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

Within-host dynamics of HTLV-2 and HIV-1 co-infection with delay

This paper formulates a mathematical model for the co-infection of HTLV-2 and HIV-1 with latent reservoirs, four types of distributed-time delays and HIV-1-specific B cells. We establish that the solutions remain bounded and nonnegative, identify the system's steady states, and derive sufficient conditions ensuring both their existence and global asymptotic stability. The system's global stability is confirmed using Lyapunov's method. We provide numerical simulations to support the stability results. Sensitivity analysis of basic reproduction numbers of HTLV-2 mono-infection (R 1 ) and HIV-1 mono-infection (R 2 ) is conducted. We examine how time delays influence the interaction between HIV-1 and HTLV-2. Including delay terms in the model reflects the influence of antiviral treatments, which help decrease R 1 and R 2 , thus limiting the spread of infection. This highlights the potential for designing therapies that prolong delay period. Incorporating such delays improves model precision and supports more effective evaluation of treatment strategies.

The roles of continuous and discontinuous proliferations on hepatitis B virus infection

The proliferation of both uninfected and infected hepatocytes, as well as the recycling effects of rcDNA-containing capsids are two key mechanisms playing significant roles in the persistence and clearance of hepatitis B virus (HBV) infection. In this study, the temporal dynamics of this viral infection is investigated through two intercellular mathematical models considering proliferation of both types of hepatocytes (uninfected and infected) and recycling effects of capsids. Both models are formulated on the basis of a key finding in the existing literature: mitosis of an infected hepatocytes yields in two uninfected progenies. In the first model (defined by P-model), we examine the continuous proliferation (which occur continuously), while the second one (defined by M-model) deals with the discontinuous proliferation (happen when the concentration of liver cells decreases to less than 70% of its initial concentration). The proposed models are calibrated with the experimental data obtained from an adult chimpanzee. Results of this study suggest that when both hepatocytes proliferate with equal rate, proliferation helps the individual in a rapid recovery from the acute infection whereas in case of chronic infection, the severity of the infection increases. On the other hand, if the infected hepatocytes proliferate at a different rate that of uninfected hepatocytes, the proliferation of uninfected hepatocytes contributes to increase the infection, but the proliferation of infected hepatocytes acts to reduce the infection from the long-term perspective. The global sensitivity analysis also shows the same results. Furthermore, it is also observed that the differences between the outcomes of continuous and discontinuous proliferations are significant and noteworthy.

Stability and optimal control of a cytokine-enhanced general HIV infection model with antibody immune response and CTLs immune response

In this article, a cytokine-enhanced viral infection model with cytotoxic T lymphocytes (CTLs) immune response and antibody immune response is proposed and analyzed. The model contains six compartments: uninfected CD4+T cells, infected CD4+T cells, inflammatory cytokines, viruses, CTLs and antibodies. Different from the previous works, this model not only considers virus-to-cell transmission and cell-to-cell transmission, but also includes a new infection mode, namely cytokine-enhanced viral infection. The incidence rates of the healthy CD4+T cells with viruses, infected cells and inflammatory cytokines are given by general functions. Moreover, the production/proliferation and removal/death rates of all compartments are represented by general functions. Firstly, we prove that all the solutions of the model are nonnegative and uniformly bounded. Then, five key parameters with strong biological significance, namely the virus basic reproduction number R0, CTLs immune response reproduction number R1, antibody immune response reproductive number R2, CTLs immune competitive reproductive number R3 and antibody immune competitive reproductive number R4 are derived. Then, by using Lyapunov's method and LaSalle's invariance principle, we have shown the global stability of each equilibrium. In addition, the numerical simulation results also show that the theoretical results are correct. Finally, we formulate an optimal control problem and solve it using Pontryagins Maximum Principle and an efficient iterative numerical methods. The results of our numerical simulation show that it is very important to control the infection between viruses and cells and between cells and inflammatory cytokines for controlling HIV.

