\section{Introduction}

Dynamic contrast enhanced MRI (DCE-MRI) in the liver is widely used for detecting hepatic lesions and in distinguishing malignant from benign lesions. However, such images often suffer from motion artifacts due to unpredictable respiration, dyspnea, or mismatches in k-space caused by rapid injection of the contrast agent\cite{motosugi2015investigation}\cite{davenport2013comparison}. In DCE-MRI, a series of T1-weighted MR images is obtained after the intravenous injection of a gadolinium-based MR contrast agent, such as gadoxetic acid. However, acquiring appropriate data sets for DCE arterial phase MR images is difficult due to the limited scan time available in the first pass of the contrast agent. Furthermore, it has been reported that transient dyspnea can be caused by gadoxetic acid at a non-negligible frequency \cite{motosugi2015investigation}\cite{davenport2013comparison}, which results in degraded image quality due to respiratory motion-related artifacts such as blurring and ghosting\cite{stadler2007artifacts}. Especially, coherent ghosting originating from the anterior abdominal wall decrease diagnostic performance of the images\cite{chavhan2013abdominal}

 Recently, many strategies have been proposed to avoid motion artifacts in DCE-MRI. Of these, fast acquisition strategies using compressed sensing may provide the simplest way to avoid motion artifacts in the liver \cite{vasanawala2010improved}\cite{zhang2014clinical}\cite{jaimes2016strategies}. Compressed sensing is an acquisition and reconstruction technique based on the sparsity of the signal, and the k-space undersampling results in a shorter scan time. Zhang et al. demonstrated that DCE-MRI with a high acceleration factor of 7.2 using compressed sensing provides significantly better image quality than conventional parallel imaging \cite{zhang2014clinical}. Other approaches include data acquisition without breath holding (free-breathing method) using respiratory triggering and respiratory triggered DCE-MRI, which is an effective technique to reduce motion artifacts in cases of patients who are unable to suspend their respiration \cite{vasanawala2010navigated}\cite{chavhan2013abdominal}.In these approaches, sequence acquisitions are triggered based on respiratory tracings or navigator echoes, and typically provide a one-dimensional projection of abdominal images. Shreyas et al. found that the image quality in acquisitions with navigator echoes under free-breathing conditions is significantly improved. Although triggering based approaches successfully reduce the motion artifacts, it is not possible to appropriately time arterial phase image acquisition due to the long scan times required to acquire an entire dataset. In addition, miss-triggers often occur in cases of unstable patient respiration, which causes artifacts and blurring of the images. Recently, a radial trajectory acquisition method with compressed sensing was proposed  \cite{feng2014golden}\cite{feng2016xd}, which enables high-temporal resolution imaging without breath holding in DCE-MRI. However, the image quality of radial acquisition without breath holding is worse than that with breath holding even though the clinical usefulness of radial trajectory acquisition has been demonstrated in many papers \cite{chandarana2011free}\cite{chandarana2013free}.\cite{chandarana2014free}

Post-processing artifact reduction techniques using deep learning approaches have been also been proposed. Deep learning, which is used in complex non-linear processing applications, is a machine learning technique that relies on a neural network with a large number of hidden layers. Han et al. proposed a denoising algorithm using a multi-resolution convolutional network called “U-Net” to remove the streak artifacts induced in images obtained via radial acquisition \cite{han2018deep}.In addition, aliasing artifact reduction has been demonstrated in several papers as an alternative to compressed sensing reconstruction \cite{lee2017deep}\cite{yang2018dagan}\cite{hyun2018deep}. The results of several feasibility studies of motion artifact reduction in the brain \cite{Karsten2018ismrm}\cite{Patricia2018ismrm}\cite{Kamlesh2018ismrm}, abdomen \cite{Daiki2018ismrm}, and cervical spine \cite{Hongpyo2018ismrm} have also been reported. Although these post-processing techniques have been studied extensively, no study has ever demonstrated practical artifact reduction in DCE-MRI of the liver.

In this study, a motion artifact reduction method was developed based on a convolutional network (MARC) for DCE-MRI of the liver that removes motion artifacts from input MR images. Both simulations and experiments were conducted to demonstrate the validity of the proposed algorithm.

