We explore the role of cellular life cycles for viruses and host cells in an infection process. For this purpose, we derive a generalized version of the basic model of virus dynamics (Nowak, M.A., Bangham, C.R.M., 1996. Population dynamics of immune responses to persistent viruses. Science 272, 74-79) from a mesoscopic description. In its final form the model can be written as a set of Volterra integrodifferential equations. We consider the role of age-distributed delays for death times and the intracellular (eclipse) phase. These processes are implemented by means of probability distribution functions. The basic reproductive ratio $R_0$ of the infection is properly defined in terms of such distributions by using an analysis of the equilibrium states and their stability. It is concluded that the introduction of distributed delays can strongly modify both the value of $R_0$ and the predictions for the virus loads, so the effects on the infection dynamics are of major importance. We also show how the model presented here can be applied to some simple situations where direct comparison with experiments is possible. Specifically, phage-bacteria interactions are analysed. The dynamics of the eclipse phase for phages is characterized analytically, which allows us to compare the performance of three different fittings proposed before for the one-step growth curve.
We construct a seven-component model of the in-host dynamics of the Human Immunodeficiency Virus Type-1 (i.e, HIV) that accounts for latent infection and the propensity of viral mutation. A dynamical analysis is conducted and a theorem is presented which characterizes the long time behavior of the model. Finally, we study the effects of an antiretroviral drug and treatment implications.
The design of protocols to suppress the propagation of viral infections is an enduring enterprise, especially hindered by limited knowledge of the mechanisms through which extinction of infection propagation comes about. We here report on a mechanism causing extinction of a propagating infection due to intraspecific competition to infect susceptible hosts. Beneficial mutations allow the pathogen to increase the production of progeny, while the host cell is allowed to develop defenses against infection. When the number of susceptible cells is unlimited, a feedback runaway co-evolution between h…
A mathematical model for fitness effects on viral persistence
Centre de Recerca Matemàtica (CRM)（数学研究中心）；Universitat Autònoma de Barcelona（巴塞罗那自治大学）；Universitat Politècnica de Catalunya - BarcelonaTech (UPC)（加泰罗尼亚理工大学）；Institute of Mathematics, UPC-BarcelonaTech (IMTech)（加泰罗尼亚理工大学数学研究所）；Centro de Biología Molecular Severo Ochoa (Consejo Superior de Investigaciones Científicas, CSIC-Universidad Autónoma de Madrid)（塞韦罗奥乔亚分子生物学中心）；Instituto de Investigación Sanitaria-Fundación Jiménez Díaz University Hospital, Universidad Autónoma de Madrid（马德里自治大学希梅内斯迪亚斯大学医院卫生研究院）；Instituto de Salud Carlos III（卡洛斯三世健康研究所）
In this paper, we study two-dimensional, three-dimensional monotonic and nonmonotonic immune responses in viral infection systems. Our results show that the viral infection systems with monotonic immune response has no bistability appear. However, the systems with nonmonotonic immune response has bistability appear under some conditions. For immune intensity, we got two important thresholds, post-treatment control threshold and elite control threshold. When immune intensity is less than post-treatment control threshold, the virus will be rebound. The virus will be under control when immune int…
RNA viruses are a widely used tool to study evolution experimentally. Many standard protocols of virus propagation and competition are done at nominally low multiplicity of infection (m.o.i.), but lead during one passage to two or more rounds of infection, of which the later ones are at high m.o.i. Here, we develop a model of the competition between wild type (wt) and a mutant under a regime of alternating m.o.i. We assume that the mutant is deleterious when it infects cells on its own, but derives a selective advantage when rare and coinfecting with wt, because it can profit from superior pro…
The coexistence of different viral strains (quasispecies) within the same host are nowadays observed for a growing number of viruses, most notably HIV, Marburg and Ebola, but the conditions for the formation and survival of new strains have not yet been understood. We present a model of HIV quasispecies competition, that describes the conditions of viral quasispecies coexistence under different immune system conditions. Our model incorporates both T and B cells responses, and we show that the role of B cells is important and additive to that of T cells. Simulations of coinfection (simultaneous…
Despite being similar in structure, functioning, and size viral pathogens enjoy very different mostly well-defined ways of life. They occupy their hosts for a few days (influenza), for a few weeks (measles), or even lifelong (HCV), which manifests in acute or chronic infections. The various transmission routes (airborne, via direct contact, etc.), degrees of infectiousness (referring to the load required for transmission), antigenic variation/immune escape and virulence define further pathogenic lifestyles. To survive pathogens must infect new hosts; the success determines their fitness. Infec…
Much of the work on modeling the spread of viral infections utilized partial differential equa- tions. Traveling-wave solutions to these PDEs are typically concentrated on velocities and their dependence on the various parameters. Most of the investigations into the dynamical interaction of virus and defective interfering particles (DIP), which are incomplete forms of the virus that replicate through co-infection, have followed the same lines. In this work we present an agent based model of viral infection with consideration of DIP and the negative feedback loop introduced by interferon produc…
