key: cord-0630091-6h2kx9ob
authors: Olivier, Laurentz E.; Botha, Stefan; Craig, Ian K.
title: Optimized lockdown strategies for curbing the spread of COVID-19: A South African case study
date: 2020-06-29
journal: nan
DOI: nan
sha: 33bd557c62ed95e9567671777755c0b4bb4a1b0f
doc_id: 630091
cord_uid: 6h2kx9ob

To curb the spread of COVID-19, many governments around the world have implemented tiered lockdowns with varying degrees of stringency. Lockdown levels are typically increased when the disease spreads and reduced when the disease abates. A predictive control approach is used to develop optimized lockdown strategies for curbing the spread of COVID-19. These strategies are then applied to South African data. The South African case is of immediate interest as the number of confirmed infectious cases does not appear to have peaked yet (at the time of writing), while at the same time the South African government is busy reducing the degree of lockdown. An epidemiological model for the spread of COVID-19 in South Africa was previously developed, and is used in conjunction with a hybrid model predictive controller to optimize lockdown management under different policy scenarios. Scenarios considered include how flatten the curve to a level that the healthcare system can cope with, how to balance lives and livelihoods, and what impact the compliance of the population to the lockdown measures has on the spread of COVID-19.

A novel coronavirus believed to be of zoonotic origin emerged in Wuhan, China towards the end of 2019. This virus, which was subsequently named SARS-CoV-2 and the disease it causes COVID-19 (World Health Organization, 2020a) , has since spread around the world. The WHO characterized COVID-19 as a pandemic on 11 March 2020 (World Health Organization, 2020b). The COVID-19 pandemic first took hold in regions of the world that share high volumes of air traffic with China (Lau et al., 2020) . The importance of "flattening the curve", i.e. reducing the number of COVID-19 infected patients needing critical care to be below the number of available beds in intensive care units or appropriately equipped field hospitals, soon became evident (Stewart et al., 2020) .

The South African National Institute for Communicable Diseases confirmed the first COVID-19 case in South Africa on 5 March 2020. Having learnt from elsewhere about the importance of "flattening the curve", the South African Government was quick to place the country under strict lockdown (what later became known as lockdown level 5) on 27 March 2020 after only 1,170 confirmed COVID-19 cases and 1 related death (Humanitarian Data Exchange, 2020).

The early strict lockdown measures in South Africa have been successful from an epidemiological point of view, but great harm was done to an economy that was already weak before the COVID-19 pandemic started (Arndt et al., 2020) . As a result, significant pressure was applied to relax the lockdown measures even though the number of infectious individuals was still growing exponentially (Harding, 2020) .

The South African government has formulated five lockdown levels with varying degrees of strictness with regards to the measures imposed in order to systematically restore economic activity. How and why lockdown measures are relaxed is clearly articulated. There is thus significant interest in determining the epidemiological impact of the lockdown levels (South African Government, 2020b). Towards this end, an epidemiological model was developed for South Africa in Olivier and Craig (2020) , and a predictive control approach to managing lockdown levels is presented in this work.

One policy approach to managing lockdown levels may be to flatten the curve so as not to overwhelm the healthcare system. Some infected individuals need hospitalization and intensive care -there are studies that show that roughly 5 % of confirmed infectious cases require admission to intensive care units (ICUs) (Guan et al., 2020) . As more and more individuals are exposed and infected, healthcare systems can easily become overwhelmed, especially in developing countries with fragile and underdeveloped healthcare systems (The Economist Intelligence Unit, 2020). The amount of ICU beds available can therefore serve as a high limit for the number of infected individuals needing intensive care (Stewart et al., 2020) .

Considering the cumulative impact of lockdown on the economy is however also relevant. Another policy approach may therefore be to not impose strict lockdown measures for too long, and to potentially reduce the lockdown level even though the healthcare capacity may be exceeded. This policy is known as "balancing lives and livelihoods" (see Panovska-Griffiths et al. (2020) ).

To illustrate the optimal implementation of these policy choices, a model predictive control approach is used (Camacho and Alba, 2013) . Lockdown levels are represented as integer values, whereas the SEIQRDP model used is continuous and dynamic. Dynamic systems that contain continuous and discrete state/input variables are known as hybrid systems (Camacho et al., 2010) , and therefore a hybrid model predictive control (HMPC) approach is required.

