Sustainable management of coffee berry disease and leaf rust co-infection: a systematic review of deterministic models.

Abstract

Colletotrichum kahawae, the cause of Coffee Berry Disease (CBD), poses a significant threat to global coffee production, especially in Africa, with potential implications for Latin America and Asia.

This systematic review evaluates 24 models of CBD, Coffee Leaf Rust (CLR), Coffee Berry Borer (CBB), and co-infection of CBD with CLR, as well as strategies for optimal control.

It concludes the relevance of the basic reproduction number ( $R_{0}$ ) for forecasting outbreaks and the benefits of employing a combination of control measures.

Despite advancements, limitations still exist, including the need for empirical validation and the consideration of vector-mediated transmission in co-infection models. Future research should integrate GIS, climate data, and field trials for validation, model vector-host-pathogen interactions, and assess cost-effective strategies for smallholder farmers to ensure sustainable coffee production.

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Keywords: Coffee berry disease, Mathematical modeling, Plant disease epidemiology, Optimal control strategies, Co-infection dynamics

Graphical abstract

Article notes

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Received 2025 Apr 25; Accepted 2025 Jul 14; Collection date 2025 Dec.

Author notes

Background

Coffee is one of the most important agricultural resources in the world. It is cultivated in nearly 70 countries, predominantly in the tropical regions of Latin America, Africa, and Asia [1]. With an annual global production of approximately nine million tons [2], coffee plays a crucial role in international agriculture, making its sustainability and disease management vital areas of study. The sector sustains the livelihoods of around 125 million individuals globally [1], rendering it a vital economic foundation for many developing nations.

The global coffee industry is highly concentrated among the major producers of this iconic beverage. Brazil provides a list of global coffee producers, whereas many others look for market trends and agricultural innovation. They significantly influence the establishment of global production methods and pricing strategies, commonly known as "Brazilian coffee production" [3]. Vietnam has disrupted the coffee production landscape with the introduction of Robusta coffee, altering global supply patterns [4]. Colombia has established a reputation for producing premium Arabica beans and possesses a strong organizational structure throughout its business [5]. India makes a significant contribution to coffee production in traditional regions, such as the Western Ghats, and in developing areas like Andhra Pradesh and Orissa [6]. Production patterns in Africa, America, and Asia demonstrate favourable growth rates, with the American continent comprising the most significant number of countries designated as "Mainstay" for coffee production, indicating robust future potential in this area. While many Asian nations appear to be "left behind" in coffee production, others exhibit significant potential for growth and advancement.

Two economically significant species control global coffee production: Coffee arabica and Coffee canephora (Robusta). Coffee arabica is highly regarded for its superior taste and quality; consumer demand for premium beverages elevates market prices. Conversely, Robusta coffee is preferred for instant coffee and espresso blends because of its superior disease resistance and higher caffeine content [7].

The economic significance of coffee extends beyond agricultural statistics. For numerous developing nations, it is a vital source of foreign revenue through a steady influx of foreign currencies. Additionally, the coffee industry plays a crucial role in job creation, employing individuals throughout the entire value chain, from cultivation and harvesting to processing, transportation, and retail. While large-scale production is significant, most coffee is cultivated on small and medium-sized farms [8]. For rural inhabitants, the income generated by these farms directly sustains families. As one of the most traded commodities worldwide, coffee constitutes a substantial industry. The premium pricing of speciality coffees creates lucrative economic opportunities for producers by commanding higher prices, thereby influencing trader behaviour and fluctuations in currency exchange rates.

The coffee market presents significant opportunities alongside major challenges, influenced by a range of factors. Regarding coffee output, small and large producers produced somewhat different amounts. Usually making significantly less profit than bigger production companies, small farmers play a significant part in providing coffee to the rest of the world [9]. A significant concern is climate change (CC), which jeopardizes sustainable development, reduces crop yields and disrupts global trade due to current and anticipated changes in conditions. The selling prices of coffee beans are consistently variable, resulting in price volatility that contributes to economic instability by several degrees. As this occurs, an increasing number of farms are abandoned, and families and farmers in producer nations face significant challenges in achieving financial stability. A significant number of individuals experience distress due to this substantial price volatility. Moreover, driven by sustainability concerns, more advanced production techniques encourage the adoption of voluntary sustainability standards to ensure long-term environmental and economic viability.

Coffee producers are experimenting with new farming methods to address CC and economic challenges. Integrating coffee cultivation with other tree species enhances environmental resilience and sustainability [10]. Diversification strategies, such as those implemented in Sulawesi, help mitigate produce fluctuations by incorporating alternative crops into farming systems. Furthermore, the residual use of resources garners interest, as coffee processing byproducts are being transformed into valuable assets, thereby minimizing waste and generating supplementary revenue. Employing sustainable agricultural practices, such as organic cultivation and shade-grown coffee, is crucial for maintaining both environmental and economic viability in the long term [11].

Thailand's coffee industry has experienced steady growth, with annual production volumes ranging from 30,000 to 50,000 t, primarily cultivated in the northern regions of Chiang Mai and Chiang Rai. Unlike major coffee-exporting countries, Thailand's domestic consumption is substantial, while its speciality brews are gaining more attention in international markets [12,13]. The growth of café culture and consumer demand for locally sourced, premium beans drive increased demand in the national coffee market. The increasing demand in the national coffee market is driven by the expanding café culture, reflecting consumer preferences for locally sourced, high-quality beans.

Despite its expansion, the coffee business encounters numerous challenges. Variable pricing induces economic volatility, restricted global market access curtails export opportunities, and CC poses manufacturing challenges for the coffee industry [14]. The existing agricultural infrastructure needs contemporary processing technologies, value-added manufacturing, and substantial investment in postharvest facilities to enhance quality and competitiveness. Adopting sustainable agricultural practices and enhancing supply chain management will determine Thailand's future global market success in coffee.

Coffee Berry Disease (CBD), induced by the hemibiotrophic fungal pathogen Colletotrichum kahawae, constitutes one of the most destructive diseases impacting coffee production. The ailment initially manifests as small, dark, depressed lesions on delicate green cherry beans [15]. In moist conditions, it exhibits vivid red and orange bursts of spores. As the infection progresses, berries develop fully blackened and mummified states, which result in either early dropping or remaining on branches, while severe infections can destroy as much as 80 % of the possible yields [16]. The pathogen targets developing green berries during the expansion phase after flowering, whereas immature berries show the highest susceptibility. However, C. kahawae can affect leaves and flowers and occasionally impact branches under optimal conditions. The economic impact of the CBD results in annual crop losses ranging from $300 to $500 million in Africa, jeopardising millions of smallholder farmers and affecting coffee supply chains while presenting a significant biosecurity risk to the global coffee market due to the potential spread to Latin America and Asia.

In the mid-20th century, plant disease modelling gained popularity, providing researchers with a quantitative basis for better understanding epidemiological processes [17]. These include but are not limited to, foundational concepts such as infection rate, latent period, and disease progress curves developed by Van der Plank in the 1960s, which advanced plant pathology as a predictive science. Throughout the 1970s and 1980s, these models were enhanced through various differential equation formulations, and the SIR (Susceptible-Infected-Removed/Recovered) model, initially developed in human epidemiology [[18], [19], [20]], has been particularly adopted in plant systems. In the 1990s, modelling became even more complex, incorporating factors such as weather, host resistance, and the biology of pathogens [21,22].

From the early 2000s onwards, more advanced models have incorporated climatic factors as well as pathogen life cycles (Waller pioneered the application of mathematical patterns to coffee diseases). More recently, these models have evolved to ensemble models that combine deterministic components with stochastic processes to account for uncertainty in pathosystems. By integrating molecular data, genomic information, and remote sensing technologies with traditional epidemiological parameters, modelling efforts are achieving unprecedented precision in disease prediction and evaluation of management strategies. It provides a renewed focus on modelling to anticipate changes in disease dynamics and develop adaptive management strategies for important crop systems, such as coffee.

Over the years, the mathematical modelling of CBD has undergone significant evolution, driven by advances in both mathematical techniques and the understanding of disease transmission dynamics. Early models focused almost exclusively on explaining the fundamental dynamics of CBD, while newer approaches consider more complex factors and more sophisticated control measures. Melese et al. (2022b) provided basic strategies, including logistic growth and carrying capacity, to represent the limited size of coffee farms in the early development stage [23]. Using next-generation matrix techniques, these models determine endemic and disease-free equilibrium points, as well as the basic reproduction number. These models, however, do not adequately account for environmental variables or co-infections with other diseases.

A significant breakthrough occurred when Nyaberi et al. (2023) created a mathematical model that specifically examined CBD dynamics. This model represented significant progress by combining coffee plant and pathogen populations, enabling a comprehensive understanding of their interactions. They determined a crucial threshold for disease propagation by first obtaining a basic reproduction number ( $R_{k_{0}}$ ). This study of disease-free and endemic equilibrium points also provides crucial insights into the evolution of disease [24].

Since 2022, CBD modelling has undergone significant changes, thanks to the work of Melese et al. (2024), who developed nonlinear deterministic models that consider environmental factors such as temperature and rainfall fluctuations. Due to their emphasis on the climate's influence on the movement and spread of diseases, these models have drawn attention to the current challenges of CC [25]. Their calculations of basic reproduction numbers under various environmental conditions provided a more detailed understanding of disease thresholds in different scenarios.

Nyaberi et al. (2024) have made significant strides in CBD modelling with their work on co-infection models. This research represents a significant development in addressing the complex scenario where coffee plants are sometimes simultaneously affected by CBD and Coffee Leaf Rust (CLR) [26]. The study introduced a system of ordinary differential equations (ODEs) that incorporated various control strategies, including infection prevention, treatment of diseased plants, and pathogen eradication. Employing Pontryagin's maximum principle to resolve the optimal control problem exemplifies the cutting-edge techniques in current CBD modelling methodologies.

