<p>A 4D system of stochastic differential equations is presented and analyzed with applications to mathematical biology and virus dynamics (called stochastic SEIR(S) models). Random fluctuations on contact, transition, and recovery rates in the deterministic SEIR(S) model with disease deaths are taken into account in a non-parametric manner. We obtain stochastic counterparts with general diffusion coefficients (i.e., functional contact, transition, and recovery rates) of the form <Equation ID="Equ23"> <EquationSource Format="TEX">$\begin{aligned} \displaystyle dS\!\!=\!\! & \!\Big (\!\!-\!\beta S I \!\!+\!\! \mu (K\!-\!S) \!\!+\!\! \alpha I \!+\! \zeta R\!\Big ) dt \!-\! \sigma _1 S I \!\cdot \! F_1\big (S,\!E,I,\!R\big ) dW_1 \!+\! \sigma _4 R \!\cdot \! F_4(S,\!E,I,\!R) dW_4\\ dE\!\!=\!\! & \!\Big (\!\beta S I - (\mu \!+\!\eta ) E\Big ) dt + \sigma _1 S I \!\cdot \! F_1\big (S,E,I,R\big ) dW_1 -\sigma _2 E \!\cdot \! F_2\big (S,E,I,R\big ) dW_2\\ dI\!\!=\!\! & \!\Big (\!\eta E -\big (\alpha \!+\!\gamma \!+\!\mu \big )I\Big ) dt + \sigma _2 E \!\cdot \! F_2\big (S,E,I,R\big ) dW_2 - \sigma _3 I \!\cdot \! F_3\big (S,E,I,R\big ) dW_3\\ dR\!=\! & \!\Big (\!\gamma I \!-\! (\mu \!+\!\zeta ) R\Big ) dt + \sigma _3 I \!\cdot \! F_3\big (S,E,I,R\big ) dW_3 - \sigma _4 R \!\cdot \! F_4\big (S,\!E,I,\!R\big ) dW_4 \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mi>d</mi> <mi>S</mi> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> </mrow> </mstyle> </mtd> <mtd columnalign="left"> <mrow> <mspace width="-0.166667em" /> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> <mo>-</mo> <mspace width="-0.166667em" /> <mi>β</mi> <mi>S</mi> <mi>I</mi> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> <mo>+</mo> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>K</mi> <mspace width="-0.166667em" /> <mo>-</mo> <mspace width="-0.166667em" /> <mi>S</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> <mo>+</mo> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> <mi>α</mi> <mi>I</mi> <mspace width="-0.166667em" /> <mo>+</mo> <mspace width="-0.166667em" /> <mi>ζ</mi> <mi>R</mi> <mspace width="-0.166667em" /> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mspace width="-0.166667em" /> <mo>-</mo> <mspace width="-0.166667em" /> <msub> <mi>σ</mi> <mn>1</mn> </msub> <mi>S</mi> <mi>I</mi> <mspace width="-0.166667em" /> <mo>·</mo> <mspace width="-0.166667em" /> <msub> <mi>F</mi> <mn>1</mn> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>S</mi> <mo>,</mo> <mspace width="-0.166667em" /> <mi>E</mi> <mo>,</mo> <mi>I</mi> <mo>,</mo> <mspace width="-0.166667em" /> <mi>R</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>d</mi> <msub> <mi>W</mi> <mn>1</mn> </msub> <mspace width="-0.166667em" /> <mo>+</mo> <mspace width="-0.166667em" /> <msub> <mi>σ</mi> <mn>4</mn> </msub> <mi>R</mi> <mspace width="-0.166667em" /> <mo>·</mo> <mspace width="-0.166667em" /> <msub> <mi>F</mi> <mn>4</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mspace width="-0.166667em" /> <mi>E</mi> <mo>,</mo> <mi>I</mi> <mo>,</mo> <mspace width="-0.166667em" /> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> <mi>d</mi> <msub> <mi>W</mi> <mn>4</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow> <mrow /> <mi>d</mi> <mi>E</mi> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="-0.166667em" /> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mspace