Entomopathogenic fungi can cause infectious diseases within host insect populations, while insects may counteract disease transmission through the strategy of isolating infected individuals. This study develops a system of differential equations to model the temporal dynamics of susceptible and infected individuals, as well as the concentration of fungal spores in the environment, to investigate the complex dynamics of disease transmission. The study focuses on three critical parameters: infection probability, population size, and isolation rate. The results show that infection probability is a key determinant of disease spread; as infection probability increases, the basic reproduction number (R_0) rises, potentially leading to periodic oscillations in disease incidence, highlighting the nonlinear response of disease transmission to changes in virulence. This underscores the importance of managing environmental factors such as temperature and humidity, which influence infection probability, to control disease spread. Furthermore, the analysis reveals that larger populations, due to higher contact frequency, exhibit more pronounced and persistent disease dynamics, suggesting that population density control is an effective strategy for managing disease outbreaks. The study also identifies isolation behavior as crucial for disease control, where moderate increases in isolation rates can lead to complete eradication under certain conditions. The timing of spore production relative to host death is also identified as a key variable affecting the efficacy of isolation. In conclusion, this study elucidates the complex interplay between infection probability, population size, and isolation behavior, providing a theoretical basis for optimizing biological control strategies using entomopathogenic fungi and managing high-density insect populations.

In this paper, we analyze a predator-predator-parasite model, which considers prey subjected to predator fear effect and two modes of harvesting (cooperation hunting by predators and artificial harvesting of prey). Firstly, mathematical analysis of the model with regard to the non-negativity, boundedness of solutions, stability of equilibria, permanence, backward bifurcation and Hopf-bifurcation of the model are analyzed. Secondly, we obtain some numerical simulation results: we conduct extensive numerical simulations to explore the roles of fear effect and other biologically related parameters (e.g. disease transmission rate of prey, hunting cooperation and artificial harvesting), we find that low levels fear and hunting cooperation can stabilize the eco-epidemiological system, and low level artificial harvesting will produce chaotic oscillations, but high level artificial harvesting will stabilize the eco-epidemiological system. In reality, we should protect vulnerable species from fear and cooperation hunting by predators, because decrease in the number of vulnerable species may lead to decrease in species diversity. Moreover, artificial harvesting of species with certain economic value is generally carried out (such as fish populations). However, if the species density is too low, the chaotic oscillation of the population is dangerous to the survival of the population, increasing the probability of population extinction, and it is impossible to obtain a constant yield from the fluctuating population. Therefore, it is desirable to promote the overall stable equilibrium state of the system by artificial harvesting. In addition, it is also observed the presence of disease in an ecosystem can promote stable coexistence of species.