Analysis of a viral infection model with delayed immune response

It is well known that the immune response plays an important role in eliminating or controlling the disease after human body is infected by virus. In this paper, we investigate the dynamical behavior of a viral infection model with retarded immune response. The effect of time delay on stability of the equilibria of the system has been studied and sufficient condition for local asymptotic stability of the infected equilibrium and global asymptotic stability of the infection-free equilibrium and the immune-exhausted equilibrium are given. By numerical simulating,we observe that the stationary solution becomes unstable at some critical immune response time, while the delay time and birth rate of susceptible host cells increase, and we also discover the occurrence of stable periodic solutions and chaotic dynamical behavior. The results can be used to explain the complexity of the immune state of patients.     

Dynamics analysis of an HTLV‐1 infection model with mitotic division of actively infected cells and delayed CTL immune response

In this paper, we propose an improved human T‐cell leukemia virus type 1 infection model with mitotic division of actively infected cells and delayed cytotoxic T lymphocyte immune response. By constructing suitable Lyapunov functional and using LaSalle invariance principle, we investigate the global stability of the infection‐free equilibrium of the system. Our results show that the time delay can change stability behavior of the infection equilibrium and lead to the existence of Hopf bifurcations. Finally, numerical simulations are conducted to illustrate the applications of the main results.     

Dynamical behavior of computer virus on Internet

In this paper, we presented a computer virus model using an SIRS model and the threshold value R₀ determining whether the disease dies out is obtained. If R₀ is less than one, the disease-free equilibrium is globally asymptotically stable. By using the time delay as a bifurcation parameter, the local stability and Hopf bifurcation for the endemic state is investigated. Numerical results demonstrate that the system has periodic solution when time delay is larger than a critical values. The obtained results may provide some new insight to prevent the computer virus.     

Bifurcation analysis of HIV infection model with antibody and cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response

In this paper, a mathematical model for HIV‐1 infection with antibody, cytotoxic T‐lymphocyte immune responses and Beddington–DeAngelis functional response is investigated. The stability of the infection‐free and infected steady states is investigated. The basic reproduction number R₀ is identified for the proposed system. If R₀ < 1, then there is an infection‐free steady state, which is locally asymptotically stable. Further, the infected steady state is locally asymptotically stable for R₀ > 1 in the absence of immune response delay. We use Nyquist criterion to estimate the length of the delay for which stability continues to hold. Also the existence of the Hopf bifurcation is investigated by using immune response delay as a bifurcation parameter. Numerical simulations are presented to justify the analytical results. Copyright © 2014 John Wiley & Sons, Ltd.     

具有时滞和毒素系统的动力学分析

In this paper, a toxin producing phytoplankton-zooplankton model with inhibitory substrate and time delay is investigated. A discrete time delay is induced to both of the consume response function and distribution of toxic substance term. Moreover, Tissiet type function is used for zooplankton grazing to account for the effect of toxication by the TPP population. The conditions to guarantee the coexistence of two species and stability of coexistence equilibrium are given. In particular, we show that there exist critical values of the delay parameters below which the coexistence equilibrium is stable and above which it is unstable. Hopf bifurcation occurs when the delay parameters cross their critical values. Some numerical simulations are executed to validate the analytical findings.     

Dynamical behavior of a delay virus dynamics model with CTL immune response

In this paper, the dynamical behavior of a virus dynamics model with CTL immune response and time delay is studied. Time delay is used to describe the time between the infected cell and the emission of viral particles on a cellular level. The effect of time delay on stability of the equilibria of the CTL immune response model has been studied and sufficient criteria for local asymptotic stability of the disease-free equilibrium, immune-free equilibrium and endemic equilibrium and global asymptotic stability of the disease-free equilibrium are given. Some conditions for Hopf bifurcation around immune-free equilibrium and endemic equilibrium to occur are also obtained by using the time delay as a bifurcation parameter. Numerical simulation with some hypothetical sets of data has been done to support the analytical findings.     

Dynamics of an HIV‐1 virus model with both virus‐to‐cell and cell‐to‐cell transmissions,general incidence rate,intracellular delay,and CTL immune responses

Direct cell‐to‐cell transmission of HIV‐1 is a more efficient means of virus infection than virus‐to‐cell transmission. In this paper, we incorporate both these transmissions into an HIV‐1 virus model with nonlinear general incidence rate, intracellular delay, and cytotoxic T lymphocyte (CTL) immune responses. This model admits three types of equilibria: infection‐free equilibrium, CTL‐inactivated equilibrium, and CTL‐activated equilibrium. By using Lyapunov functionals and LaSalle invariance principle, it is verified that global threshold dynamics of the model can be explicitly described by the basic reproduction numbers.     