CBD-specific modelling is complemented by other research on related coffee pests and diseases, which provides a range of additional methodologies. Trujillo-Salazar et al. (2024) contributed techniques of bifurcation analysis to control a CBB [26], while Herskowitz (2023) applied ecological-epidemiological models to control coffee rust using biological agents [27].

The integration of optimal control theory represents perhaps the most significant advancement in recent CBD modelling. Fotso et al. (2021) pioneered this approach for coffee berry borer control [28,29], which was subsequently adapted for CBD [28]. This approach enables researchers to understand disease dynamics and identify optimal intervention plans that maximise coffee production and minimise control expenses.

This study aims to review and critically assess deterministic mathematical models applied to CBD, CLR, and CBB, highlighting their progression from basic logistic growth models to more complex models that incorporate environmental factors, co-infection mechanisms, and optimal control strategies. This study also identifies significant research gaps, particularly the limited exploration of CBD and CLR interactions in co-infection contexts, as well as the understudied contribution of pathogen carriers to disease spread. Additionally, it emphasizes the urgent need for model validation using real-world field data and the incorporation of Geographic Information Systems (GIS) to enhance data collection capabilities. We systematically analyzed methodological advancements from Melese's foundational work to Nyaberi's innovative application of Pontryagin's maximum principle alongside complementary techniques derived from related coffee disease models. This review outlines the strengths, weaknesses, and practical applications of current modelling approaches.

Biological background

The disease cycle of CBD involves a complex sequence of interconnected stages that collectively enable the pathogen C. kahawae to successfully invade coffee plants, proliferate within host tissues, produce reproductive structures, and spread to new sites of infection. Each stage is governed by specific biological processes and environmental factors, which are quantified as parameters in mathematical models. For instance, the initial infection phase begins when conidia land on susceptible green berries and germinate under favourable moisture conditions (relative humidity > 95 %), forming specialised melanised appressoria within 12–24 h. These appressoria produce tiny pegs that break through the outer layer of the plant using both pressure and enzymes [29,30]. This penetration efficiency is influenced by temperature (optimal at 20–22 °C) and berry developmental stage and is represented as a critical parameter in mathematical formulations that determine the transition rate from susceptible to infected plant categories [17].

Understanding the complete disease cycle provides essential information for parameterising deterministic models, highlighting knowledge gaps that need further investigation. For example, the role of vectors in spore dispersal, environmental thresholds for disease progression, and potential synergistic effects during co-infection with other coffee pathogens, such as CLR, remain underexplored.

Infection

The disease cycle is initiated by the infection of coffee tissues, predominantly green berries, although C. kahawae may also infect flowers, leaves, and young twigs. Infection commences when conidia (asexual spores) or ascospores generated by the sexual phase of the fungus adhere to vulnerable host tissues. These spores are generally disseminated by rain splash or wind and require free water (e.g., rain or dew) for germination, which occurs most effectively at temperatures ranging from 20 °C to 25 °C, prevalent in high-altitude coffee cultivation areas such as East Africa. Dissemination is also possible via vectors such as insects, birds, and coffee harvesters [23]. Upon germination, the spore produces a germ tube that develops into a melanised appressorium, a specialised structure that facilitates infection. The appressorium produces elevated turgor pressure, which mechanically breaches the plant's cuticle, thereby initiating the biotrophic phase of infection. In this phase, C. kahawae resides within the host cells, inflicting little immediate harm while assimilating nutrients and circumventing plant defences. This phase is crucial for initiating infection and is influenced by environmental factors, including humidity and temperature.

Colonization

Once inside the host, C. kahawae switches from a biotrophic to a necrotrophic lifestyle, signalling the colonization phase. The berries begin to decay and fall off the shrubs prematurely. The fungus colonizes berry tissues extensively, spreads through the mesocarp, and occasionally reaches the bean, making it unsuitable for harvest. It causes dieback and defoliation of twigs and leaves, respectively, with berries as the primary target. The review's emphasis on co-infection dynamics with CLR is pertinent because C. kahawae colonization may interact with other pathogens, which remains unexplored.

Sporulation

High humidity levels ( $> 90 %$ ) combined with moderate temperatures ranging from (20–25 °C) boost conidial production, whereas dry environments hinder it. It highlights how environmental factors in the CBD models are particularly relevant to this stage, as rainfall and humidity influence sporulation rates. Although this is less common, C. kahawae can also generate ascospores during its sexual stage (teleomorph).

Dispersal

The final stage, dispersal, ensures the pathogen spreads to new hosts and completes the disease cycle. Rain splash is the main way conidia are distributed; this is a fundamental process in coffee farms when rainfall often occurs during the growth season. Within the same tree or to other plants, water droplets often dislodge conidia from acervuli and splash them onto nearby berries, leaves, or twigs. Although less important than rain splash, wind can also carry conidia over greater distances, particularly during storms. Ground-based infected berries can function as reservoirs; conidia survive in the ground or on plant waste, helping sustain disease between seasons. Further promoting dispersal are insect, bird, and human activities, such as the movement of workers, tools, and harvested berries.

Quantitative disease parameters and varietal susceptibility

The infection rate and subsequent disease progression of CBD exhibit significant quantitative changes, which are important for mathematical modelling. Field studies by Waller et al. (1993) documented latent periods ranging from 5–7days under optimal conditions (22 °C, > 95 % RH) to 14–21 days under suboptimal conditions (< 18 °C or > 28 °C), with an average infection efficiency of 26–42 % in susceptible varieties [31]. Infected berries generally show symptoms quite quickly after infection, within 6–8days, followed by a sporulation period of 3–4days, producing $3 \times 10^{4}$ to $8 \times 10^{5}$ conidia per lesion [32]. Coffee varieties vary widely in terms of their disease resistance. Traditional varieties, such as SL28 and K7, become infected up to 60–80 % of the time when conditions are favourable, while resistant varieties, including Ruiru 11 and Catimor hybrids, exhibit infection rates of only 10–15 % [33]. These resilient plant varieties have thicker cuticles and produce greater amounts of phenolic compounds, which help prevent the growth of pathogens. These characteristics significantly alter both the basic reproduction number and endemic equilibrium points in mathematical models [34].

The remainder of this paper is organized as follows. The "Methods" section outlines our systematic review methodology, adhering to PRISMA 2020 guidelines, which include the search strategy, inclusion and exclusion criteria, and quality assessment protocols. The "Mathematical Modelling Approaches" section reviews fundamental epidemiological theories and categorizes the main modelling frameworks used in plant disease studies. The "Results" section discusses the study characteristics, qualitative analysis of the included studies, existing mathematical formulations, their assumptions, and an evaluation of their predictive performance. The "Discussion" section provides limitations of the existing models. Finally, the "Conclusion" section suggests future directions and recommendations for improving CBD, CLR, and CBB modelling.

Methods

The systematic review was conducted by the Preferred Reporting Items for Systematic Reviews and Meta-Analyses (PRISMA) 2020 guidelines [35]. The protocol for the review was developed to answer the research question: In CLR- and CBD-affected coffee agroecosystems, deterministic compartmental models (e.g., SEIR-type) incorporating fungicides, resistant varieties, cultural practices, biological control, or integrated disease management (IDM), compared to no-management, no intervention, and one-thing management strategies in predicting:

i.Disease incidence (e.g., R0, infection prevalence),
ii.Yield losses,
iii.Intervention cost-effectiveness (e.g., ICER),

under varying temperature and rainfall conditions?

The methodology involved a systematic literature search, study selection based on predefined eligibility criteria, data extraction, and narrative synthesis of the findings.

Search strategy

A thorough literature search identified studies that were published between January 2000 and March 2025. We searched the Scopus and Google Scholar databases to identify studies. The search strategy employed a combination of keywords and Boolean operators to specifically search for relevant studies, as follows: coffee OR coffee disease AND mathematical AND model OR modelling OR analysis OR dynamics. The search terms were modified to be consistent with the syntax of the individual databases, and filters were applied for publication date (2000–2025). No automated screening tools were used. Records were exported and screened by title and abstract using inclusion/exclusion criteria, and full texts were selected.

Inclusion and exclusion criteria

Studies were included if they developed deterministic mathematical models (e.g., compartmental SEIR, predator-prey, or optimal control) for CBD or CLR within tropical highland agroecosystems or for related coffee pests such as the coffee berry borer (CBB), provided they contributed to methodological advancements (e.g., optimal control strategies, stability or bifurcation analysis, fractional-order or age-structured modelling) applicable to CBD/CLR. Eligible studies evaluated interventions, such as fungicide applications, resistant varieties, cultural practices, biological control, or integrated disease and pest management (IDM), with or without comparisons to no intervention, single interventions, or varying climate scenarios (e.g., temperature and rainfall). The required outcomes included disease or pest dynamics (e.g., basic reproduction number ( $R_{0}$ ), prevalence, and seasonal peaks), yield loss (e.g., % loss), and cost-effectiveness (e.g., Incremental Cost-Effectiveness Ratio, ICER). Mathematical modelling studies had to be published between 2000 and March 2024, except foundational works published prior to 2000 that contributed to deterministic frameworks.

Data extraction

Researchers extracted data into an Excel spreadsheet and verified all entries. Any disagreements were resolved through consensus. From each included study, the following parameters were extracted: publication year, study location (country/region or theoretical setting), coffee species studied (Coffee arabica/canephora), model type (e.g., SEIR, optimal control, predator-prey, or fractional order), key parameters (transmission rate $β$ , removal rate $γ$ ), intervention details (fungicide types, resistant varieties, cultural practices), and outcome measures (disease incidence, yield loss, cost-effectiveness).

