ABSTRACT A piecewise fractional differential equation (deterministicâstochastic differential equation or vice versa) has appeared in recent literature. Piecewise operators are used to study crossover real data effectively. By using stochasticâdeterministic âŠ
ABSTRACT A piecewise fractional differential equation (deterministicâstochastic differential equation or vice versa) has appeared in recent literature. Piecewise operators are used to study crossover real data effectively. By using stochasticâdeterministic piecewise hybrid fractional derivatives, with hybrid fractionalâorder and variableâorder fractional operators, this paper extends the deterministic model of immunoâchemotherapy with gene therapy and time delay. The immunoâchemotherapy with gene therapy combines traditional chemotherapy and immunotherapy with genetic engineering techniques to enhance the immune system's ability to target and destroy cancer cells. This offers a multifaceted approach to cancer treatment, potentially enhancing effectiveness while minimizing side effects. Two approximation techniques are used to solve the proposed model numerically. In the hybrid fractional deterministic models, we use the nonstandard Caputo proportional constant finite difference method, and in the stochastic models, we use the nonstandard Milstein technique. We examine the stability analysis of these methods to ensure their efficiency. Both the theoretical results and the efficiency of the methods are confirmed by numerical tests. New piecewise operators are illustrated by the curves presented. Studying the introduced piecewise fractional differential immunoâchemotherapy model with gene therapy and time delay along with a stochastic case greatly explained the dynamics of immunoâchemotherapy interaction.
Abstract The strong stabilization of the considered system is established by presenting sufficient conditions based on an observability condition expressed by the semigroup solution of the linear part of the âŠ
Abstract The strong stabilization of the considered system is established by presenting sufficient conditions based on an observability condition expressed by the semigroup solution of the linear part of the bilinear system. An explicit estimate on the convergence of the decay rate is established. Moreover, some sufficient conditions are used to discuss the weak stabilization of the considered system. Additionally, an illustrative example with numerical simulations is included.
The study of infectious diseases in humans has become increasingly important in public health. This paper extends the SEIR model to include unreported COVID-19 cases (U) and environmental white noise. âŠ
The study of infectious diseases in humans has become increasingly important in public health. This paper extends the SEIR model to include unreported COVID-19 cases (U) and environmental white noise. Dynamic analysis is conducted based on the variation of the environment. The ergodicity and stationary distribution criteria are discussed. Using a Lyapunov function, we write down some sufficient conditions for disease extinction. With different intensities of stochastic noises, we calculate the threshold of extinction for the stochastic epidemic system. In order to control the spread of disease, the stochastic noise plays an important role. A numerical simulation and a fit to real data have shown that the model and theoretical results are valid.
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 âŠ
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.
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 âŠ
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.
<abstract><p>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 âŠ
<abstract><p>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.</p></abstract>
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 âŠ
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.
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 âŠ
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 Pontryaginâs Maximum Principle. Some simulations are conducted to illustrate the investigation of the analytical work and the effect of control analysis.
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 âŠ
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.
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. âŠ
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 R0 is less than one, the disease-free equilibrium is locally asymptotically stable. If R0 greater 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.
This paper investigates the optimal control problem for the tumor-immune system with time-delay in the presence of gene therapy and immuno chemotherapy. Riemann-Liouville fractional order integrals and Caputo fractional order âŠ
This paper investigates the optimal control problem for the tumor-immune system with time-delay in the presence of gene therapy and immuno chemotherapy. Riemann-Liouville fractional order integrals and Caputo fractional order derivatives are combined to form a hybrid fractional order operator. For consistency with the physical model problem, a new parameter Ï is presented. A stability and bifurcation analysis of the proposed model is performed. The positivity, boundedness, and existence of optimal control for the proposed model are discussed. GrĂŒnwald-Letnikov nonstandard finite difference method with the discretization of the hybrid fractional order operator is constructed to solve the proposed model. Examples and comparative studies are presented to demonstrate the simplicity of the approximation approaches and the applicability of the utilized methods.
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 âŠ
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 Ï. 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.
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 âŠ
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.
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 âŠ
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.