Local and global stability of an HCV viral dynamics model with two routes of infection and adaptive immunity

The aim of this article is to formulate and study a mathematical model describing hepatitis C virus (HCV) infection dynamics. The model includes two essential modes of infection transmission, namely, virus-to-cell and cell-to-cell. The effect of therapy and adaptive immunity are incorporated in the suggested model. The adaptive immunity is represented by its two categories, namely, the humoral and cellular immune responses. Our article begins by establishing some mathematical results through proving the model's well-posedness in terms of existence, positivity and boundedness of solutions. We present all the steady states of the problem that depend on specific reproduction numbers. It moves then to the theoretical investigation of the local and global stability analysis of the free disease equilibrium and the four disease equilibria. The local and global stability analysis of the HCV mathematical model are established via the Routh-Hurwitz criteria and Lyapunov-LaSalle invariance principle, respectively. Finally, our article presents some numerical simulations to validate the analytical study of the global stability. Numerical simulations have shown the effect of the drug therapies on the system's dynamical behavior.

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

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

Stability of a diffusive-delayed HCV infection model with general cell-to-cell incidence function incorporating immune response and cell proliferation

In this work, we analyse the dynamics of a five-dimensional hepatitis C virus infection mathematical model including the spatial mobility of hepatitis C virus particles, the transmission of hepatitis C virus infection by mitosis process of infected hepatocytes with logistic growth, time delays, antibody response and cytotoxic T lymphocyte (CTL) immune response with general incidence functions for both modes of infection transmission, namely virus-to-cell as well as cell-to-cell. Firstly, we prove rigorously the existence, the uniqueness, the positivity and the boundedness of the solution of the initial value and boundary problem associated with the new constructed model. Secondly, we found that the basic reproductive number is the sum of the basic reproduction number determined by cell-free virus infection, determined by cell-to-cell infection and determined by proliferation of infected cells. It is proved the existence of five spatially homogeneous equilibria known as infection-free, immune-free, antibody response, CTL response and antibody and CTL responses. By using the linearization methods, the local stability of the latter is established under some rigorous conditions. Finally, we proved the existence of periodic solutions by highlighting the occurrence of a Hopf bifurcation for a certain threshold value of one delay.

Modelling disease spread with spatio-temporal fractional derivative equations and saturated incidence rate

The global analysis of a spatio-temporal fractional order SIR infection model with saturated incidence function is suggested and studied in this paper. The dynamics of the infection is described by three partial differential equations including a time-fractional derivative order for each one of them. The equations of our model describe the evolution of the susceptible, the infected and the recovered individuals with taking into account the spatial diffusion for each compartment. We will choose a saturated incidence rate in order to describe the nonlinear force of the infection. First, we will prove the well-posedness of our suggested model in terms of existence and uniqueness of the solution. Also in this context, the boundedness and the positivity of solutions are established. Afterward, we will give the forms of the disease-free equilibrium and the endemic one. It was demonstrated that the global stability of the each equilibrium depends mainly on the basic reproduction number. Finally, numerical simulations are performed to validate the theoretical results and to show the effect of vaccination in reducing the infection severity. It was shown that the fractional derivative order has no effect on the equilibria stability but only on the convergence speed towards the steady states. It was also observed that vaccination is amongst the good strategies in controlling the disease spread.

Hepatitis C virus fractional-order model: mathematical analysis

Mathematical analysis of epidemics is crucial for the prediction of diseases over time and helps to guide decision makers in terms of public health policy. It is in this context that the purpose of this paper is to study a fractional-order differential mathematical model of HCV infection dynamics, incorporating two fundamental modes of transmission of the infection; virus-to-cell and cell-to-cell along with a cure rate of infected cells. The model includes four compartments, namely, the susceptible hepatocytes, the infected ones, the viral load and the humoral immune response, which is activated in the host to attack the virus. Each compartment involves a long memory effect that is modeled by a Caputo fractional derivative. Our paper starts with the investigation of some basic analytical results. First, we introduce some preliminaries about the needed fractional calculus tools. Next, we establish the well-posedness of our mathematical model in terms of proving the existence, positivity and boundedness of solutions. We present the different problem steady states depending on some reproduction numbers. After that, the paper moves to the stage of proving the global stability of the three steady states. To evaluate the theoretical study of the global stability, we apply a numerical method based on the fundamental theorem of fractional calculus as well as a three-step Lagrange polynomial interpolation method. The numerical simulations show that the free-endemic equilibrium is stable if the basic reproduction number is less than unity. In addition, the numerical tests demonstrate the stability of the other endemic equilibria under some optimal conditions. It is observed that the numerical simulations and the founding theoretical results are coherents.