The role of CD4 T cells in immune system activation and viral reproduction in a simple model for HIV infection

CD4 T cells play a fundamental role in the adaptive immune response including the stimulation of cytotoxic lymphocytes (CTLs). Human immunodeficiency virus (HIV) which infects and kills CD4 T cells causes progressive failure of the immune system. However, HIV particles are also reproduced by the infected CD4 T cells. Therefore, during HIV infection, infected and healthy CD4 T cells act in opposition to each other, reproducing virus particles and activating and stimulating cellular immune responses, respectively. In this investigation, we develop and analyze a simple system of four ordinary differential equations that accounts for these two opposing roles of CD4 T cells. The model illustrates the importance of the CTL immune response during the asymptomatic stage of HIV infection. In addition, the solution behavior exhibits the two stages of infection, asymptomatic and final AIDS stages. In the model, a weak immune response results in a short asymptomatic stage and faster development of AIDS, whereas a strong immune response illustrates the long asymptomatic stage. A model with a latent stage for infected CD4 T cells is also investigated and compared numerically with the original model. The model shows that strong stimulation of CTLs by CD4 T cells is necessary to prevent progression to the AIDS stage.     

Stability analysis of a virus infection model with humoral immunity response and two time delays

In this paper, we investigate the dynamical properties for a model of delay differential equations, which describes a virus‐immune interaction in vivo. By analyzing corresponding characteristic equations, the local stability of the equilibria for infection‐free, antibody‐free, and antibody response and the existence of Hopf bifurcation with antibody response delay as a bifurcation parameter at the antibody‐activated infection equilibrium are established, respectively. Global stability of the equilibria for infection‐free, antibody‐free, and antibody response, respectively, also are established by applying the Lyapunov functionals method. The numerical simulations are performed in order to illustrate the dynamical behavior of the model. Copyright © 2016 John Wiley & Sons, Ltd.     

On the stability and Hopf bifurcation of a prey-generalist predator system with independent age-selective harvesting

Age-selective harvesting where harvesting of species after a certain age is a scientific strategy with respect to biological and economical point of views. By this method we can overcome the unexpected extinction risk of any harvested population due to random harvesting below its maturation (age, body size or weight). The objective of this paper is to study dynamic behavior of preypredator system with alternative form of time delay in harvesting. Arino et al. [2] have given alternative expression for a delayed logistic equation. Using this expression of time delay, a preypredator system with Holling type III functional response and independent age-selective harvesting is proposed and analyzed. We find out the critical values of delay parameters under different dynamical situations and observe that system is stable and unstable when the delay parameters are bellow and above the critical values respectively and there is Hopf bifurcation when delay parameters cross the critical values. System shows these interesting dynamical features under different critical parametric restrictions. Using the normal form theory and the center manifold theorem, we determine the stability and direction of the bifurcating periodic solutions. Numerical simulations illustrate the analytical results.     

Top-predator interference and gestation delay as determinants of the dynamics of a realistic model food chain

An attempt has been made to understand the role of top predator interference and gestation delay on the dynamics of a three species food chain model involving intermediate and top predator populations. Interaction between the prey and an intermediate predator follows the Volterra scheme (with Holling type IV functional response), while that between the top predator and its prey depends on Beddington–DeAngelis type functional response. Stability switches and Hopf-bifurcation occurs when the delay crosses some critical value. Model system exhibits irregular behavior when the interference is high or gestation period is larger than its critical value. Furthermore, the direction of Hopf-bifurcation and the stability of the bifurcating periodic solutions are determined using the center manifold theorem and normal form theory. Computer simulations have been carried out to illustrate the analytical findings. Different diagnostic tests, like, initial sensitivity, Lyapunov exponent, recurrence plot tests ensure the complex dynamical behavior of the model system. Finally, we observed the subcritical Hopf-bifurcation phenomena in the designed model system and the bifurcating periodic solution is unstable for the considered set of parameter values.     

Stability switches and bifurcation in a two-degrees-of-freedom nonlinear quarter-car with small time-delayed feedback control

In this paper, we investigate a two-degrees-of-freedom nonlinear quarter-car model with time-delayed feedback control. It is well known that a time delay has destabilizing effects in mathematical models. However, delays are not necessarily destabilizing. In this work we explore a system where a time delay can be both stabilizing and destabilizing. Using the generalized Sturm criterion, the critical control gain for the delay-independent stability region and critical time delays for stability switches are derived. It is shown that there is a small parameter region for delay-independently stability of the system. Once the controlled system with time delay is not delay-independently stable, the system may undergo stability switches with the variation of the time delay. These stability switches correspond to Hopf bifurcations that occur when the time delays cross critical values. Properties of Hopf bifurcation such as direction and stability of bifurcating periodic solutions are determined by using the normal form theory and centre manifold theorem. Numerical simulations are provided to support the theoretical analysis. The critical conditions can provide a theoretical guidance for the design of vehicles with significant reduction of vibration in order to increase passengers ride comfort.     