Solving the resulting constrained optimization problem is known as mixed integer programming. These problems are NP-hard (non-deterministic polynomial-time) and even to test if a feasible solution improves on the best solution to date is an NP problem (Camacho et al., 2010) . Genetic algorithms (see e.g. Fleming and Purshouse (2002) ) have been found in the past to be suitable for solving HMPC problems, and are therefore used in this work (see e.g. Muller and Craig (2017) ; Botha et al. (2018) ).

Genetic algorithms (GAs) are founded on the principles of natural selection and population genetics (Fleming and Purshouse, 2002) . A GA solves the optimization problem in a derivative-free manner using a population of potential solutions that are evolved over generations to produce better solutions. Each individual in the population is assigned a fitness value that determines how well it solves the problem and hence how likely it is to propagate its characteristics to successive populations. A GA does not guarantee optimality, but provides a feasible solution in an appropriate time frame.

Other approaches to solving HMPC problems are also possible, as illustrated in e.g. Viljoen et al. (2020) .

Once the control move is calculated, some time is required for the country to prepare for the new lockdown level. To achieve this the lockdown level to be implemented is calculated some time in advance. A fixed delay between the control move calculation and implementation is not a typical HMPC requirement. It is however required in this instance for practical implementation of the appropriate lockdown level.

An overview of the epidemiological model used in this work is given in Section 2 along with the parameters for the South African case. The controller design for different policy scenarios is presented in Section 3, the results and discussion in Section 4, and finally the conclusion in Section 5.

The SEIQRDP model is a generalized compartmental epidemiological model with 7 states. The model was proposed by Peng et al. (2020) and is an adaptation of the classical SEIR model (see e.g. Hethcote and den Driessche (1991) ). The model states and model parameters that drive transitions between them are shown in Fig. 1 . The colours used for Q, R, and D correspond to what is used in the results figures later in the article. The states are described as:

• S -Portion of the population still susceptible to getting infected,

• E -Population exposed to the virus; individuals are infected but not yet infectious,

• I -Infectious population; infectious but not yet confirmed infected,

• Q -Population quarantined; confirmed infected,

• R -Recovered,

• D -Deceased,

• P -Insusceptible population.

The model equations are given as:

where N is the total population size, α is the rate at which the population becomes insusceptible (in general either through vaccinations or medication). At present there is no vaccine that will allow an individual to transfer from the susceptible to insusceptible portion of the population (Prompetchara et al., 2020) . Consequently α should be considered to be close to zero. β(t) is the (possibly time dependent) transmission rate parameter, γ = [N lat ] −1 is the inverse of the average length of the latency period before a person becomes infectious (in days), δ = [N inf ] −1 is the inverse of the number of days that a person stays infectious without yet being diagnosed, λ(t) is the recovery rate, and κ(t) is the mortality rate. Both λ(t) and κ(t) are potentially functions of time, and Peng et al. (2020) notes that λ(t) gradually increases with time while κ(t) decreases with time. As such, the functions shown in (8) and (9) are used to model λ(t) and κ(t). In (8) it is set that λ 1 ≥ λ 2 such that λ(t) ≥ 0.

β is often considered to be constant, but is dependent on interventions like social distancing, restrictions on travel, and other lockdown measures (South African Government, 2020a) . This implies that β may also be time dependent as these interventions change over time. Given that the number of susceptible individuals one may encounter naturally decreases over time, Olivier and Craig (2020) found that β(t) can effectively be modelled using a decreasing function of time of the form

The basic reproduction number, R 0 , which is the expected number of cases directly generated by one case in the population, is given by Peng et al. (2020) as

where T is the number of days. When α ≈ 0, this can be simplified as

Interventions such as social distancing, restrictions on population movement, and wearing of masks (among others) can reduce the effective reproduction number mainly through reducing the effective number of contacts per person. This is why the imposition of varying lockdown levels may be used as a control handle to effect policy decisions.

Data are obtained from The Humanitarian Data Exchange 1 , as compiled by the Johns Hopkins University Center for Systems Science and Engineering (JHU CCSE) from various sources. The data include the number of confirmed infectious cases, recovered cases, and deceased cases per day from January 2020.

