key: cord-0816653-11j3s6rc
authors: Liu, Zonglin; Stursberg, Olaf
title: On the Use of MPC Techniques to Decide Intervention Policies against COVID-19
date: 2021-12-31
journal: IFAC-PapersOnLine
DOI: 10.1016/j.ifacol.2021.10.400
sha: 96f935427dae74d0b1386c919d84e800d8ff6922
doc_id: 816653
cord_uid: 11j3s6rc

This paper aims at demonstrating how and that model predictive control (MPC) strategies can be used to determine optimal intervention policies against the COVID-19 pandemic. Especially for the time after a first wave of infection and before a vaccine can be safely distributed to a sufficient extent, the intervention experience from the first outbreak can be utilized to guide the policy decision in this period. The MPC problem in this paper takes the pandemic in different regions of a country and its neighboring countries into account, while policies such as wearing masks or social distancing are selected as inputs to be optimized. This optimized policy balances the risk of a second outbreak and socio-economic costs, while considering that the measure should not be too severe to be rejected by the population. Effectiveness of this policy compared to standard intervention policies is compared through numerical simulations.

The outbreak of COVID-19 in 2020 has changed the life of virtually almost everyone on the planet. The rapid spread of the virus, together with the high mortality rate in the early period of the outbreak, forced the governments to deploy intervention policies such as lock-down of cities, or restricting the social life considerably. These policies have indeed shown their effect on controlling the spread of the virus, but also caused significant harm to the economies. In addition, parts of the populations of many countries have become annoyed by restrictions, and the desire for returning to normal has been getting stronger with time. However, recent observations revealed that second (or later) waves of infection can indeed be triggered once the policies are relaxed, and these waves are experienced to have higher amplitudes than the first one in several countries. As the availability and distribution of vaccines still is a certain time ahead (as of November 2020), governments are forced to deploy restrictions like lock-down policies again in order to mitigate the second or a later wave. Simple intervention scheme of the type of bang-bang control, as switching repeatedly between complete shutdown and no restrictions, quickly lets authorities loose credibility, and the effectiveness of intervention measures gradually decreases over time. This raises the question of which intervention schemes are more suitable, and this paper intends to address this question from the perspective of control theory.

This paper aims at illustrating and investigating the use of model-predictive control (MPC) to determine intervention schemes against COVID-19 which are continuously adapted to the evolution of the pandemic and optimized with respect to a chosen criterion. The goal is to use the experience gathered during the first wave of infection, such as the average infecting rate of the virus, the average healing time of the patients, effectiveness of different policies in controlling the spread of the virus, or the socio-economic costs of these policies. The intervention policies considered here include, among others, wearing masks, social distancing in different regions, reserving medical staff for COVID-19 patients, or introducing traffic limitation between different regions and countries. The geographic adjacency between regions is also taken into account, such that each region can deploy its own policy against the virus. This setting can help to evaluate whether a common intervention policy of different regions is necessary. It is stressed here that, while the models and measures are adapted to the COVID-19 pandemic, the insights and findings should be transferable with suitable modifications to other pandemics as well. The next section first provides a review on existing work on epidemic modeling and control problems, with distinction of whether the work refers to periods before or after the outbreak of COVID-19. In Sec. 3, the nonlinear epidemic model, the input and state constraints, and the cost functions of the considered MPC approach are specified. Section 4 compares strategies obtained from MPC with simpler schemes such as bang-bang policies. To address the uncertainties of modeling the spread of the virus and thus the influence to the corresponding MPC problems, different extensions are introduced in Sec. 5, before Sec. 6 concludes the paper.

To model the spread of a virus, the susceptible -infected -susceptible (SIS) model has been developed in Kermack and McKendrick (1932) ; Wang et al. (2003) : Any individual is either infected, or susceptible to infection at any time in this model, and it is assumed that a susceptible

Zonglin Liu * Olaf Stursberg * * Control and System Theory, Dept. of Electrical Engineering and Computer Science, University of Kassel, Germany. Email: {Z.Liu,stursberg}@uni-kassel.de.

This paper aims at demonstrating how and that model predictive control (MPC) strategies can be used to determine optimal intervention policies against the COVID-19 pandemic. Especially for the time after a first wave of infection and before a vaccine can be safely distributed to a sufficient extent, the intervention experience from the first outbreak can be utilized to guide the policy decision in this period. The MPC problem in this paper takes the pandemic in different regions of a country and its neighboring countries into account, while policies such as wearing masks or social distancing are selected as inputs to be optimized. This optimized policy balances the risk of a second outbreak and socio-economic costs, while considering that the measure should not be too severe to be rejected by the population. Effectiveness of this policy compared to standard intervention policies is compared through numerical simulations.

