Dynamics of a fractional-order rubella disease model with vertical transmission and saturated incidence rate

Rubella is one of the viruses responsible for rubella disease. If the rubella virus infects a pregnant woman during the first trimester of pregnancy, it causes CRS (the virus transmits vertically from mother to fetus). In this paper, we study the rubella disease model with a fractional-order derivative and saturated incidence rate. Infectious diseases have a history in their transmission dynamics, thus non-local operators such as fractional-order derivatives play a vital role in modeling the dynamics of such epidemics. First, we analyze the important mathematical features of the proposed model, such as the existence and uniqueness, the non-negativity and boundedness of solutions. Then, the equilibrium point, basic reproduction number, and stability of the equilibrium points are also investigated. The model has two equilibrium points, namely the disease-free equilibrium and endemic equilibrium. The disease-free equilibrium point always exists, while the endemic equilibrium point exists if R-0 > 1. The disease-free equilibrium point is locally asymptotically stable if R-0 < 1, while the endemic equilibrium point is locally asymptotically stable if the Routh-Hurwitz criterion is satisfied. Numerical simulation is done by using the Grunwald-Letnikov approximation method to confirm the results of analytical calculations.