Periodic dynamics of rubella epidemic under standard and fractional Caputo operator with real data from Pakistan

Abstract Memory effects of epidemics play a vital role in mathematical models of infectious diseases. In this research study, an epidemiological SEIR (Susceptible, Exposed, Infectious, Removed) type model for the rubella epidemic has been proposed via classical and fractional order Caputo differential operators while assuming the periodic transmission rate β ( t ) . The Caputo model has been investigated for the existence and uniqueness of its solutions via fixed point theory while the unique non-negative solution remains bounded within the biologically feasible region. Later, the non-fixed biological parameters of the classical and the Caputo model are obtained via nonlinear least squares fitting technique taking the real monthly cases for the rubella epidemic in Pakistan for the period 2010–2012. The performance rate of the Caputo model is 35% higher than that of the model with integer order derivative. The numerical simulations are obtained under different cases and it is highly suggested that the infectious rate σ must be controlled as much as possible to eradicate the rubella epidemic.

[1]  A. Tatem,et al.  Measles outbreak risk in Pakistan: exploring the potential of combining vaccination coverage and incidence data with novel data-streams to strengthen control , 2018, Epidemiology and Infection.

[2]  M. A. Khan,et al.  Modeling and simulation results of a fractional dengue model , 2019, The European Physical Journal Plus.

[3]  M. Inç,et al.  Transmission dynamics of varicella zoster virus modeled by classical and novel fractional operators using real statistical data , 2019, Physica A: Statistical Mechanics and its Applications.

[4]  Sania Qureshi,et al.  Effects of vaccination on measles dynamics under fractional conformable derivative with Liouville–Caputo operator , 2020 .

[5]  Sania Qureshi,et al.  Fractional derivatives applied to MSEIR problems: Comparative study with real world data , 2019, The European Physical Journal Plus.

[6]  Jue Liu,et al.  Sociodemographic and economic characteristics of susceptibility to rubella among women preparing for pregnancy in rural China. , 2017, International journal of infectious diseases : IJID : official publication of the International Society for Infectious Diseases.

[7]  J. F. Gómez‐Aguilar,et al.  Analytic solution for oxygen diffusion from capillary to tissues involving external force effects: A fractional calculus approach , 2019, Physica A: Statistical Mechanics and its Applications.

[8]  Abdon Atangana,et al.  New direction in fractional differentiation , 2017 .

[9]  Badr Saad T. Alkahtani,et al.  Modeling the spread of Rubella disease using the concept of with local derivative with fractional parameter : Beta-Derivative , 2016, Complex..

[10]  Abdon Atangana,et al.  A comparative study of convective fluid motion in rotating cavity via Atangana–Baleanu and Caputo–Fabrizio fractal–fractional differentiations , 2020 .

[11]  S. Qureshi,et al.  A Fractional Measles Model Having Monotonic Real Statistical Data for Constant Transmission Rate of the Disease , 2019, Fractal and Fractional.

[12]  Kashif Ali Abro,et al.  A fractional and analytic investigation of thermo-diffusion process on free convection flow: an application to surface modification technology , 2020 .

[13]  Xiao Wu,et al.  Seasonal transmission dynamics of measles in China , 2018, Theory in Biosciences.

[14]  Saif Ullah,et al.  A fractional model for the dynamics of TB virus , 2018, Chaos, Solitons & Fractals.

[15]  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.

[16]  Abdon Atangana,et al.  Mathematical analysis of dengue fever outbreak by novel fractional operators with field data , 2019, Physica A: Statistical Mechanics and its Applications.

[17]  Kazeem Oare Okosun,et al.  A fractional order pine wilt disease model with Caputo–Fabrizio derivative , 2018, Advances in Difference Equations.

[18]  Omar Abu Arqub,et al.  Solution of the fractional epidemic model by homotopy analysis method , 2013 .

[19]  N. Ford,et al.  A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations , 2013 .

[20]  J. Butcher.  Numerical methods for ordinary differential equations , 2003 .

[21]  Taza Gul,et al.  Modeling and analysis of Tuberculosis (TB) in Khyber Pakhtunkhwa, Pakistan , 2019, Math. Comput. Simul..

[22]  S. Qureshi,et al.  Monotonically decreasing behavior of measles epidemic well captured by Atangana–Baleanu–Caputo fractional operator under real measles data of Pakistan , 2020 .

[23]  Fatmawati,et al.  A new numerical solution of the competition model among bank data in Caputo-Fabrizio derivative , 2020 .

[24]  A. Atangana,et al.  Thermal stratification of rotational second-grade fluid through fractional differential operators , 2020, Journal of Thermal Analysis and Calorimetry.

[25]  Dumitru Baleanu,et al.  Two-strain epidemic model involving fractional derivative with Mittag-Leffler kernel. , 2018, Chaos.

[26]  M. Sababheh,et al.  A new definition of fractional derivative , 2014, J. Comput. Appl. Math..

[27]  D. L. C. Ching,et al.  Epidemic Model of Leptospirosis Containing Fractional Order , 2014 .

[28]  Ganna Rozhnova,et al.  Characterizing the dynamics of rubella relative to measles: the role of stochasticity , 2012, Journal of The Royal Society Interface.

[29]  N. Ford,et al.  Analysis of Fractional Differential Equations , 2002 .

[30]  Sania Qureshi,et al.  Modeling chickenpox disease with fractional derivatives: From caputo to atangana-baleanu , 2019, Chaos, Solitons & Fractals.

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

[32]  R M May,et al.  Vaccination against rubella and measles: quantitative investigations of different policies , 1983, Journal of Hygiene.