A parametrized nonlinear predictive control strategy for relaxing COVID-19 social distancing measures in Brazil

SIRSi-vaccine dynamical model for the Covid-19 pandemic

Covid-19, caused by severe acute respiratory syndrome coronavirus 2, broke out as a pandemic during the beginning of 2020. The rapid spread of the disease prompted an unprecedented global response involving academic institutions, regulatory agencies, and industries. Vaccination and nonpharmaceutical interventions including social distancing have proven to be the most effective strategies to combat the pandemic. In this context, it is crucial to understand the dynamic behavior of the Covid-19 spread together with possible vaccination strategies. In this study, a susceptible-infected-removed-sick model with vaccination (SIRSi-vaccine) was proposed, accounting for the unreported yet infectious. The model considered the possibility of temporary immunity following infection or vaccination. Both situations contribute toward the spread of diseases. The transcritical bifurcation diagram of alternating and mutually exclusive stabilities for both disease-free and endemic equilibria were determined in the parameter space of vaccination rate and isolation index. The existing equilibrium conditions for both points were determined in terms of the epidemiological parameters of the model. The bifurcation diagram allowed us to estimate the maximum number of confirmed cases expected for each set of parameters. The model was fitted with data from São Paulo, the state capital of SP, Brazil, which describes the number of confirmed infected cases and the isolation index for the considered data window. Furthermore, simulation results demonstrate the possibility of periodic undamped oscillatory behavior of the susceptible population and the number of confirmed cases forced by the periodic small-amplitude oscillations in the isolation index. The main contributions of the proposed model are as follows: A minimum effort was required when vaccination was combined with social isolation, while additionally ensuring the existence of equilibrium points. The model could provide valuable information for policymakers, helping define disease prevention mitigation strategies that combine vaccination and non-pharmaceutical interventions, such as social distancing and the use of masks. In addition, the SIRSi-vaccine model facilitated the qualitative assessment of information regarding the unreported infected yet infectious cases, while considering temporary immunity, vaccination, and social isolation index.

Mathematical Modeling of SARS-CoV-2 Omicron Wave under Vaccination Effects

Over the course of the COVID-19 pandemic millions of deaths and hospitalizations have been reported. Different SARS-CoV-2 variants of concern have been recognized during this pandemic and some of these variants of concern have caused uncertainty and changes in the dynamics. The Omicron variant has caused a large amount of infected cases in the US and worldwide. The average number of deaths during the Omicron wave toll increased in comparison with previous SARS-CoV-2 waves. We studied the Omicron wave by using a highly nonlinear mathematical model for the COVID-19 pandemic. The novel model includes individuals who are vaccinated and asymptomatic, which influences the dynamics of SARS-CoV-2. Moreover, the model considers the waning of the immunity and efficacy of the vaccine against the Omicron strain. This study uses the facts that the Omicron strain has a higher transmissibility than the previous circulating SARS-CoV-2 strain but is less deadly. Preliminary studies have found that Omicron has a lower case fatality rate compared to previous circulating SARS-CoV-2 strains. The simulation results show that even if the Omicron strain is less deadly it might cause more deaths, hospitalizations and infections. We provide a variety of scenarios that help to obtain insight about the Omicron wave and its consequences. The proposed mathematical model, in conjunction with the simulations, provides an explanation for a large Omicron wave under various conditions related to vaccines and transmissibility. These results provide an awareness that new SARS-CoV-2 variants can cause more deaths even if their fatality rate is lower.

Stochastic optimization for vaccine and testing kit allocation for the COVID-19 pandemic

We present a formal mathematical modeling framework for a multi-agent sequential decision problem during an epidemic. The problem is formulated as a collaboration between a vaccination agent and learning agent to allocate stockpiles of vaccines and tests to a set of zones under various types of uncertainty. The model is able to capture passive information processes and maintain beliefs over the uncertain state of the world. We designed a parameterized direct lookahead approximation which is robust and scalable under different scenarios, resource scarcity, and beliefs about the environment. We design a test allocation policy designed to capture the value of information and demonstrate that it outperforms other learning policies when there is an extreme shortage of resources (information is scarce). We simulate the model with two scenarios including a resource allocation problem to each state in the United States and another for the nursing homes in Nevada. The US example demonstrates the scalability of the model and the nursing home example demonstrates the robustness under extreme resource shortages.

