Abstract
We make two fits of an ODE system with 5 equations that model immune response by CD4 + T cells with the presence of regulatory T cells (Tregs). We fit the simulations to data regarding gp61 and NP309 epitopes from mice infected with lymphocytic choriomeningitis virus LCMV. We optimized parameters relating to: the T cell maximum growth rate; the T cell capacity; the T cell homeostatic level; and the ending time of the immune activation phase after infection. We quantitatively and qualitatively compare the obtained results with previous fits in the literature using different ODE models and we show that we are able to calibrate the model and obtain good fits describing the data. © 2021, Springer Nature Switzerland AG.

Abstract
We introduce an evolutionary dynamical model for corruption in a democratic state describing the interactions between citizens, government and officials, where the voting power of the citizens is the main mechanism to control corruption. Three main scenarios for the evolution of corruption emerge depending on the efficiency of the institutions and the social, political, and economic characteristics of the State. Efficient institutions can create a corruption intolerant self-reinforcing mechanism. The lack of political choices, weaknesses of institutions and vote buying can create a self-reinforcing mechanism of corruption. The ambition of the rulers can induce high levels of corruption that can be fought by the voting power of the citizens creating corruption cycles.

Abstract
In two-action generalized polymatrix games, Nash equilibria are support-type-symmetric, i.e., determined by supports for each type of player. We show that such a property does not generalize straightforwardly for games with at least three actions or where interaction weights have different signs (neither all positive nor negative). A non-trivial condition on interaction weights must be satisfied, which may go unnoticed as it is trivially satisfied for: (i) two-action games, (ii) conformity games, and (iii) congestion games. We derive this condition and the corresponding simplified analytic equation for mixed strategies.

Abstract
Recent years witnessed an increase in income inequality. Several explanations have been put forward. In the present paper, we consider a series of technologically related events that have been crucial for the increased income inequality, that is, public R&D incentives, increasing horizontal integration and spillover effects. We found that public R&D incentives and the increasing horizontal integration have biased the income distribution towards the top income group. In particular, the high-skilled workers involved in the R&D process have benefited enormously from this process. Similarly, capital owners have seen an increase in their profits, because of the reduction in product market competition and technological improvements in the production process. We found the effect of knowledge spillovers to be less clear-cut. We conclude discussing the implications of our results and suggesting possible solutions to the increasing income inequality. We call for the creation of supranational institutions, and for stricter legislation on competition and antitrust policy.

Abstract
We propose a refinement process of dynamic equilibria based on small random perturbations (SRPs) of the backward perfect foresight (bpf) equilibrium map in a class of one-step, forward-looking dynamic models. An equilibrium is selected if its stationary measure is the limit of the stationary measures associated with the processes generated by the SRPs of the bpf maps, as the perturbation size approaches 0. We show that, for full measure sets of parameter values of a large class of one-parameter families of unimodal bpf maps, only determinate cycles or the chaotic sunspot equilibrium defined by Araujo and Maldonado (2000) is selected. Two examples are provided illustrating such refinement process.

Abstract
We study an evolutionary dynamics for the contributions by agents to a common/public good in a generalized version of Baliga and Maskin's environmental protection model. The dynamical equilibria consist of three scenarios: a single agent contributing to preserve the good with its optimal contribution level, and all the other agents being free-riders: a group of agents with the same optimal contribution level contributing to preserve the good, and all the other agents being free-riders; one where no agents contribute. The dynamics of the contributions can be complex but we prove that each trajectory converges to the equilibrium associated to the single agent (or group of agents) with the highest preference for the good that are contributing since the beginning. We note that while the aggregate contribution is below the optimal contribution level of the agent with the smallest preference for the good, then the aggregate contribution is increasing and there is no free-riding. Hence, if the optimal contribution level of the agent with the smallest preference is enough to not exhaust the good too quickly and the optimal contribution level of the agent with the greatest preference is enough to preserve the good, then, in spite of the appearance of free-riding in the contributions, the good might not be exhausted.