Mathematical models of HIV pathogenesis and treatment

Exploring the dynamics of HIV and CD4+ T-cells with non-integer derivatives involving nonsingular and nonlocal kernel

It is important to examine and comprehend how HIV interacts with the immune system in order to manage the infection, enhance patient outcomes, advance medical research, and support global health and socioeconomic stability. In this study, we formulate the dynamics of HIV infection to investigate the intricate interactions between HIV and [Formula: see text] T-cells. The Atangana-Baleanu and Caputo-Fabrizio derivative frameworks are applied to comprehensively examine the phenomenon of HIV viral transmission. The basic concepts and results of fractional calculus are presented for the analysis of the model. In our work, we focus on the dynamical behavior of HIV and immune system. We introduce numerical schemes to elucidate the solution pathways of the recommended system of HIV. We have shown the influence of various input factors on the solution pathways of the recommended fractional system and highlighted the oscillatory behavior and chaotic nature of the dynamics. Our findings demonstrate the complexity of the system under study by revealing the existence of the chaotic and oscillatory nature in the dynamics of HIV. In order to quantitatively characterize HIV dynamics, a number of simulations are carried out, providing a visual representation of the effects of different input variables. It has been observed that the chaos and the oscillatory behaviour is strongly related to the nonlinearity of the system. The present study provides a basis for further initiatives that try to enhance interventions and policies to lessen the worldwide burden of infection.

Modelling the mechanisms of antibody mixtures in viral infections: the cases of sequential homologous and heterologous dengue infections

Antibodies play an essential role in the immune response to viral infections, vaccination or antibody therapy. Nevertheless, they can be either protective or harmful during the immune response. Moreover, competition or cooperation between mixed antibodies can enhance or reduce this protective or harmful effect. Using the laws of chemical reactions, we propose a new approach to modelling the antigen-antibody complex activity. The resulting expression covers not only purely competitive or purely independent binding but also synergistic binding which, depending on the antibodies, can promote either neutralization or enhancement of viral activity. We then integrate this expression of viral activity in a within-host model and investigate the existence of steady-states and their asymptotic stability. We complete our study with numerical simulations to illustrate different scenarios: firstly, where both antibodies are neutralizing and secondly, where one antibody is neutralizing and the other enhancing. The results indicate that efficient viral neutralization is associated with purely independent antibody binding, whereas strong viral activity enhancement is expected in the case of purely competitive antibody binding. Finally, data collected during a secondary dengue infection were used to validate the model. The dataset includes sequential measurements of virus and antibody titres during viremia in patients. Data fitting shows that the two antibodies are in strong competition, as the synergistic binding is low. This contributes to the high levels of virus titres and may explain the antibody-dependent enhancement phenomenon. Besides, the mortality of infected cells is almost twice as high as that of susceptible cells, and the heterogeneity of viral kinetics in patients is associated with variability in antibody responses between individuals. Other applications of the model may be considered, such as the efficacy of vaccines and antibody-based therapies.

Modelling the role of tourism in the spread of HIV: A case study from Malaysia

This study aimed to assess the role of tourism in the spread of human immunodeficiency virus (HIV) using Malaysian epidemiological data on HIV and acquired immunodeficiency syndrome (AIDS) incidence from 1986 to 2004. A population-level mathematical model was developed with the following compartments: the population susceptible to HIV infection, the clinically confirmed HIV-positive population, the population diagnosed with AIDS, and the tourist population. Additionally, newborns infected with HIV were considered. Sensitivity analyses and variations in fixed parameter values were used to explore the effect of changes to various parameter values on HIV incidence in the model. It was determined that variations in the rate of HIV-positive inbound tourist entries and the rate of foreign tourist exits (i.e., the duration of time tourists spent in Malaysia) significantly impacted the predicted incidence of HIV and AIDS in Malaysia. The model's fit to observed HIV and AIDS incidence was evaluated, resulting in adjusted R2 values of 53.3% and 53.2% for HIV and AIDS, respectively. Furthermore, the reproduction number (R0) was also calculated to quantify the stability of the HIV endemicity in Malaysia. The findings suggest that a steady-state level of HIV in Malaysia is achievable based on the low value ofR 0  = 0.0136, and the disease-free equilibrium was stable from the negative eigenvalues obtained, which is encouraging from a public health perspective. The Partial Rank Correlation Coefficient (PRCC) values between the proportion of newborns born HIV-positive, the rate of Malaysian tourist entries returning home after contracting HIV, and the rate of foreign tourist exits have a significant impact on theR 0 . The methods provide a framework for epidemiological modelling of HIV spread through transient population groups. The model results suggest that the role of tourism should not be overlooked within the set of available measures to mitigate the spread of HIV.

Modeling dynamics of acute HIV infection incorporating density-dependent cell death and multiplicity of infection

Understanding the dynamics of acute HIV infection can offer valuable insights into the early stages of viral behavior, potentially helping uncover various aspects of HIV pathogenesis. The standard viral dynamics model explains HIV viral dynamics during acute infection reasonably well. However, the model makes simplifying assumptions, neglecting some aspects of HIV infection. For instance, in the standard model, target cells are infected by a single HIV virion. Yet, cellular multiplicity of infection (MOI) may have considerable effects in pathogenesis and viral evolution. Further, when using the standard model, we take constant infected cell death rates, simplifying the dynamic immune responses. Here, we use four models-1) the standard viral dynamics model, 2) an alternate model incorporating cellular MOI, 3) a model assuming density-dependent death rate of infected cells and 4) a model combining (2) and (3)-to investigate acute infection dynamics in 43 people living with HIV very early after HIV exposure. We find that all models qualitatively describe the data, but none of the tested models is by itself the best to capture different kinds of heterogeneity. Instead, different models describe differing features of the dynamics more accurately. For example, while the standard viral dynamics model may be the most parsimonious across study participants by the corrected Akaike Information Criterion (AICc), we find that viral peaks are better explained by a model allowing for cellular MOI, using a linear regression analysis as analyzed by R2. These results suggest that heterogeneity in within-host viral dynamics cannot be captured by a single model. Depending on the specific aspect of interest, a corresponding model should be employed.