A stochastic inventory model of COVID-19 and robust, real-time identification of carriers at large and infection rate via asymptotic laws

A critical operations management problem in the ongoing COVID-19 pandemic is cognizance of (a) the number of all carriers at large (CaL) conveying the SARS-CoV-2, including asymptomatic ones and (b) the infection rate (IR). Both are random and unobservable, affecting the spread of the disease, patient arrivals to health care units (HCUs) and the number of deaths. A novel, inventory perspective of COVID-19 is proposed, with random inflow, random losses and retrials (recurrent cases) and delayed/distributed exit, with randomly varying fractions of the exit distribution. A minimal construal, it enables representation of COVID-19 evolution with close fit of national incidence profiles, including single and multiple pattern outbreaks, oscillatory, periodic or non-periodic evolution, followed by retraction, leveling off, or strong resurgence. Furthermore, based on asymptotic laws, the minimum number of variables that must be monitored for identifying CaL and IR is determined and a real-time identification method is presented. The method is data-driven, utilizing the entry rate to HCUs and scaled, or dimensionless variables, including the mean residence time of symptomatic carriers in CaL and the mean residence time in CaL of patients entering HCUs. As manifested by several robust case studies of national COVID-19 incidence profiles, it provides efficient identification in real-time under unbiased monitoring error, without relying on any model. The propagation factor, a stochastic process, is reconstructed from the identified trajectories of CaL and IR, enabling evaluation of control measures. The results are useful towards the design of policies restricting COVID-19 and encumbrance to HCUs and mitigating economic contraction.

Mathematical modeling to study the impact of immigration on the dynamics of the COVID-19 pandemic: A case study for Venezuela

We propose two different mathematical models to study the effect of immigration on the COVID-19 pandemic. The first model does not consider immigration, whereas the second one does. Both mathematical models consider five different subpopulations: susceptible, exposed, infected, asymptomatic carriers, and recovered. We find the basic reproduction number R₀ using the next-generation matrix method for the mathematical model without immigration. This threshold parameter is paramount because it allows us to characterize the evolution of the disease and identify what parameters substantially affect the COVID-19 pandemic outcome. We focus on the Venezuelan scenario, where immigration and emigration have been important over recent years, particularly during the pandemic. We show that the estimation of the transmission rates of the SARS-CoV-2 are affected when the immigration of infected people is considered. This has an important consequence from a public health perspective because if the basic reproduction number is less than unity, we can expect that the SARS-CoV-2 would disappear. Thus, if the basic reproduction number is slightly above one, we can predict that some mild non-pharmaceutical interventions would be enough to decrease the number of infected people. The results show that the dynamics of the spread of SARS-CoV-2 through the population must consider immigration to obtain better insight into the outcomes and create awareness in the population regarding the population flow.

Recursive state and parameter estimation of COVID-19 circulating variants dynamics

COVID-19 pandemic response with non-pharmaceutical interventions is an intrinsic control problem. Governments weigh social distancing policies to avoid overload in the health system without significant economic impact. The mutability of the SARS-CoV-2 virus, vaccination coverage, and mobility restriction measures change epidemic dynamics over time. A model-based control strategy requires reliable predictions to be efficient on a long-term basis. In this paper, a SEIR-based model is proposed considering dynamic feedback estimation. State and parameter estimations are performed on state estimators using augmented states. Three methods were implemented: constrained extended Kalman filter (CEKF), CEKF and smoother (CEKF & S), and moving horizon estimator (MHE). The parameters estimation was based on vaccine efficacy studies regarding transmissibility, severity of the disease, and lethality. Social distancing was assumed as a measured disturbance calculated using Google mobility data. Data from six federative units from Brazil were used to evaluate the proposed strategy. State and parameter estimations were performed from 1 October 2020 to 1 July 2021, during which Zeta and Gamma variants emerged. Simulation results showed that lethality increased between 11 and 30% for Zeta mutations and between 44 and 107% for Gamma mutations. In addition, transmissibility increased between 10 and 37% for the Zeta variant and between 43 and 119% for the Gamma variant. Furthermore, parameter estimation indicated temporal underreporting changes in hospitalized and deceased individuals. Overall, the estimation strategy showed to be suitable for dynamic feedback as simulation results presented an efficient detection and dynamic characterization of circulating variants.

