Center Manifold Theory Research Articles

Exploring the interplay between climate change and schistosomiasis transmission dynamics

Schistosomiasis, a neglected tropical disease caused by parasitic worms, poses a major public health challenge in economically disadvantaged regions, especially in Sub-Saharan Africa. Climate factors, such as temperature and rainfall patterns, play a crucial role in the transmission dynamics of the disease. This study presents a deterministic model that aims to evaluate the temporal and seasonal transmission dynamics of schistosomiasis by examining the influence of temperature and rainfall over time. Equilibrium states are examined to ascertain their existence and stability employing the center manifold theory, while the basic reproduction number is calculated using the next-generation technique. To validate the model's applicability, demographic and climatological data from Uganda, Kenya, and Tanzania, which are endemic East African countries situated in the tropical region, are utilized as a case study region. The findings of this study provide evidence that the transmission of schistosomiasis in human populations is significantly influenced by seasonal and monthly variations, with incidence rates varying across countries depending on the frequency of temperature and rainfall. Consequently, the region is marked by both schistosomiasis emergencies and re-emergences. Specifically, it is observed that monthly mean temperatures within the range of 22–27 °C create favorable conditions for the development of schistosomiasis and have a positive impact on the reproduction numbers. On the other hand, monthly maximum temperatures ranging from 27 to 33 °C have an adverse effect on transmission. Furthermore, through sensitivity analysis, it is projected that by the year 2050, factors such as the recruitment rate of snails, the presence of parasite egg-containing stools, and the rate of miracidia shedding per parasite egg will contribute significantly to the occurrence and control of schistosomiasis infections. This study highlights the significant influence of seasonal and monthly variations, driven by temperature and rainfall patterns, on the transmission dynamics of schistosomiasis. These findings underscore the importance of considering climate factors in the control and prevention strategies of schistosomiasis. Additionally, the projected impact of various factors on schistosomiasis infections by 2050 emphasizes the need for proactive measures to mitigate the disease's impact on vulnerable populations. Overall, this research provides valuable insights to anticipate future challenges and devise adaptive measures to address schistosomiasis transmission patterns.

Optimal control model for the infectiology of staphylococcus aureus with dual transmission pathways

Staphylococcus aureus has been increasing globally, which negatively affects food safety, human and veterinary medicine. It colonizes about 30% of the world population, causing superficial skin and soft tissue infections. The dynamics of direct (human-to-human) and indirect (environment-to-human) transmission, shedding, and treatment failure of the disease remains an open epidemiological challenge. Therefore, this study introduces a mathematical model that incorporates both direct and indirect transmission cycle, effect of shedding, treatment failure and time-dependent control parameters. We derive an expression for the basic reproduction number, R0s, under different scenarios. Using comparison, Lyapunov and center manifold theories, we establish that the non-optimal control model is globally asymptotically stable whenever R0s<1 with a forward bifurcation diagram. The shedding of pathogens into the environment and dual transmission routes are the key determinants in the model that increased the number of colonized individuals. Optimal control model was exploring to unveil the impact of time-dependent hygiene practice, treatment and hydrogen peroxide for good human health and clean environment. We show also the control efficacies of the proposed intervention strategies. The key finding is that having triple controls (as in strategy 7) is better than having two controls (as in strategies 4–6) or a single control (as strategies 1–3), but economically very expensive. In a situation where two controls are the only available option, then the use of either improved hygiene practice and treatment (strategy 4) or treatment and hydrogen peroxide (strategy 6) be implemented for controlling staph infection in humans and environment. Whereas, in a case only one control is affordable, then optimized treatment is better than either improved hygiene practice or hydrogen peroxide. We hope that the outcomes of this study have presented policy-makers and decision-makers of low-, middle-and high-income countries with appropriate alternative interventions to implement while dealing with staph infections.

