Mathematical modeling of forest ecosystems by a reaction-diffusion-advection system: impacts of climate change and deforestation.

We present an innovative mathematical model for studying the dynamics of forest ecosystems. Our model is determined by an age-structured reaction-diffusion-advection system in which the roles of the water resource and of the atmospheric activity are considered. The model is abstract but constructed in such a manner that it can be applied to real-world forest areas; thus it allows to establish an infinite number of scenarios for testing the robustness and resilience of forest ecosystems to anthropic actions or to climate change. We establish the well-posedness of the reaction-diffusion-advection model by using the method of characteristics and by reducing the initial system to a reaction-diffusion problem. The existence and stability of stationary homogeneous and stationary heterogeneous solutions are investigated, so as to prove that the model is able to reproduce relevant equilibrium states of the forest ecosystem. We show that the model fits with the principle of almost uniform precipitation over forested areas and of exponential decrease of precipitation over deforested areas. Furthermore, we present a selection of numerical simulations for an abstract forest ecosystem, in order to analyze the stability of the steady states, to investigate the impact of anthropic perturbations such as deforestation and to explore the effects of climate change on the dynamics of the forest ecosystem.

Complex network model for COVID-19: Human behavior, pseudo-periodic solutions and multiple epidemic waves.

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.

Analysis of a COVID-19 compartmental model: a mathematical and computational approach.

In this note, we consider a compartmental epidemic mathematical model given by a system of differential equations. We provide a complete toolkit for performing both a symbolic and numerical analysis of the spreading of COVID-19. By using the free and open-source programming language Python and the mathematical software SageMath, we contribute for the reproducibility of the mathematical analysis of the stability of the equilibrium points of epidemic models and their fitting to real data. The mathematical tools and codes can be adapted to a wide range of mathematical epidemic models.