The importance of lytic and nonlytic immune responses in viral infections

Human immunodeficiency virus dynamics in secondary lymphoid tissues and the evolution of cytotoxic T lymphocyte escape mutants

In secondary lymphoid tissues, human immunodeficiency virus (HIV) can replicate in both the follicular and extrafollicular compartments. Yet, virus is concentrated in the follicular compartment in the absence of antiretroviral therapy, in part due to the lack of cytotoxic T lymphocyte (CTL)-mediated activity there. CTLs home to the extrafollicular compartment, where they can suppress virus load to relatively low levels. We use mathematical models to show that this compartmentalization can explain seemingly counter-intuitive observations. First, it can explain the observed constancy of the viral decline slope during antiviral therapy in the peripheral blood, irrespective of the presence of CTL in Simian Immunodeficiency Virus (SIV)-infected macaques, under the assumption that CTL-mediated lysis significantly contributes to virus suppression. Second, it can account for the relatively long times it takes for CTL escape mutants to emerge during chronic infection even if CTL-mediated lysis is responsible for virus suppression. The reason is the heterogeneity in CTL activity and the consequent heterogeneity in selection pressure between the follicular and extrafollicular compartments. Hence, to understand HIV dynamics more thoroughly, this analysis highlights the importance of measuring virus populations separately in the extrafollicular and follicular compartments rather than using virus load in peripheral blood as an observable; this hides the heterogeneity between compartments that might be responsible for the particular patterns seen in the dynamics and evolution of the HIV in vivo.

Novel spatial profiles of some diffusive SIS epidemic models

In this paper, we are concerned with two SIS epidemic reaction-diffusion models with mass action infection mechanism of the form SI, and study the spatial profile of population distribution as the movement rate of the infected individuals is restricted to be small. For the model with a constant total population number, our results show that the susceptible population always converges to a positive constant which is indeed the minimum of the associated risk function, and the infected population either concentrates at the isolated highest-risk points or aggregates only on the highest-risk intervals once the highest-risk locations contain at least one interval. In sharp contrast, for the model with a varying total population number which is caused by the recruitment of the susceptible individuals and death of the infected individuals, our results reveal that the susceptible population converges to a positive function which is non-constant unless the associated risk function is constant, and the infected population may concentrate only at some isolated highest-risk points, or aggregate at least in a neighborhood of the highest-risk locations or occupy the whole habitat, depending on the behavior of the associated risk function and even its smoothness at the highest-risk locations. Numerical simulations are performed to support and complement our theoretical findings.

Mathematical Modeling and Analysis of the Dynamics of RNA Viruses in Presence of Immunity and Treatment: A Case Study of SARS-CoV-2

The emergence of novel RNA viruses like SARS-CoV-2 poses a greater threat to human health. Thus, the main objective of this article is to develop a new mathematical model with a view to better understand the evolutionary behavior of such viruses inside the human body and to determine control strategies to deal with this type of threat. The developed model takes into account two modes of transmission and both classes of infected cells that are latently infected cells and actively infected cells that produce virus particles. The cure of infected cells in latent period as well as the lytic and non-lytic immune response are considered into the model. We first show that the developed model is well-posed from the biological point of view by proving the non-negativity and boundedness of model's solutions. Our analytical results show that the dynamical behavior of the model is fully determined by two threshold parameters one for viral infection and the other for humoral immunity. The effect of antiviral treatment is also investigated. Furthermore, numerical simulations are presented in order to illustrate our analytical results.

A computational study of a stochastic fractal-fractional hepatitis B virus infection incorporating delayed immune reactions via the exponential decay

Recently, researchers have become interested in modelling, monitoring, and treatment of hepatitis B virus infection. Understanding the various connections between pathogens, immune systems, and general liver function is crucial. In this study, we propose a higher-order stochastically modified delay differential model for the evolution of hepatitis B virus transmission involving defensive cells. Taking into account environmental stimuli and ambiguities, we presented numerical solutions of the fractal-fractional hepatitis B virus model based on the exponential decay kernel that reviewed the hepatitis B virus immune system involving cytotoxic T lymphocyte immunological mechanisms. Furthermore, qualitative aspects of the system are analyzed such as the existence-uniqueness of the non-negative solution, where the infection endures stochastically as a result of the solution evolving within the predetermined system's equilibrium state. In certain settings, infection-free can be determined, where the illness settles down tremendously with unit probability. To predict the viability of the fractal-fractional derivative outcomes, a novel numerical approach is used, resulting in several remarkable modelling results, including a change in fractional-order δ with constant fractal-dimension ϖ, δ with changing ϖ, and δ with changing both δ and ϖ. White noise concentration has a significant impact on how bacterial infections are treated.

