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The fractional model has been developed for the thermal flow of hybrid nanofluid due to the inclined surface. The thermal investigation of the hybrid nanomaterial is predicted by utilizing the molybdenum disulphide nanoparticles and graphene oxide nanomaterials. The flow computations for mixed convection flow of nanoparticles and base fluids are performed due to vertical oscillating plate. The simulations for the formulated model have been done ρ-Laplace transform technique for Caputo fractional simulations. Definitions of Mittage–Leffler function and ρ-Laplace transform are also presented for the governing model. The application of updated definitions of ρ-Laplace transform for the Caputo fractional model is quite interesting unlike traditional Laplace transforms. The comparative investigation for both types of nanoparticles is performed for heat and mass transfer rates. It is observed that the heat enhancement rate due to water-based nanoparticles is relatively impressive compared to the kerosene oil-based nanomaterials.

Abstract COVID-19 or coronavirus is a newly emerged infectious disease that started in Wuhan, China, in December 2019 and spread worldwide very quickly. Although the recovery rate is greater than the death rate, the COVID-19 infection is becoming very harmful for the human community and causing financial loses to their economy. No proper vaccine for this infection has been introduced in the market in order to treat the infected people. Various approaches have been implemented recently to study the dynamics of this novel infection. Mathematical models are one of the effective tools in this regard to understand the transmission patterns of COVID-19. In the present paper, we formulate a fractional epidemic model in the Caputo sense with the consideration of quarantine, isolation, and environmental impacts to examine the dynamics of the COVID-19 outbreak. The fractional models are quite useful for understanding better the disease epidemics as well as capture the memory and nonlocality effects. First, we construct the model in ordinary differential equations and further consider the Caputo operator to formulate its fractional derivative. We present some of the necessary mathematical analysis for the fractional model. Furthermore, the model is fitted to the reported cases in Pakistan, one of the epicenters of COVID-19 in Asia. The estimated value of the important threshold parameter of the model, known as the basic reproduction number, is evaluated theoretically and numerically. Based on the real fitted parameters, we obtained R 0 ≈ 1.50 $\mathcal{R}_{0} \approx 1.50$ . Finally, an efficient numerical scheme of Adams–Moulton type is used in order to simulate the fractional model. The impact of some of the key model parameters on the disease dynamics and its elimination are shown graphically for various values of noninteger order of the Caputo derivative. We conclude that the use of fractional epidemic model provides a better understanding and biologically more insights about the disease dynamics.

COVID-19 or coronavirus is a newly emerged infectious disease that started in Wuhan, China, in December 2019 and spread worldwide very quickly. Although the recovery rate is greater than the death rate, the COVID-19 infection is becoming very harmful for the human community and causing financial loses to their economy. No proper vaccine for this infection has been introduced in the market in order to treat the infected people. Various approaches have been implemented recently to study the dynamics of this novel infection. Mathematical models are one of the effective tools in this regard to understand the transmission patterns of COVID-19. In the present paper, we formulate a fractional epidemic model in the Caputo sense with the consideration of quarantine, isolation, and environmental impacts to examine the dynamics of the COVID-19 outbreak. The fractional models are quite useful for understanding better the disease epidemics as well as capture the memory and nonlocality effects. First, we construct the model in ordinary differential equations and further consider the Caputo operator to formulate its fractional derivative. We present some of the necessary mathematical analysis for the fractional model. Furthermore, the model is fitted to the reported cases in Pakistan, one of the epicenters of COVID-19 in Asia. The estimated value of the important threshold parameter of the model, known as the basic reproduction number, is evaluated theoretically and numerically. Based on the real fitted parameters, we obtained R 0 ≈ 1.50 . Finally, an efficient numerical scheme of Adams–Moulton type is used in order to simulate the fractional model. The impact of some of the key model parameters on the disease dynamics and its elimination are shown graphically for various values of noninteger order of the Caputo derivative. We conclude that the use of fractional epidemic model provides a better understanding and biologically more insights about the disease dynamics. [ABSTRACT FROM AUTHOR]

In the present work, we study the COVID-19 infection through a new mathematical model using the Caputo derivative. The model has all the possible interactions that are responsible for the spread of disease in the community. We first formulate the model in classical differential equations and then extend it into fractional differential equations using the definition of the Caputo derivative. We explore in detail the stability results for the model of the disease-free case when R0<1. We show that the model is stable locally when R0<1. We give the result that the model is globally asymptotically stable whenever R0≤1. Further, to estimate the model parameters, we consider the real data of the fourth wave from Pakistan and provide a reasonable fitting to the data. We estimate the basic reproduction number for the proposed data to be R0=1.0779. Moreover, using the real parameters, we present the numerical solution by first giving a reliable scheme that can numerically handle the solution of the model. In our simulation, we give the graphical results for some sensitive parameters that have a large impact on disease elimination. Our results show that taking into consideration all the possible interactions can describe COVID-19 infection.

In the present work, we study the COVID-19 infection through a new mathematical model using the Caputo derivative. The model has all the possible interactions that are responsible for the spread of disease in the community. We first formulate the model in classical differential equations and then extend it into fractional differential equations using the definition of the Caputo derivative. We explore in detail the stability results for the model of the disease-free case when R 0 < 1 . We show that the model is stable locally when R 0 < 1 . We give the result that the model is globally asymptotically stable whenever R 0 ≤ 1 . Further, to estimate the model parameters, we consider the real data of the fourth wave from Pakistan and provide a reasonable fitting to the data. We estimate the basic reproduction number for the proposed data to be R 0 = 1.0779 . Moreover, using the real parameters, we present the numerical solution by first giving a reliable scheme that can numerically handle the solution of the model. In our simulation, we give the graphical results for some sensitive parameters that have a large impact on disease elimination. Our results show that taking into consideration all the possible interactions can describe COVID-19 infection. [ABSTRACT FROM AUTHOR]