Search

In this paper, we study the effect of stochastic noise on the virus infection model with nonlytic immune response. Firstly, the mathematical model describing the virus infection with nonlytic immune response is presented. The basic reproduction number is derived and the stability of disease-free state E0 and disease state E1 are analysed. Then the threshold conditions for extinction and persistence of the virus are derived by the rigorous theoretical proofs. It is found that when the noise is large enough, the virus will die out without constraint. When the noise is small, the virus will become extinct under the condition R 0 * 1 and persistence under R 0 * * > 1 . Besides, the upper bound and lower bound for persistence have been given. At last, some numerical simulations are carried out to support our results. The conclusion of this paper could help provide the theoretical basis for the further study of the virus infection.

Memory effects play a critical role in complex immune systems. In this paper, the recent efficient and realistic Atangana–Baleanu fractional order derivative, with non-local and non-singular kernel, was employed in two mathematical models for immune systems having multiple immune effectors. For each model, we derive the conditions under which a unique set of exact solutions exists. Stability analysis of equilibrium points of the two systems is carried out where the effects of model’s parameters and fractional derivatives are examined. Furthermore, a recent numerical scheme is utilized to solve each model numerically and to compare theoretical results with those of numerical experiments. Results depict that memory influences induce stabilization of immune systems such that the solution trajectories of the model always converge to either a single immune effector or a persistent immune effector/antigen equilibrium states.

T cells differentiate into Th1 or Th2 cells upon maturation to influence different patterns of the immune response. Th1 and Th2 cells regulate each other and their responses are inhibited by Treg cells. With noisy external stimulation, Th1/Th2 cell differentiation can be dynamically balanced. The underlying mechanisms of Th cell differentiation under Treg cell inhibition and the extrinsic noise effects are not yet completely understood. In this paper, a mathematical model of the interactions between Th1 and Th2 cells with Treg cell inhibition and stochastic effects is developed to study the preference of outcomes and the noise-induced hopping among different states. First, we provide the conditions for different asymptotic phases of Th1 and Th2 responses under Treg cell regulation. Numerical simulations are applied to calculate the switching probability and the mean residence time to study how the noise affects the attractiveness of different states. Our results support that due to the stronger inhibitory effect of Treg cells on Th1 cell development, the high-Th2-low-Th1 state is more attractive under small noise effects. Additionally, we show that the attractiveness of the states is affected mainly by the extrinsic noise in Th2 cell signaling.

A novel coronavirus disease (COVID-19) appeared in Wuhan, China in December 2019 and spread around the world at a rapid pace, taking the form of pandemic. There was an urgent need to look for the remedy and control this deadly disease. A new strain of coronavirus called Severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) was considered to be responsible for COVID-19. Novel coronavirus (SARS-CoV-2) belongs to the family of coronaviruses crowned with homotrimeric class 1 fusion spike protein (or S protein) on their surfaces. COVID-19 attacks primarily at our throat and lungs epithelial cells. In COVID-19, a stronger adaptive immune response against SARS-CoV-2 can lead to longer recovery time and leads to several complications. In this paper, we propose a mathematical model for examining the consequence of adaptive immune responses to the viral mutation to control disease transmission. We consider three populations, namely, the uninfected epithelial cells, infected cells, and the SARS-CoV-2 virus. We also take into account combination drug therapy on the dynamics of COVID-19 and its effect. We present a fractional-order model representing COVID-19/SARS-CoV-2 infection of epithelial cells. The main aim of our study is to explore the effect of adaptive immune response using fractional order operator to monitor the influence of memory on the cell-biological aspects. Also, we have studied the outcome of an antiviral drug on the system to obstruct the contact between epithelial cells and SARS-CoV-2 to restrict the COVID-19 disease. Numerical simulations have been done to illustrate our analytical findings.

Considering the targeted chemotherapy, a mathematical model of tumor-immune system was constructed on the basis of de Pillis’s model. In this paper, we conducted qualitative analysis on the mathematical model, including the positivity and boundedness of solutions, local stability and global stability of equilibrium solutions. Some numerical simulations were given to illustrate the analytic results. Comparing the targeted chemotherapy model with regular chemotherapy model, we found that the targeted chemotherapy was benefit to kill tumor cells.

