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 host resistance and progeny production occurs. However, physical space limits the advantage that the virus can obtain from increasing offspring numbers, thus infection clearance may result from an increase in host defenses beyond a finite threshold. Our results might be relevant to better understand propagation of viral infections in tissues with mobility constraints, and the implications that environments with different geometrical properties might have in devising control strategies.
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 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…
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…
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 canonical bacteriophage is obligately lytic: the virus infects a bacterium and hijacks cell functions to produce large numbers of new viruses which burst from the cell. These viruses are well-studied, but there exist a wide range of coexisting virus lifestyles that are less understood. Temperate viruses exhibit both a lytic cycle and a latent (lysogenic) cycle, in which viral genomes are integrated into the bacterial host. Meanwhile, chronic (persistent) viruses use cell functions to produce more viruses without killing the cell; chronic viruses may also exhibit a latent stage in addition …
RNA viruses comprise vast populations of closely related, but highly genetically diverse, entities known as quasispecies. Understanding the mechanisms by which this extreme diversity is generated and maintained is fundamental when approaching viral persistence and pathobiology in infected hosts. In this paper, we access quasispecies theory through a mathematical model based on the theory of multitype branching processes, to better understand the roles of mechanisms resulting in viral diversity, persistence and extinction. We accomplish this understanding by a combination of computational simul…
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…
Viral quasispecies can be regarded as a swarm of genetically related mutants or a quasispecies (QS). A common formalism to approach QS is the replicator-mutator equation (RME). However, a problem with the RME is how to quantify the interaction coefficients between viral variants. Here, this is addressed by adopting an ecological perspective and resorting to the niche theory of competing communities, which assumes that the utilization of resources primarily determines ecological segregation between competing individuals (the different viral variants that constitute the QS). Using this novel com…
Time evolution of number of species (genera, families, and others), population of them, and size distribution of present ones and life times are studied in terms of a new model, where population of each genetic taxon increases by a (random) rate and decreases by (random) division (fragmentation). Each of log-normal, exponential and power law distributions in various empirical data about mentioned events are considered for gradual, intermittent and abrupt evolution cases. Results come out in good agreement with several theoretical literatures and with available empirical data. Sum of exponentia…
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…
Competition within the immune system may degrade immune control of viral infections. We formalize the evolution that occurs in both HIV-1 and the immune system quasispecies. Inclusion of competition in the immune system leads to a novel balance between the immune response and HIV-1, in which the eventual outcome is HIV-1 escape rather than control. The analytical model reproduces the three stages of HIV-1 infection. We propose a vaccine regimen that may be able to reduce competition between T cells, potentially eliminating the third stage of HIV-1.
We consider a birth-death process with the birth rates $iλ$ and death rates $iμ+i(i-1)θ$, where $i$ is the current state of the process. A positive competition rate $θ$ is assumed to be small. In the supercritical case when $λ>μ$ this process can be viewed as a demographic model for a population with a high carrying capacity around $λ-μ\overθ$. The article reports in a self-contained manner on the asymptotic properties of the time to extinction for this logistic branching process as $θ\to0$. All three reproduction regimes $λ>μ$, $λ<μ$, and $λ=μ$ are studied.
Co-evolution of two coupled quasispecies is studied, motivated by the competition between viral evolution and adapting immune response. In this co-adaptive model, besides the classical error catastrophe for high virus mutation rates, a second ``adaptation-'' catastrophe occurs, when virus mutation rates are too small to escape immune attack. Maximizing both regimes of viral error catastrophes is a possible strategy for an optimal immune response, reducing the range of allowed viral mutation rates to a minimum. From this requirement one obtains constraints on B-cell mutation rates and receptor …
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）
As early indicated by Charles Darwin, languages behave and change very much like living species. They display high diversity, differentiate in space and time, emerge and disappear. A large body of literature has explored the role of information exchanges and communicative constraints in groups of agents under selective scenarios. These models have been very helpful in providing a rationale on how complex forms of communication emerge under evolutionary pressures. However, other patterns of large-scale organization can be described using mathematical methods ignoring communicative traits. These…
When four species compete stochastically in a cyclic way, the formation of two teams of mutually neutral partners is observed. In this paper we study through numerical simulations the extinction processes that can take place in this system both in the well mixed case as well as on different types of lattices. The different routes to extinction are revealed by the probability distribution of the domination time, i.e. the time needed for one team to fully occupy the system. If swapping is allowed between neutral partners, then the probability distribution is dominated by very long-lived states w…
We propose and study {a generalized Beverton-Holt competition model} subject to Allee effects to obtain insights on how the interplay of Allee effects and contest competition affects the persistence and the extinction of two competing species. By using {the theory of monotone dynamics} and the properties of critical curves for non-invertible maps, our analysis shows that our model has simple equilibrium dynamics for most conditions. The coexistence of two competing species occurs only if the system has four interior equilibria.} We provide an {approximation to} the basins of the boundary attra…
