The Effect of a Linear Tuning between the Antigenic Stimulations of CD4+ T Cells and CD4+ Tregs

We study the equilibria of an Ordinary Differencial Equation (ODE) system where CD4 + effector or helper T cells and Regulatory T cells (Tregs) are present. T cells trigger an immune response in the presence of their specific antigen. Regulatory T cells (Tregs) play a role in limiting auto-immune diseases due to their immune-suppressive ability. Here, we present explicit exact formulas that give the relationship between the concentration of T cells, the concentration of Tregs, and the antigenic stimulation of T cells, when the system is at equilibria, stable or unstable. We found a parameter region of bistability, limited by two thresholds of antigenic stimulation of T cells (hysteresis). Moreover, there are values of the slope parameter of the tuning for which an isola-center bifurcation appears, and, for some other values, there is a transcritical bifurcation. We also present time evolutions of the ODE system.A.A.P., B.M.P.M.O., and A.A.Y. would like to thank the financial support by the ERDF—European Regional Development Fund through the Operational Program for Competitiveness and Internationalization—COMPETE 2020 Program within project “POCI-01-0145-FEDER-006961”, and by National Funds through the FCT—Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) as part of project UID/EEA/50014/2013, project “Dynamics, Optimization and Modelling”, with reference PTDC/MAT-NAN/6890/2014, project “Modelling, Dynamics and Games”—MDG—with the reference PTDC/MAT-APL/31753/2017, and project “NanoSTIMA—Macro-to-Nano Human Sensing: Towards Integrated Multimodal Health Monitoring and Analytics”/NORTE-01-0145-FEDER-000016, financed by the North Portugal Regional Operational Program (NORTE 2020), under the PORTUGAL 2020 Partnership Agreement. A.A.P. is also thankful for the financial support received through the Special Visiting Researcher Program (Bolsa Pesquisador Visitante Especial—PVE) “Dynamics, Games and Applications”, with reference 401068/2014-5 (call: MEC/MCTI/CAPES/CNPQ/FAPS) at IMPA, Rio de Janeiro, Brasil. Acknowledgments The authors thank the anonymous referees of for their valuable comments. A.A.P., B.M.P.M.O., A.A.Y., and A.A. thank LIAAD-INESC TEC for all the support. Part of this research was done during visits by the authors to IMPA (Brazil) and University of Warwick (United Kingdom), and we thank them for their hospitality.