Optimisation of non-pharmaceutical measures in COVID-19 growth via neural networks

By mid-February, China had successfully demonstrated that the exponential growth of confirmed cases can be vastly mitigated by mass quarantine or lockdown measures [32]. Unfortunately, by then the virus was beginning to take hold in several nations. In particular, Italy would become the next epicenter of the pandemic [33, 34], with dire consequences on its healthcare system; before the onset of the epidemics, Italy had approximately 5200 intensive care beds available [35], and efforts had to be made to increase this number during the emergency [36]: on April 3rd the number of patients admitted to ICU reached an unprecedented figure of 4068 [34], posing a serious threat on the solidity of the entire system. In these extraordinary circumstances, the Italian government instigated severe lockdown measures similar to those implemented in China [37].

Meanwhile Taiwan, despite its close proximity and frequent flights to China, continued to contain the growth of confirmed cases through different yet efficient and effective government measures [38]. These actions never escalated to a nationwide lockdown. Nevertheless they successfully prevented the exponential growth predicted by unmitigated viral transmission [39], and did not overwhelm their healthcare system.

With the ultimate goal of providing a foundation that could serve policy-makers in the decision process, in regard to managing current outbreaks or preventing a future second-wave, this study aims to understand the impact and effectiveness of the possible measures a nation can employ to inhibit and constrict growth. Thus, we analyse these two different scenarios from two different regions.

To capture and describe the evolution of the pandemics in the two regions, several variables were employed describing the reaction of the respective governments, its timeliness, the behavior of the community, and the weather conditions. To gather the Italian data, numerous sources were used, namely the Ministry of Health [40], the Ministry of Interior [41] and the Department of Civil Protection [34]. As for Taiwan, the Taiwan Centers for Disease Control (TCDC) released timely reports to the public on government mitigation actions [42]. This was previously reported by [43] for data available until $24^{\rm{th}}$ February 2020. We extend this data by two months until $24^{\rm{th}}$ April 2020. Each variable was then coded to obtain a daily time series (Fig. 1). The dataset is summarized below.

The data collected are used as independent variables for the multivariate time series analysis where the slope is the dependent variable. Three neural network models are tested: 2D Convolutional Neural Network (CNN) [50], Extreme Learning Machine (ELM) [51] and Bidirectional Long Short-Term Memory Neural Network (BiLSTMNN) [52].

The optimal set of hyperparameters for the neural networks have been obtained by minimising the root mean square error on the predictions by means of a Bayesian optimisation process. Bayesian optimisation is an algorithm used in global optimisation to minimise a certain objective function, treated as a black box, by varying the value of its independent variables. The algorithm itself relies on an internal Gaussian process that approximates the objective function, and is trained by subsequent evaluations of the true objective. The approximated model is used for optimisation to reduce computational costs and for its robust nature with stochastic noise in function evaluations.

The collected data has been divided for training and testing, and reshaped as feature input for the multivariate time series analysis. Furthermore, in order to be able to use the neural network models for the subsequent analysis, the optimisation of government restrictions, the dependence of the historical values of the dependent variables is omitted from the input features space. Hence for the ELM and BiLSTMNN models the input features are modelled as the vector

where $N$ is the time window parameters, and $X(t)$ is the vector of independent variables at time $t$ as described in the Data Collection section. Conversely for the CNN model the input space is defined as

where $x_{i}(t)$ is the $i$-th independent variable at time $t$. The hyperparameter $N$ is optimised, by means of a Bayesian optimisation process, together with the networks hyperparameters.

The best trained neural network model (BiLSTMNN) is used to have a deeper understanding of government regulations and their impact on the slope function. Taking

as the neural network model, where $M=11N$ or $M=11\times N$ depending on the model used, the following optimisation problem has been formulated

where $J(\mathbf{x})$ is the objective function defined as the maximum value of the slope along the whole time frame considered.

The daily slope value is computed as the angular coefficient of the linear regression of the values of daily cumulative infected of the current day and two days ahead, as a smoother representation of the daily number of infected. $\mathbf{x}$ is a set of optimisation variables designed to model government restrictions previously discussed (border control, enforcement, testing volume, testing criteria, social distance, flight volume and people awareness), excluding weather parameters as they are not controllable by government. The complete set of optimisation variables for each territory is described in Table I. $\Omega$ is the feasible region defined by the lower and upper bounds of the set of optimisation variables with some constraints. Inequality constraints are introduced to enforce the temporal order of the variables modelling quarantine incoming and the step functions of social distance, quarantine control (only for Taiwan) and testing criteria. To avoid the optimisation converging towards a temporal delay of the first case, an equality constraint is introduced to ensure the day of first confirmed case matches reality. The optimisation problem aims to minimize the maximum value of the slope - this can be interpreted as minimising the number of daily infections so to not put the health care system under extreme strain.