Controlling epidemic diseases based only on social distancing level: General case

The COVID-19 outbreak is an epidemic disease caused by the novel severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). When a new virus emerges, generally, little is known about it, and no vaccines or other pharmaceutical interventions are available. In the case of a person-to-person transmission virus with no vaccines or other pharmaceutical interventions, the only way to control the virus outbreak is by keeping a sustained physical distancing between the individuals. However, to adjust the level of the physical distancing accurately can be so complicated. Any level above the necessary can compromise the economic activity, and any level below can collapse the health care system. This work proposes a controller to keep the number of hospitalized individuals below a limit, and a new group-structured model to describe the COVID-19 outbreak. The proposed controller is robust to the uncertainties in the parameters of the model and keeps the number of infected individuals controlled only by adjusting the social distancing level. Numerical simulations, to show the behavior of the proposed controller and model, are done.

Epidemiological Predictive Modeling of COVID-19 Infection: Development, Testing, and Implementation on the Population of the Benelux Union

Since the outbreak of coronavirus disease-2019 (COVID-19), the whole world has taken interest in the mechanisms of its spread and development. Mathematical models have been valuable instruments for the study of the spread and control of infectious diseases. For that purpose, we propose a two-way approach in modeling COVID-19 spread: a susceptible, exposed, infected, recovered, deceased (SEIRD) model based on differential equations and a long short-term memory (LSTM) deep learning model. The SEIRD model is a compartmental epidemiological model with included components: susceptible, exposed, infected, recovered, deceased. In the case of the SEIRD model, official statistical data available online for countries of Belgium, Netherlands, and Luxembourg (Benelux) in the period of March 15 2020 to March 15 2021 were used. Based on them, we have calculated key parameters and forward them to the epidemiological model, which will predict the number of infected, deceased, and recovered people. Results show that the SEIRD model is able to accurately predict several peaks for all the three countries of interest, with very small root mean square error (RMSE), except for the mild cases (maximum RMSE was 240.79 ± 90.556), which can be explained by the fact that no official data were available for mild cases, but this number was derived from other statistics. On the other hand, LSTM represents a special kind of recurrent neural network structure that can comparatively learn long-term temporal dependencies. Results show that LSTM is capable of predicting several peaks based on the position of previous peaks with low values of RMSE. Higher values of RMSE are observed in the number of infected cases in Belgium (RMSE was 535.93) and Netherlands (RMSE was 434.28), and are expected because of thousands of people getting infected per day in those countries. In future studies, we will extend the models to include mobility information, variants of concern, as well as a medical intervention, etc. A prognostic model could help us predict epidemic peaks. In that way, we could react in a timely manner by introducing new or tightening existing measures before the health system is overloaded.

Convex output feedback model predictive control for mitigation of COVID-19 pandemic

In this paper, a model predictive control approach is proposed for epidemic mitigation. The disease spreading dynamics is described by an 8-compartment smooth nonlinear model of the COVID-19 pandemic in Hungary known from the literature, where the manipulable control input is the stringency of the introduced non-pharmaceutical measures. It is assumed that only the number of hospitalized people is measured on-line, and the other state variables are computed using a state observer which is based on the dynamic inversion of a linear sub-system of the model. The objective function contains a measure of the direct harmful consequences of the restrictions, and the constraints refer to input bounds and to the capacity of the healthcare system. By exploiting the special properties of the model, the nonlinear optimization problem required by the control design is reformulated to convex tasks, allowing a computationally efficient solution. Two approaches are proposed: the first finds a suboptimal solution by geometric programming, while the second one further simplifies the problem and transforms it to a linear programming task. Simulations show that both suboptimal solutions fulfill the design specifications even in the presence of parameter uncertainties.

