This paper deals with a problem which describes tuberculosis granuloma formation <Equation ID="Equ21"> <EquationSource Format="TEX">$\begin{aligned} {\left\{ \begin{array}{ll} u_t = \Delta u - \nabla \cdot (u \nabla v) - uv - u + \beta , & x \in \Omega ,\ t>0, \\ v_t = \Delta v + v -uv + \mu w, & x \in \Omega ,\ t>0, \\ w_t = \Delta w + uv - wz - w, & x \in \Omega ,\ t>0, \\ z_t = \Delta z - \nabla \cdot (z \nabla w) + f(w)z -z, & x \in \Omega ,\ t>0 \end{array}\right. } \end{aligned}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mfenced open="{"> <mrow> <mtable> <mtr> <mtd columnalign="left"> <mrow> <msub> <mi>u</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>u</mi> <mo>-</mo> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mi mathvariant="normal">∇</mi> <mi>v</mi> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>u</mi> <mi>v</mi> <mo>-</mo> <mi>u</mi> <mo>+</mo> <mi>β</mi> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="4pt" /> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi>v</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>v</mi> <mo>+</mo> <mi>v</mi> <mo>-</mo> <mi>u</mi> <mi>v</mi> <mo>+</mo> <mi>μ</mi> <mi>w</mi> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="4pt" /> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi>w</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>w</mi> <mo>+</mo> <mi>u</mi> <mi>v</mi> <mo>-</mo> <mi>w</mi> <mi>z</mi> <mo>-</mo> <mi>w</mi> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="4pt" /> <mi>t</mi> <mo>></mo> <mn>0</mn> <mo>,</mo> </mrow> </mtd> </mtr> <mtr> <mtd columnalign="left"> <mrow> <mrow /> <msub> <mi>z</mi> <mi>t</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mi>z</mi> <mo>-</mo> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mrow> <mo stretchy="false">(</mo> <mi>z</mi> <mi mathvariant="normal">∇</mi> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mo stretchy="false">)</mo> </mrow> <mi>z</mi> <mo>-</mo> <mi>z</mi> <mo>,</mo> </mrow> </mtd> <mtd columnalign="left"> <mrow> <mi>x</mi> <mo>∈</mo> <mi mathvariant="normal">Ω</mi> <mo>,</mo> <mspace width="4pt" /> <mi>t</mi> <mo>></mo> <mn>0</mn> </mrow> </mtd> </mtr> </mtable> </mrow> </mfenced> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>under homogeneous Neumann boundary conditions and initial conditions, where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\Omega \subset \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">Ω</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n\ge 2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>) is a smooth bounded domain, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\beta ,\mu >0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>β</mi> <mo>,</mo> <mi>μ</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and f is some function, and shows that if the reproduction number <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$R_0:= \frac{\mu \beta + 1}{\beta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo>:</mo> <mo>=</mo> <mfrac> <mrow> <mi>μ</mi> <mi>β</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>β</mi> </mfrac> </mrow> </math></EquationSource> </InlineEquation> satisfies <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$R_0<1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>R</mi> <mn>0</mn> </msub> <mo><</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and initial data are small in some sense, then the solution (u, v, w, z) of the problem exists globally and converges to <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(\beta ,0,0,0)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>β</mi> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo>,</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> exponentially.

Boundedness and asymptotic stability in a model for tuberculosis granuloma formation

期刊论文
发表日期：

2026年3月21日

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Masaaki Mizukami,
Yuya Tanaka

摘要

This paper deals with a problem which describes tuberculosis granuloma formation $\begin{aligned} {\left\{ \begin{array}{ll} u_t = \Delta u - \nabla \cdot (u \nabla v) - uv - u + \beta , & x \in \Omega ,\ t>0, \\ v_t = \Delta v + v -uv + \mu w, & x \in \Omega ,\ t>0, \\ w_t = \Delta w + uv - wz - w, & x \in \Omega ,\ t>0, \\ z_t = \Delta z - \nabla \cdot (z \nabla w) + f(w)z -z, & x \in \Omega ,\ t>0 \end{array}\right. } \end{aligned}$ $\begin{matrix} \{\begin{matrix} u_{t} = Δ u - \nabla \cdot (u \nabla v) - u v - u + β, & x \in Ω, t > 0, \\ v_{t} = Δ v + v - u v + μ w, & x \in Ω, t > 0, \\ w_{t} = Δ w + u v - w z - w, & x \in Ω, t > 0, \\ z_{t} = Δ z - \nabla \cdot (z \nabla w) + f (w) z - z, & x \in Ω, t > 0 \end{matrix}) \end{matrix}$ under homogeneous Neumann boundary conditions and initial conditions, where $\Omega \subset \mathbb {R}^n$ $Ω \subset R^{n}$ ( $n\ge 2$ $n \geq 2$ ) is a smooth bounded domain, $\beta ,\mu >0$ $β, μ > 0$ and f is some function, and shows that if the reproduction number $R_0:= \frac{\mu \beta + 1}{\beta }$ $R_{0} : = \frac{μ β + 1}{β}$ satisfies $R_0<1$ $R_{0} < 1$ and initial data are small in some sense, then the solution (u, v, w, z) of the problem exists globally and converges to $(\beta ,0,0,0)$ $(β, 0, 0, 0)$ exponentially.