Search

This paper proposes a delay differential model with fractional order for glucose-insulin endocrine, metabolic regulation model, incorporating beta-cell dynamics to regulate and maintain bloodstream insulin concentration. In the model, two time delays are involved, namely δg and δι, which represent delayed insulin secretion and delayed glucose reduction. A moderate hyperglycemia results in beta-cell growth (negative feedback), while a severe hyperglycemia results in beta-cell reduction (positive feedback). When a time delay passes a bifurcation point, Hopf bifurcation occurs. It is evident from biological findings that the model exhibits periodic oscillations. Furthermore, we present an optimal control problem for external insulin infusions to minimize prolonged high blood sugar levels. Numerical simulations have validated the theoretical results.

This study aims to investigate the potential impact of inhibitors targeting the papain-like protease (PLpro) of SARS-CoV-2 on viral replication and the host immune response. A mathematical model was developed to simulate the interaction among susceptible cells, infected cells, PLpro, and immune cells, incorporating data on PLpro inhibition. Through numerical simulations using MATLAB, the model parameters were estimated based on available statistical data. The results indicate that strategically positioned inhibitors could impede the virus’s access to host cellular machinery, thereby enhancing the immune response and gradually reducing susceptible and infected cells over time. The dynamics of the viral enzyme PLpro showed reduced activity with the introduction of the inhibitor, leading to a decline in viral replication. Moreover, the immune cell population exhibited functional recovery as the inhibitor suppressed PLpro activity. These findings suggest that inhibitors targeting PLpro may serve as therapeutic interventions against SARS-CoV-2 by inhibiting viral replication and bolstering the immune response.

Symptomatic and asymptomatic individuals play a significant role in the transmission dynamics of novel Coronaviruses. By considering the dynamical behaviour of symptomatic and asymptomatic individuals, this study examines the temporal dynamics and optimal control of Coronavirus disease propagation using an epidemiological model. Biologically and mathematically, the well-posed epidemic problem is examined, as well as the threshold quantity with parameter sensitivity. Model parameters are quantified and their relative impact on the disease is evaluated. Additionally, the steady states are investigated to determine the model's stability and bifurcation. Using the dynamics and parameters sensitivity, we then introduce optimal control strategies for the elimination of the disease. Using real disease data, numerical simulations and model validation are performed to support theoretical findings and show the effects of control strategies.

The paper investigates a spatio-temporal SEIR epidemic model on a global level with fractional order. It employs four partial differential equations incorporating fractional derivatives and account for diffusion to characterize infection dynamics. By applying fixed-point theorem results, the paper establishes the existence, uniqueness, and boundedness of the solution. Equilibrium points are determined based on vaccination values and the basic reproduction number R0. Global stability of each equilibrium is confirmed using the Lyapunov direct method. Through simulations with a predictor–corrector algorithm, key insights into epidemiological dynamics are provided, elucidating the impact of vaccination on reducing disease transmission and altering R0. Trajectories of various compartments closely align with theoretical equilibrium points, affirming the model’s predictive precision. Furthermore, simulations indicate the potential for attaining disease-free equilibria with heightened vaccination rates, underscoring the pivotal role of vaccination strategies in epidemic control and disease eradication. It was shown that the fractional derivative order has no effect on the equilibrium stability but rather only on the convergence speed towards the equilibrium points.

This article examines the stochastic stability and global dynamics of a mathematical model of drug use. The model divides the population into five compartments current drug users, temporarily abstinent drug users, permanently abstinent drug users, and drug users in rehabilitation. Using Brownian motion, deterministic equations are extended to incorporate stochastic perturbations, capturing real-life uncertainties in drug use within compartments. An analysis of Lyapunov functions is used to determine the global stability of the model. By introducing stochastic elements into the model, we can examine its stability under random perturbations. A global sensitivity analysis, including PRCC results, is conducted to confirm the robustness of the model. Stable drug-free and drug-present equilibria can be maintained in both deterministic and stochastic environments. Numerical simulations illustrate the impact of various parameters on population dynamics and rehabilitation program effectiveness.

