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• To demonstrate the Galerkin method's validity, via the accuracy of its predictions using DPDE models. • To structure a process for applying the theoretical framework to find theoretical results. • To show the impacts of diffusion and immune response delay on the model, through bifurcation diagrams. • To determine the steady-state solutions, Hopf points, and the stable and unstable zones. • To illustrate comparisons of the DDE and DPDE approaches through a selection of 2-D and 3-D solution maps. This paper studies a diffusive viral infection system with delayed immune response in the one-domain system. A system of DDE equations was explored, both analytically and numerically, using the Galerkin method. A condition that helps to find Hopf bifurcation points is determined. Full maps of the Hopf bifurcation points as well regions of stability are constructed and considered in detail. It is shown that the time delay of cytotoxic T lymphocyte (CTL) response and the diffusion parameter can significantly impact upon the stability regions. Furthermore, the influences of the other free values have been examined for their effects on stability. It is found that, as diffusion increases, the CTL response delay increases, and also as the CTL response delay is increased, the Hopf points for both generation rate and activate rate are decreased, whereas the Hopf points for the infection and death rates increased. Moreover, an increase diffusion results in an increase in the Hopf points for growth rate and activation rate, while the Hopf bifurcations are decreased for the death rate of infected cells. Bifurcation diagrams are plotted to show selected examples of limit cycle behavior (periodic oscillation), and 3-D solutions for the three concentrations in the model have been plotted to corroborate all analytical results from the theoretical section. [ABSTRACT FROM AUTHOR]

The goal of this paper is to study the threshold dynamics of an HIV-1 viral infection model with cell-mediated immune responses and direct cell-to-cell transmission mechanism. The model demonstrates global threshold dynamics with respect to the reproductive numbers for viral infection ℜ 0 and for CTL immune response ℜ 1 . The proofs of main results come from suitable uses of analyzing the characteristic equation and constructing Lyapunov functionals. Specifically, if ℜ 0 < 1, the infection-free equilibrium E 0 is locally and globally asymptotically stable, and the viruses are cleared. If ℜ 1 < 1 < ℜ 0 , the CTL-inactivated equilibrium E 1 is locally and globally asymptotically stable, and the infection becomes chronic but without persistent CTLs response. If ℜ 1 > 1, the CTL-activated equilibrium E 2 is locally and globally asymptotically stable, and the infection is chronic with persistent CTLs response. Numerical simulations are performed to support our results. [ABSTRACT FROM AUTHOR]

• A model for HTLV-I infection that includes uninfected & infected CD4+T cells and HTLV-I-specific CD8+T cells. • To better understand the dynamics of HTLV-I infection, we incorporate a CTL immune response delay. • Applying Lyapunov functional procedures to determine global stability by the reproduction numbers R0 and R1. • The model with delay experiences a destabilization of the infected steady state leading to Hopf bifurcation and periodic solutions. • We estimate the length of delay to preserve the stability of bifurcating limit cycle using Nyquist criteria. This paper deals with a four dimensional mathematical model for Human T-cell leukemia virus type-I (HTLV-I) infection that includes delayed CD8+ cytotoxic T-cells (CTLs) immune response. The proposed system has three biologically feasible steady states, namely disease-free steady state, CTL-inactive steady state and an interior steady state. Our theoretical analysis demonstrates that local and global stability analysis are established by the two critical parameters R 0 and R 1 , basic reproduction numbers due to viral infection and due to CTLs immune response, respectively. The disease-free steady state E 0 is globally stable if R 0 ≤ 1 , and the HTLV-I infections are eliminated. The asymptotic-carrier steady state E 1 is globally stable if R 1 ≤ 1 < R 0 , which indicates HTLV-I infection is chronic without persistence of CTLs immune response. The interior steady state E 2 is globally asymptotic stable if R 1 > 1 , which implies that the HTLV-I infection is choric in persistence of CTLs immune response. Due to immune response delay, our proposed model undergoes a destabilization of the interior steady state leading to Hopf bifurcation and periodic oscillations. We estimate the length of time delay that preserve the stability of period-1 limit cycle. We also derived the direction and stability of Hopf bifurcation around the interior steady state by center manifold theory and normal form method. To determine the robustness of the model, we performed normalized forward sensitivity analysis with reference to R 0 and R 1. Our proposed model undergoes Hopf bifurcation with respect to the production rate of uninfected CD4+T cells h , removal rate of virus-specific CTLs d 4 , spontaneous infected CD4+T cell activation d 2 and transmissibility coefficient β. Implications of our numerical illustrations to the pathogenesis of HTLV-I infection and the development of HTLV-I related HAM/TSP are explored. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we investigate the dynamics of a five-dimensional virus model with immune responses and an intracellular delay which describes the interactions of the HIV virus, CD4 cells and CTLs within host, which is an improvement of some existing models by incorporating (i) two distributed kernels reflecting the variance of time for virus to invade into cells and the variance of time for invaded virions to reproduce within cells; (ii) a nonlinear incidence function f for virus infections, and (iii) antibody responses, which are implemented by the functioning of immunocompetent B lymphocytes, play a critical role in preventing and modulating infections. By constructing Lyapunov functionals and subtle estimates of the derivatives of these Lyapunov functionals, we show that the global dynamics of the model is determined by the reproductive numbers for viral infection , for CTL immune response , for antibody immune response , for CTL immune competition and for antibody immune competition . The global stability of the model precludes the existence of Hopf bifurcation and other complex dynamical behaviors in long time. Numerical simulations are also performed in order to illustrate the dynamical behavior. [Copyright &y& Elsevier]

