Optimizing within-host viral fitness: infected cell lifespan and virion production rate

Decay of HCoV-OC43 infectivity is lower in cell debris-containing media than in fresh culture media

In the quantitative description of viral dynamics within cell cultures and, more broadly, in modeling within-host viral infections, a question that commonly arises is whether the degradation of a fraction of the virus could be disregarded in comparison with the massive synthesis of new viral particles. Surprisingly, quantitative data on the synthesis and degradation rates of RNA viruses in cell cultures are scarce. In this study, we investigated the decay of the human betacoronavirus OC43 (HCoV-OC43) infectivity in cell culture lysates and in fresh media. Our findings revealed a significantly slower viral decay rate in the medium containing lysate cells compared to the fresh medium. This observation suggests that the presence of cellular debris from lysed cells may offer protection or stabilize virions, slowing down their degradation. Moreover, the growth rate of HCoV-OC43 infectivity is significantly higher than degradation as long as there are productive cells in the medium, suggesting that, as a first approximation, degradation can be neglected during early infection.

Mathematical Modeling of Virus-Mediated Syncytia Formation: Past Successes and Future Directions

Many viruses have the ability to cause cells to fuse into large multi-nucleated cells, known as syncytia. While the existence of syncytia has long been known and its importance in helping spread viral infection within a host has been understood, few mathematical models have incorporated syncytia formation or examined its role in viral dynamics. This review examines mathematical models that have incorporated virus-mediated cell fusion and the insights they have provided on how syncytia can change the time course of an infection. While the modeling efforts are limited, they show promise in helping us understand the consequences of syncytia formation if future modeling efforts can be coupled with appropriate experimental efforts to help validate the models.

Evolutionary dynamics within and among competing groups

Biological and social systems are structured at multiple scales, and the incentives of individuals who interact in a group may diverge from the collective incentive of the group as a whole. Mechanisms to resolve this tension are responsible for profound transitions in evolutionary history, including the origin of cellular life, multicellular life, and even societies. Here, we synthesize a growing literature that extends evolutionary game theory to describe multilevel evolutionary dynamics, using nested birth-death processes and partial differential equations to model natural selection acting on competition within and among groups of individuals. We analyze how mechanisms known to promote cooperation within a single group-including assortment, reciprocity, and population structure-alter evolutionary outcomes in the presence of competition among groups. We find that population structures most conducive to cooperation in multiscale systems can differ from those most conducive within a single group. Likewise, for competitive interactions with a continuous range of strategies we find that among-group selection may fail to produce socially optimal outcomes, but it can nonetheless produce second-best solutions that balance individual incentives to defect with the collective incentives for cooperation. We conclude by describing the broad applicability of multiscale evolutionary models to problems ranging from the production of diffusible metabolites in microbes to the management of common-pool resources in human societies.

Dynamic analysis of a cytokine-enhanced viral infection model with infection age

Recent studies reveal that pyroptosis is associated with the release of inflammatory cytokines which can attract more target cells to be infected. In this paper, a novel age-structured virus infection model incorporating cytokine-enhanced infection is investigated. The asymptotic smoothness of the semiflow is studied. With the help of characteristic equations and Lyapunov functionals, we have proved that both the local and global stabilities of the equilibria are completely determined by the threshold $ \mathcal{R}_0 $. The result shows that cytokine-enhanced viral infection also contributes to the basic reproduction number $ \mathcal{R}_0 $, implying that it may not be enough to eliminate the infection by decreasing the basic reproduction number of the model without considering the cytokine-enhanced viral infection mode. Numerical simulations are carried out to illustrate the theoretical results.

Assortment and Reciprocity Mechanisms for Promotion of Cooperation in a Model of Multilevel Selection

In the study of the evolution of cooperation, many mechanisms have been proposed to help overcome the self-interested cheating that is individually optimal in the Prisoners' Dilemma game. These mechanisms include assortative or networked social interactions, other-regarding preferences considering the payoffs of others, reciprocity rules to establish cooperation as a social norm, and multilevel selection involving simultaneous competition between individuals favoring cheaters and competition between groups favoring cooperators. In this paper, we build on recent work studying PDE replicator equations for multilevel selection to understand how within-group mechanisms of assortment, other-regarding preferences, and both direct and indirect reciprocity can help to facilitate cooperation in concert with evolutionary competition between groups. We consider a group-structured population in which interactions between individuals consist of Prisoners' Dilemma games and study the dynamics of multilevel competition determined by the payoffs individuals receive when interacting according to these within-group mechanisms. We find that the presence of each of these mechanisms acts synergistically with multilevel selection for the promotion of cooperation, decreasing the strength of between-group competition required to sustain long-time cooperation and increasing the collective payoff achieved by the population. However, we find that only other-regarding preferences allow for the achievement of socially optimal collective payoffs for Prisoners' Dilemma games in which average payoff is maximized by an intermediate mix of cooperators and defectors. For the other three mechanisms, the multilevel dynamics remain susceptible to a shadow of lower-level selection, as the collective outcome fails to exceed the payoff of the all-cooperator group.

Long-time behavior of a PDE replicator equation for multilevel selection in group-structured populations

In many biological systems, natural selection acts simultaneously on multiple levels of organization. This scenario typically presents an evolutionary conflict between the incentive of individuals to cheat and the collective incentive to establish cooperation within a group. Generalizing previous work on multilevel selection in evolutionary game theory, we consider a hyperbolic PDE model of a group-structured population, in which members within a single group compete with each other for individual-level replication; while the group also competes against other groups for group-level replication. We derive a threshold level of the relative strength of between-group competition such that defectors take over the population below the threshold while cooperation persists in the long-time population above the threshold. Under stronger assumptions on the initial distribution of group compositions, we further prove that the population converges to a steady state density supporting cooperation for between-group selection strength above the threshold. We further establish long-time bounds on the time-average of the collective payoff of the population, showing that the long-run population cannot outperform the payoff of a full-cooperator group even in the limit of infinitely-strong between-group competition. When the group replication rate is maximized by an intermediate level of within-group cooperation, individual-level selection casts a long shadow on the dynamics of multilevel selection: no level of between-group competition can erase the effects of the individual incentive to defect. We further extend our model to study the case of multiple types of groups, showing how the games that groups play can coevolve with the level of cooperation.