The co-infection of HIV and COVID-19 is a pressing health concern, carrying substantial potential consequences. This study focuses on the vital task of comprehending the dynamics of HIV-COVID-19 co-infection, a fundamental step in formulating efficacious control strategies and optimizing healthcare approaches. Here, we introduce an innovative mathematical model grounded in Caputo fractional order differential equations, specifically designed to encapsulate the intricate dynamics of co-infection. This model encompasses multiple critical facets: the transmission dynamics of both HIV and COVID-19, the host’s immune responses, and the influence of treatment interventions. Our approach embraces the complexity of these factors to offer an exhaustive portrayal of co-infection dynamics. To tackle the fractional order model, we employ the Laplace-Adomian decomposition method, a potent mathematical tool for approximating solutions in fractional order differential equations. Utilizing this technique, we simulate the intricate interactions between these variables, yielding profound insights into the propagation of co-infection. Notably, we identify pivotal contributors to its advancement. In addition, we conduct a meticulous analysis of the convergence properties inherent in the series solutions acquired through the Laplace-Adomian decomposition method. This examination assures the reliability and accuracy of our mathematical methodology in approximating solutions. Our findings hold significant implications for the formulation of effective control strategies. Policymakers, healthcare professionals, and public health authorities will benefit from this research as they endeavor to curtail the proliferation and impact of HIV-COVID-19 co-infection.

Pantograph differential equations are important types of delay differential equations. Using continuous mono-implicit RK schemes, we propose a numerical method for numerically approximating pantograph delay differential equations that are reliable and unconditionally stable. The method combines the accuracy of implicit methods with the efficiency of implementation. The method works for stiff and non-stiff initial value problems, reducing the computational cost of fully implicit methods. Some examples are provided to demonstrate the effectiveness of the numerical method.

Stochastic differential equation models are important and provide more valuable outputs to examine the dynamics of SARS-CoV-2 virus transmission than traditional models. SARS-CoV-2 virus transmission is a contagious respiratory disease that produces asymptomatically and symptomatically infected individuals who are susceptible to multiple infections. This work was purposed to introduce an epidemiological model to represent the temporal dynamics of SARS-CoV-2 virus transmission through the use of stochastic differential equations. First, we formulated the model and derived the well-posedness to show that the proposed epidemiological problem is biologically and mathematically feasible. We then calculated the stochastic reproductive parameters for the proposed stochastic epidemiological model and analyzed the model extinction and persistence. Using the stochastic reproductive parameters, we derived the condition for disease extinction and persistence. Applying these conditions, we have performed large-scale numerical simulations to visualize the asymptotic analysis of the model and show the effectiveness of the results derived.

Hepatitis B is one of the global health issues caused by the hepatitis B virus (HBV), producing 1.1 million deaths yearly. The acute and chronic phases of HBV are significant because worldwide, approximately 250 million people are infected by chronic hepatitis B. The chronic stage is a long-term, persistent infection that can cause liver damage and increase the risk of liver cancer. In the case of multiple phases of infection, a generalized saturated incidence rate model is more reasonable than a simply saturated incidence because it captures the complex dynamics of the different infection phases. In contrast, a simple saturated incidence rate model assumes a fixed shape for the incidence rate curve, which may not accurately reflect the dynamics of multiple infection phases. Considering HBV and its various phases, we constructed a model to present the dynamics and control strategies using the generalized saturated incidence. First, we proved that the model is well-posed. We then found the reproduction quantity and model equilibria to discuss the time dynamics of the model and investigate the conditions for stabilities. We also examined a control mechanism by introducing various controls to the model with the aim to increase the population of those recovered and minimize the infected people. We performed numerical experiments to check the biological significance and control implementation.

Abstract This article examines hepatitis B dynamics under distinct infection phases and multiple transmissions. We formulate the epidemic problem based on the characteristics of the disease. It is shown that the epidemiological model is mathematically and biologically meaningful of its well-posedness (positivity, boundedness, and biologically feasible region). The reproductive number is then calculated to find the equilibria and the stability analysis of the epidemic model is performed. A backward bifurcation is also investigated in the proposed epidemic problem. With the help of two control measures (treatment and vaccination), we develop control strategies to minimize the infected population (acute and chronic). To solve the proposed control problem, we utilize Pontryagin’s Maximum Principle. Some simulations are conducted to illustrate the investigation of the analytical work and the effect of control analysis.

