Some basic properties of immune selection

Antigenic cooperation in viral populations: Transformation of functions of intra-host viral variants

In this paper, we study intra-host viral adaptation by antigenic cooperation - a mechanism of immune escape that serves as an alternative to the standard mechanism of escape by continuous genomic diversification and allows to explain a number of experimental observations associated with the establishment of chronic infections by highly mutable viruses. Within this mechanism, the topology of a cross-immunoreactivity network forces intra-host viral variants to specialize for complementary roles and adapt to the host's immune response as a quasi-social ecosystem. Here we study dynamical changes in immune adaptation caused by evolutionary and epidemiological events. First, we show that the emergence of a viral variant with altered antigenic features may result in a rapid re-arrangement of the viral ecosystem and a change in the roles played by existing viral variants. In particular, it may push the population under immune escape by genomic diversification towards the stable state of adaptation by antigenic cooperation. Next, we study the effect of a viral transmission between two chronically infected hosts, which results in the merging of two intra-host viral populations in the state of stable immune-adapted equilibrium. In this case, we also describe how the newly formed viral population adapts to the host's environment by changing the functions of its members. The results are obtained analytically for minimal cross-immunoreactivity networks and numerically for larger populations.

The effect of noise in an HIV infection model with cytotoxic T-lymphocyte impairment

The human immunodeficiency virus (HIV) interacts with the immune cells within the human body, where the environment is uncertain and noisy. Stochastic models can successfully encapsulate the effect of such a noisy environment compared to their deterministic counterparts. The human immune system is complex but well-coordinated with various immune cells like C D ⁴T cells, dendritic cells, and cytotoxic T-lymphocyte (CTL) cells, among many others. The CTL can kill the antigenic cells after its recognition. However, the efficacy of CTL in removing the infected C D ⁴T cells is progressively compromised in HIV-infected individuals. This paper considers a noise-induced HIV-immune cell interaction model with immune impairment. A multiplicative white noise is introduced in the infection rate parameter to represent the fluctuations around the average value of the rate parameter as a causative effect of the noise. We analyzed the deterministic and stochastic models and prescribed sufficient conditions for infection eradication and persistence. It is determined under what parametric restrictions the asymptotic solutions of the noise-induced system will be a limiting case of the deterministic solutions. Simulation results revealed that the solutions of the deterministic system either converge to a CTL-dominated interior equilibrium or a CTL-free immunodeficient equilibrium, depending on the initial values of the system. Stochastic analysis divulged that higher noise might be helpful in the infection removal process. The extinction time of infected C D ⁴T cells for some fixed immune impairment gradually decreases with increasing noise intensity and follows the power law.

The effect of human vaccination behaviour on strain competition in an infectious disease: An imitation dynamic approach

Strain competition plays an important role in shaping the dynamics of multiple pathogen outbreaks in a population. Competition may lead to exclusion of some pathogens, while it may influence the invasion of an emerging mutant in the population. However, little emphasis has been given to understand the influence of human vaccination choice on pathogen competition or strain invasion for vaccine-preventable infectious diseases. Coupling game dynamic framework of vaccination choice and compartmental disease transmission model of two strains, we explore invasion and persistence of a mutant in the population despite having a lower reproduction rate than the resident one. We illustrate that higher perceived strain severity and lower perceived vaccine efficacy are necessary conditions for the persistence of a mutant strain. The numerical simulation also extends these invasion and persistence analyses under asymmetric cross-protective immunity of these strains. We show that the dynamics of this cross-immunity model under human vaccination choices is determined by the interplay of parameters defining the cross-immune response function, perceived risk of infection, and vaccine efficacy, and it can exhibit invasion and persistence of mutant strain, even complete exclusion of resident strain in the regime of sufficiently high perceived risk. We conclude by discussing public health implications of the results, that proper risk communication in public about the severity of the disease is an important task to reduce the chance of mutant invasion. Thus, understanding pathogen competitions under social interactions and choices may be an important component for policymakers for strategic decision-making.