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

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

Novel stochastic dynamics of a fractal-fractional immune effector response to viral infection via latently infectious tissues

In this paper, the global complexities of a stochastic virus transmission framework featuring adaptive response and Holling type II estimation are examined via the non-local fractal-fractional derivative operator in the Atangana-Baleanu perspective. Furthermore, we determine the existence-uniqueness of positivity of the appropriate solutions. Ergodicity and stationary distribution of non-negative solutions are carried out. Besides that, the infection progresses in the sense of randomization as a consequence of the response fluctuating within the predictive case's equilibria. Additionally, the extinction criteria have been established. To understand the reliability of the findings, simulation studies utilizing the fractal-fractional dynamics of the synthesized trajectory under the Atangana-Baleanu-Caputo derivative incorporating fractional-order α and fractal-dimension ℘ have also been addressed. The strength of white noise is significant in the treatment of viral pathogens. The persistence of a stationary distribution can be maintained by white noise of sufficient concentration, whereas the eradication of the infection is aided by white noise of high concentration.

SARS-CoV-2 and Influenza Viruses: Strategies to Cope with Co-infection and Bioinformatics Perspective

Almost a century after the devastating pandemic of the Spanish flu, humankind is facing the relatively comparable global outbreak of COVID‐19. COVID‐19 is an infectious disease caused by SARS‐CoV‐2 with an unprecedented transmission pattern. In the face of the recent repercussions of COVID‐19, many have argued that the clinical experience with influenza through the last century may have tremendous implications in the containment of this newly emerged viral disease. During the last 2 years, from the emergence of COVID‐19, tremendous advances have been made in diagnosing and treating coinfections. Several approved vaccines are available now for the primary prevention of COVID‐19 and specific treatments exist to alleviate symptoms. The present review article aims to discuss the pathophysiology, diagnosis, and treatment of SARS‐CoV‐2 and influenza A virus coinfection while delivering a bioinformatics‐based insight into this subject matter.

Qualitative behaviour of a stochastic hepatitis C epidemic model in cellular level

In this paper, a mathematical model describing the dynamical of the spread of hepatitis C virus (HCV) at a cellular level with a stochastic noise in the transmission rate is developed from the deterministic model. The unique time-global solution for any positive initial value is served. The Ito's Formula, the suitable Lyapunov function, and other stochastic analysis techniques are used to analyze the model dynamics. The numerical simulations are carried out to describe the analytical results. These results highlight the impact of the noise intensity accelerating the extinction of the disease.

Threshold dynamics of a HCV model with virus to cell transmission in both liver with CTL immune response and the extrahepatic tissue

In this paper, a deterministic model characterizing the within-host infection of Hepatitis C virus (HCV) in intrahepatic and extrahepatic tissues is presented. In addition, the model also includes the effect of the cytotoxic T lymphocyte (CTL) immunity described by a linear activation rate by infected cells. Firstly, the non-negativity and boundedness of solutions of the model are established. Secondly, the basic reproduction number R01 and immune reproduction number R02 are calculated, respectively. Three equilibria, namely, infection-free, CTL immune response-free and infected equilibrium with CTL immune response are discussed in terms of these two thresholds. Thirdly, the stability of these three equilibria is investigated theoretically as well as numerically. The results show that when R01<1 , the virus will be cleared out eventually and the CTL immune response will also disappear; when R02<1<R01 , the virus persists within the host, but the CTL immune response disappears eventually; when R02>1 , both of the virus and the CTL immune response persist within the host. Finally, a brief discussion will be given.