The reproduction number and its probability distribution for stochastic viral dynamics

We consider stochastic models of individual infected cells. The reproduction number, R, is understood as a random variable representing the number of new cells infected by one initial infected cell in an otherwise susceptible (target cell) population. Variability in R results partly from heterogeneity in the viral burst size (the number of viral progeny generated from an infected cell during its lifetime), which depends on the distribution of cellular lifetimes and on the mechanism of virion release. We analyse viral dynamics models with an eclipse phase: the period of time after a cell is infected but before it is capable of releasing virions. The duration of the eclipse, or the subsequent infectious, phase is non-exponential, but composed of stages. We derive the probability distribution of the reproduction number for these viral dynamics models, and show it is a negative binomial distribution in the case of constant viral release from infectious cells, and under the assumption of an excess of target cells. In a deterministic model, the ultimate in-host establishment or extinction of the viral infection depends entirely on whether the mean reproduction number is greater than, or less than, one, respectively. Here, the probability of extinction is determined by the probability distribution of R, not simply its mean value. In particular, we show that in some cases the probability of infection is not an increasing function of the mean reproduction number.

The effect of chemotaxis on T-cell regulatory dynamics

Autoimmune diseases, such as Multiple Sclerosis, are often modelled through the dynamics of T-cell interactions. However, the spatial aspect of such diseases, and how dynamics may result in spatially heterogeneous outcomes, is often overlooked. We consider the effects of diffusion and chemotaxis on T-cell regulatory dynamics using a three-species model of effector and regulator T-cell populations, along with a chemical signalling agent. While diffusion alone cannot lead to instability and spatial patterning, the inclusion of chemotaxis can result in multiple forms of instability that produce highly complicated spatiotemporal behaviour. The parameter regimes in which different instabilities occur are determined through linear stability analysis and the full dynamics is explored through numerical simulation.

Bayesian estimation of parameters in viral dynamics models with antiviral effect of interferons in a cell culture

The goal of this work is to estimate the efficacy of interferon therapy in the inhibition of infection by the human immunodeficiency virus type 1 (HIV-1) in a cell culture. For this purpose, three viral dynamics models with the antiviral effect of interferons are presented; the dynamics of cell growth differ among the models, and a variant with Gompertz-type cell dynamics is proposed. A Bayesian statistics approach is used to estimate the cell dynamics parameters, viral dynamics and interferon efficacy. The models are fitted to sets of experimental data on cell growth, HIV-1 infection without interferon therapy and HIV-1 infection with interferon therapy, respectively. The Watanabe-Akaike information criterion (WAIC) is used to determine the model that best fits the experimental data. In addition to the estimated model parameters, the average lifespan of the infected cells and the basic reproductive number are calculated.

A simple model for viral decay dynamics and the distribution of infected cell life spans in SHIV-infected infant rhesus macaques

The dynamics of HIV viral load following the initiation of antiretroviral therapy is not well-described by simple, single-phase exponential decay. Several mathematical models have been proposed to describe its more complex behavior, the most popular of which is two-phase exponential decay. The underlying assumption in two-phase exponential decay is that there are two classes of infected cells with different lifespans. However, with the exception of CD4+ T cells, there is not a consensus on all of the cell types that can become productively infected, and the fit of the two-phase exponential decay to observed data from SHIV.C.CH505 infected infant rhesus macaques was relatively poor. Therefore, we propose a new model for viral decay, inspired by the Gompertz model where the decay rate itself is a dynamic variable. We modify the Gompertz model to include a linear term that modulates the decay rate. We show that this simple model performs as well as the two-phase exponential decay model on HIV and SIV data sets, and outperforms it for the infant rhesus macaque SHIV.C.CH505 infection data set. We also show that by using a stochastic differential equation formulation, the modified Gompertz model can be interpreted as being driven by a population of infected cells with a continuous distribution of cell lifespans, and estimate this distribution for the SHIV.C.CH505-infected infant rhesus macaques. Thus, we find that the dynamics of viral decay in this model of infant HIV infection and treatment may be explained by a distribution of cell lifespans, rather than two distinct cell types.

Modelling and optimising healthcare interventions in a model with explicit within- and between-host dynamics

Given an endemic infectious disease and a budget, how do we optimally allocate interventions to control the disease? This paper shows that the optimal strategy varies depending on the budget, the type of intervention, the trajectory of pathogen load, and the objective. Using a model with explicit within- and between-host dynamics, we model isolation, supportive treatment, and specific treatment. Isolation and supportive treatment affect the transmission coefficient and the disease-induced mortality rate, respectively, in the between-host dynamics. Specific treatment affects the clearance rate of pathogens in the within-host dynamics. We study the optimisation of the three interventions for various budget levels via evaluating isolation and supportive treatment at the population level and specific treatment at both the population and individual levels. At the population level, we consider the risk of transmission, the burden of illness, and the survival probability, and to that end, we choose the population-level infection rate, the population density of infected individuals, and the total disease-induced mortality rate as objective functions. At the individual level, we consider the length of infection and the pathogen load, and to that end, we choose the maximum infection-age and the maximum pathogen load as objective functions. The objective is to minimise these functions through varying two variables that refer to when the intervention starts and when it stops for an infected individual and also indicate what kind of individuals can get the intervention from the population perspective. We find that the optimal strategy of isolation is to isolate individuals with a higher pathogen load, given a lower budget. The optimal strategy of supportive treatment can be the same as isolation or simply no treatment. The optimal strategy of specific treatment is complicated, and it can be to treat individuals with pathogen loads above a particular level until they recover or until the pathogens can decrease when treatment stops, or it can be another scenario.