Threshold Dynamics of a Non-Linear Stochastic Viral Model with Time Delay and CTL Responsiveness

This article focuses on a stochastic viral model with distributed delay and CTL responsiveness. It is shown that the viral disease will be extinct if the stochastic reproductive ratio is less than one. However, when the stochastic reproductive ratio is more than one, the viral infection system consists of an ergodic stationary distribution. Furthermore, we obtain the existence and uniqueness of the global positive solution by constructing a suitable Lyapunov function. Finally, we illustrate our results by numerical simulation.

Variable Effect of HIV Superinfection on Clinical Status: Insights From Mathematical Modeling

HIV superinfection (infection of an HIV positive individual with another strain of the virus) has been shown to result in a deterioration of clinical status in multiple case studies. However, superinfection with no (or positive) clinical outcome might easily go unnoticed, and the typical effect of superinfection is unknown. We analyzed mathematical models of HIV dynamics to assess the effect of superinfection under various assumptions. We extended the basic model of virus dynamics to explore systematically a set of model variants incorporating various details of HIV infection (homeostatic target cell dynamics, bystander killing, interference competition between viral clones, multiple target cell types, virus-induced activation of target cells). In each model, we identified the conditions for superinfection, and investigated whether and how successful invasion by a second viral strain affects the level of uninfected target cells. In the basic model, and in some of its extensions, the criteria for invasion necessarily entail a decrease in the equilibrium abundance of uninfected target cells. However, we identified three novel scenarios where superinfection can substantially increase the uninfected cell count: (i) if the rate of new infections saturates at high infectious titers (due to interference competition or cell-autonomous innate immunity); or when the invading strain is more efficient at infecting activated target cells, but less efficient at (ii) activating quiescent cells or (iii) inducing bystander killing of these cells. In addition, multiple target cell types also allow for modest increases in the total target cell count. We thus conclude that the effect of HIV superinfection on clinical status might be variable, complicated by factors that are independent of the invasion fitness of the second viral strain.

Theory of invasion extinction dynamics in minimal food webs

When food webs are exposed to species invasion, secondary extinction cascades may be set off. Although much work has gone into characterizing the structure of food webs, systematic predictions on their evolutionary dynamics are still scarce. Here we present a theoretical framework that predicts extinctions in terms of an alternating sequence of two basic processes: resource depletion by or competitive exclusion between consumers. We first propose a conceptual invasion extinction model (IEM) involving random fitness coefficients. We bolster this IEM by an analytical, recursive procedure for calculating idealized extinction cascades after any species addition and simulate the long-time evolution. Our procedure describes minimal food webs where each species interacts with only a single resource through the generalized Lotka-Volterra equations. For such food webs ex- tinction cascades are determined uniquely and the system always relaxes to a stable steady state. The dynamics and scale invariant species life time resemble the behavior of the IEM, and correctly predict an upper limit for trophic levels as observed in the field.

Dynamics of virus and immune response in multi-epitope network

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, consisting of Lotka-Volterra-like systems of ordinary differential equations. Stability of feasible equilibria and corresponding uniform persistence of distinct variants are characterized via a Lyapunov function. We specialize the model to a "binary sequence" setting, where for n epitopes there can be [Formula: see text] distinct viral variants mapped on a hypercube graph. The dynamics in several cases are analyzed and sharp polychotomies are derived characterizing persistent variants. In particular, we prove that if the viral fitness costs for gaining resistance to each epitope are equal, then the system of [Formula: see text] virus strains converges to a "perfectly nested network" with less than or equal to [Formula: see text] persistent virus strains. Overall, our results suggest that immunodominance, i.e. relative strength of immune response to an epitope, is the most important factor determining the persistent network structure.

Global properties of nested network model with application to multi-epitope HIV/CTL dynamics

Mathematical modeling and analysis can provide insight on the dynamics of ecosystems which maintain biodiversity in the face of competitive and prey-predator interactions. Of primary interests are the underlying structure and features which stabilize diverse ecological networks. Recently Korytowski and Smith (Theor Ecol 8(1):111-120, 2015) proved that a perfectly nested infection network, along with appropriate life history trade-offs, leads to coexistence and persistence of bacteria-phage communities in a chemostat model. In this article, we generalize their model in order to apply it to the within-host dynamics virus and immune response, in particular HIV and CTL (Cytotoxic T Lymphocyte) cells. Our model can describe sequential viral escape from dominant immune responses and rise in subdominant immune responses, consistent with observed patterns of HIV/CTL evolution. We find a Lyapunov function for the system which leads to rigorous characterization of persistent viral and immune variants, along with informing upon equilibria stability and global dynamics. Results are interpreted in the context of within-host HIV/CTL evolution and numerical simulations are provided.

