Narrow Results

In this paper, a mathematical model is proposed to study the combined effects of media awareness and fear-induced behavioral changes on the dynamics of infectious diseases. It is considered that in comparison to the unaware individuals, the aware individuals have a lower contact with infected ones. The number of media advertisements is assumed to increase at a rate proportional to the number of infected persons and declines as the number of aware individuals increases. The stability analysis of the model shows that an increase in the growth rate of media advertisements leads to generation of periodic oscillations in the system due to occurrence of Hopf-bifurcation at interior equilibrium. The fear factor and the decline in advertisements due to an increase in the number of aware individuals are found to have stabilizing effect on dynamics of system and their high values can eliminate the limit cycle oscillations present in the system. The rate at which awareness spreads among susceptible individuals and the behavioral response of the aware population are found to be the critical parameters which shape the overall impact of awareness on disease dynamics. It has been observed that the increase in contact rate of aware individuals with infected ones and the dissemination rate of awareness can result into emergence of multiple stability switches via double Hopf-bifurcation.

This paper presents a systematic study on the dynamics of a controlled Duffing oscillator with delayed displacement feedback, especially on the local bifurcations of periodic motions with respect to the time delay. The study begins with the analysis of the stability switches of the trivial equilibrium of the system with various parametric combinations and gives the critical values of time delay, where the trivial equilibrium may change its stability. It shows that as the time delay increases from zero to the positive infinity, the trivial equilibrium undergoes a different number of stability switches for different parametric combinations, and becomes unstable at last for all parametric combinations. Then, the method of multiple scales and the numerical computation method are jointly used to obtain a global diagram of local bifurcations of periodic motions with respect to the time delay for each type of parametric combinations. The diagrams indicate two kinds of local bifurcations. One is the saddle-node bifurcation and the other is the pitchfork bifurcation, of which the former means the sudden emerging of two periodic motions with different stability and the latter implies the Hopf bifurcation in the sense of dynamic bifurcation. A novel feature, referred to as the property of "periodicity in delay", is observed in the global diagrams of local bifurcations and used to justify the validity of infinite number of bifurcating branches in the bifurcation diagrams. The stability of the periodic motions is discussed not only from the high-order approximation of the asymptotic solution, but also from viewpoint of basin of attraction, which gives a good explanation for coexisting periodic motions and quasi-periodic motions, as well as an overall idea of basin of attraction. Afterwards, a conventional Poincaré section technique is used to reveal the abundant dynamic structures of a tori bifurcation sequence, which shows that the system will repeat similar quasi-periodic motions several times, with an increase of time delay, enroute to a chaotic motion. Finally, a new Poincaré section technique is proposed as a comparison with the conventional one, and the results show that the dynamical structures on the two kinds of Poincaré sections are topologically symmetric in rotation.

This paper presents a detailed study on the bifurcation of a controlled Duffing oscillator with a time delay involved in the feedback loop. The first objective is to determine the bifurcating periodic motions and to obtain the global diagrams of local bifurcations of periodic motions with respect to time delay. In order to determine the bifurcation point, an analysis on the stability switches of the trivial equilibrium is first performed for all possible parametric combinations. Then, by means of the method of multiple scales, an analysis on the local bifurcation of periodic motions is given. The static bifurcation diagrams on the amplitude-delay plane exhibit two kinds of local bifurcations of periodic motions, namely the saddle-node bifurcation and the pitchfork bifurcation, indicating a sudden emergence of two periodic motions with different stability and a Hopf bifurcation, respectively, in the sense of dynamic bifurcation. The second objective is to develop a shooting technique to locate both stable and unstable periodic motions of autonomous delay differential equations such that the periodic motions and their stability predicted using the method of multiple scales could be verified. The efficacy of the shooting scheme is well illustrated by some examples via phase trajectory and time history. It is shown that periodic motions located by the shooting method agree very well with those predicted on the bifurcation diagrams. Finally, the paper presents some interesting features of this simple, but dynamics-rich system.

The dynamical transitions resulting from time delay in single Hindmarsh–Rose system are investigated in the present paper. As the time delay varies, the change of the structure of slow manifold is first formulated by using the method of stability switch. Then, the delay-induced dynamical transitions are investigated through the analysis of geometric singular perturbation, and in several case studies, the mechanism of the dynamical transitions is illuminated. Numerical results demonstrate the validity of the theoretical results.