Analysis of HIV latent infection model with multiple infection stages and different drug classes

Latently infected CD4+ T cells represent one of the major obstacles to HIV eradication even after receiving prolonged highly active anti-retroviral therapy (HAART). Long-term use of HAART causes the emergence of drug-resistant virus which is then involved in HIV transmission. In this paper, we develop mathematical HIV models with staged disease progression by incorporating entry inhibitor and latently infected cells. We find that entry inhibitor has the same effect as protease inhibitor on the model dynamics and therefore would benefit HIV patients who developed resistance to many of current anti-HIV medications. Numerical simulations illustrate the theoretical results and show that the virus and latently infected cells reach an infected steady state in the absence of treatment and are eliminated under treatment whereas the model including homeostatic proliferation of latently infected cells maintains the virus at low level during suppressive treatment. Therefore, complete cure of HIV needs complete eradication of latent reservoirs.

Nonlinear Sub-optimal Control Design for Suppressing HIV Replication

Acquired Immunodeficiency Syndrome is a deadly viral disease caused by the Human Immunodeficiency Virus in vivo, and its purpose is to destroy the immune system of the human body. The disease does not currently have a definitive vaccine or treatment, but treatment with pharmaceutical interventions (antiretroviral therapy, or ART) can slow down the progression of HIV. Daily use of prophylaxis measures may also have serious side effects for the patient, so the dosage and regimen of drugs should be constantly controlled. The dynamic models formulated for HIV infection are nonlinear differential equations. Therefore, nonlinear optimal control methods can be effective in increasing the efficiency of treatment. In this study, a sub-optimal controller based on the state-dependent Riccati equation (SDRE) approach to the dynamic model of HIV is introduced. One of the advantages of the SDRE approach is that the nonlinear properties of the system are preserved in the design control procedure. Furthermore, the specific conditions of infected individuals can be considered via choosing appropriate coefficients in the cost function and limiting the amount of drug administered. In the procedure of control design, all the state variables must be available for feedback in order to use the SDRE controller. In this regard, the Extended Kalman Filter observer is also implemented. The effect of different weighting matrices on these states is examined. In addition, to assess the effectiveness of the proposed control strategy, the well-known performance indicator root mean square error is also considered. Numerical simulations confirm the high efficiency and flexibility of the proposed approach.

Optimal Control of an HIV Model with Gene Therapy and Latency Reversing Agents

In this paper, we study the dynamics of HIV under gene therapy and latency reversing agents. While previous works modeled either the use of gene therapy or latency reversing agents, we consider the effects of a combination treatment strategy. For constant treatment controls, we establish global stability of the disease-free equilibrium and endemic equilibrium based on the value of R0. We then consider time-dependent controls and formulate an associated optimal control problem that emphasizes reduction of the latent reservoir. Characterizations for the optimal control profiles are found using Pontryagin’s Maximum Principle. We perform numerical simulations of the optimal control model using the fourth-order Runge–Kutta forward-backward sweep method. We find that a combination treatment of gene therapy with latency reversing agents provides better remission times than gene therapy alone. We conclude with a discussion of our findings and future work.

Modeling within-Host SARS-CoV-2 Infection Dynamics and Potential Treatments

The goal of this study was to develop a mathematical model to simulate the actions of drugs that target SARS-CoV-2 virus infection. To accomplish that goal, we have developed a mathematical model that describes the control of a SARS-CoV-2 infection by the innate and adaptive immune components. Invasion of the virus triggers the innate immunity, whereby interferon renders some of the target cells resistant to infection, and infected cells are removed by effector cells. The adaptive immune response is represented by plasma cells and virus-specific antibodies. The model is parameterized and then validated against viral load measurements collected in COVID-19 patients. We apply the model to simulate three potential anti-SARS-CoV-2 therapies: (1) Remdesivir, a repurposed drug that has been shown to inhibit the transcription of SARS-CoV-2, (2) an alternative (hypothetical) therapy that inhibits the virus' entry into host cells, and (3) convalescent plasma transfusion therapy. Simulation results point to the importance of early intervention, i.e., for any of the three therapies to be effective, it must be administered sufficiently early, not more than a day or two after the onset of symptoms. The model can serve as a key component in integrative platforms for rapid in silico testing of potential COVID-19 therapies and vaccines.

STABILITY ANALYSIS AND OPTIMAL CONTROL OF AN INTRACELLULAR HIV INFECTION MODEL WITH ANTIRETROVIRAL TREATMENT

In this paper, a deterministic model describing the dynamics of the in-host HIV infection of CD4[Formula: see text] T-cells is proposed. The model incorporates the presence of the CD[Formula: see text] T-cells and two types of antiretroviral drugs, for disrupting new infection and for inhibiting virus production, respectively. First, the existence, boundedness and positivity of the model solutions are shown, the basic reproduction number [Formula: see text] being then derived and shown to be a threshold value as far as the stability of the equilibria is concerned. When [Formula: see text] the infection-free equilibrium point is globally stable, whereas when [Formula: see text] the system is uniformly persistent and the infected equilibrium point is globally asymptotically stable. Further, we develop an optimal control model by taking the effect of the antiretroviral drugs to be control variables in order to minimize the HIV infection in different scenarios. By using Pontryagin’s Minimum Principle and solving the model numerically, the results show that each antiretroviral drug in isolation can play a key role in reducing the count of both infected CD4[Formula: see text] T-cells and HIV viruses. However, a combination of both drugs could reduce the in-host HIV infection more significantly.

