Mathematical Modeling and Analysis of MDR-TB with Vaccination and Re-infection

Abstract

Tuberculosis (TB) is still one of the major infectious diseases in the world and a leading public health problem, particularly in developing countries where multidrug-resistant tuberculosis (MDR-TB) seriously threatens the efforts to control it. A deterministic compartmental model is developed to study TB transmission dynamics with vaccination, reinfection, treatment failure and development of MDR-TB. The total human population is stratified into six disjoint classes: susceptible unvaccinated, susceptible vaccinated, exposed, infectious with drug-susceptible TB, infectious with MDR-TB, and recovered individuals. The positivity and boundedness of solutions are proved to ensure biological feasibility of the qualitative properties of the model. We obtain the disease-free and endemic equilibrium states and calculate the basic reproduction number $\mathcal{R}_0$ using the next-generation matrix method. The stability analysis reveals that the disease-free equilibrium is locally and globally asymptotically stable if $\mathcal{R}_0 < 1$, and unstable if $\mathcal{R}_0 > 1$. Furthermore, by developing a suitable Lyapunov function and using LaSalle's invariance principle, the global asymptotic stability of the endemic equilibrium is established for $\mathcal{R}_0>1$. Numerical simulations are performed to illustrate the effects of critical epidemiological parameters on the disease progression and to verify the analytical results. Sensitivity analysis based on Partial Rank Correlation Coefficients (PRCC) shows that the effective contact rate is the most critical parameter influencing TB transmission, while the treatment and recovery parameters play a strong role in reducing the spread of the disease. The study shows that effective vaccination, better treatment adherence and strong MDR-TB case management are necessary to cut down TB burden and attain long-term control over the disease