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The existence of landscape constraints in the home range of living organisms that adopt Lévy-flight movement patterns, prevents them from making arbitrarily large displacements. Their random movements indeed occur in a finite space with an upper bound. In order to make realistic models, by introducing exponentially truncated Lévy flights, such an upper bound can thus be taken into account in the reaction-diffusion models. In this work, we have investigated the influence of the $\lambda$-truncated fractional-order diffusion operator on the spatial propagation of the epidemics caused by infectious diseases, where $\lambda$ is the truncation parameter. Analytical and numerical simulations show that depending on the value of $\lambda$, different asymptotic behaviours of the travelling-wave solutions can be identified. For small values of $\lambda$ ($\lambda \gtrsim 0$), the tails of the infective waves can decay algebraically leading to an exponential growth of the epidemic speed. In that case, the truncation has no impact on the superdiffusive epidemics. By increasing the value of $\lambda$, the algebraic decaying tails can be tamed leading to either an upper bound on the epidemic speed representing the maximum speed value or the generation of the infective waves of a constant shape propagating at a minimum constant speed as observed in the classical models (second-order diffusion epidemic models). Our findings suggest that the truncated fractional-order diffusion equations have the potential to model the epidemics of animals performing Lévy flights, as the animal diseases can spread more smoothly than the exponential acceleration of the human disease epidemics. Full article