Transmission dynamics of varicella zoster virus modeled by classical and novel fractional operators using real statistical data

Abstract

This study proposes a new epidemiological fractional order mathematical model called MSEIR (Maternally-derived immunity, Susceptible, Exposed, Infectious, and Recovered) using three most widely used operators, namely, the classical Caputo, the Caputo–Fabrizio (CF) and the Atangana–Baleanu–Caputo (ABC). During the process of fractionalization of the classical model, the dimensional consistency has been taken care of and the experimental data (for 20 weeks) available in literature for the chickenpox outbreak in 2014 among school children of the Shenzhen city of China has been employed in order to validate the fractional order model. The use of fixed point theory helps to prove the existence and the uniqueness for the solutions of each fractional order model under consideration. It is also proved that the model possesses a positively invariant region for a positive hyper-octant R+4. For the fractional models, disease free and endemic equilibria are found while computing basic reproduction number R0 which helps to determine local asymptotic stability for the steady states. Furthermore, three numerical methods recently made available in literature are used to carry out the numerical simulations for each operator under consideration. An interesting feature called the norm is obtained based upon the statistical data in which the parameter for the transmission rate (β) of the epidemic and the fractional-order parameters (λ,μ,ρ) in the models are obtained via least squares technique of optimization revealing the highest rate of performance for the ABC fractional operator.