Vaccination and treatment as control interventions in an infectious disease model with their cost optimization

In this work, an optimal control problem with vaccination and treatment as control policies is proposed and analysed for an SVIR model. We choose vaccination and treatment as control policies because both these interventions have their own practical advantage and ease in implementation. Also, they are widely applied to control or curtail a disease. The corresponding total cost incurred is considered as weighted combination of costs because of opportunity loss due to infected individuals and costs incurred in providing vaccination and treatment. The existence of optimal control paths for the problem is established and guaranteed. Further, these optimal paths are obtained analytically using Pontryagin's Maximum Principle. We analyse our results numerically to compare three important strategies of proposed controls, viz.: vaccination only; with both treatment and vaccination; and treatment only. We note that first strategy (vaccination only) is less effective as well as expensive. Though, for a highly effective vaccine, vaccination alone may also work well in comparison with treatment only strategy. Among all the strategies, we observe that implementation of both treatment and vaccination is most effective and less expensive. Moreover, in this case the infective population is found to be relatively very low. Thus, we conclude that the comprehensive effect of vaccination and treatment not only minimizes cost burden due to opportunity loss and applied control policies but also keeps a tab on infective population.