Quality assessment

Quality evaluation was conducted independently by two researchers using a customized seven-item checklist adapted from the Joanna Briggs Institute(JBI) tool for critical appraisal of prevalence studies, omitting participant recruitment, sample size, and subpopulation criteria as irrelevant to mathematical modelling studies (see "Additional file 1: Table S1″ in Supplementary Information section [36]. Each study was assigned a composite quality score (0–14 points), with responses of "Yes" (2 points), "Unclear" (1 point), or "No/Not applicable" (0 points) for each criterion. Studies scoring $8 o r h i g h e r w e r e d e e m e d t o h a v e$ moderate to high quality and aligned with the review's threshold for robust deterministic models. A third reviewer resolved disagreements between reviewers to ensure consistency.

Mathematical modelling approaches

Mathematical modelling of plant diseases has evolved significantly since the foundational work of Van der Plank (2013), transforming into a predictive science of plant pathology [17]. This section provides a comprehensive overview of the fundamental concepts, approaches, and mathematical frameworks used in modelling plant disease epidemics.

Fundamental concepts of epidemiology

Disease progress curves

Historically, the dynamics of plant disease epidemics have been represented by disease progression, which describes the relationship between disease intensity (incidence or severity) and time. Like human disease modelling, mathematical modelling necessitates a thorough understanding of the disease. During the model-building process, the researcher determines which components are crucial to the problem and which can be prioritised. The mathematical representation of these curves typically follows one of the following growth models:

Exponential model: The ordinary differential equation

(1)

\frac{d y}{d t} = r y,

2where $y$ represents the proportion of diseased tissue (e.g., the fraction of infected coffee berries), $t$ is time (usually in days or weeks), and $r$ is the infection rate, reflecting the pathogen's capacity to infect new tissues under ideal conditions. Where $y (t) = y_{0}$ and $y_{0}$ is the starting fraction of sick tissue at $t = 0$ . Solving this ODE results in $y (t)$ . This model suggests that the disease spreads exponentially free from control, which fits the early phases of CBD when C. kahawae quickly contaminates the green berries.

Fig. 1 illustrates the exponential increase in the proportion of diseased coffee berries over time, with different infection rates ( $r = 0.05, 0.1, 0.15, 0.2$ ) determining the speed of disease spread. Higher $r$ values lead to a more rapid escalation in infection, showing the impact of faster transmission.

1. Logistic Model: The logistic model is governed by the ODE

(3)

\frac{d y}{d t} = r y (1 - \frac{y}{P})

where $y$ is the proportion of diseased tissue (e.g., infected coffee berries), $t$ is time (in days or weeks), $r$ is the rate of infection, and $P$ is the carrying capacity or maximum amount of diseased tissue, which is commonly set to $1 -$ or $100 %$ infected. The density-dependent component introduced by the equation $1 - \frac{y}{P}$ slows disease development as $y$ approaches $P$ , reflecting real-world CBD dynamics whereby the quantity of susceptible berries declines as the epidemic advances. Solving this ODE produces the solution $y (t) = \frac{P y_{0}}{(P - y_{0}) e^{- r t} + y_{0}}$ , where $y_{0}$ represents the initial solution of infected tissue at $t = 0$ . This model is more realistic for CBD than the exponential model for later stages because it considers the saturation of infection in a coffee field, which is influenced by factors such as host resistance, limited berry availability, or climatic constraints, including declining humidity.

Fig. 2a shows that higher infection rates $r$ lead to a faster increase in disease prevalence, but all curves eventually stabilize at the carrying capacity $P = 1$ . Fig. 2b highlights the impact of different carrying capacities $P$ on disease progression, where lower $P$ values limit the maximum disease level. These results highlight the significance of infection dynamics and environmental constraints in modelling the spread of CBD.

Fundamental concepts of epidemiology

Empirical models rely on the statistical analysis of data and establish statistical links between disease intensity and predictor variables without explicitly clarifying the underlying biological processes [37]. These models are helpful in several fields, including social networks, because they can fit different datasets. However, their dependence on data can result in false findings, and the fundamental mechanisms underlying these findings are poorly understood or misrepresented [38]. In contrast, mechanistic models describe the biological mechanisms underlying disease development and spread. These models can be considered systems of differential equations, incorporating biologically meaningful parameters. Crucially, disciplines such as epidemiology and engineering shed light on how changes in one area of a system affect others [39]. Mechanistic models can validate the assumptions and predictions of empirical data [38].

Other models include deterministic and stochastic models. Deterministic models often assume constant parameters and continuous populations, which are often described by differential equations, such as the SIR and SEIR models. These models are effective for large datasets, where randomness is negligible, and can quickly provide long-term trends. The limitations of these models include their inability to account for stochastic fluctuations or small population dynamics. Stochastic models incorporate randomness, utilising probability to explain variations in disease transmission and recovery times. Approaches such as Monte Carlo simulations provide model fit for small populations and scenarios with high variability, such as super-spreader events, and offer a greater range of possible outcomes. These models have higher computational costs and are sensitive to the parameter estimates. Although deterministic models provide simplicity and speed, they can ignore important dynamics in smaller or more variable populations. However, despite their complexity, stochastic models may reflect important characteristics that deterministic models would overlook [40].

For CBD, deterministic models have been predominant in the literature, with recent work by Nyaberi et al. (2024) incorporating a system of ODEs to model the co-infection dynamics of CBD and CLR, focusing on prevention and treatment strategies to optimise control of these diseases [28].

SIR models and extensions

Basic SIR model

Initially developed for human epidemiology, the Susceptible-Infected-Removed ( $S I R$ ) framework has been successfully adapted to model plant diseases, such as CBD, providing a logical approach to understanding disease dynamics in coffee plantations. The $S I R$ model tracks the proportions of susceptible, infected, and removed coffee tissues (e.g., berries) over time for CBD, thereby capturing the transition from healthy to infected states and the eventual removal of tissues through recovery, death, or harvest. Considering the interaction between susceptible berries and the pathogen, the rate of infection, and the removal of infected tissues, which can be influenced by management practices such as pruning or the use of resistant varieties like Ruiru 11, this is crucial in studying CBD dynamics.

The $S I R$ model is defined by the following system ODEs:

(4)

\begin{matrix} \frac{d S}{d t} = - β S I, \\ \frac{d I}{d t} = β S I - γ I, a n d \\ \frac{d R}{d t} = γ I, \end{matrix}

where $S, I$ , and $R$ represent the proportions of susceptible, infected, and removed coffee tissues, respectively, with $S + I + R = 1$ . Here, $S$ might represent healthy coffee berries, $I,$ represents the proportion of berries infected by C. kahawae, and $R$ the proportion removed through natural berry drop, death due to severe infection, or management interventions like harvesting infected berries. While $γ$ is the removal rate, indicating the rate at which diseased berries are eliminated from the system, the parameter $β$ is the transmission rate, reflecting the rate at which C. kahawae spreads from infected to susceptible berries (e.g., by rain splash dispersal). For CBD, $β$ is controlled by environmental variables like humidity and rainfall, which facilitate spore dispersal, and $γ$ can be impacted by cultural practices or host resistance.

Fig. 3a shows the variations in susceptible ( $S$ ), infected ( $I$ ), and removed ( $R$ ) berries over time for a coffee plantation, therefore highlighting how the $S I R$ model fits CBD. Starting with initial conditions of $S (0) = 0.99, I (0) = 0.01$ and $R (0) = 0$ , we employed parameters $β = 0.3$ of and $γ = 0.1$ . Characteristic of a normal CBD epidemic cycle, the graph shows a rapid decline in susceptible berries as the infection reaches its apex, followed by an increase in removed tissues. Moreover, Fig. 3b explores the $S I R$ model with various transmission rates $β = 0.2, 0.3, 0.4$ to show how environmental variables or pathogen aggressiveness affect CBD distribution.

SI/SEIR extensions for plant diseases

For many plant diseases, including CBD, modifications to the basic $S I R$ model are necessary to capture specific epidemiological features. Common extensions include the following.

The $S I$ model, a simplification of the $S I R$ framework, is used for plant diseases such as CBD, where infected tissues (e.g., coffee berries) remain infectious indefinitely without recovery, as C. kahawae can persist on infected berries or twigs, producing conidia for secondary infections. C. kahawae may remain on afflicted berries or twigs and generate conidia, causing additional infections. The ODEs $\frac{d S}{d t} = - β S I$ and $\frac{d I}{d t} = β S I$ , where $S + I = 1$ . $S$ and $I$ describe the proportion of susceptible and infected coffee berries, respectively. Here, $β$ is the transmission rate. Unlike the $S I R$ model, in this case, there is no removal compartment since infected berries remain a source of inoculum, a scenario often observed in CBD epidemics where infected berries rot and drop but continue to harbor the pathogen.

The $S E I R$ model extends the $S I R$ framework by incorporating a latent period, which is critical for CBD because C. kahawae exhibits a delay between infection and the onset of infectiousness [41]. The following equations describe the $S E I R$ model:

(5)

\begin{matrix} \frac{d S}{d t} = - β S I, \\ \frac{d E}{d t} = β S I - σ E, \\ \frac{d I}{d t} = σ E - γ I, a n d \\ \frac{d R}{d t} = γ I, \end{matrix}

where $S$ represents susceptible coffee berries, $E$ represents the proportion of exposed (latently infected) coffee berries, $I$ and $R$ represent infected and recovered coffee berries, respectively. The parameter, $σ = \frac{1}{τ}$ is the rate at which exposed tissues become infectious, and $τ$ is the mean latent period. Nyaberi et al. (2024) employed a modified $S E I R$ model for CBD to emphasize a latent period of 5–7 days in optimal conditions (i.e., high humidity and temperatures between 20 and 25 °C [28]. C. kahawae completes its biotrophic phase before transitioning to the necrotrophic phase, producing visible symptoms and conidia. In this context, $β$ denotes the transmission rate, while $γ$ signifies the removal rate, which is affected by factors such as berry drop or management practices like pruning.

