Dynamics of an HIV Model with Multiple Infection Stages and Treatment with Different Drug Classes

Highly active antiretroviral therapy can effectively control HIV replication in infected individuals. Some clinical and modeling studies suggested that viral decay dynamics may depend on the inhibited stages of the viral replication cycle. In this paper, we develop a general mathematical model incorporating multiple infection stages and various drug classes that can interfere with specific stages of the viral life cycle. We derive the basic reproductive number and obtain the global stability results of steady states. Using several simple cases of the general model, we study the effect of various drug classes on the dynamics of HIV decay. When drugs are assumed to be 100% effective, drugs acting later in the viral life cycle lead to a faster or more rapid decay in viremia. This is consistent with some patient and experimental data, and also agrees with previous modeling results. When drugs are not 100% effective, the viral decay dynamics are more complicated. Without a second population of long-lived infected cells, the viral load decline can have two phases if drugs act at an intermediate stage of the viral replication cycle. The slopes of viral load decline depend on the drug effectiveness, the death rate of infected cells at different stages, and the transition rate of infected cells from one to the next stage. With a second population of long-lived infected cells, the viral load decline can have three distinct phases, consistent with the observation in patients receiving antiretroviral therapy containing the integrase inhibitor raltegravir. We also fit modeling prediction to patient data under efavirenz (a nonnucleoside reverse-transcriptase inhibitor) and raltegravir treatment. The first-phase viral load decline under raltegravir therapy is longer than that under efavirenz, resulting in a lower viral load at initiation of the second-phase decline in patients taking raltegravir. This explains why patients taking a raltegravir-based therapy were faster to achieve viral suppression than those taking an efavirenz-based therapy. Taken together, this work provides a quantitative and systematic comparison of the effect of different drug classes on HIV decay dynamics and can explain the viral load decline in HIV patients treated with raltegravir-containing regimens.

Modeling the Slow CD4+ T Cell Decline in HIV-Infected Individuals

The progressive loss of CD4+ T cell population is the hallmark of HIV-1 infection but the mechanism underlying the slow T cell decline remains unclear. Some recent studies suggested that pyroptosis, a form of programmed cell death triggered during abortive HIV infection, is associated with the release of inflammatory cytokines, which can attract more CD4+ T cells to be infected. In this paper, we developed mathematical models to study whether this mechanism can explain the time scale of CD4+ T cell decline during HIV infection. Simulations of the models showed that cytokine induced T cell movement can explain the very slow decline of CD4+ T cells within untreated patients. The long-term CD4+ T cell dynamics predicted by the models were shown to be consistent with available data from patients in Rio de Janeiro, Brazil. Highly active antiretroviral therapy has the potential to restore the CD4+ T cell population but CD4+ response depends on the effectiveness of the therapy, when the therapy is initiated, and whether there are drug sanctuary sites. The model also showed that chronic inflammation induced by pyroptosis may facilitate persistence of the HIV latent reservoir by promoting homeostatic proliferation of memory CD4+ cells. These results improve our understanding of the long-term T cell dynamics in HIV-1 infection, and support that new treatment strategies, such as the use of caspase-1 inhibitors that inhibit pyroptosis, may maintain the CD4+ T cell population and reduce the latent reservoir size.

The CD4+ T cell population within HIV-infected individuals declines slowly as disease progresses. When CD4+ cells drop to below 200 cells/ul, the infection is usually considered to enter the late stage, i.e., acquired immune deficiency syndrome (AIDS). CD4+ T cell depletion can take many years but the biological events underlying such slow decline are not well understood. Some studies showed that the majority of infected T cells in lymph nodes die by pyroptosis, a form of programmed cell death, which can release inflammatory signals attracting more CD4+ T cells to be infected. We developed mathematical models to describe this process and explored whether they can generate the long-term CD4+ T cell decline. We showed that pyroptosis induced cell movement can explain the slow time scale of CD4+ T cell depletion and that pyroptosis may also contribute to the persistence of latently infected cells, which represent a major obstacle to HIV eradication. The modeling prediction agrees with patient data in Rio de Janeiro, Brazil. These results suggest that a combination of current treatment regimens and caspase-1 inhibitor that can inhibit pyroptosis might provide a new way to maintain the CD4+ T cell population and eradicate the HIV latent reservoir.

