In this paper the concept of Critical Community Size (CCS) for the deterministic SIR model is introduced and its consequences for the disease dynamics are stressed. The disease can fade out after an outburst. Also the principle of competitive exclusion holds no longer true. This is exemplified for the dynamics of two competing virus strains. The virus with higher R0 can be eradicated from the population.
Mathematical models are formal and simplified representations of the knowledge related to a phenomenon. In classical epidemic models, a neglected aspect is the heterogeneity of disease transmission and progression linked to the viral load of each infectious individual. Here, we attempt to investigate the interplay between the evolution of individuals' viral load and the epidemic dynamics from a theoretical point of view. In the framework of multi-agent systems, we propose a particle stochastic model describing the infection transmission through interactions among agents and the individual phys…
The viral load is known to be a chief predictor of the risk of transmission of infectious diseases. In this work, we investigate the role of the individuals' viral load in the disease transmission by proposing a new susceptible-infectious-recovered epidemic model for the densities and mean viral loads of each compartment. To this aim, we formally derive the compartmental model from an appropriate microscopic one. Firstly, we consider a multi-agent system in which individuals are identified by the epidemiological compartment to which they belong and by their viral load. Microscopic rules descri…
The evolution of RNA viruses such as HIV, Hepatitis C and Influenza virus occurs so rapidly that the viruses' genomes contain information on past ecological dynamics. Hence, we develop a phylodynamic method that enables the joint estimation of epidemiological parameters and phylogenetic history. Based on a compartmental susceptible-infected-removed (SIR) model, this method provides separate information on incidence and prevalence of infections. Detailed information on the interaction of host population dynamics and evolutionary history can inform decisions on how to contain or entirely avoid d…
Viral mutation spread controlled by inter-host network dynamics
Galician Center for Mathematical Research and Technology (CITMAga)（加利西亚数学研究与技术中心）；Group of Nonlinear Physics, Universidade de Santiago de Compostela（圣地亚哥-德孔波斯特拉大学非线性物理研究组）；Department of Applied Mathematics, Universidade de Santiago de Compostela（圣地亚哥-德孔波斯特拉大学应用数学系）
We study the population dynamics of lytic viruses which replicate slowly in dividing host cells within an organism or cell culture, and find a range of viral replication rates that allows viruses to persist, avoiding extinction of host cells or dilution of viruses at too rapid or too slow viral replication. For the within-host competition between multiple viral strains, a strain with a "stable" replication rate could outcompete another strain with a higher or lower replication rate, therefore natural selection of viruses stabilizes the viral persistence. However, when strains with higher and l…
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…
News spread in internet media outlets can be seen as a contagious process generating temporal networks representing the influence between published articles. In this article we propose a methodology based on the application of natural language analysis of the articles to reconstruct the spread network. From the reconstructed network, we show that the dynamics of the news spread can be approximated by a classical SIR epidemiological dynamics upon the network. From the results obtained we argue that the methodology proposed can be used to make predictions about media repercussion, and also to de…
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 COVID-19 pandemic has given rise to numerous articles from different scientific fields (epidemiology, virology, immunology, airflow physics...) without any effort to link these different insights. In this review, we aim to establish relationships between epidemiological data and the characteristics of the virus strain responsible for the epidemic wave concerned. We have carried out this study on the Wuhan, Alpha, Delta and Omicron strains allowing us to illustrate the evolution of the relationships we have highlighted according to these different viral strains. We addressed the following q…
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…
The crisis caused by the COVID-19 outbreak around the globe raised an increasing concern about the ongoing emergence of variants of SARS-CoV-2 that may evade the immune response provided by vaccines. New variants appear due to mutation, and as the cases accumulate, the probability of the emergence of a variant of concern increases. In this article, we propose a modified SIR model with waning immunity that captures the competition of two strain classes of an infectious disease under the effect of vaccination with a highly contagious and deadly strain class emerging from a prior strain due to mu…
