https://orcid.org/0009-0009-1140-0046
Figures
Abstract
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 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
. We identify critical hysteresis effects governed by vaccine efficacy (fV) and bacterial shedding (
,
), 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.
Author summary
Cholera continues to cause significant illness and death in regions with limited access to clean water and healthcare. While mathematical models help us understand how cholera spreads, most fail to capture the full complexity involving both person-to-person contact and bacteria in water sources. Here we develop a new model that includes vaccination, asymptomatic carriers who can silently spread the disease, treatment failure, and bacterial growth in the environment. Our analysis reveals three important findings for public health: First, simply reducing the reproduction number below 1 may not eliminate cholera due to a phenomenon called backward bifurcation-stronger interventions are needed. Second, person-to-person transmission creates more unpredictable outbreaks than environmental transmission, making short-term forecasting challenging. Third, our sensitivity analysis shows that vaccination and sanitation efforts must be more aggressive than previously thought to achieve elimination. These insights suggest that effective cholera control requires integrated strategies targeting both direct contact and water sanitation simultaneously. Our MATLAB simulation toolkit can help health agencies adapt these findings to local conditions for better outbreak planning and resource allocation.
Citation: Welu HT, Kefela YY, Atsbaha HA, Asgedom AA (2026) Bifurcation, sensitivity, and noise: Stochastic dynamics of cholera with vaccination and sanitation controls. PLOS Complex Syst 3(4): e0000099. https://doi.org/10.1371/journal.pcsy.0000099
Editor: Joshua Kiddy K. Asamoah, Kwame Nkrumah University of Science and Technology, GHANA
Received: August 3, 2025; Accepted: March 5, 2026; Published: April 2, 2026
Copyright: © 2026 Welu et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: The data supporting the findings of this study are derived from publicly available secondary sources. All datasets utilized in this manuscript have been properly cited in the references section, and their original sources are accessible. Data extraction and processing methods are detailed in the Methods section to enable replication. Where applicable, ethical reuse permissions for restricted-access datasets were obtained and are documented.
Funding: This research was fully funded by Mekelle University, Ethiopia through a PhD scholarship awarded to Hailu Tkue Welu (H.T.W.). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
1. Introduction
Cholera remains a persistent public health threat, particularly in regions with limited access to clean water, sanitation, and healthcare. Despite global efforts, recent estimates reveal substantial disease burdens in endemic countries [1], with complex transmission mechanisms spanning both environmental and person-to-person pathways [2–4]. The bacterium Vibrio cholerae, responsible for cholera, resides in aquatic environments and can persist asymptomatically, complicating containment and prediction strategies [2,5,6]. Furthermore, climate variability, such as El Niño events, has been linked to periodic surges in cholera incidence [7–9], highlighting the importance of environmental determinants in outbreak dynamics.
Early mathematical models, such as SIR and SIRS frameworks, provided foundational insights into cholera transmission dynamics but oversimplified real-world conditions by omitting crucial features like bacterial reservoirs, asymptomatic carriers, and reinfection [3,10,11]. More advanced models have gradually incorporated food and waterborne transmission (e.g., SIBR), temporary immunity (SIBRS), and treatment compartments (SIRTS), enhancing biological realism [3,12,13]. However, these models typically focus on either deterministic or stochastic representations, without unifying both frameworks under a comprehensive structure that captures both mean behavior and outbreak variability [14–16].
To address this gap, we propose and investigate an enriched cholera transmission model that explicitly includes vaccination, asymptomatic carriers, reinfection, treatment, and environmental bacterial dynamics. Our model extends the deterministic framework to incorporate stochasticity, capturing random fluctuations in transmission rates, treatment availability, and environmental noise. Moreover, the model undergoes detailed bifurcation analysis, allowing us to identify critical thresholds where the system exhibits dramatic shifts in behavior, such as forward and backward bifurcations-phenomena that traditional basic reproduction number analysis often fails to fully explain [14,17–19].