Throughout the last few decades, fractional-order models have been used in many fields of science and engineering, applied mathematics, and biotechnology. Fractional-order differential equations are beneficial for incorporating memory and âŠ
Throughout the last few decades, fractional-order models have been used in many fields of science and engineering, applied mathematics, and biotechnology. Fractional-order differential equations are beneficial for incorporating memory and hereditary properties into systems. Our paper proposes an asymptomatic COVID-19 model with three delay terms [Formula: see text] and fractional-order [Formula: see text]. Multiple constant time delays are included in the model to account for the latency of infection in a vector. We study the necessary and sufficient criteria for stability of steady states and Hopf bifurcations based on the three constant time-delays, [Formula: see text], [Formula: see text], and [Formula: see text]. Hopf bifurcation occurs in the addressed model at the estimated bifurcation points [Formula: see text], [Formula: see text], [Formula: see text], and [Formula: see text]. The numerical simulations fit to real observations proving the effectiveness of the theoretical results. Fractional-order and time-delays successfully enhance the dynamics and strengthen the stability condition of the asymptomatic COVID-19 model.
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 âŠ
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.
Stochastic modeling predicts various outcomes from stochasticity in the data, parameters and dynamical system. Stochastic models are deemed more appropriate than deterministic models accounting in terms of essential and practical âŠ
Stochastic modeling predicts various outcomes from stochasticity in the data, parameters and dynamical system. Stochastic models are deemed more appropriate than deterministic models accounting in terms of essential and practical information about a system. The objective of the current investigation is to address the issue above through the development of a novel deep neural network referred to as a stochastic epidemiology-informed neural network. This network learns knowledge about the parameters and dynamics of a stochastic epidemic vaccine model. Our analysis centers on examining the nonlinear incidence rate of the model from the perspective of the combined effects of vaccination and stochasticity. Based on empirical evidence, stochastic models offer a more comprehensive understanding than deterministic models, mainly when we use error metrics. The findings of our study indicate that a decrease in randomness and an increase in vaccination rates are associated with a better prediction of nonlinear incidence rates. Adopting a nonlinear incidence rate enables a more comprehensive representation of the complexities of transmitting diseases. The computational analysis of the proposed method, focusing on sensitivity analysis and overfitting analysis, shows that the proposed method is efficient. Our research aims to guide policymakers on the effects of stochasticity in epidemic models, thereby aiding the development of effective vaccination and mitigation policies. Several case studies have been conducted on nonlinear incidence rates using data from Tennessee, USA.
<abstract><p>Herein, we discuss an optimal control problem (OC-P) of a stochastic delay differential model to describe the dynamics of tumor-immune interactions under stochastic white noises and external treatments. The required âŠ
<abstract><p>Herein, we discuss an optimal control problem (OC-P) of a stochastic delay differential model to describe the dynamics of tumor-immune interactions under stochastic white noises and external treatments. The required criteria for the existence of an ergodic stationary distribution and possible extinction of tumors are obtained through Lyapunov functional theory. A stochastic optimality system is developed to reduce tumor cells using some control variables. The study found that combining white noises and time delays greatly affected the dynamics of the tumor-immune interaction model. Based on numerical results, it can be shown which variables are optimal for controlling tumor growth and which controls are effective for reducing tumor growth. With some conditions, white noise reduces tumor cell growth in the optimality problem. Some numerical simulations are conducted to validate the main results.</p></abstract>
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 âŠ
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 hospitalsâ 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.
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 âŠ
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.
This paper examines the dynamics of a time-delay differential model of the tumour immune system with random noise. The model describes the interactions between healthy tissue cells, tumour cells, and âŠ
This paper examines the dynamics of a time-delay differential model of the tumour immune system with random noise. The model describes the interactions between healthy tissue cells, tumour cells, and activated immune system cells. We discuss stability and Hopf bifurcation of the deterministic system. We then explore stochastic stability, and the dynamics of the system in view of environmental fluctuations. Criteria for persistence and sustainability are discussed. Using multiple Lyapunov functions, some sufficient criteria for tumour cell persistence and extinction are obtained. Under certain circumstances, stochastic noise can suppress tumour cell growth completely. In contrast to the deterministic model which shows no stable tumour-free state, the white noise can either lead to tumour dormancy or tumour elimination. Some numerical simulations, by using Milstein's scheme, are carried out to show the effectiveness of the obtained results.
