Considering the drug treatment strategy in both virus-to-cell and cell-to-cell transmissions, this paper presents an HIV model with Filippov control. Given the threshold level $ N_t $, when the total number of infected cells is less or greater than threshold level $ N_t $, the threshold dynamics of the model are studied by using the Routh-Hurwitz Criterion. When the total number of infected cells is equal to $ N_t $, the sliding mode equations are obtained by Utkin equivalent control method, and the dynamics are studied. In addition, the optimal control strategy is introduced for the case that the number of infected cells is greater than $ N_t $. By dynamic programming, the Hamilton-Jacobi-Bellman (HJB) equation is constructed, and the optimal control is obtained. Numerical simulations are presented to illustrate the validity of our results.
Keywords: HIV model; cell-to-cell transmission; optimal control; sliding mode dynamics; threshold level.