Kinetics of Asian and African Zika virus lineages over single-cycle and multi-cycle growth in culture: Gene expression, cell killing, virus production, and mathematical modeling

Since 2014, an Asian lineage of Zika virus has caused outbreaks, and it has been associated with neurological disorders in adults and congenital defects in newborns. The resulting threat of the Zika virus to human health has prompted the development of new vaccines, which have yet to be approved for human use. Vaccines based on the attenuated or chemically inactivated virus will require large-scale production of the intact virus to meet potential global demands. Intact viruses are produced by infecting cultures of susceptible cells, a dynamic process that spans from hours to days and has yet to be optimized. Here, we infected Vero cells adhesively cultured in well-plates with two Zika virus strains: a recently isolated strain from the Asian lineage, and a cell-culture-adapted strain from the African lineage. At different time points post-infection, virus particles in the supernatant were quantified; further, microscopy images were used to quantify cell density and the proportion of cells expressing viral protein. These measurements were performed across multiple replicate samples of one-step infections every four hours over 60 h and for multi-step infections every four to 24 h over 144 h, generating a rich data set. For each set of data, mathematical models were developed to estimate parameters associated with cell infection and virus production. The African-lineage strain was found to produce a 14-fold higher yield than the Asian-lineage strain in one-step growth and a sevenfold higher titer in multi-step growth, suggesting a benefit of cell-culture adaptation for developing a vaccine strain. We found that image-based measurements were critical for discriminating among different models, and different parameters for the two strains could account for the experimentally observed differences. An exponential-distributed delay model performed best in accounting for multi-step infection of the Asian strain, and it highlighted the significant sensitivity of virus titer to the rate of viral degradation, with implications for optimization of vaccine production. More broadly, this study highlights how image-based measurements can contribute to the discrimination of virus-culture models for the optimal production of inactivated and attenuated whole-virus vaccines.

Analysis of Multilevel Replicator Dynamics for General Two-Strategy Social Dilemma

Here, we consider a game-theoretic model of multilevel selection in which individuals compete based on their payoff and groups also compete based on the average payoff of group members. Our focus is on multilevel social dilemmas: games in which individuals are best off cheating, while groups of individuals do best when composed of many cooperators. We analyze the dynamics of the two-level replicator dynamics, a nonlocal hyperbolic PDE describing deterministic birth-death dynamics for both individuals and groups. While past work on such multilevel dynamics has restricted attention to scenarios with exactly solvable within-group dynamics, we use comparison principles and an invariant property of the tail of the population distribution to extend our analysis to all possible two-player, two-strategy social dilemmas. In the Stag-Hunt and similar games with coordination thresholds, we show that any amount of between-group competition allows for fixation of cooperation in the population. For the prisoners' dilemma and Hawk-Dove game, we characterize the threshold level of between-group selection dividing a regime in which the population converges to a delta function at the equilibrium of the within-group dynamics from a regime in which between-group competition facilitates the existence of steady-state densities supporting greater levels of cooperation. In particular, we see that the threshold selection strength and average payoff at steady state depend on a tug-of-war between the individual-level incentive to be a defector in a many-cooperator group and the group-level incentive to have many cooperators over many defectors. We also find that lower-level selection casts a long shadow: If groups are best off with a mix of cooperators and defectors, then there will always be fewer cooperators than optimal at steady state, even in the limit of infinitely strong competition between groups.

The Effects of Statistical Multiplicity of Infection on Virus Quantification and Infectivity Assays

Many biological assays are employed in virology to quantify parameters of interest. Two such classes of assays, virus quantification assays (VQAs) and infectivity assays (IAs), aim to estimate the number of viruses present in a solution and the ability of a viral strain to successfully infect a host cell, respectively. VQAs operate at extremely dilute concentrations, and results can be subject to stochastic variability in virus-cell interactions. At the other extreme, high viral-particle concentrations are used in IAs, resulting in large numbers of viruses infecting each cell, enough for measurable change in total transcription activity. Furthermore, host cells can be infected at any concentration regime by multiple particles, resulting in a statistical multiplicity of infection and yielding potentially significant variability in the assay signal and parameter estimates. We develop probabilistic models for statistical multiplicity of infection at low and high viral-particle-concentration limits and apply them to the plaque (VQA), endpoint dilution (VQA), and luciferase reporter (IA) assays. A web-based tool implementing our models and analysis is also developed and presented. We test our proposed new methods for inferring experimental parameters from data using numerical simulations and show improvement on existing procedures in all limits.

The replicator dynamics for multilevel selection in evolutionary games

We consider a stochastic model for evolution of group-structured populations in which interactions between group members correspond to the Prisoner's Dilemma or the Hawk-Dove game. Selection operates at two organization levels: individuals compete with peer group members based on individual payoff, while groups also compete with other groups based on average payoff of group members. In the Prisoner's Dilemma, this creates a tension between the two levels of selection, as defectors are favored at the individual level, whereas groups with at least some cooperators outperform groups of defectors at the between-group level. In the limit of infinite group size and infinite number of groups, we derive a non-local PDE that describes the probability distribution of group compositions in the population. For special families of payoff matrices, we characterize the long-time behavior of solutions of our equation, finding a threshold intensity of between-group selection required to sustain density steady states and the survival of cooperation. When all-cooperator groups are most fit, the average and most abundant group compositions at steady state range from featuring all-defector groups when individual-level selection dominates to featuring all-cooperator groups when group-level selection dominates. When the most fit groups have a mix of cooperators and defectors, then the average and most abundant group compositions always feature a smaller fraction of cooperators than required for the optimal mix, even in the limit where group-level selection is infinitely stronger than individual-level selection. In such cases, the conflict between the two levels of selection cannot be decoupled, and cooperation cannot be sustained at all in the case where between-group competition favors an even mix of cooperators and defectors.

