Monotonically decreasing behavior of measles epidemic well captured by Atangana–Baleanu–Caputo fractional operator under real measles data of Pakistan

Abstract

Abstract Epidemics have memory characteristics and the deterministic systems in mathematical epidemiology based upon ordinary differential equations are the most suitable tools when modeled under differential operators possessing such memory effects. This paper proposes a new four compartmental based system under the Atangana–Baleanu–Caputo (ABC) fractional order operator for measles epidemic. The mutually exclusive compartments containing susceptible, exposed, infectious and recovered individuals are fractionalyzed with ABC operator while balancing dimension for each parameter of the system. The real measles monthly incidence data of Pakistan from May-December, 2018 reported by WHO is used for parameter estimation thereby getting fitted values for transmission coefficient rate β and the fractional order parameter α via nonlinear least-squares curve fitting technique. The error norm obtained in the ABC case is about 50% smaller than the classical case, that is, when α = 1 . Fixed point theory proves existence of the unique solution for the fractional order measles system with positive invariant region. Stability analysis of the fractional system shows that both measles-free and endemic equilibria are locally asymptotically stable whenever the basic reproductive number R 0 is less and greater than unity, respectively. Later, the least (λ) and the most (α) sensitive parameters are determined using the normalized sensitivity indices for R 0 . Numerical simulations for the classical and fractional measles epidemiological systems illustrate the better fit for the real data about the infected individuals with the solution obtained via fractional order ABC operator. Various values of the biological parameters are taken to observe their effects on the proposed fractional system thereby suggesting the reduction in the contact rate of measles infected individuals with susceptible population.