arXiv2017
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 (seasonal) flux of populations between a high incidence TB country (A) and a host country (B) with low TB incidence, where (B) is divided into a community (G) with high percentage of people from (A) plus the rest of the population (C). 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 semi-closed communities are beneficial for the TB control from a global viewpoint. 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. Simulations show that variations of the transmission coefficient on the origin country has a big influence on the number of infected (and infectious) individuals on the community and the host country. Moreover, 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. We also show that the TB transmission rate $β$ 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.