A New Age-Structured Multiscale Model of the Hepatitis C Virus Life-Cycle During Infection and Therapy With Direct-Acting Antiviral Agents

The dynamics of hepatitis C virus (HCV) RNA during translation and replication within infected cells were added to a previous age-structured multiscale mathematical model of HCV infection and treatment. The model allows the study of the dynamics of HCV RNA inside infected cells as well as the release of virus from infected cells and the dynamics of subsequent new cell infections. The model was used to fit in vitro data and estimate parameters characterizing HCV replication. This is the first model to our knowledge to consider both positive and negative strands of HCV RNA with an age-structured multiscale modeling approach. Using this model we also studied the effects of direct-acting antiviral agents (DAAs) in blocking HCV RNA intracellular replication and the release of new virions and fit the model to in vivo data obtained from HCV-infected subjects under therapy.

A note on the global properties of an age-structured viral dynamic model with multiple target cell populations

Some viruses can infect different classes of cells. The age of infection can affect the dynamics of infected cells and viral production. Here we develop a viral dynamic model with the age of infection and multiple target cell populations. Using the methods of semigroup and Lyapunov function, we study the global asymptotic property of the steady states of the model. The results show that when the basic reproductive number falls below 1, the infection is predicted to die out. When the basic reproductive number exceeds 1, there exists a unique infected steady state which is globally asymptotically stable. The model can be extended to study virus dynamics with multiple compartments or coinfection by multiple types of viruses. We also show that under some scenarios the age-structured model can be reduced to an ordinary differential equation system with or without time delays.

Global properties for an age-structured within-host model with Crowley–Martin functional response

Based on a multi-scale view, in this paper, we study an age-structured within-host model with Crowley–Martin functional response for the control of viral infections. By means of semigroup and Lyapunov function, the global asymptotical property of infected steady state of the model is obtained. The results show that when the basic reproductive number falls below unity, the infection dies out. However, when the basic reproductive number exceeds unity, there exists a unique positive equilibrium which is globally asymptotically stable. This model can be deduced to different viral models with or without time delay.

Dynamics of a diffusive age-structured HBV model with saturating incidence

In this paper, we propose and investigate an age-structured hepatitis B virus (HBV) model with saturating incidence and spatial diffusion where the viral contamination process is described by the age-since-infection. We first analyze the well-posedness of the initial-boundary values problem of the model in the bounded domain Ω ⊂ Rn and obtain an explicit formula for the basic reproductive number R0 of the model. Then we investigate the global behavior of the model in terms of R0: if R0 ≤ 1, then the uninfected steady state is globally asymptotically stable, whereas if R0 > 1, then the infected steady state is globally asymptotically stable. In addition, when R0> 1, by constructing a suitable Lyapunov-like functional decreasing along the travelling waves to show their convergence towards two steady states as t tends to ∞, we prove the existence of traveling wave solutions. Numerical simulations are provided to illustrate the theoretical results.

The role of pathogen shedding in linking within- and between-host pathogen dynamics

A model linking within- and between-host pathogen dynamics via pathogen shedding (emission of pathogens throughout the course of infection) is developed, and several aspects of host availability and co-infection are considered. In this model, the rate of pathogen shedding affects both the pathogen population size within a host (also affecting host mortality) and the rate of infection of new hosts. Our goal is to ascertain how the rate of shedding is likely to evolve, and what factors permit coexistence of alternative shedding rates in a pathogen population. For a constant host population size (where an increase in infected hosts necessarily decreases susceptible hosts), important differences arise depending on whether pathogens compete only for susceptible (uninfected) hosts, or whether co-infection allows for competition for infected hosts. With no co-infection, the pathogen type that can persist with the lowest number of susceptible hosts will outcompete any other, which under the assumptions of the model is the pathogen with the highest basic reproduction number. This is often a pathogen with a relatively high shedding rate (s). If within-host competition is allowed, a trade-off develops due to the conflicting effects of shedding on within- and between-host pathogen dynamics, with within-host competition favoring clones with low shedding rates while between-host competition benefits clones with higher shedding rates. With within-host competition for the same host cells, low shedding rate clones should eliminate high-s clones in a co-infected host, if equilibrium is reached. With co-infection, but no within-host competition, pathogen clones still interact by affecting the mortality of co-infected hosts; here, coexistence is more likely. With co-infection, two clones can coexist if one is the superior competitor for uninfected hosts and the other for co-infected hosts.

Opportunities and challenges of Integral Projection Models for modelling host-parasite dynamics

Epidemiological dynamics are shaped by and may in turn shape host demography. These feedbacks can result in hard to predict patterns of disease incidence. Mathematical models that integrate infection and demography are consequently a key tool for informing expectations for disease burden and identifying effective measures for control.

A major challenge is capturing the details of infection within individuals and quantifying their downstream impacts to understand population‐scale outcomes. For example, parasite loads and antibody titres may vary over the course of an infection and contribute to differences in transmission at the scale of the population. To date, to capture these subtleties, models have mostly relied on complex mechanistic frameworks, discrete categorization and/or agent‐based approaches.

Integral Projection Models (IPMs) allow variance in individual trajectories of quantitative traits and their population‐level outcomes to be captured in ways that directly reflect statistical models of trait–fate relationships. Given increasing data availability, and advances in modelling, there is considerable potential for extending this framework to traits of relevance for infectious disease dynamics.

Here, we provide an overview of host and parasite natural history contexts where IPMs could strengthen inference of population dynamics, with examples of host species ranging from mice to sheep to humans, and parasites ranging from viruses to worms. We discuss models of both parasite and host traits, provide two case studies and conclude by reviewing potential for both ecological and evolutionary research.

A multi-scale mathematical modeling framework to investigate anti-viral therapeutic opportunities in targeting HIV-1 accessory proteins

Human immunodeficiency virus-1 (HIV-1) employs accessory proteins to evade innate immune responses by neutralizing the anti-viral activity of host restriction factors. Apolipoprotein B mRNA-editing enzyme 3G (APOBEC3G, A3G) and bone marrow stromal cell antigen 2 (BST2) are host resistance factors that potentially inhibit HIV-1 infection. BST2 reduces viral production by tethering budding HIV-1 particles to virus producing cells, while A3G inhibits the reverse transcription (RT) process and induces viral genome hypermutation through cytidine deamination, generating fewer replication competent progeny virus. Two HIV-1 proteins counter these cellular restriction factors: Vpu, which reduces surface BST2, and Vif, which degrades cellular A3G. The contest between these host and viral proteins influences whether HIV-1 infection is established and progresses towards AIDS. In this work, we present an age-structured multi-scale viral dynamics model of in vivo HIV-1 infection. We integrated the intracellular dynamics of anti-viral activity of the host factors and their neutralization by HIV-1 accessory proteins into the virus/cell population dynamics model. We calculate the basic reproductive ratio (Ro) as a function of host-viral protein interaction coefficients, and numerically simulated the multi-scale model to understand HIV-1 dynamics following host factor-induced perturbations. We found that reducing the influence of Vpu triggers a drop in Ro, revealing the impact of BST2 on viral infection control. Reducing Vif׳s effect reveals the restrictive efficacy of A3G in blocking RT and in inducing lethal hypermutations, however, neither of these factors alone is sufficient to fully restrict HIV-1 infection. Interestingly, our model further predicts that BST2 and A3G function synergistically, and delineates their relative contribution in limiting HIV-1 infection and disease progression. We provide a robust modeling framework for devising novel combination therapies that target HIV-1 accessory proteins and boost antiviral activity of host factors.