Modeling the disruption of respiratory disease clinical trials by non-pharmaceutical COVID-19 interventions

Respiratory disease trials are profoundly affected by non-pharmaceutical interventions (NPIs) against COVID-19 because they perturb existing regular patterns of all seasonal viral epidemics. To address trial design with such uncertainty, we developed an epidemiological model of respiratory tract infection (RTI) coupled to a mechanistic description of viral RTI episodes. We explored the impact of reduced viral transmission (mimicking NPIs) using a virtual population and in silico trials for the bacterial lysate OM-85 as prophylaxis for RTI. Ratio-based efficacy metrics are only impacted under strict lockdown whereas absolute benefit already is with intermediate NPIs (eg. mask-wearing). Consequently, despite NPI, trials may meet their relative efficacy endpoints (provided recruitment hurdles can be overcome) but are difficult to assess with respect to clinical relevance. These results advocate to report a variety of metrics for benefit assessment, to use adaptive trial design and adapted statistical analyses. They also question eligibility criteria misaligned with the actual disease burden.

Stochastic probical strategies in a delay virus infection model to combat COVID-19

In disease model systems, random noises and time delay factors play key role in interpreting disease dynamics to comprehend deeper insights into the course of dynamics. An endeavor to forecast intercellular behavioral dynamics of SARS-CoV-2 virus via Infection model with responsive host immune mechanisms forms the crux of this research study. Incorporation of time delay factor into infection transmission rates in noisy system epitomizes spectacular view on internal viral dynamics and stability properties are rigorously analyzed around equilibrium steady states to probe feasible strategies in mitigating rapid spread. Efforts to perceive inocular view on infection dynamics are not limited to theoretical frontiers but are substantiated with empirically simulated outcomes and visualized as graphical upshots. Discussions on numerical investigations emphasized shorter incubation periods and vaccination at pertinent time intervals to restrain massive spread and exhibit total immunity against SARS-CoV-2 infections.

Role of body temperature variations in bat immune response to viral infections

The ability of bats to coexist with viruses without being harmed is an interesting issue that is still under investigation. Here we use a mathematical model to show that the pattern of body temperature variations observed in bats between day and night is responsible for their ability to keep viruses in check. From the dynamical systems point of view, our model displays an intriguing quasi-periodic behaviour that might be relevant in making the system robust by avoiding viral escape due to perturbations in the body temperature cycle.

Mathematical Model of Antiviral Immune Response against the COVID-19 Virus

This work presents a mathematical model to investigate the current outbreak of the coronavirus disease 2019 (COVID-19) worldwide. The model presents the infection dynamics and emphasizes the role of the immune system: both the humoral response as well as the adaptive immune response. We built a mathematical model of delay differential equations describing a simplified view of the mechanism between the COVID-19 virus infection and the immune system. We conduct an analysis of the model exploring different scenarios, and our numerical results indicate that some theoretical immunotherapies are successful in eradicating the COVID-19 virus.

Mathematical analysis of an HIV model with latent reservoir, delayed CTL immune response and immune impairment

In this paper, an in-host HIV infection model with latent reservoir, delayed CTL immune response and immune impairment is investigated. By using suitable Lyapunov functions and LaSalle's invariance principle, it is shown that when time delay is equal to zero, the immunity-inactivated reproduction ratio is a threshold determining the global dynamics of the model. By means of the persistence theory for infinite dimensional systems, it is proven that if the immunity-inactivated reproduction ratio is greater than unity, the model is permanent. Choosing time delay as the bifurcation parameter and analyzing the corresponding characteristic equation of the linearized system, the existence of a Hopf bifurcation at the immunity-activated equilibrium is established. Numerical simulations are carried out to illustrate the theoretical results and reveal the effects of some key parameters on viral dynamics.

Effects of randomness on viral infection model with application

Virus population disease dynamics in various species of ecosystem keep the research interests alive for many centuries. In this research article, an attempt has been made to understand the qualitative behavior of a virus infection model with Lytic and Non-Lytic Immune Responses by perturbing with randomness (white noise) via Lyapunov technique. The conditions for the extinction and permanence of the viral infection in the interacting populations has been found, analyzed and supported with numerical simulations. An application to HIV infection model has also been presented for drawing a comparative study of the model under various modeling methods. The research findings of this paper reveal that a study that includes random fluctuations of the environment prove to be the ideal way to bring out the qualitative analysis of a mathematical model that will depict the real world scenario.

Immune response and within-host viral evolution: Immune response can accelerate evolution

The objectives of this paper are to explore the impact of immune response on within-host viral evolution towards higher Darwinian fitness and, in particular, to verify a hypothesis that immune response, which is insufficient to annihilate a viral infection, can accelerate this evolution. To address this issue, a model of within-host viral evolution with immune response is formulated. This model is an extension of a continuous phenotype space model of viral evolution that was earlier suggested by A. Korobeinikov and C. Dempsey, which incorporates strain-specific immune response with cross-immunity. The model is based upon Nowak-May and Wodarz models of within-host HIV dynamics and is mechanistic (based upon first principles); this allows straightforward interpretation of the model's parameters and simulation results, as well as its further developments. In order to make the simulation results and conclusions robust and reliable and to ensure that they do not depend on the particularities of an immune response model, four different mathematical models of cell-mediated immune response are considered with the proposed model. Simulations confirmed that immune response, when it is unable to eliminate viruses, accelerates viral evolution.

