent-based model (ABM) and a partial differential equation model (PDM). We use the model to examine the viral time course for different sizes of initial inocula, generating dose-response curves for peak viral load, time of viral peak, viral growth rate, infection duration, and area under the viral titer curve. We find that large initial inocula lead to short infections, but with higher viral titer peaks; and that smaller initial inocula lower the viral titer peak, but make the infection last longer.

A detailed model and Monte Carlo simulation for predicting DIP genome length distribution in baculovirus infection of insect cells

Baculoviruses have enormous potential for use as biopesticides to control insect pest populations without the adverse environmental effects posed by the widespread use of chemical pesticides. However, continuous baculovirus production is susceptible to DNA mutation and the subsequent production of defective interfering particles (DIPs). The amount of DIPs produced and their genome length distribution are of great interest not only for baculoviruses but for many other DNA and RNA viruses. In this study, we elucidate this aspect of virus replication using baculovirus as an example system and both experimental and modeling studies. The existing mathematical models for the virus replication process consider DIPs as a lumped quantity and do not consider the genome length distribution of the DIPs. In this study, a detailed population balance model for the cell-virus culture is presented, which predicts the genome length distribution of the DIP population along with their relative proportion. The model is simulated using the kinetic Monte Carlo algorithm, and the results agree well with the experimental results. Using this model, a practical strategy to maintain the DIP fraction to near to its maximum and minimum limits has been demonstrated.

Modeling the efficiency of filovirus entry into cells in vitro: Effects of SNP mutations in the receptor molecule

Interaction between filovirus glycoprotein (GP) and the Niemann-Pick C1 (NPC1) protein is essential for membrane fusion during virus entry. Some single-nucleotide polymorphism (SNPs) in two surface-exposed loops of NPC1 are known to reduce viral infectivity. However, the dependence of differences in entry efficiency on SNPs remains unclear. Using vesicular stomatitis virus pseudotyped with Ebola and Marburg virus GPs, we investigated the cell-to-cell spread of viruses in cultured cells expressing NPC1 or SNP derivatives. Eclipse and virus-producing phases were assessed by in vitro infection experiments, and we developed a mathematical model describing spatial-temporal virus spread. This mathematical model fit the plaque radius data well from day 2 to day 6. Based on the estimated parameters, we found that SNPs causing the P424A and D508N substitutions in NPC1 most effectively reduced the entry efficiency of Ebola and Marburg viruses, respectively. Our novel approach could be broadly applied to other virus plaque assays.

Ebola (EBOV) and Marburg (MARV) viruses, which are included viruses of the family Filoviridae, cause severe hemorrhagic fever in humans. Filovirus particles is adsorbed to the cell through glycoprotein (GP), which is the only viral surface protein. Interaction between the filovirus sugar protein (GP) and the Niemann-Pick C1 (NPC1) protein plays a key role in membrane fusion during virus entry. Although some single-nucleotide polymorphism (SNPs) in two surface-exposed loops of NPC1 are known to reduce viral infectivity, the dependence of differences in entry efficiency on SNPs has not been studied. We therefore investigated the cell-to-cell spread of viruses in cultured cells expressing NPC1 or SNP derivatives. Using a mathematical model describing spatial-temporal virus spread, we quantitatively analyze viral entry efficiency and how this affected cell-to-cell spread. Our approach may be applied to not only understanding the roles of genetic polymorphisms in human susceptibility to filoviruses, but also other virus plaque assays.

An empirical saddlepoint approximation method for producing smooth survival and hazard functions under interval-censoring

We devise a new method to produce smooth estimates of baseline survival and hazard functions for incomplete data observed subject to interval-censoring, that can in principle be viewed as being nonparametric. The key idea is to start from the nonparametric maximum likelihood estimate, and to then construct an empirical moment generating function for the underlying data generating mechanism, which is subsequently inverted via a saddlepoint approximation in order to obtain smooth distributional estimates. Unlike the typical spline-based and other semiparametric methods that have thus far been devised for the same purpose, the proposed approach is unencumbered by the choice of tuning parameters. Simulation studies show that in terms of integrated squared error, the method is very close in performance to the parametric gold standard, and should generally be preferred over the well-established spline-based approach implemented in R package logspline. The methodology is illustrated on some publicly available real datasets, and its implications and limitations are discussed.