Resources, mortality, and disease ecology: Importance of positive feedbacks between host growth rate and pathogen dynamics

Resource theory and metabolic scaling theory suggest that the dynamics of a pathogen within a host should strongly depend upon the rate of host cell metabolism. Once an infection occurs, key ecological interactions occur on or within the host organism that determine whether the pathogen dies out, persists as a chronic infection, or grows to densities that lead to host death. We hypothesize that, in general, conditions favoring rapid host growth rates should amplify the replication and proliferation of both fungal and viral pathogens. If a host population experiences an increase in mortality, to persist it must have a higher growth rate, per host, often reflecting greater resource availability per capita. We hypothesize that this could indirectly foster the pathogen, which also benefits from increased within-host resource turnover. We first bring together in a short review a number of key prior studies which illustrate resource effects on viral and fungal pathogen dynamics. We then report new results from a semi-continuous cell culture experiment with SHIV, demonstrating that higher mortality rates indeed can promote viral proliferation. We develop a simple model that illustrates dynamical consequences of these resource effects, including interesting effects such as alternative stable states and oscillatory dynamics. Our paper contributes to a growing body of literature at the interface of ecology and infectious disease epidemiology, emphasizing that host abundances alone do not drive community dynamics: the physiological state and resource content of infected hosts also strongly influence host-pathogen interactions.

Contribution of HIV-1 genomes that do not integrate to the basic reproductive ratio of the virus

Recent experimental data indicate that HIV-1 DNA that fails to integrate (from now on called uDNA) can by itself successfully produce infectious offspring virions in resting T cells that become activated after infection. This scenario is likely important at the initial stages of the infection. We use mathematical models to calculate the relative contribution of unintegrated and integrated viral DNA to the basic reproductive ratio of the virus, R0, and the models are parameterized with preliminary data. This is done in the context of both free virus spread and transmission of the virus through virological synapses. For free virus transmission, we find that under preliminary parameter estimates, uDNA might contribute about 20% to the total R0. This requires that a single copy of uDNA can successfully replicate. If the presence of more than one uDNA copy is required for replication, uDNA does not contribute to R0. For synaptic transmission, uDNA can contribute to R0 regardless of the number of uDNA copies required for replication. The larger the number of viruses that are successfully transmitted per synapse, however, the lower the contribution of uDNA to R0 because this increases the chances that at least one virus integrates. Using available parameter values, uDNA can maximally contribute 20% to R0 in this case. We argue that the contribution of uDNA to virus reproduction might also be important for continued low level replication of HIV-1 in the presence of integrase inhibitor therapy. Assuming a 20% contribution of uDNA to the overall R0, our calculations suggest that R0=1.6 in the absence of virus integration. While these are rough estimates based on preliminary data that are currently available, this analysis provides a framework for future experimental work which should directly measure key parameters.

How sticky should a virus be? The impact of virus binding and release on transmission fitness using influenza as an example

Budding viruses face a trade-off: virions need to efficiently attach to and enter uninfected cells while newly generated virions need to efficiently detach from infected cells. The right balance between attachment and detachment-the right amount of stickiness-is needed for maximum fitness. Here, we design and analyse a mathematical model to study in detail the impact of attachment and detachment rates on virus fitness. We apply our model to influenza, where stickiness is determined by a balance of the haemagglutinin (HA) and neuraminidase (NA) proteins. We investigate how drugs, the adaptive immune response and vaccines impact influenza stickiness and fitness. Our model suggests that the location in the 'stickiness landscape' of the virus determines how well interventions such as drugs or vaccines are expected to work. We discuss why hypothetical NA enhancer drugs might occasionally perform better than the currently available NA inhibitors in reducing virus fitness. We show that an increased antibody or T-cell-mediated immune response leads to maximum fitness at higher stickiness. We further show that antibody-based vaccines targeting mainly HA or NA, which leads to a shift in stickiness, might reduce virus fitness above what can be achieved by the direct immunological action of the vaccine. Overall, our findings provide potentially useful conceptual insights for future vaccine and drug development and can be applied to other budding viruses beyond influenza.

Disentangling the interaction among host resources, the immune system and pathogens

The interaction between the immune system and pathogens is often characterised as a predator–prey interaction. This characterisation ignores the fact that both require host resources to reproduce. Here, we propose novel theory that considers how these resource requirements can modify the interaction between the immune system and pathogens. We derive a series of models to describe the energetic interaction between the immune system and pathogens, from fully independent resources to direct competition for the same resource. We show that increasing within-host resource supply has qualitatively distinct effects under these different scenarios. In particular, we show the conditions for which pathogen load is expected to increase, decrease or even peak at intermediate resource supply. We survey the empirical literature and find evidence for all three patterns. These patterns are not explained by previous theory, suggesting that competition for host resources can have a strong influence on the outcome of disease.

Intracellular dynamics of HIV infection

Early studies of HIV infection dynamics suggested that virus-producing HIV-infected cells had an average half-life of approximately 1 day. However, whether this average behavior is reflective of the dynamics of individual infected cells is unclear. Here, we use HIV-enhanced green fluorescent protein (EGFP) constructs and flow cytometry sorting to explore the dynamics of cell infection, viral protein production, and cell death in vitro. By following the numbers of productively infected cells expressing EGFP over time, we show that infected cell death slows down over time. Although infected cell death in vivo could be very different, our results suggest that the constant decay of cell numbers observed in vivo during antiretroviral treatment could reflect a balance of cell death and delayed viral protein production. We observe no correlation between viral protein production and death rate of productively infected cells, showing that viral protein production is not likely to be the sole determinant of the death of HIV-infected cells. Finally, we show that all observed features can be reproduced by a simple model in which infected cells have broad distributions of productive life spans, times to start viral protein production, and viral protein production rates. This broad spectrum of the level and timing of viral protein production provides new insights into the behavior and characteristics of HIV-infected cells.

