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This paper develops a mathematical model to investigate the Human Immunodeficiency Virus (HIV) infection dynamics. The model includes two transmission modes (cell-to-cell and cell-free), two adaptive immune responses (cytotoxic T-lymphocyte (CTL) and antibody), a saturated CTL immune response, and latent HIV infection. The existence and local stability of equilibria are fully characterized by four reproduction numbers. Through sensitivity analyses, we assess the partial rank correlation coefficients of these reproduction numbers and identify that the infection rate via cell-to-cell transmission, the number of new viruses produced by each infected cell during its life cycle, the clearance rate of free virions, and immune parameters have the greatest impact on the reproduction numbers. Additionally, we compare the effects of immune stimulation and cell-to-cell spread on the model's dynamics. The findings highlight the significance of adaptive immune responses in increasing the population of uninfected cells and reducing the numbers of latent cells, infected cells, and viruses. Furthermore, cell-to-cell transmission is identified as a facilitator of HIV transmission. The analytical and numerical results presented in this study contribute to a better understanding of HIV dynamics and can potentially aid in improving HIV management strategies. [ABSTRACT FROM AUTHOR]

In this paper, a reaction–diffusion–chemotaxis HIV-1 model with a cytotoxic T lymphocyte (CTL) immune response and general sensitivity is investigated. We first prove the global classical solvability and L ∞ -boundedness for the considered model in a bounded domain with arbitrary spatial dimensions, which extends the previous existing results. Then, we apply the global existence result to the case with a linear proliferation immune response and an incidence rate. We study the spatiotemporal dynamics about the three types of spatially homogeneous steady states: infection-free steady state S 0 , CTL-inactivated infection steady state S 1 , and CTL-activated infection steady state S ∗. Our analyses indicate that S 0 is globally asymptotically stable if the basic reproduction number R 0 is less than 1; if R 0 is between 1 and a threshold, then S 1 is globally asymptotically stable. However, if R 0 is larger than the threshold, then the chemoattraction and chemorepulsion can destabilize S ∗ , and thus, a spatiotemporal pattern forms as the chemotactic sensitivity crosses certain critical values. We obtain two kinds of important patterns, which are induced by chemotaxis: stationary Turing pattern and irregular oscillatory pattern. We also find that different chemotactic response functions can affect system's dynamics. Based on some empirical parameter values, numerical simulations are given to illustrate the effectiveness of the theoretical predications. [ABSTRACT FROM AUTHOR]

In this paper, the dynamical behaviors of a multiple delayed HIV-1 infection model which describes the interactions of humoral, cytotoxic T lymphocyte (CTL) immune responses, and two modes of transmission that are the classical virus-to-cell infection and the direct cell-to-cell transmission are investigated. The model incorporates three delays, including the delays of cell infection, virus production and activation of immune response. We first prove the well-posedness of the model, and calculate the biological existence of equilibria and the reproduction numbers, which contain virus infection, humoral immune response, CTL immune response, CTL immune competition, and humoral immune competition. Further, the threshold conditions for the local and global stability of the equilibria for infection-free, immune-free, antibody response, CTL response, and interior are established by utilizing linearization method and the Lyapunov functionals. The existence of Hopf bifurcation with immune delay as a bifurcation parameter is investigated by using the bifurcation theory. Numerical simulations are carried out to illustrate the theoretical results and reveal the effects of some key parameters on viral dynamics. [ABSTRACT FROM AUTHOR]

The coronavirus disease 2019 (COVID-19) is a respiratory disease that appeared in 2019 caused by a virus called severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). COVID-19 is still spreading and causing deaths around the world. There is a real concern of SARS-CoV-2 coinfection with other infectious diseases. Tuberculosis (TB) is a bacterial disease caused by Mycobacterium tuberculosis (Mtb). SARS-CoV-2 coinfection with TB has been recorded in many countries. It has been suggested that the coinfection is associated with severe disease and death. Mathematical modeling is an effective tool that can help understand the dynamics of coinfection between new diseases and well-known diseases. In this paper, we develop an in-host TB and SARS-CoV-2 coinfection model with cytotoxic T lymphocytes (CTLs). The model investigates the interactions between healthy epithelial cells (ECs), latent Mtb-infected ECs, active Mtb-infected ECs, SARS-CoV-2-infected ECs, free Mtb, free SARS-CoV-2, and CTLs. The model's solutions are proved to be nonnegative and bounded. All equilibria with their existence conditions are calculated. Proper Lyapunov functions are selected to examine the global stability of equilibria. Numerical simulations are implemented to verify the theoretical results. It is found that the model has six equilibrium points. These points reflect two states: the mono-infection state where SARS-CoV-2 or TB occurs as a single infection, and the coinfection state where the two infections occur simultaneously. The parameters that control the movement between these states should be tested in order to develop better treatments for TB and COVID-19 coinfected patients. Lymphopenia increases the concentration of SARS-CoV-2 particles and thus can worsen the health status of the coinfected patient. [ABSTRACT FROM AUTHOR]

