In this paper, a SIRP epidemic model is proposed, wherein the pathogens derive from two ways, i.e., environmental breeding, and accumulative excretion by the infected individuals. The former is characterized by Logistic growth, while the latter is in the form of infinite integral. First, the positivity and ultimate boundedness of solutions are obtained. Second, the basic reproduction number \(\mathcal {R}_{0}\) is obtained, by which the model is analyzed if either the intrinsic growth rate of environmental pathogens is lower or higher than its clearance rate. For the first case, the disease-free equilibrium is globally asymptotically stable when \(\mathcal {R}_{0}<1\) , while the endemic equilibrium is globally asymptotically stable when \(\mathcal {R}_{0}>1\) . Conversely, if the growth rate exceeds the removal rate, the disease-free equilibrium is always unstable, meanwhile, the uniform persistence of the model indicates that there could exist one or multi-endemic equilibria, and it is globally asymptotically stable if the endemic equilibrium is unique. Finally, the theoretical results are illustrated by numerical simulations. We find that the accumulative release of pathogens by the infected individuals in the form of infinite integral is more realistic and consistent with the disease spread than that of linear form by real data.
