Global dynamics of a fractional-order SIR epidemic model with memory

In this paper,an investigation and analysis of a nonlinear fractional-order SIR epidemic model with Crowley-Martin type functional response and Holling type-II treatment rate are established along the memory.The existence and stability of the equilibrium points are investigated.The sufficient conditions for the persistence of the disease are provided.First,a threshold value,Ro,is obtained which determines the stability of equilibria,then model equilibria are determined and their stability analysis is considered by using fractional Routh-Hurwitz stability criterion and fractional La-Salle invariant principle.The fractional derivative is taken in Caputo sense and the numerical solution of the model is obtained by LI scheme which involves the memory trace that can capture and integrate all past activity.Meanwhile,by using Lyapunov functional approach,the global dynamics of the endemic equilibrium point is discussed.Further,some numerical simulations are performed to illustrate the effectiveness of the theoretical results obtained.The outcome of the study reveals that the applied LI scheme is computationally very strong and effective to analyze fractional-order differential equations arising in disease dynamics.The results show that order of the fractional derivative has a significant effect on the dynamic process.Also,from the results,it is obvious that the memory effect is zero for p=1.When the fractional-order p is decreased from 1,the memory trace nonlinearly increases from 0,and its dynamics strongly depends on time.The memory effect points out the difference between the derivatives of the fractional-order and integer order.

Modeling the mechanics of calcium regulation in T lymphocyte:A finite element method approach

Changes in cellular Ca2+concentration control a variety of physiological activities including hormone and neurotransmitter release,muscular contraction,synaptic plas-ticity,ionic channel permeability,apoptosis,enzyme activity,gene transcription and reproduction process.Spatial-temporal Ca2+dynamics due to Ca2 t release,buffering and re-uptaking plays a central role in studying the Ca2+regulation in T lympho-cytes.In most cases,Ca2+has its major signaling function when it is elevated in the cytosolic compartment.In this paper,a two-dimensional mathematical model to study spatiotemporal variations of intracellular Ca2+concentration in T lymphocyte cell is proposed and investigated.The cell is assumed to be a circular shaped geomnetrical domain for the representation of properties of Ca2+dynamics within the cell includ-ing important parameters.Ca2+binding proteins for the dynamics of Ca2+are itself buffer and other physiological parameters located in Ca2+stores.The model incorpo-rates the important biophysical processes like difusion,reaction,voltage gated Ca2+channel,leak from endoplasmic reticulum(ER),efflux from cytosol to ER via sarco ER Ca2+adenosine triphosphate(SERCA)pumps,buffers and Na+/Ca2+exchanger.The proposed mathematical model is solved using a finite difference method and the finite element method.Appropriate initial and boundary conditions are incorporated in the model based on biophysical conditions of the problem.Computer simulations in MAT-LAB R2019b are employed to investigate mathematical models of reaction-diffusion equation.The effect of source,buffer,Nat/Ca2+exchanger,etc.on spatial and tempo-ral patterns of Ca2+in T lymphocyte has been studied with the help of numerical results.From the obtained results,it is observed that,the coordinated combination of the incor-porated parameters plays a significant role in Ca2+regulation in T lymphocytes.ER leak and voltage-gated Ca2+channel provides the necessary Ca2+to the cell when required for its proper functioning,while on the other side buffers,SERCA pump and Na+/Ca2+exchanger makes balance in the Ca2+concentration,so as to prevent the cell from death as higher concentration for longer time is harmful for the cell and can cause cell death.

Stability analysis of a fractional-order cancer model with chaotic dynamics

In this paper,we develop a three-dimensional fractional-order cancer model.The proposed model involves the interaction among tumor cells,healthy tissue cells and activated effector cells.The detailed analysis of the equilibrium points is studied.Also,the existence and uniqueness of the solution are investigated.The fractional derivative is considered in the Caputo sense.Numerical simulations are performed to illustrate the effectiveness of the obtained theoretical results.The outcome of the study reveals that the order of the fractional derivative has a significant effect on the dynamic process.Further,the calculated Lyapunov exponents give the existence of chaotic behavior of the proposed model.Also,it is observed from the obtained results that decrease in fractional-order p increases the chaotic behavior of the model.