width="-0.166667em" /> <mi>β</mi> <mi>S</mi> <mi>I</mi> <mo>-</mo> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mspace width="-0.166667em" /> <mo>+</mo> <mspace width="-0.166667em" /> <mi>η</mi> <mo stretchy="false">)</mo> </mrow> <mi>E</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>+</mo> <msub> <mi>σ</mi> <mn>1</mn> </msub> <mi>S</mi> <mi>I</mi> <mspace width="-0.166667em" /> <mo>·</mo> <mspace width="-0.166667em" /> <msub> <mi>F</mi> <mn>1</mn> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>S</mi> <mo>,</mo> <mi>E</mi> <mo>,</mo> <mi>I</mi> <mo>,</mo> <mi>R</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>d</mi> <msub> <mi>W</mi> <mn>1</mn> </msub> <mo>-</mo> <msub> <mi>σ</mi> <mn>2</mn> </msub> <mi>E</mi> <mspace width="-0.166667em" /> <mo>·</mo> <mspace width="-0.166667em" /> <msub> <mi>F</mi> <mn>2</mn> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>S</mi> <mo>,</mo> <mi>E</mi> <mo>,</mo> <mi>I</mi> <mo>,</mo> <mi>R</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>d</mi> <msub> <mi>W</mi> <mn>2</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow> <mrow /> <mi>d</mi> <mi>I</mi> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> <mspace width="-0.166667em" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="-0.166667em" /> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mspace width="-0.166667em" /> <mi>η</mi> <mi>E</mi> <mo>-</mo> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>α</mi> <mspace width="-0.166667em" /> <mo>+</mo> <mspace width="-0.166667em" /> <mi>γ</mi> <mspace width="-0.166667em" /> <mo>+</mo> <mspace width="-0.166667em" /> <mi>μ</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>I</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>+</mo> <msub> <mi>σ</mi> <mn>2</mn> </msub> <mi>E</mi> <mspace width="-0.166667em" /> <mo>·</mo> <mspace width="-0.166667em" /> <msub> <mi>F</mi> <mn>2</mn> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>S</mi> <mo>,</mo> <mi>E</mi> <mo>,</mo> <mi>I</mi> <mo>,</mo> <mi>R</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>d</mi> <msub> <mi>W</mi> <mn>2</mn> </msub> <mo>-</mo> <msub> <mi>σ</mi> <mn>3</mn> </msub> <mi>I</mi> <mspace width="-0.166667em" /> <mo>·</mo> <mspace width="-0.166667em" /> <msub> <mi>F</mi> <mn>3</mn> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>S</mi> <mo>,</mo> <mi>E</mi> <mo>,</mo> <mi>I</mi> <mo>,</mo> <mi>R</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>d</mi> <msub> <mi>W</mi> <mn>3</mn> </msub> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="right"> <mrow> <mrow /> <mi>d</mi> <mi>R</mi> <mspace width="-0.166667em" /> <mo>=</mo> <mspace width="-0.166667em" /> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mspace width="-0.166667em" /> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">(</mo> </mrow> <mspace width="-0.166667em" /> <mi>γ</mi> <mi>I</mi> <mspace width="-0.166667em" /> <mo>-</mo> <mspace width="-0.166667em" /> <mrow> <mo stretchy="false">(</mo> <mi>μ</mi> <mspace width="-0.166667em" /> <mo>+</mo> <mspace width="-0.166667em" /> <mi>ζ</mi> <mo stretchy="false">)</mo> </mrow> <mi>R</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">)</mo> </mrow> <mi>d</mi> <mi>t</mi> <mo>+</mo> <msub> <mi>σ</mi> <mn>3</mn> </msub> <mi>I</mi> <mspace width="-0.166667em" /> <mo>·</mo> <mspace