Abstract In this paper, we develop a fractional-order differential model for the dynamics of immune responses to SARS-CoV-2 viral load in one host. In the model, a fractional-order derivative is âŠ
Abstract In this paper, we develop a fractional-order differential model for the dynamics of immune responses to SARS-CoV-2 viral load in one host. In the model, a fractional-order derivative is incorporated to represent the effects of temporal long-run memory on immune cells and tissues for any age group of patients. The population of cytotoxic T-cells (CD8 + ), natural killer (NK) cells and infected viruses are unknown in this model. Some interesting sufficient conditions that ensure the asymptotic stability of the steady states are obtained. This model indicates some complex phenomena in COVID-19 such as âimmune exhaustionâ and âLong COVIDâ. Sensitivity analysis is also investigated for model parameters to determine the parameters that are effective in determining of the long COVID duration, disease control and future treatment as well as vaccine design. The model is verified with clinical and experimental data of 5 patients with COVID-19.
<abstract><p>The focus of the current manuscript is to provide a theoretical and computational analysis of the new nonlinear time-fractional (2+1)-dimensional modified KdV equation involving the Atangana-Baleanu Caputo ($ \mathcal{ABC} $) âŠ
<abstract><p>The focus of the current manuscript is to provide a theoretical and computational analysis of the new nonlinear time-fractional (2+1)-dimensional modified KdV equation involving the Atangana-Baleanu Caputo ($ \mathcal{ABC} $) derivative. A systematic and convergent technique known as the Laplace Adomian decomposition method (LADM) is applied to extract a semi-analytical solution for the considered equation. The notion of fixed point theory is used for the derivation of the results related to the existence of at least one and unique solution of the mKdV equation involving under $ \mathcal{ABC} $-derivative. The theorems of fixed point theory are also used to derive results regarding to the convergence and Picard's X-stability of the proposed computational method. A proper investigation is conducted through graphical representation of the achieved solution to determine that the $ \mathcal{ABC} $ operator produces better dynamics of the obtained analytic soliton solution. Finally, 2D and 3D graphs are used to compare the exact solution and approximate solution. Also, a comparison between the exact solution, solution under Caputo-Fabrizio, and solution under the $ \mathcal{ABC} $ operator of the proposed equation is provided through graphs, which reflect that $ \mathcal{ABC} $-operator produces better dynamics of the proposed equation than the Caputo-Fabrizio one.</p></abstract>
In this paper, we study the dynamics of COVIDâ19 in the UAE with an extended SEIR epidemic model with vaccination, timeâdelays, and random noise. The stationary ergodic distribution of positive âŠ
In this paper, we study the dynamics of COVIDâ19 in the UAE with an extended SEIR epidemic model with vaccination, timeâdelays, and random noise. The stationary ergodic distribution of positive solutions is examined, in which the solution fluctuates around the equilibrium of the deterministic case, causing the disease to persist stochastically. It is possible to attain infectionâfree status (extinction) in some situations, in which diseases die out exponentially and with a probability of one. The numerical simulations and fit to real observations prove the effectiveness of the theoretical results. Combining stochastic perturbations with timeâdelays enhances the dynamics of the model, and white noise intensity is an important part of the treatment of infectious diseases.
In this paper, we develop a fractionalâorder differential model for the dynamics of immune responses to SARSâCoVâ2 viral load in one host. In the model, a fractionalâorder derivative is incorporated âŠ
In this paper, we develop a fractionalâorder differential model for the dynamics of immune responses to SARSâCoVâ2 viral load in one host. In the model, a fractionalâorder derivative is incorporated to represent the effects of temporal longârun memory on immune cells and tissues for any age group of patients. The population of cytotoxic T cells (CD8 + ), natural killer (NK) cells, and infected viruses is unknown in this model. Some interesting sufficient conditions that ensure the asymptotic stability of the steady states are obtained. This model indicates some complex phenomena in COVIDâ19 such as âimmune exhaustionâ and âlong COVID.â Sensitivity analysis is also investigated for model parameters to determine the parameters that are effective in disease control and future treatment as well as vaccine design. The model is verified with clinical and experimental data of 5 patients with COVIDâ19.