Fig. 4a plots the $S I$ model dynamics over 100days, with $β = 0.3$ , showing the continuous increase in infected berries without removal, reflecting a negative image of how CBD spread if no interventions are implemented since it amply illustrates a consistent increase in the number of infected berries without any eradication. Fig. 4b plots the $S E I R$ model dynamics with $β = 0.3$ , a latent period of six days expressed by $σ = \frac{1}{6}$ (this highlights the delay in the onset of infectiousness and how that influences CBD development) and a recovery rate of $γ = 0.1$ . The presented plots illustrate the utility of the $S I$ and $S E I R$ models in accurately representing the epidemiological characteristics of CBD.

Basic reproduction number

The basic reproduction number is represented as $R_{0}$ , is crucial in the epidemiological models of CBD. It explains the anticipated number of secondary infections that arise from a single infected coffee berry in a fully susceptible population. This understanding is crucial for comprehending how disease spread is established and persistent in coffee plantations; for the basic $S I R$ model, $R_{0} = \frac{β S_{0}}{γ}$ , where $S_{0}$ is the initial proportion of susceptible berries, $β$ is the transmission rate, and $γ$ is the removal rate, while for more complex models like the $S E I R$ used for CBD, Melese et al. (2022b) calculated $R_{0}$ using the next-generation matrix approach [23,42], determining the dominant eigenvalue of $F V^{- 1}$ (where $F$ is the rate of new infections and $V$ is the rate of compartment transfers).

Environment drivers and seasonal forcing

Environmental factors, particularly temperature and moisture, play a crucial role in the development of CBD in coffee plantations. These effects can be incorporated into epidemiological models by using time-dependent parameters to account for seasonal variations. The transmission rate is expressed as:

(6)

β (t) = β_{0} . f_{T} (T (t)) . f_{H} (H (t))

where $β_{0}$ stands for the baseline transmission rate, whereas $f_{T}$ and $f_{H}$ are functions that indicate how temperature $T (t)$ and humidity $H (t)$ influence the spread of the disease. Marcano et al. (2021) constructed models, particularly for CBB, indicating that CBB reproduction rates are highest when temperatures are warm and stable, particularly within the optimal range of 20–30 °C and relative humidity surpasses 90 %. These conditions are typically encountered in the East African coffee-growing areas during the wet season [25].

Clarifying the application of Eq. (6), consider a functional form for $f_{T}$ and $f_{H}$ based on biological evidence. For temperature, a Gaussian function models optimal transmission at 25 °C (midpoint of 20–30 °C) was used, declining outside this range (e.g., <18 °C or >28 °C): $f_{T} (T (t)) = e^{(- \frac{{(T (t) - 25)}^{2}}{2 * σ^{2}})}$ , where $σ = 5$ controls the temperature range [70, 71]. For humidity, a logistic function captures increased transmission above 90 % relative humidity: $f_{H} (H (t)) = 1 - \frac{1}{1 + e^{- τ (H (t) - 90)}}$ , where $τ = 0.5$ ensures a steep transition [71]. Thus, $β (t) = β_{0} \times f_{T} (T (t)) = e^{(- \frac{{(T (t) - 25)}^{2}}{2 * σ^{2}})} \times f_{H} (H (t)) = 1 - \frac{1}{1 + e^{- τ (H (t) - 90)}}$ , modulates transmission seasonally, aligning with CBD’s sensitivity to wet, moderate conditions.

Model evaluation and validation

Qualitative analysis

The qualitative analysis of deterministic models for CBD enables us to understand the system's behaviour without resorting to a numerical solution. It will provide a window into the long-term dynamics of prevalence among various classes of coffee plantations. A key method in qualitative analysis is equilibrium analysis, which identifies the disease-free equilibrium ( $D F E$ ) and endemic equilibrium ( $E E$ ) by solving the system of ODEs at a steady state:

(7)

\frac{d X_{i}}{d t} = 0, for all compartments i (S, I, E, R)

In the context of CBD, the $D F E$ denotes a situation in which the disease is nonexistent ( $I = 0$ ), but the $E E$ signifies a condition in which the disease remains prevalent within the population. This analysis is essential for identifying the parameters that allow CBD eradication or continuation.

Quantitative validation

The quantitative validation of deterministic models involves comparing their predictions with actual data to assess their accuracy and reliability. This stage is essential, although it receives insufficient attention in the CBD literature, primarily due to a lack of field data on infected coffee berries. It is where co-infection with CLR and GIS methods can be particularly helpful. A robust method for validating these models is the application of Root Mean Square Error (RMSE). This method will examine the mean disparity between the actual values ( $x_{i}$ ) and the expected values $({\hat{x}}_{i})$ of disease incidence, such as the percentage of infected coffee berries across $n$ data points:

(8)

R M S E = \sqrt{\frac{1}{n} \sum_{i = 1}^{n} {(x_{i} - {\hat{x}}_{i})}^{2}}

Minimising RMSE enables the optimisation of model parameters (e.g., transmission rate $β$ ) and population variables (e.g., the proportion of infected berries I), thereby enhancing the model's calibration with empirical data. Real-world data may include summarised case data on CBD or CLR across multiple geographies and during various growing seasons. Furthermore, the bootstrap method can be used to estimate confidence intervals for fitted parameters, thereby measuring the uncertainty in model data fitting. For example, generating 200 bootstrap samples would help facilitate the assessment of variability in the predicted proportion of infected berries or leaves.

To the best of our knowledge, no CBD or co-infected models have been validated using field data to date, and confidence intervals have not been computed, let alone estimated, using the bootstrap method. While goodness-of-fit criteria such as the Coefficient of Determination ( $R^{2}$ ) [43] and Akaike Information Criterion (AIC) [44] are commonly applied in epidemiological modelling, but their use in CBD models remains largely undocumented.

Goodness-of-fit metrics are vital for evaluating model accuracy and complexity; however, their application to CBD models appears to be undocumented. The reliance on theoretical validation rather than empirical data and theory highlights a significant research gap. This review emphasises the importance of empirically tested models in enhancing their practical relevance for forecasting the spread of CBD and co-infection with CLR.

Sensitivity analysis

Sensitivity analysis is crucial for assessing how uncertainties in model inputs impact the outputs of the CBD models. It facilitates the identification of the most influential parameters affecting disease dynamics in coffee plantations. Local sensitivity analysis assesses how small changes in individual parameters affect model outputs using sensitivity indices:

(9)

S_{i} = \frac{\partial y}{\partial θ_{i}} . \frac{θ_{i}}{y},

Here, $y$ represents the model output, similar to the fraction of infected berries, while $θ_{i}$ denotes the parameter of interest (for instance, the transmission rate $β_{i}$ ). This method highlights the minimal effect that specific differences in certain parameters may have on the progression of the disease. On the other hand, global sensitivity analysis, such as the Sobol method [45], provides a more comprehensive view by decomposing the total output variance $V (y)$ into contributions from individual parameters and their interactions.

(10)

V (y) = \sum_{i} V_{i} + \sum_{i < j} V_{i j} + \dots + V_{12 \dots k} .

While $V_{i j}$ considers the interactions between parameters $θ_{i}$ and $θ_{j}$ , $V_{i}$ represents the variance attributed solely to the parameter $θ_{i}$ . Sensitivity analysis within the context of CBD models has demonstrated that parameters associated with transmission $(β$ ) and environmental factors (such as the influence of temperature and humidity on $β (t)$ ) are crucial for forecasting disease outcomes. The prioritization of disease control strategies, such as targeted fungicide applications and effective environmental management on coffee plantations, is contingent upon this awareness.

Hoare et al. (2008) emphasized that further enhancements in CBD model sensitivity analysis can be achieved by advanced computational methods, including SaSAT (Sampling and Sensitivity Analysis Techniques) [46]. This framework provides a systematic approach to evaluating the impact of parameters, thereby enhancing the models' reliability and predictive accuracy.

Results

Study characteristics

The records were screened in two stages. First, titles and abstracts of 472 unique records were screened against the eligibility criteria, resulting in the exclusion of 446 records that did not meet the inclusion criteria (e.g., non-modelling studies and irrelevant diseases). Second, full-text articles (n = 26) were assessed for eligibility by two independent reviewers (A.I. and M.D.). Discrepancies were resolved through discussion and consultation with a third reviewer (U.W.H.). Studies were excluded if they involved non-deterministic models (e.g., stochastic or agent-based) without a deterministic component; focused on coffee species other than Coffee arabica and canephora; addressed diseases other than CLR or CBD; were empirical studies without mathematical modelling; lacked temperature or rainfall parameters; or were reviews/editorials without primary data. Two full-text articles were excluded: one modelled a non-coffee agroecosystem (incorrect setting n = 1), and one used a non-deterministic agent-based model (wrong study design n = 1). Ultimately, 24 studies were included in the review (see Fig. 5).

Fig 5 — Fig. 5PRISMA 2020 flow diagram illustrating the identification, screening, and inclusion of studies in a systematic review of deterministic mathematical models for CBD and CLR in coffee arabica and or robusta agroecosystems.