This paper presents a time-delay epidemic model for Tuberculosis(TB), Human Immunodeficiency Virus(HIV), and Acquired Immunodeficiency Syndrome(AIDS) co-infection. Our study examines how delay impacts mathematical models of TB, HIV, and AIDS. There are four classes in the proposed system — Susceptibles, TB infectives, HIV infectives (with or without TB), and AIDS patients. A model shows four states of equilibrium: disease-free, HIV-free, TB-free, and endemic. If the reproduction number R0is less than one, the disease-free equilibrium is locally asymptotically stable. If R0greater than one, at least one infection will be present in the population Positive endemic equilibrium is always locally stable, but it can become globally stable under certain circumstances, indicating the disease has become endemic. TB and HIV infections drop as a result of recovery, and endemic equilibrium leads to TB free conditions. The number of people living with AIDS declines when TB is not associated with HIV infection. The model is also numerically analyzed to see how some important parameters affect the disease’s progression. Mathematical and numerical methods are employed to study the impact of delay.

Combining Sterile and Incompatible Insect techniques can significantly reduce mosquito populations and prevent the transmission of diseases between insects and humans. This paper describes impulsive differential equations for the control of a mosquito with Wolbachia. Several interesting conditions are created when sterile male mosquitoes are released impulsively, ensuring both open- and closed-loop control. To determine the wild mosquito population size in real-time, we propose an open-loop control system, which uses impulsive and constant releases of sterile male mosquitoes. A closed-loop control scheme is also being investigated, which specifies the release of sterile mosquitoes according to the size of the wild mosquito population. To eliminate or reduce a mosquito population below a certain threshold, the Sterile insect technique involves mass releases of sterile insects. Numerical simulations verify the theoretical results.

This paper examines the problem of improved fixed-time stability for generalized delayed fractional-order systems (FOSs). As a first step, some stability conditions are presented in two theorems to verify fixed-time stability (F-TS) of FOSs by using Lyapunov stability theory and indefinite Lyapunov functionals, where the fractional-order derivative of the indefinite functions may not exist. Furthermore, the corresponding estimated settling time of FOSs is also provided. Second, the results are extended to study fractional-order neural networks (FONNs) with time-delays in fixed-time synchronization. Using the Dirac delta functions, we propose an explicit saturated impulsive controller to synchronize the master and slave systems. Moreover, by constructing suitable indefinite Lyapunov-Krasovskii functions (LKF), we derive algebraic conditions to guarantee the fixed-time synchronization of the addressed FONNs. The simulation results demonstrate the feasibility and efficacy of the proposed method.

This article studies the finite-time (FNT) and fixed-time (FXT) stability theorems of general fractional-order impulsive discontinuous systems (FOIDSs) through an indefinite Lyapunov functional (LF) approach. Using differential inclusion theory and set-valued map concept, the general FOIDSs can be transformed into fractional-order impulsive differential inclusions (FOIDIs). Two new stability theorems on finite and FXT stability of FOIDSs with finite impulsive instants are derived using an indefinite LF and fractional derivatives of the LF on some impulsive instants, respectively. The settling time is explicitly calculated. The obtained theoretical findings are applied to fractional-order discontinuous multi-agent systems. Discrete control schemes and indefinite Lyapunov-Krasovskii functions (LKFs) are also developed for ensuring FXT synchronization in fractional-order discontinuous multiagent systems. To demonstrate the efficiency and feasibility of the proposed method, some numerical simulations are conducted.

Abstract The central concern of this paper is to study leader‐following quasi‐bipartite synchronisation of a multiple inertial signed neural networks with varying time‐delay by utilising distributed event‐triggered impulsive control scheme, where connections between adjacent nodes of the neural networks either positive or negative. The second‐order neural networks, called inertial neural networks, can be transformed into differential equations of first‐order by implementing suitable variable substitution. Under certain hypothesis about the node dynamics, signed graph theory and balanced topology of networks, some conditions are derived in terms of lower‐dimensional linear matrix inequalities (LMIs) to achieve leader‐following quasi‐bipartite synchronisation. In addition, a basic algebraic condition is derived to estimate the theoretical upper bound for the error node. Finally, some numerical simulations are provided to illustrate the correctness of the theoretical results.