This study contributes to the literature in three significant ways. First, we rigorously analyze the model’s qualitative dynamics using next-generation matrix approaches [19,20], identifying equilibrium points and deriving conditions for local and global stability. Second, we perform bifurcation analysis with respect to key transmission parameters, revealing the presence of backward bifurcation in human-to-human transmission and forward bifurcation in environmental pathways. This finding has critical policy implications, as it indicates that reducing the basic reproduction number below one may not be sufficient to eradicate cholera [18,21,22]. Finally, we implement numerical simulations and sensitivity analysis using MATLAB, along with stochastic realizations, to compare deterministic predictions against realistic epidemic variability. Our stochastic results confirm that person-to-person transmission routes introduce higher volatility, while environmental transmission paths exhibit greater predictability-echoing recent findings in climate-sensitive cholera dynamics [8,14].
The integration of bifurcation theory [18,21], stochastic simulation [15,16,23], and detailed epidemiological compartments positions our model as a novel and comprehensive framework for understanding and controlling cholera outbreaks in dynamic, resource-limited settings, building upon the rich tradition of mathematical epidemiology [17,24,25].
2. Model formulation
2.1. Model description and assumptions
The model incorporates five key epidemiological and ecological assumptions. First, the population is homogeneous and well-mixed, with no age or spatial structure, simplifying contact rates to mass-action kinetics. Second, vaccinated individuals experience waning immunity at rate , but a fraction fV retain protection due to booster effects, aligning with clinical observations of oral cholera vaccines [26,27]. Third, asymptomatic carriers (A) shed bacteria at reduced rates (
) compared to symptomatic individuals (I), reflecting lower fecal pathogen loads. Fourth, the bacterial population B follows logistic growth with carrying capacity K, capturing nutrient-limited proliferation in aquatic reservoirs. Fifth, treatment failure (fT) and natural recovery (
,
) coexist, accounting for variability in healthcare access and antibiotic efficacy. The parameter fV represents the vaccine efficacy fraction, where vaccinated individuals with waning immunity (
) have probability fV of retaining protection, while the remainder
return to susceptibility.
Based on the compartmental structure and transmission pathways illustrated in Fig 1, the deterministic SVAITRS-B cholera model is governed by the following system of nonlinear ordinary differential equations:
The framework integrates human compartments (S,V,A,I,T,R) with environmental bacteria (B) and dual transmission routes (,
).
2.3. Stochastic SVAITRS-B cholera model
Remark 2.1 (Parameter Scaling and Units). The transmission parameters (environmental) and shedding rates
are scaled to maintain biological realism. While
reflects low per-bacterium infection probability, the shedding rates
represent high bacterial output per infected individual. This scaling ensures numerical stability while preserving the correct order-of-magnitude relationships observed in empirical cholera studies [2,10] (Table 1).
3. Model analysis
3.1. Positivity and boundedness
Theorem 3.1 (Positivity). The solutions of system (1) with non-negative initial conditions remain non-negative for all t > 0.
Proof. We employ the method of integrating factors to establish positivity. Consider the susceptible compartment S(t):
From system (1), S(t) satisfies:
where . The integrating factor is
, yielding:
For the bacterial compartment B(t), we have:
The homogeneous solution , and the particular solution is non-negative due to the non-homogeneous terms. Thus
.
For the infected compartments, consider A(t):
which implies . Similar differential inequalities hold for I(t), T(t), V(t), and R(t), establishing non-negativity throughout.□
Theorem 3.2 (Boundedness). All solutions of system (1) are uniformly bounded in the invariant region:
where .
Proof. For the total human population N = S + V + A + I + T + R, we have:
Separating variables and integrating:
Solving yields:
For bacterial boundedness, from the B-equation:
Since , we have:
At equilibrium, , giving the positive root:
Thus B(t) is uniformly bounded for all .□
3.2. Existence of Equilibrium Points
Theorem 3.3 (Disease-Free Equilibrium (DFE)). The system (1) has a unique disease-free equilibrium , where:
Proof. Set A = I = T = R = B = 0 in (1). The resulting equations reduce to:
Solving this linear system yields S0 and V0 as above. The denominator is positive since for biologically realistic parameters.□
Theorem 3.4 (Endemic Equilibrium (EE)). If , the system admits at least one endemic equilibrium
with all infected compartments positive.
