Optimal control problem in an epidemic disease SIS model with stages and delays

Abstract

Based on literature [J. Q. Li, Z. E. Ma and F. Q. Zhang, Stability analysis for an epidemic model with stage structure, J. Appl. Math. Comput. 9 (2008) 1672–1679], incorporating the recovery of the infected population with the length of the infectious periods, a modified epidemic disease SIS model with delay and stage was investigated. First, the criteria keeping stability with delay were given. Next, in order to lower the level of the infected individuals and minimize the cost of treatment, mixed, early and late therapeutic strategies were introduced into our model, respectively. Then we investigated the existence and uniqueness of optimal controls. And then, we expressed the unique optimal control in terms of the solution of the optimality systems. Finally, by numerical simulations, several important results were acquired: (1) The terminal time influenced the early optimal control largely. In detail, for a shorter terminal time it was optimal to initiate treatment with maximal effort at the start of the epidemic and continue treatment with maximal effort until the switch time was arrived. But for a longer terminal time, the maximal treatment effort need not be a prerequisite at the start or end of the epidemic but it was obligatory at the metaphase of the epidemic. (2) For our SIS model, minimizing the total infectious burden of the disease can be achieved by only early optimal treatment tactics. (3) For a disease with a shorter infectious period time, more cost would be spent to control the disease in order to achieve the optimal control objective. Otherwise, a relative lower cost would be to control the disease with a longer infectious period.