For Italy the problem can be formulated mathematically as

where

is the matrix defining the linear inequality temporal constraints on the quarantine incoming variables. $T$ is the time horizon parameter, and $L,U$ are the vectors defining the lower and upper bounds of the optimisation variables. For Taiwan the problem is defined as,

with

where $B_{1}$ and $B_{2}$ are the matrices defining the linear inequality temporal constraints on the quarantine incoming variables.

The integer programming Genetic Algorithm is used to solve the optimisation problem. Genetic Algorithms are stochastic global optimisation strategies that mimic the behaviour of natural biological evolution of mutation and crossover. They are initialised with a pool of potential solutions and by evolutionary operators and applying the principle of survival of the fittest, increasingly better offspring populations are generated, which terminate when no further improvements can be made. Genetic algorithms have the main advantage that they can solve problems with integer variables and the objective function is treated as a black box.

We propose a soft computing algorithm which combines neural networks and genetic algorithms, to determine (from a purely data-driven approach), the best set of mitigating actions. The algorithm is based on two independent stages that are summarized below and outlined in Algorithm. 1.

In the first stage, a set of neural networks models are trained to estimate the daily slope increase of cumulative infected from variables such as government actions and the reactions and adherence of the population to such regulations. The best performing model is selected taking into account the reported root mean square percentages of the neural networks implemented. The choice of training a neural network model per territory is driven by the need of capturing region specific static parameters. Time-varying parameters that have an influence on the predicted growth curve, and are not among those described in Sec. 2.2, can still be captured by the network model as hidden features if they are correlated to the input variables selected.

In the second stage, the best performing neural network (BiLSTMNN) is used as a surrogate model in an optimization procedure that targets the best set of mitigating actions. The optimization procedure incorporates an ad-hoc parametrization of the time series governmental restrictions. These parameters are then optimised through genetic algorithm for integer programming to generate the new corresponding optimal time series. The algorithmic flow of this soft computing approach is provided below.

In this section, the performance of the different neural networks models for Italy and Taiwan are statistically compared. For model selection, following the recommendations reported in [60], the time series were split in two parts: (i) the training set is roughly the first 2 and a half months of data; and (ii) the generalization set is the last section of each time series. The training set was used to estimate the parameters of the models and the generalization set to assess the performance of the different neural networks in unseen data [60]. In the experiments, the generalization set included the last five points of each time series, as suggested in [61]. Table II shows the Root Mean Square Error Percentage (RMSEP) in the generalization set of the best run per model (out of 1000 runs), $\mathrm{Best}_{RMSEP}$, and the variation coefficient of the RMSEP in the generalization set for the different models implemented, $\mathrm{CV}_{\mathrm{RMSEP}}$. Furthermore, taking into account the set of RMSEPs of the different models over the different runs, the mean rankings of RMSEP, $\overline{R}_{\mathrm{RMSEP}}$, for the different neural networks models are obtained (Table II). From the analysis of the results, it can be concluded, from a purely descriptive point of view, that the BiLSTMNN method obtained the best results in the two problems considered in mean rankings. Additionally, the best BiLSTMNN achieved the best RMSEP in Taiwan and second best in Italy (more consistent performance, more homogeneous results, better CV).

In this study, hypothesis testing was used to provide statistical support for the discussion of the results. A performance analysis through parametric tests could lead to mistaken conclusions in this research study. A previous evaluation of the RMSEP values provided by the implemented methods resulted in rejecting the normality and equality of the variance hypothesis. For these reasons, nonparametric tests were implemented to determine the statistical significance of the results previously reported [62]. Specifically, two non-parametric Friedman tests were carried out with the rankings of RMSEP of the models. For the two multivariate time series considered, the $p$-values associated to the Friedman test were smaller than 0.05 ($\alpha=0.05$), and therefore, the null hypotheses stating that all algorithms perform equally in mean RMSEP rankings were rejected for both problems.

