In-host modeling

Identifiability investigation of within-host models of acute virus infection

Uncertainty in parameter estimates from fitting within-host models to empirical data limits the model's ability to uncover mechanisms of infection, disease progression, and to guide pharmaceutical interventions. Understanding the effect of model structure and data availability on model predictions is important for informing model development and experimental design. To address sources of uncertainty in parameter estimation, we use four mathematical models of influenza A infection with increased degrees of biological realism. We test the ability of each model to reveal its parameters in the presence of unlimited data by performing structural identifiability analyses. We then refine the results by predicting practical identifiability of parameters under daily influenza A virus titers alone or together with daily adaptive immune cell data. Using these approaches, we present insight into the sources of uncertainty in parameter estimation and provide guidelines for the types of model assumptions, optimal experimental design, and biological information needed for improved predictions.

Monkeypox: a review of epidemiological modelling studies and how modelling has led to mechanistic insight

Human monkeypox (mpox) virus is a viral zoonosis that belongs to the Orthopoxvirus genus of the Poxviridae family, which presents with similar symptoms as those seen in human smallpox patients. Mpox is an increasing concern globally, with over 80,000 cases in non-endemic countries as of December 2022. In this review, we provide a brief history and ecology of mpox, its basic virology, and the key differences in mpox viral fitness traits before and after 2022. We summarize and critique current knowledge from epidemiological mathematical models, within-host models, and between-host transmission models using the One Health approach, where we distinguish between models that focus on immunity from vaccination, geography, climatic variables, as well as animal models. We report various epidemiological parameters, such as the reproduction number, R0, in a condensed format to facilitate comparison between studies. We focus on how mathematical modelling studies have led to novel mechanistic insight into mpox transmission and pathogenesis. As mpox is predicted to lead to further infection peaks in many historically non-endemic countries, mathematical modelling studies of mpox can provide rapid actionable insights into viral dynamics to guide public health measures and mitigation strategies.

Computational framework to understand the clinical stages of COVID-19 and visualization of time course for various treatment strategies

Coronavirus disease 2019 is known to be regulated by multiple factors such as delayed immune response, impaired T cell activation, and elevated levels of proinflammatory cytokines. Clinical management of the disease remains challenging due to interplay of various factors as drug candidates may elicit different responses depending on the staging of the disease. In this context, we propose a computational framework which provides insights into the interaction between viral infection and immune response in lung epithelial cells, with an aim of predicting optimal treatment strategies based on infection severity. First, we formulate the model for visualizing the nonlinear dynamics during the disease progression considering the role of T cells, macrophages and proinflammatory cytokines. Here, we show that the model is capable of emulating the dynamic and static data trends of viral load, T cell, macrophage levels, interleukin (IL)-6 and TNF-α levels. Second, we demonstrate the ability of the framework to capture the dynamics corresponding to mild, moderate, severe, and critical condition. Our result shows that, at late phase (>15 days), severity of disease is directly proportional to pro-inflammatory cytokine IL6 and tumor necrosis factor (TNF)-α levels and inversely proportional to the number of T cells. Finally, the simulation framework was used to assess the effect of drug administration time as well as efficacy of single or multiple drugs on patients. The major contribution of the proposed framework is to utilize the infection progression model for clinical management and administration of drugs inhibiting virus replication and cytokine levels as well as immunosuppressant drugs at various stages of the disease.

Development of a novel mathematical model that explains SARS-CoV-2 infection dynamics in Caco-2 cells

Mathematical modeling is widely used to study within-host viral dynamics. However, to the best of our knowledge, for the case of SARS-CoV-2 such analyses were mainly conducted with the use of viral load data and for the wild type (WT) variant of the virus. In addition, only few studies analyzed models for in vitro data, which are less noisy and more reproducible. In this work we collected multiple data types for SARS-CoV-2-infected Caco-2 cell lines, including infectious virus titers, measurements of intracellular viral RNA, cell viability data and percentage of infected cells for the WT and Delta variants. We showed that standard models cannot explain some key observations given the absence of cytopathic effect in human cell lines. We propose a novel mathematical model for in vitro SARS-CoV-2 dynamics, which included explicit modeling of intracellular events such as exhaustion of cellular resources required for virus production. The model also explicitly considers innate immune response. The proposed model accurately explained experimental data. Attenuated replication of the Delta variant in Caco-2 cells could be explained by our model on the basis of just two parameters: decreased cell entry rate and increased cytokine production rate.

