Infected Equilibrium Research Articles

Mathematical analysis of affinity hemodialysis on T-Cell depletion

Investigation into the mathematical analysis of affinity hemodialysis on T-cell depletion is considered to give more significant understanding on the infection dynamics of HIV. Our model revealed the possibility of more than two infected equilibriums with the incorporation of recovery through the affinity hemodialysis. The conditions for stable infected equilibrium to prevent full-blown of AIDS are obtained and stated as hypothesis. This work has, therefore, open the way for partnerships among modelers and clinicians to strengthening the insight into the nature and process of viral multiphasic decay noticed in some treated patients. The initial growth of virus, infected cells and glycoprotein as well as possibility of total viral clearance are examined in the study.

Impact of venereal transmission on the dynamics of vertically transmitted viral diseases among mosquitoes

Despite centuries of enormous control efforts, mosquito-borne diseases continue to show upward trend of morbidity. According to WHO reports, malaria caused 438000 deaths in the year 2015 and dengue cases have been increased 30-fold over the last five decades. To control these diseases, it is necessary to understand the transmission dynamics of them among mosquitoes. There are some vertically transmitted mosquito-borne diseases which can also be spread among mosquitoes through sexual contact (e.g., dengue, zika, chikungunya). Recent experimental observations indicate that for virus persistence in mosquito population, the role of venereal transmission cannot be ignored. It is therefore important to investigate which transmission route is more responsible for the persistence of the virus when there is no host. To this aim, we propose and analyze a novel compartmental model considering mosquito population only. To the best of authors knowledge, this is the first attempt to take into account both vertical and sexual transmission of the virus in a mathematical model. Expression representing the basic reproduction number is derived using Jacobian approach. Local stability conditions for disease-free equilibrium and complete infection equilibrium are obtained. Global sensitivity analysis of the system is performed with respect to an epidemiologically important response. While investigating the impact of sexual transmission in presence of vertical transmission, we observed that sexual transmission route has the potential to drive the equilibrium from disease free to endemic states. Further numerical experiments reveal that the virus will have higher half life in fertilized infected female mosquitoes for vertical transmission only than for venereal transmission alone. Furthermore, when both transmission pathways are active, a variety of parameters indicate threshold like behavior of the infection.

Mathematical evaluation of the role of cross immunity and nonlinear incidence rate on the transmission dynamics of two dengue serotypes

Dengue fever is a common disease which can cause shock, internal bleeding, and death in patients if a second infection is involved. In this paper, a multi-serotype dengue model with nonlinear incidence rate is formulated to study the transmission of two dengue serotypes. The dynamical behaviors of the proposed model depend on the threshold value R_{{0}}^{{n}} known as the reproductive number which depends on the associated reproductive numbers with serotype-1 and serotype-2. The value of R_{{0}}^{{n}} is used to reflect whether the disease dies out or becomes endemic. It is found that the proposed model has a globally stable disease-free equilibrium if R_{{0}}^{{n}}leq 1, which indicates that if public health measures that make (and keep) the threshold to a value less than unity are carried out, the strategy in disease control is effective in the sense that the number of infected human and mosquito populations in the community will be brought to zero irrespective of the initial sizes of sub-populations. When R_{{0}}^{{n}}>1, the endemic equilibria called the co-existence primary and secondary infection equilibria are locally asymptotically stable. The effects of cross immunity and nonlinear incidence rate are explored using data from Thailand to determine the effective strategy in controlling and preventing dengue transmission and reinfection.

Asymptotic Analysis of an Age-Structured HIV Infection Model with Logistic Target-Cell Growth and Two Infecting Routes

This paper deals with an age-structured HIV infection model with logistic growth for target cells and both virus-to-cell and cell-to-cell infection routes. Based on the existence of the infection-free and infection equilibria and some rigorous analyses for the considered model, we study the asymptotic stability of these equilibria via determining the distribution of eigenvalues. We also address the persistence of the solution semi-flow by proving the existence of a global attractor. Furthermore, Hopf bifurcation occurring at the positive steady state is exploited. At last, some numerical examples are provided to illustrate the obtained results.