The pantograph equation arises in electrodynamics as a delay differential equation (DDE). In this article, we provide the Ï âmethod for numerical solutions of pantograph equations. We investigate the stability âŠ
The pantograph equation arises in electrodynamics as a delay differential equation (DDE). In this article, we provide the Ï âmethod 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 Ï âmethods, with Ï > (1/2), are effective in resolving stiff pantograph problems.
In this paper, we propose a fractional-order viral infection model, which includes latent infection, a Holling type II response function, and a time-delay representing viral production. Based on the characteristic âŠ
In this paper, we propose a fractional-order viral infection model, which includes latent infection, a Holling type II response function, and a time-delay representing viral production. Based on the characteristic equations for the model, certain sufficient conditions guarantee local asymptotic stability of infection-free and interior steady states. Whenever the time-delay crosses its critical value (threshold parameter), a Hopf bifurcation occurs. Furthermore, we use LaSalleâs invariance principle and Lyapunov functions to examine global stability for infection-free and interior steady states. Our results are illustrated by numerical simulations.
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 âŠ
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 EulerâMaruyama scheme. White noiseâs 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.
In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said âŠ
In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said problem by using the homotopy perturbation method (HPM) for the required solution. We obtain the solution in the form of infinite series. We next triggered different parametric effects (such as x, t, and so on) on the structure of the solitary wave propagation, demonstrating that the breadth and amplitude of the solitary wave potential may alter when these parameters are changed. We have demonstrated that Heâs approach is highly effective and powerful for the solution of such a higher-order nonlinear partial differential equation through our calculations and simulations. We may apply our method to an additional complicated problem, particularly on the applied side, such as astrophysics, plasma physics, and quantum mechanics, to perform complex theoretical computation. Graphical presentation of few terms approximate solutions are given at different fractional orders.
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 âŠ
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 EulerâMaruyama scheme in order to verify the results. White noise intensity is a key factor in treating infectious diseases.
Public health science is increasingly focusing on understanding how COVID-19 spreads among humans. For the dynamics of COVID-19, we propose a stochastic epidemic model, with time-delays, Susceptible-Infected-Asymptomatic-Quarantined-Recovered (SIAQR). One global âŠ
Public health science is increasingly focusing on understanding how COVID-19 spreads among humans. For the dynamics of COVID-19, we propose a stochastic epidemic model, with time-delays, Susceptible-Infected-Asymptomatic-Quarantined-Recovered (SIAQR). One global positive solution exists with probability one in the model. As a threshold condition of persistence and existence of an ergodic stationary distribution, we deduce a generalized stochastic threshold
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. âŠ
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.
Herein, we propose a fractional-order delay differential model for the dynamics of Hepatitis-C Virus (HCV), with interferon-α(IFN-α) treatment. A fractional-order derivative is considered to represent the long-run immune memory required âŠ
Herein, we propose a fractional-order delay differential model for the dynamics of Hepatitis-C Virus (HCV), with interferon-α(IFN-α) treatment. A fractional-order derivative is considered to represent the long-run immune memory required for intermediate cellular interactions. A discrete time-delay Ï is also incorporated to represent the intracellular delay between initial infection of a cell by HCV and the release of new virions. By using time-delay Ï as a bifurcation parameter, the stability of infection-free and infected steady states is investigated. The coefficients of the corresponding characteristic equation depend on Ï, and geometric stability switch criteria is used to study the stability switching properties. The fractional-order delay differential model has been verified with real observations. Incorporating fractional-order and immune memory, in the model, greatly enriches the dynamics of the system and improves the consistency of the model with the observations.
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 âŠ
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<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:msub><mml:mi>â</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math>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 Caputoâs derivative and using unconditionally stable implicit scheme. The obtained results give insight to the modelers and infectious disease specialists.