Global Stability of a Humoral Immunity COVID-19 Model with Logistic Growth and Delays

The mathematical modeling and analysis of within-host or between-host coronavirus disease 2019 (COVID-19) dynamics are considered robust tools to support scientific research. Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is the cause of COVID-19. This paper proposes and investigates a within-host COVID-19 dynamics model with latent infection, the logistic growth of healthy epithelial cells and the humoral (antibody) immune response. Time delays can affect the dynamics of SARS-CoV-2 infection predicted by mathematical models. Therefore, we incorporate four time delays into the model: (i) delay in the formation of latent infected epithelial cells, (ii) delay in the formation of active infected epithelial cells, (iii) delay in the activation of latent infected epithelial cells, and (iv) maturation delay of new SARS-CoV-2 particles. We establish that the model’s solutions are non-negative and ultimately bounded. This confirms that the concentrations of the virus and cells should not become negative or unbounded. We deduce that the model has three steady states and their existence and stability are perfectly determined by two threshold parameters. We use Lyapunov functionals to confirm the global stability of the model’s steady states. The analytical results are enhanced by numerical simulations. The effect of time delays on the SARS-CoV-2 dynamics is investigated. We observe that increasing time delay values can have the same impact as drug therapies in suppressing viral progression. This offers some insight useful to develop a new class of treatment that causes an increase in the delay periods and then may control SARS-CoV-2 replication.

A machine learning approach to differentiate between COVID-19 and influenza infection using synthetic infection and immune response data

Data analysis is widely used to generate new insights into human disease mechanisms and provide better treatment methods. In this work, we used the mechanistic models of viral infection to generate synthetic data of influenza and COVID-19 patients. We then developed and validated a supervised machine learning model that can distinguish between the two infections. Influenza and COVID-19 are contagious respiratory illnesses that are caused by different pathogenic viruses but appeared with similar initial presentations. While having the same primary signs COVID-19 can produce more severe symptoms, illnesses, and higher mortality. The predictive model performance was externally evaluated by the ROC AUC metric (area under the receiver operating characteristic curve) on 100 virtual patients from each cohort and was able to achieve at least AUC = 91% using our multiclass classifier. The current investigation highlighted the ability of machine learning models to accurately identify two different diseases based on major components of viral infection and immune response. The model predicted a dominant role for viral load and productively infected cells through the feature selection process.

A Deterministic Model for Understanding Nonlinear Viral Dynamics in Oysters

Contamination of oysters with a variety of viruses is one key pathway to trigger outbreaks of massive oyster mortality as well as human illnesses, including gastroenteritis and hepatitis. Much effort has gone into examining the fate of viruses in contaminated oysters, yet the current state of knowledge of nonlinear virus-oyster interactions is not comprehensive because most studies have focused on a limited number of processes under a narrow range of experimental conditions. A framework is needed for describing the complex nonlinear virus-oyster interactions. Here, we introduce a mathematical model that includes key processes for viral dynamics in oysters, such as oyster filtration, viral replication, the antiviral immune response, apoptosis, autophagy, and selective accumulation. We evaluate the model performance for two groups of viruses, those that replicate in oysters (e.g., ostreid herpesvirus) and those that do not (e.g., norovirus), and show that this model simulates well the viral dynamics in oysters for both groups. The model analytically explains experimental findings and predicts how changes in different physiological processes and environmental conditions nonlinearly affect in-host viral dynamics, for example, that oysters at higher temperatures may be more resistant to infection by ostreid herpesvirus. It also provides new insight into food treatment for controlling outbreaks, for example, that depuration for reducing norovirus levels is more effective in environments where oyster filtration rates are higher. This study provides the foundation of a modeling framework to guide future experiments and numerical modeling for better prediction and management of outbreaks.

