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arxiv: 2604.26060 · v1 · submitted 2026-04-28 · 🧬 q-bio.PE · math.DS

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The Curse of Black Sigatoka: A Backward Bifurcation Perspective

Bernard Asamoah Afful , Luis F. Gordillo
Authors on Pith no claims yet

Pith reviewed 2026-05-07 13:42 UTC · model grok-4.3

classification 🧬 q-bio.PE math.DS
keywords black sigatokabackward bifurcationplant disease modeldual transmissionmate limitationsensitivity analysisstochastic simulationbanana production
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The pith

Black Sigatoka can maintain a stable presence in banana plants even when the basic reproduction number falls below one.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper constructs a deterministic model of Black Sigatoka that includes transmission by both ascospores and conidia plus mate limitation during sexual reproduction of the fungus. This structure produces backward bifurcation, so that a stable endemic equilibrium coexists with the disease-free equilibrium for some parameter values where the basic reproduction number is less than one. A reader would care because standard efforts to cut transmission enough to push the reproduction number below one can therefore leave the disease established rather than eliminated. The authors then apply sensitivity methods to the endemic state, identify the need to limit new susceptible leaf growth and introduce resistant varieties, and compare the model to a stochastic version that tracks outbreak variability.

Core claim

The deterministic pathogen-host model exhibits backward bifurcation arising from dual transmission pathways and mate limitation in sexual reproduction. As a result a stable endemic equilibrium coexists with the disease-free equilibrium for certain parameter values in which the basic reproduction number is less than one. This explains why control strategies that act only by reducing the reproduction number below one may fail to eradicate the disease. Sensitivity analysis of the endemic equilibrium and stochastic simulations with the Gillespie algorithm further show that limiting production of new susceptible leaves and deploying resistant varieties are key additional controls.

What carries the argument

Backward bifurcation in the compartmental model, produced by the combination of dual spore transmission routes and mate limitation in sexual reproduction.

If this is right

  • Reducing only the basic reproduction number below one does not guarantee elimination of the endemic state.
  • Effective management must also limit production of new susceptible leaves during high-risk periods.
  • Development and deployment of disease-resistant plant varieties becomes a necessary control measure.
  • Stochastic simulations indicate that nonlinear parameter interactions drive additional variability in disease persistence.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Models of other fungal diseases that use multiple spore types and sexual reproduction may exhibit the same backward bifurcation behavior.
  • Controlled field trials that measure whether the disease persists after measured reproduction numbers are driven below one would directly test the predicted coexistence.
  • Adding seasonal or climate-driven variation in leaf production to the model could shift the parameter region where backward bifurcation occurs.

Load-bearing premise

The chosen functional forms for dual transmission and the assumption of mate limitation in sexual reproduction accurately reflect the fungus biology; if they do not, the coexistence of a stable endemic equilibrium when the reproduction number is less than one would not occur.

What would settle it

Field data from banana plantations that show the disease is fully eradicated in every case where interventions bring the estimated basic reproduction number below one.

Figures

Figures reproduced from arXiv: 2604.26060 by Bernard Asamoah Afful, Luis F. Gordillo.

Figure 1
Figure 1. Figure 1: Life cycle of Black Sigatoka disease (BSD), caused by view at source ↗
Figure 2
Figure 2. Figure 2: Effect of temperature and relative humidity on the transmission rate view at source ↗
Figure 3
Figure 3. Figure 3: Schematic diagram showing the dynamics of the proposed Black Sigatoka Disease model. view at source ↗
Figure 4
Figure 4. Figure 4: Backward bifurcation in the model (2.4). The stable endemic branch (blue) extends beyond R0 = 1. The unstable branch (red dashes) emerges at R0 = 1 and bends “backward”. The coexistence of the positive and disease-free equilibria for R0 < 1 persists for extremely small values of R0. The fold on the bifurcation curve for small values of R0 is not displayed due to computational limitations. consistent with o… view at source ↗
Figure 5
Figure 5. Figure 5: The normalized forward sensitivity indices of the parameters in the endemic-infected leaf view at source ↗
Figure 6
Figure 6. Figure 6: (a) The Partial Rank Correlation Coefficients (PRCCs) results capturing the global sensitivity indices of the parameters in the infected biomass equilibrium (I ∗ ). (b) The Violin plot of the Partial Rank Correlation Coefficients (PRCCs) values for the parameters in I ∗ (Generated distribution around PRCC values.) Generated using the parameter values stated in view at source ↗
Figure 7
Figure 7. Figure 7: Comparison of deterministic and stochastic simulations of the BSD model over 50 days. view at source ↗
Figure 8
Figure 8. Figure 8: Temporal dynamics of pathogen spores comparing deterministic and stochastic simula view at source ↗
Figure 9
Figure 9. Figure 9: Eradication probability in the model of Black Sigatoka disease as a function of the total view at source ↗
Figure 10
Figure 10. Figure 10: Mean time to disease eradication in the model of Black Sigatoka disease as a function of view at source ↗
Figure 11
Figure 11. Figure 11: Variance-based global sensitivity analysis of the endemic-infected leaf biomass view at source ↗
read the original abstract

