A malaria model with Caputo–Fabrizio and Atangana–Baleanu derivatives

Abboubakar, Hamadjam; Kumar, Pushpendra; Rangaig, Norodin A.; Kumar, Sachin

doi:10.1142/s1793962321500136

Cited by 35 publications

(22 citation statements)

References 44 publications

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“…This is the same for the compartment B. This phenomenon was also observed in a malaria fractional model studied by [38]. It is important to note that for η = 1, the solutions of the fractional model converge to the solutions of the integer model.…”

Section: Parametersupporting

confidence: 71%

Fractional Dynamics of Typhoid Fever Transmission Models with Mass Vaccination Perspectives

2021

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In this work, we formulate and mathematically study integer and fractional models of typhoid fever transmission dynamics. The models include vaccination as a control measure. After recalling some preliminary results for the integer model (determination of the epidemiological threshold denoted by Rc, asymptotic stability of the equilibrium point without disease whenever Rc<1, the existence of an equilibrium point with disease whenever Rc>1), we replace the integer derivative with the Caputo derivative. We perform a stability analysis of the disease-free equilibrium and prove the existence and uniqueness of the solution of the fractional model using fixed point theory. We construct the numerical scheme and prove its stability. Simulation results show that when the fractional-order η decreases, the peak of infected humans is delayed. To reduce the proliferation of the disease, mass vaccination combined with environmental sanitation is recommended. We then extend the previous model by replacing the mass action incidences with standard incidences. We compute the corresponding epidemiological threshold denoted by Rc☆ and ensure the uniform stability of the disease-free equilibrium, for both new models, when Rc☆<1. A new calibration of the new model is conducted with real data of Mbandjock, Cameroon, to estimate Rc☆=1.4348. We finally perform several numerical simulations that permit us to conclude that such diseases can possibly be tackled through vaccination combined with environmental sanitation.

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Section: Parametersupporting

confidence: 71%

Fractional Dynamics of Typhoid Fever Transmission Models with Mass Vaccination Perspectives

2021

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“…The numerical values for the physical parameters are given in Table 1 and are used in the computation of the calcium profile. The solution (43) of the modified calcium model ( 14) in form of Hilfer fractional derivative is used to obtain a graphical calcium profile for low and high proteins level and sodium calcium exchanger. The calcium profile simulated for the values In Figure 1 we show calcium profile versus time near the source x = 0 for the low level of proteins level and presence of sodium calcium exchanger.…”

Section: Resultsmentioning

confidence: 99%

“…The fractional calculus is a generalization of the integer-order calculus and it provides more accurate results as compared to classical calculus. Hence, it is widely used in mathematical modelling of science and engineering, medical, and almost all area of education [39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60]. Nowadays, numbers of the fractional derivative are available to deal with real-world problems such as Caputo derivative, Caputo-Fabrizio derivative, Atangana-Baleneu derivative, Hilfer derivative, Weyl derivative, Conformable derivative and many more.…”

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

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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.

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“…Some more results on existence of hilfer non-integer order neutral equations are given in [14] . In [17] , authors solved a malaria model via CF and ABC derivatives. Kumar et al [18] solved a time-delay coronavirus non-integer order model in Caputo sense.…”

Section: Introductionmentioning

confidence: 99%

Prediction studies of the epidemic peak of coronavirus disease in Brazil via new generalised Caputo type fractional derivatives

Kumar

Ertürk

Abboubakar

et al. 2021

Alexandria Engineering Journal

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The first reported case of coronavirus disease (COVID-19) in Brazil was confirmed on 25 February 2020 and then the number of symptomatic cases produced day by day. In this manuscript, we studied the epidemic peaks of the novel coronavirus (COVID-19) in Brazil by the successful application of Predictor-Corrector (P-C) scheme. For the proposed model of COVID-19, the numerical solutions are performed by a model framework of the recent generalized Caputo type non-classical derivative. Existence of unique solution of the given non-linear problem is presented in terms of theorems. A new analysis of epidemic peaks in Brazil with the help of parameter values cited from a real data is effectuated. Graphical simulations show the obtained results to classify the importance of the classes of projected model. We observed that the proposed fractional technique is smoothly work in the coding and very easy to implement for the model of non-linear equations. By this study we tried to exemplify the roll of newly proposed fractional derivatives in mathematical epidemiology. The main purpose of this paper is to predict the epidemic peak of COVID-19 in Brazil at different transmission rates. We have also attempted to give the stability analysis of the proposed numerical technique by the reminder of some important lemmas. At last we concluded that when the infection rate increases then the nature of the diseases changes by becoming more deathly to the population.

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A malaria model with Caputo–Fabrizio and Atangana–Baleanu derivatives

Cited by 35 publications

References 44 publications

Fractional Dynamics of Typhoid Fever Transmission Models with Mass Vaccination Perspectives

Fractional Dynamics of Typhoid Fever Transmission Models with Mass Vaccination Perspectives

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

Prediction studies of the epidemic peak of coronavirus disease in Brazil via new generalised Caputo type fractional derivatives

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