Backward bifurcation in within-host HIV models

The activation and proliferation of naive CD4 T cells produce helper T cells, and increase the susceptible population in the presence of HIV. This may cause backward bifurcation. To verify this, we construct a simple within-host HIV model that includes the key variables, namely healthy naive CD4 T cells, helper T cells, infected CD4 T cells and virus. When the viral basic reproduction number R₀ is less than unity, we show theoretically and numerically that bistability for R_C<R₀<1 can be caused by a backward bifurcation due to a new susceptible population produced by activation of healthy naive CD4 T cells that become helper T cells. An extended model including the CTL dynamics may also show this backward bifurcation. In the case that the homeostatic source of healthy naive CD4 T cells is large, R_C is approximately the threshold for HIV to persist independent of initial conditions. The backward bifurcation may still occur even when we consider latent infections of naive CD4 T cells. Thus to control the spread of within-host HIV, it may be necessary for treatment to reduce the reproduction number below R_C.

A mathematical study on the drug resistant virus emergence with HIV/AIDS treatment cases

HIV/AIDS drug treatments, one of which is highly active anti-retroviral ther-apy (HAART), often fail by the emergence of drug resistant virus. In this paper we study a quantitative method to evaluate the chance of resistant virus gen-eration. To this end we develop a mathematical description of the possibility of the emergence of resistant virus species against drug treatments, depend-ing on the trajectories of the state variables of HIV infection dynamic model. By simulation studies of mathematical models we apply the proposed analy-sis method to HIV/AIDS drug therapies, improved gradual dosage reduction (iGDR) and structured treatment interruption (STI). Based on the analysis it can be explained the reason why STI therapy often fails. Moreover it is con-cluded that iGDR is desirable particularly by decreasing the threat of resistant virus emergence.

Mathematical modeling of interaction between innate and adaptive immune responses in COVID-19 and implications for viral pathogenesis

We have applied mathematical modeling to investigate the infections of the ongoing coronavirus disease-2019 (COVID-19) pandemic caused by SARS-CoV-2 virus. We first validated our model using the well-studied influenza viruses and then compared the pathogenesis processes between the two viruses. The interaction between host innate and adaptive immune responses was found to be a potential cause for the higher severity and mortality in COVID-19 patients. Specifically, the timing mismatch between the two immune responses has a major impact on disease progression. The adaptive immune response of the COVID-19 patients is more likely to come before the peak of viral load, while the opposite is true for influenza patients. This difference in timing causes delayed depletion of vulnerable epithelial cells in the lungs in COVID-19 patients while enhancing viral clearance in influenza patients. Stronger adaptive immunity in COVID-19 patients can potentially lead to longer recovery time and more severe secondary complications. Based on our analysis, delaying the onset of adaptive immune responses during the early phase of infections may be a potential treatment option for high-risk COVID-19 patients. Suppressing the adaptive immune response temporarily and avoiding its interference with the innate immune response may allow the innate immunity to more efficiently clear the virus.

A note on the impact of late diagnosis on HIV/AIDS dynamics: a mathematical modelling approach

Objectives:

The global incidence of HIV infection is not significantly decreasing, especially in sub-Saharan African countries. Though there is availability and accessibility of free HIV services, people are not being diagnosed early for HIV, and hence HIV-related mortality remains significantly high. We formulate a mathematical model for the spread of HIV using non linear ordinary differential equations in order to investigate the impact of late diagnosis of HIV on the spread of HIV.

Results:

The results suggest the need to encourage early initiation into HIV treatment as well as promoting HIV self-testing programs that enable more undiagnosed people to know their HIV status in order to curtail the continued spread of HIV.

Stochastic dynamics of Francisella tularensis infection and replication

We study the pathogenesis of Francisella tularensis infection with an experimental mouse model, agent-based computation and mathematical analysis. Following inhalational exposure to Francisella tularensis SCHU S4, a small initial number of bacteria enter lung host cells and proliferate inside them, eventually destroying the host cell and releasing numerous copies that infect other cells. Our analysis of disease progression is based on a stochastic model of a population of infectious agents inside one host cell, extending the birth-and-death process by the occurrence of catastrophes: cell rupture events that affect all bacteria in a cell simultaneously. Closed expressions are obtained for the survival function of an infected cell, the number of bacteria released as a function of time after infection, and the total bacterial load. We compare our mathematical analysis with the results of agent-based computation and, making use of approximate Bayesian statistical inference, with experimental measurements carried out after murine aerosol infection with the virulent SCHU S4 strain of the bacterium Francisella tularensis, that infects alveolar macrophages. The posterior distribution of the rate of replication of intracellular bacteria is consistent with the estimate that the time between rounds of bacterial division is less than 6 hours in vivo.

Infecting a host cell is required for the replication of many types of bacteria and viruses. In some cases, infected cells release new infectious agents continuously over their lifetime. In others, such as the Francisella tularensis bacterium studied here, they are released in a single burst that coincides with the cell’s death. We show how a stochastic model, the birth-and-death process with catastrophe, can be used to characterise infection in a single cell, thereby allowing us to account for burst events and quantify the kinetics of pathogenesis in the lung, the initial site of infection, as well as in other organs that the infection spreads to. We learn about the parameters of the mathematical model of Francisella tularensis infection making use of the experimental measurements of bacterial loads, together with approximate Bayesian statistical inference methods. The most important parameter describing the pathogenesis is the rate of replication of intracellular bacteria.

