In epidemiology, the effective reproduction number $R_{e}$ R e is used to characterize the growth rate of an epidemic outbreak. If $R_{e} >1$ R e > 1 , the epidemic worsens, and if $R_{e}< 1$ R e < 1 , then it subsides and eventually dies out. In this paper, we investigate properties of $R_{e}$ R e for a modified SEIR model of COVID-19 in the city of Houston, TX USA, in which the population is divided into low-risk and high-risk subpopulations. The response of $R_{e}$ R e to two types of control measures (testing and distancing) applied to the two different subpopulations is characterized. A nonlinear cost model is used for control measures, to include the effects of diminishing returns. Lowest-cost control combinations for reducing instantaneous $R_{e}$ R e to a given value are computed. We propose three types of heuristic strategies for mitigating COVID-19 that are targeted at reducing $R_{e}$ R e , and we exhibit the tradeoffs between strategy implementation costs and number of deaths. We also consider two variants of each type of strategy: basic strategies, which consider only the effects of controls on $R_{e}$ R e , without regard to subpopulation; and high-risk prioritizing strategies, which maximize control of the high-risk subpopulation. Results showed that of the three heuristic strategy types, the most cost-effective involved setting a target value for $R_{e}$ R e and applying sufficient controls to attain that target value. This heuristic led to strategies that begin with strict distancing of the entire population, later followed by increased testing. Strategies that maximize control on high-risk individuals were less cost-effective than basic strategies that emphasize reduction of the rate of spreading of the disease. The model shows that delaying the start of control measures past a certain point greatly worsens strategy outcomes. We conclude that the effective reproduction can be a valuable real-time indicator in determining cost-effective control strategies.

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In this paper, we present a three-stage algorithm for finding numerical solutions for optimal control problems. The algorithm first performs an exhaustive search through a discrete set of widely dispersed solutions which are representative of large subregions of the search space; then, it uses the search results to initialize a Monte Carlo process that searches quasi-randomly for a best solution; then, it finally uses a Newton-type iteration to converge to a solution that satisfies mathematical conditions of local optimality. We demonstrate our methodology on an epidemiological model of the coronavirus disease with testing and distancing controls applied over a period of 180 days to two different subpopulations (low-risk and high-risk), where model parameters are chosen to fit the city of Houston, Texas, USA. In order to enable the user to select his/her preferred trade-off between (number of deaths) and (herd immunity) outcomes, the objective function includes costs for deaths and non-immunity. Optimal strategies are estimated for a grid of (death cost) × (non-immunity cost) combinations, in order to obtain a Pareto curve that represents optimum trade-offs. The levels of the four controls for the different Pareto-optimal solutions over the 180-day period are visually represented and their characteristics discussed. Three different variants of the algorithm are run in order to determine the relative importance of the three stages in the optimization. Results from the three algorithm variants are fairly consistent, indicating that solutions are robust. Results also show that the Monte Carlo stage plays an especially prominent role in the optimization, but that all three stages of the process make significant contributions towards finding lower-cost, more effective control strategies.

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In epidemiology, the effective reproduction number R e is used to characterize the growth rate of an epidemic outbreak. If R e > 1, the epidemic worsens, and if R e < 1, then it subsides and eventually dies out. In this paper, we investigate properties of R e for a modified SEIR model of COVID-19 in the city of Houston, TX USA, in which the population is divided into low-risk and high-risk subpopulations. The response of R e to two types of control measures (testing and distancing) applied to the two different subpopulations is characterized. A nonlinear cost model is used for control measures, to include the effects of diminishing returns. Lowest-cost control combinations for reducing instantaneous R e to a given value are computed. We propose three types of heuristic strategies for mitigating COVID-19 that are targeted at reducing R e , and we exhibit the tradeoffs between strategy implementation costs and number of deaths. We also consider two variants of each type of strategy: basic strategies, which consider only the effects of controls on R e , without regard to subpopulation; and high-risk prioritizing strategies, which maximize control of the high-risk subpopulation. Results showed that of the three heuristic strategy types, the most cost-effective involved setting a target value for R e and applying sufficient controls to attain that target value. This heuristic led to strategies that begin with strict distancing of the entire population, later followed by increased testing. Strategies that maximize control on high-risk individuals were less cost-effective than basic strategies that emphasize reduction of the rate of spreading of the disease. The model shows that delaying the start of control measures past a certain point greatly worsens strategy outcomes. We conclude that the effective reproduction can be a valuable real-time indicator in determining cost-effective control strategies.