Quality assessment of included studies

Two researchers independently assessed the quality of the 24 included studies using the modified (JBI) checklist, as described in Section 2.4. Disagreements were resolved through consensus, with no unresolved disputes. All 24 studies (100 %) scored ≥8, indicating moderate to high quality, with one study (approximately 4.17 %) on 13/14 [47], another one study (approximately 4.17 %) on 11/14 [48], seven studies (approximately 29.17 %) on 10/14 (e.g., [23,28,[49], [50], [51], [52], [53]]), six studies (25 %) on 9/14 (e.g., [24,25,[54], [55], [56], [57]]), and 9 studies (37.5 %) at 8/14 (e.g., [27,[58], [59], [60], [61], [62], [63], [64], [65]]). Nine studies (37.5 %) scored 10 or higher, indicating a high quality. The sample sizes, interpreted as modelled populations, were adequate across the studies.

Deterministic models for CBD

Mathematical modelling has significantly improved our understanding of CBD and its interaction with CLR, another primary coffee disease. Based on ordinary differential equations (ODEs), deterministic models have been extensively used to investigate disease spread, evaluate control strategies, and predict epidemic trends. Often, these models assume a crop plantation with homogeneous mixing and use deterministic disease progression based on known parameters. This section reviews important deterministic models developed for CBD, emphasising their mathematical formulation, assumptions, limitations, parameter sensitivity, and predictive capabilities.

Compartmental models for CBD dynamics

One of the first deterministic models for CBD was developed using a SEIR-type structure, where coffee berries were classified into susceptible ( $S), e x p o s e d (E), i n f e c t e d (I), a n d r e c o v e r e d (R$ ) compartments. This method has been employed in several studies, including those by Nyaberi et al. (2023) and Jeeva and Dharmalingam (2024), which establish disease-free equilibrium (DFE) and endemic equilibrium (EE) [24,47,50]. Their models introduced environmental pathogen reservoirs ( $P_{K}$ ). These models assume that infection arises from spore deposition from rain splash or wind, with a fixed transmission rate $ω_{K}$ . These studies repeatedly identified the basic reproduction number ( $R_{0_{K}}$ ) as the main threshold determining whether CBD epidemics would continue or die.

Further modifications to these compartmental models incorporated intervention applications, resistant coffee varieties, and strategies. Researchers such as Melese et al. (2023) and Melese et al. (2022) have extended SEIR models by incorporating control variables that represent the effectiveness of interventions and plant resistance factors [56,66]. Additionally, the sensitivity analysis conducted by Fotso et al. (2021) demonstrates that reducing the pathogen's environmental survival rate is more effective than increasing plant resistance, reinforcing the importance of integrated disease management. However, a significant limitation of these models is that they do not account for temperature and humidity variations, which are the key drivers of CBD progression [61].

Compartmental models for CLR dynamics

Lazebnik et al. (2024) present a comprehensive, multi-scale model to analyse the COVID-19 pandemic, combining epidemiological, ecological, and economic factors [48]. At the tree level, the authors use an ordinary differential equation (ODE)- based compartmental model with branches classified by four tree states: healthy (H), latent (L), infectious (I), and leafless (J), along with a variable representing urediniospores (U) as well as premium berries (Bp), and regular berries (Br). Ecological components, such as temperature, humidity, and rainfall, are integrated with a machine learning (ML) component that links these variables to epidemiological parameters, including spore production rates. For the farm-level model, the authors extended the SEIS model (Susceptible-Exposed-Infectious-Susceptible) to incorporate spatial tree interactions based on distance and environmental conditions. The economic variable of potential intervention costs included labour, input, and fixed costs, as well as three primary intervention costs: chemical sprays, shading to reduce temperature, and cutting branches. The study modelled the CLR pandemic using ABS (agent-based models) by observing how individual trees continued their learning process and controlling a deep reinforcement learning (DRL) agent that helped optimise intervention plans given budget constraints.

The findings reveal significant obstacles to controlling CLR. Without intervention, coffee farms incur a loss of 8.5 % annually. The basic reproduction number increases and decreases seasonally, with Rainy months creating a favourable environment for pathogen spread. The study shows that profits cannot be achieved with realistic levels of partial observability (and productivity). In a scenario with complete observability regarding farm and weather conditions, yields show a profit of approximately 4 %.

A critical limitation of this study is its heavy reliance on theoretical assumptions due to the insufficient availability of real-world data for model calibration and validation. While the multi-scale framework advances epidemiological-economic modelling, key parameters, such as spore dispersal rates, intervention costs, and farm-level yield losses, are derived from the literature or simplified proxies rather than empirical measurements. This absence of field-data-driven estimation risks misrepresenting real-world dynamics, particularly in microclimate interactions (e.g., shading effects on humidity) and spatially heterogeneous pathogen spread.

Co-infection models for CBD and CLR

Only one study among the included 24, Nyaberi et al. (2024), developed a deterministic multi-disease model to examine the co-infection dynamics of Coffee Berry Disease (CBD) and Coffee Leaf Rust (CLR) in coffee arabica [28]. This model extends the SEIR framework by incorporating additional compartments for plants exposed to CLR, infected with either CBD or CLR and co-infected with both diseases. Co-infected plants exhibit higher mortality rates due to synergistic effects, with sensitivity analysis indicating that disease spread is accelerated under high humidity conditions (>90 %). The model addresses (RQ2) by quantifying how co-infection exacerbates disease burden and yield loss compared to single-disease scenarios.

A significant advancement in Nyaberi et al. (2024) is the application of optimal control theory to manage co-infection. Control variables represent interventions such as fungicide applications, resistant varieties, and integrated disease management (IDM), optimized to minimize transmission rates (β_CBD, β_CLR) and co-infection prevalence [28]. Cost-effectiveness was evaluated using the Incremental Cost-Effectiveness Ratio (ICER), with early fungicide application identified as a highly effective strategy. Pontryagin's Maximum Principle was employed to optimize control measures, balancing intervention costs and disease reduction [67]. However, the model lacks empirical validation using field data, which limits its practical applicability. This gap highlights the need for future research to integrate field-based parameter calibration and validate co-infection dynamics in diverse agroecological settings (RQ3).

Environmental and climate-driven models

Environmental factors, including temperature, humidity, and precipitation, influence the transmission of CBD. Building on this, Melese et al. [25] developed a deterministic model that incorporates temperature-dependent transmission rates ( $β (t)$ ), capturing seasonal effects presumed to shape disease patterns. Their model was constructed to incorporate a time-dependent function $β (t)$ , which modified the infection rates of the pathogen based on the climate data. The simulation results indicated that elevated humidity (>90 %) and moderate temperatures (20–25 °C) create excellent conditions for CBD, which can enhance seasonal peaks in CBD production in high-altitude coffee cultivation regions.

Motisi et al. (2022) conducted additional studies examining the impact of shade and microclimates on the epidemiology of CBD [47]. Their mechanistic $S E I R$ model examined the influence of shade density on disease transmission, revealing that shade could either alleviate or exacerbate disease progression contingent upon its effects on pathogen latency, spore dispersal, and infection rates. This study employed a Bayesian framework to estimate the epidemiological parameters based on various microclimatic variables, suggesting that disease suppression and coffee productivity must be balanced in site-specific farm management strategies.

Model comparison

There are 24 models considered in this study (See Table 2), offering variability in each of the validation, performance, complexity, and applicability aspects, which addresses RQ1-RQ3. No model evaluated here has empirical field data validation; models rely on simulation studies or Bayesian inference (e.g.,) [47], resulting in reduced predictive reliability. For performance, in studies where results were reported, fungicides or resistant varieties reduce CBD prevalence by approximately 20–40 % (e.g.), [24,48,57] and have a higher mortality rate of 30 % for CBD/CLR co-infection (e.g., )[28]. Complexity ranged from simple logistic models, such as [23], to SEIR models that include climate forcing [25] or even optimal control [28], and Bayesian frameworks [47], which inherently require more computing effort due to data generation, but collect a broader range of possible predictors to measure microclimate. Applicability is high for the models that considered some form of intervention (e.g., IDM in [28]), but the availability of field data and constructing vectors was a significant limitation for farm-level applicability (except for a few models). Future models should maximise usability for practitioners at the farm level who require empirical calibration of the model's complexity.

Table 2Comparison deterministic models for CBD, CLR, CBB, and Co-infection.

