What if a disease can keep spreading even after its reproduction number falls below one? This paper argues that Black Sigatoka can do exactly that, because nonlinear spore production creates a hidden persistence threshold. Black Sigatoka disease, also known as black leaf streak disease, is an airborne fungal disease caused by Pseudocercospora fijiensis.

It is endemic in most banana- and plantain-producing regions worldwide and ranks among the most severe threats to global banana production. In susceptible cultivars, the pathogen causes extensive leaf necrosis and premature defoliation before fruit bunches mature, with yield losses frequently exceeding fifty percent without control measures.

Despite intensive management efforts, Black Sigatoka remains exceptionally difficult to eradicate or suppress effectively. The pathogen produces abundant ascospores that support long-distance wind dispersal and genetic recombination, while resistance to multiple fungicide classes has emerged after repeated applications.

The fungus is also adapted to warm, humid tropical environments that promote short disease cycles and frequent reinfection, making containment or complete eradication nearly impossible in most production systems. Frequent fungicide sprays achieve only partial suppression, at considerable economic and environmental cost.

Black Sigatoka spreads through two spore types, asexual conidia and sexual ascospores, which play complementary roles in the epidemic cycle. Conidia drive rapid local spread through rain splash, water runoff, or localized wind, sustaining multiple infection generations within plantations.

Ascospores enable broader geographic dispersal, often across tens to hundreds of kilometers, and can initiate epidemics or reinfect plantations after seasonal lows or sanitation efforts. Together, these dispersal mechanisms create a highly effective transmission system.

The study presents a quantitative compartmental pathogen-host model using differential equations to describe Black Sigatoka dynamics. The model incorporates both spore types and accounts for mate limitation in ascospore production because Pseudocercospora fijiensis requires compatible mating types.

The model derives the basic reproduction number, R0, and demonstrates a backward bifurcation, in which a stable endemic equilibrium coexists with the disease-free equilibrium when R0 is less than one. The model also includes a stochastic version using the Stochastic Simulation Algorithm to investigate how intrinsic individual randomness affects variability in disease dynamics.

Figure one maps the Black Sigatoka disease cycle on a banana plant, from conidia and ascospores landing on young leaves under high humidity or free water, through incubation and latency, to lesion development and spore production. It shows that asexual conidia form on lesion surfaces, while sexual ascospores mature in pseudothecia within dead tissue; both are then dispersed to restart infection.

This life-cycle structure motivates the model’s exposed, infected, and leaf-biomass compartments. The total leaf biomass is partitioned into four compartments: susceptible, resistant, exposed, and infected biomass. Healthy leaf biomass consists of susceptible and resistant biomass.

Susceptible leaves permit rapid spore germination and infection under favorable climatic conditions, whereas resistant leaves inhibit initial spore development. The infection rate uses two terms: mass-action incidence represents long-distance, density-independent transmission by wind-dispersed ascospores, while standard incidence represents localized, frequency-dependent spread by rain-splash-dispersed conidia.

Figure two shows transmission rate varying with temperature across humidity levels. The curves peak around the optimal temperature of twenty-seven point two degrees Celsius, then decline, while transmission increases with humidity and approaches saturation near one.

This environmental dependence matters because it represents the warm, moist conditions needed for fungal spore germination and successful infection. At high infected leaf biomass, mating opportunities are abundant, allowing ascospore production to approach its maximum rate.

At low infected biomass, the encounter rate between compatible mating types declines sharply, introducing an Allee effect due to mate limitation. The per-unit-biomass rate of successful mating therefore decreases nonlinearly as infected biomass declines.

At high infected biomass, the mating function is close to one, yielding near-maximal ascospore production. At low infected biomass, the mating function is approximately proportional to infected biomass, so ascospore production scales quadratically, reflecting the pairwise requirement for compatible mates.

Figure three maps the Black Sigatoka model as connected compartments for resistant, healthy, exposed, infected, and removed leaves, alongside sexual and asexual spore pools. Solid arrows show leaf biomass progressing through disease states, while dashed arrows represent interactions between leaves and spores, including resistance efficacy in suppressing penetration and germination.

This schematic matters because it links the biological assumptions to the differential equations, including dual spore pathways and density-dependent sexual reproduction. The basic reproduction number, R0, is a threshold parameter that quantifies the expected number of secondary infections produced by a single infected individual introduced into a fully susceptible population.