\section{Methods}

\subsection{Network architecture}

In this paper, a patch-wise motion artifact reduction method that employs a convolutional neural network with multi-channel images (MARC) is proposed, as shown in Fig. \ref{fig_network}, which is based on the network originally proposed by Zhang et al. for Gaussian denoising, JPEG deblocking, and super-resolution of natural images\cite{zhang2017beyond}. Patch-wise training has advantages in extracting large training datasets from limited images, and efficient memory usage on host PCs and GPUs. Residual learning approach was adopted to achieve effective training of the network\cite{he2016deep}. The network relies on two-dimensional convolutions, batch normalizations, and rectified linear units (ReLU) to extract artifact components from images with artifacts. To utilize the structural similarity of the multi-contrast images, a seven-layer patched image with varying contrast, was used as input to the network. Sixty-four filters using a kernel size of 3$\times$3$\times$7 in ReLU were adopted to facilitate non-linear operation. The number of convolution layers $N_{conv}$ was determined as shown in the Analysis subsection. In the last layer, seven filters with a kernel size of 3$\times$3$\times$64 were used for second to the last layers. Finally, a 7-channel image was predicted as the output of the network. The total number of parameters was 268,423. Artifact-reduced images could then be generated by subtracting the predicted image from the input.

\begin{figure}[bt]
\centering
\includegraphics[width=12cm]{Network}
\caption{Network architecture for the proposed convolutional neural network, two-dimensional convolutions, batch normalizations, and ReLU. The network predicts the artifact component from an input dataset. The network number of convolution layers in the network was determined by simulation-based method.}
\label{fig_network}
\end{figure}

\subsection{Imaging}

Following Institutional Review Board approval, patient studies were conducted. This study retrospectively included 31 patients (M/F, mean age 59, range 34–79 y.o.) who underwent DCE-MRI of the liver in our institution . MR images were acquired using a 3T MR750 system (GE Healthcare, Waukesha, WI); a whole-body coil and 32-channel torso array were used for RF transmission and receiving, and self-calibrated parallel imaging (ARC) was used with an acceleration factor of 2 $\times$ 2. A three-dimensional (3D) T1-weighted spoiled gradient echo sequence with a dual-echo bipolar readout and variable density Cartesian undersampling (DISCO: differential subsampling with cartesian ordering) was used for the acquisition \cite{saranathan2012differential}, along with an elliptical-centric trajectory with pseudo-randomized sorting in ky-kz. DIXON-based reconstruction method was used to suppress fat signals\cite{reeder2004multicoil}. A total of seven temporal phase images, including pre-contrast and six arterial phases, were obtained using gadolinium contrast with end-expiration breath-holdings of 10 and 21 s. The standard dose (0.025 mmol/kg) of contrast agent (EOB Primovist, Bayer Heathcare, Osaka, Japan) was injected at the rate of 1 ml/s followed by a 20-mL saline flush using a power injector. The arterial phase scan was started 30 s after the start of the injection. The acquired k-space datasets were reconstructed using a view-sharing approach between the phases and a two-point Dixon method to separate the water and fat components. The following imaging parameters were used: flip angle = 12 $^\circ$, receiver bandwidth = $\pm$ 167 kHz, TR = 3.9 ms, TE = 1.1/2.2 ms, acquisition matrix size = 320 $\times$ 192, FOV = 340 $\times$ 340 mm$^2$, the total number of slices = 56, slice thickness = 3.6 mm. The acquired images were cropped to a matrix size of 320 $\times$ 280 after zero-filling to 320 $\times$ 320.

\subsection{Respiration-induced noise simulation}\label{sec_resp_sim}

A respiration-induced artifact was simulated by adding simulated errors to the k-space datasets generated from the magnitude-only image. Generally, a breath-holding failure causes phase errors in the k-space, which results in the artifact along the phase-encoding direction. In this study, for simplicity, rigid motion along the anterior-posterior direction was assumed, as shown in Fig. \ref{fig_motion}. In this case, the phase error was induced in the phase-encoding direction, and was proportional to the motion shift. Motion during readout can be neglected because it is performed within a millisecond order.  Then, the in-phase and out-of-phase MR signal with phase error $\phi$ can be expressed as follows:

