A fractional model for the dynamics of TB virus

Abstract In this paper, we present a nonlinear fractional order model in the Caputo sense to explore and simulate the TB dynamics. Using the TB confirmed notified cases from the year 2002 to 2017 in Khyber Pakhtunkhwa, Pakistan, we estimate the model parameters and demonstrate that the proposed fractional model provides a good fit to the real data. Initially, we compute the basic reproduction number and the model equilibria. Then, the existence and uniqueness of the model are shown via generalized mean value theorem. Further, we explore the local and global stability of the disease free equilibria in fractional environment. Finally, numerical results are obtained in order to validate the importance of the arbitrary order derivative and theoretical results. We conclude that the fractional epidemic model is more generalized than the classical model, which give more information about the disease dynamics and give a good agreement to the real data of TB infection.

[1]  J. Watmough,et al.  Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. , 2002, Mathematical biosciences.

[2]  Soyoung Kim,et al.  Mathematical model and intervention strategies for mitigating tuberculosis in the Philippines. , 2018, Journal of theoretical biology.

[3]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[4]  C. Pinto,et al.  HIV/HCV coinfection model: a fractional-order perspective for the effect of the HIV viral load , 2018, Advances in Difference Equations.

[5]  Nasser Hassan Sweilam,et al.  Comparative Study for Multi-Strain Tuberculosis (TB) Model of Fractional Order , 2016 .

[6]  Kai Diethelm,et al.  A fractional calculus based model for the simulation of an outbreak of dengue fever , 2013 .

[7]  N. Shawagfeh,et al.  GENERALIZED TAYLORS FORMULA , 2007 .

[8]  A. Geser,et al.  The use of mathematical models in the study of the epidemiology of tuberculosis. , 1962, American journal of public health and the nation's health.

[9]  Tailei Zhang,et al.  Global stability for a tuberculosis model , 2011, Math. Comput. Model..

[10]  Wei Lin.  Global existence theory and chaos control of fractional differential equations , 2007 .

[11]  D. Baleanu,et al.  Stability analysis of Caputo fractional-order nonlinear systems revisited , 2011, Nonlinear Dynamics.

[12]  Carla M. A. Pinto,et al.  Non-integer order analysis of the impact of diabetes and resistant strains in a model for TB infection , 2018, Commun. Nonlinear Sci. Numer. Simul..

[13]  Ilknur Koca,et al.  Chaos in a simple nonlinear system with Atangana-Baleanu derivatives with fractional order , 2016 .

[14]  R. Wallis.  Mathematical Models of Tuberculosis Reactivation and Relapse , 2016, Front. Microbiol..

[15]  W. R. Lynn,et al.  Mathematical models for the economic allocation of tuberculosis control activities in developing nations. , 1967, The American review of respiratory disease.

[16]  Delfim F. M. Torres,et al.  Uniform asymptotic stability of a fractional tuberculosis model , 2018, 1801.07059.

[17]  Saif Ullah,et al.  A new fractional model for the dynamics of the hepatitis B virus using the Caputo-Fabrizio derivative , 2018, The European Physical Journal Plus.

[18]  A. Atangana,et al.  New Fractional Derivatives with Nonlocal and Non-Singular Kernel: Theory and Application to Heat Transfer Model , 2016, 1602.03408.

[19]  Zhien Ma,et al.  Global stability of two models with incomplete treatment for tuberculosis , 2010 .

[20]  Cruz Vargas de León.  Volterra-type Lyapunov functions for fractional-order epidemic systems , 2015, Commun. Nonlinear Sci. Numer. Simul..

[21]  Emma S McBryde,et al.  Construction of a mathematical model for tuberculosis transmission in highly endemic regions of the Asia-Pacific. , 2014, Journal of theoretical biology.

[22]  José António Tenreiro Machado,et al.  Fractional model for malaria transmission under control strategies , 2013, Comput. Math. Appl..

[23]  Zaid M. Odibat,et al.  Generalized Taylor's formula , 2007, Appl. Math. Comput..

[24]  Xiao-Qiang Zhao,et al.  A Tuberculosis Model with Seasonality , 2010, Bulletin of mathematical biology.

[25]  Mohammad Saleh Tavazoei,et al.  Chaotic attractors in incommensurate fractional order systems , 2008 .

[26]  Yong Li,et al.  Mathematical modeling of tuberculosis data of China. , 2015, Journal of theoretical biology.

[27]  M. Caputo,et al.  A new Definition of Fractional Derivative without Singular Kernel , 2015 .

[28]  Gavin Churchyard,et al.  What We Know About Tuberculosis Transmission: An Overview , 2017, The Journal of infectious diseases.

[29]  J. F. Gómez‐Aguilar,et al.  Decolonisation of fractional calculus rules: Breaking commutativity and associativity to capture more natural phenomena , 2018 .

[30]  Benito M. Chen-Charpentier,et al.  A fractional order epidemic model for the simulation of outbreaks of influenza A(H1N1) , 2014 .