BISTABILITY ANALYSIS OF AN HIV MODEL WITH IMMUNE RESPONSE

Some HIV-infected patients (the so-called post-treatment controllers) can control the virus after cessation of antiretroviral therapy. A small fraction of patients can even naturally maintain undetectable viral load without therapy (they are called elite controllers). The immune response may play an important role in viral control in these patients. In this paper, we analyze a within-host model including immune response to study the virus dynamics in HIV-infected patients. We derived two threshold values for the immune cell proliferation parameter. Below the lower immune proliferation rate, the model has a stable immune-free steady state, which predicts that patients have a high viral load. Above the higher immune proliferation rate, the model has a stable low infected steady state, which indicates that patients are under elite control. Between the two immune thresholds, the model exhibits the dynamic behavior of bistability, which suggests that patients either undergo viral rebound after treatment termination or achieve the post-treatment control. These results may explain the different virus dynamics in HIV-infected patients.

Analysis of a hepatitis B viral infection model with immune response delay

In this paper, a hepatitis B viral infection model with a density-dependent proliferation rate of cytotoxic T lymphocyte (CTL) cells and immune response delay is investigated. By analyzing the model, we show that the virus-free equilibrium is globally asymptotically stable, if the basic reproductive ratio is less than one and an endemic equilibrium exists if the basic reproductive ratio is greater than one. By using the stability switches criterion in the delay-differential system with delay-dependent parameters, we present that the stability of endemic equilibrium changes and eventually become stable as time delay increases. This means majority of hepatitis B infection would eventually become a chronic infection due to the immune response time delay is fairly long. Numerical simulations are carried out to explain the mathematical conclusions and biological implications.

Multi-scale problem in the model of RNA virus evolution

A mathematical or computational model in evolutionary biology should necessary combine several comparatively fast processes, which actually drive natural selection and evolution, with a very slow process of evolution. As a result, several very different time scales are simultaneously present in the model; this makes its analytical study an extremely difficult task. However, the significant difference of the time scales implies the existence of a possibility of the model order reduction through a process of time separation. In this paper we conduct the procedure of model order reduction for a reasonably simple model of RNA virus evolution reducing the original system of three integro-partial derivative equations to a single equation. Computations confirm that there is a good fit between the results for the original and reduced models.

Mathematical model of plant-virus interactions mediated by RNA interference

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. Analytical and numerical bifurcation analyses are employed to investigate the stability of all steady states of the model in order to identify parameter regions where the system exhibits synergistic or antagonistic behaviour between viral strains, as well as different types of host recovery. We show that not only viral attributes but also the propagating component of RNA-interference in plants can play an important role in determining the dynamics.

Modeling immune response and its effect on infectious disease outbreak dynamics

BACKGROUND

In recent epidemiological models, immunity is incorporated as a simplified value that determines the capacity of an individual to become infected or to transmit the disease. Moreover, the quality of the immune response determines the chances of infection and the length of time an individual is capable to infect others. We present a model that incorporates individuals' immune responses to, further, examine the role of the collective immune response of individuals in a population during an infectious outbreak.

METHODS

We constructed a contagion model that incorporates the collective immune response of individuals represented by the superposition of individual immune responses (PIR). Multiple probability distributions are used to represent the immunocompetence of different age groups, thereby modeling the concept of Population Immune Response (PIR). Multiple experiments were conducted in which the population is divided in different age groups for which each group has a unique immune response quality and thus a different length for its immune periods. Finally, we explored the effects of implementing different vaccination strategies in the population.

RESULTS

The experiments displayed important variations in the outbreak dynamics as a consequence of incorporating PIR in homogeneous and mixed populations. The experiments showed that individuals with weak immune responses and those who are immune to the pathogen play a significant role in shaping the outbreak dynamics. Finally, after implementing different vaccination strategies, the results suggest that if vaccination resources are limited, the vaccination should be targeted towards individuals that spread the disease for a longer period of time.

CONCLUSIONS

Our results suggest that it is essential for the public health establishment to increase their understanding of the characteristics of regional demographics that could impact the quality of the immune response of the individuals. The results indicate that it is necessary to further investigate mitigation strategies to limit the capacity to transmit the disease by individuals that spread the pathogen for extended periods of time. Ultimately, this study suggests that it is crucial for public health researchers to identify appropriate targeted vaccination regimes and to explore the link between PIR and outbreak dynamics to improve the monitoring and mitigating efforts of ongoing and future epidemics.

Order reduction for an RNA virus evolution model

A mathematical or computational model in evolutionary biology should necessary combine several comparatively fast processes, which actually drive natural selection and evolution, with a very slow process of evolution. As a result, several very different time scales are simultaneously present in the model; this makes its analytical study an extremely difficult task. However, the significant difference of the time scales implies the existence of a possibility of the model order reduction through a process of time separation. In this paper we conduct the procedure of model order reduction for a reasonably simple model of RNA virus evolution reducing the original system of three integro-partial derivative equations to a single equation. Computations confirm that there is a good fit between the results for the original and reduced models.