\begin{eqnarray}
	S'_I(k_x, k_y) &=& S_I (k_x, k_y) e^{-j\phi(k_y)}\\
 	S'_O(k_x, k_y) &=& S_O (k_x, k_y) e^{-j\phi(k_y)},
\end{eqnarray}
where $S_I$ and $S_O$ are the in-phase and out-of-phase signals, respectively, without the phase error; $S'_I$ and $S'_O$ are the corresponding signals with the phase error, and $k_x$, $k_y$ represent the k-space ($-\pi < k_x < \pi$, $-\pi < k_y < \pi$) in the readout and the phase-encoding directions, respectively. Finally, k-space of the water signal ($S_W$) with the phase error can be expressed as follows:

\begin{eqnarray}
	S_W &=& \frac{S'_I+S'_O}{2}\\
 	 	&=& \frac{S_I+S_O}{2}e^{-j\phi(k_y)}\\
    	&=& \mathcal{F}[I_W]e^{-j\phi(k_y)},
\end{eqnarray}
where $\mathcal{F}$ is the Fourier operator, and Iw denotes the water image. It is clear from the above equation that artifact simulation can be implemented by simply adding the phase error components to the k-space of the water image. In this study, the k-space datasets were generated from magnitude-only water images. To simulate the background $B_0$ inhomogeneity, the magnitude images were multiplied by $B_0$ distributions derived from polynomial functions up to the third order. The coefficients for the functions were determined randomly so that the peak-to-peak value of the distribution was within $\pm5$ ppm ($\pm4.4$ radian)

To generate a motion artifact in the MR images, we used two kinds of phase error patterns: periodic and random. Generally, severe coherent ghosting artifacts are observed along the phase-encoding direction. Although there are several factors that generate artifacts in the acquired images during DCE-MRI including respiratory, voluntary motion, pulsatile arterial flow, view-sharing failure, and unfolding failure\cite{stadler2007artifacts}\cite{arena1995mr}, the artifact from the abdominal wall in the phase-encoding direction is mainly recognizable. In the case of centric-order acquisitions, the phase mismatching in the k-space results in high-frequency and coherent ghosting. An error pattern using simple sine wave with random frequency, phase, and duration was used to simulate the ghosting artifact. It was assumed that motion oscillations caused by breath-hold failures occurred after a delay as the scan time proceeded. The phase error can be expressed as follows:

\[
  \phi(k_y) = \left\{ \begin{array}{ll}
    0 & (|k_y| < k_{y0} ) \\
    2\pi \frac{k_y \Delta sin(\alpha k_y + \beta)}{N} & (otherwise),
  \end{array} \right.
\]
where $\Delta$ denotes the significance of motion, $\alpha$ is the period of the sine wave which determines the frequency, $\beta$ is the phase of the sine wave, and $k_{y0}, (0 < k_{y0} < \pi)$ is the delay time for the phase error. In this study, the values of $\Delta$ (from 0 to 20 pixels, which equals 2.4-2.6 cm depending on FOV), $\alpha$ (from 0.1 to 5 Hz), $\beta$ (from 0 to $\pi /4$) and $k_{y0}$ (from $\pi/10$ to $\pi/2$) were selected randomly. The period $\alpha$ was determined such that it covered the normal respiratory frequency for adults and elderly adults, which is generally within 0.2-0.7 Hz\cite{rodriguez2013normal}. In addition to the periodic noise, random phase error pattern was also used to simulate non-periodic irregular motion as follow. First, the number of phase-encoding lines, which have the phase error, was randomly determined as between 10-50 \% of all phase-encoding lines except at the center region of the k-space ($-\pi /10 < k_{y0} < \pi /10$). Then, the significance of the error was determined randomly line-by-line in the same manner as used for periodic noise.

\begin{figure}[bt]
\centering
\includegraphics[width=11cm]{Motion_Noise}
\caption{(left) Example of a simulation of the respiratory motion artifact by adding phase errors along the phase-encoding direction in k-space. (Right) The k-space and image datasets before and after adding simulated phase errors.}
\label{fig_motion}
\end{figure}

\subsection{Network Training}

The processing was implemented in MATLAB 2018b on a workstation running Ubuntu 16.04 LTS with an Intel Xeon CPU E5-2630, 128 GB DDR3 RAM, and an NVIDIA Quadro P5000 graphics card.

