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In this paper, we present a numerical scheme and mathematical analysis for the famous three-dimensional quadratic autonomous self-govern system that happens to be chaotic with the coexistence of multi-scroll attractors. The scheme is based on the Atangana-Baleanu fractional derivative in the Caputo sense. The formulation of these schemes introduces the non-local and non-singular kernel to the fractional derivatives. The fractional derivative is then approximated using the family of the Adams–Bashforth schemes. The results are presented in both numerical and graphical as the fractional order β varies between 0 < β ⩽ 1. We study the proposed model in the both generalized case that is 0 < β < 1 and the case where β = 1 , which is the integer standard case. Due to the impact of the generalized case, the proposed model is able to maintain the coexistence of multi-scroll attractors. [ABSTRACT FROM AUTHOR]

According to the World Health Organization reports, tuberculosis (TB) remains one of the top 10 deadly diseases of recent decades in the world. In this paper, we present the modeling, analysis and simulation of a mathematical model of TB transmission in a population incorporating several factors and study their impact on the disease dynamics. The spread of TB is modeled by eight compartments including different groups, which are too often not taken into account in the projections of tuberculosis incidence. The rigorous mathematical analysis of this model is provided, the basic reproduction number ( R 0 ) is obtained and used for TB dynamics control. The results obtained show that lost to follow-up and transferred individuals constitute a risk, but less than the cases carrying germs. Rapidly evolving latent/exposed cases are responsible for the incidence increasing in the short and medium term, while slower evolving latent/exposed cases will be responsible for the persistent long-term incidence and maintenance of TB and delay elimination in the population. The numerical simulations of the model show that, with certain parameters, TB will die out or sensibly reduce in the entire Democratic Republic of the Congo (DRC) population. The strategies on which the DRC's health system is currently based to fight this disease show their weaknesses because the TB situation in the DRC remains endemic. But monitoring contact, detection of latent individuals and their treatment are actions to be taken to reduce the incidence of the disease and thus effectively control it in the population. [ABSTRACT FROM AUTHOR]