Abstract: In this paper, an HIV-1 infection model with saturation incidence and time delay due to the CTL immune response is investigated. By analyzing corresponding characteristic equations, the local stability of each of feasible equilibria and the existence of Hopf bifurcation at the CTL-activated infection equilibrium are established, respectively. By means of Lyapunov functionals and LaSalle’s invariance principle, it is shown that the infection-free equilibrium is globally asymptotically stable when the basic reproduction ratio is less than unity. When the immune response reproductive ratio is less than unity and the basic reproductive ratio is greater than unity, the CTL-inactivated infection equilibrium of the system is globally asymptotically stable. [Copyright &y& Elsevier]

Abstract: In this paper, we study a delayed six-dimensional human immunodeficiency virus (HIV) model with Cytotoxic T Lymphocytes (CTLs) immune response. Our model describes the interaction of HIV with two target cells: CD4+ T cells and macrophages. We derive that the global asymptotic attractivity of the model is completely determined by the basic reproduction number and the immune reproduction number for the viral infection. By constructing Lyapunov functionals, we have shown that the infection-free equilibrium , the immune-free equilibrium and the chronic-infection equilibrium are globally asymptotically attractive when and , respectively. [Copyright &y& Elsevier]

Abstract: In this paper, the dynamical behavior of a virus dynamics model with CTL immune response is studied. Sufficient conditions for the asymptotical stability of a disease-free equilibrium, an immune-free equilibrium and an endemic equilibrium are obtained. We prove that there exists a threshold value of the infection rate b beyond which the endemic equilibrium bifurcates from the immune-free one. Still for increasing b values, the endemic equilibrium bifurcates towards a periodic solution. We further analyze the orbital stability of the periodic orbits arising from bifurcation by applying Poore’s condition. Numerical simulation with some hypothetical sets of data has been done to support the analytical findings. [Copyright &y& Elsevier]

Oncolytic virotherapy is a promising cancer treatment that uses replication-competent viruses to target and kill tumor cells. Oncolytic alphavirus M1 is a naturally occurring virus which showed high selectivity and potent efficacy in human cancers. Our purpose in this paper is to propose and analyze a model of oncolytic M1 virotherapy with spatial effects and anti-tumor immune response. We investigate the non-negativity and boundedness of solutions for the modified model. We calculate all possible equilibrium points and determine the threshold conditions needed for their existence. One of the equilibria represents the success of the treatment, while the others represent a partial success or a complete fail. We study the global stability of the corresponding equilibrium points by constructing suitable Lyapunov functionals. We also provide the instability conditions of the equilibrium points. We perform some numerical simulations in order to verify the effect of the immune response on oncolytic virotherapy. Our results indicate that the immune response may weaken the effectiveness of oncolytic virotherapy and control the tumor. [ABSTRACT FROM AUTHOR]