Influenza remains a major global health threat, requiring models that capture both deterministic transmission and stochastic environmental effects. This study develops a seven-compartment stochastic SECIHRV dynamical system incorporating susceptible, exposed, chronic, infectious, hospitalized, recovered, and vaccinated populations. The deterministic framework defines the basic reproduction number <InlineEquation ID="IEq1"> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{R}_{0}$</EquationSource> </InlineEquation>, while the stochastic extension introduces multiplicative perturbations through independent Wiener processes <InlineEquation ID="IEq2"> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mn>1</mn> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$B_{1}(t)$</EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mn>2</mn> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$B_{2}(t)$</EquationSource> </InlineEquation>, and <InlineEquation ID="IEq4"> <EquationSource Format="MATHML"><math> <msub> <mi>B</mi> <mn>3</mn> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$B_{3}(t)$</EquationSource> </InlineEquation>, representing random fluctuations in transmission, vaccination, and progression rates with intensities <InlineEquation ID="IEq5"> <EquationSource Format="MATHML"><math> <mo stretchy="false">(</mo> <msub> <mi>ξ</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>ξ</mi> <mn>2</mn> </msub> <mo>,</mo> <msub> <mi>ξ</mi> <mn>3</mn> </msub> <mo stretchy="false">)</mo> </math></EquationSource> <EquationSource Format="TEX">$(\xi _{1},\xi _{2},\xi _{3})$</EquationSource> </InlineEquation>.Analytical results show that if <InlineEquation ID="IEq6"> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mi>s</mi> </msub> <mo><</mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{R}_{s}<1$</EquationSource> </InlineEquation>, infection dies out almost surely, whereas <InlineEquation ID="IEq7"> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mi>s</mi> </msub> <mo>></mo> <mn>1</mn> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{R}_{s}>1$</EquationSource> </InlineEquation> yields a unique stationary distribution with ergodic behavior. Numerical simulations based on a modified Euler Maruyama scheme confirm these findings and reveal the transition from deterministic predictability to stochastic variability. A three-dimensional sensitivity analysis of <InlineEquation ID="IEq8"> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">R</mi> <mi>s</mi> </msub> </math></EquationSource> <EquationSource Format="TEX">$\mathcal{R}_{s}$</EquationSource> </InlineEquation> further demonstrates that higher stochastic intensities flatten and suppress the reproduction surface, highlighting the stabilizing influence of environmental noise on epidemic persistence and control.The primary contribution of this work is the formulation and comprehensive analysis of a high-dimensional stochastic epidemic model with simultaneous noise in multiple epidemiological parameters, a detailed 3D sensitivity analysis of the stochastic reproduction number, and the demonstration of how environmental variability can suppress or sustain infection under different noise regimes.

A stochastic SECIHRV dynamical system and 3D sensitivity analysis of influenza transmission under environmental uncertainty

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2026年3月31日

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Shah Hussain,
Thoraya N. Alharthi

摘要

Influenza remains a major global health threat, requiring models that capture both deterministic transmission and stochastic environmental effects. This study develops a seven-compartment stochastic SECIHRV dynamical system incorporating susceptible, exposed, chronic, infectious, hospitalized, recovered, and vaccinated populations. The deterministic framework defines the basic reproduction number $R_{0}$ $\mathcal{R}_{0}$ , while the stochastic extension introduces multiplicative perturbations through independent Wiener processes $B_{1} (t)$ $B_{1}(t)$ , $B_{2} (t)$ $B_{2}(t)$ , and $B_{3} (t)$ $B_{3}(t)$ , representing random fluctuations in transmission, vaccination, and progression rates with intensities $(ξ_{1}, ξ_{2}, ξ_{3})$ $(\xi _{1},\xi _{2},\xi _{3})$ .

Analytical results show that if $R_{s} < 1$ $\mathcal{R}_{s}<1$ , infection dies out almost surely, whereas $R_{s} > 1$ $\mathcal{R}_{s}>1$ yields a unique stationary distribution with ergodic behavior. Numerical simulations based on a modified Euler Maruyama scheme confirm these findings and reveal the transition from deterministic predictability to stochastic variability. A three-dimensional sensitivity analysis of $R_{s}$ $\mathcal{R}_{s}$ further demonstrates that higher stochastic intensities flatten and suppress the reproduction surface, highlighting the stabilizing influence of environmental noise on epidemic persistence and control.

The primary contribution of this work is the formulation and comprehensive analysis of a high-dimensional stochastic epidemic model with simultaneous noise in multiple epidemiological parameters, a detailed 3D sensitivity analysis of the stochastic reproduction number, and the demonstration of how environmental variability can suppress or sustain infection under different noise regimes.