Modeling the effect of comprehensive interventions on Ebola virus transmission

Since the re-emergence of Ebola in West Africa in 2014, comprehensive and stringent interventions have been implemented to decelerate the spread of the disease. The effectiveness of interventions still remains unclear. In this paper, we develop an epidemiological model that includes various controlling measures to systematically evaluate their effects on the disease transmission dynamics. By fitting the model to reported cumulative cases and deaths in Guinea, Sierra Leone and Liberia until March 22, 2015, we estimate the basic reproduction number in these countries as 1.2552, 1.6093 and 1.7994, respectively. Model analysis shows that there exists a threshold of the effectiveness of isolation, below which increasing the fraction of latent individuals diagnosed prior to symptoms onset or shortening the duration between symptoms onset and isolation may lead to more Ebola infection. This challenges an existing view. Media coverage plays a substantial role in reducing the final epidemic size. The response to reported cumulative infected cases and deaths may have a different effect on the epidemic spread in different countries. Among all the interventions, we find that shortening the duration between death and burial and improving the effectiveness of isolation are two effective interventions for controlling the outbreak of Ebola virus infection.

Global stability of an infection-age structured HIV-1 model linking within-host and between-host dynamics

Although much evidence shows the inseparable interaction between the within-host progression of HIV-1 infection and the transmission of the disease at the population level, few models coupling the within-host and between-host dynamics have been developed. In this paper, we adopt the nested approach, viewing the transmission rate at each stage (primary, chronic, and AIDS stage) of HIV-1 infection as a saturated function of the viral load, to formulate an infection-age structured epidemic model. We explicitly link the individual and the host population scale, and derive the basic reproduction number R0 for the coupled system. To analyze the model and perform a detailed global dynamics analysis, two Lyapunov functionals are constructed to prove the global asymptotical stability of the disease-free and endemic equilibria. Theoretical results indicate that R0 provides a threshold value determining whether or not the disease dies out. Numerical simulations are presented to quantitatively investigate the influence of the within-host viral dynamics on between-host transmission dynamics. The results suggest that increasing the effectiveness of inhibitors can decrease the basic reproduction number, but can also increase the overall infected population because of a lower disease-induced mortality rate and a longer lifespan of HIV infected individuals.

Modeling the impact of twitter on influenza epidemics

Influenza remains a serious public-health problem worldwide. The rising popularity and scale of social networking sites such as Twitter may play an important role in detecting, affecting, and predicting influenza epidemics. In this paper, we develop a simple mathematical model including the dynamics of ``tweets'' --- short, 140-character Twitter messages that may enhance the awareness of disease, change individual's behavior, and reduce the transmission of disease among a population during an influenza season. We analyze the model by deriving the basic reproductive number and proving the stability of the steady states. A Hopf bifurcation occurs when a threshold curve is crossed, which suggests the possibility of multiple outbreaks of influenza. We also perform numerical simulations, conduct sensitivity test on a few parameters related to tweets, and compare modeling predictions with surveillance data of influenza-like illness reported cases and the percentage of tweets self-reporting flu during the 2009 H1N1 flu outbreak in England and Wales. These results show that social media programs like Twitter may serve as a good indicator of seasonal influenza epidemics and influence the emergence and spread of the disease.

Characterization and molecular mapping of stripe rust resistance gene Yr61 in winter wheat cultivar Pindong 34

We report a new stripe rust resistance gene on chromosome 7AS in wheat and molecular markers useful for transferring it to other wheat genotypes. Several new races of the stripe rust pathogen have established throughout the wheat growing regions of China in recent years. These new races are virulent to most of the designated seedling resistance genes limiting the resistance sources. It is necessary to identify new genes for diversification and for pyramiding different resistance genes in order to achieve more durable resistance. We report here the identification of a new resistance gene, designated as Yr61, in Chinese wheat cultivar Pindong 34. A mapping population of 208 F2 plants and 128 derived F2:3 lines in a cross between Mingxian 169 and Pindong 34 was evaluated for seedling stripe rust response. A genetic map consisting of eight resistance gene analog polymorphism (RGAP), two sequence-tagged site (STS) and four simple sequence repeat (SSR) markers was constructed. Yr61 was located on the short arm of chromosome 7A and flanked by RGAP markers Xwgp5467 and Xwgp5765 about 1.9 and 3.9 cM in distance, which were successfully converted into STS markers STS5467 and STS5765b, respectively. The flanking STS markers could be used for marker-assisted selection of Yr61 in breeding programs.