The data processing sequence used in this study is summarized in Fig \ref{fig_dataset}. Training datasets containing artifacts and residual patches were generated using multi-phase magnitude-only reference images (RO $\times$ PE $\times$ SL $\times$ Phase: 320 $\times$ 280 $\times$ 56 $\times$ 7) acquired from six patients selected by a radiologist from among the 26 patients in the study. The radiologist confirmed that all reference images were successfully acquired without motion artifacts. For the multi-phase slices (320 $\times$ 280 $\times$ 7) of the images, 125,273 patches 48 $\times$ 48 $\times$ 7 in size were randomly cropped. The resulting patches that contained only background signals were removed from the training datasets. Images with motion artifact (artifact images) were generated using the reference images, as explained in the previous subsection. Artifact patches, which were used as inputs to the MARC, were cropped from the artifact images using the same method as that for the reference patches. Finally, residual patches, which were used for the output of the network, were generated by subtracting the reference patches from the artifact patches. All patches were normalized by dividing them by the maximum value of the artifact images.

Network training was performed using a Keras and Tensorflow backend (Google, Mountain View, CA), and the network was optimized using the Adam algorithm with a learning rate of 0.001. The optimization was conducted with a mini-batch of 64 patches. A total of 100 epochs with an early-stopping patience of 10 epochs were completed for convergence purposes and the L1 loss function was used as the residual components between the artifact patches and outputs were assumed to be sparse.

\begin{equation}
Loss(I_{art}, I_{out}) = \frac{1}{N}\sum_i^N \| I_{art} - I_{out} \|_1,
\end{equation}
where $I_{art}$ represents the artifact patches, $I_{out}$ represents the outputs predicted using the MARC, and N is the number of data points. Validation for L1 loss was performed using K-fold cross validation (K = 5).

The $N_{conv}$ used in the network was determined by maximizing the structural similarity (SSIM) index between the reference and artifact-reduced patches of the validation datasets. Here, the SSIM index is a quality metric used for measuring the similarity between two images, and is defined as follows:

\begin{equation}
SSIM(I_{ref}, I_{den}) = \frac{(2\mu_{ref} \mu_{den}+c_1)(2\sigma_{ref,den}+c_2)}{(\mu_{ref}^2 + \mu_{den}^2 + c_1)(\sigma_{ref}^2 + \sigma_{den}^2+c_2)},
\end{equation}
where, $I_{ref}$ and $I_{den}$ are input and artifact-reduced patches, $\mu$ is the mean intensity, $\sigma$ denotes the standard deviation, and $c_1$ and $c_2$ are constants. In this study, the values of $c_1$ and $c_2$ were as described in \cite{wang2004image}.

The number of patients used for the training versus the L1 loss with 100-epoch training was plotted to investigate the relationship between the size of the training datasets and training performance. The average sample size for one patient was 7916 training patches and 3417 validation patches for 11,333 patches. A validation dataset of 37,582 patches used for these trainings was generated from the 11 patients.

\begin{figure}[bt]
\centering
\includegraphics[width=12cm]{Dataset}
\caption{Data processing for the training. The artifact images were simulated from the reference images. Residual images were calculated by subtracting the reference patches from the artifact patches. A total of 125,273 patches were generated by randomly cropping small images from the original images.}
\label{fig_dataset}
\end{figure}

\subsection{Analysis}\label{sec_anal}

To demonstrate the performance of the MARC to reduce the artifacts in the DCE-MR images acquired during unsuccessful breath holding, the following experiments were conducted using the data from the 20 remaining patients in the study. To identify biases in the intensities and liver-to-aorta contrast between the reference and artifact-reduced images, a Bland–Altman analysis, which plots the differences of the two images versus their average, was used in which the intensities were obtained from the central slice in each phase. The Bland–Altman analysis for the intensities was conducted in the subgroups of high (mean intensity $\geq$ 0.46) and low (mean intensity < 0.46) intensities. For convenience, half of the maximum mean intensity (0.46) was used as the threshold. The mean signal intensities of the liver and aorta were measured by manually placing the region-of-interest (ROI) on the MR images, and the ROI of the liver was carefully located in the right lobe to exclude vessels. The same ROIs were applied to all other phases of the images. The quality of images before and after applying the MARC were visually evaluated by a radiologist (M.K.) with three years of experience in abdominal radiology who was not told whether each image came before or after the MARC was applied. The radiologist evaluated the images using a 5-point scale based on the significance of the artifacts (1 = no artifact; 2 = mild artifacts; 3 = moderate artifacts; 4 = severe artifacts; 5 = non-diagnostic). The scores of more than 1 were analyzed statistically by using the Wilcoxon signed rank test. To confirm the validity of the anatomical structure after applying the MARC, the artifact-reduced images in the arterial phase were compared with those without the motion artifact, which were obtained from separate MR examinations performed 71 days apart in the same patients. The same sequence and imaging parameters were used for the acquisition.