The broadcast of awareness programs through TV and radio advertisements (ads) makes people aware and brings behavioral changes among the individuals regarding the risk of infection and its control mechanisms. In this paper, we propose and analyze a nonlinear mathematical model for the control of infectious diseases due to impact of TV and radio advertisements. It is assumed that susceptible individuals are vulnerable to infection as well as information through TV and radio ads and they contract infection via direct contact with infected individuals. In the model formulation, it is also assumed that the growth rates in cumulative number of TV and radio ads are proportional to the number of infected individuals with decreasing function of aware individuals. Further, it is assumed that awareness among susceptible individuals induces behavioral changes and they form separate aware classes, which are fully protected from infection as they use precautionary measures for their protection during the infection period. The feasibility of equilibria and their stability properties are discussed. It is shown that the augmentation in dissemination rate of awareness among susceptible individuals due to TV and radio ads may cause stability switches through Hopf-bifurcation. The analytical findings are supported through numerical simulations.

In this paper, a predator–prey model with age structure in predator is studied. Using maturation period as the varying parameter, we prove the existence of Hopf bifurcation for the model and calculate the bifurcation properties, such as the direction of Hopf bifurcation and the stability of bifurcated periodic solutions. The method we employed includes Hopf bifurcation theorem, center manifolds and normal form theory for the abstract Cauchy problems with nondense domain. Under a certain set of parameter values, it turns out that subcritical Hopf bifurcation may occur, indicating that the increment of maturation period could stabilize the steady state, which is initially unstable and enclosed by a stable periodic solution. In addition, stability switches will also take place. Numerical simulations are finally carried out to show the theoretical results.

In this paper, we consider an HIV infection model with virus-to-cell infection, cell-to-cell transmission, intracellular delay and mitosis of uninfected cells. The basic reproduction number is calculated by using the method of the next generation matrix. By comparison arguments, it is proved that when the basic reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, the existence of Hopf bifurcation and stability switch at the chronic-infection equilibrium of the model with or without intracellular delay is established. Further, by constructing Lyapunov functionals, sufficient conditions are obtained for the global asymptotic stability of the chronic-infection equilibrium when the cell-to-cell transmission is negligible. Numerical simulations are carried out to illustrate the main theoretical results. The normal form is calculated to determine the bifurcation direction and stability, as well as amplitude and period of bifurcating periodic solutions when the intracellular delay is absent.

In this paper, a phytoplankton and zooplankton relationship system with two delays is investigated whose coefficients are related to one of the two delays. Firstly, the dynamic behaviors of the system with one delay are given and the stability of positive equilibrium and the existence of periodic solutions are obtained. Using the fact that the system may occur, the stable switching phenomenon is verified. Under certain conditions, the periodic solutions will exist in a wide range as the delay gets away from critical values. Fixing the delay $\tau$ in the stable interval, it is revealed that the effect of ${\tau }_1$ can also cause the vibration of system. This explains that two delays play an important role in the oscillation behavior of the system. Furthermore, using the crossing curve methods, the stable changes of the positive equilibrium in two-delays plane are given, which generalizes the results of systems for which the coefficients do not depend on delay. Some numerical simulations are provided to verify the theoretical results.

A reaction–diffusion system with two delays, which describes viral infection spreading in the lymphoid tissue, is investigated. The delays promote very complex dynamics of the model. We get the Hopf bifurcation curves and the stability region for the coexisting steady state on a two-parameter plane by finding the stability switching curves and the subsets of stable region. When two delays cross the boundary of the stable region, the system will undergo stability switches. It is shown that two types of bistability are possible: the coexistence of the stable virus-free steady state and the stable coexisting steady state; the coexistence of the stable virus-free steady state and a stable periodic solution. Numerical simulations suggest that delays can also induce tristability including a steady state and two stable periodic solutions.

In this paper, a delayed-within-host-dengue infection model with mitosis and immune response is analyzed. The basic reproduction number is calculated and a detailed discussion on the local and global dynamics of the model is conducted. By using comparison arguments, it is shown that when the basic reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable. When the basic reproduction number is greater than unity, the existence of Hopf bifurcation and stability switch at the immunity-activated infection equilibrium of the model with or without delay is established. Furthermore, by means of Lyapunov functional and LaSalle’s invariance principle, sufficient conditions are obtained for the global stability of the immunity-activated infection equilibrium. Numerical simulations are given to illustrate the main theoretical results. The normal form is calculated to analyze some properties of the bifurcation periodic solution when the time delay is absent. Moreover, we carry out sensitivity analysis on basic reproduction number to determine crucial parameters that affect the stability of each of feasible equilibrium.