Stochastic population switch may explain the latent reservoir stability and intermittent viral blips in HIV patients on suppressive therapy

Highly active antiretroviral therapy can suppress plasma viral loads of HIV-1 infected individuals to below the detection limit of standard clinical assays. However, low-level viremia still persists. Many patients also have transient viral load measurements above the detection limit (the so-called "viral blips"). The latent reservoir consisting of latently infected CD4+ T cells represents a major obstacle to HIV-1 eradication. These cells can be activated to produce virions but the size of the latent reservoir is relatively stable. The mechanisms underlying low viral load persistence, emergence of intermittent viral blips and stability of the latent reservoir are not well understood. Cellular and viral transcription factors play an important role in the establishment and maintenance of HIV-1 latency. Infected cells with intermediate transcriptional activities may either revert to a latent state or become highly activated and produce virions due to intracellular perturbations. Here we develop a mathematical model that includes such stochastic population switch. We demonstrate that the model can generate a stable latent reservoir, intermittent viral blips, as well as low-level viremia persistence. Latently infected cells with intermediate transcription activities may maintain their size through a high level of homeostatic proliferation, while cells with low transcriptional activities are likely to be maintained through the reversion from cells with intermediate transcription activities. Simulations also suggest that treatment intensification or activation therapy may not help to eradicate the latent reservoir. Blocking the proliferation of latently infected cells might be a good strategy. These results provide more insights into the long-term dynamics of virus and latently infected cells in HIV patients on suppressive therapy and may help to develop novel treatment strategies.

Treatment of hepatitis C with an interferon-based lead-in phase: a perspective from mathematical modelling

BACKGROUND

The standard of care for HCV genotype 1 is a protease inhibitor (telaprevir or boceprevir) combined with pegylated interferon (PEG-IFN) and ribavirin (RBV). A lead-in phase of PEG-IFN/RBV therapy before addition of the protease inhibitor has been used, with the aim of improving response rates by reducing the development of protease inhibitor resistance. However, whether such a strategy can bring benefit to patients is unclear.

METHODS

A viral dynamic model was used to compare in silico HCV dynamics in patients treated with a period of PEG-IFN/RBV lead-in therapy followed by the addition of a protease inhibitor versus immediate triple therapy without lead-in.

RESULTS

The model predicts that both regimens result in a similar end-of-treatment viral load change (viral decline or breakthrough). Thus, the current lead-in strategy may not decrease the rate of viral breakthrough/relapse or increase the rate of sustained virological response. This agrees with available data from clinical trials of several HCV protease inhibitors, such as telaprevir, boceprevir and faldaprevir.

CONCLUSIONS

These results suggest that current PEG-IFN/RBV lead-in strategies may not improve treatment outcomes. However, viral kinetics during a period of PEG-IFN/RBV therapy, combined with other factors, such as the IL28B polymorphism and baseline viral load, can identify IFN-sensitive patients and help develop response-guided therapies.

Mathematical analysis of multiscale models for hepatitis C virus dynamics under therapy with direct-acting antiviral agents

Chronic hepatitis C virus (HCV) infection remains a world-wide public health problem. Therapy with interferon and ribavirin leads to viral elimination in less than 50% of treated patients. New treatment options aiming at a higher cure rate are focused on direct-acting antiviral agents (DAAs), which directly interfere with different steps in the HCV life cycle. In this paper, we describe and analyze a recently developed multiscale model that predicts HCV dynamics under therapy with DAAs. The model includes both intracellular viral RNA replication and extracellular viral infection. We calculate the steady states of the model and perform a detailed stability analysis. With certain assumptions we obtain analytical approximations of the viral load decline after treatment initiation. One approximation agrees well with the prediction of the model, and can conveniently be used to fit patient data and estimate parameter values. We also discuss other possible ways to incorporate intracellular viral dynamics into the multiscale model.

Analysis of hepatitis C virus decline during treatment with the protease inhibitor danoprevir using a multiscale model

The current paradigm for studying hepatitis C virus (HCV) dynamics in patients utilizes a standard viral dynamic model that keeps track of uninfected (target) cells, infected cells, and virus. The model does not account for the dynamics of intracellular viral replication, which is the major target of direct-acting antiviral agents (DAAs). Here we describe and study a recently developed multiscale age-structured model that explicitly considers the potential effects of DAAs on intracellular viral RNA production, degradation, and secretion as virus into the circulation. We show that when therapy significantly blocks both intracellular viral RNA production and virus secretion, the serum viral load decline has three phases, with slopes reflecting the rate of serum viral clearance, the rate of loss of intracellular viral RNA, and the rate of loss of intracellular replication templates and infected cells, respectively. We also derive analytical approximations of the multiscale model and use one of them to analyze data from patients treated for 14 days with the HCV protease inhibitor danoprevir. Analysis suggests that danoprevir significantly blocks intracellular viral production (with mean effectiveness 99.2%), enhances intracellular viral RNA degradation about 5-fold, and moderately inhibits viral secretion (with mean effectiveness 56%). The multiscale model can be used to study viral dynamics in patients treated with other DAAs and explore their mechanisms of action in treatment of hepatitis C.