The stochastic stability and H∞‐fuzzy control of stochastic bifurcation of a doubly‐fed induction generator

AbstractConsidering the dynamic behavior of doubly‐fed induction generators (DFIGs) under the influence of random factors, this paper not only analyzes the phenomenon of stochastic instability and bifurcation of a DFIG dynamic variable in its random space when they are affected by environmental noise, but also proposes a method based on the Tkagi–Sugneo (T–S) fuzzy control strategy to control its stochastic bifurcation. First, a four‐dimensional stochastic dynamic DFIG model is established by using multiplicative white noise to simulate the influence of environmental noise on electrical variables, and stochastic central manifold theory is used to reduce the dimensionality of a planar model in the bifurcation neighborhood. Then, the stochastic stability of the model is investigated based on singular boundary theory, while the steady‐state probability density of the stochastic DFIG is determined using the Fokker–Plank–Kolmogorov equation to obtain the location and probability density of its stochastic P‐bifurcation. Finally, the influence of stochastic bifurcation behavior is eliminated by H∞‐fuzzy output feedback control. The numerical simulation results indicate that the location and probability of stochastic bifurcation in a DFIG will vary with the change in noise intensity, and the bifurcation parameter values and stochastic stability domain are obtained. The harm caused by random factors can be solved based on H∞‐fuzzy output feedback, which provides a theoretical basis for the stable operation of the DFIG system.

Stability and optimal control of two products innovation diffusion system

The current study is aimed at analysing a four-compartmental system to examine the interaction and market dissemination of two product categories. In the first category, the product is low-priced and preferred by economically backward people. In the second category, the product is high-priced and liked by people in high-income groups. The non-user population is the potential buyer of products in the market, and frustrated people are fed up with both products. The nonnegativity and boundedness of the solution to the system help in ensuring the well-defined nature of the model. The viability of the model with UFE (user-free equilibrium), an equilibrium point in the absence of users of low-priced products, and interior equilibria is analysed. The analysis of the local and global stability of the model’s equilibria is conducted with reference to ℛ0. We examine the system’s transcritical bifurcation using the Centre Manifold Theory at ℛ0=1. The Hopf bifurcation’s occurrence condition has been identified. The frustration class population gets reduced using the optimal control theory. The extended optimum control model is used to build the Hamiltonian function, which is then solved using Pontryagin’s maximum principle to obtain the lowest cost. To realize which parameters impact the basic influence numbers most, we also run a sensitivity analysis on them. Finally, simulations are conducted to support the theoretical model.

Study of an SIQR model with optimal control techniques: A mathematical approach

In this article, a SIQR deterministic model with a non-linear incidence function for an infectious disease is formulated and analyzed. The well-posedness of the model is ensured with the help of the non-negativity and boundedness of the solution of the system. The basic reproduction number (R0) of the model is found. The sensitivity analysis of the basic reproduction number is discussed. The local and global stability of different equilibria of the model are discussed in terms of basic reproduction number. We use the center manifold theory to analyze the transcritical bifurcation exhibited by the system at R0=1 and found that the disease does not vanish even if R0<1 due to the occurrence of backward bifurcation. Furthermore, this model is extended to the optimal control model and is analyzed to determine the effect of screening and treatment as controls on the population dynamics through Pontryagin’s Maximum Principle. Moreover, we performed numerical analysis to illustrate our theoretical findings. We aim at determining the best intervention strategy to control the infectious diseases among screening, treatment and quarantine. We conclude that only screening is economically effective as compared to only treatment but the combined effort of screening and treatment is most effective and less expensive than single implementation of any control measure. Our study reveals that both the control interventions not only reduce the severity of disease burden but also minimize the economic burden to control the disease.

Dynamic Bufurcation of a Tri-molecular Autocatalytic Reaction-diffusion System

The tri-molecular autocatalytic reaction-diffusion system is a class of model, which can reveal a variety of phenomena observed in areas of encompassing physics, biology, ecology, chemistry and many other fields. In this paper, the tri-molecular autocatalytic reaction-diffusion system with Neumann boundary condition is studied. First, We have reached the following conclusion: the equilibrium of the system loses stability if the parameter is greater than a fixed value, and the corresponding principle of exchange stability condition is then verified. Second, we obtained the local asymptotic stability of trivial, transition types, expressions of bifurcated solutions, the main tools are center manifold theory. The theory is used to reduce the infinite dynamical system to a finite dimensional. Finally, the necessary explanations for the theory is provided.

Backward bifurcation and optimal control problem for a tuberculosis model incorporating LTBI detectivity and exogenous reinfection

The detection of latent tuberculosis infection (LTBI) is one of the vital means in controlling the spread of TB. The dynamical properties of a mathematical model with LTBI detectivity and exogenous reinfection are analyzed and their impacts on TB control are explored. By applying the center manifold theory, it is revealed that the model may exhibit the phenomenon of backward bifurcation caused by exogenous reinfection. Furthermore, sensitivity analysis for the basic reproduction number R0 is performed and an optimal control problem is further formulated by incorporating TB prevention and education propaganda, timely treatment and enhancing therapy efficacy. Finally, our analysis and numerical results show that an increase in detection rate of LTBI cases reduces the value of R0 as well as the possibility that backward bifurcation occurs and the joint implementation of all three strategies effectively contains TB transmission.