Keywords: Nonlinear Model Predictive Control, Biomedical Systems, Networked Systems

The outbreak of COVID-19 in 2020 has changed the life of virtually almost everyone on the planet. The rapid spread of the virus, together with the high mortality rate in the early period of the outbreak, forced the governments to deploy intervention policies such as lock-down of cities, or restricting the social life considerably. These policies have indeed shown their effect on controlling the spread of the virus, but also caused significant harm to the economies. In addition, parts of the populations of many countries have become annoyed by restrictions, and the desire for returning to normal has been getting stronger with time. However, recent observations revealed that second (or later) waves of infection can indeed be triggered once the policies are relaxed, and these waves are experienced to have higher amplitudes than the first one in several countries. As the availability and distribution of vaccines still is a certain time ahead (as of November 2020), governments are forced to deploy restrictions like lock-down policies again in order to mitigate the second or a later wave. Simple intervention scheme of the type of bang-bang control, as switching repeatedly between complete shutdown and no restrictions, quickly lets authorities loose credibility, and the effectiveness of intervention measures gradually decreases over time. This raises the question of which intervention schemes are more suitable, and this paper intends to address this question from the perspective of control theory.

This paper aims at illustrating and investigating the use of model-predictive control (MPC) to determine intervention schemes against COVID-19 which are continuously adapted to the evolution of the pandemic and optimized with respect to a chosen criterion. The goal is to use the experience gathered during the first wave of infection, such as the average infecting rate of the virus, the average healing time of the patients, effectiveness of different policies in controlling the spread of the virus, or the socio-economic costs of these policies. The intervention policies considered here include, among others, wearing masks, social distancing in different regions, reserving medical staff for COVID-19 patients, or introducing traffic limitation between different regions and countries. The geographic adjacency between regions is also taken into account, such that each region can deploy its own policy against the virus. This setting can help to evaluate whether a common intervention policy of different regions is necessary. It is stressed here that, while the models and measures are adapted to the COVID-19 pandemic, the insights and findings should be transferable with suitable modifications to other pandemics as well. The next section first provides a review on existing work on epidemic modeling and control problems, with distinction of whether the work refers to periods before or after the outbreak of COVID-19. In Sec. 3, the nonlinear epidemic model, the input and state constraints, and the cost functions of the considered MPC approach are specified. Section 4 compares strategies obtained from MPC with simpler schemes such as bang-bang policies. To address the uncertainties of modeling the spread of the virus and thus the influence to the corresponding MPC problems, different extensions are introduced in Sec. 5, before Sec. 6 concludes the paper.

To model the spread of a virus, the susceptible -infected -susceptible (SIS) model has been developed in Kermack and McKendrick (1932) ; Wang et al. (2003) {Z.Liu,stursberg}@uni-kassel.de.

This paper aims at demonstrating how and that model predictive control (MPC) strategies can be used to determine optimal intervention policies against the COVID-19 pandemic. Especially for the time after a first wave of infection and before a vaccine can be safely distributed to a sufficient extent, the intervention experience from the first outbreak can be utilized to guide the policy decision in this period. The MPC problem in this paper takes the pandemic in different regions of a country and its neighboring countries into account, while policies such as wearing masks or social distancing are selected as inputs to be optimized. This optimized policy balances the risk of a second outbreak and socio-economic costs, while considering that the measure should not be too severe to be rejected by the population. Effectiveness of this policy compared to standard intervention policies is compared through numerical simulations.

Keywords: Nonlinear Model Predictive Control, Biomedical Systems, Networked Systems

The outbreak of COVID-19 in 2020 has changed the life of virtually almost everyone on the planet. The rapid spread of the virus, together with the high mortality rate in the early period of the outbreak, forced the governments to deploy intervention policies such as lock-down of cities, or restricting the social life considerably. These policies have indeed shown their effect on controlling the spread of the virus, but also caused significant harm to the economies. In addition, parts of the populations of many countries have become annoyed by restrictions, and the desire for returning to normal has been getting stronger with time. However, recent observations revealed that second (or later) waves of infection can indeed be triggered once the policies are relaxed, and these waves are experienced to have higher amplitudes than the first one in several countries. As the availability and distribution of vaccines still is a certain time ahead (as of November 2020), governments are forced to deploy restrictions like lock-down policies again in order to mitigate the second or a later wave. Simple intervention scheme of the type of bang-bang control, as switching repeatedly between complete shutdown and no restrictions, quickly lets authorities loose credibility, and the effectiveness of intervention measures gradually decreases over time. This raises the question of which intervention schemes are more suitable, and this paper intends to address this question from the perspective of control theory.

