Mathematical modeling of epidemiological diseases, stability analysis and intervention strategies

In this work, we constructed an optimal control problem (OCP) based on a SIR model [21] where dynamics of susceptible, infected and recovered individuals are modeled as a system of ordinary differential equations under the effect of imperfect testing. As different from the classical SIR models, Villela's model includes two more compartments where the compartment S^ describes the number of tested susceptible individuals that are identified incorrectly as infected and the compartment I^ denotes the number of infected individuals whose treatment are started after testing. To construct the required OCP, we obtained the basic reproduction number R_0. We found out that contact rate and testing rate are critical parameters to decrease the number infected individuals. Therefore, we applied some intervention strategies to optimize the contact rate so that spread of the disease can be controlled [24]. Moreover, we determined the optimal testing rate. Numerical results show the usefulness of the optimization strategies and we find out that quarantine and isolation are important interventions in case of limited testing. In addition, we learned that the spread of the disease can be successfully controlled together with optimal testing strategy.