Eugénio M. Rocha, Cristiana J. Silva, Delfim F. M. Torres
Ricerche di Matematica: A Journal of Pure and Applied Mathematics, advance article, 2017
We introduce a new tuberculosis (TB) mathematical model, with 25 state-space variables where 15 are evolution disease states (EDSs), which generalises previous models and takes into account the flux of populations between a high incidence TB country (A) and a community (G) with high percentage of people from (A), plus the rest of the population (C) of a host country (B) with low TB incidence. Contrary to some beliefs, related to the fact that agglomerations of individuals increase proportionally to the disease spread, analysis of the model shows that the existence of communities are simultaneously beneficial for the TB control from a global and regional viewpoint. There is an optimal ratio for the distribution of individuals in (C) versus (G), which minimizes the reproduction number R_0. Such value does not give the minimal total number of infected individuals in all (B), since such is attained when the community (G) is completely isolated (theoretical scenario). Sensitivity analysis and curve fitting on R_0 and on EDSs are pursuit in order to understand the TB effects in the global statistics, by measuring the variability of the relevant parameters that account for the existence of (G), composed of individuals coming from a high incidence area, and the (seasonal) flux between (A) and (B). We also show that the TB transmission rate \beta does not act linearly on R_0, as is common in compartment models where system feedback or group interactions do not occur. Further, we find the most important parameters for the increase of each EDS. The model and techniques proposed are applied to a case-study with concrete parameters, which model the situation of Angola (A) and Portugal (B), in order to show its relevance and meaningfulness.
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