Stochastic dynamics of a semiconductor laser with delayed optoelectronic feedback and noise

Semiconductor lasers(SLs), versatile tools in communication, medicine, and scientific research, demand close examination of their dynamic behavior. With their unique geometric forms and diminutive sizes, these lasers possess a heightened sensitivity to noise. In this paper, we delve into time-delay semiconductor laser systems, noise included, honing in on time-delays’ effects on noisy laser equations. Carrier density (N) and laser intensity (I) constitute the core parameters that define the performance of these lasers, and we scrutinize their dynamic trajectories. A cutting-edge method fusing stochastic center manifold theory with the stochastic averaging method enables us to probe the stochastic dynamical attributes of semiconductor laser equations. We uncover that the time-delay parameter (τ 1) produces varied consequences on the laser system’s stability under both positive and negative optical feedback conditions. The negative feedback typically bolsters laser performance, fostering increased stability, whereas the positive counterpart could incite instability. By deftly managing the time-delay parameter, one can fine-tune the dynamic performance of semiconductor lasers, mitigating output power fluctuations and tightening linewidths, thereby augmenting stability and spectral purity. Moreover, in the realm of positive optical feedback scenarios, adroitly modifying the time-delay parameter can quell instability to a degree, paving the way for a novel regulatory approach to optimize laser performance. This lays the groundwork for further enhancement of semiconductor laser performance, as well as curbing the detrimental effects of noise on their operation.

Normal Form and Unfolding of Vector Field with Codimension-3 Triple Hopf Bifurcation

The universal unfolding of a normal form can be employed to reveal the general behaviors of a specific local bifurcation, while the computation of the normal form for high codimensional bifurcation still remains unsolved. This paper focuses on a vector field with codimension-3 triple Hopf bifurcation. Besides 1:1 internal resonance for two frequencies in semi-simple form, two cases are considered, corresponding to internal resonance and noninternal resonance between the first two frequencies and the third frequency, respectively. Based on a combination of center manifold and normal theory, all the coefficients in the normal form and the nonlinear transformation are derived explicitly in terms of the coefficients of the original vector field. Upon the recursive procedure established, a user friendly computer program can be easily developed using a symbolic computation language Maple to compute the coefficients up to an arbitrary order for a specific vector field with triple Hopf bifurcation. Furthermore, universal unfolding of the normal form is obtained, which can be used to display the topological structure in the neighborhood of bifurcation point. It is pointed out that different choices of the remaining terms in the nonlinear transformation may lead to different expressions of the normal form and the unfolding, which are qualitatively equivalent to each other.

Finite time SF bifurcation and stability analysis for stochastic time-varying delay system

In this paper, it is found that the systems alternate between oscillatory and steady states in studying the stability of a class of stochastic dynamical systems with fast time-varying periodic delays. We propose a new definition of bifurcation, State Flip (SF) bifurcation, by using the abrupt change in the dwell time ratio of two states. Taking the amplitude of the time-varying delay function δ as the bifurcation parameter, we investigate the dynamics of the evolution of a time-varying delay system driven by Gaussian white noise. Firstly, we convert systems with fast time-varying periodic delays into comparative systems with distributed delays, which in turn are converted into stochastic systems with constant delays for discussion. In order to reduce the stochastic delay equation to the average Itô equation, we adopted the stochastic averaging method and the generalized central manifold theory. It is used to capture the one-dimensional slow variables of a system near the critical state of the transition between two steady states during the evolution of the system. We then find that as the amplitude increases from weak to strong, the stochastic time-varying delay system changes from a two-period oscillation to alternating between oscillation and steady, and finally to a steady state. Lastly, we perform numerical simulations of the original stochastic time-varying delay system and the transformed two-dimensional stochastic constant time-delay system respectively to verify that the ideas and methods in this paper are reasonable and effective.

