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<p>This paper presents a delayed SIR model with inhibition effects, suitable for studying epidemics in which immunity is temporary. The framework integrates temporary immunity, vaccination with imperfect efficacy and a general treatment function that accounts for limited medical resources. We derive the basic reproduction number, <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal{R}_0\)</EquationSource> </InlineEquation>, and show that the disease-free equilibrium is locally stable when <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal{R}_0 &lt; 1\)</EquationSource> </InlineEquation> while an endemic equilibrium exists for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal{R}_0 &gt; 1\)</EquationSource> </InlineEquation>. Our analysis demonstrates that varying the delay parameter <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\tau\)</EquationSource> </InlineEquation> can induce a Hopf bifurcation, leading to the emergence of periodic orbits. Furthermore, an optimal control system is designed and analyzed to determine the most effective treatment rate. The sensitivity index of <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\mathcal{R}_0\)</EquationSource> </InlineEquation> to key parameters is calculated, and the results are validated through numerical simulations.</p>

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Delayed SIR model with inhibition effect, temporary immunity effects, and optimal control of treatment

  • 期刊论文

  • 发表日期：

    2026年1月12日

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  • Mohammad Ali Shakeri Barzoki,
  • Rasool Kazemi,
  • Rasoul Asheghi

摘要

This paper presents a delayed SIR model with inhibition effects, suitable for studying epidemics in which immunity is temporary. The framework integrates temporary immunity, vaccination with imperfect efficacy and a general treatment function that accounts for limited medical resources. We derive the basic reproduction number, \(\mathcal{R}_0\) , and show that the disease-free equilibrium is locally stable when \(\mathcal{R}_0 < 1\) while an endemic equilibrium exists for \(\mathcal{R}_0 > 1\) . Our analysis demonstrates that varying the delay parameter \(\tau\) can induce a Hopf bifurcation, leading to the emergence of periodic orbits. Furthermore, an optimal control system is designed and analyzed to determine the most effective treatment rate. The sensitivity index of \(\mathcal{R}_0\) to key parameters is calculated, and the results are validated through numerical simulations.