RNA viruses exist in large intra-host populations which display great genotypic and phenotypic diversity. We analyze a model of viral competition between two different viral strains infecting a constantly replenished cell pool, in which we assume a trade-off between the virus' colonization skills (cell killing ability or virulence) and its local competition skills (replication performance within coinfected cells). We characterize the conditions that allow for viral spread by means of the basic reproductive number and show that a local coexistence equilibrium exists, which is asymptotically sta…
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…
We introduce and discuss a minimal individual-based model for influenza dynamics. The model takes into account the effects of specific immunization against viral strains, but also infectivity randomness and the presence of a short-lived strain transcending immunity recently suggested in the literature. We show by simulations that the resulting model exhibits substitution of viral strains along the years, but that their divergence remains bounded. We also show that dropping any of these features results in a drastically different behavior, leading either to the extinction of the disease, to the…
COVID-19 is an emergent viral infection which rose in December 2019 in a city in the Chinese province of Hubei, Wuhan; the viral aetiology of this infection is now known as COVID-19 virus, which belongs to the Betacoronavirus genus. This virus produces the syndrome of acute respiratory stress that h as been witnessed in other coronaviruses, such as that MERS-CoV in Middle East countries or SARS-CoV which was seen in 2002 and 2003 in China. This virus mediates its entry through its spike (S) proteins interacting with ACE2 receptors in lung epithelial cells, and may promote an inflammatory respo…
Interplay between evolutionary and epidemic time scales challenges the outcome of control policies
University of Zaragoza（萨拉戈萨大学）；Institute for Biocomputation and Physics of Complex Systems (BIFI), University of Zaragoza（萨拉戈萨大学复杂系统生物计算与物理研究所）；Universitat Rovira i Virgili（罗维拉-依维尔吉里大学）；Pacific Northwest National Laboratory（太平洋西北国家实验室）；Complexity Science Hub Vienna（维也纳复杂性科学中心）；ComSCIAM, Universitat Rovira i Virgili（罗维拉-依维尔吉里大学计算科学与数学中心）；Center for Computational Social Science, University of Kobe（神户大学计算社会科学中心）
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…
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…
Detections of mutations of the SARS-CoV-2 virus gave rise to new packages of interventions. Among them, international travel restrictions have been one of the fastest and most visible responses to limit the spread of the variants. While inducing large economic losses, the epidemiological consequences of such travel restrictions are highly uncertain. They may be poorly effective when the new highly transmissible strain of the virus already circulates in many regions. Assessing the effectiveness of travel bans is difficult given the paucity of data on daily cross-border mobility and on existing …
Population dynamics and evolutionary genetics underly the structure of ecosystems, changing on the same timescale for interacting species with rapid turnover, such as virus (e.g. HIV) and immune response. Thus, an important problem in mathematical modeling is to connect ecology, evolution and genetics, which often have been treated separately. Here, extending analysis of multiple virus and immune response populations in a resource - prey (consumer) - predator model from Browne and Smith \cite{browne2018dynamics}, we show that long term dynamics of viral mutants evolving resistance at distinct …
Assessing the risk of epidemic spread on networks and developing strategies for its containment is of tremendous practical importance, both due to direct effects in public health and its impact on economies. In this work we present the numerical results of a stochastic, event-driven, hierarchical agent-based model designed to reproduce the infectious dynamics of the cattle disease called Bovine Viral Diarrhea (BVD), for which the corresponding network of movements is the main route of spreading. For the farm-node dynamics, the model takes into account a vast number of breeding, infectious and …
The evolution of many microbes and pathogens, including circulating viruses such as seasonal influenza, is driven by immune pressure from the host population. In turn, the immune systems of infected populations get updated, chasing viruses even further away. Quantitatively understanding how these dynamics result in observed patterns of rapid pathogen and immune adaptation is instrumental to epidemiological and evolutionary forecasting. Here we present a mathematical theory of co-evolution between immune systems and viruses in a finite-dimensional antigenic space, which describes the cross-reac…