Abstract Environmental factors, such as humidity, precipitation, and temperature, have significant impacts on the spread of the new strain coronavirus COVID-19 to humans. In this paper, we use a stochastic âŠ
Abstract Environmental factors, such as humidity, precipitation, and temperature, have significant impacts on the spread of the new strain coronavirus COVID-19 to humans. In this paper, we use a stochastic epidemic SIRC model, with cross-immune class and time-delay in transmission terms, for the spread of COVID-19. We analyze the model and prove the existence and uniqueness of positive global solution. We deduce the basic reproduction number ${\mathcal{R}}_{0}^{s}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mi>R</mml:mi><mml:mn>0</mml:mn><mml:mi>s</mml:mi></mml:msubsup></mml:math> for the stochastic model which is smaller than ${\mathcal{R}}_{0}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>R</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> of the corresponding deterministic model. Sufficient conditions that guarantee the existence of a unique ergodic stationary distribution, using the stochastic Lyapunov function, and conditions for the extinction of the disease are obtained. Our findings show that white noise plays an important part in controlling the spread of the disease; When the white noise is relatively large, the infectious diseases will become extinct; Re-infection and periodic outbreaks can occur due to the existence of feedback time-delay (or memory) in the transmission terms.
Abstract This paper presents a new technique for numerical treatments of Volterra delay integro-differential equations that have many applications in biological and physical sciences. The technique is based on the âŠ
Abstract This paper presents a new technique for numerical treatments of Volterra delay integro-differential equations that have many applications in biological and physical sciences. The technique is based on the mono-implicit Runge â Kutta method for treating the differential part and the collocation method (using Booleâs quadrature rule) for treating the integral part. The efficiency and stability properties of this technique have been studied. Numerical results are presented to demonstrate the effectiveness of the methodology.
We present a delay differential model with optimal control that describes the interactions of the tumour cells and immune response cells with external therapy. The intracellular delay is incorporated into âŠ
We present a delay differential model with optimal control that describes the interactions of the tumour cells and immune response cells with external therapy. The intracellular delay is incorporated into the model to justify the time required to stimulate the effector cells. The optimal control variables are incorporated to identify the best treatment strategy with minimum side effects by blocking the production of new tumour cells and keeping the number of normal cells above 75% of its carrying capacity. Existence of the optimal control pair and optimality system are established. Pontryaginâs maximum principle is applicable to characterize the optimal controls. The model displays a tumour-free steady state and up to three coexisting steady states. The numerical results show that the optimal treatment strategies reduce the tumour cells load and increase the effector cells after a few days of therapy. The performance of combination therapy protocol of immunochemotherapy is better than the standard protocol of chemotherapy alone.
(1) One of the most striking features in the study of epidemics is the difficulty of finding a causal factor which appears to be adequate to account for the magnitude âŠ
(1) One of the most striking features in the study of epidemics is the difficulty of finding a causal factor which appears to be adequate to account for the magnitude of the frequent epidemics of disease which visit almost every population. It was with a view to obtaining more insight regarding the effects of the various factors which govern the spread of contagious epidemics that the present investigation was undertaken. Reference may here be made to the work of Ross and Hudson (1915-17) in which the same problem is attacked. The problem is here carried to a further stage, and it is considered from a point of view which is in one sense more general. The problem may be summarised as follows: One (or more) infected person is introduced into a community of individuals, more or less susceptible to the disease in question. The disease spreads from the affected to the unaffected by contact infection. Each infected person runs through the course of his sickness, and finally is removed from the number of those who are sick, by recovery or by death. The chances of recovery or death vary from day to day during the course of his illness. The chances that the affected may convey infection to the unaffected are likewise dependent upon the stage of the sickness. As the epidemic spreads, the number of unaffected members of the community becomes reduced. Since the course of an epidemic is short compared with the life of an individual, the population may be considered as remaining constant, except in as far as it is modified by deaths due to the epidemic disease itself. In the course of time the epidemic may come to an end. One of the most important probems in epidemiology is to ascertain whether this termination occurs only when no susceptible individuals are left, or whether the interplay of the various factors of infectivity, recovery and mortality, may result in termination, whilst many susceptible individuals are still present in the unaffected population. It is difficult to treat this problem in its most general aspect. In the present communication discussion will be limited to the case in which all members of the community are initially equally susceptible to the disease, and it will be further assumed that complete immunity is conferred by a single infection.
