Mathematical modeling applied to epidemics: an overview

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.

Keywords: Complex networks; Disease spreading; Epidemic; Mathematical modeling; Public health; SIR model.

Fig. 1

a The evolution of density of infected, susceptible and removed individuals over time. The number of infected individuals grows exponentially fast at the beginning of the epidemic, reaches a peak and begins to decline, showing the natural behavior of an epidemic. b The graph is the same as represented in (a) but here is a simple demonstration of the effectiveness of vaccination. A small portion initially immunized (about $10\%$ of individuals at $t=0$) is already enough to drastically decrease the number of people infected over time and it is also decrease the peak of the epidemic (Color figure online)

Fig. 2

We show the difference between the infected curve over time when the contagion is reduced. The peak becomes more attenuated and consequently the epidemic lasts longer (Color figure online)

Fig. 3

An illustration of (a) Erdős and Rényi and (b) Barabàsi-Albert networks. Both with $N=20$ nodes. It is possible to observe the difference between the connectivity patterns. While the former has nodes with almost the same number of links, the latter has a few of nodes with many edges (Color figure online)

Fig. 4

The degree distribution of networks generated by Erdős and Rényi (blue circles) and Barabàsi-Albert (red circles) networks. The inset (log-log scale) shows clearly the heavy-tail of the heterogeneous distribution compared with the homogeneous one (Color figure online)

Fig. 5

Density of recovered (or removed) nodes (individuals)—${\rho }^r$ - in function of the infection rate $\lambda$, also known as control parameter. For BA network, a small value of the infection rate is enough to start the disease spreading while for ER network, it is necessary a bigger value of $\lambda$ to start the spreading of the disease to the entire network. The value of ${\lambda }_c$ is the epidemic threshold for the SIR model running on top of these different substrates. Here, we used both networks with $N={10}^4$ nodes (Color figure online)