When highly active antiretroviral therapy is administered for long periods of time to HIV-1 infected patients, most patients achieve viral loads that are ``undetectable'' by standard assay (i.e., HIV-1 RNA $ < 50$ copies/ml). Yet despite exhibiting sustained viral loads below the level of detection, a number of these patients experience unexplained episodes of transient viremia or viral "blips". We propose here that transient activation of the immune system by opportunistic infection may explain these episodes of viremia. Indeed, immune activation by opportunistic infection may spur HIV replic…
We study the early stages of viral infection, and the distribution of times to obtain a persistent infection. The virus population proliferates by entering and reproducing inside a target cell until a sufficient number of new virus particles are released via a burst, with a given burst size distribution, which results in the death of the infected cell. Starting with a 2D model describing the joint dynamics of the virus and infected cell populations, we analyze the corresponding master equation using the probability generating function formalism. Exploiting time-scale separation between the vir…
Given a mathematical model quantifying the viral infection of pandemic influenza H1N1pdm09-H275 wild type (WT) and H1N1pdm09-H275Y mutant (MUT) strains, we describe a simple method of estimating the model's constant parameters using Monte Carlo methods. Monte Carlo parameter estimation methods present certain advantages over the bootstrapping methods previously used in such studies: the result comprises actual parameter distributions (posteriors) that can be used to compare different viral strains; the recovered parameter distributions offer an exact method to compute credible intervals (simil…
In this study, we develop and analyze a nested multi-scale model for COVID -19 disease that integrates within-host scale and between-host scale sub-models. First, the well-posedness of the multi-scale model is discussed, followed by the stability analysis of the equilibrium points. The disease-free equilibrium point is shown to be globally asymptotically stable for $R_0 < 1$. When $R_0$ exceeds unity, a unique infected equilibrium exists, and the system is found to undergo a forward (trans-critical) bifurcation at $R_0=1$. Two parameter heat plots are also done to find the parameter combinatio…
Eigen's quasi-species model describes viruses as ensembles of different mutants of a high fitness "master" genotype. Mutants are assumed to have lower fitness than the master type, yet they coexist with it forming the quasi-species. When the mutation rate is sufficiently high, the master type no longer survives and gets replaced by a wide range of mutant types, thus destroying the quasi-species. It is the so-called "error catastrophe". But natural selection acts on phenotypes, not genotypes, and huge amounts of genotypes yield the same phenotype. An important consequence of this is the appeara…
In this paper we consider a delayed nonlinear model of the dynamics of the immune system against a viral infection that contains wild-type virus and one mutant. A finite response time of the immune system was considered in order which leads to sustained oscillatory behavior as well as chaotic behavior, triggered by the presence of delays. We present a thoroughly numeric analysis and some analytical results.
The vertebrate adaptive immune system provides a flexible and diverse set of molecules to neutralize pathogens. Yet, viruses such as HIV can cause chronic infections by evolving as quickly as the adaptive immune system, forming an evolutionary arms race. Here we introduce a mathematical framework to study the coevolutionary dynamics of antibodies with antigens within a host. We focus on changes in the binding interactions between the antibody and antigen populations, which result from the underlying stochastic evolution of genotype frequencies driven by mutation, selection, and drift. We ident…
In recent years the spread of the ectoparasitic mite Varroa destructor has become the most serious threat to worldwide apiculture. In the model presented here we extend the bee population dynamics with mite viral epidemiology examined in an earlier paper by allowing a bee-dependent mite population size. The results of the analysis match field observations well and give a clear explanation of how Varroa affects the epidemiology of certain naturally occurring bee viruses, causing considerable damages to colonies. The model allows only four possible stable equilibria, using known field parameters…
This paper analyzes a simplified model of viral infection and evolution using the 'grand canonical ensemble' and formalisms from statistical mechanics and thermodynamics to enumerate all possible viruses and to derive thermodynamic variables for the system. We model the infection process as a series of energy barriers determined by the genetic states of the virus and host as a function of immune response and system temperature. We find a phase transition between a positive temperature regime of normal replication and a negative temperature 'disordered' phase of the virus. These phases define d…