We provide a family of ordinary and delay differential equations to model the dynamics of tumorâgrowth and immunotherapy interactions. We explore the effects of adoptive cellular immunotherapy on the model âŠ
We provide a family of ordinary and delay differential equations to model the dynamics of tumorâgrowth and immunotherapy interactions. We explore the effects of adoptive cellular immunotherapy on the model and describe under what circumstances the tumor can be eliminated. The possibility of clearing the tumor, with a strategy, is based on two parameters in the model: the rate of influx of the effector cells and the rate of influx of ILâ2. The critical tumorâgrowth rate, below which endemic tumor does not exist, has been found. One can use the model to make predictions about tumor dormancy.
Public health science is increasingly focusing on understanding how COVID-19 spreads among humans. For the dynamics of COVID-19, we propose a stochastic epidemic model, with time-delays, Susceptible-Infected-Asymptomatic-Quarantined-Recovered (SIAQR). One global âŠ
Public health science is increasingly focusing on understanding how COVID-19 spreads among humans. For the dynamics of COVID-19, we propose a stochastic epidemic model, with time-delays, Susceptible-Infected-Asymptomatic-Quarantined-Recovered (SIAQR). One global positive solution exists with probability one in the model. As a threshold condition of persistence and existence of an ergodic stationary distribution, we deduce a generalized stochastic threshold
This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, âŠ
This paper presents a critical review of selected topics related to the modelling of cancer onset, evolution and growth, with the aim of illustrating, to a wide applied mathematical readership, some of the novel mathematical problems in the field. This review attempts to capture, from the appropriate literature, the main issues involved in the modelling of phenomena related to cancer dynamics at all scales which characterise this highly complex system: from the molecular scale up to that of tissue. The last part of the paper discusses the challenge of developing a mathematical biological theory of tumour onset and evolution.
An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time âŠ
An introduction to fractional calculus, P.L. Butzer & U. Westphal fractional time evolution, R. Hilfer fractional powers of infinitesimal generators of semigroups, U. Westphal fractional differences, derivatives and fractal time series, B.J. West and P. Grigolini fractional kinetics of Hamiltonian chaotic systems, G.M. Zaslavsky polymer science applications of path integration, integral equations, and fractional calculus, J.F. Douglas applications to problems in polymer physics and rheology, H. Schiessel et al applications of fractional calculus and regular variation in thermodynamics, R. Hilfer.
Cancer is a complex disease in which a variety of factors interact over a wide range of spatial and temporal scales with huge datasets relating to the different scales available. âŠ
Cancer is a complex disease in which a variety of factors interact over a wide range of spatial and temporal scales with huge datasets relating to the different scales available. However, these data do not always reveal the mechanisms underpinning the observed phenomena. In this paper, we explain why mathematics is a powerful tool for interpreting such data by presenting case studies that illustrate the types of insight that realistic theoretical models of solid tumour growth may yield. These range from discriminating between competing hypotheses for the formation of collagenous capsules associated with benign tumours to predicting the most likely stimulus for protease production in early breast cancer. We will also illustrate the benefits that may result when experimentalists and theoreticians collaborate by considering a novel anti-cancer therapy.
In this paper, we provide a differential mathematical model with non-integer order derivative (fractional-order) to investigate the dynamics of Hepatitis-C Virus (HCV) replication, in presence of interferon-α (IFN) treatment. The âŠ
In this paper, we provide a differential mathematical model with non-integer order derivative (fractional-order) to investigate the dynamics of Hepatitis-C Virus (HCV) replication, in presence of interferon-α (IFN) treatment. The fractional-order is considered to represent the intermediate cellular interactions and intracellular delay of the viral life cycle. We mathematically analyze and characterize the steady states and dynamical behavior of the model in presence of interferon-α treatment. We deduce a threshold parameter â0 (average number of newly infected cells produced by a single infected cell) in terms of the treatment efficacy parameter 0 †Δ < 1 and other parameters. We also provide a numerical technique for solving the fractional-order model and fitting the model to real data during treatment. The numerical simulations confirm that the fractional-order differential models have the ability to provide accurate descriptions of nonlinear biological systems with memory. The analyses presented here give an insight to understand the dynamics of HCV infection.