Chronic infection with hepatitis C virus (HCV) remains an important health-care problem worldwide despite significant progress in the development of HCV therapy since the discovery of the virus in 1989. Current treatment options are focused on direct-acting antiviral agents (DAAs) that target specific steps of the HCV life cycle. Danoprevir, one of the DAAs that inhibit the HCV NS3-4A protease, has induced substantial viral load reductions in patients receiving therapy. We study the viral decline during therapy using a multiscale age-structured model that accounts for the dynamics of intracellular viral replication, and which includes the major steps in the HCV life cycle that are targeted by DAAs. We examine the biological parameters contributing to different phases of the viral decline after treatment initiation. We also explore the mechanisms of action of danoprevir and estimate its treatment effectiveness. The multiscale model provides a theoretical framework for studying virus dynamics in hepatitis C patients treated with other DAAs currently in clinical development, and may help one to optimally combine drugs with complementary modes of action to maximize the HCV cure rate.

Modeling within-host dynamics of influenza virus infection including immune responses

Influenza virus infection remains a public health problem worldwide. The mechanisms underlying viral control during an uncomplicated influenza virus infection are not fully understood. Here, we developed a mathematical model including both innate and adaptive immune responses to study the within-host dynamics of equine influenza virus infection in horses. By comparing modeling predictions with both interferon and viral kinetic data, we examined the relative roles of target cell availability, and innate and adaptive immune responses in controlling the virus. Our results show that the rapid and substantial viral decline (about 2 to 4 logs within 1 day) after the peak can be explained by the killing of infected cells mediated by interferon activated cells, such as natural killer cells, during the innate immune response. After the viral load declines to a lower level, the loss of interferon-induced antiviral effect and an increased availability of target cells due to loss of the antiviral state can explain the observed short phase of viral plateau in which the viral level remains unchanged or even experiences a minor second peak in some animals. An adaptive immune response is needed in our model to explain the eventual viral clearance. This study provides a quantitative understanding of the biological factors that can explain the viral and interferon kinetics during a typical influenza virus infection.

Influenza, commonly referred to as the flu, is a contagious respiratory illness caused by influenza virus infections. Although most infected subjects with intact immune systems are able to clear the virus without developing serious flu complications, the mechanisms underlying viral control are not fully understood. In this paper, we address this question by developing mathematical models that include both innate and adaptive immune responses, and fitting them to experimental data from horses infected with equine influenza virus. We find that the innate immune response, such as natural killer cell-mediated infected cell killing and interferon's antiviral effect, can explain the first rapid viral decline and subsequent second peak viremia, and that the adaptive immune response is needed to eventually clear the virus. This study improves our understanding of influenza virus dynamics and may provide more information for future research in influenza pathogenesis, treatment, and vaccination.

Modeling the dynamics of virus shedding into the saliva of Epstein-Barr virus positive individuals

Epstein-Barr virus (EBV) can infect both B cells and epithelial cells. Infection of B cells enables the virus to persist within a host while infection of epithelial cells is suggested to amplify viral output. Data from a recent study have shown that the virus shedding in EBV positive individuals is relatively stable over short periods of time but varies significantly over long periods. The mechanisms underlying the regulation of virus shedding within a host are not fully understood. In this paper, we construct a model of ordinary differential equations to study the dynamics of virus shedding into the saliva of infected hosts. Infection of epithelial cells is further separated into infection by virus released from B cells and virus released from epithelial cells. We use the model to investigate whether the long-term variation and short-term stability of virus shedding can be generated by three possible factors: stochastic variations in the number of epithelial cells susceptible to virus released from infected B cells, to virus released from infected epithelial cells, or random variation in the probability that CD8(+) T cells encounter and successfully kill infected cells. The results support all three factors to explain the long-term variation but only the first and third factors to explain the short-term stability of virus shedding into saliva. Our analysis also shows that clearance of virus shedding is possible only when there is no virus reactivation from B cells.