Hybrid Model-Based Simulation Analysis on the Effects of Social Distancing Policy of the COVID-19 Epidemic

This study utilizes modeling and simulation to analyze coronavirus (COVID-19) infection trends depending on government policies. Two modeling requirements are considered for infection simulation: (1) the implementation of social distancing policies and (2) the representation of population movements. To this end, we propose an extended infection model to combine analytical models with discrete event-based simulation models in a hybrid form. Simulation parameters for social distancing policies are identified and embedded in the analytical models. Administrative districts are modeled as a fundamental simulation agent, which facilitates representing the population movements between the cities. The proposed infection model utilizes real-world data regarding suspected, infected, recovered, and deceased people in South Korea. As an application, we simulate the COVID-19 epidemic in South Korea. We use real-world data for 160 days, containing meaningful days that begin the distancing policy and adjust the distancing policy to the next stage. We expect that the proposed work plays a principal role in analyzing how social distancing effectively affects virus prevention and provides a simulation environment for the biochemical field.

The COVID-19 (SARS-CoV-2) uncertainty tripod in Brazil: Assessments on model-based predictions with large under-reporting

The COVID-19 pandemic (SARS-CoV-2 virus) is the global crisis of our time. The absence of mass testing and the relevant presence of asymptomatic individuals causes the available data of the COVID-19 pandemic in Brazil to be largely under-reported regarding the number of infected individuals and deaths. We develop an adapted Susceptible-Infected-Recovered (SIR) model, which explicitly incorporates the under-reporting and the response of the population to public health policies (confinement measures, widespread use of masks, etc). Large amounts of uncertainty could provide misleading predictions of the COVID-19 spread. In this paper, we discuss the role of uncertainty in these model-based predictions, which is illustrated regarding three key aspects: (i) Assuming that the number of infected individuals is under-reported, we demonstrate anticipation regarding the infection peak. Furthermore, while a model with a single class of infected individuals yields forecasts with increased peaks, a model that considers both symptomatic and asymptomatic infected individuals suggests a decrease of the peak of symptomatic cases. (ii) Considering that the actual amount of deaths is larger than what is being registered, we demonstrate an increase of the mortality rates. (iii) When we consider generally under-reported data, we demonstrate how the transmission and recovery rate model parameters change qualitatively and quantitatively. We also investigate the “the uncertainty tripod”: under-reporting level in terms of cases, deaths, and the true mortality rate of the disease. We demonstrate that if two of these factors are known, the remainder can be inferred, as long as proportions are kept constant. The proposed approach allows one to determine the margins of uncertainty by assessments on the observed and true mortality rates.

Optimization in the Context of COVID-19 Prediction and Control: A Literature Review

This paper presents an overview of some key results from a body of optimization studies that are specifically related to COVID-19, as reported in the literature during 2020-2021. As shown in this paper, optimization studies in the context of COVID-19 have been used for many aspects of the pandemic. From these studies, it is observed that since COVID-19 is a multifaceted problem, it cannot be studied from a single perspective or framework, and neither can the related optimization models. Four new and different frameworks are proposed that capture the essence of analyzing COVID-19 (or any pandemic for that matter) and the relevant optimization models. These are: (i) microscale vs. macroscale perspective; (ii) early stages vs. later stages perspective; (iii) aspects with direct vs. indirect relationship to COVID-19; and (iv) compartmentalized perspective. To limit the scope of the review, only optimization studies related to the prediction and control of COVID-19 are considered (public health focused), and which utilize formal optimization techniques or machine learning approaches. In this context and to the best of our knowledge, this survey paper is the first in the literature with a focus on the prediction and control related optimization studies. These studies include optimization of screening testing strategies, prediction, prevention and control, resource management, vaccination prioritization, and decision support tools. Upon reviewing the literature, this paper identifies current gaps and major challenges that hinder the closure of these gaps and provides some insights into future research directions.

