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<p>Protection measures has a crucial role in containing epidemics and limiting the spatial outbreak of epidemics. In this work, we investigate a diffusive SPIR epidemic model incorporating permanent protection and a Holling type&#xa0;II incidence function. Our results reveal how protective measures and spatial dispersion jointly influence disease dynamics and propagation. For a bounded spatial domain, the system exhibits two equilibria: a disease-free state and an endemic state. We show that the basic reproduction number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathcal{R}_0\)</EquationSource> </InlineEquation> sharply determines which equilibrium is globally stable, highlighting the critical role of protection in preventing outbreaks. In the unbounded spatial setting, we study traveling wave solutions describing the spread of infection across space. We proved the existence of a minimal wave speed <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(c^*\)</EquationSource> </InlineEquation>, which demonstrates that protection can reduce the invasion front, effectively reducing the speed at which the epidemic spreads. Finally, numerical simulations illustrate these phenomena, which proves clear insights into how protection strategies and dispersion will shape the global behavior and spatial propagation of epidemics. These findings underscore the importance of integrating behavioral and immunological protection into diffusive epidemic models for better containment and prediction of disease outbreak.</p>

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Effect of protection measure on the dynamics and traveling wave solutions of a diffusive SPIR model

  • 期刊论文

  • 发表日期：

    2026年2月16日

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  • Soumia Bessafi,
  • Amine Loumi,
  • Abdelheq Mezouaghi

摘要

Protection measures has a crucial role in containing epidemics and limiting the spatial outbreak of epidemics. In this work, we investigate a diffusive SPIR epidemic model incorporating permanent protection and a Holling type II incidence function. Our results reveal how protective measures and spatial dispersion jointly influence disease dynamics and propagation. For a bounded spatial domain, the system exhibits two equilibria: a disease-free state and an endemic state. We show that the basic reproduction number \(\mathcal{R}_0\) sharply determines which equilibrium is globally stable, highlighting the critical role of protection in preventing outbreaks. In the unbounded spatial setting, we study traveling wave solutions describing the spread of infection across space. We proved the existence of a minimal wave speed \(c^*\) , which demonstrates that protection can reduce the invasion front, effectively reducing the speed at which the epidemic spreads. Finally, numerical simulations illustrate these phenomena, which proves clear insights into how protection strategies and dispersion will shape the global behavior and spatial propagation of epidemics. These findings underscore the importance of integrating behavioral and immunological protection into diffusive epidemic models for better containment and prediction of disease outbreak.