Proof. Let A, I, T, B > 0. From the steady-state conditions:
- Solve
from V′ = 0:
- Express
from R′ = 0:
- Linearize the B equation about
:
- Substitute into remaining equations to obtain a nonlinear system in
. By the Brouwer fixed-point theorem, a solution exists when
.□
Remark 3.5. The endemic equilibrium’s explicit form is algebraically complex but can be computed numerically. Uniqueness follows if the cross-immunity and bacterial shedding terms satisfy monotonicity conditions.
3.3. Deterministic basic reproduction number 
Theorem 3.6 (Deterministic Basic Reproduction Number). The deterministic basic reproduction number for system (1) is given by [19,20,30,31]:
where the human-to-human and environmental transmission components are:
with the rate parameters: ,
,
,
, and
.
Proof. Using the Next Generation Matrix approach [17] with infected states , we define:
The next generation matrix has spectral radius:
where the explicit components are as given. The additive form follows from the block structure of K and the Perron-Frobenius theorem, representing parallel transmission pathways.□
Remark 3.7. Therefore, the complete basic reproduction number captures all transmission routes through asymptomatic carriers, symptomatic infected, treated individuals, and environmental contamination.
Remark 3.8. The environmental parameter represents the net bacterial growth rate at the disease-free equilibrium. For biological realism and to ensure
, we assume
throughout our analysis, indicating that bacterial growth exceeds natural and sanitation-induced death in the absence of disease-induced shedding.
3.4. Stochastic analysis and extinction thresholds
Theorem 3.9 (Stochastic Extinction via Lyapunov Spectrum). For the stochastic system (2), the disease dies out almost surely if the maximal Lyapunov exponent of the linearized infected subsystem satisfies
.
Proof. Consider the linearized infection dynamics near the disease-free equilibrium. Let denote the infected states. The stochastic differential system takes the form:
where J22 is the Jacobian submatrix governing infected compartments and are the noise intensity matrices.
The top Lyapunov exponent is defined by the asymptotic growth rate:
By the multiplicative ergodic theorem [16], the disease becomes extinct almost surely if , providing a stochastic stability threshold that generalizes the deterministic condition
.□
Theorem 3.10 (Monte Carlo Invasion Probability). The probability of a major outbreak can be empirically determined via Monte Carlo simulation as:
where is the extinction time and
is the simulation horizon.
Proof. Let X(i)(t) denote the i-th sample path generated by the Euler-Maruyama scheme with step size . Define the outbreak indicator function for each realization:
where is a small persistence threshold (typically
individual). The empirical invasion probability is:
Since {I(i)} are independent Bernoulli random variables with success probability Pinvasion, the strong law of large numbers guarantees:
For N = 104 realizations, the estimator converges with high precision, and the 95% confidence interval is .□
Theorem 3.11 (Noise-Induced Stability Transitions). Environmental noise can stabilize the disease-free equilibrium even when . Specifically, there exists a critical noise intensity
such that for
,
despite
.
Proof. Consider the linearized infected subsystem . The moment Lyapunov exponent yields
. Through second-order perturbation analysis,
, where
when
. Defining
, the critical threshold
emerges from
. By continuity and the limits
and
, there exists
such that
, proving noise-induced stabilization (Table 2).□
Remark 3.12. The Lyapunov exponent criterion defines the true stochastic elimination boundary, which may differ significantly from the deterministic threshold
. Our numerical results demonstrate that environmental noise (
) can suppress outbreaks even in supercritical regimes (
), highlighting the importance of stochastic analysis for accurate public health predictions.
3.5. Stability analysis
Theorem 3.13 (Local Stability of DFE). The DFE is:
- Locally asymptotically stable if
,
- Unstable if
.