Modeling T cell responses to antigenic challenge

T cell responses are a crucial part of the adaptive immune system in the fight against infections. This article discusses the use of mathematical models for understanding the dynamics of cytotoxic T lymphocyte (CTL) responses against viral infections. Complementing experimental research, mathematical models have been very useful for exploring new hypotheses, interpreting experimental data, and for defining what needs to be measured to improve understanding. This review will start with minimally parameterized models of CTL responses, which have generated some valuable insights into basic dynamics and correlates of control. Subsequently, more biological complexity is incorporated into this modeling framework, examining different mechanisms of CTL expansion, different effector activities, and the influence of T cell help. Models and results are discussed in the context of data from specific infections.

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.

Predicting the impact of CD8+ T cell polyfunctionality on HIV disease progression

During the chronic phase of HIV-1 infection, polyfunctional CD8+ T cell responses, which are characterized by a high frequency of cells able to secrete multiple cytokines simultaneously, are associated with lower virus loads and slower disease progression. This relationship may arise for different reasons. Polyfunctional responses may simply be stronger. Alternatively, it could be that the increased functional diversity in polyfunctional responses leads to lower virus loads and slower disease progression. Lastly, polyfunctional responses could contain more CD8+ T cells that mediate a specific key function that is primarily responsible for viral control. Disentangling the influences of overall strength, functional diversity, and specific function on viral control and disease progression is very relevant for the rational design of vaccines and immunotherapy using cellular immune responses. We developed a mathematical model to study how polyfunctional CD8+ T cell responses mediating lytic and nonlytic effector functions affect the CD4+ T cell count and plasma viral load. We based our model on in vitro data on the efficacy of gamma interferon (IFN-γ) and macrophage inflammatory protein 1β (MIP-1β)/RANTES against HIV. We find that the strength of the response is a good predictor of disease progression, while functional diversity has only a minor influence. In addition, our model predicts for realistic levels of cytotoxicity that immune responses dominated by nonlytic effector functions most positively influence disease outcome.

IMPORTANCE

It is an open question in HIV research why polyfunctional CD8+ T cell responses are associated with better viral control, while individual functional correlates of protection have not been identified so far. Identifying the role of CD8+ T cells in HIV-1 infection has important implications for the potential development of effective T cell-based vaccines. Our analysis provides new ways to think about a causative role of CD8+ T cells by studying different hypotheses regarding why polyfunctional CD8+ T cells might be more advantageous. We identify measurements that have to be obtained in order to evaluate the role of CD8+ T cells in HIV-1 infection. In addition, our method shows how individual cell functionality data can be used in population-based virus dynamics models.

MODELING AND OPTIMAL CONTROL FOR ANTIRETROVIRAL THERAPY

We consider a three-dimensional nonlinear control model based on the Wodarz HIV model. The model phase variables are populations of the uninfected and infected target cells and the concentration of an antiretroviral drug. The drug intake rate is assumed to be a bounded control function. An optimal control problem of minimizing the cumulative infection level (the infected cells population) on a given time interval is stated and solved, and the types of the optimal control for different model parameters are found by analytical methods. We thereby reduce the two-point boundary value problem (TPBVP) for the Pontryagin maximum principle to a problem of the finite-dimensional optimization. Numerical results are presented to illustrate the optimal solution.

Complex dynamic behavior in a viral model with state feedback control strategies

With the consideration of mechanism of prevention and control for the spread of viral diseases, in this paper, we propose two novel virus dynamics models where state feedback control strategies are introduced. The first model incorporates the density of infected cells (or free virus) as control threshold value; we analytically show the existence and orbit stability of positive periodic solution. Theoretical results imply that the density of infected cells (or free virus) can be controlled within an adequate level. The other model determines the control strategies by monitoring the density of uninfected cells when it reaches a risk threshold value. We analytically prove the existence and orbit stability of semi-trivial periodic solution, which show that the viral disease dies out. Numerical simulations are carried out to illustrate the main results.

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.

Modeling and simulating dynamics of complete- and poor-response chronic hepatitis B chinese patients for adefovir and traditional chinese medicine plus adefovir therapy

ChiCTR-TRC-11001263 study was the first large-scale double-blind randomized placebo-controlled traditional Chinese medicines (TCMs) and adefovir (ADV) antihepatitis B virus (HBV) infection trial in the world. A total of 560 hepatitis B e antigen- (HBeAg-) positive Chinese patients with chronical HBV were randomly classified, in 1 : 1 ratio, into two groups: experimental group (EXG) receiving TCMs + ADV and controlled group (CTG) receiving ADV + TCM-placebo treatment for 48 weeks. This paper introduces two models to model and simulate the evolutions of dynamics for the complete-response patients and the poor-response patients in EXG and CTG, respectively. The stimulated mean HBV DNA and alanine aminotransferase (ALT) levels were close to the patients' experimental data. Analysis and simulations suggest that the activated patients' immune functions by TCMs + ADV may not only clear infected hepatocytes, but also clear HBV, which made the complete-response patients' mean serum HBV DNA levels in EXG reduce rapidly 12 weeks' earlier than the ones in CTG. One can assume that both the TCMs and ADV have the function of preventing complete-response patients' infected hepatocytes from being injured by cytotoxic T lymphocytes (CTLs); the patients' activated immune cells may also block HBV replications.

