A nonlinear model predictive control model aimed at the epidemic spread with quarantine strategy

Highlights

• An optimal control model aimed at the quarantine strategy is built.

• We develop an NMPC algorithm with state feedback mechanism.

• We design a state observer which may avoid or reduce the observation error caused by the partial data.

Abstract

Allocating limit medicine resources by mathematical modeling to control spreading of epidemic diseases is a very promising approach. Especially, how to use the existing partial data to efficiently control epidemic diseases is a interesting problem. When an epidemic disease is spreading, it is very urgent and essential to build a prediction and control model based on the real-time and partial data in order that decision makers find and implement the optimal strategy timely. In this paper, we developed a new framework for solving the problem. Our nonlinear model predictive control (NMPC) based on a discrete time susceptible-infected-removed dynamics (SIR) gave an attempt that aims at timely dealing with the condition. Our NMPC model minimizes the total number of infectious cases and the total cost, with the treatment beds capacity constraints and other constraints, especially, with a state observer based on the system output which can be sampled more easily and more accurately. Our control policy can be updated timely according to the current statistical data because our NMPC is a kind of closed-loop control algorithm based on our observer. We also presented some theoretical results on the state observer. Finally, we gave a numerical example to illustrate our algorithm.

Introduction

Infectious diseases have always been one of the great challenges facing mankind. In 2003, the outbreak of severe acute respiratory syndrome (SARS) caused more than 8000 cases with approximately 800 deaths in 27 countries all over the world (World Health Organization, 2004). Ebola virus disease caused 28,616 cases with 11,310 deaths (World Health Organization, 2016) in Guinea, Liberia, and Sierra Leone etc. Recently, a cumulative total number of nearly 25 million cases and 800 000 deaths have been reported since the outbreak of COVID-19 (World Health Organization, 2020). In order to minimize the losses caused by various infectious diseases, resources and efforts need be carefully planned and arranged to control the outbreaks.

Mathematical modeling has been one of the important tools that are used to describe the dynamics of epidemic process (Keeling and Rohani, 2008). Among various mathematical models, compartment models are the most commonly studied. Such models typically divide population into several sub-populations based on the possible state of individuals in the chain of disease transmission, and describe the dynamics of epidemic with differential or difference equations. Kermack and McKendrick (Kermack and McKendrick, 1927) proposed an SIR (Susceptible-Infected-Removed) model as early as 1927, and then various compartment models were further developed according to the transmission characteristics of epidemic diseases. For example, some researchers built the SEIR (Susceptible-Exposed-Infected-Removed) model to describe the dynamic behaviors of infectious diseases that have an obvious incubation period (influenza, hepatitis B,

AIDS, etc.) (Brauer, 2006). And an SEIRD model of Ebola in (Althaus et al., 2015) was presented, in which compartment D denoted death and the individuals in the state can be still highly infectious. Considering the complexity of spread and control process, Pandey et al. established a stochastic model of Ebola transmission between and within the general community, hospitals, and funerals to assess the effectiveness of containment strategies (Pandey et al., 2014). In (Giordano et al., 2020), authors established an SIDARTHE model to study the spread of COVID-19 in Italy.

Many researchers analyzed the spread and control of epidemic process through studying the epidemic threshold and basic reproductive number. For instance, Fraster et al. elaborated the change of basic reproductive number after implementing two basic public health policies (quarantine and contact tracking) (Fraser et al., 2004), and pointed out that the infectious diseases are controllable when their model parameters meet some specific conditions. In (Chowell et al., 2004, Chan, 2014), authors used a transmission rate depending on time to capture the effect of control measure. To a certain extent, these studies revealed an aspect of epidemic control, but their strategies are not convenient to be implemented for government and hospitals.

Comparatively, some strategies given by optimal control can be implemented more conveniently than the approaches mentioned above. For example, Lemos-Paiao et al. proposed the SIQRB model of cholera transmission and solved it as an optimal control problem with isolation as the control measure (Lemos-Paião et al., 2017). Reference (Bakare et al., 2014) dealt with the optimal strategy of the classical SIR process containment under the education propaganda and clinical treatment. Besides, optimal control theory was also applied to containment of computer viruses (Chen et al., 2015) and malware (Zhang et al., 2017). Taking the relationship between individuals into account, Xu et al. studied the optimal control of infectious diseases in complex networks (Xu et al., 2017). Nevertheless, model-plant mismatch and perturbations are neglected in these methods, which make it difficult to achieve the desired control effect.

