Abstract
COVID-19, a pandemic that affected the whole world, claimed the lives of almost 1.1 million people in the United States and 7 million worldwide. Prior to the discovery of vaccines, many countries resorted to implementing lock downs to reduce the spread of the virus. Most economies have implemented this policy, except in essential sectors such as public health and safety. Different states in the US have imposed lock downs at different times, based on the severity of the outbreak in their respective regions. Lock downs involve reducing social interactions, leading to a decrease in the transmission of the virus. However, if lock downs remain in effect for too long, people may become hesitant to resume social activities for fear of contracting COVID-19 [3]. Hence, businesses are facing a reduction in the number of consumers and employees, resulting in a decrease in sustainable long-term protability. Furthermore, if a business fails to have enough inventory to with stand the crisis, it may eventually shut down. Since the government is not providing nancial support, it is easy to shut down a business, but dicult to restore it to its original employment levels [3]. It is recommended by the Centers for Disease Control and Prevention (CDC) that anyone infected with Omicron should isolate themselves for five days. This is because a person infected with the virus can spread it to others, so isolation helps reduce transmission. Similarly, if more people are vaccinated, the virus will spread less and fewer people will be affected, thus saving more lives. In their study, Pramanik (2023) [10] determined the best way to decide when to shut down an economy and what rate of vaccination is optimal. They used a healthcare cost function that was minimized while taking into account a stochastic susceptible infectious-recovered (SIR) dynamic, which was first introduced in Aron et al (1984) [1]. Most models of infectious disease transmission are based on the SIR model. Pramanik's construction can be extended to a generalized random surface to investigate unprecedented shocks, such as the emergence of a new COVID-19 variant, sudden infection due to random interactions caused by travel, and environmental calamities resulting in more exposure to the pandemic. The random surface replaces the jump diffusion of the stochastic differential equations.