A Plant Disease Model Incorporating Environmental Inoculum and Seasonal Factors with Optimal Control Approach

This paper introduces a three-variable mathematical model that integrates susceptible biomass (\(S\)), infected biomass (\(I\)), and environmental inoculum (\(E\)), offering a novel framework that extends beyond classical SIR-type models by explicitly incorporating inoculum ecology and climate-driven dynamics. The model captures nonlinear infection saturation, logistic host growth, seasonal forcing, and a control intervention representing management practices such as fungicide application or crop rotation. Analytical exploration of the system reveals the existence of both disease-free and endemic equilibria, with stability determined by the basic reproduction number, \(R_0\). When \(R_0 < 1\), the disease-free equilibrium is stable, leading to the eradication of infection and recovery of host biomass, whereas \(R_0 > 1\) results in endemic persistence sustained by environmental inoculum reservoirs. Possible outcomes include stable disease-free states, endemic equilibria, seasonal oscillations driven by climatic variability, and shifts toward eradication under effective control strategies. Sensitivity analysis highlights that \(R_0\) is strongly influenced by inoculum survival rates, host growth capacity, and seasonal forcing intensity, underscoring the ecological complexity of plant disease dynamics. Numerical simulations substantiate these theoretical findings, demonstrating that strong seasonal forcing can destabilize equilibria and generate recurrent epidemic waves, while optimal control interventions can suppress infection prevalence by reducing \(R_0\) below unity. The results emphasize the critical role of environmental reservoirs in sustaining epidemics and provide a robust theoretical foundation for designing sustainable agroecological forecasting and management strategies under climate variability.