Bifurcation, sensitivity, and noise: Stochastic dynamics of cholera with vaccination and sanitation controls

This study develops the novel SVAITRS-B model, unifying deterministic and stochastic frameworks to capture cholera dynamics with vaccination, asymptomatic carriers, and environmental pathways. We demonstrate analytically that person-to-person transmission exhibits backward bifurcation, while environmental transmission follows classical forward bifurcation-establishing distinct elimination thresholds that explain disease persistence even when the basic reproduction number <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="M1"> <mml:mrow> <mml:msubsup> <mml:mi>ℛ</mml:mi> <mml:mn>0</mml:mn> <mml:mi>D</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> falls below one. Stochastic simulations reveal that human-contact transmission generates 30% greater outbreak variability than environmental routes, highlighting its role in unpredictable epidemics. Environmental transmission, however, dominates long-term endemicity, contributing 68% to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="M2"> <mml:mrow> <mml:msubsup> <mml:mi>ℛ</mml:mi> <mml:mn>0</mml:mn> <mml:mi>D</mml:mi> </mml:msubsup> </mml:mrow> </mml:math> . We identify critical hysteresis effects governed by vaccine efficacy ( f V ) and bacterial shedding ( <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="M3"> <mml:mrow> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mi>A</mml:mi> </mml:msub> </mml:mrow> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="M4"> <mml:mrow> <mml:msub> <mml:mi>ξ</mml:mi> <mml:mi>I</mml:mi> </mml:msub> </mml:mrow> </mml:math> ), and uncover a logarithmic sensitivity of bacterial concentration to sanitation-indicating that standard intervention targets may underestimate effort by 15–20%. These results provide a mathematical foundation for dual-pathway control strategies, combining human-focused interventions with environmental management. Our accompanying computational toolkit enables scenario testing for public health planning, though field validation of spatial heterogeneity remains essential for localized application.