HOW SELF-PROLIFERATION OF CD4+T CELLS AFFECT THE HIV DYNAMICS IN AN IN-HOST TARGET-CELL LIMITED HIV MODEL WITH SATURATION INFECTION RATE: A QUASI-STEADY-STATE APPROXIMATION ANALYSIS

In this study, we consider two target-cell limited models with saturation type infection rate and intracellular delay: one without self-proliferation and the other with self-proliferation of activated CD4+T cells. We discuss about the local and global behavior of both the systems in presence and absence of intracellular delay. It is shown that the endemic equilibrium of a target-cell limited model would be unstable in presence and absence of intracellular delay only when self-proliferation of activated CD4+T cell is considered. Otherwise, all positive solutions converge to the endemic equilibrium or disease-free equilibrium depending on whether the basic reproduction ratio is greater than or less than unity. Our study suggests that amplitude of oscillation is negatively correlated with the constant input rate of CD4+T cell when intracellular delay is absent or low. However, they are positively correlated if the delay is too high. Amplitude of oscillation, on the other hand, is always positively correlated with the proliferation rate of CD4+T cell for all delay. Our mathematical and simulation analysis also suggest that there are many potential contributors who are responsible for the variation of CD4+T cells and virus particles in the blood plasma of HIV patients.

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.

Lethal mutagenesis in viruses and bacteria

In this work we study how mutations that change physical properties of cell proteins (stability) affect population survival and growth. We present a model in which the genotype is presented as a set folding free energies of cell proteins. Mutations occur upon replication, so stabilities of some proteins in daughter cells differ from those in the parent cell by amounts deduced from the distribution of mutational effects on protein stability. The genotype-phenotype relationship posits that the cell's fitness (replication rate) is proportional to the concentration of its folded proteins and that unstable essential proteins result in lethality. Simulations reveal that lethal mutagenesis occurs at a mutation rate close to seven mutations in each replication of the genome for RNA viruses and at about half that rate for DNA-based organisms, in accord with earlier predictions from analytical theory and experimental results. This number appears somewhat dependent on the number of genes in the organisms and the organism's natural death rate. Further, our model reproduces the distribution of stabilities of natural proteins, in excellent agreement with experiments. We find that species with high mutation rates tend to have less stable proteins compared to species with low mutation rates.

A chronic viral infection model with immune impairment

In this paper, a chronic viral infection model with cell-mediated immunity and immune impairment is proposed and studied, under the assumption that the presence of the antigen can both stimulate and impair immunity. It is shown that the virus persists in the host if the basic reproductive ratio of the virus is greater than 1. The immune cells persist when there is only one positive equilibrium. The system can exhibit two positive equilibria if the basic reproductive ratio of the virus is above a threshold. This allows a bistable behavior, and the immune cells persist or die out, i.e., infection will result in disease or immune control outcome, depending on the initial conditions. By theoretical analysis and numerical simulations, we show that therapy could shift the patient from a disease progression to an immune control outcome, despite that the therapy is not necessarily lifelong. This would allow the immune response to control the virus in the long term even in the absence of continued therapy.

Dynamics of an HBV model with diffusion and delay

In this paper we model and analyze the hepatitis B virus (HBV) infection in a diffusion model confined to a finite domain, induced by intracellular time delay between infection of a cell and production of new virus particles. The equilibrium solutions are obtained and the stability is analyzed if the space is assumed as homogeneous. When the space is inhomogeneous, the effects of diffusion and intracellular time delay are obtained by computer simulations.

Human immunodeficiency virus evolution towards reduced replicative fitness in vivo and the development of AIDS

Human immunodeficiency virus (HIV) infection progresses to AIDS following an asymptomatic period during which the virus is thought to evolve towards increased fitness and pathogenicity. We show mathematically that progression to the strongest HIV-induced pathology requires evolution of the virus towards reduced replicative fitness in vivo. This counter-intuitive outcome can happen if multiple viruses co-infect the same cell frequently, which has been shown to occur in recent experiments. According to our model, in the absence of frequent co-infection, the less fit AIDS-inducing strains might never emerge. The frequency of co-infection can correlate with virus load, which in turn is determined by immune responses. Thus, at the beginning of infection when immunity is strong and virus load is low, co-infection is rare and pathogenic virus variants with reduced replicative fitness go extinct. At later stages of infection when immunity is less efficient and virus load is higher, co-infection occurs more frequently and pathogenic virus variants with reduced replicative fitness can emerge, resulting in T-cell depletion. In support of these notions, recent data indicate that pathogenic simian immunodeficiency virus (SIV) strains occurring late in the infection are less fit in specific in vitro experiments than those isolated at earlier stages. If co-infection is blocked, the model predicts the absence of any disease even if virus loads are high. We hypothesize that non-pathogenic SIV infection within its natural hosts, which is characterized by the absence of disease even in the presence of high virus loads, could be explained by a reduced occurrence of co-infection in this system.