Dengue is the most prevalent arthropod-borne viral disease. Clinical symptoms of dengue virus (DENV) infection range from classical mild dengue fever to severe, life-threatening dengue shock syndrome. However, most DENV infections cause few or no symptoms. Asymptomatic DENV-infected patients provide a unique opportunity to decipher the host immune responses leading to virus elimination without negative impact on t v 'health. We used an integrated approach of transcriptional profiling and immunological analysis comparing a Cambodian population of strictly asymptomatic viremic individuals with c…
In this paper, a mathematical analysis of the global dynamics of a viral infection model in vivo is carried out. We study the dynamics of a hepatitis C virus (HCV) model, under therapy, that considers both extracellular and intracellular levels of infection. At present most mathematical modeling of viral kinetics after treatment only addresses the process of infection of a cell by the virus and the release of virions by the cell, while the processes taking place inside the cell are not included. We prove that the solutions of the new model with positive initial values are positive, exist globa…
It has been known for some time that human autoimmune diseases can be triggered by viral infections. Several possible mechanisms of interactions between a virus and immune system have been analysed, with a prevailing opinion being that the onset of autoimmunity can in many cases be attributed to "molecular mimicry", where linear peptide epitopes, processed from viral proteins, mimic normal host self proteins, thus leading to a cross-reaction of immune response against virus with host cells. In this paper we present a mathematical model for the dynamics of an immune response to a viral infectio…
RNA viruses exist as genetically diverse populations displaying different phenotypes, including diverse degrees of virulence. The evolution of virulence in viral populations is, however, poorly understood. Based on the experimental observation of an RNA virus clone in cell culture diversifying into two subpopulations of different virulence, we study the dynamics of heterogeneous virus populations and the evolution of virulence. We introduce a competition-colonization trade-off into standard mathematical models of intra-host viral infection. Colonizers are fast spreading, virulent strains, wher…
Intrahost simian immunodeficiency virus (SIV) evolution is marked by repeated viral escape from cytotoxic T-lymphocyte (CTLs) response. Typically, the first such CTL escape occurs in a matter of days, starting around the time of peak viral load. Many authors have developed methods to quantify the strength of CTL response by measuring the rate at which CTL escape occurs, but such methods usually depend on sampling at two or more timepoints, while many datasets capture the dynamics of the first CTL escape at only a single timepoint. Here, we develop inference methods for CTL escape rates applica…
Dengue viral infections show unique infection patterns arising from its four serot- ypes, (DENV-1,2,3,4). Its effects range from simple fever in primary infections to potentially fatal secondary infections. We analytically and numerically analyse virus dynamics and humoral response in a host during primary and secondary dengue infection for long periods using micro-epidemic models. The models presented here incorporate time delays, antibody dependent enhancement (ADE), a dynamic switch and a correlation factor between different DENV serotypes. We find that the viral load goes down to undetecta…
Human Papillomaviruses (HPV) are one of the most prevalent sexually transmitted infections (STI) and the most oncogenic viruses known to humans. The vast majority of HPV infections clear in less than 3 years, but the underlying mechanisms, especially the involvement of the immune response, are still poorly known. Building on earlier work stressing the potential importance of stochasticity in HPV infection clearance, we develop a stochastic mathematical model of HPV dynamics with an explicit description of the intracellular level. We show that the random partitioning of virus episomes upon stem…
Human immunodeficiency virus (HIV-1 or simply HIV) induces a persistent infection, which in the absence of treatment leads to AIDS and death in almost all infected individuals. HIV infection elicits a vigorous immune response starting about 2-3 weeks post infection that can lower the amount of virus in the body, but which cannot eradicate the virus. How HIV establishes a chronic infection in the face of a strong immune response remains poorly understood. It has been shown that HIV is able to rapidly change its proteins via mutation to evade recognition by virus-specific cytotoxic T lymphocytes…
One way in which the human immunodeficiency virus (HIV-1) replicates within a host is by infecting activated CD4+ T-cells, which then produce additional copies of the virus. Even with the introduction of antiretroviral drug therapy, which has been very successful over the past decade, a large obstacle to the complete eradication of the virus is the presence of viral reservoirs in the form of latently infected CD4+ T-cells. We consider a model of HIV infection that describes T-cell and viral interactions, as well as, the production and activation of latently infected T-cells. Upon determining e…
The extent to which ART (anti-retroviral therapy) reduces HIV transmission has received attention in recent years. Using data on the relationship between transmission and viral load we show that transmission saturates at high viral loads. We fit a power-law model and an exponential converging to an asymptote. Using data on the viral load in HIV-positive people we show that ART is likely to reduce transmission by 91.6% (81.7%-96.2%) under the first and 99.5% (98.5%-99.8%) under the second model. The role of the acute phase in HIV transmission is still debated. High levels of transmission during…