\section{Results}

The changes in the mean and standard deviation ($\mu$) of the SSIM index between the reference and artifact-reduced images are plotted against $N_{conv}$ in Fig. \ref{fig_SSIM} (a), and the results show that the network with an $N_{conv}$ of more than four exhibited a better SSIM index, while networks with an $N_{conv}$ of four or below had a poor SSIM index. In this study, an $N_{conv}$ of seven was adopted in the experiments as this value maximized the SSIM index (mean: 0.87, $\mu$: 0.05). The training was successfully terminated by early stopping in 70 epochs as shown in Fig. \ref{fig_SSIM} (b). Figure \ref{fig_SSIM} (c) shows the number of patients used for the training versus the training and validation losses, and the sample size. The results implied that stable convergence was achieved when the sample size was 3 or more although the training with few patients gave inappropriate convergence. The features using the trained network extracted from the 1st, 4th, and 8th intermediate layers corresponding to specific input and output are shown in Fig. \ref{fig_Features}. Higher frequency ghosting-like patterns were extracted from the input in the 8th layer.

Figures \ref{fig_BAplot} (a) and (b) show the Bland–Altman plots of the intensities and liver-to-aorta contrast ratios between the reference and artifact-reduced images. The differences in the intensities between the two images (mean difference = 0.01 (95 \% CI, -0.05-0.04) for mean intensity < 0.46 and mean difference = -0.05 (95 \% CI, -0.19-0.01) for mean intensity $\geq$ 0.46) were heterogeneously distributed, depending on the mean intensity. The intensities of the artifact-reduced images were lower than that of the references by ~15 \% on average, which can be seen in the high signal intensity areas shown in Fig. \ref{fig_BAplot} (a). A Bland–Altman plot of the liver-to-aorta contrast ratio (Fig. \ref{fig_BAplot} (b)) showed no systematic errors in contrast between the two images.

The image quality of the artifact-reduced images (mean (SD) score = 3.2(0.63)) were significantly better (P < 0.05) than that of the reference images (mean (SD) score =2.7(0.77)), and the respiratory motion-related artifacts (Fig. \ref{fig_results} top row) were reduced by applying MARC (Fig. \ref{fig_results} bottom row). The middle row in Fig. \ref{fig_results} shows the extracted residual components for the input images.

The images with and without breath-hold failure are shown in Fig. 8 (a, b). The motion artifact in Fig. \ref{fig_Comparison} (b) was partially reduced by using MARC, as shown in Fig. \ref{fig_Comparison} (c). This result indicated that there was no loss of critical anatomical details, and additional blurring was observed although moderate artifact on the right lobe remained.

\begin{figure}[bt]
\centering
\includegraphics[width=7cm]{SSIM}
\caption{(a) SSIM changes depending on the number of layers ($N_{conv}$). The highest SSIM (0.89) was obtained with an $N_{conv}$ of 7. (b) The L1 loss decreased in both the training, and validation datasets as the number of epochs increased. No further decrease was visually observed in > 70 epochs. The training was terminated by early stopping in 80 epochs.Error bars on the validation loss represent the standard deviation for K-fold cross validation. (c) Validation loss, training loss, and sample size were plotted against the number of patients. Smaller loss was observed as the sample size and number of patients increased.}
\label{fig_SSIM}
\end{figure}

\begin{figure}[bt]
\centering
\includegraphics[width=11cm]{Features}
\caption{Features extracted from 1st, 4th, and 8th layers of the developed network corresponding to specific input and output. Low- and high-frequency components were observed in the lower layers. On the other hands, an artifact-like pattern was extracted from the higher layer.}
\label{fig_Features}
\end{figure}

