{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:43:03Z","timestamp":1760146983433,"version":"build-2065373602"},"reference-count":29,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T00:00:00Z","timestamp":1735257600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In the realm of dynamical systems described by deterministic differential equations used in biomathematical modeling, two types of random events influence the populations involved in the model: the first one is called environmental noise, due to factors external to the system; the second one is called demographic noise, deriving from the inherent randomness of the modeled phenomenon. When the populations are small, only space-discrete stochastic models are capable of describing demographic noise; when the populations are large, these discrete models converge to continuous models described by stochastic ordinary differential systems, maintaining the essence of intrinsic noise. Moving forward again from a continuous stochastic framework, we get to the continuous deterministic setting described by ordinary differential equations if we assume that noise can be neglected. The inverse process has recently been explored in the literature by means of the so-called \u201cbackward technique\u201d in a biological context, starting from a system of continuous ordinary differential equations and going \u201cbackward\u201d to the reconstruction and numerical simulation of the underlying discrete stochastic process, that models the demographic noise intrinsic to the biological phenomenon. In this study, starting from a predictable, deterministic system, we move beyond biology and explore the effects of demographic noise in a novel model arising from the social sciences. Our field will be psychosocial, that is, the connections and processes that support social relationships between individuals. We consider a group of individuals having three personality types: altruistic, selfish, and susceptible (neutral). Applying the backward technique to this model built on ordinary differential equations, we demonstrate how demographic noise can act as a switching factor, i.e., moving backward from the deterministic continuous model to the discrete stochastic process using the same parameter values, a given equilibrium switches to a different one. This highlights the importance of addressing demographic noise when studying complex social interactions. To our knowledge, this is also the first time that the backward technique has been applied in social contexts.<\/jats:p>","DOI":"10.3390\/axioms14010009","type":"journal-article","created":{"date-parts":[[2024,12,27]],"date-time":"2024-12-27T09:23:36Z","timestamp":1735291416000},"page":"9","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A First Application of the Backward Technique in Social Sciences: Exploring Demographic Noise in a Model with Three Personality Types"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3382-3153","authenticated-orcid":false,"given":"Roberto","family":"Macrelli","sequence":"first","affiliation":[{"name":"Department of Biomolecular Sciences, University of Urbino \u201cCarlo Bo\u201d, Via Piazza S. Andrea, 34, 61029 Urbino, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2708-8503","authenticated-orcid":false,"given":"Margherita","family":"Carletti","sequence":"additional","affiliation":[{"name":"Department of Pure and Applied Sciences, University of Urbino \u201cCarlo Bo\u201d, Via Piazza della Repubblica, 13, 61029 Urbino, Italy"},{"name":"National Institute for High Mathematics (INdAM-GNCS), Citt\u00e1 Universitaria, Piazzale Aldo Moro 5, 00185 Rome, Italy"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0277-6369","authenticated-orcid":false,"given":"Vincenzo","family":"Fano","sequence":"additional","affiliation":[{"name":"Department of Pure and Applied Sciences, University of Urbino \u201cCarlo Bo\u201d, Via Piazza della Repubblica, 13, 61029 Urbino, Italy"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/j.pbiomolbio.2004.01.014","article-title":"A multi-scaled approach for simulating chemical reaction systems","volume":"85","author":"Burrage","year":"2004","journal-title":"Prog. Biophys. Mol. Biol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"403","DOI":"10.1016\/0021-9991(76)90041-3","article-title":"A general method for numerically simulating the stochastic time evolution of coupled chemical reactions","volume":"22","author":"Gillespie","year":"1976","journal-title":"J. Comput. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1016\/0378-4371(92)90283-V","article-title":"A rigorous derivation of the chemical master equation","volume":"188","author":"Gillespie","year":"1992","journal-title":"Phys. A"},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Carletti, M., and Banerjee, M. (2019). A Backward Technique for Demographic Noise in Biological Ordinary Differential Equation Models. Mathematics, 7.","DOI":"10.3390\/math7121204"},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Allen, L.J.S. (2010). An Introduction to Stochastic Processes with Applications to Biology, CRC Press. [2nd ed.].","DOI":"10.1201\/b12537"},{"key":"ref_6","unstructured":"Rice, S.H. (2004). Evolutionary Theory: Mathematical and Conceptual Foundations, Sinaurer."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1016\/S0003-3472(84)80272-9","article-title":"Social transmission of behavioral traditions in a coral reef fish","volume":"32","author":"Helfman","year":"1984","journal-title":"Anim. Behav."},{"key":"ref_8","first-page":"347","article-title":"The opening of milk bottles by birds","volume":"42","author":"Fisher","year":"1949","journal-title":"Br. Birds"},{"key":"ref_9","unstructured":"Cavalli-Sforza, L.L., and Feldman, W. (1981). Cultural Trasmission and Ecolution: A Quantitative Approach, Princeton University Press."