This paper aims at illustrating and investigating the use of model-predictive control (MPC) to determine intervention schemes against COVID-19 which are continuously adapted to the evolution of the pandemic and optimized with respect to a chosen criterion. The goal is to use the experience gathered during the first wave of infection, such as the average infecting rate of the virus, the average healing time of the patients, effectiveness of different policies in controlling the spread of the virus, or the socio-economic costs of these policies. The intervention policies considered here include, among others, wearing masks, social distancing in different regions, reserving medical staff for COVID-19 patients, or introducing traffic limitation between different regions and countries. The geographic adjacency between regions is also taken into account, such that each region can deploy its own policy against the virus. This setting can help to evaluate whether a common intervention policy of different regions is necessary. It is stressed here that, while the models and measures are adapted to the COVID-19 pandemic, the insights and findings should be transferable with suitable modifications to other pandemics as well. The next section first provides a review on existing work on epidemic modeling and control problems, with distinction of whether the work refers to periods before or after the outbreak of COVID-19. In Sec. 3, the nonlinear epidemic model, the input and state constraints, and the cost functions of the considered MPC approach are specified. Section 4 compares strategies obtained from MPC with simpler schemes such as bang-bang policies. To address the uncertainties of modeling the spread of the virus and thus the influence to the corresponding MPC problems, different extensions are introduced in Sec. 5, before Sec. 6 concludes the paper.

To model the spread of a virus, the susceptible -infected -susceptible (SIS) model has been developed in Kermack and McKendrick (1932) ; Wang et al. (2003) : Any individual is either infected, or susceptible to infection at any time in this model, and it is assumed that a susceptible

Zonglin Liu * Olaf Stursberg * * Control and System Theory, Dept. of Electrical Engineering and Computer Science, University of Kassel, Germany. Email:

{Z.Liu,stursberg}@uni-kassel.de.

This paper aims at demonstrating how and that model predictive control (MPC) strategies can be used to determine optimal intervention policies against the COVID-19 pandemic. Especially for the time after a first wave of infection and before a vaccine can be safely distributed to a sufficient extent, the intervention experience from the first outbreak can be utilized to guide the policy decision in this period. The MPC problem in this paper takes the pandemic in different regions of a country and its neighboring countries into account, while policies such as wearing masks or social distancing are selected as inputs to be optimized. This optimized policy balances the risk of a second outbreak and socio-economic costs, while considering that the measure should not be too severe to be rejected by the population. Effectiveness of this policy compared to standard intervention policies is compared through numerical simulations.

Keywords: Nonlinear Model Predictive Control, Biomedical Systems, Networked Systems

The outbreak of COVID-19 in 2020 has changed the life of virtually almost everyone on the planet. The rapid spread of the virus, together with the high mortality rate in the early period of the outbreak, forced the governments to deploy intervention policies such as lock-down of cities, or restricting the social life considerably. These policies have indeed shown their effect on controlling the spread of the virus, but also caused significant harm to the economies. In addition, parts of the populations of many countries have become annoyed by restrictions, and the desire for returning to normal has been getting stronger with time. However, recent observations revealed that second (or later) waves of infection can indeed be triggered once the policies are relaxed, and these waves are experienced to have higher amplitudes than the first one in several countries. As the availability and distribution of vaccines still is a certain time ahead (as of November 2020), governments are forced to deploy restrictions like lock-down policies again in order to mitigate the second or a later wave. Simple intervention scheme of the type of bang-bang control, as switching repeatedly between complete shutdown and no restrictions, quickly lets authorities loose credibility, and the effectiveness of intervention measures gradually decreases over time. This raises the question of which intervention schemes are more suitable, and this paper intends to address this question from the perspective of control theory.

This paper aims at illustrating and investigating the use of model-predictive control (MPC) to determine intervention schemes against COVID-19 which are continuously adapted to the evolution of the pandemic and optimized with respect to a chosen criterion. The goal is to use the experience gathered during the first wave of infection, such as the average infecting rate of the virus, the average healing time of the patients, effectiveness of different policies in controlling the spread of the virus, or the socio-economic costs of these policies. The intervention policies considered here include, among others, wearing masks, social distancing in different regions, reserving medical staff for COVID-19 patients, or introducing traffic limitation between different regions and countries. The geographic adjacency between regions is also taken into account, such that each region can deploy its own policy against the virus. This setting can help to evaluate whether a common intervention policy of different regions is necessary. It is stressed here that, while the models and measures are adapted to the COVID-19 pandemic, the insights and findings should be transferable with suitable modifications to other pandemics as well. The next section first provides a review on existing work on epidemic modeling and control problems, with distinction of whether the work refers to periods before or after the outbreak of COVID-19. In Sec. 3, the nonlinear epidemic model, the input and state constraints, and the cost functions of the considered MPC approach are specified. Section 4 compares strategies obtained from MPC with simpler schemes such as bang-bang policies. To address the uncertainties of modeling the spread of the virus and thus the influence to the corresponding MPC problems, different extensions are introduced in Sec. 5, before Sec. 6 concludes the paper.