Modeling the effect of delay strategy on transmission dynamics of HIV/AIDS disease

In this manuscript, we investigate a nonlinear delayed model to study the dynamics of human-immunodeficiency-virus in the population. For analysis, we find the equilibria of a susceptible-infectious-immune system with a delay term. The well-established tools such as the Routh-Hurwitz criterion, Volterra-Lyapunov function, and Lasalle invariance principle are presented to investigate the stability of the model. The reproduction number and sensitivity of parameters are investigated. If the delay tactics are decreased, then the disease is endemic. On the other hand, if the delay tactics are increased then the disease is controlled in the population. The effect of the delay tactics with subpopulations is investigated. More precisely, all parameters are dependent on delay terms. In the end, to give the strength to a theoretical analysis of the model, a computer simulation is presented.

Immunological Paradigms, Mechanisms, and Models: Conceptual Understanding Is a Prerequisite to Effective Modeling

Most mathematical models that describe the individual or collective actions of cells aim at creating faithful representations of limited sets of data in a self-consistent manner. Consistency with relevant physiological rules pertaining to the greater picture is rarely imposed. By themselves, such models have limited predictive or even explanatory value, contrary to standard claims. Here I try to show that a more critical examination of currently held paradigms is necessary and could potentially lead to models that pass the test of time. In considering the evolution of paradigms over the past decades I focus on the “smart surveillance” theory of how T cells can respond differentially, individually and collectively, to both self- and foreign antigens depending on various “contextual” parameters. The overall perspective is that physiological messages to cells are encoded not only in the biochemical connections of signaling molecules to the cellular machinery but also in the magnitude, kinetics, and in the time- and space-contingencies, of sets of stimuli. By rationalizing the feasibility of subthreshold interactions, the “dynamic tuning hypothesis,” a central component of the theory, set the ground for further theoretical and experimental explorations of dynamically regulated immune tolerance, homeostasis and diversity, and of the notion that lymphocytes participate in nonclassical physiological functions. Some of these efforts are reviewed. Another focus of this review is the concomitant regulation of immune activation and homeostasis through the operation of a feedback mechanism controlling the balance between renewal and differentiation of activated cells. Different perspectives on the nature and regulation of chronic immune activation in HIV infection have led to conflicting models of HIV pathogenesis—a major area of research for theoretical immunologists over almost three decades—and can have profound impact on ongoing HIV cure strategies. Altogether, this critical review is intended to constructively influence the outlook of prospective model builders and of interested immunologists on the state of the art and to encourage conceptual work.

Global Properties of a Delay-Distributed HIV Dynamics Model Including Impairment of B-Cell Functions

In this paper, we construct an Human immunodeficiency virus (HIV) dynamics model with impairment of B-cell functions and the general incidence rate. We incorporate three types of infected cells, (i) latently-infected cells, which contain the virus, but do not generate HIV particles, (ii) short-lived productively-infected cells, which live for a short time and generate large numbers of HIV particles, and (iii) long-lived productively-infected cells, which live for a long time and generate small numbers of HIV particles. The model considers five distributed time delays to characterize the time between the HIV contact of an uninfected CD4 + T-cell and the creation of mature HIV. The nonnegativity and boundedness of the solutions are proven. The model admits two equilibria, infection-free equilibrium E P 0 and endemic equilibrium E P 1 . We derive the basic reproduction number R 0 , which determines the existence and stability of the two equilibria. The global stability of each equilibrium is proven by utilizing the Lyapunov function and LaSalle’s invariance principle. We prove that if R 0 < 1 , then E P 0 is globally asymptotically stable, and if R 0 > 1 , then E P 1 is globally asymptotically stable. These theoretical results are illustrated by numerical simulations. The effect of impairment of B-cell functions, time delays, and antiviral treatment on the HIV dynamics are studied. We show that if the functions of B-cells are impaired, then the concentration of HIV is increased in the plasma. Moreover, we observe that the time delay has a similar effect to drug efficacy. This gives some impression for developing a new class of treatments to increase the delay period and then suppress the HIV replication.

CMPK2 and BCL-G are associated with type 1 interferon-induced HIV restriction in humans

Type 1 interferons (IFN) are critical for host control of HIV and simian immunodeficiency virus. However, it is unknown which of the hundreds of interferon-stimulated genes (ISGs) restrict HIV in vivo. We sequenced RNA from cells that support HIV replication (activated CD4+ T cells) in 19 HIV-infected people before and after interferon-α2b (IFN-α2b) injection. IFN-α2b administration reduced plasma HIV RNA and induced mRNA expression in activated CD4+ T cells: The IFN-α2b-induced change of each mRNA was compared to the change in plasma HIV RNA. Of 99 ISGs, 13 were associated in magnitude with plasma HIV RNA decline. In addition to well-known restriction factors among the 13 ISGs, two novel genes, CMPK2 and BCL-G, were identified and confirmed for their ability to restrict HIV in vitro: The effect of IFN on HIV restriction in culture was attenuated with RNA interference to CMPK2, and overexpression of BCL-G diminished HIV replication. These studies reveal novel antiviral molecules that are linked with IFN-mediated restriction of HIV in humans.