IMPORTANCE The fate of viruses in contaminated oysters has received a significant amount of attention in the fields of oyster aquaculture, food quality control, and public health. However, intensive studies through laboratory experiments and in situ observations are often conducted under a narrow range of experimental conditions and for a specific purpose in their respective fields. Given the complex interactions of various processes and nonlinear viral responses to changes in physiological and environmental conditions, a theoretical framework fully describing the viral dynamics in oysters is warranted to guide future studies from a top-down design. Here, we developed a process-based, in-host modeling framework that builds a bridge for better communications between different disciplines studying virus-oyster interactions.

Effectiveness of Antiviral Therapy in Highly-Transmissible Variants of SARS-CoV-2: A Modeling and Simulation Study

As of October 2021, neither established agents (e.g., hydroxychloroquine) nor experimental drugs have lived up to their initial promise as antiviral treatment against SARS-CoV-2 infection. While vaccines are being globally deployed, variants of concern (VOCs) are emerging with the potential for vaccine escape. VOCs are characterized by a higher within-host transmissibility, and this may alter their susceptibility to antiviral treatment. Here we describe a model to understand the effect of changes in within-host reproduction number R₀, as proxy for transmissibility, of VOCs on the effectiveness of antiviral therapy with molnupiravir through modeling and simulation. Molnupiravir (EIDD-2801 or MK 4482) is an orally bioavailable antiviral drug inhibiting viral replication through lethal mutagenesis, ultimately leading to viral extinction. We simulated 800 mg molnupiravir treatment every 12 h for 5 days, with treatment initiated at different time points before and after infection. Modeled viral mutations range from 1.25 to 2-fold greater transmissibility than wild type, but also include putative co-adapted variants with lower transmissibility (0.75-fold). Antiviral efficacy was correlated with R₀, making highly transmissible VOCs more sensitive to antiviral therapy. Total viral load was reduced by up to 70% in highly transmissible variants compared to 30% in wild type if treatment was started in the first 1–3 days post inoculation. Less transmissible variants appear less susceptible. Our findings suggest there may be a role for pre- or post-exposure prophylactic antiviral treatment in areas with presence of highly transmissible SARS-CoV-2 variants. Furthermore, clinical trials with borderline efficacious results should consider identifying VOCs and examine their impact in post-hoc analysis.

Evolution during primary HIV infection does not require adaptive immune selection

Modern HIV research depends crucially on both viral sequencing and population measurements. To directly link mechanistic biological processes and evolutionary dynamics during HIV infection, we developed multiple within-host phylodynamic models of HIV primary infection for comparative validation against viral load and evolutionary dynamics data. The optimal model of primary infection required no positive selection, suggesting that the host adaptive immune system reduces viral load but surprisingly does not drive observed viral evolution. Rather, the fitness (infectivity) of mutant variants is drawn from an exponential distribution in which most variants are slightly less infectious than their parents (nearly neutral evolution). This distribution was not largely different from either in vivo fitness distributions recorded beyond primary infection or in vitro distributions that are observed without adaptive immunity, suggesting the intrinsic viral fitness distribution may drive evolution. Simulated phylogenetic trees also agree with independent data and illuminate how phylogenetic inference must consider viral and immune-cell population dynamics to gain accurate mechanistic insights.

How Physical Factors Coordinate Virus Infection: A Perspective From Mechanobiology

Pandemics caused by viruses have threatened lives of thousands of people. Understanding the complicated process of viral infection provides significantly directive implication to epidemic prevention and control. Viral infection is a complex and diverse process, and substantial studies have been complemented in exploring the biochemical and molecular interactions between viruses and hosts. However, the physical microenvironment where infections implement is often less considered, and the role of mechanobiology in viral infection remains elusive. Mechanobiology focuses on sensation, transduction, and response to intracellular and extracellular physical factors by tissues, cells, and extracellular matrix. The intracellular cytoskeleton and mechanosensors have been proven to be extensively involved in the virus life cycle. Furthermore, innovative methods based on micro- and nanofabrication techniques are being utilized to control and modulate the physical and chemical cell microenvironment, and to explore how extracellular factors including stiffness, forces, and topography regulate viral infection. Our current review covers how physical factors in the microenvironment coordinate viral infection. Moreover, we will discuss how this knowledge can be harnessed in future research on cross-fields of mechanobiology and virology.