An infection can invade and establish itself when R0 is greater than one, while it is expected to die out when R0 is less than one. Computing R0 is essential for assessing invasion risk, designing control strategies, and identifying interventions that reduce disease transmission.

In this model, the infected compartments used to compute R0 are exposed leaves, infected leaves, and asexual spores, or conidia; sexual spores, or ascospores, are excluded. Near the disease-free equilibrium, infected leaf biomass is close to zero, so sexual spore production is a higher-order term that vanishes quadratically.

Therefore, sexual spore production does not contribute to the leading-order invasion dynamics in the linearized system. For small infected leaf biomass, the force of infection from ascospores is nonlinear because ascospore production scales quadratically with infected biomass.

The disease can therefore successfully invade a healthy population despite having R0 less than one. When R0 is less than one but infected biomass is above the threshold Ic, and D is less than one, disease invasion occurs. The nonlinear effects of sexual ascospore production may trigger an outbreak once the threshold of infected biomass Ic is crossed and ascospore production is sufficiently rapid.

These conditions suggest bistability through a backward bifurcation, which is a subcritical transcritical bifurcation. Numerical computations of the equilibria clearly show the occurrence of a backward bifurcation. In this Black Sigatoka model, the basic reproduction number is an inadequate predictor of disease eradication.

Reducing R0 below one may not achieve eradication because the pathogen can persist on a stable branch when R0 is less than one, supported by its polycyclic life cycle, dual spore pathways, reinfection capacity, and long-distance dispersal. The bistable behavior arises predominantly from nonlinear dynamics in the sexual reproduction cycle, which creates an unstable equilibrium branch in the bifurcation diagram.

In the plot’s lower region, the red dashed branch stretches leftward from the dashed line at R-zero equals one, while remaining close to the disease-free horizontal line. The blue branch stays visibly separated above it across the displayed range, and the caption notes that the turning fold at very small R-zero values lies beyond the computationally displayed portion.

The fraction of newly recruited susceptible leaves is the most influential parameter, with a sensitivity index of one point four seven. A one percent increase in that fraction leads to a one point four seven percent increase in infected equilibrium biomass, while several other parameters have indices close to zero.

The recruitment structure and host susceptibility drive infected equilibrium biomass far more strongly than most other biological processes do. In the bar chart, the κ column rises far above the baseline, while nearly all other parameter bars remain clustered around zero.

The only visibly notable exceptions are the small positive bar for Γ and the small negative bar for γ, both much shorter than κ. The host recruitment parameters have the strongest positive correlations, indicating that infected equilibrium biomass is primarily driven by the continuous supply of susceptible leaf biomass.

The progression rate also has a moderate global positive effect, while the cultural practices parameter has a negative correlation, indicating that removing infected biomass is the most effective control mechanism. The natural leaf decay rate also contributes negatively by reducing available host biomass.

In panel (a), the bars for Lambda and kappa rise to nearly one, while the bar for rho extends to about negative zero point nine, making the strongest visual contrast in the chart. Panel (b) preserves these separated clusters: their violin shapes are tight and centered well away from zero, whereas most remaining parameters are compressed around the horizontal zero line.

Figures seven and eight compare leaf and spore biomass over time under deterministic and stochastic frameworks. The stochastic simulations show individual realizations, their mean across four hundred realizations, and the corresponding standard deviation, while deterministic trajectories are shown separately.

Stochastic paths fluctuate especially during early stages and at low population levels, reflecting demographic stochasticity. Sexual spores display highly intermittent, burst-like behavior due to mate limitation, a feature not captured by the deterministic model. In the bottom-left exposed-leaf panel, the blue realization rises to roughly one hundred eighty biomass units around day fifteen, then drops near one hundred twenty, while the red mean and black trajectory remain around one hundred fifty.

In the infected-leaf panel at bottom right, isolated blue spikes reach about fifty, well beyond the surrounding mean and deterministic curves near thirty. The deterministic compartmental model incorporates dual transmission through ascospores and conidia, together with mate limitation in sexual reproduction, to describe Black Sigatoka dynamics.

The model exhibits a backward bifurcation: a stable endemic equilibrium coexists with the disease-free equilibrium for parameter values where R0 is less than one. This bistability provides a theoretical explanation for Black Sigatoka’s resilience.

Management strategies focused only on reducing R0 below one may be insufficient because the system can remain trapped in the endemic state despite subcritical transmission potential. The central message is that reducing the basic reproduction number below one may not be enough for Black Sigatoka control.

The model points instead toward limiting susceptible leaves and deploying resistant varieties.