Contagious respiratory diseases, such as COVID-19, depend on sufficiently prolonged exposures for the successful transmission of the underlying pathogen. It is important for organizations to evaluate the efficacy of interventions aiming at mitigating viral transmission among their personnel. We have developed a operational risk assessment simulation framework that couples a spatial agent-based model of movement with a SIR epidemiological model to assess the relative risks of different intervention strategies. By applying our model on MIT's STATA building, we assess the impacts of three possibl…
Rapidly evolving pathogens like influenza viruses can persist by accumulating antigenic novelty fast enough to evade the adaptive immunity of the host population, yet without continuous accumulation of genetic diversity. This dynamical state is often compared to the Red Queen evolving as fast as it can just to maintain its foothold in the host population: Accumulation of antigenic novelty is balanced by the build-up of host immunity. Such Red Queen States (RQS) of continuous adaptation in large rapidly mutating populations are well understood in terms of Traveling Wave (TW) theories of populat…
The diversity of virus populations within single infected hosts presents a major difficulty for the natural immune response as well as for vaccine design and antiviral drug therapy. Recently developed pyrophosphate based sequencing technologies (pyrosequencing) can be used for quantifying this diversity by ultra-deep sequencing of virus samples. We present computational methods for the analysis of such sequence data and apply these techniques to pyrosequencing data obtained from HIV populations within patients harboring drug resistant virus strains. Our main result is the estimation of the pop…
Mutational escape from vaccine induced immune responses has thwarted the development of a successful vaccine against AIDS, whose causative agent is HIV, a highly mutable virus. Knowing the virus' fitness as a function of its proteomic sequence can enable rational design of potent vaccines, as this information can focus vaccine induced immune responses to target mutational vulnerabilities of the virus. Spin models have been proposed as a means to infer intrinsic fitness landscapes of HIV proteins from patient-derived viral protein sequences. These sequences are the product of non-equilibrium vi…
Broadly neutralizing antibodies are promising candidates for treatment and prevention of HIV-1 infections. Such antibodies can temporarily suppress viral load in infected individuals; however, the virus often rebounds by escape mutants that have evolved resistance. In this paper, we map an in vivo fitness landscape of HIV-1 interacting with broadly neutralizing antibodies, using data from a recent clinical trial. We identify two fitness factors, antibody dosage and viral load, that determine viral reproduction rates reproducibly across different hosts. The model successfully predicts the escap…
Currently, due to the COVID-19 pandemic the public life in most European countries stopped almost completely due to measures against the spread of the virus. Efforts to limit the number of new infections are threatened by the advent of new variants of the SARS-COV-2 virus, most prominent the B.1.1.7 strain with higher infectivity. In this article we consider a basic two-strain SIR model to explain the spread of those variants in Germany on small time scales. For a linearized version of the model we calculate relevant variables like the time of minimal infections or the dynamics of the share of…
For many infectious diseases, a small-world network on an underlying regular lattice is a suitable simplified model for the contact structure of the host population. It is well known that the contact network, described in this setting by a single parameter, the small-world parameter $p$, plays an important role both in the short term and in the long term dynamics of epidemic spread. We have studied the effect of the network structure on models of immune for life diseases and found that in addition to the reduction of the effective transmission rate, through the screening of infectives, spatial…
Cross-protection, which refers to a process whereby artificially inoculating a plant with a mild strain provides protection against a more aggressive isolate of the virus, is known to be an effective tool of disease control in plants. In this paper we derive and analyse a new mathematical model of the interactions between two competing viruses with particular account for RNA interference. Our results show that co-infection of the host can either increase or decrease the potency of individual infections depending on the levels of cross-protection or cross-enhancement between different viruses. …
In this work we study the stability properties of the equilibrium points of deterministic epidemic models with nonconstant population size. Models with nonconstant population have been studied in the past only in particular cases, two of which we review and combine. Our main result shows that for simple "matrix epidemic models" introduced in [1], an explicit general formula for the reproduction number and the corresponding "weak stability alternative" still holds, under small modifications, for models with nonconstant population size, and even when the model allows for vaccination and loss of …
