Modelling and analysis of dynamics of viral infection of cells and of interferon resistance

Interferons are active biomolecules, which help fight viral infections by spreading from infected to uninfected cells and activate effector molecules, which confer resistance from the virus on cells. We propose a new model of dynamics of viral infection, including endocytosis, cell death, production of interferon and development of resistance. The novel element is a specific biologically justified mechanism of interferon action, which results in dynamics different from other infection models. The model reflects conditions prevailing in liquid cultures (ideal mixing), and the absence of cells or virus influx from outside. The basic model is a nonlinear system of five ordinary differential equations. For this variant, it is possible to characterise global behaviour, using a conservation law. Analytic results are supplemented by computational studies. The second variant of the model includes age-of-infection structure of infected cells, which is described by a transport-type partial differential equation for infected cells. The conclusions are: (i) If virus mortality is included, the virus becomes eventually extinct and subpopulations of uninfected and resistant cells are established. (ii) If virus mortality is not included, the dynamics may lead to extinction of uninfected cells. (iii) Switching off the interferon defense results in a decrease of the sum total of uninfected and resistant cells. (iv) Infection-age structure of infected cells may result in stabilisation or destabilisation of the system, depending on detailed assumptions. Our work seems to constitute the first comprehensive mathematical analysis of the cell-virus-interferon system based on biologically plausible hypotheses.     

Stability and Hopf bifurcation for a viral infection model with delayed non-lytic immune response

In this paper, a class of more general viral infection model with delayed non-lytic immune response is proposed based on some important biological meanings. The sufficient criteria for local and global asymptotic stabilities of the viral free equilibrium are given. And the stability and Hopf bifurcation of the infected equilibrium have been studied. Numerical simulations are carried out to explain the mathematical conclusions, and the effects of the birth rate of susceptible T cells and the efficacy of the non-lytic component on the stabilities of the positive equilibrium $\bar{E}$ are also studied by numerical simulations.     

A viral infection model with immune circadian rhythms

A viral infection model with immune circadian rhythms is investigated in this paper. By employing the persistence theory, we establish a threshold between the extinction and the uniform persistence of the disease. These results can be used to explain the oscillation behaviors of virus population, which were observed in chronic HBV or HCV carriers. Further, numerical simulations indicate that the dynamics of the lytic component of cytotoxicity T cells (CTLs) is crucial to the outcome of a viral infection.     

Global dynamics of an age‐structured virus model with saturation effects

An infection‐age virus dynamics model for human immunodeficiency virus (or hepatitis B virus) infections with saturation effects of infection rate and immune response is investigated in this paper. It is shown that the global dynamics of the model is completely determined by two critical values R ₀, the basic reproductive number for viral infection, and R ₁, the viral reproductive number at the immune‐free infection steady state (R ₁<R ₀). If R ₀<1, the uninfected steady state E ⁰ is globally asymptotically stable; if R ₀>1 > R ₁, the immune‐free infected steady state E ^? is globally asymptotically stable; while if R ₁>1, the antibody immune infected steady state is globally asymptotically stable. Moreover, our results show that ignoring the saturation effects of antibody immune response or infection rate will result in an overestimate of the antibody immune reproductive number. Copyright © 2016 John Wiley & Sons, Ltd.     

Probability of a major infection in a stochastic within-host model with multiple stages

Establishment or spread of a viral infection within healthy individuals depends on exposure to a viral source, either through virus particles or through cells that have been infected. We assume that a potential infection has reached the site of the healthy target cells and we apply stochastic within-host models and multitype branching processes to investigate whether a major infection becomes established. The model includes multiple latent and actively infected stages. It is shown that the probability of a major infection is generally more likely after the virus has entered the target cell and the cell is actively infected. In some cases, the probability of a major infection is less likely if the burst size of actively infected cells is small.     

Global stability of humoral immunity virus dynamics models with nonlinear infection rate and removal

In this paper, we investigate the dynamical behavior of two nonlinear models for viral infection with humoral immune response. The first model contains four compartments; uninfected target cells, actively infected cells, free virus particles and B cells. The intrinsic growth rate of uninfected cells, incidence rate of infection, removal rate of infected cells, production rate of viruses, neutralization rate of viruses, activation rate of B cells and removal rate of B cells are given by more general nonlinear functions. The second model is a modification of the first one by including an eclipse stage of infected cells. We assume that the latent-to-active conversion rate is also given by a more general nonlinear function. For each model we derive two threshold parameters and establish a set of conditions on the general functions which are sufficient to determine the global dynamics of the models. By using suitable Lyapunov functions and LaSalle’s invariance principle, we prove the global asymptotic stability of the all equilibria of the models. We perform some numerical simulations for the models with specific forms of the general functions and show that the numerical results are consistent with the theoretical results.     