Propagation of HBV with spatial dependence

A mathematical model is proposed to simulate the hepatitis B virus (HBV) infection with spatial dependence. The existence of traveling waves is established via the geometric singular perturbation method. Numerical simulations show that the model admits non-monotone traveling profiles. Influences of various parameters on the minimum wave speed are also discussed.

Dynamics of killer T cell inflation in viral infections

Upon acute viral infection, a typical cytotoxic T lymphocyte (CTL) response is characterized by a phase of expansion and contraction after which it settles at a relatively stable memory level. Recently, experimental data from mice infected with murine cytomegalovirus (MCMV) showed different and unusual dynamics. After acute infection had resolved, some antigen specific CTL started to expand over time despite the fact that no replicative virus was detectable. This phenomenon has been termed as "CTL memory inflation". In order to examine the dynamics of this system further, we developed a mathematical model analysing the impact of innate and adaptive immune responses. According to this model, a potentially important contributor to CTL inflation is competition between the specific CTL response and an innate natural killer (NK) cell response. Inflation occurs most readily if the NK cell response is more efficient than the CTL at reducing virus load during acute infection, but thereafter maintains a chronic virus load which is sufficient to induce CTL proliferation. The model further suggests that weaker NK cell mediated protection can correlate with more pronounced CTL inflation dynamics over time. We present experimental data from mice infected with MCMV which are consistent with the theoretical predictions. This model provides valuable information and may help to explain the inflation of CMV specific CD8+T cells seen in humans as they age.

Dynamical modeling of viral spread in spatially distributed populations: stochastic origins of oscillations and density dependence

In order to understand the spatio-temporal structure of epidemics beyond that permitted with classical SIR (susceptible-infective-recovered)-type models, a new mathematical model for the spread of a viral disease in a population of spatially distributed hosts is described. The positions of the hosts are randomly generated in a rectangular habitat. Encounters between any pair of individuals are according to a Poisson process with a mean rate that declines exponentially as the distance between them increases. The contact rate allows the mean rates to be set at a certain number of encounters per day on average. The relevant state variables for each individual at any time are given by the solution of a pair of coupled differential equations for the viral load and the quantity of general immune system effectors which reduce the viral load. The parameters describing within-host viral-immune system dynamics are generated randomly to reflect variability across a population. Transmission is assumed to depend on the viral loads in donors and occurs with a probability ptrans. The initial conditions are such that one randomly chosen individual carries a randomly chosen amount of the virus, whereas the rest of the population is uninfected. Simulations reveal local or whole-population responses. Whole-population disease spread may be in the form of isolated or multiple occurrences, the latter often being approximately periodic. The mechanisms of this oscillatory behaviour are analyzed in terms of several parameters and the distribution of critical points in the host dynamical systems. Increased contact rate, increased probability of transmission and decreased threshold for viral transmission, decreased immune strength and increased viral growth rate all increase the probability of multiple outbreaks and the distribution of the critical points also plays a role.

Ecological and evolutionary principles in immunology

Experimental immunology has given rise to detailed insights into how immune cells react to infectious agents and fight pathogens. At the same time, however, the interplay between infectious agents and immune responses can be viewed as an ecological system in vivo. This is characterized by complex interactions between species of immune cells and populations of pathogens. This review discusses how an understanding of the immune system can be aided by the application of ecological and evolutionary principles: competition, predation, and the evolution of viruses in vivo. These concepts can shed light onto important immunological concepts such as the correlates of efficient virus control, immunodominance, the relationship between viral evolution and the development of pathology, as well as the ability of the immune system to control immunosuppressive infections.

Perforin-deficient CD8+ T cells mediate fatal lymphocytic choriomeningitis despite impaired cytokine production