Proof. The Jacobian matrix at the DFE is block-triangular:
where: - J11 corresponds to (S, V, R) with eigenvalues ,
,
, all negative. - J22 governs the infected subsystem (A, I, T, B):
The eigenvalues of J22 determine stability. By the Routh-Hurwitz criterion, all eigenvalues have negative real parts iff .□
Theorem 3.14 (Global Stability of DFE). If , the DFE is globally asymptotically stable in the feasible region
.
Proof. We construct a Lyapunov function to prove global stability of the disease-free equilibrium. Consider the candidate Lyapunov function:
where the positive constants c1 and c2 are chosen as:
with and
.
Differentiating L along solutions of system (1) and using the inequalities for all
:
For , we have
with equality only at the DFE
. By LaSalle’s invariance principle, the DFE is globally asymptotically stable in
.□
Theorem 3.15 (Local Stability of EE). For , the endemic equilibrium
is locally asymptotically stable if:
where ai are coefficients of the characteristic polynomial of .
Proof. Linearize (1) around . The Jacobian
satisfies:
By the Routh-Hurwitz conditions, all eigenvalues have negative real parts if:
These hold when the transmission terms are sufficiently small relative to recovery rates
.□
3.6. Numerical Bifurcation Results
Numerical continuation methods are applied to map equilibrium stability and bifurcation structures across parameter space. The analysis focuses on transitions between disease-free and endemic states as functions of ,
, and
. We identify forward and backward bifurcation regimes dictated by waning immunity rate
. These numerical findings quantify the parameter ranges where bistability occurs, informing targeted intervention strategies.
The comparison in Fig 2 shows two distinct epidemiological scenarios governed by waning immunity rate. When immunity loss is slow (, left), forward bifurcation occurs: the disease-free equilibrium (DFE) remains stable for
and transitions smoothly to a stable endemic equilibrium (EE) as
exceeds 1, making
sufficient for elimination. When immunity loss is fast (
, right), backward bifurcation emerges: a subcritical branch creates bistability where both DFE and EE coexist for
, allowing cholera persistence below the classical threshold and requiring
to be pushed significantly below 1 for eradication.
Fig 3 illustrates the global stability of the endemic equilibrium through both geometric and temporal perspectives. Panel (A) displays a phase portrait in the (I,B) plane, where trajectories from varied initial conditions-spanning low to very high initial pathogen concentrations (B0)-converge to a single equilibrium point , confirming asymptotic stability. Panel (B) shows corresponding time series for infected population I(t) and pathogen concentration B(t), demonstrating that despite different starting points, both variables approach steady-state values within 600 days. This consistent convergence regardless of initial magnitude reinforces that the endemic equilibrium acts as a global attractor when
, validating the model’s long-term predictive reliability under endemic conditions.
Fig 4 shows the two-parameter stability diagram in the space, delineating regions of disease-free equilibrium (DFE), endemic equilibrium (EE), and bistability. The green bistable region corresponds to
, where backward bifurcation allows cholera to persist despite
. This region is bounded by the elimination threshold (
, orange line) and the classical epidemiological threshold (
, violet dashed line), illustrating the narrow parameter combinations in human (
) and environmental (
) transmission rates that permit both disease elimination and endemic coexistence.
4. Bifurcation analysis
4.1. Analytical bifurcation analysis
Theorem 4.1 (Type of Bifurcation at ). The dynamical system exhibits a backward bifurcation at
.
Proof. We apply the center manifold theory [21]. Let be the bifurcation parameter. The critical value
, where
, is obtained by solving:
At , the Jacobian has a simple zero eigenvalue. Let w and v be the right and left eigenvectors corresponding to this eigenvalue, respectively, with
.
A non-zero right eigenvector w is given by:
A corresponding left eigenvector v (with v1 = 1) is given by:
The bifurcation coefficients are:
The coefficient b is positive:
The coefficient a simplifies to:
Under the biologically reasonable condition that nonlinear forcing terms dominate stabilizing terms, we have a > 0. Since b > 0 and a > 0, the system undergoes a backward bifurcation at [21].□
Remark 4.2 (Explicit Condition for Backward Bifurcation). The backward bifurcation occurs when:
This condition quantifies when asymptomatic transmission and progression to symptoms outweigh direct symptomatic transmission, creating the bistability region.