Complex Dynamics of a Predator-Prey System With Gompertz Growth and Herd Behavior

The complex dynamics of a predator-prey system in discrete time are studied. In this system, we consider the prey’s Gompertz growth and the square-root functional response. The existence of fixed points and stability are examined. Using the center manifold and bifurcation theory, we found that the system undergoes transcritical bifurcation, period-doubling bifurcation, and Neimark-Sacker bifurcation. In addition, numerical examples are presented to illustrate the consistency of the analytical findings. The bifurcation diagrams show that the positive fixed point is stable if the death rate of the predator is greater than a threshold value. Biologically, it means that to prevent the predator population from growing uncontrollably and stability of the positive fixed point, the predator’s death rate should be greater than the threshold value.

Complex patterns in a space–time discrete mathematical model of antibiotic resistance in hospitals with self-diffusion

In this work, we extend an antibiotic resistance mathematical model in hospitals spatially on a two-dimensional coupled map lattice. The complex dynamics of the resultant space–time discrete model are investigated. An invasion reproduction number Rar and two control parameters Rsc and Rrc for sensitive bacteria and resistant bacteria, respectively, are defined. The three numbers play an important role in the existence and stability of fixed points. According to the normal form theorem and the center manifold theory, we show that the discrete model exhibits both the flip and the Turing bifurcations. We prove that the existence of the interior fixed point excludes the possibility of the occurrence of a Neimark–Sacker bifurcation. Numerical simulations, including phase diagrams, Lyapunov exponents, bifurcation diagrams, and complex patterns, are enrolled to verify the analyses. The complex patterns induced here are either a result of the pure Turing instability, or the flip instability (non-Turing), while no Turing-flip patterns are formed. The results of the current study show that the space–time discrete version of the model of antibiotic resistance in hospitals captures complex and rich nonlinear dynamics and adds to our understanding of the complicated pattern formation of the model.

Bifurcation analysis of a delayed reaction–diffusion–advection Nicholson’s blowflies equation

In this paper, we investigate the dynamics of a reaction-diffusion Nicholson’s blowflies equation with advection. The stability of positive steady state and existence of Hopf bifurcation are obtained by analyzing the distribution of the eigenvalues. Moreover, by using the center manifold theory and normal form method, an explicit algorithm for determining the direction and stability of the Hopf bifurcation is derived. Meanwhile, we find out that the bifurcation value is increasing with respect to the advection rate. Finally, numerical results demonstrate that the advection term causes the population to move from upstream to downstream, which also indicates that advection term plays a key role in the description and interpretation of some common natural phenomena.

On the qualitative study of a discrete fractional order prey–predator model with the effects of harvesting on predator population

This research investigates the discrete prey–predator model by including harvesting on the predator population, in the sense of Caputo fractional derivative. We define the topological categories of the model fixed points. We demonstrate mathematically that, under certain parametric conditions, a fractional order prey-predator model undergoes both a Neimark–Sacker (NS) and a Period-doubling (PD) bifurcations. Using the central manifold and bifurcation theory, we present proof for NS and PD bifurcations. It has been discovered that the fractional order prey-predator model’s dynamical behavior is significantly influenced by the parameter values and the initial conditions. Two chaos control techniques have been used to eliminate the chaos in the model. In order to support our theoretical and analytical results and to illustrate complex and chaotic behavior, numerical simulations have been shown.

MATHEMATICAL MODELING OF INFECTION MECHANISM BETWEEN MILD AND SEVERE COVID-19 PATIENTS

The SARS-CoV-2 leads to a worldwide COVID-19 pandemic, which has caused tremendous damage to the world. In this paper, we develop a dynamic model in vivo, fitting and estimating parameters for T lymphocytes and pro-inflammatory cytokines IL-6 in patients with mild and severe COVID-19 at Yale New Haven Hospital through the GWMCMC algorithm. Meanwhile, we also analyze the structural identifiability and practical identifiability of the model. Further, we add time-varying parameters to the model, using the least squares method to perform data fitting and parameter estimation on survivors and non-survivors of the Italian infectious disease hospital. Then analyze the similarities and differences in immune response mechanisms between the two countries. Finally, we demonstrate the existence and stability of the equilibrium state of the model and analyze the Hopf bifurcation at the positive equilibrium state by using the central manifold theory and normal form theory. This result may explain the recurrence of infection in some COVID-19 patients.