The Role of Immune Response in Optimal HIV Treatment Interventions

A mathematical model for the transmission dynamics of human immunodeficiency virus (HIV) within a host is developed. Our model focuses on the roles of immune response cells or cytotoxic lymphocytes (CTLs). The model includes active and inactive cytotoxic immune cells. The basic reproduction number and the global stability of the virus free equilibrium is carried out. The model is modified to include anti-retroviral treatment interventions and the controlled reproduction number is explored. Their effects on the HIV infection dynamics are investigated. Two different disease stage scenarios are assessed: early-stage and advanced-stage of the disease. Furthermore, optimal control theory is employed to enhance healthy CD4+ T cells, active cytotoxic immune cells and minimize the total cost of anti-retroviral treatment interventions. Two different anti-retroviral treatment interventions (RTI and PI) are incorporated. The results highlight the key roles of cytotoxic immune response in the HIV infection dynamics and corresponding optimal treatment strategies. It turns out that the combined control (both RTI and PI) and stronger immune response is the best intervention to maximize healthy CD4+ T cells at a minimal cost of treatments.

Mathematical Analysis of Viral Replication Dynamics and Antiviral Treatment Strategies: From Basic Models to Age-Based Multi-Scale Modeling

Viral infectious diseases are a global health concern, as is evident by recent outbreaks of the middle east respiratory syndrome, Ebola virus disease, and re-emerging zika, dengue, and chikungunya fevers. Viral epidemics are a socio-economic burden that causes short- and long-term costs for disease diagnosis and treatment as well as a loss in productivity by absenteeism. These outbreaks and their socio-economic costs underline the necessity for a precise analysis of virus-host interactions, which would help to understand disease mechanisms and to develop therapeutic interventions. The combination of quantitative measurements and dynamic mathematical modeling has increased our understanding of the within-host infection dynamics and has led to important insights into viral pathogenesis, transmission, and disease progression. Furthermore, virus-host models helped to identify drug targets, to predict the treatment duration to achieve cure, and to reduce treatment costs. In this article, we review important achievements made by mathematical modeling of viral kinetics on the extracellular, intracellular, and multi-scale level for Human Immunodeficiency Virus, Hepatitis C Virus, Influenza A Virus, Ebola Virus, Dengue Virus, and Zika Virus. Herein, we focus on basic mathematical models on the population scale (so-called target cell-limited models), detailed models regarding the most important steps in the viral life cycle, and the combination of both. For this purpose, we review how mathematical modeling of viral dynamics helped to understand the virus-host interactions and disease progression or clearance. Additionally, we review different types and effects of therapeutic strategies and how mathematical modeling has been used to predict new treatment regimens.

Differential T cell response against BK virus regulatory and structural antigens: A viral dynamics modelling approach

BK virus (BKV) associated nephropathy affects 1-10% of kidney transplant recipients, leading to graft failure in about 50% of cases. Immune responses against different BKV antigens have been shown to have a prognostic value for disease development. Data currently suggest that the structural antigens and regulatory antigens of BKV might each trigger a different mode of action of the immune response. To study the influence of different modes of action of the cellular immune response on BKV clearance dynamics, we have analysed the kinetics of BKV plasma load and anti-BKV T cell response (Elispot) in six patients with BKV associated nephropathy using ODE modelling. The results show that only a small number of hypotheses on the mode of action are compatible with the empirical data. The hypothesis with the highest empirical support is that structural antigens trigger blocking of virus production from infected cells, whereas regulatory antigens trigger an acceleration of death of infected cells. These differential modes of action could be important for our understanding of BKV resolution, as according to the hypothesis, only regulatory antigens would trigger a fast and continuous clearance of the viral load. Other hypotheses showed a lower degree of empirical support, but could potentially explain the clearing mechanisms of individual patients. Our results highlight the heterogeneity of the dynamics, including the delay between immune response against structural versus regulatory antigens, and its relevance for BKV clearance. Our modelling approach is the first that studies the process of BKV clearance by bringing together viral and immune kinetics and can provide a framework for personalised hypotheses generation on the interrelations between cellular immunity and viral dynamics.

THE DYNAMICS OF HIV MODELS WITH SWITCHING PARAMETERS AND PULSE CONTROL

This paper studies some human immunodeficiency virus (HIV) models with switching parameters and pulse control. The classical virus dynamics model is first modified by incorporating switching parameters which are assumed to be time-varying. Some threshold conditions are derived to guarantee the virus elimination by utilizing a Razumikhin-type approach. The results show that the proper switching conditions chosen can increase the counts of CD4+T-cells while reducing viral load. Pulse control strategies are then applied to the above model. More precisely, the treatment strategy and the vaccination strategy are applied to infected cells and uninfected cells, respectively. Each control strategy is analyzed to gauge its success in achieving viral suppression. Numerical simulations are performed to complement the analytical results and motivate future directions.

An evaluation of the emergence of drug resistant virus for HIV/AIDS drug treatments

HIV/AIDS drug treatment, such as highly active anti-retroviral therapy (HAART), often fails due to the emergence of drug resistant species. In this paper we investigate a new estimation method for the possibility of emergence of drug resistant mutation. To the best knowledge of the author this work is the first study to try to describe quantitatively the possibility of drug resistance emergence for HIV/AIDS drug treatments. In simulation studies we compare HIV/AIDS treatment methods, such as structured treatment interruption (STI) and improved gradual dosage reduction (iGDR), based on the proposed analysis. From the analysis we can explain why STI treatment often fails and also can show that iGDR is desirable rather than STI particularly in terms of the decrease of the possibility of emergence of drug resistant virus.