Quantifying antiviral effects against simian/human immunodeficiency virus induced by host immune response

Chimeric simian and human immunodeficiency viruses (SHIVs) are appropriate animal models for the human immunodeficiency virus (HIV) because HIV has quite a narrow host range. Additionally, SHIVs that encode the HIV-1 Env protein and are infectious to macaques have many strains that show different pathogenesis, such as the highly pathogenic SHIV-KS661 and the less pathogenic SHIV-#64. Therefore, we used SHIVs to understand different aspects of AIDS pathogenesis. In a previous study, we established a mathematical model of in vivo early SHIV infection dynamics, which revealed the expected uninfected and infected dynamics in Rhesus macaques. In concrete, the number of uninfected CD4⁺ T cells in SHIV-KS661-infected Rhesus macaques decreased more significantly and rapidly than that of SHIV-#64 Rhesus macaques, and these Rhesus macaques did not any induce host immune response. In contrast, the number of uninfected CD4⁺ T cells in SHIV-#64-infected Rhesus macaques is maintained, and host immune response developed. Although we considered that the peak viral load might determine whether systemic CD4⁺ T cell depletion occurs or host immune responses develop, we could not investigate this because our model quantified only SHIV infection prior to the development of the pathogenicity or host immune responses. Therefore, we developed a new mathematical model to investigate why SHIV-#64 and SHIV-KS661 showed different long-term viral dynamics. We fitted our new model considering antibody responses to our experimental datasets that included antibody titers, CD4⁺ T cells, and viral load data. We performed a maximum likelihood estimation using a non-linear mixed effect model. From the results, we derived the basic reproduction numbers of SHIV-#64 and SHIV-KS661 from intravenous infection (IV) and SHIV-KS661 from intrarectal infection (IR), as well as the antiviral effects of antibodies against SHIV-#64(IV) and SHIV-KS661(IR). We found significant differences between the basic reproduction number of SHIV-#64(IV) or -KS661(IR) and that of SHIV-KS661(IV). We found no clear difference between the antiviral effects of SHIV-#64(IV) and SHIV-KS661(IR), and revealed that an antiviral effect more than 90% of that of maximum antibody responses was induced from initial antibody responses (i.e., antibody response just after its inducement). In conclusion, we found that the basic reproduction number, rather than SHIV strains determines whether systemic CD4⁺ T cell depletion develops, and the subsequent antibody responses occurs.

Modelling Degradation and Replication Kinetics of the Zika Virus In Vitro Infection

Mathematical models of in vitro viral kinetics help us understand and quantify the main determinants underlying the virus–host cell interactions. We aimed to provide a numerical characterization of the Zika virus (ZIKV) in vitro infection kinetics, an arthropod-borne emerging virus that has gained public recognition due to its association with microcephaly in newborns. The mathematical model of in vitro viral infection typically assumes that degradation of extracellular infectious virus proceeds in an exponential manner, that is, each viral particle has the same probability of losing infectivity at any given time. We incubated ZIKV stock in the cell culture media and sampled with high frequency for quantification over the course of 96 h. The data showed a delay in the virus degradation in the first 24 h followed by a decline, which could not be captured by the model with exponentially distributed decay time of infectious virus. Thus, we proposed a model, in which inactivation of infectious ZIKV is gamma distributed and fit the model to the temporal measurements of infectious virus remaining in the media. The model was able to reproduce the data well and yielded the decay time of infectious ZIKV to be 40 h. We studied the in vitro ZIKV infection kinetics by conducting cell infection at two distinct multiplicity of infection and measuring viral loads over time. We fit the mathematical model of in vitro viral infection with gamma distributed degradation time of infectious virus to the viral growth data and identified the timespans and rates involved within the ZIKV-host cell interplay. Our mathematical analysis combined with the data provides a well-described example of non-exponential viral decay dynamics and presents numerical characterization of in vitro infection with ZIKV.