Modeling quasispecies and drug resistance in hepatitis C patients treated with a protease inhibitor

Telaprevir, a novel hepatitis C virus (HCV) NS3-4A serine protease inhibitor, has demonstrated substantial antiviral activity in patients infected with HCV. However, drug-resistant HCV variants were detected in vivo at relatively high frequency a few days after drug administration. Here we use a two-strain mathematical model to explain the rapid emergence of drug resistance in HCV patients treated with telaprevir monotherapy. We examine the effects of backward mutation and liver cell proliferation on the preexistence of the mutant virus and the competition between wild-type and drug-resistant virus during therapy. We also extend the two-strain model to a general model with multiple viral strains. Mutations during therapy only have a minor effect on the dynamics of various viral strains, although they are capable of generating low levels of HCV variants that would otherwise be completely suppressed because of fitness disadvantages. Liver cell proliferation may not affect the pretreatment frequency of mutant variants, but is able to influence the quasispecies dynamics during therapy. It is the relative fitness of each mutant strain compared with wild-type that determines which strain(s) will dominate the virus population. This study provides a theoretical framework for exploring the prevalence of preexisting mutant variants and the evolution of drug resistance during treatment with other HCV protease inhibitors or polymerase inhibitors.

Combination antiviral therapy for influenza: predictions from modeling of human infections

Emergence of resistance is a major concern in influenza antiviral treatment and prophylaxis. Combination antiviral therapy might overcome this problem. Here, we estimate that all possible single mutants and a sizeable fraction of double mutants are generated during an uncomplicated influenza infection. While most of them may sustain a fitness cost, some variants may confer drug resistance and be selected during therapy. We argue that a triple combination regimen would markedly reduce the risk of antiviral resistance emergence in seasonal and pandemic influenza viruses, especially in seriously ill or immunocompromised hosts.

A perspective on modelling hepatitis C virus infection

By mathematically describing early hepatitis C virus (HCV) RNA decay after initiation of interferon (IFN)-based antiviral therapy, crucial parameters of the in vivo viral kinetics have been estimated, such as the rate of production and clearance of free virus, and the rate of loss of infected cells. Furthermore, by suggesting mechanisms of action for IFN and ribavirin mathematical modelling has provided a means for evaluating and optimizing treatment strategies. Here, we review recent modelling developments for understanding complex viral kinetics patterns, such as triphasic HCV RNA declines and viral rebounds observed in patients treated with pegylated interferon and ribavirin. Moreover, we discuss new modelling approaches developed to interpret the viral kinetics observed in clinical trials with direct-acting antiviral agents, which induce a rapid decline of wild-type virus but also engender a higher risk for emergence of drug-resistant variants. Lastly, as in vitro systems have allowed a better characterization of the virus lifecycle, we discuss new modelling approaches that combine the intracellular and the extracellular viral dynamics.

Treatment-mediated alterations in HIV fitness preserve CD4+ T cell counts but have minimal effects on viral load

For most HIV-infected patients, antiretroviral therapy controls viral replication. However, in some patients drug resistance can cause therapy to fail. Nonetheless, continued therapy with a failing regimen can preserve or even lead to increases in CD4⁺ T cell counts. To understand the biological basis of these observations, we used mathematical models to explain observations made in patients with drug-resistant HIV treated with enfuvirtide (ENF/T-20), an HIV-1 fusion inhibitor. Due to resistance emergence, ENF was removed from the drug regimen, drug-sensitive virus regrown, and ENF was re-administered. We used our model to study the dynamics of plasma-viral RNA and CD4⁺ T cell levels, and the competition between drug-sensitive and resistant viruses during therapy interruption and re-administration. Focusing on resistant viruses carrying the V38A mutation in gp41, we found ENF-resistant virus to be 17±3% less fit than ENF-sensitive virus in the absence of the drug, and that the loss of resistant virus during therapy interruption was primarily due to this fitness cost. Using viral dynamic parameters estimated from these patients, we show that although re-administration of ENF cannot suppress viral load, it can, in the presence of resistant virus, increase CD4⁺ T cell counts, which should yield clinical benefits. This study provides a framework to investigate HIV and T cell dynamics in patients who develop drug resistance to other antiretroviral agents and may help to develop more effective strategies for treatment.

The impact of antiretroviral drug-resistance on viral load, CD4⁺ T cells, and clinical outcomes is complex. We used mathematical models to evaluate the benefits of HIV drug therapy in the presence of drug-resistant virus. As an example, we considered resistance to enfuvirtide, the first FDA-approved fusion inhibitor. If viral load increases on drug therapy due to drug resistance, therapy with this drug may be stopped. We found that the drug resistant virus is less fit than the drug-sensitive virus in the absence of drug, and this fitness disadvantage causes the loss of drug-resistant virus during drug interruption. After the drug-sensitive virus replaces resistant virus, enfuvirtide therapy was re-administered. Analyzing the resulting viral kinetics, we demonstrate that despite the inability of the re-administered drug to suppress viral load because of the continued presence of drug resistant virus, therapy still provides benefit to the patient by preserving or increasing peripheral blood CD4⁺ T cell levels.