Mathematical assessment of the optimal intervention strategy for Cercospora fungal disease in orange plants

Abstract

Cercospora fragariae is a fungal plant pathogen. Citrus trees and shrubs are particularly vulnerable to Cercospora infections, which can result in significant crop losses. This paper presented a new mathematical model of Cercospora fungal disease transmission dynamics in orange plants, which incorporates two control interventions – plant inoculation or immunization for resistance to the disease, $\nu$, and quarantine of infective orange plants, $\alpha$. The model was qualitatively analysed to establish the disease-free and endemic equilibria associated with it. The next generation matrix method was used to compute the effective reproduction number, $R_e$, of the model. The model was later extended to an optimal control model by considering the two interventions as time-dependent control functions $\nu \left(t\right)$ and $\alpha \left(t\right)$, respectively. These time-dependent controls were characterized using Pontryagin’s maximum principle. Qualitative analysis of the autonomous model reveals that the disease-free equilibrium is both locally and globally asymptotically stable if $R_e<1$, and unstable otherwise, whereas the endemic equilibrium is globally asymptotically stable in the absence of re-infection whenever $R_e>1$. Under the situation of no re-infection, additional analysis of the model establishes the existence of forward bifurcation. Simulations of the autonomous model show that the constant control rate $\alpha$ is more sensitive than the constant control rate $\nu$ as small increase in $\alpha$ value influences the dynamics of the disease transmission as compared to high increase in $\nu$ value. Simulated results of the optimal control model suggest that the use of only optimal quarantine of infective orange plants, $\alpha \left(t\right)$, is as good as the combination of the two optimal controls as an intervention strategy when compared to the use of only optimal inoculation of orange plant for resistance to disease, $\nu \left(t\right)$. A cost analysis was conducted to identify the most cost-effective strategy for the prevention and control of disease spread with limited resources.