Based on this rejection, the nonparametric Bonferroni Dunn-Sidak test was implemented to compare all neural network methods to the BiLSTMNN method (which was used as the control method) [63]. Table II shows the p-value results of the Bonferroni Dunn-Sidak test for the two problems considered. Thus, the Bonferroni Dunn-Sidak’s tests indicate that the control method (BiLSTMNN) statistically outperforms all the remaining models in Italy and also outperforms, in statistical terms, the ELM model for Taiwain. Furthermore, the BiLSTMNN method achieves the best mean ranking in the two problems considered, which, in our opinion, justifies our decision to use it as the base model for the next stage (optimization study). An analogous study was also performed to determine the model to be analyzed within all the different BiLSTMNN configurations.

The best BiLSTMNN model for each region is obtained for the optimal value of hyperparameters reported in Table III, the network visualisation model in Fig. 2 and the prediction model in Fig. 3.

The multivariate time series model captures the trend of the slope in both cases, with a more accurate overall error in the case of Italy (due to the less noisy nature of the Italian time series). It is important to clarify, as mentioned previously, that the lagged values of the slope, cumulative cases or daily cases were not included in the model to reduce dependence between input variables and assist the interpretation of the optimum values. These variables are usually the most discriminant in times series analysis, which highlights even more the competitive results yielded. Thus, the BiLSTMNN is the appealing approach for this problem.

A genetic algorithm has been used to solve the corresponding optimisation problem for each region and promising results have been found that do not contradict previous findings of epidemiological models (Figures 4 and 5). All variables are considered integer and optimised within the bounds reported in Table IV, together with their optimal value. The genetic algorithm, with population size of 100 and the maximum number of generations set to 1000, has been run with the stopping criteria set as the maximum number of generations reached or the average relative change in the best fitness function value over 100 generations being less than or equal to a tolerance of $1e-10$.

The optimisation process for Italy terminates after almost 600 iterations when the change in the penalty fitness value is less than the allowable tolerance. All constraints are satisfied and the final objective function is 3755, with respect to the original value of 6272. This is almost a $40\%$ reduction with respect to the original value and equates to a reduction of almost 21,700 infected individuals at the end of the prediction horizon, corresponding to $16\%$ of the actual total value.

The optimisation process for Taiwan terminates after almost 300 iterations again when the change in the penalty fitness value is less than the allowable tolerance. All constraints are satisfied and the final objective function is 7.6, with respect to the original value of 23. This is almost a $63\%$ reduction with respect to the original vale and a reduction of 102 infected individuals at the end of the prediction horizon, corresponding to $28\%$ of the real value.

As illustrated in Fig. 4, an earlier and more extensive testing campaign (+56$\%$) could have been combined with an earlier creation of red zones in Northern Italy and, interestingly, with a milder implementation of the government mitigation plan on the rest of the country: the localized restrictions could have been maintained as sufficient measures until $14^{\mbox{th}}$ March. Furthermore, in this scenario the country-wide suspension of manufacturing businesses, a measure burdened with a substantial socio-economic impact, is delayed and limited to a shorter period of time, while the volume of police checks, aimed at monitoring the population compliance, is larger and commences earlier. This is all combined with an earlier and more pervasive consciousness of the public about the risk posed by the epidemics and thus on the benefits of a temporary reduction of social contacts.

The containment of the growth is also optimally managed in Italy by earlier restrictive regulations on the intra- and extra- Schengen air traffic, and by entry restrictions that span beyond the mere suspension of direct flights to and from China. As for the national air traffic, it could have been maintained unrestricted for a longer timeframe: a finding to be read in the context of larger and earlier testing, larger and earlier volume of police checks, better awareness of the public, increased control on the infections potentially coming from abroad.

Although government actions in Taiwan successfully prevented exponential growth of confirmed cases, the optimised model predicts earlier actions would have damped growth even further as outlined in Fig. 5. By allowing movable dates on border control decisions, Taiwan would have benefited from even earlier actions against China (by a few days) and earlier self-health management and home quarantine regulations against selected nations (those shaded in Table 7 of Supplementary Material in February and March, prior to $19^{\mbox{th}}$ March by the maximum 10 days allowed). This allows a later date for the travel restriction on all foreign nationals that was implemented on $19^{\mbox{th}}$ March, as well as a slight delay in the decrease of international flight volume. The optimizer increases testing volume 2 weeks earlier than reality and suggests community surveillance and mandatory testing for incoming travellers in the first week of the observatory period. Actions regarding mitigating internal spread are also suggested earlier - heightened hygiene practices, self-health management of positive cases’ contact and social distance guidelines should have occurred at least 3 weeks earlier (comparison of Fig. 5 and Table 12 in Supplementary Material). In addition, national flight volume should be kept low to inhibit internal spread. This allows less people to stay at home as illustrated by the reduced online presence. Finally we see people awareness and quarantine control can both be delayed, the latter significantly. This suggests the Taiwan public awareness is sufficient, and enforcement measures like fines and mandatory quarantine may not be necessary.