Within Host Dynamics of SARS-CoV-2 in Humans: Modeling Immune Responses and Antiviral Treatments

In December 2019, a newly discovered SARS-CoV-2 virus was emerged from China and propagated worldwide as a pandemic, resulting in about 3–5% mortality. Mathematical models can provide useful scientific insights about transmission patterns and targets for drug development. In this study, we propose a within-host mathematical model of SARS-CoV-2 infection considering innate and adaptive immune responses. We analyze the equilibrium points of the proposed model and obtain an expression of the basic reproduction number. We then numerically show the existence of a transcritical bifurcation. The proposed model is calibrated to real viral load data of two COVID-19 patients. Using the estimated parameters, we perform global sensitivity analysis with respect to the peak of viral load. Finally, we study the efficacy of antiviral drugs and vaccination on the dynamics of SARS-CoV-2 infection. Results suggest that blocking the virus production from infected cells can be an effective target for antiviral drug development. Finally, it is found that vaccination is more effective intervention as compared to the antiviral treatments.

Common Themes in Zoonotic Spillover and Disease Emergence: Lessons Learned from Bat- and Rodent-Borne RNA Viruses

Rodents (order Rodentia), followed by bats (order Chiroptera), comprise the largest percentage of living mammals on earth. Thus, it is not surprising that these two orders account for many of the reservoirs of the zoonotic RNA viruses discovered to date. The spillover of these viruses from wildlife to human do not typically result in pandemics but rather geographically confined outbreaks of human infection and disease. While limited geographically, these viruses cause thousands of cases of human disease each year. In this review, we focus on three questions regarding zoonotic viruses that originate in bats and rodents. First, what biological strategies have evolved that allow RNA viruses to reside in bats and rodents? Second, what are the environmental and ecological causes that drive viral spillover? Third, how does virus spillover occur from bats and rodents to humans?

Modeling the Relationship Between Antibody-Dependent Enhancement and Disease Severity in Secondary Dengue Infection

Sequential infections with different dengue serotypes (DENV-1, 4) significantly increase the risk of a severe disease outcome (fever, shock, and hemorrhagic disorders). Two hypotheses have been proposed to explain the severity of the disease: (1) antibody-dependent enhancement (ADE) and (2) original T cell antigenic sin. In this work, we explored the first hypothesis through mathematical modeling. The proposed model reproduces the dynamic of susceptible and infected target cells and dengue virus in scenarios of infection-neutralizing and infection-enhancing antibody competition induced by two distinct serotypes of the dengue virus during secondary infection. The enhancement and neutralization functions are derived from basic concepts of chemical reactions and used to mimic binding to the virus by two distinct populations of antibodies. The analytic study of the model showed the existence of two equilibriums: a disease-free equilibrium and an endemic one. Using the concept of the basic reproduction number [Formula: see text], we performed the asymptotic stability analysis for the two equilibriums. To measure the severity of the disease, we considered the maximum value of infected cells as well as the time when this maximum is reached. We observed that it corresponds to the time when the maximum enhancing activity for the infection occurs. This critical time was calculated from the model to be a few days after the occurrence of the infection, which corresponds to what is observed in the literature. Finally, using as output [Formula: see text], we were able to rank the contribution of each parameter of the model. In particular, we highlighted that the cross-reactive antibody responses may be responsible for the disease enhancement during secondary heterologous dengue infection.

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.

