Bifurcation Analysis of a Mathematical Model for Tuberculosis Transmission

The total population of the model is divided into five mutually exclusive classes: Vaccinated V ( t), susceptible S ( t), latent Lt(t)infected It (t ) and Recovery Rt(t ). Invariant region of the model were obtained and the result shows that whenever, NT > ^/N , the population reduces asymptotically to the carrying capacity and whenever NT  <= ^/N , every solution with initial condition inohms 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 ROT with respect to the model parameters were carried out

Mathematical Model for the Transmission Dynamics of Lassa Fever Disease with Contact Tracing and Effective Quarantine

In this paper, we developed a mathematical model to study transmission dynamics of Lassa fever with contact tracing and effective quarantine as control strategies. The model is sub divided into eight mutually exclusive classes namely: Susceptible human SH, Humans suspected to have had contact with the infected CH , Infected humans IH, Quarantined humans QH, , Dead humans D, Susceptible rodents SR, and Infected rodents IR. We established the positivity of the solution for the model equations, more also, we obtained the invariant region of the system. The disease-free equilibrium point was established and the model threshold parameter (Reproduction Number) was examined using next-generation operator method. The model analytical result shows that diseases free equilibrium is local asymptotically stable at R0 < 1 and unstable at R0 > 1, We also established that the model is globally asymptotically stable using Castillo-Chavez method. The simulated results show that effective contact tracing and quarantining reduces the spread of Lassa fever. We therefore conclude that every effort must be put in place by the agencies concern to ensure that early contact tracing and effective quarantining is encouraged as this play a very vital role in controlling the spread of the virus