Care must be taken in comparing the growth and government actions for two different territories. All data is not equal; different testing methods, diagnostic criteria and methods of counting confirmed cases can affect comparisons. There may also be underlying themes behind data / variables, e.g., the availability of personal protection equipment like masks to enable adherence to control measures and keep health workers safe (3 workers from healthcare settings where reported positive by TCDC [42], while in Italy more than 28,000 healthcare workers contracted the virus by the end of April [64, 65]), the method and extent of a region’s track and trace program in locating high risk individuals, cultural / social response of a population to government actions, healthcare capacity and quality in early isolation and treatment (0.86 vs 3.02 ICU beds per 10,000 population in Italy [35] and Taiwan [66] respectively). To overcome such inequity of data in this study, Italy and Taiwan were simulated independently. Variables were chosen to be those for which there was data readily available and have a) known or suspected effect on growth curves or b) an ability to highlight any differences between the two territories. Any overarching themes are then considered.

The different levels of action applied to travellers entering the country were equated to a numeric scale. This ranges from 0 to 5 for national citizens and 0 to 7 for non-nationals as depicted in Table V, where 0 represents zero constraints and 7 entry banned.

Level 1 involves quarantine inspections that targets symptomatic travellers on direct flights from an infected area. Level 2, whilst still targeting symptomatic passengers, opens to those with a travel history to an infected area within 14 days. Level 3 requires health declaration forms from an infected area by all travellers, therefore expanding the target to asymptomatic travellers [44].

Level 4 indicates a 14-day self-health management; in Tawian the TCDC describes this as: hand and respiratory health, record temperature and activities twice daily, avoid public places (wearing a face mask if necessary), and contact local health bureaus if fever or respiratory symptoms develop (inform physician of any history of travel, occupation, contact, and cluster) [72].

Level 5 requires individuals to a 14-day home quarantine; again for Taiwan the TCDC required travellers to follow several measures [44]; mask on arrival, private transportation from airport/point of entry, hand and respiratory health, stay at home, keep one meter away from those sharing the same household and avoid contact with them. Quarantine/isolation hotels can be availed of if one’s home/destination does not qualify for home quarantine. Groceries or necessary products are to be obtained via help of household or family members, alternatively one can contact the care and support center for quarantined/isolated individuals. Non-urgent medical care should be postponed and if symptoms develop, individuals are not to seek medical attention by themselves but contact health authorities to arrange their medical care.

Level 6 has a restriction on entry visas for specified foreign nationals while level 7 involves a ban on entry for non-residents.

On January 31st, the Italian Presidency of the Council of Ministers, the Ministry of Health issued the suspension of direct flights between Italy and China. After this decision, other measures of increasing severity have been taken to control the cross-border traffic. Data regarding these legal acts was collected from the archive of regulations issued by the Ministry of Health [40]. Dates and detail are listed below in Table VI along with the suitable quarantine levels as described in Table V.

Taiwan initiated on-board quarantine inspections on the same day China reported cases of pneumonia of an unknown etiology in the city of Wuhan to the World Health Organisation (WHO), $31^{\mbox{st}}$ December 2019. From there they adapted to the global spread of the virus, with different requirements rolled out for different regions of the world as outlined in Table VII. As the epidemic in China worsened, Taiwan increased entry requirements for different provinces of China. Initially they focused on symptomatic travellers from Wuhan, escalating to an entry ban for Wuhan residents on $23^{\mbox{th}}$ January, with severe restrictions on all Chinese citizens following on $26^{\mbox{th}}$ January. Early February saw an entry ban for Chinese citizens and foreign nationals with a travel history in China, Macau or Hong Kong, with home quarantine for Taiwan, Macau and Hong Kong residents. By end of February, as the virus spread globally, Taiwan initiated a variation of self-health management and home quarantine for travellers from Thailand, Italy, Iran, Singapore, Japan and South Korea. In March, as the situation in Europe worsened, Taiwan further enhanced its measures by mandatory quarantine for incoming travellers from Schengen, U.K., Ireland, and Dubai. This was quickly followed by several Asian countries, Moldova, U.S., New Zealand, Australia and Canada. On $19^{\mbox{th}}$ March, Taiwan announced banned entry for all foreign national travellers without documents of granted entry.