Effect of topology and time window on probability distribution underlying baclofen induced Ca2+ response in hippocampal neurons

Cytosolic Ca²⁺ oscillation in neurons regulate a wide range of cellular functions starting from cell division to apoptosis. The major challenge in analyzing the large-scale heterogeneous calcium data obtained from hippocampal neurons is that there is no specific tool available for probability density function (pdf) fitting and model ranking. First, we focus on the ranking of various pdf and selection of a particular pdf using maximum log-likelihood. Five pdfs were analyzed in this study, exponential, gamma, log-normal, Rayleigh, and Weibull. Next, we used the statistical models to find the effect of two factors, the network topology and time window for the calcium response. The robustness of the best pdf was validated using multiple datasets obtained through random sampling of neurons from a neuron pool. GPCR targeting drug, baclofen was chosen as the model drug to inhibit Ca²⁺ response. Strongly-connected neurons show a significant change in Ca²⁺ oscillations after the addition of drug in comparison to weakly-connected neurons. The proposed technique can be used to study the dose-response from a large number of calcium imaging videos having heterogeneous responses.

Model Averaging in Viral Dynamic Models

The paucity of experimental data makes both inference and prediction particularly challenging in viral dynamic models. In the presence of several candidate models, a common strategy is model selection (MS), in which models are fitted to the data but only results obtained with the "best model" are presented. However, this approach ignores model uncertainty, which may lead to inaccurate predictions. When several models provide a good fit to the data, another approach is model averaging (MA) that weights the predictions of each model according to its consistency to the data. Here, we evaluated by simulations in a nonlinear mixed-effect model framework the performances of MS and MA in two realistic cases of acute viral infection, i.e., (1) inference in the presence of poorly identifiable parameters, namely, initial viral inoculum and eclipse phase duration, (2) uncertainty on the mechanisms of action of the immune response. MS was associated in some scenarios with a large rate of false selection. This led to a coverage rate lower than the nominal coverage rate of 0.95 in the majority of cases and below 0.50 in some scenarios. In contrast, MA provided better estimation of parameter uncertainty, with coverage rates ranging from 0.72 to 0.98 and mostly comprised within the nominal coverage rate. Finally, MA provided similar predictions than those obtained with MS. In conclusion, parameter estimates obtained with MS should be taken with caution, especially when several models well describe the data. In this situation, MA has better performances and could be performed to account for model uncertainty.

Statistical modeling of cell-to-cell variability in viral infection during passaging in suspension cell culture: Application in Monte-Carlo simulation

Packaging during the passaging of viruses in cell cultures yields various phenotypes and is regulated by viral protein expression in infected cells. Although such a packaging mechanism has a profound effect in controlling the virus yield, little is known about the underlying statistical models followed by virus packaging and protein expression among cells infected with the virus. A predictive framework combining identification of the probability density function (PDF) based on log-likelihood and using the PDF for Monte-Carlo simulations is developed. The Birnbaum-Saunders distribution was found to be consistent with all three-virus packaging levels, including nucleocapsids/occlusion-derived virus (ODV), ODVs/polyhedra, and polyhedra/cell for both wild-type and genetically modified AcMNPV. Next, it was demonstrated that PDF fitting could be used to compare two viruses having distinctly different genetic configurations. Finally, the identified PDF can be incorporated in RNA synthesis parameters for baculovirus infection to predict the cell-to-cell variability in protein expression using Monte-Carlo simulations. The proposed tool can be used for the estimation of uncertainty in the kinetic parameter and prediction of cell-to-cell variability for other biological systems.

A mathematical model for assessing the effectiveness of controlling relapse in Plasmodium vivax malaria endemic in the Republic of Korea

Malaria has persisted as an endemic near the Demilitarized Zone in the Republic of Korea since the re-emergence of Plasmodium vivax malaria in 1993. The number of patients affected by malaria has increased recently despite many controls tools, one of the reasons behind which is the relapse of malaria via liver hypnozoites. Tafenoquine, a new drug approved by the United States Food and Drug Administration in 2018, is expected to reduce the rate of relapse of malaria hypnozoites and thereby decrease the prevalence of malaria among the population. In this work, we have developed a new transmission model for Plasmodium vivax that takes into account a more realistic intrinsic distribution from existing literature to quantify the current values of relapse parameters and to evaluate the effectiveness of the anti-relapse therapy. The model is especially suitable for estimating parameters near the Demilitarized Zone in Korea, in which the disease follows a distinguishable seasonality. Results were shown that radical cure could significantly reduce the prevalence level of malaria. However, eradication would still take a long time (over 10 years) even if the high-level treatment were to persist. In addition, considering that the vector's behavior is manipulated by the malaria parasite, relapse repression through vector control at the current level may result in a negative effect in containing the disease. We conclude that the use of effective drugs should be considered together with the increased level of the vector control to reduce malaria prevalence.