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"238","DOI":"10.1016\/0040-5809(76)90047-2","article-title":"Cultural and biological evolutionary processes, selection for a trait under complex transmission","volume":"9","author":"Feldman","year":"1976","journal-title":"Theor. Popul. Biol."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"1604","DOI":"10.1073\/pnas.81.5.1604","article-title":"Cultural and biological evolutionary processes: Gene-culture disequilibrium","volume":"81","author":"Feldman","year":"1984","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Mondaini, R.P. (2023). Cultural and Biological Transmission: A Simple Case of Evolutionary Discrete Dynamics. Trends in Biomathematics: Modeling Epidemiological, Neuronal, and Social Dynamics, Springer. BIOMAT 2022.","DOI":"10.1007\/978-3-031-33050-6"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1038\/nature02043","article-title":"The nature of human altruism","volume":"425","author":"Fehr","year":"2003","journal-title":"Nature"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"4059","DOI":"10.1063\/1.1778376","article-title":"Efficient formulation of the stochastic simulation algorithm for chemically reacting systems","volume":"121","author":"Cao","year":"2004","journal-title":"J. Chem. Phys."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1633","DOI":"10.1093\/genetics\/149.4.1633","article-title":"Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells","volume":"149","author":"Arkin","year":"1998","journal-title":"Genetics"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"335","DOI":"10.1038\/35002125","article-title":"A synthetic oscillatory network of transcriptional regulators","volume":"403","author":"Elowitz","year":"2000","journal-title":"Nature"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"673","DOI":"10.1073\/pnas.022628299","article-title":"Robustness of circadian rythms with respect to molecular noise","volume":"99","author":"Gonze","year":"2002","journal-title":"Proc. Natl. Acad. Sci. USA"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"83","DOI":"10.1016\/0025-5564(94)90025-6","article-title":"Some discrete-time SI, SIR, and SIS epidemic models","volume":"124","author":"Allen","year":"1994","journal-title":"Math. Biosci."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1142\/S0218339024400023","article-title":"A Simple Problem for Simulating Demographic Noise in Biological Differential Equation Models: A discrepancy effect","volume":"0 0:0","author":"Macrelli","year":"2024","journal-title":"J. Biol. Syst."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"167","DOI":"10.1016\/S0168-9274(01)00034-4","article-title":"Implicit Taylor methods for stiff stochastic differential equations","volume":"38","author":"Tian","year":"2001","journal-title":"Appl. Numer. Math."},{"key":"ref_21","first-page":"311","article-title":"Effective simulation techniques for biological systems, in fluctuations and noise in biological, biophysical, and biomedical systems II","volume":"5467","author":"Burrage","year":"2004","journal-title":"Proc. SPIE\u2014Int. Soc. Opt. Eng."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Holcman, D. (2017). A review of stochastic and delay simulation approaches in both time and space in computational cell biology. Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology, Springer.","DOI":"10.1007\/978-3-319-62627-7"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"965","DOI":"10.3934\/mbe.2015.12.965","article-title":"Stochastic modelling of PTEN regulation in brain tumors: A model for glioblastoma multiforme","volume":"12","author":"Carletti","year":"2015","journal-title":"Math. Biosci. Eng."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"184102","DOI":"10.1063\/1.1889434","article-title":"Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems","volume":"122","author":"Goutsias","year":"2005","journal-title":"J. Chem. Phys."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Szallasi, Z., Stelling, Z., and Periwal, V. (2006). Numerical Simulation for Biochemical Kinetics. System Modeling in Cellular Biology: From Concepts to Nuts and Bolts, The MIT Press.","DOI":"10.7551\/mitpress\/9780262195485.001.0001"},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Horsthemke, W., and Lefever, R. (1984). Noise-Induced Transitions in Physics, Chemistry, and Biology. Noise-Induced Transitions, Springer Series in Synergetics, Springer.","DOI":"10.1007\/978-3-642-70196-2_23"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"537","DOI":"10.1111\/j.1461-0248.2006.00903.x","article-title":"Experimental support of the scaling rule for demographic stochasticity","volume":"9","author":"Desharnais","year":"2006","journal-title":"Ecol. Lett."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"R5875","DOI":"10.1103\/PhysRevE.62.R5875","article-title":"Field Theory of Absorbing Phase Transitions with a Nondiffusive Conserved Field","volume":"62","author":"Vespignani","year":"2000","journal-title":"Phys. Rev. E"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"971","DOI":"10.1007\/BF02179861","article-title":"Renormalization group study of the A+B\u2192\u2298 diffusion-limited reaction","volume":"80","author":"Lee","year":"1995","journal-title":"J. Stat. Phys."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/9\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T17:01:21Z","timestamp":1760115681000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,27]]},"references-count":29,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010009"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010009","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,12,27]]}}}