In addition to these actions, in rare cases where it was acknowledged that mandatory quarantine orders had been issued later than ideal, retrospective health monitoring was put in place for recently arrived travellers from specified regions. Other scenarios that required spacial attention to which the Central Epidemic Command Center (CECC) reacted include: the docking of the Princess Diamond where the use of smart technology assisted in identifying and following up with 627,386 individuals [73]; special requirements in dealing with Taiwan residents on other cruise ships; evacuating Taiwan residents from Wuhan [74]; and an outbreak on a three-ship fleet, Dunmu (敦睦) Fleet.

This is summarized in Table VII, along with the allocated level as described in Table V. Some information was not included in the analysis, yet still included in Table VII for completeness.

To quantify these actions for both Taiwan and Italy, each territory is allocated a quality of control between 0 and 1 each day, thus creating a time series for both regions as illustrated in Figure. 6. To compute this level of control on a given day for either territory, the following procedure was followed using available data from John Hopkins University [33].

The daily cases of every country reported globally have been collected and smoothed by a rolling 3-day average (previous, current, day after). The percentage of daily cases associated with each country is then computed, with respect to the global daily cases, and excluding daily cases in Taiwan and Italy respectively (as it is their protection from incoming cases being gauged). For Taiwan, areas associated with China that were treated differently, i.e., different regulations on different dates, include Hubei, Guangdong, Zhejiang, China (rest), Hong Kong and Macau. When actions were directed by cities, provincial data was used. In addition, some states of the United States (U.S.) were treated differently, namely Washington, New York and California. As actions differed only by a day, the date of regulations affecting these states have been taken to represent all of the U.S.

As different levels of quarantine are implemented for residents and non-residents (see Table VII and VI for Taiwan and Italy respectively). Statistics from the Taiwan Tourism Bureau [75] for 2019 have been used to encapsulate the ratio of travellers that are residents and visitors as outbound and inbound respectively. For China, the national statistics were used for each province. Similar information was obtained from the World Tourism Organisation for Italy visitors [76] and returning residents [77] in 2018. This data has been used to obtain a weighted-level time series by dot product of the ratios (residents, non-residents) and the levels associated for that day for each country. For example, with travellers from South Korea to Taiwan

where $L^{(SK)}(n)$ is the level on day n from South Korea (on day 37 or $27^{\rm{th}}$ February we have $L^{(SK)}(37)=(\frac{5}{5},\frac{5}{7})$ from Table VII) and $RT^{(SK)}$ is the ratio of travellers (outbound, inbound) with $RT^{(SK)}=(0.4932,0.5068)$ from [75].

The time series quantifying the border control for say Taiwan entry from each country is calculated by multiplying the percentage of daily cases for a given country on a given day by the weighted-level calculated for that country on that day. Again, for the above example of South Korea on day 37,

where $D^{(SK)}(n)$ is the smoothed daily global percentage of cases associated with South Korea on day $n$ (with Taiwan cases removed) obtained using John Hopkins data [33]. By summing all the time series for each country, the daily total quarantine incoming measure for Taiwan (or Italy) on each day is retrieved. This summing of reactions is illustrated by a stacked bar graph in Figures 6(a) and 6(b) for Italy and Taiwan respectively.

To encourage compliance with quarantine and isolation orders in Taiwan, fines were introduced early of NT$150,000 and NT$300,000 respectively (USD$5,000 and USD$10,000) and raised to NT$1 million (USD$33,000) on $27^{\mbox{th}}$ March and mandatory group quarantine at a designated centre. Health Declaration Cards were required by travellers from high-risk areas from $24^{\mbox{th}}$ January with smart technologies announced on $28^{\mbox{th}}$; this evolved into the Entry Quarantine System announced on $14^{\mbox{th}}$ February - an online system where travellers could obtain a mobile health declaration pass on their phones. This hastened the immigration process, allowed the Taiwan healthcare system immediate access to travel history of ‘perspective’ future patients and further enabled mobile tracking of quarantined individuals. Inaccurate information earned travellers a NT$150,000 fine (USD$5,000). By $3^{\mbox{rd}}$ April, the LINE Bot system was announced to track quarantined individuals, however the newer system also allowed voluntary health reporting and updated information on prevention measures. These were reported to the public [42] (ongoing) and previously noted in [43] (until $24^{\rm{th}}$ February 2020). Table VIII lists and quantifies these consequences as a step function.

The progression of testing in who qualified for tests for both Italy and Taiwan are summarised in Tables IX and X respectively below.

Actions taken by both governments to implement social distancing and health promotion are listed and quantified for both Italy and Taiwan in Tables XI and XII respectively.

The list of keywords used for quantifying public awareness via Google Trends are listed in Table XIII