Threshold dynamics of an HIV infection model with two distinct cell subsets

In this paper, we study an HIV infection model with two distinct cell subsets, which extends the one proposed by Hill et al. (2018). With the approach of Lyapunov functions, we establish a threshold dynamics with the basic reproduction number R0 being the threshold parameter. When R0<1 the infection-free equilibrium is globally asymptotically stable while when R0>1 the infectious equilibrium is globally asymptotically stable.

Stability of a general discrete-time viral infection model with humoral immunity and cellular infection

This paper studies the global stability of a general discrete-time viral infection model with virus-to-cell and cell-to-cell transmissions and with humoral immune response. We consider both latently and actively infected cells. The model incorporates three types of intracellular time delays. The production and clearance rates of all compartments as well as incidence rates of infection are modeled by general nonlinear functions. We use the nonstandard finite difference method to discretize the continuous-time model. We show that the solutions of the discrete-time model are positive and ultimately bounded. We derive two threshold parameters, the basic reproduction number R0 and the humoral immune response activation number R1, which completely determine the existence and stability of the model’s equilibria. By using Lyapunov functions, we have proven that if R0≤1, then the virus-free equilibrium Q0 is globally asymptotically stable; if R1≤1&amp;lt; R0, then the persistent infection equilibrium without immune response Q* is globally asymptotically stable; and if R1&amp;gt;1, then the persistent infection equilibrium with immune response Q¯ is globally asymptotically stable. We illustrate our theoretical results by using numerical simulations. The effects of antiretroviral drug therapy and time delay on the virus dynamics are also studied. We have shown that the time delay has a similar effect as the antiretroviral drug therapy.

GLOBAL STABILITY ANALYSIS AND PERMANENCE FOR AN HIV-1 DYNAMICS MODEL WITH DISTRIBUTED DELAYS

This paper mainly investigates the global asymptotic stabilities of two HIV dynamics models with two distributed intracellular delays incorporating Beddington-DeAngelis functional response infection rate. An eclipse stage of infected cells (i.e. latently infected cells), not yet producing virus, is included in our models. For the first model, it is proven that if the basic reproduction number $R_0$ is less than unity, then the infection-free equilibrium is globally asymptotically stable, and if $R_0 $ is greater than unity, then the infected equilibrium is globally asymptotically stable. We also obtain that the disease is always present when $R_0 $ is greater than unity by using a permanence theorem for infinite dimensional systems. What is more, a n-stage-structured HIV model with two distributed intracellular delays, which is the extensions to the first model, is developed and analyzed. We also prove the global asymptotical stabilities of two equilibria by constructing suitable Lyapunov functionals.

Some models with repulsion effect on superinfecting viruses by infected cells

In this paper, we study some models with repulsion effect on superinfecting viruses by infected cells [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text] are the density of uninfected cells, infected cells and viruses at time [Formula: see text] at location [Formula: see text], respectively. The functions [Formula: see text] and [Formula: see text] are assumed to be positive, continuous and bounded. [Formula: see text] denotes the production rate of uninfected cells. The infection rate is [Formula: see text] and the function [Formula: see text] is the production rate of free viruses. And [Formula: see text] is the rate of transfer from uninfected cells to infected cells. The positive constants [Formula: see text] and [Formula: see text] denote the death rate of uninfected cells, infected cells and viruses, respectively. The stability of the infection-free equilibrium solution and infection equilibrium solution is discussed. It is shown that if the basic reproduction number [Formula: see text] then the chemotaxis has no effect, that is, the infection-free constant solution is stable. For the system with chemotactic sensitivity [Formula: see text], if [Formula: see text], then the infection constant solution will be unstable under some conditions.

