ABSTRACT
We propose a mathematical model for the transmission dynamics of SARS-CoV-2 in a homogeneously mixing non constant population, and generalize it to a model where the parameters are given by piecewise constant functions. This allows us to model the human behavior and the impact of public health policies on the dynamics of the curve of active infected individuals during a COVID-19 epidemic outbreak. After proving the existence and global asymptotic stability of the disease-free and endemic equilibrium points of the model with constant parameters, we consider a family of Cauchy problems, with piecewise constant parameters, and prove the existence of pseudo-oscillations between a neighborhood of the disease-free equilibrium and a neighborhood of the endemic equilibrium, in a biologically feasible region. In the context of the COVID-19 pandemic, this pseudo-periodic solutions are related to the emergence of epidemic waves. Then, to capture the impact of mobility in the dynamics of COVID-19 epidemics, we propose a complex network with six distinct regions based on COVID-19 real data from Portugal. We perform numerical simulations for the complex network model, where the objective is to determine a topology that minimizes the level of active infected individuals and the existence of topologies that are likely to worsen the level of infection. We claim that this methodology is a tool with enormous potential in the current pandemic context, and can be applied in the management of outbreaks (in regional terms) but also to manage the opening/closing of borders.
ABSTRACT
The COVID-19 pandemic has forced policy makers to decree urgent confinements to stop a rapid and massive contagion. However, after that stage, societies are being forced to find an equilibrium between the need to reduce contagion rates and the need to reopen their economies. The experience hitherto lived has provided data on the evolution of the pandemic, in particular the population dynamics as a result of the public health measures enacted. This allows the formulation of forecasting mathematical models to anticipate the consequences of political decisions. Here we propose a model to do so and apply it to the case of Portugal. With a mathematical deterministic model, described by a system of ordinary differential equations, we fit the real evolution of COVID-19 in this country. After identification of the population readiness to follow social restrictions, by analyzing the social media, we incorporate this effect in a version of the model that allow us to check different scenarios. This is realized by considering a Monte Carlo discrete version of the previous model coupled via a complex network. Then, we apply optimal control theory to maximize the number of people returning to "normal life" and minimizing the number of active infected individuals with minimal economical costs while warranting a low level of hospitalizations. This work allows testing various scenarios of pandemic management (closure of sectors of the economy, partial/total compliance with protection measures by citizens, number of beds in intensive care units, etc.), ensuring the responsiveness of the health system, thus being a public health decision support tool.