Fractional order modeling of parasite-produced marine diseases with memory effect

Marine parasites are strongly linked to marine diseases; they negatively impact fish farming by reducing production and hindering sustainable industry development. However, there is less research on infectious diseases in marine environments compared to terrestrial systems. In this paper, we investigate the mathematical modeling and dynamics of marine diseases caused by parasites, incorporating memory effects. We propose a fractional-order Susceptible-Infectious-Parasite (SIP) model featuring a non-linear general incidence rate. The computation of the basic reproduction number is provided, along with discussions on positivity, boundedness, and the existence and uniqueness of solution. We also explore the local stability of equilibrium states and conduct a sensitivity analysis of parameters to identify those most significant in the model. Our findings show that when the basic reproduction number is less than 1, the disease-free equilibrium is locally asymptotically stable, and when it exceeds 1, the endemic equilibrium point is asymptotically stable. Key parameters influencing disease transmission by marine parasites include the rates of parasite production by infected hosts, and transmission from susceptible to infectious individuals. Numerical simulations validate theoretical results, demonstrating that incorporating memory effects provides more realistic disease dynamics, influencing the time required to reach stable states. We find that the number of marine parasites peaks in correlation with the formula of the incidence rate.