In this paper, we propose and study a Middle East respiratory syndrome coronavirus (MERS-CoV) infection model with cytotoxic T lymphocyte (CTL) immune response and intracellular delay. This model includes five compartments: uninfected cells, infected cells, viruses, dipeptidyl peptidase 4 (DPP4), and CTL immune cells. We obtained an immunity-inactivated reproduction number R 0 and an immunity-activated reproduction number R 1 . By analyzing the distributions of roots of the corresponding characteristic equations, the local stability results of the infection-free equilibrium, the immunity-inactivated equilibrium, and the immunity-activated equilibrium were obtained. Moreover, by constructing suitable Lyapunov functionals and combining LaSalle's invariance principle and Barbalat's lemma, some sufficient conditions for the global stability of the three types of equilibria were obtained. It was found that the infection-free equilibrium is globally asymptotically stable if R 0 ≤ 1 and unstable if R 0 > 1 ; the immunity-inactivated equilibrium is locally asymptotically stable if R 0 > 1 > R 1 and globally asymptotically stable if R 0 > 1 > R 1 and condition (H1) holds, but unstable if R 1 > 1 ; and the immunity-activated equilibrium is locally asymptotically stable if R 1 > 1 and is globally asymptotically stable if R 1 > 1 and condition (H1) holds. [ABSTRACT FROM AUTHOR]

In this paper, we develop a diffusive and delayed hepatitis B virus (HBV) infection model, which incorporates both virus‐to‐cell and cell‐to‐cell transmissions along with cytotoxic T lymphocyte (CTL) immune response. Two threshold parameters ℜ0$$ {\mathfrak{\Re}}_0 $$ (basic reproduction number) and ℜ1$$ {\mathfrak{\Re}}_1 $$ (immune response reproduction number) are defined, which determine the dynamic behavior of the model. Employing the approach of Lyapunov functionals, we show that the infection‐free steady state is globally asymptotically stable when ℜ0≤1$$ {\mathfrak{\Re}}_0\le 1 $$; the immune‐free infected steady state is globally asymptotically stable when ℜ1≤1<ℜ0$$ {\mathfrak{\Re}}_1\le 1<{\mathfrak{\Re}}_0 $$; and the immune‐present steady state is globally asymptotically stable when ℜ0>1,ℜ1>1$$ {\mathfrak{\Re}}_0>1,{\mathfrak{\Re}}_1>1 $$, and φ(I)=I$$ \varphi (I)=I $$. We perform the sensitivity and uncertainly analysis on ℜ0$$ {\mathfrak{\Re}}_0 $$ with respect to parameters and find that the most important parameter that drives the HBV infection is the mortality of HBV DNA capsids. Moreover, the theoretical results are supported with numerical simulations. [ABSTRACT FROM AUTHOR]

A mathematical model for the quantitative analysis of the interaction between cancer cells and cell-mediated immune system with two discrete-time delays, considering the role of antibodies, is studied in this paper. The model is analyzed both analytically and numerically to understand the dynamics of interaction delay and proliferation enhancement effect delay in the eradication of cancer. Hopf bifurcation occurs when one-time delay crosses the critical value in the absence of the other. It is also observed that as we increase any of the delays, it will increase the cancer burden. The analysis shows that a cancer-free state cannot be obtained solely by the killing rate of cytotoxic T-lymphocytes, but humoral response in combination with cell-mediated immune responses plays an important role to eradicate cancer. [ABSTRACT FROM AUTHOR]