Cell-to-Cell Variability in Protein Expression during Viral Infection: Monte-Carlo Simulation and Validation based on Confocal Imaging

One of the major challenges is to identify the statistical model underlying the heterogeneity in viral protein expression in single cells. In this endeavor, we propose a computational tool to address the cell-to-cell variability in protein expression by random variate generation following probability distributions. Here, we show that statistical modeling using the probability density function of various distribution offers considerable potential for providing stochastic inputs to Monte Carlo simulation. Specifically, we present the ranking between three distribution families including gamma, normal and Weibull distribution using a comparison of cumulative frequency obtained from experiment and simulation. The major contribution of the proposed simulation method is to identify the underlying statistical model in kinetic parameters that capture the variability in protein expression in single cells obtained through imaging using confocal microscopy.

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.

Mathematical modelling identifies the role of adaptive immunity as a key controller of respiratory syncytial virus in cotton rats

Respiratory syncytial virus (RSV) is a common virus that can have varying effects ranging from mild cold-like symptoms to mortality depending on the age and immune status of the individual. We combined mathematical modelling using ordinary differential equations (ODEs) with measurement of RSV infection kinetics in primary well-differentiated human bronchial epithelial cultures in vitro and in immunocompetent and immunosuppressed cotton rats to glean mechanistic details that underlie RSV infection kinetics in the lung. Quantitative analysis of viral titre kinetics in our mathematical model showed that the elimination of infected cells by the adaptive immune response generates unique RSV titre kinetic features including a faster timescale of viral titre clearance than viral production, and a monotonic decrease in the peak RSV titre with decreasing inoculum dose. Parameter estimation in the ODE model using a nonlinear mixed effects approach revealed a very low rate (average single-cell lifetime > 10 days) of cell lysis by RSV before the adaptive immune response is initiated. Our model predicted negligible changes in the RSV titre kinetics at early times post-infection (less than 5 dpi) but a slower decay in RSV titre in immunosuppressed cotton rats compared to that in non-suppressed cotton rats at later times (greater than 5 dpi) in silico. These predictions were in excellent agreement with the experimental results. Our combined approach quantified the importance of the adaptive immune response in suppressing RSV infection in cotton rats, which could be useful in testing RSV vaccine candidates.

Revealing uninfected and infected target cell dynamics from peripheral blood data in highly and less pathogenic simian/human immunodeficiency virus infected Rhesus macaque

Since chimeric simian and human immunodeficiency viruses (SHIVs) used here, that is, SHIV-#64 and -KS661 utilize both CCR5 and CXCR4 chemokine receptors, they have broad target cell properties. A highly pathogenic SHIV strain, SHIV-KS661, causes an infection that systemically depletes the CD4⁺ T cells of Rhesus macaques, while a less pathogenic strain, SHIV-#64, does not cause severe symptoms in the macaques. In our previous studies, we established in vitro quantification system for virus infection dynamics, and concluded that SHIV-KS661 effectively produces infectious virions compared with SHIV-#64 in the HSC-F cell culture. However, in vivo dynamics of SHIV infection have not been well understood. To quantify SHIV-#64 and -KS661 infection dynamics in Rhesus macaques, we developed a novel approach and analyzed total CD4⁺ T cells and viral load in peripheral blood, and reproduced the expected dynamics for the uninfected and infected CD4⁺ T cells in silico. Using our previous cell culture experimental datasets, we revealed that an infection rate constant is different between SHIV-#64 and -KS661, but the viral production rate and the death rate are similar for the both strains. Thus, here, we assumed these relations in our in vivo data and carried out the data fitting. We performed Bayesian estimation for the whole dataset using MCMC sampling, and simultaneously fitted our novel model to total CD4⁺ T cells and viral load of SHIV-#64 and -KS661 infection. Our analyses explained that the Malthusian parameter (i.e., fitness of virus infection) and the basic reproduction number (i.e., potential of virus infection) for SHIV-KS661 are significantly higher than those of SHIV-#64. In addition, we demonstrated that the number of uninfected CD4⁺ T cells in SHIV-KS661 infected Rhesus macaques decreases to the significantly lower value than that before the inoculation several days earlier compared with SHIV-#64 infection. Taken together, the differences between SHIV-#64 and -KS661 infection before the peak viral load might determine the subsequent destiny, that is, whether the systemic CD4⁺ T cell depletion occurs or the host immune response develop.