Linking immunological and epidemiological dynamics of HIV: the case of super-infection

In this paper, a two-strain model that links immunological and epidemiological dynamics across scales is formulated. On the within-host scale, the two strains eliminate each other with the strain with the larger immunological reproduction persisting. However, on the population scale superinfection is possible, with the strain with larger immunological reproduction number super-infecting the strain with the smaller immunological reproduction number. The two models are linked through the age-since-infection structure of the epidemiological variables. In addition, the between-host transmission and the disease-induced death rate depend on the within-host viral load. The immunological reproduction numbers, the epidemiological reproduction numbers and invasion reproduction numbers are computed. Besides the disease-free equilibrium, there are two population-level strain one and strain two isolated equilibria, as well as a population-level coexistence equilibrium when both invasion reproduction numbers are greater than one. The single-strain population-level equilibria are locally asymptotically stable suggesting that in the absence of superinfection oscillations do not occur, a result contrasting previous studies of HIV age-since-infection structured models. Simulations suggest that the epidemiological reproduction number and HIV population prevalence are monotone functions of the within-host parameters with reciprocal trends. In particular, HIV medications that decrease within-host viral load also increase overall population prevalence. The effect of the immunological parameters on the population reproduction number and prevalence is more pronounced when the initial viral load is lower.

Evolutionary analysis of human immunodeficiency virus type 1 therapies based on conditionally replicating vectors

Efforts to reduce the viral load of human immunodeficiency virus type 1 (HIV-1) during long-term treatment are challenged by the evolution of anti-viral resistance mutants. Recent studies have shown that gene therapy approaches based on conditionally replicating vectors (CRVs) could have many advantages over anti-viral drugs and other approaches to therapy, potentially including the ability to circumvent the problem of evolved resistance. However, research to date has not explored the evolutionary consequences of long-term treatment of HIV-1 infections with conditionally replicating vectors. In this study, we analyze a computational model of the within-host co-evolutionary dynamics of HIV-1 and conditionally replicating vectors, using the recently proposed ‘therapeutic interfering particle’ as an example. The model keeps track of the stochastic process of viral mutation, and the deterministic population dynamics of T cells as well as different strains of CRV and HIV-1 particles. We show that early in the co-infection, mutant HIV-1 genotypes that escape suppression by CRV therapy appear; this is similar to the dynamics observed in drug treatments and other gene therapies. In contrast to other treatments, however, the CRV population is able to evolve and catch up with the dominant HIV-1 escape mutant and persist long-term in most cases. On evolutionary grounds, gene therapies based on CRVs appear to be a promising tool for long-term treatment of HIV-1. Our model allows us to propose design principles to optimize the efficacy of this class of gene therapies. In addition, because of the analogy between CRVs and naturally-occurring defective interfering particles, our results also shed light on the co-evolutionary dynamics of wild-type viruses and their defective interfering particles during natural infections.

A long-standing challenge in efforts to control human immunodeficiency virus type 1 (HIV-1) is the rapid evolution of the virus. Any effective therapy quickly gives rise to so-called escape mutants of the virus, potentially resulting in treatment failure. A distinct class of gene therapy based on conditionally replicating vectors has been suggested to have potential to circumvent the problem of viral evolutionary escape. A conditionally replicating vector cannot replicate on its own, but when it coinfects the same cell with HIV-1, it is packaged into a virion-like particle and can be transmitted from cell to cell. Importantly, these vectors replicate using the same machinery that HIV-1 uses, and so they mutate at the same rate. This opens the possibility that conditionally replicating vectors could ‘keep up’ with HIV-1 evolution and prevent HIV-1 escape. In this study, we present mathematical analyses of the co-evolutionary dynamics of HIV-1 and conditionally replicating vectors within a patient. Our results show that with proper genetic design, conditionally replicating vectors can keep pace with HIV-1 evolution, leading to persistent reduction in HIV-1 viral loads. Therefore, this class of gene therapies shows potential for ‘evolution-proof’ control of HIV-1, and merits further investigation in laboratory trials.

Modelling the course of an HIV infection: insights from ecology and evolution

The Human Immunodeficiency Virus (HIV) is one of the most threatening viral agents. This virus infects approximately 33 million people, many of whom are unaware of their status because, except for flu-like symptoms right at the beginning of the infection during the acute phase, the disease progresses more or less symptom-free for 5 to 10 years. During this asymptomatic phase, the virus slowly destroys the immune system until the onset of AIDS when opportunistic infections like pneumonia or Kaposi’s sarcoma can overcome immune defenses. Mathematical models have played a decisive role in estimating important parameters (e.g., virion clearance rate or life-span of infected cells). However, most models only account for the acute and asymptomatic latency phase and cannot explain the progression to AIDS. Models that account for the whole course of the infection rely on different hypotheses to explain the progression to AIDS. The aim of this study is to review these models, present their technical approaches and discuss the robustness of their biological hypotheses. Among the few models capturing all three phases of an HIV infection, we can distinguish between those that mainly rely on population dynamics and those that involve virus evolution. Overall, the modeling quest to capture the dynamics of an HIV infection has improved our understanding of the progression to AIDS but, more generally, it has also led to the insight that population dynamics and evolutionary processes can be necessary to explain the course of an infection.

