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The stability conditions of the fractional differential equations described by the Caputo generalized fractional derivative have been addressed. The generalized asymptotic stability of a class of the fractional differential equations has been investigated. The fractional input stability in the context of the fractional differential equations described by the Caputo generalized fractional derivative has been introduced. The Lyapunov characterizations of the generalized asymptotic stability and the generalized fractional input stability of the fractional differential equations with or without inputs have been provided. Several examples illustrating the main results of the paper have been proposed. The Caputo generalized fractional derivative and the generalized Gronwall lemma have been used.

In this work, a fractional predator-prey model with the harvesting rate is considered. Besides the existence and uniqueness of the solution to the model, local stability and global stability are experienced. A novel discretization depending on the numerical discretization of the Riemann–Liouville integral was introduced and the corresponding numerical discretization of the predator–prey fractional model was obtained. The net reproduction number R 0 was obtained for the prediction and persistence of the disease. The dynamical behavior of the equilibria was examined by using the stability criteria. Furthermore, numerical simulations of the model were performed and their graphical representations are shown to support the numerical discretizations, to visualize the effectiveness of our theoretical results and to monitor the effect of arbitrary order derivative. In our investigations, the fractional operator is understood in the Caputo sense.

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In this note, we present a global asymptotic stability criterion for the fractional differential equations in triangular form. We use the Caputo generalized fractional derivative in our investigations. In our note, we introduce a new procedure to study the global asymptotic stability of the fractional differential equations.Keywords: Caputo left generalized fractional derivative, fractional differential equations with input, Mittag-Leffler input stability.2010 MSC: 93D25, 93D05, 26A33.

The new coronavirus disease 2019 (COVID-19) infection is a double challenge for people infected with comorbidities such as cardiovascular and cerebrovascular diseases and diabetes. Comorbidities have been reported to be risk factors for the complications of COVID-19. In this work, we develop and analyze a mathematical model for the dynamics of COVID-19 infection in order to assess the impacts of prior comorbidity on COVID-19 complications and COVID-19 re-infection. The model is simulated using data relevant to the dynamics of the diseases in Lagos, Nigeria, making predictions for the attainment of peak periods in the presence or absence of comorbidity. The model is shown to undergo the phenomenon of backward bifurcation caused by the parameter accounting for increased susceptibility to COVID-19 infection by comorbid susceptibles as well as the rate of re-infection by those who have recovered from a previous COVID-19 infection. Sensitivity analysis of the model when the population of individuals co-infected with COVID-19 and comorbidity is used as response function revealed that the top ranked parameters that drive the dynamics of the co-infection model are the effective contact rate for COVID-19 transmission, $\beta\sst{cv}$, the parameter accounting for increased susceptibility to COVID-19 by comorbid susceptibles, $\chi\sst{cm}$, the comorbidity development rate, $\theta\sst{cm}$, the detection rate for singly infected and co-infected individuals, $\eta_1$ and $\eta_2$, as well as the recovery rate from COVID-19 for co-infected individuals, $\varphi\sst{i2}$. Simulations of the model reveal that the cumulative confirmed cases (without comorbidity) may get up to 180,000 after 200 days, if the hyper susceptibility rate of comorbid susceptibles is as high as 1.2 per day. Also, the cumulative confirmed cases (including those co-infected with comorbidity) may be as high as 1000,000 cases by the end of November, 2020 if the re-infection rates for COVID-19 is 0.1 per day. It may be worse than this if the re-infection rates increase higher. Moreover, if policies are strictly put in place to step down the probability of COVID-19 infection by comorbid susceptibles to as low as 0.4 per day and step up the detection rate for singly infected individuals to 0.7 per day, then the reproduction number can be brought very low below one, and COVID-19 infection eliminated from the population. In addition, optimal control and cost-effectiveness analysis of the model reveal that the the strategy that prevents COVID-19 infection by comorbid susceptibles has the least ICER and is the most cost-effective of all the control strategies for the prevention of COVID-19.

This paper deals with fractional input stability, and contributes to introducing a new stability notion in the stability analysis of fractional differential equations (FDEs) with exogenous inputs using the Caputo fractional derivative. In particular, we study the fractional input stability of FDEs with exogenous inputs. A Lyapunov characterization of this notion is proposed and several examples are provided to explain the fractional input stability of FDEs with exogenous inputs. The applicability and simulation of this method are illustrated by studying the particular class of fractional neutral networks.