The rate of viral transfer between upper and lower respiratory tracts determines RSV illness duration

Respiratory syncytial virus can lead to serious lower respiratory infection (LRI), particularly in children and the elderly. LRI can cause longer infections, lingering respiratory problems, and higher incidence of hospitalization. In this paper, we use a simplified ordinary differential equation model of viral dynamics to study the role of transport mechanisms in the occurrence of LRI. Our model uses two compartments to simulate the upper respiratory tract and the lower respiratory tract (LRT) and assumes two distinct types of viral transfer between the two compartments: diffusion and advection. We find that a range of diffusion and advection values lead to long-lasting infections in the LRT, elucidating a possible mechanism for the severe LRI infections observed in humans.

Modeling to capture bystander-killing effect by released payload in target positive tumor cells

Background

Antibody-drug conjugates (ADCs) are intended to bind to specific positive target antigens and eradicate only tumor cells from an intracellular released payload through the lysosomal protease. Payloads, such as MMAE, have the capacity to kill adjacent antigen-negative (Ag–) tumor cells, which is called the bystander-killing effect, as well as directly kill antigen-positive (Ag+) tumor cells. We propose that a dose-response curve should be independently considered to account for target antigen-positive/negative tumor cells.

Methods

A model was developed to account for the payload in Ag+/Ag– cells and the associated parameters were applied. A tumor growth inhibition (TGI) effect was explored based on an ordinary differential equation (ODE) after substituting the payload concentration in Ag+/Ag– cells into an Emax model, which accounts for the dose-response curve. To observe the bystander-killing effects based on the amount of Ag+/Ag– cells, the Emax model is used independently. TGI models based on ODE are unsuitable for describing the initial delay through a tumor–drug interaction. This was solved using an age-structured model based on the stochastic process.

Results

β∈(0,1] is a fraction parameter that determines the proportion of cells that consist of Ag+/Ag– cells. The payload concentration decreases when the ratio of efflux to influx increases. The bystander-killing effect differs with varying amounts of Ag+ cells. The larger β is, the less bystander-killing effect. The decrease of the bystander-killing effect becomes stronger as Ag+ cells become larger than the Ag– cells. Overall, the ratio of efflux to influx, the amount of released payload, and the proportion of Ag+ cells determine the efficacy of the ADC. The tumor inhibition delay through a payload-tumor interaction, which goes through several stages, may be solved using an age-structured model.

Conclusions

The bystander-killing effect, one of the most important topics of ADCs, has been explored in several studies without the use of modeling. We propose that the bystander-killing effect can be captured through a mathematical model when considering the Ag+ and Ag– cells. In addition, the TGI model based on the age-structure can capture the initial delay through a drug interaction as well as the bystander-killing effect.

Electronic supplementary material

The online version of this article (10.1186/s12885-019-5336-7) contains supplementary material, which is available to authorized users.

Investigating Different Mechanisms of Action in Combination Therapy for Influenza

Combination therapy for influenza can have several benefits, from reducing the emergence of drug resistant virus strains to decreasing the cost of antivirals. However, there are currently only two classes of antivirals approved for use against influenza, limiting the possible combinations that can be considered for treatment. However, new antivirals are being developed that target different parts of the viral replication cycle, and their potential for use in combination therapy should be considered. The role of antiviral mechanism of action in the effectiveness of combination therapy has not yet been systematically investigated to determine whether certain antiviral mechanisms of action pair well in combination. Here, we use a mathematical model of influenza to model combination treatment with antivirals having different mechanisms of action to measure peak viral load, infection duration, and synergy of different drug combinations. We find that antivirals that lower the infection rate and antivirals that increase the duration of the eclipse phase perform poorly in combination with other antivirals.

Mathematical modeling of within-host Zika virus dynamics

Recent Zika virus outbreaks have been associated with severe outcomes, especially during pregnancy. A great deal of effort has been put toward understanding this virus, particularly the immune mechanisms responsible for rapid viral control in the majority of infections. Identifying and understanding the key mechanisms of immune control will provide the foundation for the development of effective vaccines and antiviral therapy. Here, we outline a mathematical modeling approach for analyzing the within-host dynamics of Zika virus, and we describe how these models can be used to understand key aspects of the viral life cycle and to predict antiviral efficacy.

