Mansoor Davoodi

The COVID-19 pandemic has significantly impacted long-term care facilities, with retirement homes being particularly vulnerable due to the high mortality risk among infected elderly residents. Once an outbreak occurs, containing the virus is challenging due to frequent resident interactions and limited isolation measures. While regular testing has proven effective in preventing outbreaks, high-frequency testing can strain staff resources, creating a trade-off between testing efforts and essential care provision. This paper addresses this challenge by proposing two novel optimization models for testing schedules that minimize infection risk while balancing staff workload. Using a probabilistic approach, the models incorporate factors such as contact rates, incidence status, and infection probabilities among residents. To solve these models, we introduce an enhanced local search algorithm that leverages the symmetry property of optimal solutions. Experimental results demonstrate the effectiveness of the proposed approach, outperforming a genetic algorithm in deriving optimal testing strategies.

We study the problem of perfect tiling in the plane and explore the possibility of tiling a rectangle using integral distinct squares. Assume a set of distinguishable squares (or equivalently a set of distinct natural numbers) is given, and one has to decide whether it can tile the plane or a rectangle or not. Previously, it has been proved that tiling the plane is not feasible using a set of odd numbers or an infinite sequence of natural numbers including exactly two odd numbers. The problem is open for different situations in which the number of odd numbers is arbitrary. In addition to providing a solution to this special case, we discuss some open problems to tile the plane and rectangles in this paper.

The optimal placement of healthcare facilities, including the placement of diagnostic test centers, plays a pivotal role in ensuring efficient and equitable access to healthcare services. However, the emergence of unique complexities in the context of a pandemic, exemplified by the COVID-19 crisis, has necessitated the development of customized solutions. This paper introduces a bi-objective integer linear programming model designed to achieve two key objectives: minimizing average travel time for individuals visiting testing centers and maximizing an equitable workload distribution among testing centers. To address this problem, we propose a customized local search algorithm based on the Voronoi diagram. Additionally, we employ an $ε$-constraint approach, which leverages the Gurobi solver. We rigorously examine the effectiveness of the model and the algorithms through numerical experiments and demonstrate their capability to identify Pareto-optimal solutions. We show that while the Gurobi performs efficiently in small-size instances, our proposed algorithm outperforms it in large-size instances of the problem.

Background: Long-term care facilities have been widely affected by the COVID-19 pandemic. Empirical evidence demonstrated that older people are the most impacted and are at higher risk of mortality after being infected. Regularly testing care facility residents is a practical approach to detecting infections proactively. In many cases, the care staff must perform the tests on the residents while also providing essential care, which in turn causes imbalances in their working time. Once an outbreak occurs, suppressing the spread of the virus in retirement homes (RHs) is challenging because the residents are in contact with each other, and isolation measures cannot be widely enforced. Regular testing strategies, on the other hand, have been shown to effectively prevent outbreaks in RHs. However, high-frequency testing may consume substantial staff working time, which results in a trade-off between the time invested in testing and the time spent providing essential care to residents. Objective: We developed a web application (Retirement Home Testing Optimizer) to assist RH managers in identifying effective testing schedules for residents. The outcome of the app, called the “testing strategy,” is based on dividing facility residents into groups and then testing no more than 1 group per day. Methods: We created the web application by incorporating influential factors such as the number of residents and staff, the average rate of contacts, the amount of time spent to test, and constraints on the test interval and size of groups. We developed mixed integer nonlinear programming models for balancing staff workload in long-term care facilities while minimizing the expected detection time of a probable infection inside the facility. Additionally, by leveraging symmetries in the problem, we proposed a fast and efficient local search method to find the optimal solution. Results: Considering the number of residents and staff and other practical constraints of the facilities, the proposed application computes the optimal trade-off testing strategy and suggests the corresponding grouping and testing schedule for residents. The current version of the application is deployed on the server of the Where2Test project and is accessible on their website. The application is open source, and all contents are offered in English and German. We provide comprehensive instructions and guidelines for easy use and understanding of the application’s functionalities. The application was launched in July 2022, and it is currently being tested in RHs in Saxony, Germany. Conclusions: Recommended testing strategies by our application are tailored to each RH and the goals set by the managers. We advise the users of the application that the proposed model and approach focus on the expected scenarios, that is, the expected risk of infection, and they do not guarantee the avoidance of worst-case scenarios.

Effective personnel scheduling is crucial for organizations to match workload demands. However, staff scheduling is sometimes affected by unexpected events, such as the COVID-19 pandemic, that disrupt regular operations. Limiting the number of on-site staff in the workplace together with regular testing is an effective strategy to minimize the spread of infectious diseases like COVID-19 because they spread mostly through close contact with people. Therefore, choosing the best scheduling and testing plan that satisfies the goals of the organization and prevents the virus’s spread is essential during disease outbreaks. In this paper, we formulate these challenges in the framework of two Mixed Integer Non-linear Programming (MINLP) models. The first model aims to derive optimal staff occupancy and testing strategies to minimize the risk of infection among employees, while the second is aimed only at optimal staff occupancy under a random testing strategy. To solve the problems expressed in the models, we propose a canonical genetic algorithm as well as two commercial solvers. Using both real and synthetic contact networks of employees, our results show that following the recommended occupancy and testing strategy reduces the risk of infection 25–60% under different scenarios. The minimum risk of infection can be achieved when the employees follow a planned testing strategy. Further, vaccination status and interaction rate of employees are important factors in developing scheduling strategies that minimize the risk of infection.

During pandemics like COVID-19, both the quality and quantity of services offered by businesses and organizations have been severely impacted. They often have applied a hybrid home office setup to overcome this problem, although in some situations, working from home lowers employee productivity. So, increasing the rate of presence in the office is frequently desired from the manager’s standpoint. On the other hand, as the virus spreads through interpersonal contact, the risk of infection increases when workplace occupancy rises. Motivated by this trade-off, in this paper, we model this problem as a bi-objective optimization problem and propose a practical approach to find the trade-off solutions. We present a new probabilistic framework to compute the expected number of infected employees for a setting of the influential parameters, such as the incidence level in the neighborhood of the company, transmission rate of the virus, number of employees, rate of vaccination, testing frequency, and rate of contacts among the employees. The results show a wide range of trade-offs between the expected number of infections and productivity, for example, from 1 to 6 weekly infections in 100 employees and a productivity level of 65% to 85%. This depends on the configuration of influential parameters and the occupancy level. We implement the model and the algorithm and perform several experiments with different settings of the parameters. Moreover, we developed an online application based on the result in this paper which can be used as a recommender for the optimal rate of occupancy in companies/workplaces.