The impact of continuously distributed delay has been investigated in this paper to describe the interaction among tumor cells, tumor-specific CD8+T cells, helper T cells and immuno-stimulatory cytokine interleukin-2 (IL-2) through a system of coupled nonlinear ordinary differential equations. We analyze the qualitative properties of the model such as positivity of the solutions and the existence of biologically feasible equilibrium points. Next, we discuss the local asymptotic stability for the delayed and non-delayed system. Our model system experiences Hopf bifurcation with respect to the activation rate λ 1 of tumor-specific CD8+T cells. The effect of continuously distributed delay involved in immune-activation on the system dynamics of the tumor is demonstrated. Our study reveals that the activation rate of CD8+T cells can prevent the oscillation of tumor-presence equilibria as well as tumor-free equilibria of the system. Then we performed some numerical results and interpret their biological implications to validate our analytical findings.

A tumor is most dangerous disease of medical science which is a mass or lumps of tissue that’s formed by an accumulation of abnormal cells. A famous fractional tumor-immune model is interpreting the dynamics of tumor and effector cells. In this work, we provide a comparative and chaotic study of tumor and effector cells through fractional tumor-immune dynamical model. A new arbitrary operator based on the Mittag-Leffler law is assumed for this study. Again, we examine the interactions among distinct tumor cell inhabitants and immune structure through a model of real world problem of medical science. We First investigate the dynamical effect of the activation of the effector immune and tumor cells by using Adams-Bashforth-Moulton and Toufik-Atangana methods. Furthermore, this paper analyses the existence and uniqueness of given tumor-immune model of arbitrary order. Further, we have examined the dynamical behaviors of the fractional tumor-immunne model and obtained results are compared with exiting results by other methods. Numerical simulations are executed by Adams-Bashforth-Moulton and Toufik-Atangana methods using popular Atangana-Baleanu fractional derivative. Our obtained results will be useful for biologists to the treatment of cancer disease.

This paper investigates the global stability of discrete-time pathogen infection model with pathogen-to-cell and cell-to-cell transmissions and humoral immunity. We consider both latently and actively infected cells. The model incorporates three types of intracellular time delays. We use nonstandard finite difference method to discretize the continuous-time model. We establish by using Lyapunov method, the global stability of equilibria in terms of the basic reproduction number R 0 and the humoral immune response activation number R 1 . We have proven that if R 0 ≤ 1 , then the pathogen-free equilibrium Q0 is globally asymptotically stable, if R 1 ≤ 1 R 0 , then the persistent pathogen equilibrium without immune response Q* is globally asymptotically stable, and if R 1 > 1 , then the persistent pathogen equilibrium with immune response Q ¯ is globally asymptotically stable. We illustrate our theoretical results by using numerical simulations. The effect of time delay on the pathogen dynamics is also studied. We have shown that the time delay has similar effect as the drug therapy. This gives some impression to develop new class of treatment to increase the delay period and then suppress the pathogen replication.

This paper studies the global dynamics of a viral infection model that takes into account circadian rhythm and time delay in the CTL response. It is shown that the basic reproduction numbers, R 0 and R 1 , determine the outcome of viral infection. Numerical simulations demonstrate that the changes in the amplitude of lytic component can generate a variety of dynamical patterns, ranging from simple daily oscillation to multi-day dynamics and eventually chaos, whereas time delay can alter the period of oscillation for the larger level of periodic forcing. These results can help to explain the viral oscillation behaviors, which were observed in chronic HBV and HCV infection patients.

In this paper, we studied the global dynamics of a SEIR epidemic model in which the latent and immune state were infective. The basic reproductive rate, R0, is derived. If R0 6 1, the disease-free equilibrium is globally stable and the disease always dies out. If R0 > 1, there exists a unique endemic equilibrium which is locally stable. Furthermore, we proved the global stability of the unique endemic equilibrium when a1 = a2 = 0 and the disease persists at an endemic equilibrium state if it initially exists. � 2005 Elsevier Ltd. All rights reserved.

It is still unknown why the adaptive immune response in the natural immune system based on clonal proliferation of lymphocytes requires interaction of at least two different cell types with the same antigen. We present a simple mathematical model illustrating that the system with separate types of cells for antigen recognition and patogen destruction provides more robust adaptive immunity than the system where just one cell type is responsible for both recognition and destruction. The model is over-simplified as we did not have an intention of describing the natural immune system. However, our model provides a tool for testing the proposed approach through qualitative analysis of the immune system dynamics in order to construct more sophisticated models of the immune systems that exist in the living nature. It also opens a possibility to explore specific features of highly non-linear dynamics in nature-inspired computational paradigms like artificial immune systems [1] and immunocomputing [2] . We expect this paper to be of interest not only for mathematicians but also for biologists; therefore we made effort to explain mathematics in sufficient detail for readers without professional mathematical background.