Controlling the Spread of COVID-19: Optimal Control Analysis

Abstract

Coronavirus disease 2019 (COVID-19) is a disease caused by severe acute respiratory syndrome coronavirus 2 (SARS CoV-2). It was declared on March 11, 2020, by the World Health Organization as pandemic disease. The disease has neither approved medicine nor vaccine and has made governments and scholars search for drastic measures in combating the pandemic. Regrettably, the spread of the virus and mortality due to COVID-19 has continued to increase daily. Hence, it is imperative to control the spread of the disease particularly using nonpharmacological strategies such as quarantine, isolation, and public health education. This work studied the effect of these different control strategies as time-dependent interventions using mathematical modeling and optimal control approach to ascertain their contributions in the dynamic transmission of COVID-19. The model was proven to have an invariant region and was well-posed. The basic reproduction number and effective reproduction numbers were computed with and without interventions, respectively, and were used to carry out the sensitivity analysis that identified the critical parameters contributing to the spread of COVID-19. The optimal control analysis was carried out using the Pontryagin's maximum principle to figure out the optimal strategy necessary to curtail the disease. The findings of the optimal control analysis and numerical simulations revealed that time-dependent interventions reduced the number of exposed and infected individuals compared to time-independent interventions. These interventions were time-bound and best implemented within the first 100 days of the outbreak. Again, the combined implementation of only two of these interventions produced a good result in reducing infection in the population. While, the combined implementation of all three interventions performed better, even though zero infection was not achieved in the population. This implied that multiple interventions need to be deployed early in order to reduce the virus to the barest minimum.

Figure 1

The systematic diagram of the COVID-19 model.

Figure 2

The effect of η, τ, on the basic reproduction number, R₀. Here, other parameters are kept constant.

Figure 3

Optimal solutions for (a) exposed population, (E), and infected population, (I), when different combinations of control efforts are implemented. All the parameter values used are in Table 2.

Figure 4

Control profiles for different control efforts. (a) All the three control profile: (b) public health education and quarantine control profile, (c) public health education and isolation control profile, and (d) quarantine and the infected population control profile. All the parameter values used are in Table 2.

Figure 5

Optimal solutions for (a) exposed population, (E), and infected population, (I), when the control efforts are optimal control efforts and constant control efforts. All the parameter values used are in Table 2.

Figure 6

Optimal solutions for (a) exposed population, (E), and infected population, (I), with and without control. All the parameter values used are in Table 2.

Figure 7

Optimal solutions for (a) exposed population,(E), and infected population, (I), when more people are quarantined.