To model the spread of a virus, the susceptible -infected -susceptible (SIS) model has been developed in Kermack and McKendrick (1932) ; Wang et al. (2003) {Z.Liu,stursberg}@uni-kassel.de.

This paper aims at demonstrating how and that model predictive control (MPC) strategies can be used to determine optimal intervention policies against the COVID-19 pandemic. Especially for the time after a first wave of infection and before a vaccine can be safely distributed to a sufficient extent, the intervention experience from the first outbreak can be utilized to guide the policy decision in this period. The MPC problem in this paper takes the pandemic in different regions of a country and its neighboring countries into account, while policies such as wearing masks or social distancing are selected as inputs to be optimized. This optimized policy balances the risk of a second outbreak and socio-economic costs, while considering that the measure should not be too severe to be rejected by the population. Effectiveness of this policy compared to standard intervention policies is compared through numerical simulations.

Keywords: Nonlinear Model Predictive Control, Biomedical Systems, Networked Systems

The outbreak of COVID-19 in 2020 has changed the life of virtually almost everyone on the planet. The rapid spread of the virus, together with the high mortality rate in the early period of the outbreak, forced the governments to deploy intervention policies such as lock-down of cities, or restricting the social life considerably. These policies have indeed shown their effect on controlling the spread of the virus, but also caused significant harm to the economies. In addition, parts of the populations of many countries have become annoyed by restrictions, and the desire for returning to normal has been getting stronger with time. However, recent observations revealed that second (or later) waves of infection can indeed be triggered once the policies are relaxed, and these waves are experienced to have higher amplitudes than the first one in several countries. As the availability and distribution of vaccines still is a certain time ahead (as of November 2020), governments are forced to deploy restrictions like lock-down policies again in order to mitigate the second or a later wave. Simple intervention scheme of the type of bang-bang control, as switching repeatedly between complete shutdown and no restrictions, quickly lets authorities loose credibility, and the effectiveness of intervention measures gradually decreases over time. This raises the question of which intervention schemes are more suitable, and this paper intends to address this question from the perspective of control theory.

This paper aims at illustrating and investigating the use of model-predictive control (MPC) to determine intervention schemes against COVID-19 which are continuously adapted to the evolution of the pandemic and optimized with respect to a chosen criterion. The goal is to use the experience gathered during the first wave of infection, such as the average infecting rate of the virus, the average healing time of the patients, effectiveness of different policies in controlling the spread of the virus, or the socio-economic costs of these policies. The intervention policies considered here include, among others, wearing masks, social distancing in different regions, reserving medical staff for COVID-19 patients, or introducing traffic limitation between different regions and countries. The geographic adjacency between regions is also taken into account, such that each region can deploy its own policy against the virus. This setting can help to evaluate whether a common intervention policy of different regions is necessary. It is stressed here that, while the models and measures are adapted to the COVID-19 pandemic, the insights and findings should be transferable with suitable modifications to other pandemics as well. The next section first provides a review on existing work on epidemic modeling and control problems, with distinction of whether the work refers to periods before or after the outbreak of COVID-19. In Sec. 3, the nonlinear epidemic model, the input and state constraints, and the cost functions of the considered MPC approach are specified. Section 4 compares strategies obtained from MPC with simpler schemes such as bang-bang policies. To address the uncertainties of modeling the spread of the virus and thus the influence to the corresponding MPC problems, different extensions are introduced in Sec. 5, before Sec. 6 concludes the paper.

To model the spread of a virus, the susceptible -infected -susceptible (SIS) model has been developed in Kermack and McKendrick (1932) ; Wang et al. (2003) {Z.Liu,stursberg}@uni-kassel.de.

This paper aims at demonstrating how and that model predictive control (MPC) strategies can be used to determine optimal intervention policies against the COVID-19 pandemic. Especially for the time after a first wave of infection and before a vaccine can be safely distributed to a sufficient extent, the intervention experience from the first outbreak can be utilized to guide the policy decision in this period. The MPC problem in this paper takes the pandemic in different regions of a country and its neighboring countries into account, while policies such as wearing masks or social distancing are selected as inputs to be optimized. This optimized policy balances the risk of a second outbreak and socio-economic costs, while considering that the measure should not be too severe to be rejected by the population. Effectiveness of this policy compared to standard intervention policies is compared through numerical simulations.