Mathematical model reveals how regulating the three phases of T-cell response could counteract immune evasion

T cells are key players in immune action against the invasion of target cells expressing non-self antigens. During an immune response, antigen-specific T cells dynamically sculpt the antigenic distribution of target cells, and target cells concurrently shape the host's repertoire of antigen-specific T cells. The succession of these reciprocal selective sweeps can result in 'chase-and-escape' dynamics and lead to immune evasion. It has been proposed that immune evasion can be countered by immunotherapy strategies aimed at regulating the three phases of the immune response orchestrated by antigen-specific T cells: expansion, contraction and memory. Here, we test this hypothesis with a mathematical model that considers the immune response as a selection contest between T cells and target cells. The outcomes of our model suggest that shortening the duration of the contraction phase and stabilizing as many T cells as possible inside the long-lived memory reservoir, using dual immunotherapies based on the cytokines interleukin-7 and/or interleukin-15 in combination with molecular factors that can keep the immunomodulatory action of these interleukins under control, should be an important focus of future immunotherapy research.

Antigenic cooperation among intrahost HCV variants organized into a complex network of cross-immunoreactivity

Hepatitis C virus (HCV) has the propensity to cause chronic infection. Continuous immune escape has been proposed as a mechanism of intrahost viral evolution contributing to HCV persistence. Although the pronounced genetic diversity of intrahost HCV populations supports this hypothesis, recent observations of long-term persistence of individual HCV variants, negative selection increase, and complex dynamics of viral subpopulations during infection as well as broad cross-immunoreactivity (CR) among variants are inconsistent with the immune-escape hypothesis. Here, we present a mathematical model of intrahost viral population dynamics under the condition of a complex CR network (CRN) of viral variants and examine the contribution of CR to establishing persistent HCV infection. The model suggests a mechanism of viral adaptation by antigenic cooperation (AC), with immune responses against one variant protecting other variants. AC reduces the capacity of the host's immune system to neutralize certain viral variants. CRN structure determines specific roles for each viral variant in host adaptation, with variants eliciting broad-CR antibodies facilitating persistence of other variants immunoreacting with these antibodies. The proposed mechanism is supported by empirical observations of intrahost HCV evolution. Interference with AC is a potential strategy for interruption and prevention of chronic HCV infection.

GLOBAL PROPERTIES FOR A VIRUS DYNAMICS MODEL WITH LYTIC AND NON-LYTIC IMMUNE RESPONSES, AND NONLINEAR IMMUNE ATTACK RATES

We consider a mathematical model that describes a viral infection with lytic and non-lytic immune responses. One of the main features of the model is that it includes a rate of linear activation of cytotoxic T lymphocytes (CTLs) immune response, a constant production rate of CTLs export from thymus, and a nonlinear attack rate for each immune effector mechanism. Stability of the infection-free equilibrium, and existence, uniqueness and stability of an immune-controlled equilibrium, are investigated. The stability results are given in terms of the basic reproductive number. We use the method of Lyapunov functions to study the global stability of the infection-free equilibrium and the immune-controlled equilibrium. We give a sufficient condition on the non-lytic-immune attack rate for the global asymptotic stability of the immune-controlled equilibrium. By theoretical analysis and numerical simulations, we show that the lytic and non-lytic activities are required to combat a viral infection.

Improving the estimation of the death rate of infected cells from time course data during the acute phase of virus infections: application to acute HIV-1 infection in a humanized mouse model

Background

Mathematical modeling of virus dynamics has provided quantitative insights into viral infections such as influenza, the simian immunodeficiency virus/human immunodeficiency virus, hepatitis B, and hepatitis C. Through modeling, we can estimate the half-life of infected cells, the exponential growth rate, and the basic reproduction number (R₀). To calculate R₀ from virus load data, the death rate of productively infected cells is required. This can be readily estimated from treatment data collected during the chronic phase, but is difficult to determine from acute infection data. Here, we propose two new models that can reliably estimate the average life span of infected cells from acute-phase data, and apply both methods to experimental data from humanized mice infected with HIV-1.

