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

Bifurcation and stability analysis of within host HIV dynamics with multiple infections and intracellular delay

Human immunodeficiency virus (HIV) manifests multiple infections in CD4+ T cells, by binding its envelope proteins to CD4 receptors. Understanding these biological processes is crucial for effective interventions against HIV/AIDS. Here, we propose a mathematical model that accounts for the multiple infections of CD4+ T cells and an intracellular delay in the dynamics of HIV infection. We study the model system and establish the conditions under which the disease-free equilibrium point and the endemic equilibrium point are locally and globally asymptotically stable. We further provide the conditions under which these equilibrium points undergo forward or backward transcritical bifurcations for the autonomous model and Hopf bifurcation for both the delay model and autonomous models. Our simulation results show that an increase in the rate of multiple infections of CD4+ T cells stabilizes the endemic equilibrium point through Hopf bifurcation. However, in the presence of an intracellular delay, the model system evinces three types of stability scenarios at the endemic equilibrium point-instability switch, stability switch, and stability invariance and is demonstrated using bi-parameter diagrams. One of the novel aspects of this study is exhibiting all these interesting nonlinear dynamical results within a single model incorporating a single time delay.

Incorporating Intracellular Processes in Virus Dynamics Models

In-host models have been essential for understanding the dynamics of virus infection inside an infected individual. When used together with biological data, they provide insight into viral life cycle, intracellular and cellular virus-host interactions, and the role, efficacy, and mode of action of therapeutics. In this review, we present the standard model of virus dynamics and highlight situations where added model complexity accounting for intracellular processes is needed. We present several examples from acute and chronic viral infections where such inclusion in explicit and implicit manner has led to improvement in parameter estimates, unification of conclusions, guidance for targeted therapeutics, and crossover among model systems. We also discuss trade-offs between model realism and predictive power and highlight the need of increased data collection at finer scale of resolution to better validate complex models.

Stability and Hopf bifurcation of an HIV infection model with two time delays

This work focuses on an HIV infection model with intracellular delay and immune response delay, in which the former delay refers to the time it takes for healthy cells to become infectious after infection, and the latter delay refers to the time when immune cells are activated and induced by infected cells. By investigating the properties of the associated characteristic equation, we derive sufficient criteria for the asymptotic stability of the equilibria and the existence of Hopf bifurcation to the delayed model. Based on normal form theory and center manifold theorem, the stability and the direction of the Hopf bifurcating periodic solutions are studied. The results reveal that the intracellular delay cannot affect the stability of the immunity-present equilibrium, but the immune response delay can destabilize the stable immunity-present equilibrium through the Hopf bifurcation. Numerical simulations are provided to support the theoretical results.

Dynamics in a reaction-diffusion epidemic model via environmental driven infection in heterogenous space

In this paper, a reaction-diffusion SIR epidemic model via environmental driven infection in heterogeneous space is proposed. To reflect the prevention and control measures of disease in allusion to the susceptible in the model, the nonlinear incidence function Ef(S) is applied to describe the protective measures of susceptible. In the general spatially heterogeneous case of the model, the well-posedness of solutions is obtained. The basic reproduction number R0 is calculated. When R0≤1 the global asymptotical stability of the disease-free equilibrium is obtained, while when R0>1 the model is uniformly persistent. Furthermore, in the spatially homogeneous case of the model, when R0>1 the global asymptotic stability of the endemic equilibrium is obtained. Lastly, the numerical examples are enrolled to verify the open problems.

Stability and bifurcation for a stochastic differential algebraic Holling-II predator–prey model with nonlinear harvesting and delay

In this paper, a stochastic delayed differential algebraic predator–prey model with Michaelis–Menten-type prey harvesting is proposed. Due to the influence of gestation delay and stochastic fluctuations, the proposed model displays a complex dynamics. Criteria on the local stability of the interior equilibrium are established, and the effect of gestation delay on the model dynamics is discussed. Taking the gestation delay and economic profit as bifurcation parameters, Hopf bifurcation and singularity induced bifurcation can occur as they cross through some critical values, respectively. Moreover, the solution of the model will blow up in a limited time when delay [Formula: see text]. Then, we calculate the fluctuation intensity of the stochastic fluctuations by Fourier transform method, which is the key to illustrate the effect of stochastic fluctuations. Finally, we demonstrate our theoretical results by numerical simulations.

Viral dynamics of a latent HIV infection model with Beddington-DeAngelis incidence function, B-cell immune response and multiple delays

In this paper, an HIV infection model with latent infection, Beddington-DeAngelis infection function, B-cell immune response and four time delays is formulated. The well-posedness of the model solution is rigorously derived, and the basic reproduction number $\mathcal{R}_0$ and the B-cell immune response reproduction number $\mathcal{R}_1$ are also obtained. By analyzing the modulus of the characteristic equation and constructing suitable Lyapunov functions, we establish the global asymptotic stability of the uninfected and the B-cell-inactivated equilibria for the four time delays, respectively. Hopf bifurcation occurs at the B-cell-activated equilibrium when the model includes the immune delay, and the B-cell-activated equilibrium is globally asymptotically stable if the model does not include it. Numerical simulations indicate that the increase of the latency delay, the cell infection delay and the virus maturation delay can cause the B-cell-activated equilibrium stabilize, while the increase of the immune delay can cause it destabilize.

Effects of delay in immunological response of HIV infection

In this paper, a human immunodeficiency virus (HIV) infection model with both the types of immune responses, the antibody and the killer cell immune responses has been introduced. The model has been made more logical by including two delays in the activation of both the immune responses, along with the combination drug therapy. The inclusion of both the delayed immune responses provides a greater understanding of long-term dynamics of the disease. The dependence of the stability of the steady states of the model on the reproduction number [Formula: see text] has been explored through stability theory. Moreover, the global stability analysis of the infection-free steady state and the infected steady state has been proved with respect to [Formula: see text]. The bifurcation analysis of the infected steady state with respect to both delays has been performed. Numerical simulations have been carried out to justify the results proved. This model is capable of explaining the long-term dynamics of HIV infection to a greater extent than that of the existing model as it captures some basic parameters involved in the system such as immunological delay and immune response. Similarly, the model also explains the basic understanding of the disease dynamics as a result of activation of the immune response toward the virus.