THE STRUCTURE OF SINGULAR CONTROLS OF VSIR MODEL

33 [ ] and [ ] We first analyze the control u, i.e., vaccination schedules. Applying Proposition 2 once more to , it follows that [ [ ]] 124 and [ [ ]] 125 The switching function of 13 is 126 Implies that [ [ ]] 127 and gives that and must be positive along a singular arc. Hence we have that [ [ ]] . Singular controls of this type , i.e., for which [ [ ]] does not vanish, are said to be of order 1 and it is a second-order necessary condition for minimality, the are so called legendre-Clebsch condition, that this quantity be negative [9]. Thus for this model singular controls u are locally optimal. Furthermore, we can compute the singular control as [ [ ]] [ [ ]] 128 To evaluate the vector fields, this equation can be simplified. A direct, but some what lengthy computation shows 34 [ [ ]] and [ [ ]] 129 we can write [ [ ]] [ ] [ [ ]] where Since [ ] , it follows from 16 that Once more using 14, we simplified the second term to and we obtain the following result: Proposition 7 . A singular control u is of order 1 and satisfied the Legendre-Clebsch condition for minimality. The singular control is given as a function depending both on the state and the multiplier in the form Based on the structure of singular control we apply the same way to analysis treatment control . Let switching function give The first derivative of is [ ] and the second derivative is given by [ [ ]] . Furthermore, a direct calculation verifies that 35 [ [ ]] [ ] Since and [ ] commute, it follows from the Jacobi identity that [ [ ]] [ [ ]] . we found [ ] and thus also [ [ ]] and [ ] and [ [ ]] . Thus there is no singular on we obtain the following result Proposition 8 . The control v cannot be singular.

4.9 FORMULATION AS AN OPTIMAL CONTROL PROBLEM OF VSEIR

MODEL Let the population sizes of all there classes, and are given, find the best strategy in terms of combined efforts of vaccination and treatment that would minimize the number of people who die from the infection while at the same time also minimizing the cost vaccination and treatment of the population. In this paper, we consider the following objective for a fixed terminal time : ∫ 130 The first term in the objective, represents the number of exposed who are infected but are yet to show any sign of symptoms at time , , represents the number of people who are exposed and infected at time and are taken as measure for the deaths associated with the outbreak. The terms, and represent the cost of vaccination and treatment, respectively, and are assumed to be proportional to the vaccination and treatment rates. We shall apply methods of geometric optimal control theory to analyse the relations between optimal vaccination and treatment schedules. These techniques become more transparent if the problem is formulated as a Mayer –type optimal control problem : that is , one where we only minimize a penalty term at the terminal point. Such a structure can easily 36 be achieved at the cost of one more dimension if the objective is adjoined as an extra variable. Defining . 132 We therefore consider the following optimal control problem. For a fixed terminal time, minimize the value subject to the dynamics 133 134 135 136 137 where . Over all Lebesque measurable function [ ] [ ] and [ ] [ ] Introducing the state , the dynamics of the system is a multiinput control- affine system of the form 138 with drift vector field given by 139 and control vector fields and given by and . 140 We call an admissible control pair with corresponding solution a controlled trajectory of the system.

4.10 NECESSARY CONDITIONS FOR OPTIMALITY OF VSEIR MODEL