Time-dependent and time-independent SIR models applied to the COVID-19 outbreak in Argentina, Brazil, Colombia, Mexico and South Africa

Abstract

We consider the SIR epidemiological model applied to the evolution of COVID-19 with two approaches. In the first place we fit a global SIR model, with time delay, and constant parameters throughout the outbreak, including the contagion rate. The contention measures are reflected on an effective reduced susceptible population $N_{eff}$. In the second approach we consider a time-dependent contagion rate that reflects the contention measures either through a step by step fitting process or by following an exponential decay. In this last model the population is considered the one of the country $N$. In the linear region of the differential equations, when the total population $N$ is large the predictions are independent of $N$. We apply these methodologies to study the spread of the pandemic in Argentina, Brazil, Colombia, Mexico, and South Africa for which the infection peaks are yet to be reached. In all of these cases we provide estimates for the reproduction and recovery rates. The scenario for a time varying contagion rate is optimistic, considering that reasonable measures are taken such that the reproduction factor $R_0$ decreases exponentially. The measured values for the recovery rate $\gamma$ are reported finding a universality of this parameter over various countries. We discuss the correspondence between the global SIR with effective population $N_{eff}$ and the evolution of the time local SIR.