A control framework to optimize public health policies in the course of the COVID-19 pandemic

The SARS-CoV-2 pandemic triggered substantial economic and social disruptions. Mitigation policies varied across countries based on resources, political conditions, and human behavior. In the absence of widespread vaccination able to induce herd immunity, strategies to coexist with the virus while minimizing risks of surges are paramount, which should work in parallel with reopening societies. To support these strategies, we present a predictive control system coupled with a nonlinear model able to optimize the level of policies to stop epidemic growth. We applied this system to study the unfolding of COVID-19 in Bahia, Brazil, also assessing the effects of varying population compliance. We show the importance of finely tuning the levels of enforced measures to achieve SARS-CoV-2 containment, with periodic interventions emerging as an optimal control strategy in the long-term.

The Quixotic Task of Forecasting Peaks of COVID-19: Rather Focus on Forward and Backward Projections

Over the last months, mathematical models have been extensively used to help control the COVID-19 pandemic worldwide. Although extremely useful in many tasks, most models have performed poorly in forecasting the pandemic peaks. We investigate this common pitfall by forecasting four countries' pandemic peak: Austria, Germany, Italy, and South Korea. Far from the peaks, our models can forecast the pandemic dynamics 20 days ahead. Nevertheless, when calibrating our models close to the day of the pandemic peak, all forecasts fail. Uncertainty quantification and sensitivity analysis revealed the main obstacle: the misestimation of the transmission rate. Inverse uncertainty quantification has shown that significant changes in transmission rate commonly precede a peak. These changes are a key factor in forecasting the pandemic peak. Long forecasts of the pandemic peak are therefore undermined by the lack of models that can forecast changes in the transmission rate, i.e., how a particular society behaves, changes of mitigation policies, or how society chooses to respond to them. In addition, our studies revealed that even short forecasts of the pandemic peak are challenging. Backward projections have shown us that the correct estimation of any temporal change in the transmission rate is only possible many days ahead. Our results suggest that the distance between a change in the transmission rate and its correct identification in the curve of active infected cases can be as long as 15 days. This is intrinsic to the phenomenon and how it affects epidemic data: a new case is usually only reported after an incubation period followed by a delay associated with the test. In summary, our results suggest the phenomenon itself challenges the task of forecasting the peak of the COVID-19 pandemic when only epidemic data is available. Nevertheless, we show that exciting results can be obtained when using the same models to project different scenarios of reduced transmission rates. Therefore, our results highlight that mathematical modeling can help control COVID-19 pandemic by backward projections that characterize the phenomena' essential features and forward projections when different scenarios and strategies can be tested and used for decision-making.

Modeling and control of epidemics through testing policies

Testing is a crucial control mechanism in the beginning phase of an epidemic when the vaccines are not yet available. It enables the public health authority to detect and isolate the infected cases from the population, thereby limiting the disease transmission to susceptible people. However, despite the significance of testing in epidemic control, the recent literature on the subject lacks a control-theoretic perspective. In this paper, an epidemic model is proposed that incorporates the testing rate as a control input and differentiates the undetected infected from the detected infected cases, who are assumed to be removed from the disease spreading process in the population. After estimating the model on the data corresponding to the beginning phase of COVID-19 in France, two testing policies are proposed: the so-called best-effort strategy for testing (BEST) and constant optimal strategy for testing (COST). The BEST policy is a suppression strategy that provides a minimum testing rate that stops the growth of the epidemic when implemented. The COST policy, on the other hand, is a mitigation strategy that provides an optimal value of testing rate minimizing the peak value of the infected population when the total stockpile of tests is limited. Both testing policies are evaluated by their impact on the number of active intensive care unit (ICU) cases and the cumulative number of deaths for the COVID-19 case of France.