Modeling of fusion inhibitor treatment of RSV in African green monkeys

Respiratory syncytial virus (RSV) is a respiratory infection that can cause serious illness, particularly in infants. In this study, we test four different model implementations for the effect of a fusion inhibitor, including one model that combines different drug effects, by fitting the models to data from a study of TMC353121 in African green monkeys. We use mathematical modeling to estimate the drug efficacy parameters, ε_max, the maximum efficacy of the drug, and EC₅₀, the drug concentration needed to achieve half the maximum effect. We find that if TMC353121 is having multiple effects on viral kinetics, more detailed data, using different treatment delays, is needed to detect this effect.

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.

Influenza Virus Infection Model With Density Dependence Supports Biphasic Viral Decay

Mathematical models that describe infection kinetics help elucidate the time scales, effectiveness, and mechanisms underlying viral growth and infection resolution. For influenza A virus (IAV) infections, the standard viral kinetic model has been used to investigate the effect of different IAV proteins, immune mechanisms, antiviral actions, and bacterial coinfection, among others. We sought to further define the kinetics of IAV infections by infecting mice with influenza A/PR8 and measuring viral loads with high frequency and precision over the course of infection. The data highlighted dynamics that were not previously noted, including viral titers that remain elevated for several days during mid-infection and a sharp 4–5 log₁₀ decline in virus within 1 day as the infection resolves. The standard viral kinetic model, which has been widely used within the field, could not capture these dynamics. Thus, we developed a new model that could simultaneously quantify the different phases of viral growth and decay with high accuracy. The model suggests that the slow and fast phases of virus decay are due to the infected cell clearance rate changing as the density of infected cells changes. To characterize this model, we fit the model to the viral load data, examined the parameter behavior, and connected the results and parameters to linear regression estimates. The resulting parameters and model dynamics revealed that the rate of viral clearance during resolution occurs 25 times faster than the clearance during mid-infection and that small decreases to this rate can significantly prolong the infection. This likely reflects the high efficiency of the adaptive immune response. The new model provides a well-characterized representation of IAV infection dynamics, is useful for analyzing and interpreting viral load dynamics in the absence of immunological data, and gives further insight into the regulation of viral control.

Intermittent treatment of severe influenza

Severe, long-lasting influenza infections are often caused by new strains of the virus. The long duration of these infections leads to an increased opportunity for the emergence of drug resistant mutants. This is particularly problematic since for new strains there is often no vaccine, so drug treatment is the first line of defense. One strategy for trying to minimize drug resistance is to apply drugs periodically. During treatment phases the wild-type virus decreases, but resistant virus might increase; when there is no treatment, wild-type virus will hopefully out-compete the resistant virus, driving down the number of resistant virus. A stochastic model of severe influenza is combined with a model of drug resistance to simulate long-lasting infections and intermittent treatment with two types of antivirals: neuraminidase inhibitors, which block release of virions; and adamantanes, which block replication of virions. Each drug's ability to reduce emergence of drug resistant mutants is investigated. We find that cell regeneration is required for successful implementation of intermittent treatment and that the optimal cycling parameters change with regeneration rate.

A PDE multiscale model of hepatitis C virus infection can be transformed to a system of ODEs

Direct-acting antivirals (DAAs) treat hepatitis C virus (HCV) by targeting its intracellular viral replication. DAAs are effective and deliver high clinical performance against HCV infection, but optimization of the DAA treatment regimen is ongoing. Different classes of DAAs are currently under development, and HCV treatments that combine two or three DAAs with different action mechanisms are being improved. To accurately quantify the antiviral effect of these DAA treatments and optimize multi-drug combinations, we must describe the intracellular viral replication processes corresponding to the action mechanisms by multiscale mathematical models. Previous multiscale models of HCV treatment have been formulated by partial differential equations (PDEs). However, estimating the parameters from clinical datasets requires comprehensive numerical PDE computations that are time consuming and often converge poorly. Here, we propose a user-friendly approach that transforms a standard PDE multiscale model of HCV infection (Guedj J et al., Proc. Natl. Acad. Sci. USA 2013; 110(10):3991-6) to mathematically identical ordinary differential equations (ODEs) without any assumptions. We also confirm consistency between the numerical solutions of our transformed ODE model and the original PDE model. This relationship between a detailed structured model and a simple model is called ``model aggregation problem'' and a fundamental important in theoretical biology. In particular, as the parameters of ODEs can be estimated by already established methods, our transformed ODE model and its modified version avoid the time-consuming computations and are broadly available for further data analysis.