In order to get a sense of the applicability of the model and what the parameter values should be, parameter estimations were first carried out to determine SEIQRDP models for Germany, Italy, and South Korea in Olivier and Craig (2020) . These countries are selected as their outbreaks started earlier than that of South Africa, and consequently their parameter estimation should be more accurate. They have also had differing approaches under different circumstances, which means that the different parameters obtained should illustrate how the model behaves. Epidemiological considerations largely drove the constraints on parameter values used in the South African model. Parameter values obtained for β from the models for other countries however guided the constraints on each parameter in the South African case.

The number of cases in South Africa only really started to increase in March 2020. As such, the data taken for fitting is only from 23 March 2020. The South African government relaxed the lockdown to level 4 on 1 May 2020, and given the incubation period data up to 8 May 2020 was used for the initial "level 5" lockdown model derived in Olivier and Craig (2020) . Level 3 was instituted from 1 June 2020. Pulliam et al. (2020) notes that a pessimistic scenario for South Africa is that level 5 lockdown reduces the contact rate by 40 %, level 4 by 25 %, and level 3 by 10 %. In order to test the contact rate reduction assumptions after the lockdown was progressively reduced, the parameter fitting simulation in Olivier and Craig (2020) is repeated with varying β-values. β(t) in Olivier and Craig (2020) is however time-varying. In order to vary the contact rate without varying the time dynamics of β(t) a level-based multiplier is used to determine the final β-value as: where β m (l) is the β-multiplier as a function of the lockdown level (l).

The pre-lockdown β-multiplier is set such that a 40 % reduction yields the β-value found in the original level 5 model; β is decreased by 25 % from the pre-lockdown value during lockdown level 4, and decreased by 15 % from the pre-lockdown value during level 3 (15 % instead of the 10 % proposed by Pulliam et al. (2020) produces a better fit). The β-multiplier per lockdown level is shown in Table 1 . The simulation result is shown in Fig. 2 . The progression of cases during level 4 and level 3 still fits the data well with the proposed β-multipliers, but it was found that the time decay of β in Olivier and Craig (2020) needed to be increased to still fit the data well (β 3 = 0.005 instead of 0.0027 found in Olivier and Craig (2020)). The recovery rate also seems to increase from early May compared to what was found in Olivier and Craig (2020) . λ is also time-varying and a multiplier of 1.6 is used from the start of lockdown level 4.

There is a very big difference between the final number of cases predicted by the level 5 model and the varying level version. This sensitivity is similar to what was found in Olivier and Craig (2020) for varying β(t).

One criticism of deterministic epidemiological models (see e.g. Britton (2010) ) is that if R 0 < 1 there will only be a small outbreak, and if R 0 > 1 there will be a major outbreak. This is because the model assumes that the community is homogeneous and that individuals mix uniformly. In reality however individuals will not mix uniformly, especially if regional travel is prohibited. This means that the effective reproduction number does decrease over time in practice, as modelled in Olivier and Craig (2020) . Varying β however leads to large model sensitivity, meaning that the total number of cases predicted may be higher than what occurs in practice. The varying β version of the model is however still useful to illustrate the effect of policy decisions on the relative number of cases recorded.

An HMPC controller using a GA to solve for the optimal control action (similar to Botha et al. (2018) ) is implemented to determine optimal lockdown levels for different policy scenarios. The controlled variable is the active number of confirmed infectious cases: Q as given by (4).

The control problem is formulated as:

where x : R → R nx is the state trajectory, u : R → R nu is the control trajectory, x k is the state at time step k, θ c (·) is a possibly nonlinear constraint function, f (·) is the state transitions as given in (1) -(7), g(·) = I 7 · x is the output function, u k contains the exogenous input (the lockdown level in this case), θ k represents the system parameters, and d k ∈ D represents the disturbances. The performance index (or objective function) to be minimized, J(·), depends on the policy in place as presented in the rest of this section, and is given in (15) and (17).

A flattening the curve policy is one where lockdown is implemented in order to ensure that the healthcare system is not overwhelmed by keeping the maximum number of cases requiring intensive care below the number of ICU beds available (Kissler et al., 2020) . In South Africa the number of ICU beds available was stated by the Minister of Health to be 10,500 (as reported in Cowan (2020)). Using the number of beds for the country as a whole may be somewhat crude. Regional values might be preferred, but as the model is for the country as a whole the high limit is considered in that fashion as well. Guan et al. (2020) found that roughly 5 % of confirmed infectious cases required admission to ICU. With 10,500 ICU beds available, this implies that the active confirmed infectious case number should remain below 210,000.

