The understanding of infectious diseases is a priority in the field of public health. This has generated the inclusion of several disciplines and tools that allow for analyzing the dissemination of infectious diseases. The aim of this manuscript is to model the spreading of a disease in a population that is registered in a database. From this database, we obtain an edge-weighted graph. The spreading was modeled with the classic SIR model. The model proposed with edge-weighted graph allows for identifying the most important variables in the dissemination of epidemics. Moreover, a deterministic approximation is provided. With database COVID-19 from a city in Chile, we analyzed our model with relationship variables between people. We obtained a graph with 3866 vertices and 6,841,470 edges. We fitted the curve of the real data and we have done some simulations on the obtained graph. Our model is adjusted to the spread of the disease. The model proposed with edge-weighted graph allows for identifying the most important variables in the dissemination of epidemics, in this case with real data of COVID-19. This valuable information allows us to also include/understand the networks of dissemination of epidemics diseases as well as the implementation of preventive measures of public health. These findings are important in COVID-19’s pandemic context.

The COVID-19 pandemic declared by the World Health Organization (WHO) has generated a wide-ranging debate regarding epidemiological forecasts and the global implications. With the data obtained from thc Chilean Ministry of Health (MINSAL), a prospective study was carried out using the generalized SEIR model to estimate the course of COVID-19 in Chile. Three scenarios were estimated: Scenario 1 with official MINSAL data; scenario 2 with official MINSAL data and recovery criteria proposed by international organizations of health; and scenario 3 with official MINSAL data, recovery criteria proposed by international organizations of health, and without considering deaths in thc total recovered. There are considerable differences between scenario 1 compared to 2 and 3 in the number of deaths, active patients, and duration of thc disease. Scenario 3, considered thc most adverse, estimates a total of 11,000 infected people, 1,151 deaths, and that thc peak of the disease will occur in the first days of May. We concluded that the concept of "recovered" may be decisive for the epidemiological forecasts of COVID-19 in Chile.

Establishing a node importance ranking is a problem that has attracted the attention of many researchers in recent decades. For unweighted networks where the edges do not have any attached weight, many proposals have been presented, considering local or global information of the networks. On the contrary, it occurs in undirected edge-weighted networks, where the proposals to address this problem have been more scarce. In this paper, a ranking method of node importance for undirected and edge-weighted is provided, generalizing the measure of line importance (DIL) based on the centrality degree proposed by Opsahl. The experimentation was done on five real networks and the results illustrate the benefits of our proposal.

Among the diverse and important applications that networks currently have is the modeling of infectious diseases. Immunization, or the process of protecting nodes in the network, plays a key role in stopping diseases from spreading. Hence the importance of having tools or strategies that allow the solving of this challenge. In this paper, we evaluate the effectiveness of the DIL-W${}^{\alpha }$ ranking in immunizing nodes in an edge-weighted network with 3866 nodes and 6,841,470 edges. The network is obtained from a real database and the spread of COVID-19 was modeled with the classic SIR model. We apply the protection to the network, according to the importance ranking list produced by DIL-W${}^{\alpha }$, considering different protection budgets. Furthermore, we consider three different values for $\alpha$; in this way, we compare how the protection performs according to the value of $\alpha$.

Fake news, viruses on computer systems or infectious diseases on communities are some of the problems that are addressed by researchers dedicated to study complex networks. The immunization process is the solution to these challenges and hence the importance of obtaining immunization strategies that control these spreads. In this paper, we evaluate the effectiveness of the DIL-W${}^{\alpha }$ ranking in the immunization of nodes that are attacked by an infectious disease that spreads on an edge-weighted graph using a graph-based SIR model. The experimentation was done on real and scale-free networks and the results illustrate the benefits of this ranking.

In this paper the linear representation of a graph is defined. A linear representation of a graph is a subgroup of $GL(p,\mathbb{R})$, the group of invertible matrices of order $ p $ and real coefficients. It will be demonstrated that every graph admits a linear representation. In this paper, simple and finite graphs will be used, framed in the graphs theory's area