Bifurcation Analysis of a Mathematical Model for Tuberculosis Transmission

Abstract

The total population of the model is divided into five mutually exclusive classes: Vaccinated V ( t), susceptible S ( t), latent L_t(t)
infected I_t (t ) and Recovery R_t(t ). Invariant region of the model were obtained and the result shows that whenever, N_T > ^/N , the population reduces asymptotically to the carrying capacity and whenever N_T  <= ^/N , every solution with initial condition in
ohms remains in that region for t > 0, so the model is well posed in ohms. We obtained positivity of Solution, it shows that all the solutions of the system are all positive for all t > 0; disease free and endemic equilibrium of the model were obtained, more, also the model threshold parameter (Reproduction Number) was examined using next-generation operator method. The model result shows that diseases free equilibrium is local asymptotically stable at R₀ < 1 and unstable at R₀ > 1. we also tested to know if the system will exhibits a backward bifurcation and the result show that ^_T = 0 correspond to disease free equilibrium point (PDE) and ^_T > 0 correspond to a situation when the disease persists (endemic), then we applied transcriptical bifurcation and the result show that a > 0 and b > 0 hence backward bifurcation; we also obtained Hopf Bifurcation for the system and the result show that B = B(complement) and dT/dB != 0 which means that the susceptible, latent and Recovery have to be control or else the endemic will occur and finally sensitivity analysis of R_OT with respect to the model parameters were carried out.