A comparison of RSV and influenza in vitro kinetic parameters reveals differences in infecting time

Influenza and respiratory syncytial virus (RSV) cause acute infections of the respiratory tract. Since the viruses both cause illnesses with similar symptoms, researchers often try to apply knowledge gleaned from study of one virus to the other virus. This can be an effective and efficient strategy for understanding viral dynamics or developing treatment strategies, but only if we have a full understanding of the similarities and differences between the two viruses. This study used mathematical modeling to quantitatively compare the viral kinetics of in vitro RSV and influenza virus infections. Specifically, we determined the viral kinetics parameters for RSV A2 and three strains of influenza virus, A/WSN/33 (H1N1), A/Puerto Rico/8/1934 (H1N1), and pandemic H1N1 influenza virus. We found that RSV viral titer increases at a slower rate and reaches its peak value later than influenza virus. Our analysis indicated that the slower increase of RSV viral titer is caused by slower spreading of the virus from one cell to another. These results provide estimates of dynamical differences between influenza virus and RSV and help provide insight into the virus-host interactions that cause observed differences in the time courses of the two illnesses in patients.

Fighting Cancer with Mathematics and Viruses

After decades of research, oncolytic virotherapy has recently advanced to clinical application, and currently a multitude of novel agents and combination treatments are being evaluated for cancer therapy. Oncolytic agents preferentially replicate in tumor cells, inducing tumor cell lysis and complex antitumor effects, such as innate and adaptive immune responses and the destruction of tumor vasculature. With the availability of different vector platforms and the potential of both genetic engineering and combination regimens to enhance particular aspects of safety and efficacy, the identification of optimal treatments for patient subpopulations or even individual patients becomes a top priority. Mathematical modeling can provide support in this arena by making use of experimental and clinical data to generate hypotheses about the mechanisms underlying complex biology and, ultimately, predict optimal treatment protocols. Increasingly complex models can be applied to account for therapeutically relevant parameters such as components of the immune system. In this review, we describe current developments in oncolytic virotherapy and mathematical modeling to discuss the benefit of integrating different modeling approaches into biological and clinical experimentation. Conclusively, we propose a mutual combination of these research fields to increase the value of the preclinical development and the therapeutic efficacy of the resulting treatments.

Number of infection events per cell during HIV-1 cell-free infection

HIV-1 accumulates changes in its genome through both recombination and mutation during the course of infection. For recombination to occur, a single cell must be infected by two HIV strains. These coinfection events were experimentally demonstrated to occur more frequently than would be expected for independent infection events and do not follow a random distribution. Previous mathematical modeling approaches demonstrated that differences in target cell susceptibility can explain the non-randomness, both in the context of direct cell-to-cell transmission, and in the context of free virus transmission (Q. Dang et al., Proc. Natl. Acad. Sci. USA 101:632-7, 2004: K. M. Law et al., Cell reports 15:2711-83, 2016). Here, we build on these notions and provide a more detailed and extensive quantitative framework. We developed a novel mathematical model explicitly considering the heterogeneity of target cells and analysed datasets of cell-free HIV-1 single and double infection experiments in cell culture. Particularly, in contrast to the previous studies, we took into account the different susceptibility of the target cells as a continuous distribution. Interestingly, we showed that the number of infection events per cell during cell-free HIV-1 infection follows a negative-binomial distribution, and our model reproduces these datasets.

In-host modeling

Understanding the mechanisms governing host-pathogen kinetics is important and can guide human interventions. In-host mathematical models, together with biological data, have been used in this endeavor. In this review, we present basic models used to describe acute and chronic pathogenic infections. We highlight the power of model predictions, the role of drug therapy, and advantage of considering the dynamics of immune responses. We also present the limitations of these models due in part to the trade-off between the complexity of the model and their predictive power, and the challenges a modeler faces in determining the appropriate formulation for a given problem.