Keywords: Nonlinear Model Predictive Control, Biomedical Systems, Networked Systems

The outbreak of COVID-19 in 2020 has changed the life of virtually almost everyone on the planet. The rapid spread of the virus, together with the high mortality rate in the early period of the outbreak, forced the governments to deploy intervention policies such as lock-down of cities, or restricting the social life considerably. These policies have indeed shown their effect on controlling the spread of the virus, but also caused significant harm to the economies. In addition, parts of the populations of many countries have become annoyed by restrictions, and the desire for returning to normal has been getting stronger with time. However, recent observations revealed that second (or later) waves of infection can indeed be triggered once the policies are relaxed, and these waves are experienced to have higher amplitudes than the first one in several countries. As the availability and distribution of vaccines still is a certain time ahead (as of November 2020), governments are forced to deploy restrictions like lock-down policies again in order to mitigate the second or a later wave. Simple intervention scheme of the type of bang-bang control, as switching repeatedly between complete shutdown and no restrictions, quickly lets authorities loose credibility, and the effectiveness of intervention measures gradually decreases over time. This raises the question of which intervention schemes are more suitable, and this paper intends to address this question from the perspective of control theory.

This paper aims at illustrating and investigating the use of model-predictive control (MPC) to determine intervention schemes against COVID-19 which are continuously adapted to the evolution of the pandemic and optimized with respect to a chosen criterion. The goal is to use the experience gathered during the first wave of infection, such as the average infecting rate of the virus, the average healing time of the patients, effectiveness of different policies in controlling the spread of the virus, or the socio-economic costs of these policies. The intervention policies considered here include, among others, wearing masks, social distancing in different regions, reserving medical staff for COVID-19 patients, or introducing traffic limitation between different regions and countries. The geographic adjacency between regions is also taken into account, such that each region can deploy its own policy against the virus. This setting can help to evaluate whether a common intervention policy of different regions is necessary. It is stressed here that, while the models and measures are adapted to the COVID-19 pandemic, the insights and findings should be transferable with suitable modifications to other pandemics as well. The next section first provides a review on existing work on epidemic modeling and control problems, with distinction of whether the work refers to periods before or after the outbreak of COVID-19. In Sec. 3, the nonlinear epidemic model, the input and state constraints, and the cost functions of the considered MPC approach are specified. Section 4 compares strategies obtained from MPC with simpler schemes such as bang-bang policies. To address the uncertainties of modeling the spread of the virus and thus the influence to the corresponding MPC problems, different extensions are introduced in Sec. 5, before Sec. 6 concludes the paper. (2008), where the number 2 denotes the two states of a person, susceptible or infected, and n denotes the total number of people in the graph. The healing and infection of any person i ∈ N = {1, · · · , n} of a Markov model are described by two independent Poisson processes with rates δ i and β i , respectively. Although the 2 n Markov model describes the spread of a virus disease often fairly well, its shortcoming is obvious: the state-space size grows exponentially with the number n. Accordingly, an N − Intertwined model has been proposed in Gourdin et al. (2011) , in which a mean-field type approximation technique is applied to approximate the 2 n Markov model. This approximation enables one to only use a set of n nonlinear ordinary differential equations to describe the dynamics of the epidemic, i.e., by using p i (t) to denote the probability of person i being infected in time t, the following dynamics applies:

(1)

In addition to δ i and β i representing the local healing and infection rate of person i, the α ij in (1) corresponds to the connectivity between two persons i and j in a connection graph (while each node of the graph represents a person). In general, an adjacency matrix A can be introduced to describe the connection graph among the people, where α ij denotes an entry of matrix A, and α ij = 1 applies if i and j are connected, while α ij = 0 otherwise, and α ii = 1 for all i ∈ N . Note that if the index i refers to a group of persons instead of an individual (which is the case in the rest of this paper), then the p i (t) in (1) represents the percentage of infection of a group with index i. The dynamics (1) can also be written in matrix form:

(2) where p(t) = [p 1 (t), · · · , p n (t)] T , B = diag(β i ) and D = diag(δ i ). As the extinction of the disease implies p(t) = 0 in (2), research has focused on determining stability conditions for the origin of (2). First of all, one of the most important indices in epidemiology, the basic reproduction number R 0 = ρ(D −1 A T B), is derived from this model, where ρ denotes the spectral radius of a matrix, see Khanafer et al. (2016) . It has been proven that if R 0 ≤ 1, then the disease-free equilibrium (DFE) is asymptotically stable. In case R 0 > 1, there also exists a unique endemic equilibrium which is asymptotically stable according to Fall et al. (2007) . This endemic equilibrium also refers to herd immunity.