Strong Inference in Mathematical Modeling: A Method for Robust Science in the Twenty-First Century

While there are many opinions on what mathematical modeling in biology is, in essence, modeling is a mathematical tool, like a microscope, which allows consequences to logically follow from a set of assumptions. Only when this tool is applied appropriately, as microscope is used to look at small items, it may allow to understand importance of specific mechanisms/assumptions in biological processes. Mathematical modeling can be less useful or even misleading if used inappropriately, for example, when a microscope is used to study stars. According to some philosophers (Oreskes et al., 1994), the best use of mathematical models is not when a model is used to confirm a hypothesis but rather when a model shows inconsistency of the model (defined by a specific set of assumptions) and data. Following the principle of strong inference for experimental sciences proposed by Platt (1964), I suggest “strong inference in mathematical modeling” as an effective and robust way of using mathematical modeling to understand mechanisms driving dynamics of biological systems. The major steps of strong inference in mathematical modeling are (1) to develop multiple alternative models for the phenomenon in question; (2) to compare the models with available experimental data and to determine which of the models are not consistent with the data; (3) to determine reasons why rejected models failed to explain the data, and (4) to suggest experiments which would allow to discriminate between remaining alternative models. The use of strong inference is likely to provide better robustness of predictions of mathematical models and it should be strongly encouraged in mathematical modeling-based publications in the Twenty-First century.

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.

HIV-1 CCR5 gene therapy will fail unless it is combined with a suicide gene

Highly active antiretroviral therapy (ART) has successfully turned Human immunodeficiency virus type 1 (HIV-1) from a deadly pathogen into a manageable chronic infection. ART is a lifelong therapy which is both expensive and toxic, and HIV can become resistant to it. An alternative to lifelong ART is gene therapy that targets the CCR5 co-receptor and creates a population of genetically modified host cells that are less susceptible to viral infection. With generic mathematical models we show that gene therapy that only targets the CCR5 co-receptor fails to suppress HIV-1 (which is in agreement with current data). We predict that the same gene therapy can be markedly improved if it is combined with a suicide gene that is only expressed upon HIV-1 infection.

Outcome Prediction in Mathematical Models of Immune Response to Infection

Clinicians need to predict patient outcomes with high accuracy as early as possible after disease inception. In this manuscript, we show that patient-to-patient variability sets a fundamental limit on outcome prediction accuracy for a general class of mathematical models for the immune response to infection. However, accuracy can be increased at the expense of delayed prognosis. We investigate several systems of ordinary differential equations (ODEs) that model the host immune response to a pathogen load. Advantages of systems of ODEs for investigating the immune response to infection include the ability to collect data on large numbers of ‘virtual patients’, each with a given set of model parameters, and obtain many time points during the course of the infection. We implement patient-to-patient variability v in the ODE models by randomly selecting the model parameters from distributions with coefficients of variation v that are centered on physiological values. We use logistic regression with one-versus-all classification to predict the discrete steady-state outcomes of the system. We find that the prediction algorithm achieves near 100% accuracy for v = 0, and the accuracy decreases with increasing v for all ODE models studied. The fact that multiple steady-state outcomes can be obtained for a given initial condition, i.e. the basins of attraction overlap in the space of initial conditions, limits the prediction accuracy for v > 0. Increasing the elapsed time of the variables used to train and test the classifier, increases the prediction accuracy, while adding explicit external noise to the ODE models decreases the prediction accuracy. Our results quantify the competition between early prognosis and high prediction accuracy that is frequently encountered by clinicians.

A mechanistic theory to explain the efficacy of antiretroviral therapy

In the early years of the AIDS epidemic, a diagnosis of HIV-1 infection was equivalent to a death sentence. The development of combination antiretroviral therapy (cART) in the 1990s to combat HIV-1 infection was one of the most impressive achievements of medical science. Today, patients who are treated early with cART can expect a near-normal lifespan. In this Opinion article, we propose a fundamental theory to explain the mechanistic basis of cART and why it works so well, including a model to assess and predict the efficacy of antiretroviral drugs alone or in combination.

MODELING AND OPTIMAL CONTROL FOR ANTIRETROVIRAL THERAPY

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

Mathematical models of HIV replication and pathogenesis

This review outlines how mathematical models have been helpful, and continue to be so, for obtaining insights into the in vivo dynamics of HIV infection. The review starts with a discussion of a basic mathematical model that has been frequently used to study HIV dynamics. Some crucial results are described, including the estimation of key parameters that characterize the infection, and the generation of influential theories which argued that in vivo virus evolution is a key player in HIV pathogenesis. Subsequently, more recent concepts are reviewed that have relevance for disease progression, including the multiple infection of cells and the direct cell-to-cell transmission of the virus through the formation of virological synapses. These are important mechanisms that can influence the rate at which HIV spreads through its target cell population, which is tightly linked to the rate at which the disease progresses towards AIDS.

HIV-1 eradication strategies: design and assessment

PURPOSE OF REVIEW

Recent developments have generated renewed interest in the possibility of curing HIV-1 infection. This review describes some of the practical challenges that will need to be overcome if curative strategies are to be successful.

RECENT FINDINGS

The latent reservoir for HIV-1 in resting memory CD4 T cells is the major barrier to curing the infection. The most widely discussed approach to curing the infection involves finding agents that reverse latency in resting CD4 T cells, with the assumption that the cells will then die from viral cytopathic effects or be lysed by host cytolytic T lymphocytes (CTLs). A major challenge is the development of in-vitro models that can be used to explore mechanisms and identify latency-reversing agents (LRAs). Although several models have been developed, including primary cell models, none of them may fully capture the quiescent state of the cells that harbour latent HIV-1 in vivo. An additional problem is that LRAs that do not cause T-cell activation may not lead to the death of infected cells. Finally, measuring the effects of LRAs in vivo is complicated by the lack of correlation between different assays for the latent reservoir.

SUMMARY

Progress on these practical issues is essential to finding a cure.

