Real life application of Caputo fractional derivative for measles epidemiological autonomous dynamical system

Abstract

Abstract Various infectious diseases primarily contain the characteristics of memory and non-locality. Keeping this in view; we have, in the present paper, proposed a new epidemiological system for the measles epidemic via Caputo fractional order operator which is not only a non-local operator but also contains all characteristics concerned with memory of the epidemic. The proposed system is autonomous in nature which has been made physically meaningful by dimensional homogeneity among the fractional order Caputo derivative and the biological parameters used in the system. Caputo fractional order parameter “τ” and the disease transmission rate “β” are fitted with nonlinear least-squares curve fitting technique while using real confirmed measles incidence cases for May-December, 2018 in Pakistan as reported by World Health Organization (WHO). Coming to mathematical analysis, the system is found to have unique solution under fixed point theory with biologically feasible region which is shown as positively invariant. Steady-states are determined to be locally asymptotically stable under different conditions on the basic reproduction number whereas the least and the most influential parameters for the system are computed via forward sensitivity index analysis. Finally, the Caputo system is simulated using Adams technique devised for finding approximate solutions of fractional Caputo ordinary differential equations. Comparative analysis is carried out and effects of different biological parameters on dynamics and transmission of the measles epidemic have been thoroughly investigated.