The research is then extended to account for homogeneous or heterogeneous cases, i.e., whether the healing and infection rate are identical or different for all people. Given the epidemic model (2), the following optimal control tasks have been proposed in literature: 1.) An optimal vaccine allocation plan is considered in Wan et al. (2007) ; Preciado et al. (2013) and the task there is to investigate how to effectively distribute the vaccine; 2.) The work in Sahneh and Scoglio (2012) proposed an optimal information dissemination strategy to control the epidemic, in which the susceptible group may receive an alert before being infected. The optimal control task is to construct the alert network, such that the susceptible persons are always warned in time, thus reducing the R 0 value; 3.) The work in Wan et al. (2008) proposed a spatially heterogeneous strategy for controlling the spread of the virus, where the population is categorized into different groups sharing the same or similar healing and infection rate.

After the outbreak of COVID-19, existing epidemic modeling and control techniques have been quickly adapted to capture particular characteristics of this virus. A quite detailed review to these adaptations is given in Kantner and Koprucki (2020) . The work in Giordano et al. (2020) extended the SIS model to a more explicit one, the socalled SIDARTHE model accounting for the pandemic in Italy, which includes more disease-related status of the population at the beginning of COVID-19 . Based on the SIDARTHE model, the work in Kantner and Koprucki (2020) proposed an optimal non-pharmaceutical intervention strategy for the case that a vaccine is never found. For the work by Köhler et al. (2020) , which also considers the pandemic in Germany, the social distancing requirements are optimized through an MPC strategy in order to minimize the fatalities over a fixed period of time. The above work has all been written in the mid of the first wave of COVID-19, thus the main concern was limited to either finding a suitable model to reproduce the evolution of the pandemic during the first outbreak, or to evaluate whether the virus can be erased through a given policy. After the first wave, however, a fast elimination of the virus without vaccine has proven to be impossible in most countries (whereas the work by Liu and Stursberg (2021) has started to consider an optimized distribution of limited amount of available vaccines). The focus thus should be shifted to the question of how to properly utilize the experience gathered during the first wave, such that better intervention policies can be developed to mitigate the pandemic, until a vaccine is distributed to significant extent.

As the geographic position of a country plays an important role in analyzing the spread of diseases, a connection graph is constructed to model the adjacency to other countries, as well as the adjacency among different regions within a country. For one country to be investigated, let an overall set of nodes N = {1, · · · , n} = N s ∪ N f be given, where the nodes in N s represent the regions of the country, while the nodes in N f represent the neighboring countries. A directed graph G = {N, E} is constructed, where E models the set of edges connecting the nodes. For the connection of any two local regions i, j ∈ N s , two edges {e i,j , e j,i } ⊂ E are introduced (direction irrelevant). In case a region i ∈ N s shares border with a different country j ∈ N f , a directed edge e i,j ∈ E (from j to i) is assigned. It is emphasized that the intervention policy to be addressed will take into account the pandemic in both local regions and neighboring countries, but only the control in the local regions is deployed. An adjacency matrix A can be determined for the graph, in which α ij = 1 if the edge e i,j exists and α ii = 1 for all i ∈ N s , see Fig. 1 .

In addition to the standard SIS model in (1), the Susceptible-Infected-Recovered (SIR) model, or the more complicated SIDARTHE model have been suggested to model and study the epidemic. The use of the SIR model is suitable if the recovered sub-population has significantly changed the population structure, and if this sub-population is constantly immune to reinfection. For COVID-19, however, the recovered sub-population still represents a relatively small share in many countries as of mid November 2020 and the risk of a reinfection can also not be excluded. The SIDARTHE model can cover more disease-related states and is suitable to model the epidemic in the first outbreak. For the period after the first wave, however, some states such as Diagnosed, Recognized or Threatened, may be no more crucial for long term policy making. The true critical factors after the first wave have actually been reduced to the daily number of active cases and death cases.

Note that the p i (t) in (1) represents the percentage of infected persons in the region i, i.e., as the number of population can be regarded as constant within the considered duration, and p i (t) changes with the number of active cases in region i. For the daily number of death cases (the mortality rate f ), it has been observed that it grows rapidly if the number of critically infected patients exceeds the available intensive care units. By assuming that the number of critically infected patients also scales with the number of active cases, an active-case-based mortality curve is plotted in Fig. 2 . Based on the relation in Fig.  2 , the goal of minimizing the death cases in region i can be cast into a constraint p i (t) ≤ p c i , and a minimization of p i (t) in the latter problem. Now, as both active and death cases in region i are related to p i (t), the SIS model in (1) is reformulated into the following form for each region i ∈ N s : Fig. 1 . Nodes in black represent local regions in the considered country, whereas nodes in red are neighboring countries (and the adjacency matrix on the right).

