Abstract
ABSTRACTThe 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 is proved, where is the reproduction number. We prove also that when 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.
Highlights
Tuberculosis (TB) is one of the top 10 causes of death worldwide
We demonstrate that the model exhibits threshold behaviour with a globally stable disease-free equilibrium (DFE) if the basic reproduction number is less than unity and a globally stable endemic equilibrium if the basic reproduction number is greater than unity
In order to investigate the impact of the effective contact rate on the propagation of TB, we carry out some numerical simulations to show the contribution of effective contact rate during the whole infection
Summary
This paper deals with the weakness of the TB strategy through mass vaccination and the multidimensional poverty in the spread of TB via a SV ELI transmission model. An are constants, xi is the population of ith compartment and xi∗ is the equilibrium level. In order to study the stability of a positive endemic equilibrium state, we use Lyapunov’s direct method and LaSalle’s Invariance Principle with a Lyapunov function of Goh–Volterra type: V(x1, x2, . Lyapunov functions of this type have proven to be useful for Lotka–Voltera predator–prey systems [1], and it appears that they can be useful for more complex compartmental epidemic.
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