Shivam Kumar Mishra, Syed Abbas, Juan Jose Nieto
Abstract:
In epidemiology, more than one infectious disease can pose a risk to the host
population. This area of study has attracted researchers in recent times.
In this article, we have considered an epidemic model that incorporates two different
transmission techniques, namely SIR and SIRS. The considered deterministic model has
been perturbed stochastically at transmission rates. The analysis has been done for
the resulting stochastic model. Firstly, we explore the existence of the positive
\(T\)-periodic solution for the stochastic system. We determine that the non-autonomous
periodic version of the system with white noise has a positive periodic solution
using the Lyapunov function and Khasminskii theory. In addition, we analyze the positive
recurrence of the system. The results obtained in this article give the idea that
reducing the white noise in the stochastic model is critical for observing positive
\(T\)-periodic solutions and positive recurrence. Finally, we present some examples and
perform numerical simulations to validate the theoretical results established in this
study.
Submitted October 15, 2025. Published December 16, 2025.
Math Subject Classifications: 45J05, 34K50, 92D25.
Key Words: Double epidemic hypothesis; periodic solution; positive recurrence.
DOI: 10.58997/ejde.2025.116
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Shivam Kumar Mishra School of Mathematical and Statistical Sciences Indian Institute of Technology Mandi Mandi, H.P., 175005, India email: shivammishra1807@gmail.com |
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Syed Abbas School of Mathematical and Statistical Sciences Indian Institute of Technology Mandi Mandi, H.P., 175005, India email: abbas@iitmandi.ac.in, sabbas.iitk@gmail.com |
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Juan Jose Nieto CITMAga, Departamento de EstatÃstica, Análise Matemática e Optimización Universidade de Santiago de Compostela 15782, Santiago de Compostela, Spain email: juanjose.nieto.roig@usc.es |
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