Global Dynamics of a Stochastic Viral Infection Model with Latently Infected Cells

In this paper, we study the global dynamics of a stochastic viral infection model with humoral immunity and Holling type II response functions. The existence and uniqueness of non-negative global solutions are derived. Stationary ergodic distribution of positive solutions is investigated. The solution fluctuates around the equilibrium of the deterministic case, resulting in the disease persisting stochastically. The extinction conditions are also determined. To verify the accuracy of the results, numerical simulations were carried out using the Euler–Maruyama scheme. White noise’s intensity plays a key role in treating viral infectious diseases. The small intensity of white noises can maintain the existence of a stationary distribution, while the large intensity of white noises is beneficial to the extinction of the virus.

Optimal Control of a Cell-to-Cell Fractional-Order Model with Periodic Immune Response for HCV

In this paper, a Caputo fractional-order HCV Periodic immune response model with saturation incidence, cell-to-cell and drug control was proposed. We derive two different basic reproductive numbers and their relation with infection-free equilibrium and the immune-exhausted equilibrium. Moreover, there exists some symmetry in the relationship between the two equilibria and the basic reproduction numbers. We obtain the global stability of the infection-free equilibrium by using Lyapunov function and the local stability of the immune-exhausted equilibrium. The optimal control problem is also considered and two control strategies are given; one is for ITX5061 monotherapy, the other is for ITX5061 and DAAs combination therapy. Matlab numerical simulation shows that combination therapy has lower objective function value; therefore, it is worth trying to use combination therapy to treat HCV infection.

Lyapunov function and global asymptotic stability for a new multiscale viral dynamics model incorporating the immune system response: Implemented upon HCV

In this paper, a new mathematical model is formulated that describes the interaction between uninfected cells, infected cells, viruses, intracellular viral RNA, Cytotoxic T-lymphocytes (CTLs), and antibodies. Hence, the model contains certain biological relations that are thought to be key factors driving this interaction which allow us to obtain precise logical conclusions. Therefore, it improves our perception, that would otherwise not be possible, to comprehend the pathogenesis, to interpret clinical data, to control treatment, and to suggest new relations. This model can be used to study viral dynamics in patients for a wide range of infectious diseases like HIV, HPV, HBV, HCV, and Covid-19. Though, analysis of a new multiscale HCV model incorporating the immune system response is considered in detail, the analysis and results can be applied for all other viruses. The model utilizes a transformed multiscale model in the form of ordinary differential equations (ODE) and incorporates into it the interaction of the immune system. The role of CTLs and the role of antibody responses are investigated. The positivity of the solutions is proven, the basic reproduction number is obtained, and the equilibrium points are specified. The stability at the equilibrium points is analyzed based on the Lyapunov invariance principle. By using appropriate Lyapunov functions, the uninfected equilibrium point is proven to be globally asymptotically stable when the reproduction number is less than one and unstable otherwise. Global stability of the infected equilibrium points is considered, and it has been found that each equilibrium point has a specific domain of stability. Stability regions could be overlapped and a bistable equilibria could be found, which means the coexistence of two stable equilibrium points. Hence, the solution converges to one of them depending on the initial conditions.

Ecological Approach to Understanding Superinfection Inhibition in Bacteriophage

In microbial communities, viruses compete with each other for host cells to infect. As a consequence of competition for hosts, viruses evolve inhibitory mechanisms to suppress their competitors. One such mechanism is superinfection exclusion, in which a preexisting viral infection prevents a secondary infection. The bacteriophage ΦX174 exhibits a potential superinfection inhibition mechanism (in which secondary infections are either blocked or resisted) known as the reduction effect. In this auto-inhibitory phenomenon, a plasmid containing a fragment of the ΦX174 genome confers resistance to infection among cells that were once permissive to ΦX174. Taking advantage of this plasmid system, we examine the inhibitory properties of the ΦX174 reduction effect on a range of wild ΦX174-like phages. We then assess how closely the reduction effect in the plasmid system mimics natural superinfection inhibition by carrying out phage-phage competitions in continuous culture, and we evaluate whether the overall competitive advantage can be predicted by phage fitness or by a combination of fitness and reduction effect inhibition. Our results show that viral fitness often correctly predicts the winner. However, a phage's reduction sequence also provides an advantage to the phage in some cases, modulating phage-phage competition and allowing for persistence where competitive exclusion was expected. These findings provide strong evidence for more complex dynamics than were previously thought, in which the reduction effect may inhibit fast-growing viruses, thereby helping to facilitate coexistence.

