ABSTRACT The present work has as main objective to investigate the use of fractional calculus in the modeling of epidemic outbreaks of interacting populations. In particular, we propose a generalization of the SIR model with fractional derivatives to describe the dynamics of the COVID-19 epidemic outbreak in two cities with interacting populations. In special, we consider the dynamics of COVID-19 in the municipalities of Pelotas and Rio Grande, which are neighboring cities and are relatively geographically isolated from the rest of the state of Rio Grande do Sul.

RESUMO O presente trabalho analisa e desenvolve um método para resolver equações diferenciais fracionárias utilizando o Método de Monte Carlo. Uma simulação numérica é realizada para algumas equações diferenciais, comparando os resultados com o que existe na literatura matemática. A linguagem Python é usada para criar modelos computacionais.

ABSTRACT This article analyzes and develops a method to solve fractional ordinary differential equations using the Monte Carlo Method. A numerical simulation is performed for some differential equations, comparing the results with what exists in the mathematical literature. The Python language is used to create computational models.

ABSTRACT In this contribution we analyze a discretized SIR (Susceptible, Infectious, and Removed) compartmental model, to investigate the role of individual interactions in the spread of diseases. The compartments S i,j , I i , j and R i , j ( i , j , = 1 , 2 , . . . , n ) are spatially distributed in a two-dimensional n × n network. We assume that the dynamics follow the well-known SIR-like iteration within the population in each (i, j) site. Moreover, the dynamics are enriched by considering a multi-population interaction following a Gaussian spatial distribution. Therefore, the mobility of individuals between distinct networks is measured from the width α of the Gaussian distribution. The interaction of individuals between distinct sites, responsible for the contagion between different populations, is assumed to occur in a time interval smaller than a fixed interval h so that, the total population in each site (i, j), given by N i , j = S i , j + I i , j + R i , j, remained constant (for example, individual leaves his home site (i, j) to work at a neighboring site and returns to his home in a time interval less than h). We numerically explore some scenarios of population interaction, based on distinct choices of width α, that include a hypothetically rapidly closedness and reopening of the economy. The results found show interesting dynamics in the infected population due to the interaction parameter α(t) between the populations. Finally, the model can be applied to evaluate the spread of diseases such as COVID-19 enabling decision making in different contexts.

RESUMO Este manuscrito propõem um modelo compartimental estruturado SEIR com n populações distintas interagindo para descrever a dissiminação da pandemia de COVID-19. O modelo proposto possui a flexibilidade de incluir comunidades geograficamente separadas, bem como levar em conta grupos populacionais com idades distintas e sua interações. Mostramos que hipóteses distintas sobre a dinâmica do modelo proposto leva a uma estrutura de platô na curva da população infectada, o que está em correspondência com os dados coletados em países de grandes dimensões como o Brasil. Tais observações sugerem a formulação da seguinte conjectura: “o efeito da difusão da COVID-19 das grandes cidades para o interior do Brasil, como especulado pelo modelo, é o resposável pela forma de platô dos casos reportados de infecção no país”. Apresentamos cenários numéricos e comparações com os dados de casos reportados de infecções por COVID-19 no Brasil que colaboram com as conclusões acima mensionada.

ABSTRACT This manuscript proposes a model of n distinct populations interaction structured SEIRcompartmental model to describe the spread of COVID-19 pandemic diseases. The proposed model has the flexibility to include geographically separated communities as well as taking into account aging population groups and their interactions. We show that distinct assumptions on the dynamics of the proposed model lead to a plateau-like curve of the infected population, reflecting collected data from large countries like Brazil. Such observations aim to the following conjecture: “COVID-19 diseased diffusion from the capitals to the Brazil interior, as reflected by the proposed model, is responsible for plateau-like reported cases in the country”. We present numerical simulations of some scenarios and comparison with reported data from Brazil that corroborate with the aforementioned conclusions.