A highly pathogenic simian/human immunodeficiency virus effectively produces infectious virions compared with a less pathogenic virus in cell culture

BACKGROUND

The host range of human immunodeficiency virus (HIV) is quite narrow. Therefore, analyzing HIV-1 pathogenesis in vivo has been limited owing to lack of appropriate animal model systems. To overcome this, chimeric simian and human immunodeficiency viruses (SHIVs) that encode HIV-1 Env and are infectious to macaques have been developed and used to investigate the pathogenicity of HIV-1 in vivo. So far, we have many SHIV strains that show different pathogenesis in macaque experiments. However, dynamic aspects of SHIV infection have not been well understood. To fully understand the dynamic properties of SHIVs, we focused on two representative strains-the highly pathogenic SHIV, SHIV-KS661, and the less pathogenic SHIV, SHIV-#64-and measured the time-course of experimental data in cell culture.

METHODS

We infected HSC-F with SHIV-KS661 and -#64 and measured the concentration of Nef-negative (target) and Nef-positive (infected) HSC-F cells, the total viral load, and the infectious viral load daily for 9 days. The experiments were repeated at two different multiplicities of infection, and a previously developed mathematical model incorporating the infectious and non-infectious viruses was fitted to the full dataset of each strain simultaneously to characterize the infection dynamics of these two strains.

RESULTS AND CONCLUSIONS

We quantified virological indices including virus burst sizes and basic reproduction number of both SHIV-KS661 and -#64. Comparing the burst size of total and infectious viruses (viral RNA copies and TCID₅₀, respectively), we found that there was a statistically significant difference between the infectious virus burst size of SHIV-KS661 and -#64, while there was no significant difference between the total virus burst size. Furthermore, our analyses showed that the fraction of infectious virus among the produced SHIV-KS661 viruses, which is defined as the infectious viral load (TCID₅₀/ml) divided by the total viral load (RNA copies/ml), is more than 10-fold higher than that of SHIV-#64 during overall infection (i.e., for 9 days). Taken together, we conclude that the highly pathogenic SHIV produces infectious virions more effectively than the less pathogenic SHIV in cell culture.

Duration of SHIV production by infected cells is not exponentially distributed: Implications for estimates of infection parameters and antiviral efficacy

The duration of the eclipse phase, from cell infection to the production and release of the first virion progeny, immediately followed by the virus-production phase, from the first to the last virion progeny, are important steps in a viral infection, by setting the pace of infection progression and modulating the response to antiviral therapy. Using a mathematical model (MM) and data for the infection of HSC-F cells with SHIV in vitro, we reconfirm our earlier finding that the eclipse phase duration follows a fat-tailed distribution, lasting 19 h (18–20 h). Most importantly, for the first time, we show that the virus-producing phase duration, which lasts 11 h (9.8–12 h), follows a normal-like distribution, and not an exponential distribution as is typically assumed. We explore the significance of this finding and its impact on analysis of plasma viral load decays in HIV patients under antiviral therapy. We find that incorrect assumptions about the eclipse and virus-producing phase distributions can lead to an overestimation of antiviral efficacy. Additionally, our predictions for the rate of plasma HIV decay under integrase inhibitor therapy offer an opportunity to confirm whether HIV production duration in vivo also follows a normal distribution, as demonstrated here for SHIV infections in vitro.

Avian influenza viruses that cause highly virulent infections in humans exhibit distinct replicative properties in contrast to human H1N1 viruses

Avian influenza viruses present an emerging epidemiological concern as some strains of H5N1 avian influenza can cause severe infections in humans with lethality rates of up to 60%. These have been in circulation since 1997 and recently a novel H7N9-subtyped virus has been causing epizootics in China with lethality rates around 20%. To better understand the replication kinetics of these viruses, we combined several extensive viral kinetics experiments with mathematical modelling of in vitro infections in human A549 cells. We extracted fundamental replication parameters revealing that, while both the H5N1 and H7N9 viruses replicate faster and to higher titers than two low-pathogenicity H1N1 strains, they accomplish this via different mechanisms. While the H7N9 virions exhibit a faster rate of infection, the H5N1 virions are produced at a higher rate. Of the two H1N1 strains studied, the 2009 pandemic H1N1 strain exhibits the longest eclipse phase, possibly indicative of a less effective neuraminidase activity, but causes infection more rapidly than the seasonal strain. This explains, in part, the pandemic strain’s generally slower growth kinetics and permissiveness to accept mutations causing neuraminidase inhibitor resistance without significant loss in fitness. Our results highlight differential growth properties of H1N1, H5N1 and H7N9 influenza viruses.