Asymptotic behavior of a stochastic hybrid SIQRS model with vertical transmission and nonlinear incidence

Shan Wang; Feng Wang; Shan Wang; Feng Wang

doi:10.3934/math.2024613

AIMS Mathematics

2024, Volume 9, Issue 5: 12529-12549. doi: 10.3934/math.2024613

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Research article

Asymptotic behavior of a stochastic hybrid SIQRS model with vertical transmission and nonlinear incidence

Shan Wang ¹,
Feng Wang ^{1,2
,
,}

1.
Department of Mathematics, Pingxiang University, Pingxiang 337000, China
2.
School of Mathematics and Statistics, Central South University, Changsha 410083, China

Received: 10 February 2024 Revised: 25 March 2024 Accepted: 27 March 2024 Published: 01 April 2024
MSC : 60H10, 92D30

We studied a class of a stochastic hybrid SIQRS model with nonlinear incidence and vertical transmission and obtained a threshold $ \Delta $ to distinguish behaviors of the model. Concretely, the disease was extinct exponentially when $ \Delta < 0 $. If $ \Delta > 0 $, the model we discussed admitted an invariant measure. A new class of the Lyapunov function was constructed in proving the latter conclusion. Some remarks were presented to shed light on the major results. Finally, several numerical simulations were provided to test the reached results.
- stochastic SIQRS model,
- vertical transmission,
- extinction,
- invariant measure
Citation: Shan Wang, Feng Wang. Asymptotic behavior of a stochastic hybrid SIQRS model with vertical transmission and nonlinear incidence[J]. AIMS Mathematics, 2024, 9(5): 12529-12549. doi: 10.3934/math.2024613

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Abstract

We studied a class of a stochastic hybrid SIQRS model with nonlinear incidence and vertical transmission and obtained a threshold $ \Delta $ to distinguish behaviors of the model. Concretely, the disease was extinct exponentially when $ \Delta < 0 $. If $ \Delta > 0 $, the model we discussed admitted an invariant measure. A new class of the Lyapunov function was constructed in proving the latter conclusion. Some remarks were presented to shed light on the major results. Finally, several numerical simulations were provided to test the reached results.

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