Analysis of a stochastic HBV infection model with delayed immune response

Considering the environmental factors and uncertainties, we propose, in this paper, a higher-order stochastically perturbed delay differential model for the dynamics of hepatitis B virus (HBV) infection with immune system. Existence and uniqueness of an ergodic stationary distribution of positive solution to the system are investigated, where the solution fluctuates around the endemic equilibrium of the deterministic model and leads to the stochastic persistence of the disease. Under some conditions, infection-free can be obtained in which the disease dies out exponentially with probability one. Some numerical simulations, by using Milstein's scheme, are carried out to show the effectiveness of the obtained results. The intensity of white noise plays an important role in the treatment of infectious diseases.

Quantitative differences between intra-host HCV populations from persons with recently established and persistent infections

Detection of incident hepatitis C virus (HCV) infections is crucial for identification of outbreaks and development of public health interventions. However, there is no single diagnostic assay for distinguishing recent and persistent HCV infections. HCV exists in each infected host as a heterogeneous population of genomic variants, whose evolutionary dynamics remain incompletely understood. Genetic analysis of such viral populations can be applied to the detection of incident HCV infections and used to understand intra-host viral evolution. We studied intra-host HCV populations sampled using next-generation sequencing from 98 recently and 256 persistently infected individuals. Genetic structure of the populations was evaluated using 245,878 viral sequences from these individuals and a set of selected features measuring their diversity, topological structure, complexity, strength of selection, epistasis, evolutionary dynamics, and physico-chemical properties. Distributions of the viral population features differ significantly between recent and persistent infections. A general increase in viral genetic diversity from recent to persistent infections is frequently accompanied by decline in genomic complexity and increase in structuredness of the HCV population, likely reflecting a high level of intra-host adaptation at later stages of infection. Using these findings, we developed a machine learning classifier for the infection staging, which yielded a detection accuracy of 95.22 per cent, thus providing a higher accuracy than other genomic-based models. The detection of a strong association between several HCV genetic factors and stages of infection suggests that intra-host HCV population develops in a complex but regular and predictable manner in the course of infection. The proposed models may serve as a foundation of cyber-molecular assays for staging infection, which could potentially complement and/or substitute standard laboratory assays.

Some stability results for a model of Hepatitis C including alanine aminotransferase and immune system

In clinical practice, many cirrhosis scores based on alanine aminotransferase (ALT) levels exist. Although the most recent direct acting antivirals (DAAs) reduce fibrosis and ALT levels, the Hepatitis C virus (HCV) is not always removed. In this paper, we study a mathematical model of the HCV virus, which takes into account the role of the immune system, to investigate the ALT behavior during therapy. We find five equilibrium points and analyze their stability. A sufficient condition for global asymptotical stability of the infection-free equilibrium is obtained and local asymptotical stability conditions are given for the immune-free infection and cytotoxic T lymphocytes (CTL) response equilibria. The stability of the infection equilibrium with the full immune response is numerically performed.

Stability and Hopf bifurcation of a delayed virus infection model with latently infected cells and Beddington–DeAngelis incidence

In this paper, the dynamical behaviors for a five-dimensional virus infection model with Latently Infected Cells and Beddington–DeAngelis incidence are investigated. In the model, four delays which denote the latently infected delay, the intracellular delay, virus production period and CTL response delay are considered. We define the basic reproductive number and the CTL immune reproductive number. By using Lyapunov functionals, LaSalle’s invariance principle and linearization method, the threshold conditions on the stability of each equilibrium are established. It is proved that when the basic reproductive number is less than or equal to unity, the infection-free equilibrium is globally asymptotically stable; when the CTL immune reproductive number is less than or equal to unity and the basic reproductive number is greater than unity, the immune-free infection equilibrium is globally asymptotically stable; when the CTL immune reproductive number is greater than unity and immune response delay is equal to zero, the immune infection equilibrium is globally asymptotically stable. The results show that immune response delay may destabilize the steady state of infection and lead to Hopf bifurcation. The existence of the Hopf bifurcation is discussed by using immune response delay as a bifurcation parameter. Numerical simulations are carried out to justify the analytical results.

