Mathematical model for Covid-19 with “protected susceptible” in the post-lockdown era

Lockdown is one of the drastic measures implemented by governments to curtail the spread of the Covid-19 pandemic and save lives. However, it has caused unprecedented damages to the economy. This paper provides a quantitative approach to assess the impact of a gradual, post-lockdown context concerning the spread of the disease. We propose to create a special class of individuals called “protected” who are risk-free to be infected. Such individuals could also be the vaccinated when an effective vaccine will be available. We developed a mathematical epidemic model for Covid-19 which describes the interactions between susceptible and infected individuals. We investigate the various and optimal strategies to curtail the spread of the infection at a minimum cost. As a case study on South Africa, the sensitivity analysis shows that investing on the special class “protected” is a better approach to reducing new secondary infections as opposed to reducing the contact rate between susceptible and infected individuals, or having more recovered patients. The simulations reveal that the peak could be reached in September 2020. This is consistent with the projection of the South African government as the winter season is expected to be over in mid August. Moreover, if 1 out of 1000 susceptible (cumulatively) join the special class, we project a maximum of 400,000 active cases. The number of infected and deaths could drastically increase as the proportion of individuals joining the special class decreases.