Mathematical model to assess vaccination and effective contact rate impact in the spread of tuberculosis

Abstract

The long and binding treatment of tuberculosis (TB) at least 6–8 months for the new cases, the partial immunity given by BCG vaccine, the loss of immunity after a few years doing that strategy of TB control via vaccination and treatment of infectious are not sufficient to eradicate TB. TB is an infectious disease caused by the bacillus Mycobacterium tuberculosis. Adults are principally attacked. In this work, we assess the impact of vaccination in the spread of TB via a deterministic epidemic model (SV ELI) (Susceptible, Vaccinated, Early latent, Late latent, Infectious). Using the Lyapunov–Lasalle method, we analyse the stability of epidemic system (SV ELI) around the equilibriums (disease-free and endemic). The global asymptotic stability of the unique endemic equilibrium whenever $ \mathcal R_{0} \gt 1 $ is proved, where $ \mathcal R_{0} $ is the reproduction number. We prove also that when $ \mathcal R_{0} $ is less than 1, TB can be eradicated. Numerical simulations, using some TB data found in the literature in relation with Cameroon, are conducted to approve analytic results, and to show that vaccination coverage is not sufficient to control TB, effective contact rate has a high impact in the spread of TB.

Keywords

• effective contact rate
• vaccination
• tuberculosis
• basic reproduction number
• global stability