Sensitivity analysis of chronic hepatitis C virus infection with immune response and cell proliferation

A new mathematical model of chronic hepatitis C virus (HCV) infection incorporating humoral and cell-mediated immune responses, distinct cell proliferation rates for both uninfected and infected hepatocytes, and antiviral treatment all at once, is formulated and analyzed meticulously. Analysis of the model elucidates the existence of multiple equilibrium states. Moreover, the model has a locally asymptotically stable disease-free equilibrium (DFE) whenever the basic reproduction number is less than unity. Local sensitivity analysis (LSA) result exhibits that the most influential (negatively sensitive) parameters on the epidemic threshold are the drug efficacy of blocking virus production and the drug efficacy of removing infection. However, LSA does not accurately assess uncertainty and sensitivity in the system and may mislead us since by default this technique holds all other parameters fixed at baseline values. To overcome this pitfall, one of the most robust and efficient global sensitivity analysis (GSA) methods, which is Latin hypercube sampling-partial rank correlation coefficient technique, elucidates that the proliferation rate of infected hepatocytes and the drug efficacy of killing infected hepatocytes are highly sensitive parameters that affect the transmission dynamics of HCV in any population. Our study suggests that cell proliferation of the infected hepatocytes can be very crucial in controlling disease outbreak. Thus, a future HCV drug that boosts cell-mediated immune response is expected to be quite effective in controlling disease outbreak.

Hopf Bifurcation and Stability Switches Induced by Humoral Immune Delay in Hepatitis C

The role of humoral immune delay on the dynamics of HCV infection incorporating both the modes of infection transmission, namely, viral and cellular transmissions with a non-cytolytic cure of infected hepatocytes is studied. The local and global asymptotic stability of the boundary equilibria, namely, infection-free and immune-free equilibrium are analyzed theoretically as well as numerically under the conditions on the basic reproduction number and the humoral immune reproduction number. The existence of Hopf bifurcation and consequent occurrence of bifurcating periodic orbits around the humoral immune activated equilibrium are illustrated. The findings show that Hopf bifurcation and stability switches occur under certain conditions as the bifurcation parameter crosses the critical values. Furthermore, the dynamical effect of the development rate of B cells is investigated numerically. The results obtained show that the system becomes unstable from stable and regains stability from instability depending on the development rate of B cells for a fixed delay value. Further, the results suggest that a high antigenic stimulation in humoral immunity is beneficial for uninfected hepatocytes with a significant reduction in virions density.

Impact of adaptive immune response and cellular infection on delayed virus dynamics with multi-stages of infected cells

In this investigation, we propose and analyze a virus dynamics model with multi-stages of infected cells. The model incorporates the effect of both humoral and cell-mediated immune responses. We consider two modes of transmissions, virus-to-cell and cell-to-cell. Multiple intracellular discrete-time delays have been integrated into the model. The incidence rate of infection as well as the generation and removal rates of all compartments are described by general nonlinear functions. We derive five threshold parameters which determine the existence of the equilibria of the model under consideration. A set of conditions on the general functions has been established which is sufficient to investigate the global stability of the five equilibria of the model. The global asymptotic stability of all equilibria is proven by utilizing Lyapunov function and LaSalle’s invariance principle. The theoretical results are illustrated by numerical simulations of the model with specific forms of the general functions.

Chemotactic effects in reaction-diffusion equations for inflammation

Predator-prey systems are used to model time-dependent virus and lymphocyte population during a liver infection and to discuss the influence of chemotactic behavior on the chronification tendency of such infections. Therefore, a model family of reaction-diffusion equations is presented, and the long-term behavior of the solutions is estimated by a critical value containing the reaction strength, the diffusion rate, and the extension of the liver domain. Fourier techniques are applied to evaluate the influence of chemotactic behavior of the immune response to the long-term behavior of locally linearized models. It turns out that the chemotaxis is a subordinated influence with respect to the chronification of liver infections.