How does within‐host dynamics affect population‐level dynamics? Insights from an immuno‐epidemiological model of malaria

Malaria is one of the most common mosquito‐borne diseases widespread in the tropical and subtropical regions. Few models coupling the within‐host malaria dynamics with the between‐host mosquito‐human dynamics have been developed. In this paper, by adopting the nested approach, a malaria transmission model with immune response of the host is formulated. Applying age‐structured partial differential equations for the between‐host dynamics, we describe the asymptomatic and symptomatic infectious host population for malaria transmission. The basic reproduction numbers for the within‐host model and for the coupled system are derived, respectively. The existence and stability of the equilibria of the coupled model are analyzed. We show numerically that the within‐host model can exhibit complex dynamical behavior, possibly even chaos. In contrast, equilibria in the immuno‐epidemiological model are globally stable and their stabilities are determined by the reproduction number. Increasing the activation rate of the within‐host immune response “dampens” the sensitivity of the population level reproduction number and prevalence to the increase of the within‐host reproduction of the pathogen. From public health perspective this means that treatment in a population with higher immunity has less impact on the population‐level reproduction number and prevalence than in a population with less immunity.     

A fractional order HIV/AIDS model based on the effect of screening of unaware infectives

In this paper, a fractional order model for the spread of human immunodeficiency virus (HIV) infection is proposed to study the effect of screening of unaware infected individuals on the spread of the HIV virus. For this purpose, local asymptotic stability analysis of the disease‐free equilibrium is investigated. In addition, the model is studied for different values of the fractional order to show the relation between the variations of the reproduction number and the order of the proposed model. Finally, numerical solutions are simulated by using a predictor‐corrector method to illustrate the dynamics between susceptible individuals and unaware infected individuals.     

Hopf bifurcation of a bacteria-immunity system with delayed quorum sensing

On the basis of Zhang’s model (see [P. Fergola, J. Zhang, M. Cerasuolo, Z.E. Ma, On the influence of quorum sensing in the competition between bacteria and immune system of invertebrates, in: Collective Dynamic: Topics on Competition and Cooperation in the Biosciences: A Selection of Papers in the Proceedings of the BIOCOMP2007 International Conference, AIP Conference Proceedings, vol. 1028, 2008, pp. 215-232] for more details), we formulate a bacteria-immunity model to describe the competition between bacteria and immune cells with bacterial quorum sensing mechanism. A time delay is introduced to characterize the time in which bacteria receive signal molecules and then combat with immune cells. Subsequently, the length of delay which preserves the stability of the positive equilibrium is estimated and Hopf bifurcation occurs when time delay crosses through a critical value are researched. Further, by using the normal form theory and center manifold theory, the explicit formulaes are calculated which determine the stability, the direction and the period of bifurcating periodical solutions. Finally, numerical simulations are employed to verify the mathematical conclusions.     

Global stability of a virus dynamics model with intracellular delay and CTL immune response

In this paper, the global stability of a virus dynamics model with intracellular delay, Crowley–Martin functional response of the infection rate, and CTL immune response is studied. By constructing suitable Lyapunov functions and using LaSalles invariance principle, the global dynamics is established; it is proved that if the basic reproductive number, R₀, is less than or equal to one, the infection‐free equilibrium is globally asymptotically stable; if R₀ is more than one, and if immune response reproductive number, R⁰, is less than one, the immune‐free equilibrium is globally asymptotically stable, and if R⁰ is more than one, the endemic equilibrium is globally asymptotically stable. Copyright © 2014 John Wiley & Sons, Ltd.     

Dynamics of a discrete food-limited population model with time delay

In this paper, we consider a discrete food-limited population model with time delay. Firstly, the stability of the equilibrium of the system is investigated by analyzing the characteristic equation. By choosing the time delay as a bifurcation parameter, we prove that Neimark-Sacker bifurcations occur when the delay passes a sequence of critical values. Then the explicit algorithm for determining the direction of the Neimark-Sacker bifurcations and the stability of the bifurcating periodic solutions are derived. Finally, some numerical simulations are given to verify the theoretical analysis.