Considering the environmental factors and uncertainties, we propose, in this paper, a higher-order stochastically perturbed delay differential model for the dynamics of hepatitis B virus (HBV) infection with immune system. Existence and uniqueness of an ergodic stationary distribution of positive solution to the system are investigated, where the solution fluctuates around the endemic equilibrium of the deterministic model and leads to the stochastic persistence of the disease. Under some conditions, infection-free can be obtained in which the disease dies out exponentially with probability one. Some numerical simulations, by using Milstein's scheme, are carried out to show the effectiveness of the obtained results. The intensity of white noise plays an important role in the treatment of infectious diseases.

The modeling of biological processes has increasingly been based on fractional calculus. In this paper, a novel fractional-order model is used to investigate the epidemiological impact of vaccination measures on the co-dynamics of viral hepatitis B and COVID-19. To investigate the existence and stability of the new model, we use some fixed point theory results. The COVID-19 and viral hepatitis B thresholds are estimated using the model fitting. The vaccine parameters are plotted against transmission coefficients. The effect of non-integer derivatives on the solution paths for each epidemiological state and the trajectory diagram for infected classes are also examined numerically. An infection-free steady state and an infection-present equilibrium are achieved when R0<1 and R0>1, respectively. Similarly, phase portraits confirm the behaviour of the infected components, showing that, regardless of the order of the fractional derivative, the trajectories of the disease classes always converge toward infection-free steady states over time, no matter what initial conditions are assumed for the diseases. The model has been verified using real observations.

Abstract Stochastic differential models provide an additional degree of realism compared to their corresponding deterministic counterparts because of the randomness and stochasticity of real life. In this work, we study the dynamics of a stochastic delay differential model for prey–predator system with hunting cooperation in predators. Existence and uniqueness of global positive solution and stochastically ultimate boundedness are investigated. Some sufficient conditions for persistence and extinction, using Lyapunov functional, are obtained. Illustrative examples and numerical simulations, using Milstein’s scheme, are carried out to validate our analytical findings. It is observed that a small scale of white noise can promote the survival of both species; while large noises can lead to extinction of the predator population.

A fractional order COVID-19 model consisting of six compartments in Caputo sense is constructed. The indirect transmission of the virus through susceptible populations by the shedding effect is studied. Equilibrium solutions are calculated, and basic reproduction ratio (that depends both on direct and indirect mode of transmission), existence and uniqueness, as well as stability analysis of the solution of the model, are studied. The paper studies the effect of optimal control policy applied to shedding effect. The control is the observation of standard hygiene practices and chemical disinfectants in public spaces. Numerical simulations are carried out to support the analytic result and to show the significance of the fractional order from the biological viewpoint.

In this study, we provide a fractional-order mathematical model that considers the effect of vaccination on COVID-19 spread dynamics. The model accounts for the latent period of intervention strategies by incorporating a time delay τ. A basic reproduction number, R0, is determined for the model, and prerequisites for endemic equilibrium are discussed. The model’s endemic equilibrium point also exhibits local asymptotic stability (under certain conditions), and a Hopf bifurcation condition is established. Different scenarios of vaccination efficacy are simulated. As a result of the vaccination efforts, the number of deaths and those affected have decreased. COVID-19 may not be effectively controlled by vaccination alone. To control infections, several non-pharmacological interventions are necessary. Based on numerical simulations and fitting to real observations, the theoretical results are proven to be effective.

The pantograph equation arises in electrodynamics as a delay differential equation (DDE). In this article, we provide the ϑ-method for numerical solutions of pantograph equations. We investigate the stability conditions for the numerical schemes. The theoretical results are verified by numerical simulations. The theoretical results and numerical simulations show that implicit or partially implicit ϑ-methods, with ϑ>1/2, are effective in resolving stiff pantograph problems.