4.2 Forward bifurcation in environmental transmission
Theorem 4.3 (Forward Bifurcation for Environmental Transmission). The system exhibits a forward (transcritical) bifurcation at when varying the environmental transmission rate
.
Proof. We employ the same center manifold framework used for , now treating
as the bifurcation parameter. Let
be the critical value satisfying
.
The Jacobian structure remains identical to Theorem 4.1, but the bifurcation coefficients now reflect the environmental transmission pathway:
The critical coefficient a demonstrates the fundamental difference between transmission pathways:
For environmental transmission, the stabilizing terms dominate the nonlinear forcing terms, yielding a < 0. Since b > 0 and a < 0, the system undergoes a forward bifurcation at [21].□
Remark 4.4. The contrasting bifurcation behaviors for (backward) and
(forward) indicate fundamental differences in transmission pathway dynamics. Human-to-human transmission generates complex elimination barriers through reinfection and asymptomatic carriage, while environmental transmission follows predictable threshold dynamics. This mathematical distinction underscores the need for pathway-specific intervention strategies.
4.3. Sensitivity analysis
To quantify parameter influence on disease dynamics, we conduct comprehensive local and global sensitivity analyses. The normalized forward sensitivity index measures the percentage change in per 1% parameter increase:
For endemic prevalence , sensitivity indices are computed numerically via Latin Hypercube Sampling with Partial Rank Correlation Coefficient analysis across 104 parameter combinations.
Global sensitivity analysis employs variance-based Sobol’ indices to quantify parameter importance across the entire feasible space:
where Y denotes model outputs (peak infection, endemic level) and indices capture main (Si) and total effects (STi) including interactions Table 3.
4.3.1. Policy implications from sensitivity analysis.
The sensitivity analysis shows a clear intervention hierarchy: environmental transmission () demonstrates the highest
sensitivity (+0.72), establishing water sanitation as the most efficient control strategy, while human-to-human transmission (
) dominates endemic prevalence sensitivity (+0.82), emphasizing hygiene education and contact reduction for outbreak containment. Critically, bacterial shedding rates (
,
) emerge as pivotal factors, suggesting that asymptomatic carrier identification could substantially reduce environmental contamination and break persistent transmission cycles (Table 4).
Fig 5 compares parameter sensitivities for the basic reproduction number (A) and endemic prevalence
(B). Direct transmission rate
and environmental transmission rate
exhibit the strongest positive influence on both metrics, highlighting dual-pathway transmission as a critical amplification mechanism. Recovery rate
and sanitation-related parameters (
,
) show substantial negative sensitivity, confirming that treatment and environmental hygiene are potent control levers. Interestingly, some parameters (e.g., waning immunity rate
) affect
minimally but influence endemic prevalence significantly, indicating context-dependent intervention priorities. These rankings guide resource allocation by identifying which parameters, when modified, yield the greatest reduction in disease burden.
As shown in Fig 6 global sensitivity analysis, human transmission () contributes most to model uncertainty, followed by environmental transmission (
) and asymptomatic shedding (
). The substantial gaps between first-order and total-effect indices indicate strong parameter interactions, particularly for
and
, where combined effects exceed individual impacts. This highlights the importance of considering parameter interdependencies in cholera intervention planning.
4.3.2. Intervention strategy.
The analysis supports a tiered approach where sanitation and water treatment target the environmentally-driven component, case management provides rapid prevalence reduction, vaccination builds population resilience, and environmental controls ensure long-term stability. This integrated strategy exploits the mathematical decomposition
by simultaneously addressing both transmission pathways for optimal outbreak control.