Nonlinear model predictive control (NMPC), with the ability of correcting model-plant mismatch timely, is widely used in many fields (Xi et al., 2013), including epidemic process control. By taking samples from a plant at certain times, NMPC controller is able to adjust the optimal policy according to actual situation. In addition, the solution of NMPC is carried out timely (Diehl et al., 2002, Findeisen and Allgower, 2002), which makes it easier to concretize control measures that need to be taken at every current time point. For example, when SARS, Ebola and COVID-19 were breaking out, WHO or the countries should give optimal control policies according to current situation to guide the population and medical teams. However, the research for this is still in its initial stage. Recently, Watkins et al. studied the model predictive control of SEIR model in complex networks with isolation as the control strategy (Watkins et al., 2020). Their control strategy is quite specific and easy to implement for decision makers. However, too many details are needed to build the model and implement the optimal strategy. For practical epidemic context, the details are not clear because of a large number of population and complex real situation.

In order to correct systematic deviation and model-plant mismatch on time, sampling and state feedback are the important steps of NMPC so as to achieve the desired control effect. Yet not all of the state variables could be obtained directly during the operation of the system, thus we can use state observer to estimate the states of the system based on output measurements and model structure (Dawson et al., 1992). In some practical epidemic spreads, the number of susceptible and removed cases are not available from official data. Under this circumstance, a state observer is needed to estimate these states. Several valuable accomplishments had been achieved on state observer design for epidemic models (De la Sen et al., 2011, Alonso-Quesada et al., 2012, Ibeas et al., 2015, Iggidr and Souza, 2019). They took the number of infected individuals as the output variable of the system, which implied that the value of infectious state I(k) can be observed accurately. Nevertheless, the observation of I(k) depends greatly on partial actual data, which usually differ from its exact values due to misdiagnosis, case concealment, or data absence. This may enlarge the disturbances imposed on the system and cause the increase of observation error.

Most recently, several papers studied the optimal control on the COVID-19 by using NMPC. In (Köhler et al., 2021), aimed at the social distancing measures of Germany, authors proposed a robust MPC-based feedback policy. Their model could lead to a minimum number of fatalities even if measurements are inaccurate and the infection rates cannot be precisely specified by social distancing. Authors of (Morato et al., 2020) presented an optimal predictive control strategy for COVID-19 social distancing policy in Brazil. First, they presented two modified versions of SIR model. Second, they concerned the fact that large error margins have been reported regarding the available COVID-19 data and statistics in Brazil and performed an uncertainty-weighted least squares criterion to estimate the parameters. Finally, they proposed an MPC-based control framework to determine whether to apply the social distancing policy or not in real time.

In this study, we proposed a discrete-time SI_gI_qR model with hospital isolation as control strategy. In order to obtain optimal control strategy, an NMPC algorithm was given under a performance index with two objectives, that is, the number of infected cases in general community and the cost of control process. As for state feedback, we designed a reduced-order state observer with an output variable (the number of infectious cases under quarantine, this can be counted timely and accurately) whose error converges asymptotically to zero. Our work has the following advantages compared with the existing researches on epidemic process control:

• An optimal control model aimed at the quarantine strategy is built and the constraints about the treatment capacity are given.

• We develop an NMPC algorithm with state feedback mechanism, such that the optimal policy can be updated timely according to the current data.

• As for the design of state observer, our state output chosen could be measured more accurately by the reported data from hospitals or isolation centers, which may avoid or reduce the observation error caused by the partial data.

This paper is organized as follows. Section 2 describes the model construction of an epidemic process. Section 3 introduces some basic knowledge of nonlinear model predictive control. In section 4, we propose the NMPC algorithm for the epidemic process model introduced in section 2 with an asymptotically stable state observer. Section 5 provides numerical simulations of the results introduced in the preceding sections. In section 6, we present conclusions and perspectives on future work.