Intracerebral (i.c.) infection with lymphocytic choriomeningitis virus (LCMV) is one of the most studied models for virus-induced immunopathology, and based on results from perforin-deficient mice, it is currently assumed that fatal disease directly reflects perforin-mediated cell lysis. However, recent studies have revealed additional functional defects within the effector T cells of LCMV-infected perforin-deficient mice, raising the possibility that perforin may not be directly involved in mediating lethal disease. For this reason, we decided to reevaluate the role of perforin in determining the outcome of i.c. infection with LCMV. We confirmed that the expansion of virus-specific CD8(+) T cells is unimpaired in perforin-deficient mice. However, despite the fact that the virus-specific CD8(+) effector T cells in perforin-deficient mice are broadly impaired in their effector function, these mice invariably succumb to i.c. infection with LCMV strain Armstrong, although a few days later than matched wild-type mice. Upon further investigation, we found that this delay correlates with the delayed recruitment of inflammatory cells to the central nervous system (CNS). However, CD8(+) effector T cells were not kept from the CNS by sequestering in infected extraneural organ sites such as liver or lungs. Thus, the observed dysfunctionality regarding the production of proinflammatory mediators probably results in the delayed recruitment of effector cells to the CNS, and this appears to be the main explanation for the delayed onset of fatal disease in perforin-deficient mice. However, once accumulated in the CNS, virus-specific CD8(+) T cells can induce fatal CNS pathology despite the absence of perforin-mediated lysis and reduced capacity to produce several key cytokines.

CD27 expression promotes long-term survival of functional effector-memory CD8+ cytotoxic T lymphocytes in HIV-infected patients

Human immunodeficiency virus (HIV)-specific CD8(+) T cells persist in high frequencies in HIV-infected patients despite impaired CD4(+) T helper response to the virus, but, unlike other differentiated effector cytotoxic T lymphocytes, most continue to express the tumor necrosis factor receptor family member CD27. Because the ligand for CD27 (CD70) is also overexpressed in HIV-infected hosts, we examined the nature of expression and potential functional consequences of CD27 expression on HIV-specific CD8(+) T cells. Analysis of CD27(+) and CD27(-) T cells derived from the same HIV-specific clone revealed that retention of CD27 did not interfere with acquisition of effector functions, and that after T cell receptor stimulation, CD27(+) cells that concurrently were triggered via CD27 exhibited more resistance to apoptosis, interleukin 2 production, and proliferation than CD27(-) T cells. After transfer back into an HIV-infected patient, autologous HIV-specific CD27(-) T cells rapidly disappeared, but CD27(+) T cells derived from the same clone persisted at high frequency. Our findings suggest that the CD27-CD70 interaction in HIV infection may provide CD27(+) CD8(+) T cells with a survival advantage and compensate for limiting or absent CD4(+) T help to maintain the CD8 response.

The Role of T Cells in Corona-Virus-Induced Demyelination

Mice infected with neurotropic strains of coronavirus develop acute encephalomyelitis and eliminate infectious virus. However, control of acute infection is incomplete resulting in persistence of viral RNA in the central nervous system (CNS) associated with ongoing primary demyelination. A high prevalence of virus specific CD8 and CD4 T cells within the CNS correlates with ex vivo cytolytic activity and IFN-γ secretion, which are both required for virus reduction during the acute infection. Although most infected cell types are susceptible to perforin mediated clearance, IFN-γ is required for controlling infection of oligodendrocytes. Furthermore, by enhancing class I expression and inducing class II expression within resident CNS cells IFN-γ optimizes T cell receptor dependent functions. In addition to its direct anti viral activity, these multifactorial effects make IFN-γ more essential than perforin for viral control. CD4 T cells enhance CD8 T cell expansion, survival and effectiveness. Although both CD8 and CD4 T cells are retained within the CNS during persistence, they cannot control viral recrudescence in the absence of humoral immunity. Demyelination can be mediated by either CD8 or CD4 T cells; however, although a variety of effector molecules have been excluded, a dominant common denominator remains elusive. Thus concerted efforts to control infection coincide with a variety of potential mechanisms causing chronic demyelinating disease.

Can we really learn from model pathogens?

Cytolytic T (Tc) lymphocytes are the first order response of the adaptive immune system in the recovery from primary viral infections. These effector cells execute their function either by direct cytotoxicity through the Fas or perforin pathway and/or by the release of cytokines that either directly or indirectly exert antiviral activity. Mice respond to infection by closely related viruses with a vigorous Tc response, which is characterized by extensive crossreactivity on target cells infected with these viruses. However, the action of these cells can be beneficial, detrimental or neutral. From our current state of knowledge, no generalizations as to protective or detrimental effects of cytolytic effector functions in recovery from virus infections can be made. Thus, virus-host immune interactions have to be assessed individually and cannot be generalized.

Perforin and gamma interferon-mediated control of coronavirus central nervous system infection by CD8 T cells in the absence of CD4 T cells

