Abstract
This work presents an overview of the evolution of mathematical modeling applied to the context of epidemics and the advances in modeling in epidemiological studies. In fact, mathematical treatments have contributed substantially in the epidemiology area since the formulation of the famous SIR (susceptible-infected-recovered) model, in the beginning of the 20th century. We presented the SIR deterministic model and we also showed a more realistic application of this model applying a stochastic approach in complex networks. Nowadays, computational tools, such as big data and complex networks, in addition to mathematical modeling and statistical analysis, have been shown to be essential to understand the developing of the disease and the scale of the emerging outbreak. These issues are fundamental concerns to guide public health policies. Lately, the current pandemic caused by the new coronavirus further enlightened the importance of mathematical modeling associated with computational and statistical tools. For this reason, we intend to bring basic knowledge of mathematical modeling applied to epidemiology to a broad audience. We show the progress of this field of knowledge over the years, as well as the technical part involving several numerical tools.





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Here we are considering an approximation to simplify the mathematical calculations. We know that there are several biological factors that can, for example, make a fraction of the population naturally immune to some new epidemic. However, to attend our aim, this assumption is quite reasonable.
In fact, any realization of the epidemic dynamics in finite networks reaches the absorbing state sooner or later because of dynamic fluctuations inherent to stochastic process, even above the critical point. This simulation difficulty was traditionally overcome by quasi stationary methods. For further details, the reader can consult the following references: [54, 61]
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Angélica S. Mata acknowledges the support from FAPEMIG (Grant No. APQ-02482-18) and CNPq (Grant No. 423185/2018-7).
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Mata, A.S., Dourado, S.M.P. Mathematical modeling applied to epidemics: an overview. São Paulo J. Math. Sci. 15, 1025–1044 (2021). https://doi.org/10.1007/s40863-021-00268-7
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DOI: https://doi.org/10.1007/s40863-021-00268-7