<p>We study a stochastic tumor–immune interaction model with a saturating response. The stochastic perturbation is driven by one Brownian motion acting through the tumor–immune interaction term, so random fluctuations modify effective immune contact rather than the two populations independently. For positive initial data, the system admits a unique global positive solution. If immune loss dominates baseline immune proliferation, <InlineEquation ID="IEq1"><EquationSource Format="TEX">\(\delta>r_2\)</EquationSource></InlineEquation>, then the total population is ultimately bounded in first moment. Tumor extinction requires sustained immune pressure. If <InlineEquation ID="IEq1000"><EquationSource Format="TEX">\({{\Phi \left( {T\left( t \right),I\left( t \right)} \right)} \mathord{\left/ {\vphantom {{\Phi \left( {T\left( t \right),I\left( t \right)} \right)} {T(t)}}} \right. \kern-\nulldelimiterspace} {T(t)}} \ge q> 0\)</EquationSource></InlineEquation> for all sufficiently large <InlineEquation ID="IEq2"><EquationSource Format="TEX">\(t\)</EquationSource></InlineEquation> almost surely, and if <InlineEquation ID="IEq3"><EquationSource Format="TEX">\(r_1<\beta _1q\)</EquationSource></InlineEquation>, then <InlineEquation ID="IEq4"><EquationSource Format="TEX">\(T(t)\rightarrow 0\)</EquationSource></InlineEquation> almost surely and in mean. Persistence is governed by the saturated upper bound <InlineEquation ID="IEq4001"><EquationSource Format="TEX">\({{\Phi \left( {T,I} \right)} \mathord{\left/ {\vphantom {{\Phi \left( {T,I} \right)} T}} \right. \kern-\nulldelimiterspace} T} \le {1 \mathord{\left/ {\vphantom {1 {\alpha _{2} }}} \right. \kern-\nulldelimiterspace} {\alpha _{2} }}.\)</EquationSource></InlineEquation>The resulting threshold is stochastic: weak persistence in mean follows when <InlineEquation ID="IEq4002"><EquationSource Format="TEX">\(r_{1}> {{\beta _{1} } \mathord{\left/ {\vphantom {{\beta _{1} } {\alpha _{2} }}} \right. \kern-\nulldelimiterspace} {\alpha _{2} }} + {{\sigma ^{2} } \mathord{\left/ {\vphantom {{\sigma ^{2} } {2\alpha _{2}^{2} }}} \right. \kern-\nulldelimiterspace} {2\alpha _{2}^{2} }}.\)</EquationSource></InlineEquation>The term <InlineEquation ID="IEq5"><EquationSource Format="TEX">\(\sigma ^2/(2\alpha _2^2)\)</EquationSource></InlineEquation> is the Itô correction produced by interaction noise, and the deterministic saturated threshold is recovered when <InlineEquation ID="IEq6"><EquationSource Format="TEX">\(\sigma =0\)</EquationSource></InlineEquation>. Krylov–Bogoliubov averaging gives existence of invariant probability measures on the closed state space under a Feller assumption. Uniqueness, ergodicity, and exponential convergence remain open for the present rank-one degenerate diffusion. Numerical experiments with dimensional parameters from tumor–immune calibrations support the persistence threshold, saturation sensitivity, bounded empirical occupation measures, time-average consistency, and step-size reliability.</p>

Stochastic tumor immune dynamics with saturating response

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2026年6月26日

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Javed Hussain,
Tareq Saeed,
Lianlian Zhou,
Ali Jan Sohu

摘要

We study a stochastic tumor–immune interaction model with a saturating response. The stochastic perturbation is driven by one Brownian motion acting through the tumor–immune interaction term, so random fluctuations modify effective immune contact rather than the two populations independently. For positive initial data, the system admits a unique global positive solution. If immune loss dominates baseline immune proliferation, \(\delta>r_2\), then the total population is ultimately bounded in first moment. Tumor extinction requires sustained immune pressure. If \({{\Phi \left( {T\left( t \right),I\left( t \right)} \right)} \mathord{\left/ {\vphantom {{\Phi \left( {T\left( t \right),I\left( t \right)} \right)} {T(t)}}} \right. \kern-\nulldelimiterspace} {T(t)}} \ge q> 0\) for all sufficiently large \(t\) almost surely, and if \(r_1<\beta _1q\), then \(T(t)\rightarrow 0\) almost surely and in mean. Persistence is governed by the saturated upper bound \({{\Phi \left( {T,I} \right)} \mathord{\left/ {\vphantom {{\Phi \left( {T,I} \right)} T}} \right. \kern-\nulldelimiterspace} T} \le {1 \mathord{\left/ {\vphantom {1 {\alpha _{2} }}} \right. \kern-\nulldelimiterspace} {\alpha _{2} }}.\)The resulting threshold is stochastic: weak persistence in mean follows when \(r_{1}> {{\beta _{1} } \mathord{\left/ {\vphantom {{\beta _{1} } {\alpha _{2} }}} \right. \kern-\nulldelimiterspace} {\alpha _{2} }} + {{\sigma ^{2} } \mathord{\left/ {\vphantom {{\sigma ^{2} } {2\alpha _{2}^{2} }}} \right. \kern-\nulldelimiterspace} {2\alpha _{2}^{2} }}.\)The term \(\sigma ^2/(2\alpha _2^2)\) is the Itô correction produced by interaction noise, and the deterministic saturated threshold is recovered when \(\sigma =0\). Krylov–Bogoliubov averaging gives existence of invariant probability measures on the closed state space under a Feller assumption. Uniqueness, ergodicity, and exponential convergence remain open for the present rank-one degenerate diffusion. Numerical experiments with dimensional parameters from tumor–immune calibrations support the persistence threshold, saturation sensitivity, bounded empirical occupation measures, time-average consistency, and step-size reliability.