A fractional-order cholera model in the Caputo sense is constructed. The model is an extension of the Susceptible–Infected–Recovered (SIR) epidemic model. The transmission dynamics of the disease are studied by incorporating the saturated incidence rate into the model. This is particularly important since assuming that the increase in incidence for a large number of infected individualsis equivalent to a small number of infected individualsdoes not make much sense. The positivity, boundedness, existence, and uniqueness of the solution of the model are also studied. Equilibrium solutions are computed, and their stability analyses are shown to depend on a threshold quantity, the basic reproduction ratio (R0). It is clearly shown that if R0<1, the disease-free equilibrium is locally asymptotically stable, whereas if R0>1, the endemic equilibrium exists and is locally asymptotically stable. Numerical simulations are carried out to support the analytic results and to show the significance of the fractional order from the biological point of view. Furthermore, the significance of awareness is studied in the numerical section.

In this paper, we present a fractional-order mathematical model in the Caputo sense to investigate the significance of vaccines in controlling COVID-19. The Banach contraction mapping principle is used to prove the existence and uniqueness of the solution. Based on the magnitude of the basic reproduction number, we show that the model consists of two equilibrium solutions that are stable. The disease-free and endemic equilibrium points are locally stably when R0<1 and R0>1 respectively. We perform numerical simulations, with the significance of the vaccine clearly shown. The changes that occur due to the variation of the fractional order α are also shown. The model has been validated by fitting it to four months of real COVID-19 infection data in Thailand. Predictions for a longer period are provided by the model, which provides a good fit for the data.

A fractional-order model consisting of a system of four equations in a Caputo–Fabrizio sense is constructed. This paper investigates the role of negative and positive attitudes towards vaccination in relation to infectious disease proliferation. Two equilibrium points, i.e., disease-free and endemic, are computed. Basic reproduction ratio is also deducted. The existence and uniqueness properties of the model are established. Stability analysis of the solutions of the model is carried out. Numerical simulations are carried out and the effects of negative and positive attitudes towards vaccination areclearly shown; the significance of the fractional-order from the biological point of view is also established. The positive effect of increasing awareness, which in turn increases positive attitudes towards vaccination, is also shown numerically.The results show that negative attitudes towards vaccination increase infectious disease proliferation and this can only be limited by mounting awareness campaigns in the population. It is also clear from our findings that the high vaccine hesitancy during the COVID-19 pandemicisan important problem, and further efforts should be madeto support people and give them correct information about vaccines.

It is a known fact that there are a particular set of people who are at higher risk of getting COVID-19 infection. Typically, these high-risk individuals are recommended to take more preventive measures. The use of non-pharmaceutical interventions (NPIs) and the vaccine are playing a major role in the dynamics of the transmission of COVID-19. We propose a COVID-19 model with high-risk and low-risk susceptible individuals and their respective intervention strategies. We find two equilibrium solutions and we investigate the basic reproduction number. We also carry out the stability analysis of the equilibria. Further, this model is extended by considering the vaccination of some non-vaccinated individuals in the high-risk population. Sensitivity analyses and numerical simulations are carried out. From the results, we are able to obtain disease-free and endemic equilibrium solutions by solving the system of equations in the model and show their global stabilities using the Lyapunov function technique. The results obtained from the sensitivity analysis shows that reducing the hospitals’ imperfect efficacy can have a positive impact on the control of COVID-19. Finally, simulations of the extended model demonstrate that vaccination could adequately control or eliminate COVID-19.

In this paper, we study the global dynamics of a stochastic viral infection model with humoral immunity and Holling type II response functions. The existence and uniqueness of non-negative global solutions are derived. Stationary ergodic distribution of positive solutions is investigated. The solution fluctuates around the equilibrium of the deterministic case, resulting in the disease persisting stochastically. The extinction conditions are also determined. To verify the accuracy of the results, numerical simulations were carried out using the Euler–Maruyama scheme. White noise’s intensity plays a key role in treating viral infectious diseases. The small intensity of white noises can maintain the existence of a stationary distribution, while the large intensity of white noises is beneficial to the extinction of the virus.