As antibiotic resistance grows more frequent for common bacterial infections, alternative treatment strategies such as phage therapy have become more widely studied in the medical field. While many studies have explored the efficacy of antibiotics, phage therapy, or synergistic combinations of phages and antibiotics, the impact of virus competition on the efficacy of antibiotic treatment has not yet been considered. Here, we model the synergy between antibiotics and two viral types, temperate and chronic, in controlling bacterial infections. We demonstrate that while combinations of antibiotic…
We study the stochastic evolution of four species in cyclic competition in a well mixed environment. In systems composed of a finite number $N$ of particles these simple interaction rules result in a rich variety of extinction scenarios, from single species domination to coexistence between non-interacting species. Using exact results and numerical simulations we discuss the temporal evolution of the system for different values of $N$, for different values of the reaction rates, as well as for different initial conditions. As expected, the stochastic evolution is found to closely follow the me…
Background. In a number of recent experiments with food-and-mouth disease virus, a deleterious mutant, was found to avoid extinction and remain in the population for long periods of time. This observation was called quasispecies memory. The origin of quasispecies memory is not fully understood. Results. We propose and analyze a simple model of complementation between the wild type virus and a mutant that has an impaired ability of cell entry. The mutant will go extinct unless it is recreated from the wild type through mutations. However, under phenotypic mixing-and-hiding as a mechanism of com…
Cytotoxic T-lymphocytes (CTLs) recognize viral protein fragments displayed by major histocompatibility complex (MHC) molecules on the surface of virally infected cells and generate an anti-viral response that can kill the infected cells. Virus variants whose protein fragments are not efficiently presented on infected cells or whose fragments are presented but not recognized by CTLs therefore have a competitive advantage and spread rapidly through the population. We present a method that allows a more robust estimation of these escape rates from serially sampled sequence data. The proposed meth…
One of the most remarkable aspects of our species is that while we show surprisingly little genetic diversity, we demonstrate astonishing amounts of cultural diversity. Perhaps most impressive is the diversity of our technologies, broadly defined as all the physical objects we produce and the skills we use to produce them. Despite considerable focus on the evolution of technology by social scientists and philosophers, there have been few attempts to systematically quantify technological diversity and therefore the dynamics of technological change remain poorly understood. Here we show a novel …
We study a generic reaction-diffusion model for single-species population dynamics that includes reproduction, death, and competition. The population is assumed to be confined in a refuge beyond which conditions are so harsh that they lead to certain extinction. Standard continuum mean field models in one dimension yield a critical refuge length L_c such that a population in a refuge larger than this is assured survival. Herein we extend the model to take into account the discreteness and finiteness of the population, which leads us to a stochastic description. We present a particular critical…
We propose a novel network growth model coupled with the competition interaction to simulate macroevolution. Our work shows that the competition plays an important role in macroevolution and it is more rational to describe the interaction between species by network structures. Our model presents a complete picture of the development of phyla and the splitting process. It is found that periodic mass extinction occurred in our networks without any extraterrestrial factors and the lifetime distribution of species is very close to fossil record. We also perturb networks with two scenarios of mass …
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…
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…
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…
We study the competition and the evolution of nodes embedded in Euclidean restricted spaces. The population evolves by a branching process in which new nodes are generated when up to two new nodes are attached to the previous ones at each time unit. The competition in the population is introduced by considering the effect of overcrowding of nodes in the embedding space. The branching process is suppressed if the newborn node is closer than a distance $ξ$ of the previous nodes. This rule may be relevant to describe a competition for resources, limiting the density of individuals and therefore t…
In $1993$, Holt and Lawton introduced a stochastic model of two host species parasitized by a common parasitoid species. We introduce and analyze a generalization of these stochastic difference equations with any number of host species, stochastically varying parasitism rates, stochastically varying host intrinsic fitnesses, and stochastic immigration of parasitoids. Despite the lack of direct, host density-dependence, we show that this system is dissipative i.e. enters a compact set in finite time for all initial conditions. When there is a single host species, stochastic persistence and exti…
Multipartite viruses replicate through a puzzling evolutionary strategy. Their genome is segmented into two or more parts, and encapsidated in separate particles that appear to propagate independently. Completing the replication cycle, however, requires the full genome, so that a systemic infection of a host requires the concurrent presence of several particles. This represents an apparent evolutionary drawback of multipartitism, while its advantages remain unclear. A transition from monopartite to multipartite viral forms has been described in vitro under conditions of high multiplicity of in…
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…
The dynamics leading to extinction or coexistence of competing species is of great interest in ecology and related fields. Recently a model of intra- and interspecific competition between two species was proposed by Gabel et al. [Phys. Rev. E 87 (2013) 010101], in which the scarcer species (i.e., with smaller stationary population size) can be more resistant to extinction when it holds a competitive advantage; the latter study considered populations without spatial variation. Here we verify this phenomenon in populations distributed in space. We extend the model of Gabel et al. to a d-dimensio…
Mechanisms of immunity, and of the host-pathogen interactions in general are among the most fundamental problems of medicine, ecology, and evolution studies. Here, we present a microscopic, protein-level, sequence-based model of immune system, with explicitly defined interactions between host and pathogen proteins.. Simulations of this model show that possible outcomes of the infection (extinction of cells, survival with complete elimination of viruses, or chronic infection with continuous coexistence of cells and viruses) crucially depend on mutation rates of the viral and immunoglobulin prot…