Analysis of a diffusive virus infection model with humoral immunity, cell-to-cell transmission and nonlinear incidence

In this paper, we investigate a reaction–diffusion virus infection model with humoral immunity and nonlinear incidence in heterogeneous and homogenous environments. The model includes simultaneously the virus-to-cell and cell-to-cell transmissions of the infection. The well-posedness of solutions, including the existence of global solutions and the ultimate boundedness, is established. The virus infection reproduction number R0 is calculated. When R0≤1, it is proved that the infection-free steady state is globally asymptotically stable. Otherwise, when R0>1 then the infection with antibody-free response is uniformly persistent. Furthermore, the antibody-free infection equilibrium is also globally asymptotically stable in spatially homogeneous case. In order to investigate the effect of humoral immunity, the virus infection with antibody response model in the spatially homogeneous environment is considered. The antibody response reproduction number R1 is calculated. The threshold criteria on the uniform persistence of total model and the global asymptotic stability of the antibody-response infection equilibrium are established. Finally, the numerical examples are presented to illustrate the theoretical results and verify the conjectures.

Studi Numerik Model Virus Hepatitis B dengan Pengaruh Penyembuhan dan Absorpsi

The phenomenon of Hepatitis B outbreak almost occurs in all developing countries including Indonesia. Hepatitis B infection can develop into acute or chronic. In the chronic stage, the infection can cause liver complications such as liver cirrhosis or liver cancer or even death. Mathematical modeling have been widely used to study the Hepatitis B virus infection. In this study a mathematical model is constructed by considering non-cytolytic immune response and pathogen absorption. The model is analyzed by determining the equilibrium point of the model, determining the existence of the equilibrium point, and analyzing the stability of the equilibrium point of the model with numerical simulation. In this case, numerical analysis is used to illustrate the conditions of infection-free and infected. Furthermore, the relation of the stability requirements of each equilibrium point is studied. The results show that there are two equilibrium points, uninfected and infected equilibrium point. Both of the uninfected equilibrium point and infected equilibrium point is asymptotically stable if a certain condition are met. Based on these results, the causes of a persistent infection are studied.

Global Dynamics of an Hepatitis C Virus Mathematical Cellular Model with a Logistic Term

In this paper, the aim is to analyze the global dynamics of Hepatitis C Virus (HCV) cellular mathematical model under therapy with uninfected hepatocytes proliferation. We prove that the solution of the model with positive initial values are global, positive and bounded. In addition, firstly we show that the model is locally asymptotically stable at free virus equilibrium and also at infected equilibrium. Secondly we show that the model is globally asymptotically stable at the free virus equilibrium by using an appropriate lyapunov function.

Dynamique d'un modèle d'infection HBV avec transmission cellule-à-cellule et réaction immunitaire CTL

In this work, we propose a mathematical model to describe the dynamics of the hepatitis B virus (HBV) infection by taking into account the cure of infected cells, the export of precursor cytotoxic T lympho-cytes (CTL) cells from the thymus and both modes of transmission that are the virus-to-cell infection and the cell-to-cell transmission. The local stability of the disease-free equilibrium and the chronic infection equilibrium is obtained via characteristic equations. Furthermore, the global stability of both equilibria is established by using two techniques, the direct Lyapunov method for the disease-free equilibrium and the geometrical approach for the chronic infection equilibrium. Dans ce travail, nous proposons un modèle mathématique pour décrire la dynamique du virus d'hépatite B (HBV) en prenant en compte le taux de guérison de cellules infectées, l'exportation de précurseur cytotoxic des lymphocytes T (CTL) des cellules du thymus et les deux modes de transmission qui sont l'infection virus-à-cellule et la transmission cellule-à-cellule.La stabilité locale de l'équilibre libre et l'équilibre d'infection chronique est obtenue via des équations caractéristiques. En outre, la stabilité globale des deux équilibres est établie en utilisant deux techniques, la méthode directe de Lyapunov pour l'équilibre libre et l'approche géométrique pour l'équilibre d'infection chronique.