\begin{figure}[bt]
\centering
\includegraphics[width=11cm]{BA_plot}
\caption{Bland–Altman plots for (a) the intensities and (b)the liver-to-aorta contrast ratio between the reference and artifact-reduced images in the validation dataset. The mean difference in the intensities was 0.01 (95 \% CI, -0.05-0.04) in the areas corresponding to mean intensity of < 0.46 and -0.05 (95 \% CI, -0.19-0.01) in the parts of mean intensity of $\geq$ 0.4. Mead difference in the contrast ratio was 0.00 (95 \% CI, -0.02-0.02). These results indicated that there were no systematic errors in the contrast ratios, whereas the intensities of the artifact-reduced images were lower than that of the reference images due to the effect of artifact reduction especially in the area with high signal intensities. }
\label{fig_BAplot}
\end{figure}

\begin{figure}[bt]
\centering
\includegraphics[width=12cm]{Filter_Results}
\caption{Examples of artifact reduction with MARC for a patient from the validation dataset. The motion artifacts in the images (upper row) were reduced (lower row) by using the MARC.The residual components are shown in the middle row.}
\label{fig_results}
\end{figure}

\begin{figure}[bt]
\centering
\includegraphics[width=12cm]{Comparison}
\caption{(a, b) MR image in the arterial phase with and without breath-hold failure. (c) The artifact-reduced image of (b). The images were acquired in difference studies with same imaging parameters.}
\label{fig_Comparison}
\end{figure}

\section{Discussion}

In this paper, an algorithm to reduce the number of motion-related artifacts after data acquisition was developed using a deep convolutional network, and was then used to extract artifacts from local multi-channel patch images. The network was trained using reference MR images acquired with appropriate breath-holding, and noisy images were generated by adding phase error to the reference images. The number of convolution layers in the network was semi-optimized in the simulation. Once trained, the network was applied to MR images of patients who failed to hold their breath during the data acquisition. The results of the experimental studies demonstrate that the MARC successfully extracted the residual components of the images and reduced the amount of motion artifacts and blurring. No study has ever attempted to demonstrate blind artifact reduction in abdominal imaging, although many motion correction algorithms with navigator echoes or respiratory signal have been proposed \cite{vasanawala2010navigated}\cite{brau2006generalized}\cite{cheng2012nonrigid}. In these approaches, additional RF pulses and/or longer scan time will be required to fill the k-space signal whereas MARC enables motion reduction without sequence modification or additional scan time. The processing time for one slice was 4 ms, resulting in about 650 ms for all slices of one patient. This computational cost is acceptable for practical clinical use.

In MRI of the liver, DCE-MRI is mandatory in order to identify hypervascular lesions, including hepatocellular carcinoma \cite{tang2017evidence}\cite{chen2016added}, and to distinguish malignant from benign lesions. At present, almost all DCE-MR images of the liver are acquired with a 3D gradient echo sequence due to its high spatial resolution and fast acquisition time within a single breath hold. Despite recent advances in imaging techniques that improve the image quality \cite{yang2016sparse}\cite{ogasawara2017image}, it remains difficult to acquire uniformly high quality DCE-MRI images without respiratory motion-related artifacts. In terms of reducing motion artifacts,  the unpredictability of patients is the biggest challenge to overcome, as the patients who will fail to hold their breath are not known in advance. One advantage of the proposed MARC algorithm is that it is able to reduce the magnitude of artifacts in images that have been already acquired, which will have a significant impact on the efficacy of clinical MR.

In the current study, an optimal $N_{conv}$ of seven was selected based on the SSIM indexes of the reference image and the artifact-reduced image after applying MARC. The low SSIM index observed for small values of $N_{conv}$was thought to be due to the difficulty of modeling the features of the input datasets with only a small number of layers. On the other hand, a slight decrease in the SSIM index were observed for $N_{conv}$ of >12. This result implies that overfitting of the network occurred by using too many layers. To overcome this problem, a larger number of learning datasets and/or regularization and optimization of a more complicated network will be required.