width="-0.166667em" /> <msub> <mi>F</mi> <mn>3</mn> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>S</mi> <mo>,</mo> <mi>E</mi> <mo>,</mo> <mi>I</mi> <mo>,</mo> <mi>R</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>d</mi> <msub> <mi>W</mi> <mn>3</mn> </msub> <mo>-</mo> <msub> <mi>σ</mi> <mn>4</mn> </msub> <mi>R</mi> <mspace width="-0.166667em" /> <mo>·</mo> <mspace width="-0.166667em" /> <msub> <mi>F</mi> <mn>4</mn> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>S</mi> <mo>,</mo> <mspace width="-0.166667em" /> <mi>E</mi> <mo>,</mo> <mi>I</mi> <mo>,</mo> <mspace width="-0.166667em" /> <mi>R</mi> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mi>d</mi> <msub> <mi>W</mi> <mn>4</mn> </msub> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>with intensity (variance) parameters <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\sigma _i\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>σ</mi> <mi>i</mi> </msub> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and positive constants <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\alpha ,\beta ,\eta ,\gamma ,\zeta ,\mu \ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> <mo>,</mo> <mi>η</mi> <mo>,</mo> <mi>γ</mi> <mo>,</mo> <mi>ζ</mi> <mo>,</mo> <mi>μ</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. The introduced stochastic model has functional diffusion coefficients which involve arbitrary local Lipschitz-continuous functions <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$F_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>’s defined on 4D prism <Equation ID="Equ24"> <EquationSource Format="TEX">$\mathbb {D}=\Big \{(S,E,I,R) \in \mathbb {R}^4_+: ~0< S,E,I,R< K, ~S+E+I+R< K\Big \}.$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mi mathvariant="double-struck">D</mi> <mo>=</mo> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">{</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>S</mi> <mo>,</mo> <mi>E</mi> <mo>,</mo> <mi>I</mi> <mo>,</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mrow> <mo>∈</mo> <msubsup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mo>+</mo> <mn>4</mn> </msubsup> <mo>:</mo> <mspace width="3.33333pt" /> <mn>0</mn> <mo><</mo> <mi>S</mi> <mo>,</mo> <mi>E</mi> <mo>,</mo> <mi>I</mi> <mo>,</mo> <mi>R</mi> <mo><</mo> <mi>K</mi> <mo>,</mo> <mspace width="3.33333pt" /> <mi>S</mi> <mo>+</mo> <mi>E</mi> <mo>+</mo> <mi>I</mi> <mo>+</mo> <mi>R</mi> <mo><</mo> <mi>K</mi> <mrow> <mo maxsize="1.623em" minsize="1.623em" stretchy="true">}</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </Equation>We prove the global existence of a bounded, unique strong solution, discuss asymptotic stochastic and moment stability of disease-free and endemic equilibria, and visualize our results with some simulations. Under appropriate conditions on <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$F_i$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>F</mi> <mi>i</mi> </msub> </math></EquationSource> </InlineEquation>’s, all the stochastic dynamics takes place inside a compact, nonempty domain <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\overline{\mathbb {D}} \subset \mathbb {R}^4$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mover> <mi mathvariant="double-struck">D</mi> <mo>¯</mo> </mover> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>4</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> with triangular-shaped boundary surfaces with natural saturation parameter <i>K</i> for the maximum population within the positive cone of <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {R}^4$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mn>4</mn> </msup> </math></EquationSource> </InlineEquation>.</p>