Local immunodeficiency: Minimal networks and stability

Some basic aspects of the recently discovered phenomenon of local immunodeficiency (Skums et al. [1]) generated by antigenic cooperation in cross-immunoreactivity (CR) networks are investigated. We prove that local immunodeficiency (LI) that is stable under perturbations already occurs in very small networks and under general conditions on their parameters. Therefore our results are applicable not only to Hepatitis C where CR networks are known to be large (Skums et al. [1]), but also to other diseases with CR. A major necessary feature of such networks is the non-homogeneity of their topology. It is also shown that one can construct larger CR networks with stable LI by using small networks with stable LI as their building blocks. Our results imply that stable LI occurs in networks with quite general topology. In particular, the scale-free property of a CR network, assumed in Skums et al. [1], is not required.

Effects of randomness on viral infection model with application

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

Stability and Hopf bifurcation for a five-dimensional virus infection model with Beddington-DeAngelis incidence and three delays

In this paper, the dynamical behaviours for a five-dimensional virus infection model with three delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses and Beddington-DeAngelis incidence are investigated. The reproduction numbers for virus infection, antibody immune response, CTL immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By using the Lyapunov functionals and linearization method, the threshold conditions on the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response and interior, respectively, are established. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are presented to justify the analytical results.

Global dynamics of a delay virus model with recruitment and saturation effects of immune responses

In this paper, we formulate a virus dynamics model with the recruitment of immune responses, saturation effects and an intracellular time delay. With the help of uniform persistence theory and Lyapunov method, we show that the global stability of the model is totally determined by the basic reproductive number R0. Furthermore, we analyze the effects of the recruitment of immune responses on virus infection by numerical simulation. The results show ignoring the recruitment of immune responses will result in overestimation of the basic reproductive number and the severity of viral infection.

Stability of latent pathogen infection model with adaptive immunity and delays

In this paper we propose and analyze a pathogen dynamics model with antibody and Cytotoxic T Lymphocyte (CTL) immune responses. We incorporate latently infected cells and three distributed time delays into the model. We show that the solutions of the proposed model are nonnegative and ultimately bounded. We derive four threshold parameters which fully determine the existence and stability of the five steady states of the model. Using Lyapunov functionals, we established the global stability of the steady states of the model. The theoretical results are confirmed by numerical simulations.

Global stability of a diffusive and delayed virus infection model with general incidence function and adaptive immune response

In this paper, the dynamical behaviors for a five-dimensional virus infection model with diffusion and two delays which describes the interactions of antibody, cytotoxic T-lymphocyte (CTL) immune responses and a general incidence function are investigated. The reproduction numbers for virus infection, antibody immune response, CTL immune response, CTL immune competition and antibody immune competition, respectively, are calculated. By using the Lyapunov functionals and linearization methods, the threshold conditions on the global stability of the equilibria for infection-free, immune-free, antibody response, CTL response and antibody and CTL responses, respectively, are established if the space is assumed as homogeneous. When the space is inhomogeneous, the effects of diffusion, intracellular delay and production delay are obtained by the numerical simulations.

Global dynamics for discrete-time analog of viral infection model with nonlinear incidence and CTL immune response

In this paper, a discrete-time analog of a viral infection model with nonlinear incidence and CTL immune response is established by using the Micken non-standard finite difference scheme. The two basic reproduction numbers R 0 and R 1 are defined. The basic properties on the positivity and boundedness of solutions and the existence of the virus-free, the no-immune, and the infected equilibria are established. By using the Lyapunov functions and linearization methods, the global stability of the equilibria for the model is established. That is, when R 0 ≤ 1 then the virus-free equilibrium is globally asymptotically stable, and under the additional assumption ( A 4 ) when R 0 > 1 and R 1 ≤ 1 then the no-immune equilibrium is globally asymptotically stable and when R 0 > 1 and R 1 > 1 then the infected equilibrium is globally asymptotically stable. Furthermore, the numerical simulations show that even if assumption ( A 4 ) does not hold, the no-immune equilibrium and the infected equilibrium also may be globally asymptotically stable.

