On a reaction–diffusion model for calcium dynamics in neurons with Mittag–Leffler memory

Joshi, Hardik; Jha, Brajesh Kumar

doi:10.1140/epjp/s13360-021-01610-w

Cited by 27 publications

(16 citation statements)

References 49 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The obtained protocols u and w, provide from the single-valued selection c θ by (9) to the set-valued map of regulation F θ by (6a), which should be strict on the subset D θ by (5c) : ∀(p, q, r, w) ∈ D θ , F θ (p, q, r, w) = ∅, and they rend the model (1) globally viable on the subset D θ , as it is demonstrated in the Proof 4 of the Lemma 1. The linear dynamics (1c) and (3) of the tumor repair r and the radiation control w respectively, allow to get the useful expression (7a) of the set-valued map of regulation F θ , as it is proved in the Proof 4 of the Lemma 3, and the single-valued selection c θ is a solution to the following problem of minimization : min ||(u, v)|| such that (u, v) ∈ [0, u max ] × [0, w max ] by (5a), and θ (p, q, r, w) + h(p, q, r, w), (u, v) ≤ 0 by (8), which is numerically approached by the method of Uzawa in the last Section 5, and implemented into the discretized model (11) by the method of Euler, to get the numerical simulations of Figure 2, which are in perfect conformity with the theoretical results of the preceding Section 4.…”

Section: Discussionmentioning

confidence: 99%

“…[10] Develops a Hilfer fractional model related to Parkinson's disease, and obtains a closed form solution in the terms of Wright function and Mittag-Leffler function, by using Sumudu transform technique. [11] Uses the Laplace transform and exponential Fourier transform of Atangana-Baleanu-Caputo (ABC) derivative, to obtain the approximate analytical solutions of a reaction-diffusion model for calcium dynamics in neurons, in terms of generalized Mittag-Leffler function. [12] Presents a two-dimensional fractional-order reaction-diffusion model to develop a control mechanism of Calcium in nerve cells, and uses the integral transform technique of arbitrary order to find the solution of the model.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Set-valued analysis of anti-angiogenic therapy and radiotherapy

MOUSTAFİD

2022

mmnsa

View full text Add to dashboard Cite

The aim of the paper is to study a cancer model based on anti-angiogenic therapy and radiotherapy. A set-valued analysis is carried out to control the tumor and carrying capacity of the vasculature, so in order to reverse tumor growth and augment tumor repair. The viability technique is used on an augmented model to solve the control problem. Obtained control is a selection of set-valued map of regulation and reduces tumor volume to around zero. A numerical simulation scheme with graphical representations and biological interpretations are given.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Set-valued analysis of anti-angiogenic therapy and radiotherapy

MOUSTAFİD

2022

mmnsa

View full text Add to dashboard Cite

show abstract

“…Clinicians can, thus, use the information (in terms of behavior's predictions) of fractional‐order systems to fit patients data with the most appropriate non‐integer‐order index. From the literature review, we have also seen that fractional‐order derivatives provide the best tool for real‐life modeling systems that are more accurate than integer‐order cases; see previous studies 46–54 …”

Section: Introductionmentioning

confidence: 89%

“…From the literature review, we have also seen that fractional-order derivatives provide the best tool for real-life modeling systems that are more accurate than integer-order cases; see previous studies. [46][47][48][49][50][51][52][53][54] The remaining work is organized as follows: Section 1 presents the study's introduction, followed by Section 2, which discusses some basics of fractional derivatives. Section 3 gives the formulation of the proposed SEIR model.…”

Section: Introductionmentioning

confidence: 99%

Global dynamics and bifurcation analysis of a fractional‐order SEIR epidemic model with saturation incidence rate

Ghori

Naik

et al. 2022

Math Methods in App Sciences

View full text Add to dashboard Cite

The present paper studies a fractional‐order SEIR epidemic model for the transmission dynamics of infectious diseases such as HIV and HBV that spreads in the host population. The total host population is considered bounded, and Holling type‐II saturation incidence rate is involved as the infection term. Using the proposed SEIR epidemic model, the threshold quantity, namely, basic reproduction number scriptR0, is obtained that determines the status of the disease, whether it dies out or persists in the whole population. The model's analysis shows that two equilibria exist, namely, disease‐free equilibrium (DFE) and endemic equilibrium (EE). The global stability of the equilibria is determined using a Lyapunov functional approach. The disease status can be verified based on obtained threshold quantity scriptR0. If scriptR0<1, then DFE is globally stable, leading to eradicating the population's disease. If scriptR0>1, a unique EE exists, and that is globally stable under certain conditions in the feasible region. The Caputo type fractional derivative is taken as the fractional operator. The bifurcation and sensitivity analyses are also performed for the proposed model that determines the relative importance of the parameters in disease transmission. The numerical solution of the model is obtained by the generalized Adams–Bashforth–Moulton method. Finally, numerical simulations are performed to illustrate and verify the analytical results. From the obtained results, it is concluded that the order of the fractional derivative plays a significant role in the dynamic process. Also, from the sensitivity analysis results, it is seen that the parameter δ ( infection rate) is positively correlated to scriptR0, while the parameter μ (inhibition rate) is negatively correlated to scriptR0, which means these parameters are more sensitive than others in disease dynamics.

show abstract

“…Also, there have been several experimental attempts that were performed in the past to identify the role and physiological impact of sodium calcium exchangers on various cells [13,14,15,16]. Beside this a researches has explored the role of parameters of calcium toolkits on astrocytes [17,18,19,20,21,22], neuron [23,24,25,26,27,28,29,30], oocytes [31,32], myocytes [33,34,35], hepatocytes [36], and T lymphocytes cells [37,38]. Thus a very little amount of work has attempted to study parameters of calcium toolkits by using the fractional calculus approach.…”

Section: Introductionmentioning

confidence: 99%

Chaos of calcium diffusion in Parkinson s infectious disease model and treatment mechanism via Hilfer fractional derivative

Joshi¹,

Jha²

2021

mmnsa

Self Cite

View full text Add to dashboard Cite

Calcium is a vital element in our body and plays a crucial role to moderate the calcium signalling process. Calcium-dependent protein and flux through the sodium-calcium exchanger are also involved in signalling process to perform and execute necessary cellular activities. The loss or alteration in this cellular activity starts the early progress of Parkinson's disease. A mathematical calcium model is developed in the form of the Hilfer fractional reaction-diffusion equation to examine the calcium diffusion in the cells. The effect of calcium-dependent protein and flux through the sodium-calcium exchanger is incorporated in the model. The solution of the Hilfer fractional calcium model is obtained by using the Sumudu transform technique in the form of the Wright function and Mittag-Leffler function. The graphical results are obtained for the different amounts of proteins, presence, and absence of sodium-calcium exchanger, and various orders of Hilfer derivative. The obtained results show that the modified calcium model is a function of time, position, and Hilfer fractional derivative. Thus the modified Hilfer calcium model provides a rich physical interpretation of a calcium model as compared to the classical calcium model.

show abstract

On a reaction–diffusion model for calcium dynamics in neurons with Mittag–Leffler memory

Cited by 27 publications

References 49 publications

Set-valued analysis of anti-angiogenic therapy and radiotherapy

Set-valued analysis of anti-angiogenic therapy and radiotherapy

Global dynamics and bifurcation analysis of a fractional‐order SEIR epidemic model with saturation incidence rate

Chaos of calcium diffusion in Parkinson s infectious disease model and treatment mechanism via Hilfer fractional derivative

Contact Info

Product

Resources

About