Optimal quarantine‐related strategies for COVID‐19 control models

Abstract

At the time when this paper was written, quarantine‐related strategies (from full lockdown to some relaxed preventive measures) were the only available measure to control coronavirus disease 2019 (COVID‐19) epidemic. However, long‐term quarantine and especially full lockdown is an extremely expensive measure. To explore the possibility of controlling and suppressing the COVID‐19 epidemic at the lowest possible cost, we apply optimal control theory. In this paper, we create two controlled Susceptible‐Exposed‐Infectious‐Removed (SEIR) type models describing the spread of COVID‐19 in a human population. For each model, we solve an optimal control problem and find the optimal quarantine strategy that ensures the minimal level of the infected population at the lowest possible cost. The properties of the corresponding optimal controls are established analytically using the Pontryagin maximum principle. The optimal solutions, obtained numerically, validate our analytical results. Additionally, for both controlled models, we find explicit formulas for the basic reproductive ratios in the presence of a constant control and show that while the epidemic can be eventually stopped under long‐term quarantine measures of maximum strength (full lockdown), the strength of quarantine can be reduced under the optimal quarantine policies. The behavior of the appropriate optimal solutions and their dependence on the basic reproductive ratio, population density, and the duration of quarantine are discussed, and practically relevant conclusions are made.

Backward bifurcation and permanence of a disease‐severity‐structured epidemic model with treatment

“…To obtain positive solutions for ( 6), solving (6) in terms of 𝐼 𝑚 and 𝐼 𝑠 with 𝑆 and substituting the result into the first equation, we have…”

“…Treatment, quarantine, and vaccination are important methods that decrease the spread of diseases, such as measles, tuberculosis, flu, and the coronavirus disease 2019 (COVID-19) (see, e.g., Refs. [5,6,12,17]). For the COVID-19 epidemic, medicine and vaccine development has been rapidly promoted by medical teams worldwide during a few years since it appeared.…”

A Novel Compartmental Approach to Modeling COVID-19 Disease Dynamics and Analyzing the Effect of Common Preventative Measures

“…Governments also have limited resources, and citizens have a finite tolerance for preventative measures [12,13]. Still, such measures have been found to significantly reduce spread [14,15,16]. Thus, in addition to forecasting the virus's spread, identifying and analyzing common preventative measures are key to ending the pandemic.…”

Spatial dynamics for an SIRE epidemic model with diffusion and prevention in contaminated environments

“…Human beings are always more seriously threatened by diseases than war, poverty, disaster, and so on, especially, some fatal epidemics, for example, Bubonic plague, HIV, Malaria, Cholera, Ebola, COVID‐19, and so on, which have deprived a great many of lives all the time in the world 1–8 . During the past of couple decades, a great deal of work on the infectious dynamics of diseases 9–15 has been done. The main issues are focused on the persistence, extinction of diseases, stability of equilibria (e.g., disease‐free and endemic, etc.…”

“…COVID-19, and so on, which have deprived a great many of lives all the time in the world. [1][2][3][4][5][6][7][8] During the past of couple decades, a great deal of work on the infectious dynamics of diseases [9][10][11][12][13][14][15] has been done. The main issues are focused on the persistence, extinction of diseases, stability of equilibria (e.g., disease-free and endemic, etc.…”