Single injection of CD8+ T lymphocytes derived from hematopoietic stem cells - Mathematical and numerical insights

Recently, hematopoietic stem cell (HSC) based therapy is being discussed as a possible treatment for HIV infection. The main advantage of this approach is that it limits the immune impairing effect of infection by introducing an independent influx of antigen-specific cytotoxic T lymphocytes (CTL). In this paper, we present a mathematical approach to predict the dynamics of HSC based therapy. We use a modification of a basic mathematical model for virus induced impairment of help to study how virus - immune system dynamics can be influenced by a single injection of CD8+ T lymphocytes derived from hematopoietic stem cells. Our mathematical and numerical results indicate that a single, large enough dose of genetically derived CTL may lead to restoration of the cellular immune response and result in long-term control of infection.

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

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

Dynamical analysis on a chronic hepatitis C virus infection model with immune response

A mathematical model for HCV infection is established, in which the effect of dendritic cells (DC) and cytotoxic T lymphocytes (CTL) on HCV infection is considered. The basic reproduction numbers of chronic HCV infection and immune control are found. The obtained results show that the infection dies out finally as the basic reproduction number of HCV infection is less than unity, and the infection becomes chronic as it is greater than unity. In the presence of chronic infection, the existence of immune control equilibrium is discussed completely, which illustrates that the backward bifurcation may occur under certain conditions, and that the two quantities, the sizes of the activated DC and the removed CTL during their average life-terms, play a critical role in controlling chronic HCV infection and immune response. The occurrence of backward bifurcation implies that there may be bistability for the model, i.e., the outcome of infection depends on the initial situation. By choosing the activated rate of non-activated DC or the cross-representation rate of activated DC as bifurcation number, Hopf bifurcation for certain condition shows the existence of periodic solution of the model. Again, numerical simulations suggest the dynamical complexity of the model including the instability of immune control equilibrium and the existence of stable periodic solution.

HIV evolution and progression of the infection to AIDS

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

Modelling hepatitis C virus infection and the development of hepatocellular carcinoma

While mathematical models exist describing the dynamics of hepatitis C virus (HCV), most of them focus on the short term dynamics after the commencement of antiviral therapy. This work is the first attempt at mathematically modelling the full course of HCV infection and the impact that these viral and immune processes have on the progression to hepatocellular carcinoma (HCC). This model is based on the premise that these long term conditions are ultimately random and likely driven by the cell-mediated immune response. The risk of cancer arising is modelled through a stochastic model that incorporates the dynamics of HCV over the course of infection. Our model simulations produce approximately 9% prevalence of HCC in individuals after 40 years, consistent with the literature estimates. We find that higher viral infectivity leads to a greater likelihood of developing HCC (p<0.0001), but it does not determine the speed with which it arises. This infectivity drives the level of immune response, the amount of hepatocyte proliferation, and the risk of a mutational event. In our simulations the probability of developing HCC increases approximately linearly with duration of infection at the rate of 2.4 incident cases per thousand HCV-infected person years. This indicates that the sooner viral replication can be suppressed through antiviral therapy, the greater the chance of forestalling HCC.

Noninvasive monitoring of hepatic damage from hepatitis C virus infection

The mathematical model for the dynamics of the hepatitis C proposed in Avendaño et al. (2002), with four populations (healthy and unhealthy hepatocytes, the viral load of the hepatitis C virus, and T killer cells), is revised. Showing that the reduced model obtained by considering only the first three of these populations, known as basic model, has two possible equilibrium states: the uninfected one where viruses are not present in the individual, and the endemic one where viruses and infected cells are present. A threshold parameter (the basic reproductive virus number) is introduced, and in terms of it, the global stability of both two possible equilibrium states is established. Other central result consists in showing, by model numerical simulations, the feasibility of monitoring liver damage caused by HCV, avoiding unnecessary biopsies and the undesirable related inconveniences/imponderables to the patient; another result gives a mathematical modelling basis to recently developed techniques for the disease assessment based essentially on viral load measurements.