To explore the effect of toxins produced by phytoplankton in algal blooms, a diffusive toxic phytoplankton–zooplankton system with delay is studied through detailed bifurcation analysis. The local stability of the positive equilibrium is studied by analyzing the characteristic equation. The critical values of delay at which Hopf bifurcations occur are obtained and ordered, and it is proved that the delay can induce stability switches, which means that the delay has a significant impact on both the formation and the termination of algal blooms. During the process of stability switching, double Hopf bifurcation arises. Taking the rate of toxin liberation and time delay as the bifurcation parameters, the normal form for double Hopf bifurcation is derived, from which the classification for dynamics and bifurcation sets near double Hopf bifurcation point is obtained. It is proved that the system has complex dynamics, such as periodic oscillations, quasi-periodic oscillations on two- or three-torus and even chaos. Numerical simulations are presented to support the theoretical results.

The Investment Savings-Liquidity preference Money supply (IS-LM) model is represented as a graph depicting the intersection of products and the money market. It elaborates how an equilibrium of money supply versus interest rates may keep the economy in control. In this paper, we combine the basic business cycle IS-LM model with Kaldor’s growth model in order to create an augmented model. The IS-LM model, when coupled with a certain economics expansion (in our instance, the Kaldor–Kalecki Business Cycle Model), provides a comprehensive description of a developing but robust economy. Right after the introduction of capital stock into the system, it cannot be employed and also, while making some investment choices, this requires some time in execution, which ultimately alters resources, i.e. capital. Thus, in the capital accumulation, we will be incorporating double time delays in Gross product and Capital Stock. These time delays represent the time periods during which investment decisions were made and executed and the time spent in order for the capital to be put to productive use. After formulating a mathematical model using delayed differential equations, dynamic functioning of the system around equilibrium point is examined where three instances appeared based on time delays. These cases are: when both delays are not in action, when only one delay is in action and when both delays are in action. It is shown that time delay affects the stability of the equilibrium point and, as the delay crosses a critical point, Hopf bifurcation exists. It is observed that by using Kaldor type investment function, the delay residing in capital stock only will destabilize in less time as compared to when both the delays are present in the system. The system is sensitive to certain parameters which is also analyzed in this work.

In real-world networks, due to complex topological structures and uncertainties such as time delays, uncontrolled systems may generate instability and complexity, thereby degrading network performance. This paper provides a detailed analysis of the stability, Hopf bifurcation, and complex dynamics of a networked system under delayed feedback control. Based on the linear stability method and Hopf bifurcation theorem, the stability of the equilibrium of the error system and the existence of Hopf bifurcation are studied. The stability of periodic solutions bifurcating from the trivial equilibrium is analyzed using normal form theory and central manifold theorem. Special focus is on the effects of the network topology and time delays on the stability and Hopf bifurcation. The theoretical results are also extended to the complex networks with asymmetric adjacent matrices. In addition, the controlled model exhibits complicated dynamical behavior via three types of codimension two bifurcations and period-doubling bifurcations that eventually lead to chaos. Numerical experiments have validated the theoretical results and indicated that delayed feedback control can effectively generate or annihilate the complicated behavior of complex networks.

In this work, a dengue transmission model with logistic growth and time delay $(\tau )$ is investigated. Through detailed mathematical analysis, the local stability of a disease-free equilibrium and an endemic equilibrium is discussed, the existence of Hopf bifurcation and stability switch is established, and it is proved that the system is permanent if the basic reproduction number is greater than 1. On the basis of Lyapunov functional and LaSalle’s invariance principle, sufficient conditions are derived for the global stability of the endemic equilibrium. The primary theoretical results are simulated numerically. In addition, when $\tau =0$, relevant properties of the Hopf bifurcation are analyzed. Finally, sensitivity analysis is given and data fitting is carried out to predict the epidemic development trend of dengue fever in Singapore in 2020.

A two-degree-of-freedom nonlinear high-speed railway wheelset model with two time delays in the lateral and yaw dampers is studied. The aim is to investigate the effect of time delays on stability and Hopf bifurcation characteristics of the wheelset model. The local stability of the trivial equilibrium under different time delay conditions is qualitatively analyzed. Analytical studies reveal that the wheelset model undergoes stability switches with the variation of the time delays. The stability switches correspond to Hopf bifurcations that occur when the time delays cross critical values. Furthermore, properties of Hopf bifurcation including direction and stability of bifurcating limit cycles are studied by using the normal form theory and the center manifold theorem. Our findings indicate that time delays in both lateral dampers and yaw dampers influence the stability and direction of Hopf bifurcation. Additionally, the numerical results show that time delays in the lateral and yaw dampers not only affect the amplitude of the hunting motion of the wheelset but also the periodic and chaotic motions. If the time delays gradually increase, the wheelset will vibrate irregularly with large lateral displacements. The analytical results presented in this paper offer a theoretical reference for the stability design of wheelsets.