ModeLang: a new approach for experts-friendly viral infections modeling

Computational modeling is an important element of systems biology. One of its important applications is modeling complex, dynamical, and biological systems, including viral infections. This type of modeling usually requires close cooperation between biologists and mathematicians. However, such cooperation often faces communication problems because biologists do not have sufficient knowledge to understand mathematical description of the models, and mathematicians do not have sufficient knowledge to define and verify these models. In many areas of systems biology, this problem has already been solved; however, in some of these areas there are still certain problematic aspects. The goal of the presented research was to facilitate this cooperation by designing seminatural formal language for describing viral infection models that will be easy to understand for biologists and easy to use by mathematicians and computer scientists. The ModeLang language was designed in cooperation with biologists and its computer implementation was prepared. Tests proved that it can be successfully used to describe commonly used viral infection models and then to simulate and verify them. As a result, it can make cooperation between biologists and mathematicians modeling viral infections much easier, speeding up computational verification of formulated hypotheses.

Identification and experimental validation of an HIV model for HAART treated patients

The objective of this paper is to identify the parameters of a human immunodeficiency virus (HIV) evolution model from a clinical data set of patients treated with two different highly active antiretroviral therapy (HAART) protocols. After introducing a model with six state variables, a preliminary step considers the reduction of the number of parameters to be identified by means of sensitivity analysis, and then identifiability items are discussed. A nonlinear optimization-based procedure for identification is developed, which divides the unknown parameters into two families: the group dependent and the patient dependent parameters. Numerical results show that the identified model can be individually adapted to each patient and this result is promising for predicting the effects (e.g., failures or successes) of therapeutic actions.

Multi-agent model of hepatitis C virus infection

OBJECTIVES

The objective of this study is to design a method for modeling hepatitis C virus (HCV) infection using multi-agent simulation and to verify it in practice.

METHODS AND MATERIALS

In this paper, first, the modeling of HCV infection using a multi-agent system is compared with the most commonly used model type, which is based on differential equations. Then, the implementation and results of the model using a multi-agent simulation is presented. To find the values of the parameters used in the model, a method using inverted simulation flow and genetic algorithm is proposed. All of the data regarding HCV infection are taken from the paper describing the model based on the differential equation to which the proposed method is compared.

RESULTS

Important advantages of the proposed method are noted and demonstrated: these include flexibility, clarity, re-usability and the possibility to model more complex dependencies. Then, the simulation framework that uses the proposed approach is successfully implemented in C++ and is verified by comparing it to the approach based on differential equations. The verification proves that an objective function that performs the best is the function that minimizes the maximal differences in the data. Finally, an analysis of one of the already known models is performed, and it is proved that it incorrectly models a decay in the hepatocytes number by 40%.

CONCLUSIONS

The proposed method has many advantages in comparison to the currently used model types and can be used successfully for analyzing HCV infection. With almost no modifications, it can also be used for other types of viral infections.

APOBEC3G-Augmented Stem Cell Therapy to Modulate HIV Replication: A Computational Study

The interplay between the innate immune system restriction factor APOBEC3G and the HIV protein Vif is a key host-retrovirus interaction. APOBEC3G can counteract HIV infection in at least two ways: by inducing lethal mutations on the viral cDNA; and by blocking steps in reverse transcription and viral integration into the host genome. HIV-Vif blocks these antiviral functions of APOBEC3G by impeding its encapsulation. Nonetheless, it has been shown that overexpression of APOBEC3G, or interfering with APOBEC3G-Vif binding, can efficiently block in vitro HIV replication. Some clinical studies have also suggested that high levels of APOBEC3G expression in HIV patients are correlated with increased CD4+ T cell count and low levels of viral load; however, other studies have reported contradictory results and challenged this observation. Stem cell therapy to replace a patient's immune cells with cells that are more HIV-resistant is a promising approach. Pre-implantation gene transfection of these stem cells can augment the HIV-resistance of progeny CD4+ T cells. As a protein, APOBEC3G has the advantage that it can be genetically encoded, while small molecules cannot. We have developed a mathematical model to quantitatively study the effects on in vivo HIV replication of therapeutic delivery of CD34+ stem cells transfected to overexpress APOBEC3G. Our model suggests that stem cell therapy resulting in a high fraction of APOBEC3G-overexpressing CD4+ T cells can effectively inhibit in vivo HIV replication. We extended our model to simulate the combination of APOBEC3G therapy with other biological activities, to estimate the likelihood of improved outcomes.

Investigating mathematical models of immuno-interactions with early-stage cancer under an agent-based modelling perspective

Many advances in research regarding immuno-interactions with cancer were developed with the help of ordinary differential equation (ODE) models. These models, however, are not effectively capable of representing problems involving individual localisation, memory and emerging properties, which are common characteristics of cells and molecules of the immune system. Agent-based modelling and simulation is an alternative paradigm to ODE models that overcomes these limitations. In this paper we investigate the potential contribution of agent-based modelling and simulation when compared to ODE modelling and simulation. We seek answers to the following questions: Is it possible to obtain an equivalent agent-based model from the ODE formulation? Do the outcomes differ? Are there any benefits of using one method compared to the other? To answer these questions, we have considered three case studies using established mathematical models of immune interactions with early-stage cancer. These case studies were re-conceptualised under an agent-based perspective and the simulation results were then compared with those from the ODE models. Our results show that it is possible to obtain equivalent agent-based models (i.e. implementing the same mechanisms); the simulation output of both types of models however might differ depending on the attributes of the system to be modelled. In some cases, additional insight from using agent-based modelling was obtained. Overall, we can confirm that agent-based modelling is a useful addition to the tool set of immunologists, as it has extra features that allow for simulations with characteristics that are closer to the biological phenomena.