In this paper we study some deterministic mathematical models that seek to explain the expansion of zika virus, as a viral epidemic, using published data for Brazil. SIR type models are proposed and validated using the epidemic data found, considering several aspects in the spread of the disease. Finally, we confirmed that the crucial epidemic parameter such as $R_0$ is consistent with those previously reported in the literature for other areas. We also explored variations of the parameters within Brazil for different federal entities. We concluded that a parsimonious model that includes both …
Different extensions of the classical single-strain SIR model for the host population, motivated by modeling dengue fever epidemiology, have reported a rich dynamic structure including deterministic chaos which was able to explain the large fluctuations of disease incidences. A comparison between the basic two-strain dengue model, which already captures differences between primary and secondary infections, with the four-strain dengue model, that introduces the idea of competition of multiple strains in dengue epidemics shows that the difference between first and secondary infections drives the…
Chronic infections of the human immunodeficiency virus (HIV) create a very complex co-evolutionary process, where the virus tries to escape the continuously adapting host immune system. Quantitative details of this process are largely unknown and could help in disease treatment and vaccine development. Here we study a longitudinal dataset of ten HIV-infected people, where both the B-cell receptors and the virus are deeply sequenced. We focus on simple measures of turnover, which quantify how much the composition of the viral strains and the immune repertoire change between time points. At the …
Since the end of 2019 an outbreak of a new strain of coronavirus, called 2019--nCoV, is reported from China and later other parts of the world. Since January 21, WHO reports daily data on confirmed cases and deaths from both China and other countries. In this work we present some discrete and continuous models to discribe the disease dynamics in China and estimate the needed epidemiological parameters. Good agreement with the current dynamics has be found for both a discrete transmission model and a slightly modified SIR-model.
We develop a new perturbation method for studying quasi-neutral competition in a broad class of stochastic competition models, and apply it to the analysis of fixation of competing strains in two epidemic models. The first model is a two-strain generalization of the stochastic Susceptible-Infected-Susceptible (SIS) model. Here we extend previous results due to Parsons and Quince (2007), Parsons et al (2008) and Lin, Kim and Doering (2012). The second model, a two-strain generalization of the stochastic Susceptible-Infected-Recovered (SIR) model with population turnover, has not been studied pr…
During pandemic events, strategies such as social distancing can be fundamental to curb viral spreading. Such actions can reduce the number of simultaneous infections and mitigate the disease spreading, which is relevant to the risk of a healthcare system collapse. Although these strategies can be suggested, their actual implementation may depend on the population perception of the disease risk. The current COVID-19 crisis, for instance, is showing that some individuals are much more prone than others to remain isolated, avoiding unnecessary contacts. With this motivation, we propose an epidem…
We develop a multiple compartment Susceptible-Infected-Recovered (SIR) model to analyze the spread of several infectious diseases through different geographic areas. Additionally, we propose a data-quality sensitive optimization framework for fitting this model to observed data. We fit the model to the temporal profile of the number of people infected by one of six influenza strains in Europe over $7$ influenza seasons. In addition to describing the temporal and spatial spread of influenza, the model provides an estimate of the inter-country and intra-country infection and recovery rates of ea…
Influenza viruses undergo continual antigenic evolution allowing mutant viruses to evade host immunity acquired to previous virus strains. Antigenic phenotype is often assessed through pairwise measurement of cross-reactivity between influenza strains using the hemagglutination inhibition (HI) assay. Here, we extend previous approaches to antigenic cartography, and simultaneously characterize antigenic and genetic evolution by modeling the diffusion of antigenic phenotype over a shared virus phylogeny. Using HI data from influenza lineages A/H3N2, A/H1N1, B/Victoria and B/Yamagata, we determin…