Mathematical modelling of acute virus influenza A infections

This article models the immune system and the virus dynamics of acute influenza infection mathematically. We use the model to study the virus dynamics of some well-known and severe and mild types of viruses. Linkages to well-known models in the literature are illustrated. Simulations are compared with experimental results in vivo by comparing with results from infected ferrets where infection closely resembles those in humans. Good agreement is achieved between the model calculations and the experimental values for influenza A viruses. For the Spanish flu virus H1N1 peak virus load is high and virtually all cells are infected in the nostril. In general, the H1N1 viruses show much more prolonged infections than the H3N2 in the nostril. We suggest that the reason is that unspecific immunity attacks H3N2-budded viruses but not H1N1 viruses.     

GLOBAL STABILITY IN A VIRAL INFECTION MODEL

This paper investigates the global stability of a viral infection model with lytic immune response. If the basic reproductive ratio of the virus is less than or equal to one, by the LaSalle's invariance principle, the disease-free steady state is globally asymptotically stable. If the basic reproductive ratio of the virus is greater than one but less than or equal to a constant, which is defined by the parameters of the model, then the immune-exhausted steady state is globally asymptotically stable. The endemic steady state is globally asymptotically stable if the inverse is valid.     

Analysis of a viral infection model with delayed immune response

It is well known that the immune response plays an important role in eliminating or controlling the disease after human body is infected by virus. In this paper, we investigate the dynamical behavior of a viral infection model with retarded immune response. The effect of time delay on stability of the equilibria of the system has been studied and sufficient condition for local asymptotic stability of the infected equilibrium and global asymptotic stability of the infection-free equilibrium and the immune-exhausted equilibrium are given. By numerical simulating,we observe that the stationary solution becomes unstable at some critical immune response time, while the delay time and birth rate of susceptible host cells increase, and we also discover the occurrence of stable periodic solutions and chaotic dynamical behavior. The results can be used to explain the complexity of the immune state of patients.     

Dynamics analysis of a delayed viral infection model with immune impairment

In this paper, the dynamical behavior of a delayed viral infection model with immune impairment is studied. It is shown that if the basic reproductive number of the virus is less than one, then the uninfected equilibrium is globally asymptotically stable for both ODE and DDE model. And the effect of time delay on stabilities of the equilibria of the DDE model has been studied. By theoretical analysis and numerical simulations, we show that the immune impairment rate has no effect on the stability of the ODE model, while it has a dramatic effect on the infected equilibrium of the DDE model.     

Lytic cycle: A defining process in oncolytic virotherapy

The viral lytic cycle is an important process in oncolytic virotherapy. Most mathematical models for oncolytic virotherapy do not incorporate this process. In this article, we propose a mathematical model with the viral lytic cycle based on the basic mathematical model for oncolytic virotherapy. The viral lytic cycle is characterized by two parameters, the time period of the viral lytic cycle and the viral burst size. The time period of the viral lytic cycle is modeled as a delay parameter. The model is a nonlinear system of delay differential equations. The model reveals a striking feature that the critical value of the period of the viral lytic cycle is determined by the viral burst size. There are two threshold values for the burst size. Below the first threshold, the system has an unstable trivial equilibrium and a globally stable virus free equilibrium for any nonnegative delay, while the system has a third positive equilibrium when the burst size is greater than the first threshold. When the burst size is above the second threshold, there is a functional relation between the bifurcation value of the delay parameter for the period of the viral lytic cycle and the burst size. If the burst size is greater than the second threshold, the positive equilibrium is stable when the period of the viral lytic cycle is smaller than the bifurcation value, while the system has orbitally stable periodic solutions when the period of the lytic cycle is longer than the bifurcation value. However, this bifurcation value becomes smaller when the burst size becomes bigger. The viral lytic cycle may explain the oscillation phenomena observed in many studies. An important clinic implication is that the burst size should be carefully modified according to its effect on the lytic cycle when a type of a virus is modified for virotherapy, so that the period of the viral lytic cycle is in a suitable range which can break away the stability of the positive equilibria or periodic solutions.     

The role of CD4 T cells in immune system activation and viral reproduction in a simple model for HIV infection

CD4 T cells play a fundamental role in the adaptive immune response including the stimulation of cytotoxic lymphocytes (CTLs). Human immunodeficiency virus (HIV) which infects and kills CD4 T cells causes progressive failure of the immune system. However, HIV particles are also reproduced by the infected CD4 T cells. Therefore, during HIV infection, infected and healthy CD4 T cells act in opposition to each other, reproducing virus particles and activating and stimulating cellular immune responses, respectively. In this investigation, we develop and analyze a simple system of four ordinary differential equations that accounts for these two opposing roles of CD4 T cells. The model illustrates the importance of the CTL immune response during the asymptomatic stage of HIV infection. In addition, the solution behavior exhibits the two stages of infection, asymptomatic and final AIDS stages. In the model, a weak immune response results in a short asymptomatic stage and faster development of AIDS, whereas a strong immune response illustrates the long asymptomatic stage. A model with a latent stage for infected CD4 T cells is also investigated and compared numerically with the original model. The model shows that strong stimulation of CTLs by CD4 T cells is necessary to prevent progression to the AIDS stage.     

