A statistical moment-based spectral approach to the chance-constrained stochastic optimal control of epidemic models

This paper presents a spectral approach to the uncertainty management in epidemic models through the formulation of chance-constrained stochastic optimal control problems. Specifically, a statistical moment-based polynomial expansion is used to calculate surrogate models of the stochastic state variables of the problem that allow for the efficient computation of their main statistics as well as their marginal and joint probability density functions at each time instant, which enable the uncertainty management in the epidemic model. Moreover, the surrogate models are employed to perform the corresponding sensitivity and risk analyses. The proposed methodology provides the designers of the optimal control policies with the capability to increase the predictability of the outcomes by adding suitable chance constraints to the epidemic model and formulating a proper cost functional. The chance-constrained optimal control of a COVID-19 epidemic model is considered in order to illustrate the practical application of the proposed methodology.

Robust optimal control of compartmental models in epidemiology: Application to the COVID-19 pandemic.

In this paper, a spectral approach is used to formulate and solve robust optimal control problems for compartmental epidemic models, allowing the uncertainty propagation through the optimal control model to be represented by a polynomial expansion of its stochastic state variables. More specifically, a statistical moment-based polynomial chaos expansion is employed. The spectral expansion of the stochastic state variables allows the computation of their main statistics to be carried out, resulting in a compact and efficient representation of the variability of the optimal control model with respect to its random parameters. The proposed robust formulation provides the designers of the optimal control strategy of the epidemic model the capability to increase the predictability of the results by simply adding upper bounds on the variability of the state variables. Moreover, this approach yields a way to efficiently estimate the probability distributions of the stochastic state variables and conduct a global sensitivity analysis. To show the practical implementation of the proposed approach, a mathematical model of COVID-19 transmission is considered. The numerical results show that the spectral approach proposed to formulate and solve robust optimal control problems for compartmental epidemic models provides healthcare systems with a valuable tool to mitigate and control the impact of infectious diseases.

Optimal Control Applied to Vaccination and Testing Policies for COVID-19

In this paper, several policies for controlling the spread of SARS-CoV-2 are determined under the assumption that a limited number of effective COVID-19 vaccines and tests are available. These policies are calculated for different vaccination scenarios representing vaccine supply and administration restrictions, plus their impacts on the disease transmission are analyzed. The policies are determined by solving optimal control problems of a compartmental epidemic model, in which the control variables are the vaccination rate and the testing rate for the detection of asymptomatic infected people. A combination of the proportion of threatened and deceased people together with the cost of vaccination of susceptible people, and detection of asymptomatic infected people, is taken as the objective functional to be minimized, whereas different types of algebraic constraints are considered to represent several vaccination scenarios. A direct transcription method is employed to solve these optimal control problems. More specifically, the Hermite–Simpson collocation technique is used. The results of the numerical experiments show that the optimal control approach offers healthcare system managers a helpful resource for designing vaccination programs and testing plans to prevent COVID-19 transmission.

Optimal control-based vaccination and testing strategies for COVID-19.

BACKGROUND AND OBJECTIVE: Assuming the availability of a limited amount of effective COVID-19 rapid tests, the effects of various vaccination strategies on SARS-CoV-2 virus transmission are compared for different vaccination scenarios characterized by distinct limitations associated with vaccine supply and administration. METHODS: The vaccination strategies are defined by solving optimal control problems of a compartmental epidemic model in which the daily vaccination rate and the daily testing rate for the identification and isolation of asymptomatic subjects are the control variables. Different kinds of algebraic constraints are considered, representing different vaccination scenarios in which the total amount of vaccines available during the time period under consideration is limited or the number of daily available vaccines is limited. These optimal control problems are numerically solved by means of a direct transcription technique, which allows both equality and inequality constraints to be straightforwardly included in the formulation of the optimal control problems. RESULTS: Several numerical experiments are conducted, in which the objective functional to be minimized is a combination of the number of symptomatic and asymptomatic infectious subjects with the cost of vaccination of susceptible subjects and testing of asymptomatic infectious subjects. The results confirm the hypothesis that the implementation of early control measures significantly reduces the number of symptomatic infected subjects, which is a key aspect for the resilience of the healthcare system. The sensitivity analysis of the solutions to the weighting parameters of the objective functional reveals that it is possible to obtain a vaccination strategy that allows vaccination supplies to be saved while keeping the same number of symptomatic infected subjects. Furthermore, it indicates that if the vaccination plan is not supported by a sufficient rate of testing, the number of symptomatic infected subjects could increase. Finally, the sensitivity analysis shows that a significant reduction in the efficacy of the vaccines could also lead to a relevant increase in the number of symptomatic infected subjects. CONCLUSIONS: The numerical experiments show that the proposed approach, which is based on optimal control of compartmental epidemic models, provides healthcare systems with a suitable method for scheduling vaccination plans and testing policies to control the spread of the SARS-CoV-2 virus.

Uncertainty quantification of a mathematical model of COVID-19 transmission dynamics with mass vaccination strategy.

In this paper, the uncertainty quantification and sensitivity analysis of a mathematical model of the SARS-CoV-2 virus transmission dynamics with mass vaccination strategy has been carried out. More specifically, a compartmental epidemic model has been considered, in which vaccination, social distance measures, and testing of susceptible individuals have been included. Since the application of these mitigation measures entails a degree of uncertainty, the effects of the uncertainty about the application of social distance actions and testing of susceptible individuals on the disease transmission have been quantified, under the assumption of a mass vaccination program deployment. A spectral approach has been employed, which allows the uncertainty propagation through the epidemic model to be represented by means of the polynomial chaos expansion of the output random variables. In particular, a statistical moment-based polynomial chaos expansion has been implemented, which provides a surrogate model for the compartments of the epidemic model, and allows the statistics, the probability distributions of the interesting output variables of the model at a given time instant to be estimated and the sensitivity analysis to be conducted. The purpose of the sensitivity analysis is to understand which uncertain parameters have most influence on a given output random variable of the model at a given time instant. Several numerical experiments have been conducted whose results show that the proposed spectral approach to uncertainty quantification and sensitivity analysis of epidemic models provides a useful tool to control and mitigate the effects of the COVID-19 pandemic, when it comes to healthcare resource planning.