Evolution of viral life-cycle in response to cytotoxic T lymphocyte-mediated immunity

Viruses in mammals are constantly faced with the problem of elimination by the host immunity. Cytotoxic T lymphocyte (CTL) responses are thought to play a major role in the control and clearance of several viral infections in mice and humans. It is therefore expected that over evolutionary time, viruses would be forced to evolve to avoid recognition by CTLs. Indeed, a number of studies have documented the accumulation of viral variants with escape mutations. These mutations allow viruses to hide from CTL responses common in the host population. CTLs recognize viruses by short protein sequences, named epitopes, derived from viral proteins. The efficiency of viral recognition by epitope-specific CTL responses depends on the expression pattern of the proteins carrying these epitopes, and the total amount of that protein (and thus epitopes) in the cell. When a virus replicates in a cell, some viral genes are expressed early in the life cycle of the virus, while other proteins are expressed late. For example, HIV infected cells first express Rev and Tat proteins, and the Gag proteins are expressed late. Here we propose a dynamical model of the viral life cycle to study how expression level of early vs. late genes may affect viral dynamics within the host and virus transmission over the course of infection. We find that for acute and chronic viral infections lower expression of early genes than that of the late genes is expected to give selective advantage and higher transmission to viruses.

AN EPIDEMIC MODEL STRUCTURED BY HOST IMMUNITY

In this paper we introduce a physiologically structured SIR epidemic model where the individuals are distributed according to their immune status. An individual immune status is assumed to increase during the infectious period and remain unchanged after the recovery. Recovered individuals can become reinfected at a rate which is a decreasing function of their immune status. We find that the possibility of reinfection of recovered individuals results in subthreshold endemic equilibria. The differential immunity of the infectious individuals leads to multiple nontrivial equilibria in the superthreshold case. We present an example that has exactly three nontrivial equilibria. We also analyze the local stability of equilibria.

A strong case for viral genetic factors in HIV virulence

HIV infections show great variation in the rate of progression to disease, and the role of viral genetic factors in this variation had remained poorly characterized until recently. Now a series of four studies [1-4] published within a year has filled this important gap and has demonstrated a robust effect of the viral genotype on HIV virulence.

Stochastic theory of early viral infection: continuous versus burst production of virions

Viral production from infected cells can occur continuously or in a burst that generally kills the cell. For HIV infection, both modes of production have been suggested. Standard viral dynamic models formulated as sets of ordinary differential equations can not distinguish between these two modes of viral production, as the predicted dynamics is identical as long as infected cells produce the same total number of virions over their lifespan. Here we show that in stochastic models of viral infection the two modes of viral production yield different early term dynamics. Further, we analytically determine the probability that infections initiated with any number of virions and infected cells reach extinction, the state when both the population of virions and infected cells vanish, and show this too has different solutions for continuous and burst production. We also compute the distributions of times to establish infection as well as the distribution of times to extinction starting from both a single virion as well as from a single infected cell for both modes of virion production.

The dynamics of HIV infection and treatment has been extensively studied using ordinary differential equation models. Recent work on HIV transmission has suggested that most sexually transmitted infections are started by a single virus or infected cell. This observation coupled with the fact that successful HIV transmission only occurs in 1 per 100 to 1 per 1000 coital acts suggests that early events in infection are stochastic. Here we develop a stochastic model of HIV infection and use it to characterize the dynamics of early infection when virus is released from cells either continuously or in a burst. We show that these mechanisms of viral production produce different early dynamics, with different probabilities of extinction and different distributions of time to establish infection. In deterministic models, these modes of viral production are indistinguishable.

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.

Epidemiological and clinical consequences of within-host evolution

Many viruses and bacteria are known to evolve rapidly over the course of an infection. However, epidemiological studies generally assume that within-host evolution is an instantaneous process. We argue that the dynamics of within-host evolution has implications at the within-host and at the between-host levels. We first show that epidemiologists should consider within-host evolution, notably because it affects the genotype of the pathogen that is transmitted. We then present studies that investigate evolution at the within-host level and examine the extent to which these studies can help to understand infection traits involved in the epidemiology (e.g. transmission rate, virulence, recovery rate). Finally, we discuss how new techniques for data acquisition can open new perspectives for empirical and theoretical research.

Identifying genetic markers of adaptation for surveillance of viral host jumps

Adaptation is often thought to affect the likelihood that a virus will be able to successfully emerge in a new host species. If so, surveillance for genetic markers of adaptation could help to predict the risk of disease emergence. However, adaptation is difficult to distinguish conclusively from the other processes that generate genetic change. In this Review we survey the research on the host jumps of influenza A, severe acute respiratory syndrome-coronavirus, canine parvovirus and Venezuelan equine encephalitis virus to illustrate the insights that can arise from combining genetic surveillance with microbiological experimentation in the context of epidemiological data. We argue that using a multidisciplinary approach for surveillance will provide a better understanding of when adaptations are required for host jumps and thus when predictive genetic markers may be present.

Trade-offs in resource allocation in the intracellular life-cycle of hepatitis C virus

Positive sense single-stranded RNA viruses undergo three mutually exclusive processes to replicate within a cell. These are translation to produce proteins, replication to produce RNA viral genomes, and packaging to form virions. The allocation of newly synthesised viral genomes to these processes, which can be regarded as life-history traits, may be subject to natural selection for efficient reproduction. Here, we develop a mathematical model of the process of intracellular viral replication to study alternative strategies for the allocation and reallocation of viral genomes to these processes. We explore four cases of the model: (1) Free Movement, in which viral genomes can freely be allocated and reallocated among translation, replication and packaging; (2) Unidirectional Reallocation, in which allocation occurs freely but reallocation can only proceed from translation to replication to packaging; (3) Conveyor Belt, in which viral genomes are first allocated to translation, then passed on to replication and finally to packaging; and (4) Permanent Allocation in which new genomes are allocated to the three processes but not reallocated between them. We apply this model to hepatitis C virus and study changes in the production of virus as the rates of allocation and reallocation are varied. We find that high viral production occurs when allocation and reallocation of the genome are weighted towards the translation and replication processes. The replication process in particular is favoured. The most productive strategy is a form of the Free Movement model in which genomes are allocated entirely to the replication-translation cycle while allowing some genomes to be packaged through reallocation.

