<p>Seasonal influenza occurs annually and is one of the most common infectious diseases in the world, posing a threat to public health security. Therefore, it is essential to study the dynamics of seasonal influenza to raise public awareness and implement scientific prevention measures. Huo et al. studied a deterministic seasonal influenza model in Gansu, China; in this paper, we extend their study starting from proving the local asymptotic stability of the endemic equilibrium. In addition, considering the non-negligible effects of environmental disturbance on the transmission of influenza viruses, we assume that the transmission rate in the deterministic model follows a lognormal Ornstein–Uhlenbeck process; therefore, we formulate the corresponding stochastic model. To analyze the dynamics of the stochastic model, we first verify the existence and uniqueness of the global positive solution. Next, by constructing suitable Lyapunov functions, we obtain sufficient conditions for the stationary distribution and the extinction of the disease. More precisely, we deduce that the seasonal influenza persists when <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10132_Article_IEq1.gif" Format="GIF" Height="18" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">$R_{0}^{s}>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>R</mi> <mrow> <mn>0</mn> </mrow> <mi>s</mi> </msubsup> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> while it dies out when <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="332_2025_10132_Article_IEq2.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">$R_{0}^{E}<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>R</mi> <mrow> <mn>0</mn> </mrow> <mi>E</mi> </msubsup> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we derive the exact expression of probability density function around the quasi-endemic equilibrium. Finally, we carry out numerical simulations to explore the effects of environmental noise on virus dynamics.</p>

A Study of a Seasonal Influenza Model in Deterministic and Stochastic Environments

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发表日期：

2025年2月2日

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Xiaoshan Zhang,
Xinhong Zhang,
Daqing Jiang

摘要

Seasonal influenza occurs annually and is one of the most common infectious diseases in the world, posing a threat to public health security. Therefore, it is essential to study the dynamics of seasonal influenza to raise public awareness and implement scientific prevention measures. Huo et al. studied a deterministic seasonal influenza model in Gansu, China; in this paper, we extend their study starting from proving the local asymptotic stability of the endemic equilibrium. In addition, considering the non-negligible effects of environmental disturbance on the transmission of influenza viruses, we assume that the transmission rate in the deterministic model follows a lognormal Ornstein–Uhlenbeck process; therefore, we formulate the corresponding stochastic model. To analyze the dynamics of the stochastic model, we first verify the existence and uniqueness of the global positive solution. Next, by constructing suitable Lyapunov functions, we obtain sufficient conditions for the stationary distribution and the extinction of the disease. More precisely, we deduce that the seasonal influenza persists when $R_{0}^{s}>1$ $R_{0}^{s} > 1$ while it dies out when $R_{0}^{E}<1$ $R_{0}^{E} < 1$ . Furthermore, we derive the exact expression of probability density function around the quasi-endemic equilibrium. Finally, we carry out numerical simulations to explore the effects of environmental noise on virus dynamics.