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In the present paper, we study the dynamics of tuberculosis model using fractional order derivative in Caputo-Fabrizio sense. The number of confirmed notified cases reported by national TB program Khyber Pakhtunkhwa, Pakistan, from the year 2002 to 2017 are used for our analysis and estimation of the model biological parameters. The threshold quantity R 0 and equilibria of the model are determined. We prove the existence of the solution via fixedpoint theory and further examine the uniqueness of the model variables. An iterative solution of the model is computed using fractional Adams-Bashforth technique. Finally, the numerical results are presented by using the estimated values of model parameters to justify the significance of the arbitrary fractional order derivative. The graphical results show that the fractional model of TB in Caputo-Fabrizio sense gives useful information about the complexity of the model and one can get reliable information about the model at any integer or non-integer case.

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In this paper, we produced a new family of distribution called Gull Alpha Power Family of distributions (GAPF). A Special case of GAPF is derived by considering the Weibull distribution as a baseline distribution called Gull Alpha Power Weibull distribution (GAPW). The suitability of the proposed distribution derives from its ability to model both the monotonic and nonmonotonic hazard rate functions which are a common practice in survival analysis and reliability engineering. Various statistical properties were derived in addition to their special cases. The unknown parameters of the model are estimated using the maximum likelihood method. Moreover, the usefulness of the proposed distribution is supported by using two real lifetime data sets as well as simulated data.