Initial size sensitivity in Huanglongbing dynamics: How branching processes challenge deterministic outbreak predictions

Huanglongbing is a serious threat to the citrus industry. In the early stage of Huanglongbing infection, when the number of infected citrus trees and psyllids is sufficiently small, the deterministic system fails to accurately depict the transmission dynamics. However, continuous-time Markov chain (CTMC) can effectively describe these dynamic characteristics through state discretization for small population sizes. To this end, we first study a stochastic system based on CTMC and then estimate the probability of disease extinction and outbreak by applying the Galton–Watson branching process (GWbp). A comparison analysis between the deterministic and stochastic systems of Huanglongbing yields the following insights: $\left(1\right)$ when $R_0>1$, the deterministic system predicts Huanglongbing infection outbreak, whereas the stochastic system indicates a nonzero probability of Huanglongbing infection elimination; $\left(2\right)$ The initial number of infected citrus trees and infected psyllids does not affect the dynamics of the deterministic system, while the stochastic system’s dynamics are highly sensitive to the initial sizes and cannot be ignored. These findings have significant implications for understanding the epidemiological characteristics of Huanglongbing and provide a theoretical foundation for designing more precise and effective control strategies.