Recent clinical studies have shown that HIV disease pathogenesis can depend strongly on many factors at the time of transmission, including the strength of the initial viral load and the local availability of CD4+ T-cells. In this article, a new within-host model of HIV infection that incorporates the homeostatic proliferation of T-cells is formulated and analyzed. Due to the effects of this biological process, the influence of initial conditions on the proliferation of HIV infection is further elucidated. The identifiability of parameters within the model is investigated and a local stability…
Inferring infectiousness: a joint model of the within-host viral kinetics of SARS-CoV-2
Cleveland Clinic（克利夫兰诊所）；Cleveland Clinic Lerner College of Medicine of Case Western Reserve University（凯斯西储大学克利夫兰诊所勒纳医学院）；University of Colorado Boulder（科罗拉多大学博尔德分校）；Colorado School of Public Health（科罗拉多公共卫生学院）；Imperial College London（伦敦帝国理工学院）；Stanford University（斯坦福大学）
The origin of the unusual incubation period distribution in the development of AIDS is largely unresolved. A key factor in understanding the observed distribution of latency periods, as well as the occurrence of infected individuals not developing AIDS at all, is the dynamics of the long lasting struggle between HIV and the immune system. Using a computer simulation, we study the diversification of viral genomes under mutation and the selective pressure of the immune system.In common infections vast spreading of viral genomes usually does not takes place. In the case of an HIV infection this m…
We measure the effect of different public health regulations to the spread of COVID-19, based on a SEIRA model -- a SEIR model including asymptomatic transmissions. The cumulative confirmed cases and death show nonlinear positive relationship with the value of asymptomatic rate. Based on this model, we analyze the inhibit effects to COVID-19 of three types of public health policies, i.e. isolation of laboratory confirmed cases, general personal protection and quarantine (lock-down). The simulations conclude that the isolation display limited effects to the asymptomatic viral carriers. The gene…
Marine viruses shape the structure of the microbial community. They are, thus, a key determinant of the most important biogeochemical cycles in the planet. Therefore, a correct description of the ecological and evolutionary behavior of these viruses is essential to make reliable predictions about their role in marine ecosystems. The infection cycle, for example, is indistinctly modeled in two very different ways. In one representation, the process is described including explicitly a fixed delay between infection and offspring release. In the other, the offspring are released at exponentially d…
Bacteria and their viruses ("bacteriophages") coexist in natural environments forming complex infection networks. Recent empirical findings suggest that phage-bacteria infection networks often possess a nested structure such that there is a hierarchical relationship among who can infect whom. Here we consider how nested infection networks may affect phage and bacteria dynamics using a multi-type Lotka-Volterra framework with cross-infection. Analysis of similar models have, in the past, assumed simpler interaction structures as a first step towards tractability. We solve the proposed model, fi…
Viruses are incapable of autonomous energy production. Although many experimental studies make it clear that viruses are parasitic entities that hijack the host's molecular resources, a detailed estimate for the energetic cost of viral synthesis is largely lacking. To quantify the energetic cost of viruses to their hosts, we enumerated the costs associated with two very distinct but representative DNA and RNA viruses, namely T4 and influenza. We found that for these viruses, translation of viral proteins is the most energetically expensive process. Interestingly, the cost of building a T4 phag…
In this work we have investigated the evolutionary dynamics of a generalist pathogen, e.g. a virus population, that evolves towards specialisation in an environment with multiple host types. We have particularly explored under which conditions generalist viral strains may rise in frequency and coexist with specialist strains or even dominate the population. By means of a nonlinear mathematical model and bifurcation analysis, we have determined the theoretical conditions for stability of nine identified equilibria and provided biological interpretation in terms of the infection rates for the vi…
A current challenge for disease modeling and public health is understanding pathogen dynamics across scales since their ecology and evolution ultimately operate on several coupled scales. This is particularly true for vector-borne diseases, where within-vector, within-host, and between vector-host populations all play crucial roles in diversity and distribution of the pathogen. Despite recent modeling efforts to determine the effect of within-host virus-immune response dynamics on between-host transmission, the role of within-vector viral dynamics on disease spread is overlooked. Here we formu…
In December 2019, a newly discovered SARS-CoV-2 virus was emerged from China and propagated worldwide as a pandemic. In the absence of preventive medicine or a ready to use vaccine, 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 trans…
