Fractal–fractional order mathematical modelling of co–infection dynamics of tuberculosis and diabetes mellitus with vaccination - Nawaz, Rashid; Kainat, Saba; Nik Long, Nik Mohd Asri; Shah, Kamal; Abdeljawad, Thabet (Springer Nature, 2026)
This manuscript presents a Caputo-type fractal-fractional order model for tuberculosis and diabetes mellitus co-infection. Fundamental properties of the model are examined, including existence and uniqueness of solutions, the basic reproduction number, and Hyers–Ulam stability via fixed point theory. Sensitivity analysis is conducted to identify the most sensitive parameters. The model is further extended to an optimal control problem incorporating two controls: vaccination and the influence of TB-vaccinated infected individuals on the co-infected population. The TB vaccine can help to control TB and co-infection. Numerical results are obtained via the modified Euler method to illustrate the effects of TB vaccine on population classes, while numerical simulations performed via MATLAB to demonstrate the influence of fractional and fractal orders on the transmission dynamics of TB and co-infection with diabetes mellitus. Our findings indicate that when more co-infected patients seek preventative treatment, the infection rate among co-infected individuals will decline. It means an appropriate treatment with the tuberculosis vaccine can help to control co-infection.