Mathematical modeling of respiratory viral infection and applications to SARS-CoV-2 progression

Viral infection in cell culture and tissue is modeled with delay reaction-diffusion equations. It is shown that progression of viral infection can be characterized by the viral replication number, time-dependent viral load, and the speed of infection spreading. These three characteristics are determined through the original model parameters including the rates of cell infection and of virus production in the infected cells. The clinical manifestations of viral infection, depending on tissue damage, correlate with the speed of infection spreading, while the infectivity of a respiratory infection depends on the viral load in the upper respiratory tract. Parameter determination from the experiments on Delta and Omicron variants allows the estimation of the infection spreading speed and viral load. Different variants of the SARS-CoV-2 infection are compared confirming that Omicron is more infectious and has less severe symptoms than Delta variant. Within the same variant, spreading speed (symptoms) correlates with viral load allowing prognosis of disease progression.     

Dynamics of an HIV infection model with virus diffusion and latently infected cell activation

Effective combination therapy usually reduces the plasma viral load of HIV to below the detection limit, but it cannot eradicate the virus. The latently infected cell activation is considered to be the main obstacle to completely eradicating HIV infection. In this paper, we consider an HIV infection model with latently infected cell activation, virus diffusion and spatial heterogeneity under Neumann boundary condition. The basic reproduction ratio is characterized by the principal eigenvalue of the related elliptic eigenvalue problem. Besides, by constructing Lyapunov functionals and using Green’s first identity, the global threshold dynamics of the system are completely established. Numerical simulations are carried out to illustrate the theoretical results, in particular, the influence of virus diffusion rate on the basic reproduction ratio is addressed.     

A with-in host Dengue infection model with immune response

A model of viral infection of monocytes population by Dengue virus is formulated here. The model can capture phenomena that dengue virus is quickly cleared in approximately 7 days after the onset of the symptoms. The model takes into account the immune response. It is shown that the quantity of free virus is decreasing when the viral invasion rate is increasing. The basic reproduction ratio of model without immune response is reduced significantly by adding the immune response. Numerical simulations indicate that the growth of immune response and the invasion rate are very crucial in identification of the intensity of infection.     

Dynamics of virus infection models with density‐dependent diffusion and Beddington–DeAngelis functional response

In this work, we integrate both density‐dependent diffusion process and Beddington–DeAngelis functional response into virus infection models to consider their combined effects on viral infection and its control. We perform global analysis by constructing Lyapunov functions and prove that the system is well posed. We investigated the viral dynamics for scenarios of single‐strain and multi‐strain viruses and find that, for the multi‐strain model, if the basic reproduction number for all viral strains is greater than 1, then each strain persists in the host. Our investigation indicates that treating a patient using only a single type of therapy may cause competitive exclusion, which is disadvantageous to the patient's health. For patients infected with several viral strains, the combination of several therapies is a better choice. Copyright © 2017 John Wiley & Sons, Ltd.     

Dynamics of an HIV‐1 virus model with both virus‐to‐cell and cell‐to‐cell transmissions,general incidence rate,intracellular delay,and CTL immune responses

Direct cell‐to‐cell transmission of HIV‐1 is a more efficient means of virus infection than virus‐to‐cell transmission. In this paper, we incorporate both these transmissions into an HIV‐1 virus model with nonlinear general incidence rate, intracellular delay, and cytotoxic T lymphocyte (CTL) immune responses. This model admits three types of equilibria: infection‐free equilibrium, CTL‐inactivated equilibrium, and CTL‐activated equilibrium. By using Lyapunov functionals and LaSalle invariance principle, it is verified that global threshold dynamics of the model can be explicitly described by the basic reproduction numbers.     

Global dynamics of a viral infection model with a latent period and Beddington-DeAngelis response

In this paper, we study the global dynamics of a viral infection model with a latent period. The model has a nonlinear function which denotes the incidence rate of the virus infection in vivo. The basic reproduction number of the virus is identified and it is shown that the uninfected equilibrium is globally asymptotically stable if the basic reproduction number is equal to or less than unity. Moreover, the virus and infected cells eventually persist and there exists a unique infected equilibrium which is globally asymptotically stable if the basic reproduction number is greater than unity. The basic reproduction number determines the equilibrium that is globally asymptotically stable, even if there is a time delay in the infection.