A Mathematical Study of a Coronavirus Model with the Caputo Fractional-Order Derivative

Belgaid, Youcef; Helal, Mohamed; Lakmeche, Abdelkader; Venturino, Ezio

doi:10.3390/fractalfract5030087

“…Proof. We follow the approach used by [11], to prove the boundedness of the solution of the fractional model. Consider, N (t) = S(t) + I(t) + Q(t) is a function and for each µ 1 > 0, we have…”

Section: Non-negativity and Boundednessmentioning

confidence: 99%

“…Fractional-order conditions are required for Riemann-Liouville fractional operators to solve mathematical models. The Caputo operator overcomes this limitation and the initial conditions with integer-order derivatives with physical relevance may be used with the Caputo fractional operator [42,11,2]. [33,14] is used in most of the COVID-19 mathematical models that simulate the dynamics of infectious diseases among a community of humans.…”

Section: Introductionmentioning

confidence: 99%

Numerical Simulations of the Fractional-Order SIQ Mathematical Model of Corona Virus Disease Using the Nonstandard Finite Difference Scheme

Raza¹,

Bakar²,

Khan³

et al. 2022

MJMS

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This paper proposes a novel nonlinear fractional-order pandemic model with Caputo derivative for corona virus disease. A nonstandard finite difference (NSFD) approach is presented to solve this model numerically. This strategy preserves some of the most significant physical properties of the solution such as non-negativity, boundedness and stability or convergence to a stable steady state. The equilibrium points of the model are analyzed and it is determined that the proposed fractional model is locally asymptotically stable at these points. Non-negativity and boundedness of the solution are proved for the considered model. Fixed point theory is employed for the existence and uniqueness of the solution. The basic reproduction number is computed to investigate the dynamics of corona virus disease. It is worth mentioning that the non-integer derivative gives significantly more insight into the dynamic complexity of the corona model. The suggested technique produces dynamically consistent outcomes and excellently matches the analytical works. To illustrate our results, we conduct a comprehensive quantitative study of the proposed model at various quarantine levels. Numerical simulations show that can eradicate a pandemic quickly if a human population implements obligatory quarantine measures at varying coverage levels while maintaining sufficient knowledge.

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“…This section looks at the existence and uniqueness of the solution for the fractional model. The following are the sufficient conditions for the existence and uniqueness of a solution [11,26].…”

Section: Existence and Uniqueness Of Solutionmentioning

confidence: 99%

Untitled

2022

MJMS

0

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This paper proposes a novel nonlinear fractional-order pandemic model with Caputo derivative for corona virus disease. A nonstandard finite difference (NSFD) approach is presented to solve this model numerically. This strategy preserves some of the most significant physical properties of the solution such as non-negativity, boundedness and stability or convergence to a stable steady state. The equilibrium points of the model are analyzed and it is determined that the proposed fractional model is locally asymptotically stable at these points. Non-negativity and boundedness of the solution are proved for the considered model. Fixed point theory is employed for the existence and uniqueness of the solution. The basic reproduction number is computed to investigate the dynamics of corona virus disease. It is worth mentioning that the non-integer derivative gives significantly more insight into the dynamic complexity of the corona model. The suggested technique produces dynamically consistent outcomes and excellently matches the analytical works. To illustrate our results, we conduct a comprehensive quantitative study of the proposed model at various quarantine levels. Numerical simulations show that can eradicate a pandemic quickly if a human population implements obligatory quarantine measures at varying coverage levels while maintaining sufficient knowledge.

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“…Besides, nowadays the researchers are devoting their research work to the fractional-order COVID-19 mathematical models. A huge number of good research papers related to fractional-order COVID-19 mathematical models can be found in the literature, some of them are the following: Caputo fractional-order [ 46 , 47 ], Caputo–Fabrizio fractional-order [ 48 , 49 ], Atangana–Baleanu fractional-order [ 50 , 51 ], fuzzy fractional-order [ 52 , 53 ], fractal-fractional order [ 54 , 55 ].…”

Section: Introductionmentioning

confidence: 99%

A fractional-order mathematical model for COVID-19 outbreak with the effect of symptomatic and asymptomatic transmissions

Ali

¹

,

Rabiei

²

,

Rashidi

³

et al. 2022

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The purpose of this paper is to investigate the transmission dynamics of a fractional-order mathematical model of COVID-19 including susceptible ($$\textsc {S}$$ S ), exposed ($$\textsc {E}$$ E ), asymptomatic infected ($$\textsc {I}_1$$ I 1 ), symptomatic infected ($$\textsc {I}_2$$ I 2 ), and recovered ($$\textsc {R}$$ R ) classes named $$\mathrm {SEI_{1}I_{2}R}$$ SEI 1 I 2 R model, using the Caputo fractional derivative. Here, $$\mathrm {SEI_{1}I_{2}R}$$ SEI 1 I 2 R model describes the effect of asymptomatic and symptomatic transmissions on coronavirus disease outbreak. The existence and uniqueness of the solution are studied with the help of Schaefer- and Banach-type fixed point theorems. Sensitivity analysis of the model in terms of the variance of each parameter is examined, and the basic reproduction number $$(R_{0})$$ ( R 0 ) to discuss the local stability at two equilibrium points is proposed. Using the Routh–Hurwitz criterion of stability, it is found that the disease-free equilibrium will be stable for $$R_{0} < 1$$ R 0 < 1 whereas the endemic equilibrium becomes stable for $$R_{0} > 1$$ R 0 > 1 and unstable otherwise. Moreover, the numerical simulations for various values of fractional-order are carried out with the help of the fractional Euler method. The numerical results show that asymptomatic transmission has a lower impact on the disease outbreak rather than symptomatic transmission. Finally, the simulated graph of total infected population by proposed model here is compared with the real data of second-wave infected population of COVID-19 outbreak in India.

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A Mathematical Study of a Coronavirus Model with the Caputo Fractional-Order Derivative

Cited by 14 publications

References 21 publications

Numerical Simulations of the Fractional-Order SIQ Mathematical Model of Corona Virus Disease Using the Nonstandard Finite Difference Scheme

Numerical Simulations of the Fractional-Order SIQ Mathematical Model of Corona Virus Disease Using the Nonstandard Finite Difference Scheme

Untitled

A fractional-order mathematical model for COVID-19 outbreak with the effect of symptomatic and asymptomatic transmissions

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