PSfrag replacements Fig. 2 . Relation between f (t) and p i (t) in region i, where a threshold value p c i exists according to Kantner and Koprucki (2020) .ṗ

(3) Compared to (1), the effect from other nodes to the local node i is divided into the one from the neighboring regions and the one from the neighboring countries. This is because only the p i (t) of local regions i ∈ N s are assumed to be controllable by the policies, while the p j (t), j ∈ N f , of neighboring countries are obtained from prediction.

In this paper, the following inputs (or policies) and their impact are assumed to be at the disposal of the authorities: 1.) For each edge e i,j of the graph G, the entry α ij ∈ R ≥0 in the adjacency matrix A can be selected from α min to 1, where α ij = 1 implies that no restriction is imposed on the connection, while α min implies the most strict limitation. 2.) The requirement to wear masks or social distancing in region i will reduce the infection rate β i ∈ [β min , β max ], β i ∈ R ≥0 . 3.) The hospital staff allocated for COVID-19 patients in region i will increase the healing rate δ

It is assumed that the mapping from different policies to the α ij , β i , and δ i values can be estimated according to the experiences from the first outbreak. These policies, however, will also lead to socio-economic costs. For a given planning horizon [t 0 , t 0 + H], the policy is assumed to be adjusted every T days (thus changed for totally H c = H T times within this horizon and being constantly for T days). Then the following socio-economic costs are considered: 1.) Decreasing the α ij (k) in step k ∈ {0, · · · , H c − 1}, implies a reduction of the interaction between different countries and regions, leading to a cost of:

2.) Decreasing the infection rate β i leads to a higher load for local people, which is formulated by:

3.) Increasing the healing rate δ i implies that more hospital staff is reserved for COVID-19 patients, thus the patients due to other diseases cannot be properly handled, i.e.:

The cost function terms in (4) -(6) are chosen to be linear in order to directly reflect the effect of the corresponding policy. In addition, as a frequent change of the policy should be avoided, the following cost function term is introduced to penalize large changes of the policy:

The overall cost function of i thus takes the form of:

where the last term records the active cases over the considered horizon (each term of J all,i can also be weighted differently). The constraints on policy decisions consist of input constraints for α ij , β i and δ i , and the state constraint p i (t) ≤ p c i , i.e., the mortality threshold must not be exceeded. Finally, for decision time t 0 and for a given p i (t 0 ) in each local region i ∈ N s , as well as a prediction p j (t), t ∈ [t 0 , t 0 + H], on how the virus will spread in neighboring countries j ∈ N f , the following MPC problem is solved to find the optimal intervention policy: Problem 1. min αij ,βi,δi,∀i∈Ns i∈Ns J all,i (12) Note that the threshold p c i here should not be exceeded in any time t, instead of only in decision time t k·T (leading to a continuous-time optimization problem). The solution can be carried out, e.g. by using the multiple-shooting methods in Bock et al. (2000) . Note that a high accuracy of the solution is desired as it refers to social costs and life of patients, while the solution time is not critical for H being chosen to several weeks. By using the recedinghorizon scheme of MPC, the optimized policy of step k is applied, and the optimization is repeated after T days in the next step.

To demonstrate which solutions are obtained from the considered MPC strategy, the selected scenarios are presented: Consider the connection graph in Fig. 1 , and the percentages of infection p j (t), j ∈ N f , for all neighboring countries for totally 450 days as in Fig. 3 . For the investigated country with five regions as in Fig. 1 , the weight α ij , i � = j, can take values in the interval [0.2, 1], and the infection rate β i varies within [5.4 · 10 −3 , 9 · 10 −3 ], while the healing rate δ i is in [0.014, 0.017]. The infection rate β i has a larger range than δ i , since the latter value cannot be significantly changed without efficient medicine. Default values of these variables are α ij = 1, β i = 9 · 10 −3 and δ i = 0.014. The threshold value is chosen to p c = 0.0001. Now, if the bang-bang intervention scheme is introduced in the studied country, i.e., whenever p i (t) ≥ p c applies in region i, this region is locked down in the next decision time: thus the weight α ij of each edge connecting region i and the infection rate β i are reduced to the minimum, and the healing rate β i is increased to the maximum. The lock-down status then lasts for T = 30 days up to the next decision time. The outcome by deploying this policy is illustrated in Fig. 4 and the lock-down sequences in all five regions are plotted in Fig. 5 . One can notice that the p i (t) are beyond the threshold p c in all regions for almost all the time, despite the frequent lock-downs and large fluctuations of p j (t) in other countries.

