Revista de la
Unión Matemática Argentina
A stochastic simplicial SIS model driven by two independent noises
Ángel Tocino, Juan Hernández-Serrano, and Daniel Hernández Serrano

Volume 69, no. 1 (2026), pp. 227–246    

Published online (final version): February 12, 2026

https://doi.org/10.33044/revuma.4994

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Abstract

We propose a stochastic simplicial SIS model where two independent sources of noise are utilized to perturb the individual and collective infection rates. After proving that the model has a unique global solution, two sets of conditions on the parameters that give exponential stability of the trivial solution are presented. We then find conditions for persistence and show that the solution of the SDE oscillates infinitely often around a point under such requirements. We validate the theoretical statements by performing numerical experiments, as well as simulations on both real and synthetic simplicial networks, with results that align with the theoretical and numerical predictions of the model.

References

  1. J. A. D. Appleby, X. Mao, and A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control 53 no. 3 (2008), 683–691.  DOI  MR  Zbl
  2. L. Arnold, Stochastic differential equations: Theory and applications, John Wiley & Sons, New York, 1974.  MR  Zbl
  3. G. F. de Arruda, G. Petri, and Y. Moreno, Social contagion models on hypergraphs, Phys. Rev. Res. 2 no. 2 (2020), Paper No. 023032.  DOI
  4. G. F. de Arruda, M. Tizzani, and Y. Moreno, Phase transitions and stability of dynamical processes on hypergraphs, Commun. Phys. 4 (2021), Paper No. 24.  DOI
  5. A. Barrat, G. F. de Arruda, I. Iacopini, and Y. Moreno, Social contagion on higher-order structures, in Higher-order systems, Underst. Complex Syst., Springer, Cham, 2022, pp. 329–346.  DOI  MR  Zbl
  6. F. Battiston, G. Cencetti, I. Iacopini, V. Latora, M. Lucas, A. Patania, J.-G. Young, and G. Petri, Networks beyond pairwise interactions: Structure and dynamics, Phys. Rep. 874 (2020), 1–92.  DOI  MR  Zbl
  7. G. Burgio, A. Arenas, S. Gómez, and J. T. Matamalas, Network clique cover approximation to analyze complex contagions through group interactions, Commun. Phys. 4 (2021), Paper No. 111.  DOI
  8. S. Cai, Y. Cai, and X. Mao, A stochastic differential equation SIS epidemic model with two independent Brownian motions, J. Math. Anal. Appl. 474 no. 2 (2019), 1536–1550.  DOI  MR  Zbl
  9. N. Dalal, D. Greenhalgh, and X. Mao, A stochastic model of AIDS and condom use, J. Math. Anal. Appl. 325 no. 1 (2007), 36–53.  DOI  MR  Zbl
  10. I. Dzhalladova, M. Růžičková, and V. Š. Růžičková, Stability of the zero solution of nonlinear differential equations under the influence of white noise, Adv. Difference Equ. 2015 (2015), Paper No. 143.  DOI  MR  Zbl
  11. E. Estrada and G. J. Ross, Centralities in simplicial complexes. Applications to protein interaction networks, J. Theoret. Biol. 438 (2018), 46–60.  DOI  MR  Zbl
  12. C. Giusti, R. Ghrist, and D. S. Bassett, Two's company, three (or more) is a simplex, J. Comput. Neurosci. 41 no. 1 (2016), 1–14.  DOI  MR
  13. C. Granell, S. Gómez, and A. Arenas, Dynamical interplay between awareness and epidemic spreading in multiplex networks, Phys. Rev. Lett. 111 no. 12 (2013), Paper No. 128701.  DOI
  14. A. Gray, D. Greenhalgh, L. Hu, X. Mao, and J. Pan, A stochastic differential equation SIS epidemic model, SIAM J. Appl. Math. 71 no. 3 (2011), 876–902.  DOI  MR  Zbl
  15. D. Hernández Serrano, J. Hernández-Serrano, and D. Sánchez Gómez, Simplicial degree in complex networks. Applications of topological data analysis to network science, Chaos Solitons Fractals 137 (2020), Paper No. 109839.  DOI  MR  Zbl
