Robustness of behaviorally induced oscillations in epidemic models under a low rate of imported cases

D.J. and D.R. have been partially supported by the Agencia Estatal de Investigación and Ministerio de Ciencia, Innovación y Universidades Grant No. MTM2017-86795-C3-1-P, and D.J. and J.S. have been partially supported by Grant No. PID2019-104437GB-I00 of the Ministerio de Ciencia e Innovación of the Spanish government. D.J. and D.R. are members of the Consolidated Research Group 2017 SGR 1617 funded by AGAUR (Generalitat de Catalunya). J.S. is member of the Consolidated Research Group 2017 SGR 01392 of AGAUR (Generalitat de Catalunya).This paper is concerned with the robustness of the sustained oscillations predicted by an epidemic ODE model defined on contact networks. The model incorporates the spread of awareness among individuals and, moreover, a small inflow of imported cases. These cases prevent stochastic extinctions when we simulate the epidemics and, hence, they allow to check whether the average dynamics for the fraction of infected individuals are accurately predicted by the ODE model. Stochastic simulations confirm the existence of sustained oscillations for different types of random networks, with a sharp transition from a nonoscillatory asymptotic regime to a periodic one as the alerting rate of susceptible individuals increases from very small values. This abrupt transition to periodic epidemics of high amplitude is quite accurately predicted by the Hopf-bifurcation curve computed from the ODE model using the alerting rate and the infection transmission rate for aware individuals as tuning parameters