The objective function used to implement this policy penalizes an output (confirmed infectious cases Q) above the number of ICU bed imposed high limit as well as the magnitude of the control move (the lockdown level). This ensures that the output value will tend to remain below the high limit without setting the lockdown level needlessly high. The objective function is stated as:

where N p and N c are the prediction and control horizons respectively; · 2 W is the W -weighted 2-norm; W s and W u are weighting matrices corresponding to the relative importance of penalizing slack variables for constraint violations and control values respectively. The slack variables are represented by s i and are defined to be:

where y h is the output high limit. ∆T = 1 day (which was found to produce a sufficient resolution for progressing the simulation numerically), N p = 280 (which, based on the modelling results of Olivier and Craig (2020) , is a long enough horizon to capture the dynamics of the spread of the virus), and N c = 3 (which is an often used control horizon value that provides a good middle ground between controller aggressiveness and resolution to solve the control problem, see e.g. Bemporad et al. (2010) ); N c is however implemented using a blocking vector of N b = [7, 7, 7] implying that the lockdown level may only change at most every 7 days (1 week).

When implementing the "flattening the curve" policy the lockdown might end up being extremely long, which has an economic impact in itself. Preventing economic activity and therefore preventing certain people from earning a living will likely increase poverty, which in itself leads to life years lost. As such the cumulative economic impact of lockdown should also be considered, which is done by adapting the objective function to be:

where E C is the cumulative economic impact of lockdown levels that have already been implemented, E i is the marginal economic impact of lockdown as implemented over the control horizon, and W E is the weighting matrix relating to the economic impact. At each simulation step the cumulative economic impact is increased as:

For E i the quantified relative economic impact per lockdown level is required. One may wish to consider something like the gross domestic product (GDP) or the value add of the industries that may operate during each lockdown level as in Arndt et al. (2020) . GDP figures are often only reported at least one quarter after the fact which does not help in this case. The percentage of each industry that will be affected per lockdown level is not known directly and estimating these may add too much uncertainty. A more frequently updated indicator, which is used here instead, is the BankservAfrica Economic Transactions Index (BETI) which show the volumes and values of inter-bank transfers (BankservAfrica, 2020). Bankserv-Africa states that BETI is a leading indicator for the South African GDP -it correlates well with GDP figures while appearing a quarter earlier.

There is however some seasonality connected to economic transaction volumes, and as such year-on-year values are used to gauge the impact of lockdown levels. The relative economic impact for level 1, 4, and 5 are determined using the year-on-year decrease in the BETI value for March to May 2020. The values for levels 2 and 3 are interpolated from the other levels. The values and corresponding months are shown in Table 2 as well as Fig. 3 to highlight the interpolation results.

The relative economic impact values per level are normalized to be between 0 and 1. As they are weighted in (17) their relative values are important, and because they are included in a 2-norm calculation the absolute values need to be monotonically increasing.

This type of policy may reduce lockdown levels to curb economic losses, even though infection rates may be considered to be unacceptably high. This scenario is presently happening in various countries in the developing world (Oanh Ha et al., 2020).

Some residents cannot or will not endure living under continual lockdown regulations (for various reasons). BusinessTech (2020) notes that 230,000 cases have been opened against South African residents for violating lockdown rules by 22 May 2020. A significant proportion of these cases were opened in the last couple of weeks before the BusinessTech (2020) article was written, showing that violations become more prevalent over time. Compliance Lockdown level to lockdown regulations can therefore be considered as an unmeasured disturbance that impact on the parameters of the SEIQRDP model. In this scenario (17) is still minimized, but the compliance disturbance is introduced as a multiplicative disturbance on the expected number of contacts between people by altering the value of β; (13) is altered to produce:

where ζ ∈ [0, 1] is the level of compliance to the stipulated lockdown level. When ζ = 1 the intended β-value for the associated lockdown level is obtained as in (13). When ζ = 0 the β-value obtained is as if lockdown level 1 had been implemented.

