Analytical and numerical explorations of optimal control techniques for the bi-modal dynamics of Covid-19

The emergence of coronavirus disease (Covid-19) triggered a global pandemic with profound health, social, and economic impacts. Despite extensive governmental efforts, the virus remains a persistent worldwide threat. In this research, we introduce a nonlinear bi-susceptible model, also called bimodal dynamical system to study the dynamics of Covid-19. We delve into its transmission modes, risk factors, and potential long-term effects. Using analytical mathematical techniques, we ascertain the behaviors exhibited by the dynamic system at two main equilibrium states by imposing essential conditions on threshold parameter, thereby validating and affirming its inherent properties. To validate findings, we apply the nonstandard finite difference (NSFD) technique. By adjusting vaccination and hospitalization rates through constant control methods, we perform quantitative analysis and find that combining these measures with awareness expedites pandemic elimination. We identify influential parameters and formulate an optimal control problem with associated optimality conditions, determining effective time-dependent controls. Our study provides an evidence of the effectiveness of control strategies in achieving the desired outcome of reducing both financial costs and infection spread. The novelty of this research lies in utilizing a structure-preserving NSFD numerical scheme, backward in time, to analyze optimally the developed bi-susceptible Corona model.