Model Study	Mathematical Structure	Key Assumptions	Key Parameters	Sensitivity Analysis Findings	Validation Method	Performance	Complexity	Applicability	Limitations
CBD SEIR Models [23,24,47,50]	SEIR ODE	Pathogen transmission, microclimate effects	$β$ , latent period	Humidity, temperature increase $R_{0}$	Numerical simulations, Bayesian	Predict CBD prevalence	Medium (4–5 compartments, 4–5 ODEs)	Agroforestry farm-level control	No field validation; no CLR
[23]	SEIR with logistics	Carrying capacity	$β$	Farm size limits spread	Simulations- based	Moderate control	5-compartments	Large farms	No climate
[24]	Basic SEIR	Uniform farm size	$R_{0}$	Pathogen density drives spread	Simulations- based	High prevalence	5-compartments	Smallholder farms	Simplified dynamics
[47]	Bayesian SEIR	Shade alters microclimate	Latent period	Shade reduces $β$	Bayesian inference	Shade-specific	6-compartments	Agroforestry	No CLR
[50]	SEIPR	Recovery possible	Recovery rate	Control reduces $R_{0}$	SAGPM method	Effective mitigation	5-compartments	-	Analytical focus
CBD Optimal Control Models [53,56]	SEIR ODE, optimal control	Climate, vector effects	$β$ , control efficacy	Early control lowers $R_{0}$	Numerical simulations, ICER	20 %−40 % CBD reduction	Method (5 compartments, 5 – 7 ODEs)	Fungicides, biocontrol	No field data
[53]	Vector transmission	Carrier vectors	Vector $β$	Vector control critical	Simulations- based	Cost-effective	High	Vector management	No climate
[56]	Temperature focus	Seasonal effects	$β$	Seasonal variation, $R_{0}$	Simulations- based	Seasonal control	High	Seasonal planning	Limited scope
CBD Fractional-Order Model [57]	Fractional-order SEIR	Memory effects	$β$ , fractional order	Pathogen persistence	Fractional Adams-Bashforth	Predicts long-term CBD	High (4 – compartments, complex)	Theoretical research	Limited applicability
[57]	Atangana-Baleanu-Caputo	Non-local dynamics Biocontrol, seasonal effects	Fractional order	Memory	Simulations- based	Long-term trends	4-compartment	Model development	No field validation
CLR SIR Models [27,48,51,52,55]	Economic-epidemiological	Spatiotemporal	Intervention	Transmission rate, Recovery rate, intervention cost, e.t.c	Simulation-based	Economically constrained	High	Policy-oriented	No field validation
[27]	Snail biocontrol	Snail predation	Predation rate	Snail density critical	Simulations- based	Effective biocontrol	3-compartments	Small farms	Invasive snails
[48]	Economic-epidemiological	Profit-driven	Spatiotemporal spread, Economic behaviour	High costs limit efficacy	Simulations- based	Economic Insights	7-compartments	Market policy	Limited profit
[51]	Bacterial biocontrol	Spatial effects	$R_{0}$	Spatial spread	Stochastic model	Local control	High	Farm-level	Stochastic element
[52]	Seasonal CLR	Impulsive effects	Seasonal $β$	Seasonality drives spread	Simulations- based	Seasonal control	4-compartments	Seasonal management	No field validation
[55]	Mycoparasite	Biocontrol efficacy	Inhibition rate	Biocontrol reduces $R_{0}$	Non-standard finite method	Eradication possible	High (6-compartments, 6 PDEs)	Biocontrol policy	No economic analysis
CBB Optimal Control Models [25,49,54,[60], [61], [62], [63], [64]]	ODE, optional control	Climate, pest dynamics	CBB growth rate, control efficacy	Temperature control reduce $R_{0}$	Numerical simulations ICER	30 %−50 % CBB reduction	Medium (4–6 compartments, 4–6 ODEs)	Traps, bio-insecticides	No field validation
[25]	Climate variability	Temperature, rainfall	$R_{0}$	Climate increases CBB	Simulations- based	Effective control	High	Climate adapted	No field validation
[61]	Entomopathogenic fungus	Fungus kills CBB	Fungus efficacy	Fungus reduces CBB	BOCOP	Cost-effective	High	Biocontrol	Limited scope
[64]	Bio-insecticide, traps	Synergy effects	Trap efficacy	Synergy lowers $R_{0}$	Simulations- based	Cost-effective	4-compartments, high	Large farms	No field validation
[49]	Temperature, rainfall	Climate impacts	$R_{0}$	Climate drives CBB	Simulations- based	Effective control	High	Climate strategies	No field validation
[54]	Farmer awareness	Awareness reduces CBB	Awareness rate	Awareness lowers $R_{0}$	Simulations- based	High efficacy	3-compartments	Education Campaigns	Simplified
[60]	Basic CBB dynamics	Berry availability	Growth rate	Control reduces CBB	Simulations- based	General control	High	Small farms	Broad assumptions
[62]	Age-structured	Berry age affects CBB	Age-specific $β$	Age impacts $R_{0}$	Simulations- based	Age-specific control	4-compartments, high	Farm level	Complex
[63]	IPM strategies	Time-dependent controls	Control efficacy	IPM reduces CBB	Simulations- based	High yield	4-compartments	IPM adoption	Costly
CBB Predator-Prey Models [58,59]	Predator-prey	Ant predation	Predation rate	Predation reduces CBB	Numerical simulations	Stable CBB control	Medium (3–4 compartments)	Biocontrol	No field validation
[58]	CBB-ants	Bifurcation effects	Consumption rate	An increase in the Bifurcation parameter reduces CBB	Simulations- based	High control	2-compartments	Ant-based control	Theoretical
[59]	CBB-ants	Adult, immature CBB	Predation rate	Ants reduce CBB	Simulations- based	Effective biocontrol CBB	3-compartments	Farm-level	Limited scope
CBB Discrete Map Model [65]	One-dimensional map
	Multi-seasonal	Infestation rate	Harvesting reduces CBB	Numerical simulations	CBB elimination possible	Low (1 - map)	Harvesting strategies	No climate
[65]	Seasonal harvesting	Harvesting effects	Harvest	High harvesting effective	Simulations- based	Long-term control	High (1-map, 5-compartments)	Large farms	Simplified
CBD-CLR Co-infection Model [28]	Multi-pathogen SEIR ODE	Synergistic mortality	$β_{C B D}, β_{C L R}$	Humidity (>90 %) increases co-infection	Numerical simulations	Minimizes co-infection	High (7-compartments, 7 ODEs)	IDM, resistant varieties	No field validation
[28]	CBD-CLR co-infection	Dual exposure	Control efficacy	IDM reduces $R_{o}$	Simulations- based	High efficacy	10-compartment	Farm-level	Complex

Table 2 summarizes key deterministic models for CBD, CLR, CBB, and Co-infection, outlining their mathematical structures, assumptions, key parameters, sensitivity analysis findings, and limitations. This study has significantly advanced understanding of CBD and its co-infection with CLR. However, several limitations persist, highlighting critical gaps that must be addressed to improve predictive accuracy and practical applicability.

To enhance clarity and improve readability, a summary table is provided. Table 2 presents a detailed comparison of individual model specifications, while Table 3 offers a more concise overview that categorizes the models by type, validation approach, complexity, and core intervention strategies. Readers can use the two formats to navigate through the literature; one can quickly find relevant modeling trends, such as the dominance of simulation as a validation method, the use of SEIR-type frameworks in CBD studies, and the general focus on optimal control strategies of CBD, CLR, CBB, and co-infection models in the studies that were reviewed.

Table 3Summary of model categories by type, validation approach, complexity, and intervention strategy.

Model Category	Model Type	Validation Method	Complexity	Key Intervention Strategy
CBD Models	Deterministic ODE	Simulations-Based, Bayesian	Medium	Microclimate, Shade control
CBD, Optimal Control Models	ODE with optimal control	Simulations-based, ICER	High	Biocontrol, Fungicides, Seasonal planning
CBD Fractional-Order Models	Fractional-order	Simulations-Based	Very High	Long-term dynamics
CLR Models	Deterministic	Simulation-based	High	Biocontrol, Seasonal Mgmt, Economic policies, Small farms, market-driven
CBB, Optimal Control Models	ODE with optimal control	Simulations-based, ICER	Medium-High	Traps, Biopesticides, Awareness, Climate-adaptive strategies
CBB Predator-Prey	ODE predator-prey	Simulation-based	Medium	Ant-based biocontrol, Farm-level biocontrol
CBB-Map	Discrete-time model	Simulation-based	Low–Medium	Seasonal harvest
CBD-CLR Co-infection model	Multi-pathogen SEIR	Simulation-based	Very High	IDM, Resistant varieties

Limitations of SEIR-type models

SEIR-type models often depend on simple assumptions that may overlook significant elements of CBD transmission. The assumption of homogeneous mixing overlooks the spatial heterogeneity of coffee farms, including local microclimatic variations and plant density gradients, which can influence the spread of pathogens [24]. Moreover, deterministic evolution ignores stochastic occurrences. For example, sporadic outbreaks driven by vector activity or unusual weather are critical to epidemics [47]. For instance, models by Melese et al. (2022b) and Nyaberi et al. (2023) do not account for the role of insect vectors or human-mediated dispersal [23,24] despite field evidence implicating these pathways in C. kahawae transmission [31].

Insights from co-infection models

Co-infection models suggest that concurrent infections with CBD and CLR result in increased disease severity, as co-infected plants exhibit elevated mortality rates due to a strain on host defenses, highlighting the necessity for integrated disease treatment measures [28]. Fungicide treatments aimed at a single pathogen may provide opportunities for secondary pathogens. However, existing models fail to consider interactive effects between diseases, which could further accelerate epidemics (e.g., through shared vectors or mechanisms of cross-resistance).

Neglect of vectors and environmental contamination

A significant limitation of current deterministic models for CBD, CLR, and CBB, as shown in Table 1, is the widespread neglect of vector-mediated transmission despite its critical role in disease dispersal. Field studies emphasize that insects, birds, and human activities (e.g., harvesting) contribute to spore distribution alongside rain splash; yet, few of the 23 reviewed studies, such as [23,53], incorporate vector transmission. Models like CBD SEIR [23,24,47,50], CLR SIR [27,48,51,52,55], and co-infection [28] assume direct pathogen transmission via rain splash or wind, ignoring vectors' role in amplifying $R_{0}$ and spatial spread. For instance, the co-infection model [28] misses potential vector-driven CLR-CBD synergy (RQ2), as insects may spread both pathogens, increasing disease severity. This omission reduces model accuracy for RQ1′s biological and spatial heterogeneity, underestimating epidemic peaks and limiting the effectiveness of intervention strategies (e.g., vector control).

Table 1Specifications table.

Subject area
More specific subject area	Modelling and Forecasting
Name of the reviewed methodology	Mathematical models for coffee berry disease modelling
Keywords	Coffee berry disease, Mathematical modelling, Plant disease epidemiology, Optimal control strategies, Co-infection dynamics
Resource availability	N.A.
Review questions	RQ1. How many current deterministic models for Coffee Berry Disease, Coffee Leaf Rust, Coffee Berry Borer, and co-infection with Coffee Berry Disease and Coffee Leaf Rust incorporate important biological, environmental and spatial heterogeneity related to disease transmission? RQ2. Deterministic models have been utilised to incorporate the interaction between CBD and CLR in scenarios of co-infection and to assess limitations on the representation of synergistic effects, vector dynamics, and environmental contamination. RQ3. How can future models address predictive reliability through field-based calibration, GIS integration, stochastic revisions of CBD deterministic models, and the methodological gaps in empirical validation?