5. Numerical simulation
Numerical simulations validate our analytical findings and quantify outbreak dynamics under realistic conditions [32]. Using MATLAB for time-domain analysis and MatCont for bifurcation tracking, we examine both deterministic trajectories and stochastic realizations. The simulations confirm backward bifurcation behavior and quantify noise-induced variability across transmission pathways. Our computational framework provides actionable insights for public health intervention planning under uncertainty.
Fig 7 compares outbreak predictions from deterministic and stochastic models through time-course trajectories. Panel (A) shows human case dynamics where stochastic realizations (red) demonstrate 34% variability around the median compared to the single deterministic curve (blue), with a 1.7-day delay in peak timing. Panel (B) illustrates environmental pathogen concentrations with smaller variability (24.1%) in stochastic trajectories (green) versus deterministic prediction (black). Panel (C) quantifies transmission pathway uncertainty, revealing person-to-person spread (red) exhibits 2.3 times more unpredictability (42.3%) than environmental transmission (green, 18.7%). Panel (D) displays cumulative burden, where stochastic projections (orange) show 6% higher total cases than deterministic estimates (purple) with substantial outcome variability. The curve-based visualization highlights how stochastic modeling captures outbreak randomness that deterministic approaches overlook, particularly in human transmission dynamics and timing uncertainty.
Fig 8 presents cholera transmission dynamics through threshold analysis and temporal convergence. The left panel shows scaling with
, where the
threshold (black dashed line) separates elimination (green zone) from endemic (red zone) regimes. The right panel displays outbreak trajectories converging to equilibrium, with color progression from green (low transmission) to red (high transmission) indicating increasing convergence times from 34 to 74 days. These dynamics reveal that higher transmission intensities not only produce larger endemic equilibria but also slower system relaxation, requiring extended intervention periods. The predictable relationship between transmission rate and convergence time informs optimal timing for control measures in different epidemiological settings.
6. Conclusion
This study presents a comprehensive analysis of cholera transmission dynamics through the SVAITRS-B model, integrating deterministic and stochastic approaches with vaccination, asymptomatic carriage, and environmental transmission. Our findings reveal fundamental insights into cholera persistence and control.
Fig 8 demonstrates that cholera elimination depends critically on crossing transmission thresholds rather than merely reducing the reproduction number. The backward bifurcation creates a bistable regime where disease persists even when , necessitating more aggressive interventions than conventional wisdom suggests. This mathematical phenomenon explains cholera’s stubborn endemicity in resource-limited settings where partial control measures fail to push transmission below the subcritical threshold.
Our bifurcation analysis quantifies the intervention gap: achieving elimination requires 66.7% greater sanitation effort than maintaining control. This hysteresis effect, visualized through phase portraits showing global convergence to endemic equilibria from diverse initial conditions, underscores the need for sustained, intensive interventions rather than temporary measures.
The comparative stochastic-deterministic analysis reveals that human-to-human transmission introduces 2.3 times more unpredictability than environmental spread, with case variability reaching 42% compared to 19% for bacterial concentrations. This explains why outbreak timing and magnitude remain difficult to predict despite understanding transmission mechanisms.
Sensitivity analysis identifies dual leverage points: reducing direct transmission through hygiene education and vaccination while simultaneously controlling environmental reservoirs through sanitation. The logarithmic relationship between bacterial levels and sanitation effort suggests conventional targets may underestimate required efforts by 15–20%. The accompanying computational toolkit provides public health planners with practical resources for scenario analysis, intervention planning, and outbreak forecasting. By integrating threshold analysis, convergence dynamics, and uncertainty quantification, this framework offers a comprehensive approach to cholera control that respects both mathematical rigor and practical implementation constraints.
Future research should focus on validating spatially heterogeneous parameters, incorporating climate variability effects on bacterial survival, and extending the framework to include antimicrobial resistance dynamics. These refinements will enhance the model’s utility for local public health decision-making while preserving its fundamental insights into cholera transmission and control.
Acknowledgments
This research was supported by Mekelle University through a PhD scholarship awarded to H.T.W. The authors gratefully acknowledge this institutional support.
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