A unified model for the coevolution of resistance, tolerance, and virulence

We present a general host-parasite model that unifies previous theory by investigating the coevolution of virulence, resistance, and tolerance, with respect to multiple physiological, epidemiological, and environmental parameters. Four sets of new predictions emerge. First, compared to virulence coevolving with resistance or tolerance, three-trait coevolution promotes more virulence and less tolerance, and broadens conditions under which pure defenses evolve. Second, the cost and efficiency of virulence and the epidemiological rates are the key factors of virulence coevolving with resistance and tolerance. Maximum virulence evolves for intermediate infection rate, at which coevolved levels of resistance and tolerance are both high. The influence of host and parasite background mortalities is strong on the evolution of defenses and weak on the coevolution of virulence. Third, evolutionary correlations between defenses can switch sign along single-parameter gradients. The evolutionary trade-off between resistance and tolerance may coevolve with virulence that either increases or decreases monotonically, depending on the underlying parameter gradient. Fourth, despite global attractiveness and stability of coevolutionary equilibria, not-so-rare and not-so-small mutations can beget large variation in virulence and defenses around equilibrium, in the form of transient "evolutionary spikes." Implications for evolutionary management of infections are discussed and directions for future research are outlined.

The relationship between mutation frequency and replication strategy in positive-sense single-stranded RNA viruses

For positive-sense single-stranded RNA virus genomes, there is a trade-off between the mutually exclusive tasks of transcription, translation and encapsidation. The replication strategy that maximizes the intracellular growth rate of the virus requires iterative genome transcription from positive to negative, and back to positive sense. However, RNA viruses experience high mutation rates, and the proportion of genomes with lethal mutations increases with the number of replication cycles. Thus, intracellular mutant frequency will depend on the replication strategy. Introducing apparently realistic mutation rates into a model of viral replication demonstrates that strategies that maximize viral growth rate could result in an average of 26 mutations per genome by the time plausible numbers of positive strands have been generated, and that virus viability could be as low as 0.1 per cent. At high mutation rates or when a high proportion of mutations are deleterious, the optimal strategy shifts towards synthesizing more negative strands per positive strand, and in extremis towards a 'stamping-machine' replication mode where all the encapsidated genomes come from only two transcriptional steps. We conclude that if viral mutation rates are as high as current estimates suggest, either mutation frequency must be considerably higher than generally anticipated and the proportion of viable viruses produced extremely small, or replication strategies cannot be optimized to maximize viral growth rate. Mechanistic models linking mutation frequency to replication mechanisms coupled with data generated through new deep-sequencing technologies could play an important role in improving the estimates of viral mutation rate.

The evolutionary dynamics of a rapidly mutating virus within and between hosts: the case of hepatitis C virus

Many pathogens associated with chronic infections evolve so rapidly that strains found late in an infection have little in common with the initial strain. This raises questions at different levels of analysis because rapid within-host evolution affects the course of an infection, but it can also affect the possibility for natural selection to act at the between-host level. We present a nested approach that incorporates within-host evolutionary dynamics of a rapidly mutating virus (hepatitis C virus) targeted by a cellular cross-reactive immune response, into an epidemiological perspective. The viral trait we follow is the replication rate of the strain initiating the infection. We find that, even for rapidly evolving viruses, the replication rate of the initial strain has a strong effect on the fitness of an infection. Moreover, infections caused by slowly replicating viruses have the highest infection fitness (i.e., lead to more secondary infections), but strains with higher replication rates tend to dominate within a host in the long-term. We also study the effect of cross-reactive immunity and viral mutation rate on infection life history traits. For instance, because of the stochastic nature of our approach, we can identify factors affecting the outcome of the infection (acute or chronic infections). Finally, we show that anti-viral treatments modify the value of the optimal initial replication rate and that the timing of the treatment administration can have public health consequences due to within-host evolution. Our results support the idea that natural selection can act on the replication rate of rapidly evolving viruses at the between-host level. It also provides a mechanistic description of within-host constraints, such as cross-reactive immunity, and shows how these constraints affect the infection fitness. This model raises questions that can be tested experimentally and underlines the necessity to consider the evolution of quantitative traits to understand the outcome and the fitness of an infection.

Rapidly mutating viruses, such as hepatitis C virus, can escape host immunity by generating new strains that avoid the immune system. Existing data support the idea that such within-host evolution affects the outcome of the infection. Few theoretical models address this question and most follow viral diversity or qualitative traits, such as drug resistance. Here, we study the evolution of two virus quantitative traits—the replication rate and the ability to be recognised by the immune response—during an infection. We develop an epidemiological framework where transmission events are driven by within-host dynamics. We find that the replication rate of the virus that initially infects the host has a strong influence on the epidemiological success of the disease. Furthermore, we show that the cross-reactive immune response is key to determining the outcome of the infection (acute or chronic). Finally, we show that the timing of the start of an anti-viral treatment has a strong effect on viral evolution, which impacts the efficiency of the treatment. Our analysis suggests a new mechanism to explain infection outcomes and proposes testable predictions that can drive future experimental approaches.

Exploring the impact of target cell heterogeneity on HIV loads in a within-host model

HIV can be transmitted from blood plasma or semen of an infected male. Viral loads in blood plasma are routinely measured, but the same is not true of semen. Even before drug treatment, viral loads have been shown to be different in the two body compartments (blood and genital tract), and this heterogeneity may be exacerbated by treatments using those antiretroviral drugs which have different efficacies in the two compartments. In addition to this heterogeneity, and despite highly effective drugs (in the blood) low-level viral replication is commonly reported for HIV patients as are differences in drug resistant mutation patterns in the two compartments. In this paper we investigate the effect of target cell heterogeneity between compartments on HIV viral loads using a within-host model that includes wildtype and drug resistant strains of HIV. We find that modelling target cell heterogeneity in the blood and male genital tract gives different viral loads in the two compartments prior to treatment and allows low-level viral loads to persist during therapy even if drug penetration is good. The model also allows coexistence of the two viral strains (in the absence of a mutation mechanism) with different dominance patterns in each body compartment. Our results suggest that monitoring of blood plasma viral strains may not give an accurate picture of the strains of HIV being transmitted between individuals and that continued research into the nature of HIV target cells in the male genital tract would be beneficial.