Infection of the central nervous system (CNS) with the neurotropic JHM strain of mouse hepatitis virus produces acute and chronic demyelination. The contributions of perforin-mediated cytolysis and gamma interferon (IFN-gamma) secretion by CD8(+) T cells to the control of infection and the induction of demyelination were examined by adoptive transfer into infected SCID recipients. Untreated SCID mice exhibited uncontrolled virus replication in all CNS cell types but had little or no demyelination. Memory CD8(+) T cells from syngeneic wild-type (wt), perforin-deficient, or IFN-gamma-deficient (GKO) donors all trafficked into the infected CNS in the absence of CD4(+) T cells and localized to similar areas. Although CD8(+) T cells from all three donors suppressed virus replication in the CNS, GKO CD8(+) T cells expressed the least antiviral activity. A distinct viral antigen distribution in specific CNS cell types revealed different mechanisms of viral control. While wt CD8(+) T cells inhibited virus replication in all CNS cell types, cytolytic activity in the absence of IFN-gamma suppressed the infection of astrocytes, but not oligodendroglia. In contrast, cells that secreted IFN-gamma but lacked cytolytic activity inhibited replication in oligodendroglia, but not astrocytes. Demyelination was most severe following viral control by wt CD8(+) T cells but was independent of macrophage infiltration. These data demonstrate the effective control of virus replication by CD8(+) T cells in the absence of CD4(+) T cells and support the necessity for the expression of distinct effector mechanisms in the control of viral replication in distinct CNS glial cell types.

The race between initial T-helper expansion and virus growth upon HIV infection influences polyclonality of the response and viral set-point

Infection with HIV is characterized by very diverse disease-progression patterns across patients, associated with a wide variation in viral set-points. Progression is a multifactorial process, but an important role has been attributed to the HIV-specific T-cell response. To explore the conditions under which different set-points may be explained by differences in initial CD4 and CD8 T-cell responses and virus inoculum, we have formulated a model assuming that HIV-specific CD4 cells are both targets for infection and mediators of a monoclonal or polyclonal immune response. Clones differ in functional avidity for HIV epitopes. Importantly, in contrast to previous models, in this model we obtained coexistence of multiple clones at steady-state viral set-point, as seen in HIV infection. We found that, for certain parameter conditions, multiple steady states are possible: with few initial CD4 helper cells and high virus inoculum, no immune response is established and target-cell-limited infection follows, with associated high viral load; when CD4 clones are initially large and virus inoculum is low, infection can be controlled by several clones. The conditions for the dependence of viral set-point on initial inoculum and CD4 T-helper clone availability are investigated in terms of the effector mechanism of the clones involved.

Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses

This paper investigates the role of CTL and antibody responses in hepatitis C virus (HCV) dynamics and pathology. Mathematical models suggest that a strong CTL response is required for resolution of HCV infection and that a weak CTL response can result in persistent infection. According to the model, establishment of persistent infection is accompanied mainly by an ongoing antibody response, while CTLs are not maintained at high levels. In the model, this outcome correlates with absence of pathology. Persistent infection in the face of an ongoing antibody response can result in evolution of antigenic escape. According to the model, evolution towards escape from antibodies can shift the balance of immune responses so that the weak CTL levels become increasingly more dominant relative to antibodies. This shift results in onset of liver pathology as the virus evolves towards increased levels of antigenic escape. Therefore, the relative balance of the immune response can be a decisive factor that determines whether patients are asymptomatic or whether pathology is observed. Virus evolution can shift this balance towards pathology over time. Theoretical results are discussed in the context of published data.

Modelling the dynamics of LCMV infection in mice: II. Compartmental structure and immunopathology

In this study, we develop a mathematical model for analysis of the compartmental aspects and immunopathology of lymphocytic choriomeningitis virus (LCMV) infection in mice. We used sets of original and published data on systemic (extrasplenic) virus distribution to estimate the parameters of virus growth and elimination for spleen and other anatomical compartments, such as the liver, kidney, thymus and lung as well as transfer rates between blood and the above organs. A mathematical model quantitatively integrating the virus distribution kinetics in the host, the specific cytotoxic T lymphocyte (CTL) response in spleen and the re-circulation of effector CTL between spleen, blood and liver is advanced to describe the CTL-mediated immunopathology (hepatitis) in mice infected with LCMV. For intravenous and "peripheral" routes of infection we examine the severity of the liver disease, as a function of the virus dose and the host's immune status characterized by the numbers of precursor and/or cytolytic effector CTL. The model is used to predict the efficacy of protection against virus persistence and disease in a localized viral infection as a function of the composition of CTL population. The modelling analysis suggests quantitative demands to CTL memory for maximal protection against a wide range of doses of infection with a primarily peripheral site of virus replication without the risk of favoring immunopathology. It specifies objectives for CTL vaccination to ensure virus elimination with minimal immunopathology vs. vaccination for disease.

Failing immune control as a result of impaired CD8+ T-cell maturation: CD27 might provide a clue

Despite readily detectable virus-specific CD8(+) T cells in most HIV-infected patients, immune surveillance is eventually lost, leading to progression to AIDS. Recently developed insights into human T-cell differentiation have been used to study the phenotype of virus-specific T cells in HIV-infected individuals. Based on these results, we propose that failing immune control in human viral infection could be a result of impaired cytotoxic T-lymphocyte (CTL) maturation into fully differentiated effector T cells. Impaired maturation is not confined to HIV-specific CD8(+) T cells but could also be involved in failing immunity to Epstein-Barr virus and other viral infections. We postulate that CD27(-) effector CD8(+) T cells might be required for adequate control of chronic viral infection and prevention of disease development.