Crime and corruptions are two major problems in developing nations. They spread like infectious disease in the population. Here, a nonlinear mathematical model is formulated and analyzed to study the interaction between criminal population and non-criminal population by considering non-monotone incidence rate. Non-monotone incidence rate shows a behavioral change in non-criminal populations. First, we consider constant recruitment and death type demography and then we take logistic growth type demography. For both the models the basic reproduction number R is computed. All possible equilibria of the models are obtained and stabilities of these equilibria are discussed in details. Further these models are extended to optimal control problem to study the impact of time-dependent law enforcement rate on the size of criminal population. Numerical simulations are presented to illustrate our analytical findings. © 2020, Springer-Verlag GmbH Austria, part of Springer Nature.