Within-host model of respiratory virus shedding and antibody response to H9N2 avian influenza virus vaccination and infection in chickens

Avian influenza virus (AIV) H9N2 subtype is an infectious pathogen that can affect both the respiratory and gastrointestinal systems in chickens and continues to have an important economic impact on the poultry industry. While the host innate immune response provides control of virus replication in early infection, the adaptive immune response aids to clear infections and prevent future invasion. Modelling virus-innate immune response pathways can improve our understanding of early infection dynamics and help to guide our understanding of virus shedding dynamics that could lead to reduced transmission between hosts. While some countries use vaccines for the prevention of H9N2 AIV in poultry, the virus continues to be endemic in regions of Eurasia and Africa, indicating a need for improved vaccine efficacy or vaccination strategies. Here we explored how three type-I interferon (IFN) pathways affect respiratory virus shedding patterns in infected chickens using a within-host model. Additionally, prime and boost vaccination strategies for a candidate H9N2 AIV vaccine are assessed for the ability to elicit seroprotective antibody titres. The model demonstrates that inclusion of virus sensitivity to intracellular type-I IFN pathways results in a shedding pattern most consistent with virus titres observed in infected chickens, and the inclusion of a cellular latent period does not improve model fit. Furthermore, early administration of a booster dose two weeks after the initial vaccine is administered results in seroprotective titres for the greatest length of time for both broilers and layers. These results demonstrate that type-I IFN intracellular mechanisms are required in a model of respiratory virus shedding in H9N2 AIV infected chickens, and also highlights the need for improved vaccination strategies for laying hens.

Early events in hepatitis B infection: the role of inoculum dose

The relationship between the inoculum dose and the ability of the pathogen to invade the host is poorly understood. Experimental studies in non-human primates infected with different inoculum doses of hepatitis B virus have shown a non-monotonic relationship between dose magnitude and infection outcome, with high and low doses leading to 100% liver infection and intermediate doses leading to less than 0.1% liver infection, corresponding to CD4 T-cell priming. Since hepatitis B clearance is CD8 T-cell mediated, the question of whether the inoculum dose influences CD8 T-cell dynamics arises. To help answer this question, we developed a mathematical model of virus-host interaction following hepatitis B virus infection. Our model explains the experimental data well, and predicts that the inoculum dose affects both the timing of the CD8 T-cell expansion and the quality of its response, especially the non-cytotoxic function. We find that a low-dose challenge leads to slow CD8 T-cell expansion, weak non-cytotoxic functions, and virus persistence; high- and medium-dose challenges lead to fast CD8 T-cell expansion, strong cytotoxic and non-cytotoxic function, and virus clearance; while a super-low-dose challenge leads to delayed CD8 T-cell expansion, strong cytotoxic and non-cytotoxic function, and virus clearance. These results are useful for designing immune cell-based interventions.

Dynamical characterization of antiviral effects in COVID-19

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

Structure and Hierarchy of SARS-CoV-2 Infection Dynamics Models Revealed by Reaction Network Analysis

This work provides a mathematical technique for analyzing and comparing infection dynamics models with respect to their potential long-term behavior, resulting in a hierarchy integrating all models. We apply our technique to coupled ordinary and partial differential equation models of SARS-CoV-2 infection dynamics operating on different scales, that is, within a single organism and between several hosts. The structure of a model is assessed by the theory of chemical organizations, not requiring quantitative kinetic information. We present the Hasse diagrams of organizations for the twelve virus models analyzed within this study. For comparing models, each organization is characterized by the types of species it contains. For this, each species is mapped to one out of four types, representing uninfected, infected, immune system, and bacterial species, respectively. Subsequently, we can integrate these results with those of our former work on Influenza-A virus resulting in a single joint hierarchy of 24 models. It appears that the SARS-CoV-2 models are simpler with respect to their long term behavior and thus display a simpler hierarchy with little dependencies compared to the Influenza-A models. Our results can support further development towards more complex SARS-CoV-2 models targeting the higher levels of the hierarchy.

