<p>In this paper, we investigate the emergence of backward bifurcation in a treatment-dependent cholera transmission model incorporating time delays. We develop a delay differential equation framework that captures key aspects of cholera dynamics, including delayed treatment responses, degradation of water quality (modeled through bacterial concentration dynamics with natural decay), and nonlinear infection processes. A thorough mathematical analysis is conducted to derive the basic reproduction number and explore the local and global stability of the equilibrium points. Special emphasis is placed on identifying parameter regimes that give rise to backward bifurcation, revealing the possibility of multiple endemic equilibria even when the basic reproduction number is below unity. In addition, we incorporate optimal control strategies by introducing a time-dependent control variable <i>u</i>(<i>t</i>), which represents the intensity of treatment interventions. The role of <i>u</i>(<i>t</i>) in mitigating disease transmission and improving public health outcomes is rigorously analyzed through Pontryagin’s Maximum Principle. Numerical simulations highlight the impact of treatment efficacy and intervention delays on epidemic trajectories. The findings underscore the critical importance of timely and sustained control efforts in preventing severe cholera outbreaks and reducing the burden of disease within affected populations.</p>

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Modeling backward bifurcation cholera diseases with time delays: insights into treatment impact

  • Samia Bushnaq,
  • Anwar Zeb,
  • Ayesha Bashir,
  • Hijaz Ahmad,
  • Shumaila Irum,
  • Taha Radwan

摘要

In this paper, we investigate the emergence of backward bifurcation in a treatment-dependent cholera transmission model incorporating time delays. We develop a delay differential equation framework that captures key aspects of cholera dynamics, including delayed treatment responses, degradation of water quality (modeled through bacterial concentration dynamics with natural decay), and nonlinear infection processes. A thorough mathematical analysis is conducted to derive the basic reproduction number and explore the local and global stability of the equilibrium points. Special emphasis is placed on identifying parameter regimes that give rise to backward bifurcation, revealing the possibility of multiple endemic equilibria even when the basic reproduction number is below unity. In addition, we incorporate optimal control strategies by introducing a time-dependent control variable u(t), which represents the intensity of treatment interventions. The role of u(t) in mitigating disease transmission and improving public health outcomes is rigorously analyzed through Pontryagin’s Maximum Principle. Numerical simulations highlight the impact of treatment efficacy and intervention delays on epidemic trajectories. The findings underscore the critical importance of timely and sustained control efforts in preventing severe cholera outbreaks and reducing the burden of disease within affected populations.