We analyze a within-host model of virus infection with antibody and CD8+ cytotoxic T lymphocyte (CTL) responses proposed by Schwartz et al. (2013). The goal of this work is to gain an overview of the stability of the biologically-relevant equilibria as a function of the model's immune response parameters. We show that the equilibria undergo at most two forward transcritical bifurcations. The model is also explored numerically and results are applied to equine infectious anemia virus infection. In order to arrive at stability of the biologically-relevant endemic equilibrium characterized by coe…
Virophages are viruses that rely on the replication machinery of other viruses to reproduce within eukaryotic hosts. Two different modes of coinfection have been posited based on experimental observation. In one mode, the virophage and virus enter the host independently. In the other mode, the virophage adheres to the virus so both virophage and virus enter the host together. Here we ask: what are the ecological effects of these different modes of coinfection? In particular, what ecological effects are common to both infection modes, and what are the differences particular to each mode? We dev…
Accurate and reliable forecasting of emerging dominant severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) variants enables policymakers and vaccine makers to get prepared for future waves of infections. The last three waves of SARS-CoV-2 infections caused by dominant variants Omicron (BA.1), BA.2, and BA.4/BA.5 were accurately foretold by our artificial intelligence (AI) models built with biophysics, genotyping of viral genomes, experimental data, algebraic topology, and deep learning. Based on newly available experimental data, we analyzed the impacts of all possible viral spike (S)…
Viral zoonoses have emerged as the key drivers of recent pandemics. Human infection by zoonotic viruses are either spillover events -- isolated infections that fail to cause a widespread contagion -- or species jumps, where successful adaptation to the new host leads to a pandemic. Despite expensive bio-surveillance efforts, historically emergence response has been reactive, and post-hoc. Here we use machine inference to demonstrate a high accuracy predictive bio-surveillance capability, designed to pro-actively localize an impending species jump via automated interrogation of massive sequence…
The host immune response can often efficiently suppress a virus infection, which may lead to selection for immune-resistant viral variants within the host. For example, during HIV infection, an array of CTL immune response populations recognize specific epitopes (viral proteins) presented on the surface of infected cells to effectively mediate their killing. However HIV can rapidly evolve resistance to CTL attack at different epitopes, inducing a dynamic network of interacting viral and immune response variants. We consider models for the network of virus and immune response populations, consi…
We consider a general mathematical model of a within-host viral infection with $n$ virus strains and explicit age-since-infection structure for infected cells. In the model, multiple virus strains compete for a population of target cells. Cells infected with virus strain $i\in\left\{1,...,n\right\}$ die at per-capita rate $δ_i(a)$ and produce virions at per-capita rate $p_i(a)$, where $δ_i(a)$ and $p_i(a)$ are functions of the age-since-infection of the cell. Viral strain $i$ has a basic reproduction number, $\mathcal{R}_i$, and a corresponding positive single strain equilibrium, $E_i$, when $…
The flux of visitors through popular places undoubtedly influences viral spreading -- from H1N1 and Zika viruses spreading through physical spaces such as airports, to rumors and ideas spreading though online spaces such as chatrooms and social media. However there is a lack of understanding of the types of viral dynamics that can result. Here we present a minimal dynamical model which focuses on the time-dependent interplay between the {\em mobility through} and the {\em occupancy of} such spaces. Our generic model permits analytic analysis while producing a rich diversity of infection profil…
Measles is an infectious disease caused by the Morbilivirus Measles Virus which has accompanied the human race since the 4th millennium BC, it is a disease usually concerning the paediatric population and in the past, before the advent of vaccination, almost all the population suffered from it, and in some cases the complications derived from this disease, such as central nervous involvement. Vaccination changed the course of the disease worldwide and diminished the associated comorbidities and mortality; in Mexico the vaccination program commenced in the decade of the 1970s and was successful…
Viruses can amplify their genomes following different replication modes (RMs) ranging from the stamping machine replication (SMR) model to the geometric replication (GR) model. Different RMs are expected to produce different evolutionary and dynamical outcomes in viral quasispecies due to differences in the mutations accumulation rate. Theoretical and computational models revealed that while SMR may provide RNA viruses with mutational robustness, GR may confer a dynamical advantage against genomes degradation. Here, recent advances in the investigation of the RM in positive-sense single-strand…
The aftermath of influenza infection is determined by a complex set of host-pathogen interactions, where genomic variability on both viral and host sides influences the final outcome. Although there exists large body of literature describing influenza virus variability, only a very small fraction covers the issue of host variance. The goal of this review is to explore the variability of host genes responsible for host-pathogen interactions, paying particular attention to genes responsible for the presence of sialylated glycans in the host endothelial membrane, mucus, genes used by viral immune…