In this study, we consider two target-cell limited models with saturation type infection rate and intracellular delay: one without self-proliferation and the other with self-proliferation of activated CD4⁺T cells. We discuss about the local and global behavior of both the systems in presence and absence of intracellular delay. It is shown that the endemic equilibrium of a target-cell limited model would be unstable in presence and absence of intracellular delay only when self-proliferation of activated CD4⁺T cell is considered. Otherwise, all positive solutions converge to the endemic equilibrium or disease-free equilibrium depending on whether the basic reproduction ratio is greater than or less than unity. Our study suggests that amplitude of oscillation is negatively correlated with the constant input rate of CD4⁺T cell when intracellular delay is absent or low. However, they are positively correlated if the delay is too high. Amplitude of oscillation, on the other hand, is always positively correlated with the proliferation rate of CD4⁺T cell for all delay. Our mathematical and simulation analysis also suggest that there are many potential contributors who are responsible for the variation of CD4⁺T cells and virus particles in the blood plasma of HIV patients.

In this paper, a hepatitis B viral infection model with a density-dependent proliferation rate of cytotoxic T lymphocyte (CTL) cells and immune response delay is investigated. By analyzing the model, we show that the virus-free equilibrium is globally asymptotically stable, if the basic reproductive ratio is less than one and an endemic equilibrium exists if the basic reproductive ratio is greater than one. By using the stability switches criterion in the delay-differential system with delay-dependent parameters, we present that the stability of endemic equilibrium changes and eventually become stable as time delay increases. This means majority of hepatitis B infection would eventually become a chronic infection due to the immune response time delay is fairly long. Numerical simulations are carried out to explain the mathematical conclusions and biological implications.

Cholera is a serious threat to the health of human-kind all over the world and its control is a problem of great concern. In this context, a nonlinear mathematical model to control the prevalence of cholera disease is proposed and analyzed by incorporating TV and social media advertisements as a dynamic variable. It is considered that TV and social media ads propagate the knowledge among the people regarding the severe effects of cholera disease on human health along with its precautionary measures. It is also assumed that the mode of transmission of cholera disease among susceptible individuals is due to consumption of contaminated drinking water containing it Vibrio cholerae. Moreover, the propagation of knowledge through TV and social media ads makes the people aware to adopt precautionary measures and also the aware people make some effectual efforts to washout the bacteria from the aquatic environment. Model analysis reveals that increase in the washout rate of bacteria due to aware individuals causes the stability switch. It is found that TV and social media ads have the potential to reduce the number of infectives in the region and thus control the cholera epidemic. Numerical simulation is performed for a particular set of parameter values to support the analytical findings.

In this paper, a nonlinear mathematical model for tuberculosis transmission, which incorporates multiple saturated exogenous reinfections, is proposed and explored. The existence of disease-free and endemic steady states is investigated. Disease-free equilibrium (DFE) is locally asymptotically stable (LAS) but not globally asymptotically stable (GAS) when the basic reproduction number, ${{ℛ}}_0<1$. However, it is GAS only when there is no exogenous reinfection. The local asymptotic stability and global asymptotic stability of the unique endemic equilibrium point (EEP) are established under certain conditions when ${{ℛ}}_0>1$. Further, the EEP is GAS when ${{ℛ}}_0>1$, provided there is no exogenous reinfection. When ${{ℛ}}_0$ is below unity, the presence of multiple endemic equilibria is found which leads to backward bifurcation. It is demonstrated that the system encounters a Hopf-bifurcation when the transmission rate $\beta$ crosses a critical value, resulting in the formation of limit cycles, i.e. periodic solutions bifurcate around the EEP when $\beta$ passes a critical value. The stability and direction of Hopf-bifurcation are also studied. The results of the analytical work are validated through numerical simulations. A numerical simulation illustrates that EEP losses its stability via Hopf-bifurcation for specific parameters. However, when the bifurcation parameter $\beta$ is increased further, the EEP regains its stability. In addition, Hopf-bifurcation occurs due to exogenous reinfection rates $p$ and $𝜃$. Thus, our model shows some important nonlinear dynamical behaviors.