Global stability of a fractional order HIV infection model with cure of infected cells in eclipse stage

Modeling by fractional order differential equations has more advantages to describe the dynamics of phenomena with memory which exists in many biological systems. In this paper, we propose a fractional order model for human immunodeficiency virus (HIV) infection by including a class of infected cells that are not yet producing virus, i.e., cells in the eclipse stage. We first prove the positivity and bound-edness of solutions in order to ensure the well-posedness of the proposed model. By constructing appropriate Lyapunov functionals, the global stability of the disease-free equilibrium and the chronic infection equilibrium is established. Numerical simulations are presented in order to validate our theoretical results.

Global dynamics of a reaction–diffusion virus infection model with humoral immunity and nonlinear incidence

In this paper, we propose and investigate a reaction–diffusion virus infection model with humoral immunity and nonlinear incidence. In spatially heterogeneous case, the basic reproduction number of virus infection R0 is calculated, when R0≤1 the global asymptotical stability of the infection-free steady state is established, and when R0>1 the uniform persistence of infected cells and viruses, as well as the existence of antibody-free infection steady state are also obtained. In spatially homogeneous case, the antibody response basic reproduction number R1 is calculated, by using the Lyapunov functions method and the persistence theory of dynamical systems we obtain that when R0>1 and R1≤1 the antibody-free infection equilibrium is globally asymptotically stable, and when R0>1 and R1>1 the model is uniformly persistent and the infection equilibrium exists and is also globally asymptotically stable. Finally, the numerical examples are presented in order to verify the validity of our theoretical results.

Delayed information induces oscillations in a dynamical model for infectious disease

During disease outbreak, it has been observed that information about the disease prevalence induces the individual’s behavioral changes. This information is usually assumed to be generated by the density of infective individuals and active mass media. The delay in reporting of these infective individuals may have its impact on generated information. Hence, to study the impact of delay on information generation, and therefore on the disease dynamics, a delay differential equation model is proposed and analyzed. The dynamics of information with delay effect is also modeled by a separate rate equation. Model analysis is performed and a unique infected equilibrium is obtained when the basic reproduction number ([Formula: see text]) is greater than one, whereas the disease free equilibrium always exists. When [Formula: see text], the disease free equilibrium is found to be locally stable independent of delay effect. The unique infected equilibrium is found to be locally stable till delay reaches a threshold value. The global stability of the unique infected equilibrium is also established under some parametric conditions by constructing a suitable Lyapunov function. The occurrence of Hopf bifurcation is observed when the delay in information crosses the threshold value. Analytically, the direction and stability of bifurcating periodic solutions is established. Further, we observed the occurrence of Hopf-Hopf bifurcation at two different delays. At first delay threshold, the endemic equilibrium loses its stability and produces periodic oscillations via Hopf bifurcation. It further regains its stability at second delay threshold via another Hopf bifurcation. Hence, the delay effect on information shows possibility of stability switches. Numerical experiments are carried out to support the obtained analytical results. Our study infers that the disease will show persistent oscillations if there is a significant time lag in reporting of infective after the disease outbreak. Thus, the delay in dissemination of information shows rich and complex dynamics in the model and provides important insights. We also observe numerically that the saturation in information plays a significant role on stability of infected equilibrium in presence of delay.

DYNAMICS OF AN INTRA-HOST MODEL OF MALARIA WITH A CONSTANT DRUG EFFICIENCY

In this paper, we investigate the dynamics of an intra-host model of malaria with logistic red blood growth, treatment and immune response. We provide a theoretical study of the model. We derive the basic reproduction number $\mathcal R_f$ which determines the extinction and the persistence of malaria within the body of a host. We compute equilibria and study their stability. More precisely, we show that there exists a threshold parameter $\zeta$ such that if $\mathcal R_f\leq\zeta\leq1$, the disease-free equilibrium is globally asymptotically stable. However, if $\mathcal R_f&gt;1$, there exist two malaria infection equilibria which are locally asymptotically stable: one malaria infection equilibrium without immune response and one malaria infection equilibrium with immune response. The sensitivity analysis of the model has been performed in order to determine the impact of related parameters on outbreak severity. The theory is supported by numerical simulations. We also derive a spatio-temporal model, using Diffusion-Reaction equations to model parasites dispersal. Finally, we provide numerical simulations for parasites spreading, and test different treatment scenarios.

