In this work, a swarming computational procedure is presented for the numerical treatment of the dynamical model of the susceptible, exposed, infected, and recovered (SEIR) classes that portrayed the spreading of Zika virus. The artificial neural network procedures (ANNPs) have been applied to solve the SEIR mathematical model for spreading of the Zika virus together with the hybridization efficiency of global swarming and local search schemes. The global particle swarm optimization (PSO) and local search active-set algorithm (ASA) have been proposed to solve the model. An error based objective function is presented for the SEIR differential model and then optimized by the hybrid computing efficiency of PSO-ASA. Five neurons, fifteen variables of each class and ten numbers of trials have been used to solve the SEIR mathematical model for spreading of the Zika virus. The correctness of the proposed computing ANNPs-PSO-ASA is observed by using the comparison of the obtained and reference solutions along with the performances of the absolute error, ranges around 10−06 to 10−08. The reliability of the designed computing ANNPs-PSO-ASA technique is observed by using the statistical operator performances on single/multiple trials for the SEIR system for spreading of the Zika virus dynamics.