Threshold-based impulsive biocontrol for coffee leaf rust

TL;DR

This work addresses threshold-based impulsive biocontrol for Coffee Leaf Rust (CLR) by developing deterministic ODE/IDE models and stochastic CTMC/MBPA frameworks. It derives the basic reproduction number $\\mathcal{R}_0$ for the ODE system and a periodic-stability criterion $\\mathcal{R}$ for the IDE with annual impulses, establishing conditions for global stability of disease-free states and the impact of harvest-driven impulses. It further explores prevalence-based control where releases occur when infection crosses $I_s$, revealing rich dynamics and sometimes counterintuitive outcomes across different $\\mathcal{R}_0$ values, supported by extensive numerical simulations and heatmaps. The stochastic analyses show that CLR extinction is possible in low-infection regimes but becomes unlikely with high $\\mathcal{R}_0$ or large initial spore loads, highlighting the need for timely and appropriately calibrated interventions with smallholders’ limited resources.

Abstract

Coffee leaf rust (CLR) severely affects coffee production worldwide, leading to reduced yields and economic losses. To reduce the cost of control, small-scale farmers often only apply control measures once a noticeable level of infection is reached. In this work, we develop mathematical models to better understand CLR dynamics and impulsive biocontrol with threshold-based interventions. We first use ordinary and impulsive differential equations to describe disease spread and the application of control measures once a certain infection level is detected. These models help determine when and how often interventions should occur. To capture the early stages of the disease and the chance that it might die out by itself, we then use a continuous-time Markov chain approach. This stochastic model allows us to estimate the probability that the pathogen fails to establish, thereby avoiding serious outbreaks.

Figures (6)

• Figure 1: Partial Rank Correlation Coefficient (PRCC) sensitivity analysis to model parameters of the basic reproduction number $\mathcal{R}_0$.
• Figure 2: Dynamics of infected leaves $I$ (top panel) and predators $P$ (bottom panel) as a function of the threshold for activation of biocontrol (blue dashed line). Years shown by bands of alternating colours. 10-year simulations of model \ref{['sys:imp']} using initial conditions \ref{['eq:int-cond']}. (a) dynamics when $I_s=10$, (b) $I_s=50$ and (c) $I_s=70$. All parameter values as in Table \ref{['tab:model-parameters']} except the basic reproduction number $\mathcal{R}_0=1.5$ and $\omega$ computed from $\mathcal{R}_0$.
• Figure 3: Dynamics of infected leaves $I$ (top panel) and predators $P$ (bottom panel) as a function of the threshold for biocontrol $I_s$ (blue dashed lines) and $\mathcal{R}_0$. Years shown by bands of alternating colours. 10-year simulations of model \ref{['sys:imp']} using initial conditions \ref{['eq:int-cond']}. (a) $I_s=10$; (b) $I_s=30$; (c) $I_s=40$; (d) $I_s=50$. All parameter values as in Table \ref{['tab:model-parameters']} except $\omega$ computed from $\mathcal{R}_0$.
• Figure 4: Total number $I_{\text{total}}(10\text{ years})$ of newly infected leaves as a function of (a,b) the initial number $U_0$ of spores, (c,d) the basic reproduction $\mathcal{R}_0$ and (e,f) the death rate $\mu_P$ of predators, all versus the biocontrol threshold $I_s$. All parameter values are given in Table \ref{['tab:model-parameters']}, except for $\omega$, which is determined for each value of $\mathcal{R}_0$.
• Figure 5: Partial rank correlation coefficients (PRCC) for the sensitivity of the probability of an outbreak to changes in parameters. Two initial conditions are shown, one where the infection starts with one infected leaf $(i_0, u_0) = (1, 0)$ and the other with one spore $(i_0, u_0) = (0, 1)$.

Theorems & Definitions (9)

• Lemma 1
• Lemma 2
• proof
• Lemma 3
• Theorem 4
• Theorem 5
• proof
• proof : Proof of Lemma \ref{['lm:LAS-DFE']}
• proof : Proof of Theorem \ref{['thm:gloal-sta-PDFS']}