<p>Prominent airborne particles that can influence the spread of influenza include nitrogen dioxide (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$NO_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <msub> <mi>O</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>), sulfur dioxide (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$SO_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>S</mi> <msub> <mi>O</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation>) and ozone <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$(O_{3}).$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <msub> <mi>O</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> This research introduces a novel non-autonomous stochastic differential equation (SDE) framework that considers the impacts of immunization and seasonal variations while incorporating age heterogeneity. Initially, the global positive solution and the periodic solution of the stochastic model are determined, and the basic reproduction number for the corresponding impulsive system is identified. Furthermore, it is demonstrated that the system is uniformly persistent. To analyze the stochastic influenza model, one of the techniques employed is the application of Poisson random measure noise, which facilitates a broader comprehension of the mechanisms of flu propagation and optimal control mechanisms without requiring verbose second-order adjoint equations. A threshold parameter, <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathbb {R}_{0\textrm{k}}^{s}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">R</mi> <mrow> <mn>0</mn> <mtext>k</mtext> </mrow> <mi>s</mi> </msubsup> </math></EquationSource> </InlineEquation>, is introduced to examine infection dynamics. Specifically, if <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {R}_{0\textrm{k}}^{s}<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="double-struck">R</mi> <mrow> <mn>0</mn> <mtext>k</mtext> </mrow> <mi>s</mi> </msubsup> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the virus will die out; whereas if <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {R}_{0\textrm{k}}^{s}>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="double-struck">R</mi> <mrow> <mn>0</mn> <mtext>k</mtext> </mrow> <mi>s</mi> </msubsup> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, the virus will persist. Influenza statistics from Quebec Province are analyzed using the Monte-Carlo-Markov-Chain (MCMC) technique to estimate the most suitable parameter values. Our estimated basic reproduction number for Quebec, <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\mathbb {R}_{0\textrm{k}}=0.5645$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">R</mi> <mrow> <mn>0</mn> <mtext>k</mtext> </mrow> </msub> <mo>=</mo> <mn>0.5645</mn> </mrow> </math></EquationSource> </InlineEquation> (95% C-I: 0.3261, 0.7239), indicates that the influenza outbreak is likely to decline. However, our data also suggest that the WHO’s 2040 target is unlikely to be met, as projections show the annual incidence of new influenza cases will reach 1,061 (95% C-I: 136, 7,040) by that time. In light of this, we evaluate the feasibility of initiating adult immunization programs and reducing clinical testing delays starting in 2025 to align with WHO’s objectives. According to our findings, WHO’s 2040 benchmark can be achieved either by vaccinating 10% of the adult population or by reducing testing delays from four months to three weeks. These measures would prevent 74,127 (95% C-I: 24,700, 237,090) and 56,812 (95% C-I: 15,723, 212,476) influenza cases, respectively, between 2024 and 2040. Additionally, under some mild conditions regarding a convex control set, the stochastic process and its associated adjoint formulations are derived. Using various mathematical approaches, including Ekeland’s variational principle, sufficient and necessary near-optimality conditions are established without requiring the second-order adjoint equations. Lastly, the validity of the theoretical analysis is demonstrated using a biological mathematical model. The simulation and modeling approaches outlined in this study provide valuable insights for policymakers to design effective control strategies aimed at reducing influenza prevalence.</p>

Decoding influenza virus heterogeneity and seasonal perturbations through integrated Poisson random measure and near-optimal control analysis involving Markovian switching

期刊论文
发表日期：

2026年2月10日

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Ayesha Siddiqa,
Ilyas Ali,
Rehana Ashraf,
Saima Rashid,
Fathea M. O. Birkea

摘要

Prominent airborne particles that can influence the spread of influenza include nitrogen dioxide ( $NO_{2}$ $N O_{2}$ ), sulfur dioxide ( $SO_{2}$ $S O_{2}$ ) and ozone $(O_{3}).$ $(O_{3}) .$ This research introduces a novel non-autonomous stochastic differential equation (SDE) framework that considers the impacts of immunization and seasonal variations while incorporating age heterogeneity. Initially, the global positive solution and the periodic solution of the stochastic model are determined, and the basic reproduction number for the corresponding impulsive system is identified. Furthermore, it is demonstrated that the system is uniformly persistent. To analyze the stochastic influenza model, one of the techniques employed is the application of Poisson random measure noise, which facilitates a broader comprehension of the mechanisms of flu propagation and optimal control mechanisms without requiring verbose second-order adjoint equations. A threshold parameter, $\mathbb {R}_{0\textrm{k}}^{s}$ $R_{0 k}^{s}$ , is introduced to examine infection dynamics. Specifically, if $\mathbb {R}_{0\textrm{k}}^{s}<1$ $R_{0 k}^{s} < 1$ , the virus will die out; whereas if $\mathbb {R}_{0\textrm{k}}^{s}>1$ $R_{0 k}^{s} > 1$ , the virus will persist. Influenza statistics from Quebec Province are analyzed using the Monte-Carlo-Markov-Chain (MCMC) technique to estimate the most suitable parameter values. Our estimated basic reproduction number for Quebec, $\mathbb {R}_{0\textrm{k}}=0.5645$ $R_{0 k} = 0.5645$ (95% C-I: 0.3261, 0.7239), indicates that the influenza outbreak is likely to decline. However, our data also suggest that the WHO’s 2040 target is unlikely to be met, as projections show the annual incidence of new influenza cases will reach 1,061 (95% C-I: 136, 7,040) by that time. In light of this, we evaluate the feasibility of initiating adult immunization programs and reducing clinical testing delays starting in 2025 to align with WHO’s objectives. According to our findings, WHO’s 2040 benchmark can be achieved either by vaccinating 10% of the adult population or by reducing testing delays from four months to three weeks. These measures would prevent 74,127 (95% C-I: 24,700, 237,090) and 56,812 (95% C-I: 15,723, 212,476) influenza cases, respectively, between 2024 and 2040. Additionally, under some mild conditions regarding a convex control set, the stochastic process and its associated adjoint formulations are derived. Using various mathematical approaches, including Ekeland’s variational principle, sufficient and necessary near-optimality conditions are established without requiring the second-order adjoint equations. Lastly, the validity of the theoretical analysis is demonstrated using a biological mathematical model. The simulation and modeling approaches outlined in this study provide valuable insights for policymakers to design effective control strategies aimed at reducing influenza prevalence.