There is a very delicate balance to be maintained when implementing policy decisions. Due regard has to be given to the effect that policies may have on the citizens of the country concerned. If not, compliance may reduce and disgruntled citizens may deliberately violate regulations or fail to keep track of which lockdown level is currently in force.

To prevent this scenario, a dynamic constraint is placed on the controller which ensures that from the present lockdown level to the end of the control horizon, the control moves must be monotonically increasing or monotonically decreasing. This constraint is enforced as:

(20) This prevents excessive fluctuations of the lockdown level that the controller may otherwise seek to exploit in order to minimize the objective function. Excessive fluctuations of the lockdown levels also make it difficult to enforce and regulate the lockdown rules on a national scale as time is needed to implement any policy changes.

Once the control move (lockdown level) has been calculated, some time is needed for the nation to prepare to implement that lockdown level. Here, 1 week (7 days) of preparation time is provided. Given the blocking vector of N b = [7, 7, 7] , this is practically achieved by fixing the first element of the control vector to the previously calculated value and then using the second element as the control move to be implemented. This is contrary to regular receding horizon control where the first element is implemented.

This implementation approach is no different from the regular receding horizon approach in the absence of modelplant mismatch and/or disturbances. No difference should therefore be expected for the initial flattening the curve and balancing lives and livelihoods policy simulations. Once the compliance disturbance is introduced however this implementation does however have some impact, given that the level of compliance when the control move is calculated may not be the same as the level of compliance when it is implemented.

The controller design discussed in this section relies on a solver that can solve an objective function based on a hybrid model which contains continuous time dynamics, time varying model parameters, discontinuities in the form of discrete lockdown levels, and dynamic non-linear constraints.

The GA is a solver that has shown good results with such mixed integer models, non-linearities and complex non-linear constraint functions, and convergence to a global minimum in the presence of many local minima (Muller and Craig, 2017; Mitra and Gopinath, 2004) . It has also been successfully used as a solver for HMPC (Muller and Craig, 2017; Botha et al., 2018) . Owing to these advantages a GA is used in this study.

The Matlab ga function in the Global Optimization Toolbox was used to implement the GA. The Matlab ga function uses a set of solutions, called the population, that are calculated to minimize a fitness function. With each iteration in the GA a new generation is calculated from the old population through a mutation function (while always adhering to the upper and lower bounds as well as the inequality, equality, and non-linear constraints). The mutation function keeps the variables in the population that minimised the fitness function the most, and also defines new values according to a stochastic function. The GA is terminated when the fitness function value, fitness tolerance (i.e. the change in the fitness function value between iterations), or the maximum number of generations is exceeded (Whitley, 1994) .

In this work the HMPC problem solves within approximately 5 seconds on a standard computing platform. As a solution is only required once a day, the execution time is not an issue. Therefore the fitness tolerance, fitness limit, population size and maximum number of generations were left as default in the Matlab ga function. It is important to note that these parameters may be altered when the controller execution time needs to be reduced. The ga function was set up using the following options and parameters:

• The fitness function is the objective function policy in (15) or (17).

• The lower bound (LB) is the hard low limit for the three control moves, which is [u k−1 , 1, 1].

• The upped bound (UB) is the hard high limit for the three control moves, which is [u k−1 , 5, 5].

• The non-linear constraint function is implemented using (20).

The simulation results for each scenario described in Section 3 is presented and discussed here. The controller design parameters and result metrics for each scenario are shown in Table 3 . The metrics shown are the maximum number of active confirmed infectious cases, i.e. the maximum of Q, the number of days (from the start of the simulation on 20 March 2020) before the lockdown level is raised above 1, the number of days after the lockdown level has been raised above 1 until it is returned to 1, and the number of days in each lockdown level. The number of days in level 1 is only taken up to the point where the lockdown is finally reduced back to level 1. The number of days before lockdown is implemented is an important metric to consider as it will improve the nation's readiness for lockdown and in turn improve long term compliance. 

The simulation result for the flattening the curve policy scenario described in Section 3.1 is shown in Fig. 4 .

From Fig. 4 it is visible that implementing lockdown at the correct time and to the correct level allows the active number of confirmed infectious cases to hardly violate the ICU capacity limit. The lockdown is however implemented relatively early (44 days after the start of the simulation on 20 March 2020) and lasts very long (308 days).