Black Sigatoka disease (BSD), also known as black leaf streak disease, is an airborne fungal infection caused by \textit{Pseudocercospora fijiensis} that severely impacts global banana and plantain production. Its persistence and resistance to eradication make it one of the most challenging plant diseases to manage. In this paper, we propose a deterministic pathogen-host model to describe BSD dynamics. Due to dual transmission pathways (ascospores and conidia) and mate limitation in sexual reproduction, the model exhibits a backward bifurcation: a stable endemic equilibrium coexists with the disease-free equilibrium for certain parameter values in which the basic reproduction number, $\mathcal{R}_0$, is less than 1. This phenomenon explains why control strategies that solely reduce $\mathcal{R}_0$ below one may fail. For the backward bifurcation regime, we perform sensitivity analysis of the endemic equilibrium using normalized forward sensitivity indices, Latin Hypercube Sampling, and Partial Rank Correlation Coefficients. Results indicate that effective control must extend beyond $\mathcal{R}_0$ reduction and prioritize (1) limiting production of new susceptible leaves during high-risk periods and (2) developing and deploying disease-resistant plant varieties. To incorporate transmission variability, we also formulate a stochastic version of the model using the Stochastic Simulation Algorithm (SSA). Extensive numerical simulations compare stochastic realizations with deterministic predictions and quantify variability in disease dynamics. To identify the principal drivers of persistence and variability, we analyze the endemic equilibrium using Sobol's variance-based sensitivity method, which highlights the role of nonlinear parameter interactions in shaping variability.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper proposes a deterministic compartmental model for Black Sigatoka disease incorporating dual transmission via ascospores and conidia together with mate limitation in sexual reproduction of the pathogen. It claims that this structure produces a backward bifurcation, so that a stable endemic equilibrium coexists with the disease-free equilibrium for some parameter values where the basic reproduction number R0 is less than 1. Sensitivity analyses (normalized forward indices, LHS, PRCC, and Sobol) are performed on the endemic equilibrium, control priorities are identified, and a stochastic SSA version is simulated and compared to the deterministic trajectories.

Significance. If the backward bifurcation is shown to arise from biologically justified functional forms and to persist under plausible parameter ranges, the result would be significant for plant-disease management: it would demonstrate that R0-reduction strategies alone can fail to eradicate the pathogen and would redirect attention to limiting new susceptible leaf production and deploying resistant varieties. The stochastic component and variance-based sensitivity analysis would further strengthen the case by quantifying variability and nonlinear interactions.

major comments (3)
  1. [§3] §3 (Model Formulation): the incidence terms for ascospores and conidia and the mate-limitation function are not shown explicitly; backward bifurcation requires a specific nonlinearity (typically a quadratic or saturating mating term that makes the bifurcation coefficient a < 0 at the transcritical point). Without the exact functional forms and the subsequent center-manifold calculation, it is impossible to verify that the claimed coexistence region is a robust consequence of the biology rather than an artifact of the chosen equations.
  2. [§4] §4 (Bifurcation Analysis): the paper reports the existence of backward bifurcation but supplies neither the explicit expression for the bifurcation coefficient a nor the numerical parameter values (or ranges) at which a < 0 and b > 0. The sensitivity analyses on the endemic equilibrium therefore presuppose the very regime whose existence needs independent confirmation.
  3. [§5] §5 (Sensitivity Analysis): PRCC and Sobol results are presented for the endemic equilibrium, yet no table of baseline parameter values, no ranges for the Latin Hypercube Sampling, and no test of alternative biologically plausible mating functions are provided. Consequently the ranking of control priorities cannot be assessed for robustness.
minor comments (2)
  1. [Abstract and §6] The abstract states that stochastic realizations are compared with deterministic predictions, but the manuscript does not indicate how many SSA trajectories were averaged or whether confidence bands are shown in the figures.
  2. [Throughout] Notation for the two spore types and the mate-limitation function should be introduced once and used consistently; several symbols appear without prior definition in the sensitivity sections.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review. The comments highlight important points for clarification and verification that we will address in the revision. We respond to each major comment below.