Several other network architectures have been proposed for the denoising of MRI images. For example, U-Net \cite{ronneberger2015u}, which consists of upsampling and downsampling layers with skipped connections, is a widely used fully convolutional network for the segmentation \cite{dalmics2017using}, reconstruction, and denoising\cite{yu2017deep} of medical images. This architecture, which was originally designed for biomedical image segmentation, uses multi-resolution features instead of a max-pooling approach to implement segmentation with high localization accuracy. Most of the artifacts observed in MR images, such as motion, aliasing, or streak artifacts, are distributed globally in the image domain because the noise and errors usually contaminate the k-space domain. It is known that because U-Net has a large receptive field, these artifacts can be effectively removed using global structural information. Generative adversarial networks (GANs) \cite{goodfellow2014generative}, which are comprised of two networks, called the generator and discriminator, is another promising approach for denoising MR images. Yang et al. proposed a network to remove aliasing artifacts in compressed sensing MRI using a GAN-based network with a U-Net generator\cite{yang2018dagan}. We used patched images instead of a full-size image, because it was difficult to implement appropriate training with limited number of datasets as well as owing to computational limitation. However, we believe this approach is reasonable because the pattern of artifact due to respiratory motion looks similar in every patch, even though the respiratory artifact is distributed globally. Although it should be studied further in the future, we consider that MARC from the patched image can be generalized to a full-size image from our results. Recently, the AUtomated TransfOrm by Manifold APproximation (AUTOMAP) method, which uses full connection and convolution layers, has been proposed for MRI reconstruction \cite{zhu2018image}. The AUTOMAP method directly transforms the domain from the k-space to the image space, and thus enables highly flexible reconstruction for arbitrary k-space trajectories. Three-dimensional CNNs , which are network architectures for 3D images \cite{kamnitsas2017efficient}\cite{chen2018efficient} are also promising method. However, these networks require large number of parameters, huge memory on GPUs and host computers, and long computational time for training and hyperparameter tuning. Therefore, it is still challenging to apply these approaches in practical applications. These network architectures may be combined to achieve more spatial and temporal resolution. It is anticipated that further studies will be conducted on the use of deep learning strategies in MRI.

The limitations in the current study were as follows. First, clinical significance was not fully assessed. While the image quality appeared to improve in almost all cases, it will be necessary to confirm that no anatomical/pathological details were removed by MARC before this approach can be clinically applied. Second, simple centric acquisition ordering was assumed when generating the training datasets, which means that MARC can only be applied for a limited sequence. Additional training will be necessary before MARC can be generalized to more pulse sequences. In addition, realistic simulation can offer further improvement of our algorithm because noise simulation in this study was based on the assumption that ghosting originates from simple rigid motion. Moreover, the artifact was simulated in the generated k-space data from images for clinical use. Simulation in the original k-space data may offer different results. We need further researches to reveal which approach would be appropriate for artifact simulation.

The research on diagnostic performance using deep learning-based filters has not been performed sufficiently in spite of considerable effort spent for the development of algorithms. Our approach can provide additional structures and texture to the input images using the information learned from the trained datasets. Therefore, no essential information was added using MARC although image quality based on visual assessment was improved. However, even non-essential improvement may help non-expert or inexperienced readers to find lesions in the images. Further research on diagnostic performance will be required to demonstrate its clinical usefulness.

\section{Conclusion}

In this study, a deep learning-based network was developed to remove motion artifacts in DCE-MRI images. The results of experiments showed the proposed network effectively removed the motion artifacts from the images. These results indicate the deep learning-based network has the potential to also remove unpredictable motion artifacts from images.


\section{Introduction}
Pandemics have caused death and destruction several times throughout human history, and have caused large and lasting impacts on society: for example the Black Death in 14\textsuperscript{th} century Europe \citep{herlihy_black_1997}, the Spanish flu in 1918-1920 \citep{johnson_updating_2002}, HIV in the 1980s \citep{pope_transmission_2003}, and H1N1 in 2009 \citep{trifonov_origin_2009}. In 2019 and 2020, policymakers have struggled to design effective policy responses to the COVID-19 pandemic, as authorities have resorted to unprecedented and widespread use of temporary emergency measures such as lockdowns, mass quarantines, and other ``social distancing'' measures, despite that the human and economic consequences of these \textit{ad-hoc} interventions are poorly understood. It is clear that more research is needed on effective policy intervention during pandemics, and how to mitigate its human and economic impacts \citep{bauch_covid-19_2020}.

Therefore, we incorporate a model of disease transmission into a model of economic growth, considering that policymakers can implement temporary policies that simultaneously slow the spread of the disease and lower the economic output. We then select model parameters so as to represent the global economy under the impact of COVID-19, and perform numerical simulations of various policies to provide insights into the trade-offs between short- and long-term human and economic outcomes. The policy simulations will help policymakers understand how altering the starting time, intensity, and duration of the policy intervention can impact the outcomes, and contribute to the understanding of how to design effective policies to confront rapidly spreading pandemics.