Characterization of SARS-CoV-2 dynamics in the host

While many epidemiological models were proposed to understand and handle COVID-19 pandemic, too little has been invested to understand human viral replication and the potential use of novel antivirals to tackle the infection. In this work, using a control theoretical approach, validated mathematical models of SARS-CoV-2 in humans are characterized. A complete analysis of the main dynamic characteristic is developed based on the reproduction number. The equilibrium regions of the system are fully characterized, and the stability of such regions is formally established. Mathematical analysis highlights critical conditions to decrease monotonically SARS-CoV-2 in the host, as such conditions are relevant to tailor future antiviral treatments. Simulation results show the aforementioned system characterization.

In-host Mathematical Modelling of COVID-19 in Humans

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

A multiscale model of virus pandemic: Heterogeneous interactive entities in a globally connected world

This paper is devoted to the multidisciplinary modelling of a pandemic initiated by an aggressive virus, specifically the so-called SARS-CoV-2 Severe Acute Respiratory Syndrome, corona virus n.2. The study is developed within a multiscale framework accounting for the interaction of different spatial scales, from the small scale of the virus itself and cells, to the large scale of individuals and further up to the collective behaviour of populations. An interdisciplinary vision is developed thanks to the contributions of epidemiologists, immunologists and economists as well as those of mathematical modellers. The first part of the contents is devoted to understanding the complex features of the system and to the design of a modelling rationale. The modelling approach is treated in the second part of the paper by showing both how the virus propagates into infected individuals, successfully and not successfully recovered, and also the spatial patterns, which are subsequently studied by kinetic and lattice models. The third part reports the contribution of research in the fields of virology, epidemiology, immune competition, and economy focussed also on social behaviours. Finally, a critical analysis is proposed looking ahead to research perspectives.

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.

An Evolutionary Computing Model for the Study of Within-Host Evolution

Evolution of an individual within another individual is known as within-host dynamics (WHD). The most common modeling technique to study WHD involves ordinary differential equations (ODEs). In the field of biology, models of this kind assume, for example, that both the number of viruses and the number of mouse cells susceptible to being infected change according to their interaction as stated in the ODE model. However, viruses can undergo mutations and, consequently, evolve inside the mouse, whereas the mouse, in turn, displays evolutionary mechanisms through its immune system (e.g., clonal selection), defending against the invading virus. In this work, as the main novelty, we propose an evolutionary WHD model simulating the coexistence of an evolving invader within a host. In addition, instead of using ODEs we developed an alternative methodology consisting of the hybridization of a genetic algorithm with an artificial immune system. Aside from the model, interest in biology, and its potential clinical use, the proposed WHD model may be useful in those cases where the invader exhibits evolutionary changes, for instance, in the design of anti-virus software, intrusion detection algorithms in a corporation’s computer systems, etc. The model successfully simulates two intruder detection paradigms (i.e., humoral detection, danger detection) in which the intruder represents an evolving invader or guest (e.g., virus, computer program,) that infects a host (e.g., mouse, computer memory). The obtained results open up the possibility of simulating environments in which two entities (guest versus host) compete evolutionarily with each other when occupying the same space (e.g., organ cells, computer memory, network).

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.

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

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

Duration of a minor epidemic

Disease outbreaks in stochastic SIR epidemic models are characterized as either minor or major. When ℛ0<1, all epidemics are minor, whereas if ℛ0>1, they can be minor or major. In 1955, Whittle derived formulas for the probability of a minor or a major epidemic. A minor epidemic is distinguished from a major one in that a minor epidemic is generally of shorter duration and has substantially fewer cases than a major epidemic. In this investigation, analytical formulas are derived that approximate the probability density, the mean, and the higher-order moments for the duration of a minor epidemic. These analytical results are applicable to minor epidemics in stochastic SIR, SIS, and SIRS models with a single infected class. The probability density for minor epidemics in more complex epidemic models can be computed numerically applying multitype branching processes and the backward Kolmogorov differential equations. When ℛ0 is close to one, minor epidemics are more common than major epidemics and their duration is significantly longer than when ℛ0≪1 or ℛ0≫1.