Next, assume that the development of p j (t) in the neighboring countries can be exactly predicted, i.e.,p j (t) = p j (t), for all j ∈ N f , t ∈ R ≥0 (In case a prediction error exists, one can use the method in Liu and Stursberg (2019) to handle uncertain prediction problems in MPC). In the first test, the prediction horizon is selected to be H = 30 days (only one decision step), and the term (7) of the cost function is not considered. Then, by solving Problem 1 in a receding-horizon scheme in each decision time, the development of p i (t) in Fig. 6 is obtained, as well as the corresponding β i and δ i sequences, see Fig. 7 and 8 (The sequences of α ij are not shown due to space limitations). Compared to the bang-bang intervention scheme, an overshoot over the threshold p c has been successfully avoided, but a frequent change of the intervention policy occurs, which is certainly undesired and also unrealistic.

Accordingly, if the prediction horizon is extended to H = 90 days (three decision steps) in Problem 1, and the term (7) is considered in the cost function, the development of p i (t) in Fig. 9 is obtained. Compared to the last test, the threshold p c is reached after 270 days, while it was 120 days in Fig. 6 , i.e. the spread of the virus has been slowed down. For the corresponding α ij , β i and δ i sequences plotted in Fig. 9 to Fig. 12 , they also appear to be more smoothly changed than in the last test. In general, the above simulation results establish the following fact: If the development of the pandemic can be well predicted based on the experiences from the first wave, and a sufficiently large horizon is taken into account when deciding upon the policy, a smooth strategy balancing the socio-economic costs and the risk of a new outbreak can be found, while the mortality rate is controlled to a relative low level.

Note that the Problem 1 is a deterministic optimization problem without uncertainties. This setting, however, seems barely realistic if, e.g., intensive testing is not always possible. In particular, exact values of the infection rate β i and healing rate δ i , or the edge weight α ij can hardly be determined in practice. More often, one can only identify possible ranges of these rates and weights. This challenges the MPC strategy which should be able to robustly satisfy the constraints in Problem 1. To this end, the min-max approaches tailored to robust MPC, see e.g. Campo and Morari (1987), can be applied to satisfy the constraints even in the worst case. Another type of uncertainty often reported in media is a sudden outbreak in a hotspot of region i, which can be represented by a dramatical increase of p i (t) in short time. This hotspot scenario can be modeled by adding a bounded stochastic disturbance ω i (t) to the local dynamics (3), which follows a distribution that can be estimated from evolutions in the first year of the outbreak. Now, by replacing the dynamics (3) in Problem 1 by the stochastic model, one can either further apply the min-max approaches to ensure the robust satisfaction of the constraints. or adopt e.g. the technique of scenariobased MPC, see Bernardini and Bemporad (2009) , to ensure that the constraints are satisfied in a mean-square sense. Furthermore, one can use the stochastic model to represent insufficient measurements of infected cases due to a lack of testing. In addition, it should be mentioned that Problem 1 only considers typical epidemic-related social-costs and constraints, while additional types are conceivable. For example, the economic costs such as the damage caused by lock-down, or constraints like available hospital staff can be included. Depending on the measurable data, Problem 1 should thus be adapted in every decision step k by identifying new instances of dynamics, costs, and constraints.

The objective of this paper was to illustrate that MPC strategies can make a meaningful contribution to determine intervention policies for a country against pandemics, and thus may serve as a tool for authorities. The background of the problem is that simple bang-bang schemes are not guaranteed to avoid a second wave of the pandemic, as shown by simulation and recent experiences. In addition, a frequent change of policies will also lead to negative effects on life and economy, and thus decreases the effectiveness. This paper has outlined how to determine ,,smoother " strategies by using MPC, taking the geographic position, the mortality rate, and the socio-econonmic costs into account. Simulation results have confirmed significant advantages compared to bangbang schemes. Upcoming work will take prediction and modeling uncertainties into account, in order to enhance robustness of the obtained policies. 

Scenario-based model predictive control of stochastic constrained linear systems

A direct multiple shooting method for real-time optimization of nonlinear DAE processes

Robust model predictive control

Epidemiological models and lyapunov functions

Modelling the Covid-19 epidemic and implementation of population-wide interventions in Italy

Optimization of network protection against virus spread

Beyond just flattening the curve: Optimal control of epidemics with purely non-pharmaceutical interventions

Contributions to the mathematical theory of epidemics. II.the problem of endemicity

Stability of epidemic models over directed graphs: A positive systems approach

Robust and optimal predictive control of the Covid-19 outbreak

Recursive feasibility and stability of MPC with time-varying and uncertain state constraints

A study on model-based optimization of vaccination strategies against epidemic virus spread

Optimal vaccine allocation to control epidemic outbreaks in arbitrary networks

Optimal information dissemination in epidemic networks

Virus spread in networks

Network design problems for controlling virus spread

Designing spatially heterogeneous strategies for control of virus spread

Epidemic spreading in real networks: An eigenvalue viewpoint