EXISTENCE, UNIQUENESS, BOUNDEDNESS, AND ASYMPTOTIC STABILITY OF STOCHASTIC SEIR(S) MODEL WITH VARIABLE DIFFUSION RATES AND RANDOM TRANSITIONS

期刊论文
发表日期：

2025年3月21日

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Henri Schurz,
Shanika Chandrasena,
Taniya Chandrasena

摘要

A 4D system of stochastic differential equations is presented and analyzed with applications to mathematical biology and virus dynamics (called stochastic SEIR(S) models). Random fluctuations on contact, transition, and recovery rates in the deterministic SEIR(S) model with disease deaths are taken into account in a non-parametric manner. We obtain stochastic counterparts with general diffusion coefficients (i.e., functional contact, transition, and recovery rates) of the form $\begin{aligned} \displaystyle dS\!\!=\!\! & \!\Big (\!\!-\!\beta S I \!\!+\!\! \mu (K\!-\!S) \!\!+\!\! \alpha I \!+\! \zeta R\!\Big ) dt \!-\! \sigma _1 S I \!\cdot \! F_1\big (S,\!E,I,\!R\big ) dW_1 \!+\! \sigma _4 R \!\cdot \! F_4(S,\!E,I,\!R) dW_4\\ dE\!\!=\!\! & \!\Big (\!\beta S I - (\mu \!+\!\eta ) E\Big ) dt + \sigma _1 S I \!\cdot \! F_1\big (S,E,I,R\big ) dW_1 -\sigma _2 E \!\cdot \! F_2\big (S,E,I,R\big ) dW_2\\ dI\!\!=\!\! & \!\Big (\!\eta E -\big (\alpha \!+\!\gamma \!+\!\mu \big )I\Big ) dt + \sigma _2 E \!\cdot \! F_2\big (S,E,I,R\big ) dW_2 - \sigma _3 I \!\cdot \! F_3\big (S,E,I,R\big ) dW_3\\ dR\!=\! & \!\Big (\!\gamma I \!-\! (\mu \!+\!\zeta ) R\Big ) dt + \sigma _3 I \!\cdot \! F_3\big (S,E,I,R\big ) dW_3 - \sigma _4 R \!\cdot \! F_4\big (S,\!E,I,\!R\big ) dW_4 \end{aligned}$ $\begin{matrix} d S = & (- β S I + μ (K - S) + α I + ζ R) d t - σ_{1} S I \cdot F_{1} (S, E, I, R) d W_{1} + σ_{4} R \cdot F_{4} (S, E, I, R) d W_{4} \\ d E = & (β S I - (μ + η) E) d t + σ_{1} S I \cdot F_{1} (S, E, I, R) d W_{1} - σ_{2} E \cdot F_{2} (S, E, I, R) d W_{2} \\ d I = & (η E - (α + γ + μ) I) d t + σ_{2} E \cdot F_{2} (S, E, I, R) d W_{2} - σ_{3} I \cdot F_{3} (S, E, I, R) d W_{3} \\ d R = & (γ I - (μ + ζ) R) d t + σ_{3} I \cdot F_{3} (S, E, I, R) d W_{3} - σ_{4} R \cdot F_{4} (S, E, I, R) d W_{4} \end{matrix}$ with intensity (variance) parameters $\sigma _i\ge 0$ $σ_{i} \geq 0$ and positive constants $\alpha ,\beta ,\eta ,\gamma ,\zeta ,\mu \ge 0$ $α, β, η, γ, ζ, μ \geq 0$ . The introduced stochastic model has functional diffusion coefficients which involve arbitrary local Lipschitz-continuous functions $F_i$ $F_{i}$ ’s defined on 4D prism $\mathbb {D}=\Big \{(S,E,I,R) \in \mathbb {R}^4_+: ~0< S,E,I,R< K, ~S+E+I+R< K\Big \}.$ $D = {(S, E, I, R) \in R_{+}^{4} : 0 < S, E, I, R < K, S + E + I + R < K} .$ We prove the global existence of a bounded, unique strong solution, discuss asymptotic stochastic and moment stability of disease-free and endemic equilibria, and visualize our results with some simulations. Under appropriate conditions on $F_i$ $F_{i}$ ’s, all the stochastic dynamics takes place inside a compact, nonempty domain $\overline{\mathbb {D}} \subset \mathbb {R}^4$ $\bar{D} \subset R^{4}$ with triangular-shaped boundary surfaces with natural saturation parameter K for the maximum population within the positive cone of $\mathbb {R}^4$ $R^{4}$ .