Reassessing the human immunodeficiency virus type 1 life cycle through age-structured modeling: life span of infected cells, viral generation time, and basic reproductive number, R0

The rapid decay of the viral load after drug treatment in patients infected with human immunodeficiency virus type 1 (HIV-1) has been shown to result from the rapid loss of infected cells due to their high turnover, with a generation time of around 1 to 2 days. Traditionally, viral decay dynamics after drug treatment is investigated using models of differential equations in which both the death rate of infected cells and the viral production rate are assumed to be constant. Here, we describe age-structured models of the viral decay dynamics in which viral production rates and death rates depend on the age of the infected cells. In order to investigate the effects of age-dependent rates, we compared these models with earlier descriptions of the viral load decay and fitted them to previously published data. We have found no supporting evidence that infected-cell death rates increase, but cannot reject the possibility that viral production rates increase, with the age of the cells. In particular, we demonstrate that an exponential increase in viral production with infected-cell age is perfectly consistent with the data. Since an exponential increase in virus production can compensate for the exponential loss of infected cells, the death rates of HIV-1-infected cells may be higher than previously anticipated. We discuss the implications of these findings for the life span of infected cells, the viral generation time, and the basic reproductive number, R0.

Virulence evolution and the trade-off hypothesis: history, current state of affairs and the future

It has been more than two decades since the formulation of the so-called 'trade-off' hypothesis as an alternative to the then commonly accepted idea that parasites should always evolve towards avirulence (the 'avirulence hypothesis'). The trade-off hypothesis states that virulence is an unavoidable consequence of parasite transmission; however, since the 1990s, this hypothesis has been increasingly challenged. We discuss the history of the study of virulence evolution and the development of theories towards the trade-off hypothesis in order to illustrate the context of the debate. We investigate the arguments raised against the trade-off hypothesis and argue that trade-offs exist, but may not be of the simple form that is usually assumed, involving other mechanisms (and life-history traits) than those originally considered. Many processes such as pathogen adaptation to within-host competition, interactions with the immune system and shifting transmission routes, will all be interrelated making sweeping evolutionary predictions harder to obtain. We argue that this is the heart of the current debate in the field and while species-specific models may be better predictive tools, the trade-off hypothesis and its basic extensions are necessary to assess the qualitative impacts of virulence management strategies.

Acute or chronic? Within-host models with immune dynamics, infection outcome, and parasite evolution

There is ample theoretical and experimental evidence that virulence evolution depends on the immune response of the host. In this article, we review a number of recent studies that attempt to explicitly incorporate the dynamics of the immune system (instead of merely representing it by a single black box parameter) in models for the evolution of parasite virulence. A striking observation is that the type of infection (acute or chronic) is invariably considered to be a constraint that model assumptions have to satisfy rather than as a potential outcome of the interaction of the parasite with its host's immune system. We argue that avoiding making assumptions about the type of infection will lead to a better understanding of infectious diseases, even though a number of fundamental and technical problems remain. Dynamical modeling of the immune system opens a wide range of perspectives: for understanding how the immune system eradicates a parasite (which it does for most pathogens but not for all, HIV being a notorious example of a virus that is not completely eliminated), for studying multiple infections through concomitant immunity, for understanding the emergence and evolution of the immune system in animals, and for evolutionary epidemiology in general (e.g., predicting evolutionary consequences of new therapies and public health policies). We conclude by discussing new approaches based on embedded (or nested) models and identify future perspectives for the modeling of infectious diseases.

Implications of decoupling the intracellular and extracellular levels in multi-level models of virus growth

Virus infections are characterized by two distinct levels of detail: the intracellular level describing how viruses hijack the host machinery to replicate, and the extracellular level describing how populations of virus and host cells interact. Deterministic, population balance models for viral infections permit incorporation of both the intracellular and extracellular levels of information. In this work, we identify assumptions that lead to exact, selective decoupling of the interaction between the intracellular and extracellular levels, effectively permitting solution of first the intracellular level, and subsequently the extracellular level. This decoupling leads to (1) intracellular and extracellular models of viral infections that have been previously reported and (2) a significant reduction in the computational expense required to solve the model. However, the decoupling restricts the behaviors that can be modeled. Simulation of a previously reported multi-level model demonstrates this decomposition when the intracellular level of description consists of numerous reaction events. Additionally, examples demonstrate that viruses can persist even when the intracellular level of description cannot sustain a steady-state production of virus (i.e., has only a trivial equilibrium). We expect the combination of this modeling framework with experimental data to result in a quantitative, systems-level understanding of viral infections and cellular antiviral strategies that will facilitate controlling both these infections and antiviral strategies.

Evaluating the importance of within- and between-host selection pressures on the evolution of chronic pathogens

Infectious pathogens compete and are subject to natural selection at multiple levels. For example, viral strains compete for access to host resources within an infected host and, at the same time, compete for access to susceptible hosts within the host population. Here we propose a novel approach to study the interplay between within- and between-host competition. This approach allows for a single host to be infected by and transmit two strains of the same pathogen. We do this by nesting a model for the host-pathogen dynamics within each infected host into an epidemiological model. The nesting of models allows the between-host infectivity and mortality rates suffered by infected hosts to be functions of the disease progression at the within-host level. We present a general method for computing the basic reproduction ratio of a pathogen in such a model. We then illustrate our method using a basic model for the within-host dynamics of viral infections, embedded within the simplest susceptible-infected (SI) epidemiological model. Within this nested framework, we show that the virion production rate at the level of the cell-virus interaction leads, via within-host competition, to the presence or absence of between-host level competitive exclusion. In particular, we find that in the absence of mutation the strain that maximizes between-host fitness can outcompete all other strains. In the presence of mutation we observe a complex invasion landscape showing the possibility of coexistence. Although we emphasize the application to human viral diseases, we expect this methodology to be applicable to be many host-parasite systems.

Modeling Within-Host Evolution of HIV: Mutation, Competition and Strain Replacement

Virus evolution during infection of a single individual is a well-known feature of disease progression in chronic viral diseases. However, the simplest models of virus competition for host resources show the existence of a single dominant strain that grows most rapidly during the initial period of infection and competitively excludes all other virus strains. Here, we examine the dynamics of strain replacement in a simple model that includes a convex trade-off between rapid virus reproduction and long-term host cell survival. Strains are structured according to their within-cell replication rate. Over the course of infection, we find a progression in the dominant strain from fast- to moderately-replicating virus strains featuring distinct jumps in the replication rate of the dominant strain over time. We completely analyze the model and provide estimates for the replication rate of the initial dominant strain and its successors. Our model lays the groundwork for more detailed models of HIV selection and mutation. We outline future directions and application of related models to other biological situations.