The SIR epidemic model from a PDE point of view

We present a derivation of the classical SIR model through a mean-field approximation from a discrete version of SIR. We then obtain a hyperbolic forward Kolmogorov equation, and show that its projected characteristics recover the standard SIR model. Moreover, we show that the long time limit of the evolution will be a Dirac measure. The exact position will depend on the well-know $R_0$ parameter, and it will be supported on the corresponding stable SIR equilibrium

Insuperable Strategies in Two-Player and Reducible Multi-Player Games

Part of this work was done during the stay of FACCC at Carnegie Mellon University, supported by the CMU-Portugal Program. Furthermore, part of this work was done during FACC stays at City University, London (UK), Universidade Federal do Cear\u00E1 (Brazil), and Instituto de Matem\u00E1tica Pura e Aplicada (Rio de Janeiro, Brazil). FACCC also acknowledges discussions on the concept of Hamiltonian spite with Andr\u00E9 d\u2019Almeida, which eventually led to preliminary ideas in the concept of insuperable strategies. MOS also acknowledges the support of CAPES/BR - Finance code 01 and FAPERJ through grant E-26/210.440.2019. All authors acknowledge the support of the CAPES PRINT program at UFF through grant 88881.310210/2018-01. Last, but not least, FACCC acknowledges the input of his daughter, Alice, to explain how the N-player generalization of the zerinho-ou-um game is played nowadays by school children. Publisher Copyright: © The Author(s) 2025.Real populations are seldom found at the Nash equilibrium strategy. The present work focuses on how population size can be a relevant evolutionary force diverting the population from its expected Nash equilibrium. We introduce the concept of insuperable strategy, a strategy that guarantees that no other player can have a larger payoff than the player that adopts it. We show that this concept is different from the rationality assumption frequently used in game theory and that for small populations the insuperable strategy is the most probable evolutionary outcome for any dynamics that equal game payoff and reproductive fitness. We support our ideas with several examples and numerical simulations. We finally discuss how to extend the concept to multiplayer games, introducing, in a limited way, the concept of game reduction.publishersversionpublishe

Aspects of the color evolution after the imaginal molt of Pachycoris torridus (Scopoli, 1772) (Hemiptera: Scutelleridae)

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Laboratório de Citogenética e Molecular de Insetos Departamento de Biologia Instituto de Biociências Letras e Ciências Exatas – IBILCE Universidade Estadual Paulista “Júlio de Mesquita Filho” – UNESP, Rua Cristóvão Colombo, 2265, Jardim NazarethLaboratório de Biologia Celular Departamento de Biologia Instituto de Biociências Letras e Ciências Exatas – IBILCE Universidade Estadual Paulista “Júlio de Mesquita Filho” – UNESP, Rua Cristóvão Colombo, 2265, Jardim NazarethLaboratório de Citogenética e Molecular de Insetos Departamento de Biologia Instituto de Biociências Letras e Ciências Exatas – IBILCE Universidade Estadual Paulista “Júlio de Mesquita Filho” – UNESP, Rua Cristóvão Colombo, 2265, Jardim NazarethLaboratório de Biologia Celular Departamento de Biologia Instituto de Biociências Letras e Ciências Exatas – IBILCE Universidade Estadual Paulista “Júlio de Mesquita Filho” – UNESP, Rua Cristóvão Colombo, 2265, Jardim NazarethFAPESP: 2008/09166-0FAPESP: 2010/16080-5FAPESP: 2011/11064-

From discrete to continuous evolution models: a unifying approach to drift-diffusion and replicator dynamics

We study the large population limit of the Moran process, assuming weak-selection, and for different scalings. Depending on the particular choice of scalings, we obtain a continuous model that may highlight the genetic-drift (neutral evolution) or natural selection; for one precise scaling, both effects are present. For the scalings that take the genetic-drift into account, the continuous model is given by a singular diffusion equation, together with two conservation laws that are already present at the discrete level. For scalings that take into account only natural selection, we obtain a hyperbolic singular equation that embeds the Replicator Dynamics and satisfies only one conservation law. The derivation is made in two steps: a formal one, where the candidate limit model is obtained, and a rigorous one, where convergence of the probability density is proved. Additional results on the fixation probabilities are also presented.Comment: 18 pages, 3 figure