Stability and Hopf Bifurcation Analysis for a Phage Therapy Model with and without Time Delay

This study proposes a mathematical model that accounts for the interaction of bacteria, phages, and the innate immune response with a discrete time delay. First, for the non-delayed model we determine the local and global stability of various equilibria and the existence of Hopf bifurcation at the positive equilibrium. Second, for the delayed model we provide sufficient conditions for the local stability of the positive equilibrium by selecting the discrete time delay as a bifurcation parameter; Hopf bifurcation happens when the time delay crosses a critical threshold. Third, based on the normal form method and center manifold theory, we derive precise expressions for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions. Finally, numerical simulations are performed to verify our theoretical analysis.

Stability and Bautin bifurcation of four-wheel-steering vehicle system with driver steering control.

In this paper, the stability and Bautin bifurcation of a four-wheel-steering (4WS) vehicle system, by considering driver steering control, are investigated. By using the central manifold theory and projection method, the first and second Lyapunov coefficients are calculated to predict the type of Hopf bifurcation of the vehicle system. The topological structure of Bautin bifurcation, a degenerate Hopf bifurcation of the 4WS vehicle system, is presented in parameter space, and it reveals the dynamics of the vehicle system of different choices of control parameters. The influences of system parameters on critical values of the bifurcation parameter are also analyzed. It is shown that with the increase in the frontal visibility distance of the driver control strategy coefficient and the cornering stiffness coefficients of rear wheels, the critical speed increases. Nevertheless, the critical speed decreases with the increase in the distance from the center of gravity of the vehicle to the front axles, Driver's perceptual time delay, and cornering stiffness coefficients of the front wheels.

Bifurcation Analysis and 0-1 Chaos Test of a Discrete T System

This study examines discrete-time T system. We begin by listing the topological divisions of the system's fixed points. Then, we analytically demonstrate that a discrete T system sits at the foundation of a Neimark Sacker(NS) bifurcation under specific parametric circumstances. With the use of the explicit Flip-NS bifurcation criterion, we establish the flip-NS bifurcation's reality. Center manifold theory is then used to establish the direction of both bifurcations. We do numerical simulations to validate our theoretical findings. Additionally, we employ the $0-1$ test for chaos to demonstrate whether or not chaos exists in the system. In order to stop the system's chaotic trajectory, we ultimately employ a hybrid control method.

Mathematical modeling and analysis of COVID-19 and TB co-dynamics

This study proposes a mathematical model for examining the COVID-19 and tuberculosis (TB) co-dynamics thoroughly. First, the single infection dynamics: COVID-19 infection and TB infection models are taken into consideration and examined. Following that, the co-dynamics with TB and COVID-19 is also investigated. In order to comprehend the developed model dynamics, the basic system attributes including the region of definition, theory of nonnegativity and boundedness of solution are investigated. Further, a qualitative analysis of the equilibria of the formulated model equations is performed. The equilibria of both infection models are globally asymptotically stable if their respective basic reproductive number is smaller than one. As the associated reproductive number reaches unity, they experience the forward bifurcation phenomenon. Additionally, it is demonstrated that the formulated co-dynamics model would not experience backward bifurcation by applying the center manifold theory. Moreover, model fitting is done by using daily reported COVID-19 cumulative data in Ethiopia between March 13, 2020, and May 31, 2022. For instance, the non-linear least squares approach of fitting a function to data was performed in the fitting process using scipy.optimize.curve_fit from the Python. Finally, to corroborate the analytical findings of the model equation, numerical simulations were conducted.

Global Stability and Bifurcation Analysis of a Virus Infection Model with Nonlinear Incidence and Multiple Delays

In order to investigate the impact of general nonlinear incidence, cellular infection, and multiple time delays on the dynamical behaviors of a virus infection model, a within-host model describing the virus infection is formulated and studied by taking these factors into account in a single model. Qualitative analysis of the global properties of the equilibria is carried out by utilizing the methods of Lyapunov functionals. The existence and properties of local and global Hopf bifurcations are discussed by regarding immune delay as the bifurcation parameter via the normal form, center manifold theory, and global Hopf bifurcation theorem. This work reveals that the immune delay is mainly responsible for the existence of the Hopf bifurcation and rich dynamics rather than the intracellular delays, and the general nonlinear incidence does not change the global stability of the equilibria. Moreover, ignoring the cell-to-cell infection may underevaluate the infection risk. Numerical simulations are carried out for three kinds of incidence function forms to show the rich dynamics of the model. The bifurcation diagrams and the identification of the stability region show that increasing the immune delay can destabilize the immunity-activated equilibrium and induce a Hopf bifurcation, stability switches, and oscillation solutions. The obtained results are a generalization of some existing models.