Traditional observer design methods rely on certain properties of the system's nonlinearity, such as Lipschitz continuity, one-sided Lipschitzness, a bounded Jacobian, or quadratic boundedness. These properties are described by parameterized inequalities. However, enforcing these inequalities globally can lead to very large parameters, resulting in overly conservative observer design criteria. These criteria become infeasible for highly nonlinear applications, such as networked epidemic processes. In this paper, we present an observer design approach for estimating the state of nonlinear systems, without requiring any parameterization of the system's nonlinearities. The proposed observer design depends only on systems' matrices and applies to systems with any nonlinearity. We establish different design criteria for ensuring both asymptotic and exponential convergence of the estimation error to zero. To demonstrate the efficacy of our approach, we employ it for estimating the state of a networked SIR epidemic model. We show that, even in the presence of measurement noise, the observer can accurately estimate the epidemic state of each node in the network. To the best of our knowledge, the proposed observer is the first that is capable of estimating the state of networked SIR models.