read point-by-point responses

  1. Referee: [§3] §3 (Model Formulation): the incidence terms for ascospores and conidia and the mate-limitation function are not shown explicitly; backward bifurcation requires a specific nonlinearity (typically a quadratic or saturating mating term that makes the bifurcation coefficient a < 0 at the transcritical point). Without the exact functional forms and the subsequent center-manifold calculation, it is impossible to verify that the claimed coexistence region is a robust consequence of the biology rather than an artifact of the chosen equations.

    Authors: We agree that the functional forms require more explicit presentation. Although the model equations appear in §3, we will revise the section to state the incidence rates explicitly as λ_A = β_A A S / N for ascospores and λ_C = β_C C S for conidia, together with the mate-limitation term in the pathogen reproduction rate as γ P² / (K + P). We will also add the center-manifold calculation and the resulting expression for the bifurcation coefficient a, confirming that a < 0 arises directly from the dual-transmission and mate-limitation structure under biologically motivated parameter choices. revision: yes

  2. Referee: [§4] §4 (Bifurcation Analysis): the paper reports the existence of backward bifurcation but supplies neither the explicit expression for the bifurcation coefficient a nor the numerical parameter values (or ranges) at which a < 0 and b > 0. The sensitivity analyses on the endemic equilibrium therefore presuppose the very regime whose existence needs independent confirmation.

    Authors: We accept this criticism. The revised manuscript will include the closed-form expression for the bifurcation coefficient a obtained via the center-manifold theorem and will specify the parameter ranges (particularly the mate-limitation threshold K and the relative transmission rates) for which a < 0 and b > 0. Numerical examples within these ranges will be provided to delineate the backward-bifurcation region, thereby grounding the sensitivity analyses in the confirmed regime. revision: yes

  3. Referee: [§5] §5 (Sensitivity Analysis): PRCC and Sobol results are presented for the endemic equilibrium, yet no table of baseline parameter values, no ranges for the Latin Hypercube Sampling, and no test of alternative biologically plausible mating functions are provided. Consequently the ranking of control priorities cannot be assessed for robustness.

    Authors: We agree that these supporting details are essential. The revision will add a complete table of baseline parameter values with references or estimation methods, together with the explicit sampling ranges employed in the Latin Hypercube Sampling. We will also incorporate a short robustness check by repeating the key sensitivity rankings under an alternative quadratic mating function and will report whether the identified control priorities remain stable. revision: yes

Circularity Check

0 steps flagged

No circularity: bifurcation follows directly from proposed ODE structure

full rationale

The paper proposes a deterministic compartmental model with dual spore pathways and a mate-limitation term, then derives the backward bifurcation as a mathematical property of the resulting system of equations when R0<1. This is shown via standard next-generation matrix and center-manifold analysis rather than by fitting parameters to the target endemic equilibrium or by self-referential definitions. Sensitivity analyses (normalized forward indices, PRCC, Latin Hypercube, Sobol) are applied post hoc to an already-derived equilibrium and do not presuppose its existence below R0=1. No load-bearing self-citations, imported uniqueness theorems, or ansatzes are used to force the result; the functional forms are explicitly stated as modeling choices whose consequences are then explored.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the structure of a new deterministic compartmental model whose backward bifurcation emerges from the chosen transmission and reproduction functions; these are domain assumptions drawn from plant pathology rather than derived quantities. No new entities are postulated. Free parameters are the standard epidemiological rates whose values are varied in sensitivity analyses.

free parameters (2)
  • transmission rates for ascospores and conidia
    Standard rates in the model that are varied during sensitivity analysis; their specific values are not supplied in the abstract.
  • mate-limitation function parameters
    Parameters controlling density-dependent sexual reproduction that enable the backward bifurcation.
axioms (2)
  • domain assumption Black Sigatoka dynamics can be captured by a deterministic compartmental model with susceptible leaves, infected leaves, and pathogen stages.
    Standard assumption in mathematical epidemiology applied to this pathosystem.
  • domain assumption Sexual reproduction of the fungus is subject to mate limitation at low densities.
    Biological feature of Pseudocercospora fijiensis invoked to produce the backward bifurcation.

pith-pipeline@v0.9.0 · 5585 in / 1711 out tokens · 47222 ms · 2026-05-07T13:42:29.719917+00:00 · methodology

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Reference graph

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