The Global Stability Analysis of a Mathematical Cellular Model of Hepatitis C Virus Infection with Non-Cytolytic Process

The aim of this work is to analyse the global dynamics of an extended mathematical model of Hepatitis C virus (HCV) infection in vivo with cellular proliferation, spontaneous cure and hepatocyte homeostasis. We firstly prove the existence of local and global solutions of the model and establish some properties of this solution as positivity and asymptotic behaviour. Secondly we show, by the construction of appropriate Lyapunov functions, that the uninfected equilibrium and the unique infected equilibrium of the mathematical model of HCV are globally asymptotically stable respectively when the threshold number and when .

Dynamical behaviors of stochastic virus dynamic models with saturation responses

We consider a stochastic virus model with saturation functional responses, in which the death rate of the uninfected CD4+T, the infected CD4+T, and the HIV virus particles are influenced by white noises. We prove the global existence of a positive solution of the system. Although this system does not have any equilibrium points compared with the corresponding deterministic model, by Itô’s formula and by constructing a proper Lyapunov function, we prove that the solutions of the stochastic virus model fluctuate randomly around the uninfected equilibrium of the deterministic virus model if the crucial value R0 < 1. Further, we assume that the crucial value R0 > 1 and that the stochastic perturbations (of standard white noise type) influence the rate of change of the uninfected CD4+T, the infected CD4+T, and the HIV virus particles directly by strength proportional to the distances between T¯ and T, T¯* and T*(t), and V¯ and V(t), respectively. Here (T¯,T¯*,V¯) is the infected equilibrium of not only the stochastic system but also the corresponding deterministic system. We obtain the sufficient conditions which guarantee the stochastic asymptotical stability of the infected equilibrium. Finally, we present some numerical simulations to verify our results, and discuss our results.

Dynamics of a diffusive virus model with general incidence function, cell-to-cell transmission and time delay

In this paper, we revisit a diffusive virus dynamics model with general incidence function and time delay. One novelty of our model is that we introduce cell-to-cell transmission via formation of virological synapses to reflect the fact that it may play a more important role in virus spreading in addition to virus-to-cell infection. We justify the well-posedness of the model and identify the basic reproduction number ℜ0 for the model to be a sharp threshold value. The global stability of equilibria is determined by constructing suitable Lyapunov functionals in the sense that: the infection-free equilibrium is globally asymptotically stable if ℜ0≤1, and when ℜ0>1, the global asymptotic stability of infection equilibrium implies that the infection will persist. A significant impact of the cell-to-cell transmission is that they increase the basic reproduction number. If one neglects either the cell-to-cell transmission or virus-to-cell infection, the basic reproduction number of the model that is under-evaluated. Last, we perform numerical simulation to support our theoretic results. We set the domain of the viruses to be a two-dimensional square domain with the homogeneous Neumann boundary conditions to reflect the spatial spreading.

Dynamics of two time delays differential equation model to HIV latent infection

In this paper, we studied a HIV latent infection model with two time delays, where one delay is the time between viral entry into a cell and establishment of HIV latency and the other delay is the time between cell infection and viral production. The infection usually considered is linear, but in this article we consider that the infection rate of modeling HIV infection is nonlinear, where the rate of infection is βTV1+bV, and logistic growth of the uninfected target cells T. We defined the basic reproductive number and showed the local and global stability of the disease-free equilibrium and the permanence of the infected equilibrium. Furthermore, we discussed the dynamics of system under the three conditions: (1) τ1=τ2=0, (2) τ1=0,τ2>0, (3) τ1>0,τ2∈[0,τ2∗).