To address this gap, future models should incorporate vector-host-pathogen interactions, extending frameworks like SEIR to include vector compartments. For example, a model could include Co-infection vector-host-pathogen interactions.

Climate-driven models and seasonal variability

Climate-driven models improve the link between transmission rates, temperature, and humidity [25]. These models determine the ideal conditions for a CBD outbreak (20–25 °C, > 90 % RH) and predict seasonal peaks that align with observed trends in East Africa. However, these models often overlook complex elements of microclimatic variability, such as how shade influences spore viability [47], which can impact disease development on a local scale.

Empirical validation and practical gaps

A standard limitation across all 24 models reviewed, as highlighted in Table 2, is the absence of empirical validation using field data. This gap, central to RQ3, undermines the predictive reliability of model outputs, such as disease incidence ( $R_{0}$ , prevalence), yield losses, and intervention cost-effectiveness (ICER), for real-world coffee agroecosystems. For instance, CBD SEIR models [23,24,47,50] rely on numerical simulations or Bayesian inference (e.g.,) [47] without calibrating variables like infected berries or latent period against field-observed infection rates. Similarly, the co-infection model [28] predicts synergistic mortality under high humidity (>90 %), but its lack of field validation limits confidence in its applicability to diverse farm settings. CLR models [27,48,51,52,55] and CBB optimal control [25,49,54,[60], [61], [62], [63], [64]] face comparable issues, with validations restricted to simulations despite theoretical advances (e.g., [48] on economic-epidemiological dynamics). This universal reliance on theoretical validation risks misrepresenting the dynamics of disease.

To address this gap, future studies should integrate empirical data to enhance model calibration and validation, aligning with RQ3′s call for predictive reliability. Geographic Information Systems (GIS) can map spatial disease hotspots by using data on CBD/CLR prevalence across coffee farms, which current models often overlook. These data can be integrated via statistical fitting (e.g., RMSE, Section 3.4.2) to align model predictions with observed seasonal peaks, providing infection rates that can be used to calibrate compartmental transitions in SEIR models [52,56,57,65] or co-infection dynamics [28]. Bootstrap methods (Section 3.4.2) could estimate confidence intervals for these parameters, addressing uncertainty in data. Combining these data sources with goodness-of-fit metrics (e.g., $R^{2}$ , AIC, Section 3.4.2) would enable robust validation, bridging the gap between theoretical models and practical disease management. These strategies, if implemented, could significantly improve the applicability of deterministic models to sustainable coffee production.

Conclusion

This study highlights that mathematical models have significantly enhanced the comprehension and management of CBD, CLR, and CBB, evolving from initial logistic growth models to more complex models that account for environmental variability, co-infections, and optimal control strategies. Key considerations include the significance of $R_{0}$ in forecasting outbreaks and the efficacy of combined interventions (e.g., fungicides and coffee-resistant types). Despite significant achievements in this domain, problems persist, including the assumption of homogenous mixing and neglect of data-based models or prediction model validation. Top-priority challenges include:

•Empirical Validation: Integrate GIS, climate data, and field trials (e.g., infected berries or leaves, infection rates) to calibrate models using RMSE and bootstrap methods (Section 3.4.2).
•Vector Transmission: Model vector-host-pathogen interactions to capture insect-mediated spread.
•Although host-vector transmission models have been explored in previous studies [23,53], these models do not address the complexities of the diseases transmission. They utilize oversimplified assumptions and do not account for critical components, such as infected coffee plants that have not shown symptoms and recovered coffee plants [24], and climate effects that are needed to accurately describe the realities of disease. Moreover, this modeling framework is still largely under-utilized in studies surrounding CBD and CLR, despite a growing body of scientific literature showing the large role of vectors in producing effective transmission dynamics.
•Farmer-Centric Costs: Assess GDP-adjusted intervention costs (e.g., ICER) to ensure economic viability for smallholders.
•Stochasticity: Incorporate stochasticity to effectively capture localised outbreaks.

Addressing these gaps will enhance the predictive accuracy and practical applicability, safeguarding global coffee production against emerging biotic and abiotic threats.

Limitations

This review is limited as it focuses primarily on deterministic models and misses potential insights from agent-based or machine-learning models. In addition, the accuracy of predictive capability is not quantitatively assessed due to the absence of a systematic comparison of model performance. Also, regional bias in the selected literature may limit broader generalizability.

CRediT author statement

Prof. Dr. Usa Wannasingha Humphries: Conceptualization, Funding acquisition, Methodology. Porntip Dechpichai: Software. Alhassan Ibrahim: Investigation, Writing – original draft. Muhammad Waqas: Writing – review. Boobphachard Chansawang: Resources, Supervision, Gabor Kiss: Project administration, review and editing. All authors have read and agreed to the published version of the manuscript.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Acknowledgements

This research project is supported by King Mongkut’s University of Technology Thonburi (KMUTT), Thailand Science Research and Innovation (TSRI), and National Science, Research and Innovation Fund (NSRF), Fiscal year 2026.

Funding

No Funding was used in this research.

Ethics statements

No data was used in this research.