In this paper, the dynamics of stochastic human T‐cell leukemia virus type I (HTLV‐I) infection model with cytotoxic T lymphocyte (CTL) immune response is investigated. First, we show that the stochastic model exists as a unique positive global solution originating from the positive initial value. Second, we demonstrate that the stochastic model is stochastically permanent and stochastically ultimately bounded for any positive initial value. Third, we establish sufficient conditions for the existence of ergodic stationary distribution of the stochastic model. Fourth, the threshold R0∗ between extinction and persistence of the virus is obtained. Finally, numerical simulations are carried out to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

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 R 01 and immune reproduction number R 02 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 R 01 < 1 , the virus will be cleared out eventually and the CTL immune response will also disappear; when R 02 < 1 < R 01 , the virus persists within the host, but the CTL immune response disappears eventually; when R 02 > 1 , both of the virus and the CTL immune response persist within the host. Finally, a brief discussion will be given. [ABSTRACT FROM AUTHOR]

HIV and HTLV-I are two retroviruses that infect the CD 4 + T cells. HIV causes acquired immunodeficiency syndrome, while HTLV-I is the causative agent for adult T-cell leukemia and HTLV-I-associated myelopathy/tropical spastic paraparesis. It is not surprising that a mono-infected patient with one of these viruses can be exposure to co-infect with the other virus since they share the same way of transmissions between individuals, through direct contact with certain contaminated body fluids. Mathematical modeling and analysis of HIV and HTLV-I mono-infections have received considerable attention during the last decades. However, the dynamics of HIV/HTLV-I co-infection has not been formulated. In the present paper, we formulate a new HIV/HTLV-I co-infection model with cytotoxic T lymphocytes (CTLs) immune response. The model describes the interaction between susceptible CD 4 + T cells, HIV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIV-specific CTLs and HTLV-specific CTLs. The HIV can spread by virus-to-cell and cell-to-cell transmissions, while the HTLV-I can only spread via cell-to-cell transmission. The well-posedness of the model is established by showing that the solutions of the model are nonnegative and bounded. We derive the threshold parameters which govern the existence and stability of all equilibria of the model. We prove the global asymptotic stability of all equilibria by utilizing Lyapunov function and applying Lyapunov–LaSalle asymptotic stability theorem. We present numerical simulations to illustrate the effectiveness of our main results. In addition, we discuss the effect of HTLV-I infection on the HIV-infected patients and vice versa. [ABSTRACT FROM AUTHOR]

Individuals infected by human immunodeficiency virus (HIV) are under oxidative stress due to the imbalance between reactive oxygen species (ROS) production and elimination. This paper presents a mathematical model with the cytotoxic T lymphocytes (CTL) immune response to examine the role of ROS in the dynamics of HIV infection. We classify the equilibria of the model and study the stability of these equilibria. Numerical simulations show that incorporating ROS and CTL immune response into the model leads to very rich dynamics, including bistable phenomena and periodic solutions. Although the current antiretroviral therapy can suppress viral load to the undetectable level, it cannot eradicate the virus. A high level of ROS may be a factor for HIV persistence in patients despite suppressive therapy. These results suggest that oxidative damage and anti-oxidant therapy should be considered in the study of HIV infection and treatment. [ABSTRACT FROM AUTHOR]

• 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]

Malaria is a life-threatening disease caused by infection with Plasmodium parasites. The goal of developing an effective malaria vaccine is yet to be reached despite decades of massive research efforts. CD4+ helper T cells, CD8+ cytotoxic T cells, and γδ T cells are associated with immune responses to both liver-stage and blood-stage Plasmodium infection. The immune responses of T cell-lineages to Plasmodium infection are associated with both protection and immunopathology. Studies with mouse model of malaria contribute to our understanding of host immune response. In this paper, we focus primarily on mouse malaria model with blood-stage Plasmodium berghei infection and review our knowledge of T cell immune responses against Plasmodium infection. Moreover, we also discuss findings of experimental human studies. Uncovering the precise mechanisms of T cell-mediated immunity to Plasmodium infection can be accomplished through further investigations using mouse models of malaria with rodent Plasmodium parasites. Those findings would be invaluable to advance the efforts for development of an effective malaria vaccine. [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]