Multiobjective Optimal Control of the COVID Virus Outbreak

The number of fatalities caused by the COVID virus is not only extremely high but also increasing at an alarming rate. The many strategies that are being used throughout the world, to control the pandemic, are being overwhelmed mercilessly by the global pandemic. In this paper three different optimization strategies are used to determine the best strategy that can minimize the damage. It is also demonstrated that for one value of the number of infected subjects two values of recovered and perished subjects are possible. This is an important result because one can take steps to ensure that the number of perished subjects is the lowest possible while the number of recovered subjects is the highest possible. ReSeaRch aRticLe *Corresponding author: Lakshmi Sridhar, Professor, Chemical Engineering, University of Puerto Rico, Mayaguez, Puerto Rico Check for updates as an objective function (cost function) and the optimal trajectories are obtained. However, all this work involves single objective optimal control. In this work a rigorous multiobjective optimal control (MOOC) strategy that does not involve weighting functions or additional constraint equations is used to generate optimal trajectories of the variables in the ODE that comprises the COVID pandemic model. The model used is the SEAIR model described in Calvin Tsay [22].

as an objective function (cost function) and the optimal trajectories are obtained. However, all this work involves single objective optimal control.
In this work a rigorous multiobjective optimal control (MOOC) strategy that does not involve weighting functions or additional constraint equations is used to generate optimal trajectories of the variables in the ODE that comprises the COVID pandemic model. The model used is the SEAIR model described in Calvin Tsay [22].

SEAIR model
The SEAIR (susceptible, exposed asymptotic/unconfirmed, Infected/confirmed, recovered) Involves the following variables, • Number of subjects susceptible to the infection S(t) • Number exposed to the virus e(t) • Number that is infected but asymptotic/unconfirmed a(t) • Number of confirmed infections i(t) • Number of subjects recovered r(t)

Modeling of the COVID-19 outbreak
The non-stop rise in the number of lives lost in the COVID virus has triggered research where dynamic optimization is used to develop control strategies to minimize the number of infected people and the number of deaths while maximizing the number of people who have recovered. Modeling and optimization are important strategies that can be useful to control the damages done by diseases [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16].
Epidemological models comprise of ordinary differential equations (ODE) and recently there has been research involving dynamic optimization (optimal control) of COVID models [17][18][19][20][21]. Optimal control and parameter estimation was performed [22] using a COVID model demonstrating the effect of social distancing and quarantining. In this article a combination of the rates of the exposure to the virus to from the population of the asymptotic/unconfirmed and confirmed subjects and the rate of unconfirmed cases becoming confirmed is used by i α social distancing and shelter in place determine . a α The effect of testing and Screening is reflected by the value of κ The equations that constitute the SEAIR model are as follows:

s t a t t s t i t r t dt
The parameter values obtained from the article Calvin Tsay, et al. [22] are The inverse of the latent period of the virus

Multiobjective optimal control
The multiobjective nonlinear optimal control (MOOC) method was first proposed by Flores Tlacuahuaz, et al. [23] and used by Sridhar [24]. This method is rigorous and it does not involve the use of weighting functions not does it impose additional parameters or additional constraints on the problem unlike the weighted function or the epsilon correction method [25]. For a problem that is posed as The MOOC method first solves dynamic optimization problems independently minimizing each i φ (i = 1,2,3…n) individually. This will lead to minimized values * i φ (i = 1,2,3,..n) . Then the optimization problem that will be solved is The optimization package in Python, Pyomo [26], where the differential equations are automatically converted to a Nonlinear Program (NLP) using the orthogonal collocation method [27]. The Lagrange-Radau quadrature with three collocation points is used and 10 finite elements are chosen to solve the optimal control problems. The resulting nonlinear optimization problem was solved using the solver BARON 19.3 [28], accessed through the Pyomo-GAMS27. 2 [29] interface. BARON implements a Branch-and-reduce strategy to provide valid lower and upper bounds for the optimal solution and provides a guaranteed global optimal solution. The following three MOOC problems are solved in this paper.
The rate of exposure to virus from infected subjects In the first optimal control problem, the objective function that is minimized is This problem is referred to as OPTCON1 While in the second the optimal control problem, This problem is referred to as OPTCON2. The third optimal control problem is a combination of the first two. Here the minimized objective function is  (17) This problem is referred to as OPTCON3.
All the minimizations are done subject to the equations 1-6.

Results and Discussion
Comparison of the effectiveness of the three optimization strategies ered subjects than OPTCON1 and the Figure 1d, Figure  2d and Figure 3d show that the strategies OPTCON2 and OPTCON3 result in a lower number of perished subjects than OPTCON1. The OPTCON1 strategy leads to a) Higher number of infected subjects, b) Higher number of subjects that have perished and c) A lower number of recovered subjects as compared to using the strategies OPTCON2 and OPTCON3. Hence it is preferable to use the strategies OPTCON2 or OPTCON3. There is no significant difference between the performance of the two strategies OPTCON2 and OPTCON3. However it is better to use OPTCON3 since the objective function involves more variables than OPTCON2.
First, the effectiveness of the three optimal control strategies will be compared. The Figure 1a, Figure 1b Figure 3f demonstrate that for one value of I (infected) subjects two values of R (recovered subjects) and two values of P (perished subjects) are possible. This is an important fact because it gives the health officials the opportunity to take steps that, for a given value of I, can keep the value of R higher and the value of P lower and consequently saving lives.

Conclusions
The main results of this paper indicate that in trying to control the COVID pandemic, minimizing the rate of exposure to virus from infected subjects, rate exposure to virus from asymptotic/unconfirmed subjects and maximizing the rate at which unconfirmed cases become confirmed is less beneficial than minimizing the number of subjects infected I and the number of subjects perished while maximizing the number of subjects that have recovered. It is demonstrated that two values of the number of subjects recovered or perished is possible for the same value of the number of infected cases. This can enable us to take steps to ensure that less people perish and more people recover.

Acknowledgement
Dr. Sridhar thanks Dr. Saif R Kazi of CMU for his help with the graphics.