Mathematical Modeling on the Strategies of Imperfect Vaccination, Quarantine, and Natural Immunity in a Future Epidemic

In this paper, we present a mathematical analysis of the extent to which contact measures such as masks and distancing need to be balanced with vaccine efficacy, vaccination rate, natural immunity, and quarantine to mitigate the outbreak in the event of a future pandemic and formulate a method to answer the question of what to implement, when, and how. At the end, we have simulated various strategies from the time that there is no vaccine for different time periods, one after the other, as a course of action in the fight against the pandemic. The model is written as a system of nonlinear ordinary differential equations of nonvaccinated, vaccinated, exposed, quarantined, infected, and recovered individuals. Basic reproduction number (Formula presented.) and vaccine and quarantine reproduction numbers (Formula presented.) are determined. It is proven that the disease-free equilibrium point is locally asymptotically stable if (Formula presented.) and globally asymptotically stable if there exists such (Formula presented.) so that (Formula presented.). The sensitivity analysis has also been investigated. The derived conditions between constrains state the balance of the effects of vaccine rate, efficiency, contact rate, mask, lockdown policy, and the quarantine rate by the various nature of the virus. The theoretical results are confirmed by simulation of the numerical results. For a future pandemic, the conditions in which natural immunity could play a role in controlling the outbreak is also discussed by simulations.