The lockdown scenario illustrated in Fig. 4 will likely have devastating effects on a South African economy that was already fragile before the onset of COVID-19. The result for the balancing lives and livelihodds policy, as described in Section 3.2, is shown in Fig. 5 . In this case lockdown is implemented up to level 5 to try and curb the spread of the virus, albeit somewhat later than in the flattening the curve scenario to alleviate some of the economic impact of lockdown. After a couple of weeks on level 5 the cumulative economic impact increases so much that the level is reduced in spite of the ICU capacity limit being exceeded. The dotted line in the top panel of Fig. 5 shows the cumulative economic impact of the lockdown that has already been implemented, weighted to fit onto the same scale as Q. It is clear that as the economic impact grows, the ICU limit is allowed to be violated further in order to reduce the lockdown level and curb economic losses.

This policy balances lives and livelihoods by delaying the start of lockdown with a week and reducing the total time in lockdown by 8 weeks. Furthermore, while in lockdown fewer days are spent in each level. The peak number in Fig. 5 of active confirmed infectious cases is nearly double that of Fig. 4 . The advantage of this policy however is that the peak in the maximum number of cases is delayed as opposed to having no policy in place. Besides winning time, this scenario also provides an estimate of the additional number of COVID-19 specific intensive care beds required, something that healthcare authorities can use to plan their response. This implies that if the economic impact of each level of lockdown can be quantified beforehand, temporary ICU facilities can be procured to the point where lockdown might be eased earlier to limit the cumulative economic impact while the ICU limit is still respected.

Given that compliance to lockdown regulations has seemingly waned over time in South Africa, the compliance parameter (ζ) in (19) is initiated with a value of 1, and reduced linearly after a number of weeks in lockdown down to a value of 0.3 (as seen in the bottom panel of Fig. 6 ). After the initial decrease the level of compliance is set to increase every time that the lockdown level is reduced. This is because there are fewer regulations on lower levels, and the level of compliance to those regulations will likely be higher initially. After the initial surge, compliance will again reduce in a linear fashion.

It is visible from Fig. 6 that the lockdown is imposed at the same time and with the same magnitude as what it was in Fig. 5 . Lockdown is also reduced from level 5 to 4 at the same time. After a couple of weeks in lockdown however the population starts to deviate from the rules and the effective number of contacts per person increases, which consequently increases the spread of the virus. To try and curb this phenomenon the controller moves the lockdown level back to 5 as it applies feedback to try and balance the ICU bed imposed limit with the economic impact of the lockdown. After another period at level 5 the cumulative economic impact has however ballooned while compliance remains relatively low. Left with large economic losses and a non-compliant population, the controller ramps down the lockdown level from 5 to 1 in a relatively short time. The compliance does increase at each reduction of the lockdown level, but this has little impact given the magnitude of the cumulative economic losses.

An epidemiological model was developed for the spread of COVID-19 in South Africa in Olivier and Craig (2020) using data for the period from 23 March to 8 May 2020 while the country was mainly under lockdown level 5. The model was adapted here with varying values for the spread rate (β) under varying levels of the lockdown using more recent data. An HMPC controller was then implemented to determine the optimal lockdown level over time for different policy scenarios.

Under a scenario where "flattening the curve" is the goal, the healthcare capacity, expressed in terms of the number of available ICU beds, is largely respected, but the lockdown is long and severe. The detrimental cumulative economic impact of such a long and severe lockdown is very high. A balancing lives and livelihoods policy was therefore introduced to allow for increased economic activity by reducing the lockdown level earlier. This has the effect that the ICU bed imposed limit is violated, but that more livelihoods are saved.

Lastly, compliance to lockdown regulations is added as an unmeasured disturbance. The effect of the compliance is seen through the higher number of confirmed infectious cases. To curb waning compliance the lockdown level is increased and extended to the point where the economic loss is too great and lockdown is ended rather abruptly.

Since the spread of the virus is ongoing and the peak number of active cases has not yet been reached, future work to consider from this paper are:

• Better granularity may be achieved by developing regional models,

• Should a vaccine become available the epidemiological model presented in this paper can be adapted (by increasing α) which will affect the lockdown levels that the controller subsequently implements should the predicted peak number of active cases be later than the vaccine release date.

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