With deterministic differential equations, we can understand the dynamics of tumor-immune interactions. Cancer-immune interactions can, however, be greatly disrupted by random factors, such as physiological rhythms, environmental factors, and cell-to-cell communication. The present study introduces a stochastic differential model in infectious diseases and immunology of the dynamics of a tumor-immune system with random noise. Stationary ergodic distribution of positive solutions to the system is investigated in which the solution fluctuates around the equilibrium of the deterministic case and causes the disease to persist stochastically. In some conditions, it may be possible to attain infection-free status, where diseases die out exponentially with a probability of one. Some numerical simulations are conducted with the Euler–Maruyama scheme in order to verify the results. White noise intensity is a key factor in treating infectious diseases.

In this paper, we study the dynamics of a fractional-order epidemic model with general nonlinear incidence rate functionals and time-delay. We investigate the local and global stability of the steady-states. We deduce the basic reproductive threshold parameter, so that if R0<1, the disease-free steady-state is locally and globally asymptotically stable. However, for R0>1, there exists a positive (endemic) steady-state which is locally and globally asymptotically stable. A Holling type III response function is considered in the numerical simulations to illustrate the effectiveness of the theoretical results.

Time delays and fractional order play a vital role in biological systems with memory. In this paper, we propose an epidemic model for Zika virus infection using delay differential equations with fractional order. Multiple time delays are incorporated in the model to consider the latency of the infection in a vector and the latency of the infection in the infected host. We investigate the necessary and sufficient conditions for stability of the steady states and Hopf bifurcation with respect to three time delays τ1, τ2, and τ3. The model undergoes a Hopf bifurcation at the threshold parameters τ1∗, τ2∗, and τ3∗. Some numerical simulations are given to show the effectiveness of obtained results. The numerical simulations confirm that combination of fractional order and time delays in the epidemic model effectively enriches the dynamics and strengthens the stability condition of the model.

In this work, we consider a class of fractional stochastic differential system with Hilfer fractional derivative and Poisson jumps in Hilbert space. We study the existence and uniqueness of mild solutions of such a class of fractional stochastic system, using successive approximation theory, stochastic analysis techniques, and fractional calculus. Further, we study the existence of optimal control pairs for the system, using general mild conditions of cost functional. Finally, we provide an example to illustrate the obtained results.

This paper deals with stability and Hopf bifurcation analyses of a mathematical model of HIV infection of CD4+ T-cells. The model is based on a system of delay differential equations with logistic growth term and antiretroviral treatment with a discrete time delay, which plays a main role in changing the stability of each steady state. By fixing the time delay as a bifurcation parameter, we get a limit cycle bifurcation about the infected steady state. We study the effect of the time delay on the stability of the endemically infected equilibrium. We derive explicit formulae to determine the stability and direction of the limit cycles by using center manifold theory and normal form method. Numerical simulations are presented to illustrate the results.

We propose a fractional order SIRC epidemic model to describe the dynamics of Salmonella bacterial infection in animal herds. The infection-free and endemic steady sates, of such model, are asymptotically stable under some conditions. The basic reproduction number ℛ0 is calculated, using next-generation matrix method, in terms of contact rate, recovery rate, and other parameters in the model. The numerical simulations of the fractional order SIRC model are performed by Caputo’s derivative and using unconditionally stable implicit scheme. The obtained results give insight to the modelers and infectious disease specialists.

We provide a class of fractional-order differential models of biological systems with memory, such as dynamics of tumor-immune system and dynamics of HIV infection of CD4+ T cells. Stability and nonstability conditions for disease-free equilibrium and positive equilibria are obtained in terms of a threshold parameter (minimum infection parameter) for each model. We provide unconditionally stable method, using the Caputo fractional derivative of order and implicit Euler’s approximation, to find a numerical solution of the resulting systems. The numerical simulations confirm the advantages of the numerical technique and using fractional-order differential models in biological systems over the differential equations with integer order. The results may give insight to infectious disease specialists.

We consider a delayed SIR epidemic model in which the susceptibles are assumed to satisfy the logistic equation and the incidence term is of saturated form with the susceptible. We investigate the qualitative behaviour of the model and find the conditions that guarantee the asymptotic stability of corresponding steady states. We present the conditions in the time lag in which the DDE model is stable. Hopf bifurcation analysis is also addressed. Numerical simulations are provided in order to illustrate the theoretical results and gain further insight into the behaviour of this system.