Optimal control problem arising from COVID-19 transmission model with rapid-test

The world health organization (WHO) has declared the Coronavirus (COVID-19) a pandemic. In light of this ongoing global issue, different health and safety measure has been recommended by the WHO to ensure the proactive, comprehensive, and coordinated steps to bring back the whole world into a normal situation. This is an infectious disease and can be modeled as a system of non-linear differential equations with reaction rates which consider the rapid-test as the intervention program. Therefore, we have developed the biologically feasible region, i.e., positively invariant for the model and boundedness solution of the system. Our system becomes well-posed mathematically and epidemiologically for sensitive analysis and our analytical result shows an occurrence of a forward bifurcation when the basic reproduction number is equal to unity. Further, the local sensitivities for each model state concerning the model parameters are computed using three different techniques: non-normalizations, half-normalizations, and full normalizations. The numerical approximations have been measured by using System Biology Toolbox (SBedit) with MATLAB, and the model is analyzed graphically. Our result on the sensitivity analysis shows a potential of rapid-test for the eradication program of COVID-19. Therefore, we continue our result by reconstructing our model as an optimal control problem. Our numerical simulation shows a time-dependent rapid test intervention succeeded in suppressing the spread of COVID-19 effectively with a low cost of the intervention. Finally, we forecast three COVID-19 incidence data from China, Italy, and Pakistan. Our result suggests that Italy already shows a decreasing trend of cases, while Pakistan is getting closer to the peak of COVID-19.