Methods

Both new models, called as the reduced quasi-steady state (RQS) model and the piece-wise regression (PWR) model, are derived by simplification of a standard model for the acute-phase dynamics of target cells, viruses and infected cells. By having only a limited number of parameters, both models allow us to reliably estimate the death rate of productively infected cells. Simulated datasets with plausible parameter values are generated with the standard model to compare the performance of the new models with that of the major previous model (i.e., the simple exponential model). Finally, we fit models to time course data from HIV-1 infected humanized mice to estimate the several important parameters characterizing their acute infection.

Results and conclusions

The new models provided much better estimates than the previous model because they more precisely capture the de novo infection process. Both models describe the acute phase of HIV-1 infected humanized mice reasonably well, and we estimated an average death rate of infected cells of 0.61 and 0.61, an average exponential growth rate of 0.69 and 0.76, and an average basic reproduction number of 2.30 and 2.38 in the RQS model and the PWR model, respectively. These estimates are fairly close to those obtained in humans.

Can Horton hear the whos? The importance of scale in mosquito-borne disease

The epidemiology of vector-borne pathogens is determined by mechanisms and interactions at different scales of biological organization, from individual-level cellular processes to community interactions between species and with the environment. Most research, however, focuses on one scale or level with little integration between scales or levels within scales. Understanding the interactions between levels and how they influence our perception of vector-borne pathogens is critical. Here two examples of biological scales (pathogen transmission and mosquito mortality) are presented to illustrate some of the issues of scale and to explore how processes on different levels may interact to influence mosquito-borne pathogen transmission cycles. Individual variation in survival, vector competence, and other traits affect population abundance, transmission potential, and community structure. Community structure affects interactions between individuals such as competition and predation, and thus influences the individual-level dynamics and transmission potential. Modeling is a valuable tool to assess interactions between scales and how processes at different levels can affect transmission dynamics. We expand an existing model to illustrate the types of studies needed, showing that individual-level variation in viral dose acquired or needed for infection can influence the number of infectious vectors. It is critical that interactions within and among biological scales and levels of biological organization are understood for greater understanding of pathogen transmission with the ultimate goal of improving control of vector-borne pathogens.

Apoptosis in virus infection dynamics models

In this paper, on the basis of the simplified two-dimensional virus infection dynamics model, we propose two extended models that aim at incorporating the influence of activation-induced apoptosis which directly affects the population of uninfected cells. The theoretical analysis shows that increasing apoptosis plays a positive role in control of virus infection. However, after being included the third population of cytotoxic T lymphocytes immune response in HIV-infected patients, it shows that depending on intensity of the apoptosis of healthy cells, the apoptosis can either promote or comfort the long-term evolution of HIV infection. Further, the discrete-time delay of apoptosis is incorporated into the pervious model. Stability switching occurs as the time delay in apoptosis increases. Numerical simulations are performed to illustrate the theoretical results and display the different impacts of a delay in apoptosis.

Effects of replicative fitness on competing HIV strains

We develop an n-strain model to show the effects of replicative fitness of competing viral strains exerting selective density-dependant infective pressure on each other. A two strain model is used to illustrate the results. A perturbation technique and numerical simulations were used to establish the existence and stability of steady states. More than one infected steady states governed by the replicative fitness resulted from the model exhibiting either strain replacement or co-infection. We found that the presence of two or more HIV strains could result in a disease-free state that, in general, is not globally stable.