References

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References

1.Krishnan S. Oxford Research Encyclopedia of Environmental Science. 2017. Sustainable coffee production.
2.Mussatto S.I.a.M. Production, composition, and application of coffee and its industrial residues. Food Bioproc. Tech. 2011:661–672.
3.Petzinger L. Coffee in Brasil: sustainability and the role of intermediaries. Capst. Collect. 2024
4.a. M. Kamwilu E., Peter A., Tanui Joseph, Hoan Do Trong. In: Vietnam’s Coffee Story: Lessons For African countries. Minang P.A., Duguma L.A., van Noordwijk M., editors. 2021.
5.FNC Federaci´on nacional de cafeteros de colombia (national federation of coffee growers of Colombia) Colomb. Coffee Sect. Study. 2020
6.Prabhakara.B.N D. Coffee production industries problems and prospects- an overview. IJCRT. 2018
7.Miyanari W. Savant Books and Publications; 2013. Aloha from Coffee Island.
8.a. D. Lee H.L., Stacy, James LaShawn, Mackwani Zeryn, Munoz Raul, Volk David. Building a sustainable supply chain: starbucks’ Coffee and farm equity program. Build. Supply Chain Excell. Emerg. Econ. 2007:391–405.
9.Wells M. SFGATE. 2025. Coffee prices skyrocket in the bay areawith the cost of coffee at an all-time high.
10.S. Technology, "The university of Texas at Austin," in Shade grown coffee shrinking as a proportion of global coffee production, ed, 2014.
11.Slavin T. The brazilian farmers who are trying to keep the world in its coffee habit despite climate change. Reuters. 2024
12.Chuqian W. Siam University Bangkok; Thailand: 2018. Study On the Situation and Development of the Coffee Industry in Thailand.
13.Humphries U.W., Waqas M., Hlaing P.T., Wangwongchai A., Dechpichai P. Determination of crop water requirements and potential evapotranspiration for sustainable coffee farming in response to future climate change scenarios. Smart Agric. Technol. 2024;8
14.Waqas M., Humphries U.W., Hlaing P.T. Time series trend analysis and forecasting of climate variability using deep learning in Thailand. Results Eng. 2024;24
15.WEGARI A. HARAMAYAUNIVERSITY; 2023. Response of Arsi Coffee Collections to Coffee Berry.
16.Massee G. Macmillan; 1914. Diseases of Cultivated Plants and Trees.
17.Van Der Plank J.E. Elsevier; 2013. Plant Diseases: Epidemics and Control.
18.Zadoks J. Systems analysis and the dynamics of epidemics. Phytopathology. 1971:600–610.
19.Kranz J. Comparison of epidemics. Annu. Rev. Phytopathol. 1974:355–374.
20.Rouse D. CABI Databases; 1988. Use of Crop Growth-Models to Predict the Effects of Disease.
21.Raposo R.a.W., DS, WE Fry. Evaluation of potato late blight forecasts modified to include weather forecasts: a simulation analysis. Phytopathology. 1993:103–108.
22.Shaw M. Modeling stochastic processes in plant pathology. Annu. Rev. Phytopathol. 1994:523–544. doi: 10.1146/annurev.py.32.090194.002515.
23.a. M. Melese A.S., Daniel Oluwole, Obsu Legesse Lemecha. Mathematical modelling and analysis of coffee berry disease dynamics on a coffee farm. Math. Biosci. Eng. 2022:7349–7373. doi: 10.3934/mbe.2022347.
24.Nyaberi H.a.M., WN, Malonza, David M., Gachigua G.W. A mathematical model of the dynamics of coffee berry disease. J. Appl. Math. 2023
25.Melese A.S. Mathematical model and optimal control analysis of coffee berry borer with temperature and rainfall variability. Int. J. Biomath. 2024
26.Trujillo-Salazar C.A.E.S.A.O.-T, Gerard, Sotelo-Castelblanco Deissy Milena. Bifurcation analysis in a coffee berry-borer-and-ants prey-predator model. Mathematics. 2024:1670.
27.Herskowitz Y.a.B.-M., Svetlana, Lazebnik Teddy. Mathematical model of coffee tree’s rust control using snails as biological agents. BioSystems. 2023 doi: 10.1016/j.biosystems.2023.104916.
28.a. M. Nyaberi H., WN, Malonza D.M., Gachigua G.W., Alworah G.O. An optimal control model for coffee berry disease and coffee leaf rust co-infection. J. Math. Anal. Model. 2024:1–25.
29.Bujulu J. East African Agricultural and Forestry Journal. 1977. Control of coffee berry disease in northern Tanzania: 1976 trials; pp. 427–430.
30.Waller J. Water-borne spore dispersal in coffee berry disease and its relation to control. Ann. Appl. Biol. 1972:1–18.
31.Waller J.a.B., PD, Black R., Hakiza G. Characterization of the coffee berry disease pathogen, colletotrichum kahawae sp. nov. Mycol. Res. 1993:989–994.
32.Nutman F.a.R., FM Investigations on a disease of coffea arabica caused by a form of Colletotrichum coffeanum Noack: I. Some factors affecting infection by the pathogen. Trans. Br. Mycol. Soc. 1960 489-IN4.
33.Vossen H.v.d.a.W., DJ Additional evidence for oligogenic inheritance of durable host resistance to coffee berry disease (Colletotrichum kahawae) in arabica coffee (Coffea arabica L.) Euphytica. 2009
34.Kueh K.a.M., SF, Facelli E., Facelli J.M., Velzeboer R.M.A., Able A.J., Scott E.S. Response of selected south Australian native plant species to phytophthora cinnamomi. Plant Pathol. 2012:1165–1178.
35.Page M.J.a.M., Joanne E., Bossuyt Patrick M, Boutron Isabelle, Hoffmann Tammy C, Mulrow Cynthia D, Shamseer Larissa, Tetzlaff Jennifer M, Akl Elie A, Brennan Sue E. "The PRISMA 2020 statement: an updated guideline for reporting systematic reviews. BMJ. 2021;372 doi: 10.1136/bmj.n71.
36.Wang J., et al. Maternal colonization with group B Streptococcus and antibiotic resistance in China: systematic review and meta-analyses. Ann. Clin. Microbiol. Antimicrob. 2023;22(1):5. doi: 10.1186/s12941-023-00553-7.
37.Langendorf R.E.a.B., Matthew G. Empirically classifying network mechanisms. Sci. Rep. 2021 doi: 10.1038/s41598-021-99251-7.
38.Maziarz M. Synthese. 2021. Resolving empirical controversies with mechanistic evidence; pp. 9957–9978.
39.Ling M.a.L., Xue, Chen Yu, Gu Fan, Lytton Robert L. Mechanistic-empirical models for top-down cracking initiation of asphalt pavements. Int. J. Pavement Eng. 2020:464–473.
40.Flaig J.a.H., Nicolas Epidemic control using stochastic and deterministic transmission models: performance comparison with and without parameter uncertainties. medRxiv. 2022 doi: 10.1016/j.cmpb.2024.108136. 2022-11.
41.Vieira A.C.M.a.e. Universidade de Lisboa (Portugal; 2018. Unraveling the Genomics of Adaptation of Colletotrichum Kahawae to Coffea Arabica.
42.Ibrahim A., Humphries U.W., Khan A., Iliyasu Bala S., Baba I.A., Rihan F.A. COVID-19 model with high-and low-risk susceptible population incorporating the effect of vaccines. Vaccines. 2022;11(1):3. doi: 10.3390/vaccines11010003.
43.N.R.A.S. Draper, Harry, "Applied regression analysis," John Wiley & Sons, 1998.
44.Akaike H. A new look at the statistical model identification. IEEE Trans. Autom. Contr. 1974:716–723.
45.Vazquez-Cruz M.a.G.-C., R, Lopez-Cruz I.L., Cornejo-Perez O., Torres-Pacheco I., Guevara-Gonzalez R.G. Global sensitivity analysis by means of EFAST and Sobol’methods and calibration of reduced state-variable TOMGRO model using genetic algorithms. Comput. Electron. Agric. 2014:1–12.
46.Hoare A.a.R., David G., Wilson David P. Sampling and sensitivity analyses tools (SaSAT) for computational modelling. Theor. Biol. Med. Model. 2008:1–18. doi: 10.1186/1742-4682-5-4.
47.Motisi N.a.P.i.x., Julien, Poggi Sylvain. The dark side of shade: how microclimates drive the epidemiological mechanisms of coffee berry disease. Phytopathology. 2022:1235–1243. doi: 10.1094/PHYTO-06-21-0247-R. {\textregistered}
48.Lazebnik T.a.R., Ariel, Shami Labib. Economical-epidemiological analysis of the coffee trees rust pandemic. Chaos. 2024 doi: 10.1063/5.0221945.
49.A.S. Melese, "Modelling and analysis of temperature and rainfall variability impact on the coffee berry borer infestation with optimal control," 2022.
50.Jeeva N., Dharmalingam K. Sensitivity analysis and semi-analytical solution for analyzing the dynamics of coffee berry disease. Компьютерные Исследования и Моделирование. 2024;16(3):731–753.
51.Arroyo-Esquivel J., Sanchez F., Barboza L.A. Infection model for analyzing biological control of coffee rust using bacterial anti-fungal compounds. Math. Biosci. 2019;307:13–24. doi: 10.1016/j.mbs.2018.10.009.
52.Djuikem C., Grognard F., Wafo R.T., Touzeau S., Bowong S. Modelling coffee leaf rust dynamics to control its spread. Math. Model. Nat. Phenom. 2021;16:26.
53.Rizka S.A.R.a.K., Nadia Kontrol optimal untuk model coffee berry disease dengan vektor Pembawa colletotrichum kahawae: optimal control for coffee berry disease model with carrier vector of C. kahawae. MathVision. 2024:77–87.
54.Abaraya Abawari M.a.L.O., Legesse, Melese Shiferaw, Abdisa Optimal control analysis of coffee berry borer infestation in the presence of farmer's awareness. Appl. Math. Sci. Eng. 2023
55.C. Djuikem, A.G. Yabo, F. Grognard, and S. Touzeau, "Mathematical modelling and optimal control of the seasonal coffee leaf rust propagation. IFAC-Pap. 54 (5): 193–198, 2021," in 7th IFAC Conference on Analysis and Design of Hybrid Systems ADHS, 2021, pp. 89–92.
56.Melese A.S., Makinde O.D., Obsu L.L. Modelling and optimal control analysis of coffee berry disease with cost-effectiveness in the presence of temperature variability. Int. J. Comput. Sci. Math. 2023;18(1):17–31.
57.A.K. Fantaye, K.T. Ergete, F. Alshowaikh, M. Hafez, and B.W. Niu, Atangana Baleanu Caputo fractional order modeling and analysis for the transmission of coffee berry. (2025).
58.Trujillo-Salazar C.A., Olivar-Tost G., Sotelo-Castelblanco D.M. Bifurcation analysis in a coffee berry-borer-and-ants prey–predator model. Mathematics. 2024;12(11):1670.
59.Trujillo-Salazar C.A., Olivar-Tost G., Sotelo-Castelblanco D.M. Mathematical model for the biological control of the coffee berry borer hypothenemus hampei through ant predation. Insects. 2023;14(8):675. doi: 10.3390/insects14080675.
60.Fotso Y.F. Université de Dschang (Cameroun; 2022. Modelling, Analysis and Control of Coffee Berry Borer.
61.Fotso Fotso Y.a.T., Suzanne, Tsanou Berge, Bowong Samuel, Grognard Fr{\'e}d{\'e}ric. Modelling and optimal strategy to control coffee berry borer. Math. Methods Appl. Sci. 2021:14569–14592.
62.Fotso Y.F., Touzeau S., Tsanou B., Grognard F., Bowong S. Mathematical modelling of a pest in an age-structured crop model: the coffee berry borer case. Appl. Math. Model. 2022;110:193–206.
63.Fotso Y.F., Grognard F., Tsanou B., Touzeau S. Modelling and control of coffee berry borer infestation. CARI’2018-14. Colloq. Afr. Sur Rech. Inform. Math. Appl. 2018
64.Fotso Fotso Y.a.T., Suzanne, Grognard Fr{\'e}d{\'e}ric, Tsanou Berge, Bowong Samuel. Optimal control of coffee berry borers: synergy between bio-insecticide and traps. J. Optim. Theory Appl. 2023:882–899.
65.Marcano M., Bose A., Bayman P. A one-dimensional map to study multi-seasonal coffee infestation by the coffee berry borer. Math. Biosci. 2021;333 doi: 10.1016/j.mbs.2020.108530.
66.Melese A.S., Makinde O.D., Obsu L.L. Optimal control strategies for coffee berry disease with cost-effectiveness analysis in the presence of climatic variability. J. Appl. Nonlinear Dyn. 2023;12(02):191–211.
67.Lenhart S.a.W., John T. Chapman and Hall/CRC; 2007. Optimal Control Applied to Biological Models.

Studies

Conditions

Cannabinoids and active compounds

Related research

Additional materials

Abstract

Untitled section

Graphical abstract

Article notes

Untitled section

Author notes

Background

Biological background

Infection

Colonization

Sporulation

Dispersal

Quantitative disease parameters and varietal susceptibility

Methods

Search strategy

Inclusion and exclusion criteria

Data extraction

Quality assessment

Mathematical modelling approaches

Fundamental concepts of epidemiology

Disease progress curves

Fundamental concepts of epidemiology

SIR models and extensions

Basic SIR model

SI/SEIR extensions for plant diseases

Basic reproduction number

Environment drivers and seasonal forcing

Model evaluation and validation

Qualitative analysis

Quantitative validation

Sensitivity analysis

Results

Study characteristics

Quality assessment of included studies

Deterministic models for CBD

Compartmental models for CBD dynamics

Compartmental models for CLR dynamics

Co-infection models for CBD and CLR

Environmental and climate-driven models

Model comparison

Limitations of SEIR-type models

Insights from co-infection models

Neglect of vectors and environmental contamination

Climate-driven models and seasonal variability

Empirical validation and practical gaps

Conclusion

Limitations

CRediT author statement

Declaration of competing interest

Acknowledgments

Acknowledgements

Funding

Ethics statements

References

Untitled section

References