  16. D. Hernández Serrano and D. Sánchez Gómez, Centrality measures in simplicial complexes: Applications of topological data analysis to network science, Appl. Math. Comput. 382 (2020), Paper No. 125331.  DOI  MR  Zbl
  17. D. Hernández Serrano, J. Villarroel, J. Hernández-Serrano, and A. Tocino, Stochastic simplicial contagion model, Chaos Solitons Fractals 167 (2023), Paper No. 113008.  DOI  MR
  18. H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 no. 4 (2000), 599–653.  DOI  MR  Zbl
  19. I. Iacopini, G. Petri, A. Barrat, and V. Latora, Simplicial models of social contagion, Nat. Commun. 10 (2019), Article No. 2485.  DOI
  20. C. Ji, D. Jiang, and N. Shi, The behavior of an SIR epidemic model with stochastic perturbation, Stoch. Anal. Appl. 30 no. 5 (2012), 755–773.  DOI  MR  Zbl
  21. A. P. Kartun-Giles and G. Bianconi, Beyond the clustering coefficient: A topological analysis of node neighbourhoods in complex networks, Chaos Solitons Fractals X 1 (2019), Paper No. 100004.  DOI
  22. W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A 115 (1927), 700–721.  DOI  Zbl
  23. A. Lahrouz, L. Omari, and D. Kiouach, Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model, Nonlinear Anal. Model. Control 16 no. 1 (2011), 59–76.  DOI  MR  Zbl
  24. A. Lahrouz, A. Settati, M. El Fatini, and A. Tridane, The effect of a generalized nonlinear incidence rate on the stochastic SIS epidemic model, Math. Methods Appl. Sci. 44 no. 1 (2021), 1137–1146.  DOI  MR  Zbl
  25. Q. Lu, Stability of SIRS system with random perturbations, Phys. A 388 no. 18 (2009), 3677–3686.  DOI  MR
  26. X. Mao, Stochastic differential equations and their applications, Horwood Ser. Math. Appl., Horwood, Chichester, 1997.  MR  Zbl
  27. J. T. Matamalas, S. Gómez, and A. Arenas, Abrupt phase transition of epidemic spreading in simplicial complexes, Phys. Rev. Res. 2 (2020), Paper No. 012049(R).  DOI
  28. A. Patania, G. Petri, and F. Vaccarino, The shape of collaborations, EPJ Data Sci. 6 (2017), Paper No. 18.  DOI
  29. G. Petri, M. Scolamiero, I. Donato, and F. Vaccarino, Topological strata of weighted complex networks, PLOS One 8 no. 6 (2013), Paper No. e66506.  DOI
  30. L. Shaikhet and A. Korobeinikov, Stability of a stochastic model for HIV-1 dynamics within a host, Appl. Anal. 95 no. 6 (2016), 1228–1238.  DOI  MR  Zbl
  31. G. T. Tilahun, M. T. Belachew, and Z. Gebreselassie, Stochastic model of tuberculosis with vaccination of newborns, Adv. Difference Equ. (2020), Paper No. 658.  DOI  MR
  32. A. Tocino, D. Hernández Serrano, J. Hernández-Serrano, and J. Villarroel, A stochastic simplicial SIS model for complex networks, Commun. Nonlinear Sci. Numer. Simul. 120 (2023), Paper No. 107161.  DOI  MR  Zbl
  33. A. Tocino and A. Martín del Rey, Local stochastic stability of SIRS models without Lyapunov functions, Commun. Nonlinear Sci. Numer. Simul. 103 (2021), Paper No. 105956.  DOI  MR  Zbl
  34. E. Tornatore, S. M. Buccellato, and P. Vetro, Stability of a stochastic SIR system, Phys. A 354 (2005), 111–126.  DOI
  35. L. Torres, A. S. Blevins, D. Bassett, and T. Eliassi-Rad, The why, how, and when of representations for complex systems, SIAM Rev. 63 no. 3 (2021), 435–485.  DOI  MR  Zbl
  36. P. J. Witbooi, Stability of an SEIR epidemic model with independent stochastic perturbations, Phys. A 392 no. 20 (2013), 4928–4936.  DOI  MR  Zbl
  37. X. Zhang, D. Jiang, T. Hayat, and B. Ahmad, Dynamical behavior of a stochastic SVIR epidemic model with vaccination, Phys. A 483 (2017), 94–108.  DOI  MR  Zbl
  38. Y. Zhao, D. Jiang, and D. O'Regan, The extinction and persistence of the stochastic SIS epidemic model with vaccination, Phys. A 392 no. 20 (2013), 4916–4927.  DOI  MR  Zbl