HIV evolution and progression of the infection to AIDS

In this paper, we propose and discuss a possible mechanism, which, via continuous mutations and evolution, eventually enables HIV to break from immune control. In order to investigate this mechanism, we employ a simple mathematical model, which describes the relationship between evolving HIV and the specific CTL response and explicitly takes into consideration the role of CD4(+)T cells (helper T cells) in the activation of the CTL response. Based on the assumption that HIV evolves towards higher replication rates, we quantitatively analyze the dynamical properties of this model. The model exhibits the existence of two thresholds, defined as the immune activation threshold and the immunodeficiency threshold, which are critical for the activation and persistence of the specific cell-mediated immune response: the specific CTL response can be established and is able to effectively control an infection when the virus replication rate is between these two thresholds. If the replication rate is below the immune activation threshold, then the specific immune response cannot be reliably established due to the shortage of antigen-presenting cells. Besides, the specific immune response cannot be established when the virus replication rate is above the immunodeficiency threshold due to low levels of CD4(+)T cells. The latter case implies the collapse of the immune system and beginning of AIDS. The interval between these two thresholds roughly corresponds to the asymptomatic stage of HIV infection. The model shows that the duration of the asymptomatic stage and progression of the disease are very sensitive to variations in the model parameters. In particularly, the rate of production of the naive lymphocytes appears to be crucial.

Two optimal mutation rates in obligate pathogens subject to deleterious mutation

Pathogen species with high mutation rates are likely to accumulate deleterious mutations that reduce their reproductive potential within the host. By altering the within-host growth rate of the pathogen, the deleterious mutation load has the potential to affect epidemiological properties such as prevalence, mean pathogen load, and the mean duration of infections. Here, I examine an epidemiological model that allows for multiple segregating mutations that affect within-host replication efficiency. The model demonstrates a complex range of outcomes depending on pathogen mutation rate, including two distinct, widely separated mutation rates associated with high pathogen prevalence. The low mutation rate prevalence peak is associated with small amounts of genetic diversity within the pathogen population, relatively stable prevalence and infection dynamics, and genetic variation partitioned between hosts. The high mutation rate peak is characterized by considerable genetic diversity both within and between hosts, relatively frequent invasions by more virulent types, and is qualitatively similar to an RNA virus quasispecies. The two prevalence peaks are separated by a valley where natural selection favors evolution toward the optimal within-host state, which is associated with high virulence and relatively rapid host mortality. Both chronic and acute infections are examined using stochastic forward simulations.

Global stability of models of humoral immunity against multiple viral strains

We analyse, from a mathematical point of view, the global stability of equilibria for models describing the interaction between infectious agents and humoral immunity. We consider the models that contain the variables of pathogens explicitly. The first model considers the situation where only a single strain exists. For the single strain model, the disease steady state is globally asymptotically stable if the basic reproductive ratio is greater than one. The other models consider the situations where multiple strains exist. For the multi-strain models, the disease steady state is globally asymptotically stable. In the model that does not explicitly contain an immune variable, only one strain with the maximum basic reproductive ratio can survive at the steady state. However, in our models explicitly involving the immune system, multiple strains coexist at the steady state.

Viral diversity limits immune diversity in asymptomatic phase of HIV infection

We propose a new diversity threshold theory which states that the specific CTLs to the viral strain become inactivated (that is, some HIV strain can escape from its specific immune response) when the diversity of HIV strains exceeds some threshold number. We call this number "immune diversity threshold". Our theory can explain the inactivation of specific immune response and a limit of maximum immune diversity. We can conclude that the accumulation of viral diversity eventually leads to AIDS.

Virus evolution within patients increases pathogenicity

Viruses like the human immunodeficiency virus (HIV), the hepatitis B virus (HBV), the hepatitis C virus (HCV) and many others undergo numerous rounds of inaccurate reproduction within an infected host. The resulting viral quasispecies is heterogeneous and sensitive to any selection pressure. Here we extend earlier work by showing that for a wide class of models describing the interaction between the virus population and the immune system, virus evolution has a well-defined direction toward increased pathogenicity. In particular, we study virus-induced impairment of the immune response and certain cross-reactive stimulation of specific immune responses. For eight different mathematical models, we show that virus evolution reduces the equilibrium abundance of uninfected cells and increases the rate at which uninfected cells are infected. Thus, in general, virus evolution makes things worse. An idea for combating HIV infection, however, is constructing a virus mutant that could outcompete the existing infection without being pathogenic itself.