A growing body of literature has been devoted to studying the economic impacts of social distancing measures and the design of emergency policies during pandemics. For instance, \cite{eichenbaum_macroeconomics_2020} constructed a mathematical model and ran numerical experiments that resulted in important insights for policymakers during a pandemic, although the study abstracts from forces that affect the long-term economic development. \cite{andersson_optimal_2020} use a similar framework to study policy responses and the trade-off between output and health during a pandemic, operating with a welfare function that only includes the period of the epidemic, and will also not account for lasting impacts of the pandemic on the economy. Our study is similar to these studies, although our model includes capital stock and population growth, which enables us to look at possible long-term effects beyond the duration of the pandemic. \cite{guan_global_2020} consider the global economic impacts of the COVID-19 pandemic in a model with multiple countries and productive sectors, suggesting that the economic impacts propagate through supply chains, and that earlier, stricter, and shorter lockdowns will minimise the economic damages. \cite{acemoglu_optimal_2020} include a simplified evaluation of economic loss into an SIR-type epidemiological model that distinguishes between age groups, and find that lockdown policies that are tailored to the different age groups are more efficient. \cite{la_torre_optimal_2020} model the social costs of an epidemic, in which a social planner must choose to allocate tax revenues between prevention and treatment, showing that the optimal allocation depends on the infectivity rate of the epidemic. \cite{alvarez_simple_2020} studied the optimal lockdown policy in a linear economy, where each fatality incurs a cost.

However, even before the onset of the COVID-19 pandemic, several studies had focused on the determinants and incentives of social distancing during epidemics \citep{fenichel_economic_2013, perrings_merging_2014, kleczkowski_spontaneous_2015, toxvaerd_equilibrium_2020, quaas_social_2020}. The models in these studies mainly apply economic insights to improve epidemiological models by examining and modelling individual contact decisions.

On the relationship between epidemics and economic growth, \cite{herlihy_black_1997} and \cite{hansen_malthus_2002} argue that the increased mortality caused by the Black Death resulted in great economic damage by decreasing labour supply, which induced the substitution of labour for capital and triggered an economic modernisation that eventually lead to greater economic growth. \cite{delfino_positive_2000}, however, argue that there are causal links in both directions in the interaction between the economy and disease transmission, and proposed a model that combines disease transmission into a model of economic growth, although the model is not used to explore temporary policy interventions during specific rapidly-spreading pandemics. \cite{bonds_poverty_2010}, \cite{ngonghala_poverty_2014} and \cite{ngonghala_general_2017} show that poverty traps\footnote{That is, in this particular case, that a greater disease burden leads to more poverty, and more poverty leads to a greater disease burden, so as to create a self-perpetuating effect.} frequently arise when combining models of infectious diseases and of economic development, and that these may help explain the differences in economic development between countries. \cite{goenka_infectious_2010}, \cite{goenka_infectious_2014} and \cite{goenka_infectious_2019} also integrate disease transmission directly into an economic growth model, and allow investments in health -- building \textit{health capital} -- to affect the epidemiological parameters. Optimal investments in health and the accumulation of health capital, however, are different from designing temporary policies during a pandemic. Most of these studies are based on disease models that allow the individual to contract the disease multiple times, which is consistent with some diseases that are common in the developing world, such as malaria or dengue fever, but may not be applicable to COVID-19. Furthermore, most of these models do not include disease-related mortality, one of the main channels through which serious pandemics affect the economy \citep{hansen_malthus_2002}. These studies, however, show that integrating disease transmission in economic growth models can result in multiple steady states, and we are therefore careful to choose an approach that is not sensitive to this: we use our model to run numerical experiments, solving the numerical optimisation problem using backwards induction.

The macroeconomic impact of the HIV/AIDS epidemic has also received much research attention, altough many studies assume the disease transmission is exogenous \citep{cuddington_modeling_1993, cuddington_further_1993, cuddington_assessing_1994, haacker_modeling_2002, arndt_hiv-aids_2003, cuesta_how_2010}. \cite{azomahou_hivaids_2016} allow mortality rate to depend on health expenditure, but the disease transmission still remains exogenous to their model. \cite{bell_macroeconomics_2009} studied government investments in health and education in an overlapping generations model of HIV/AIDS in Africa. However, the HIV/AIDS epidemic and the COVID-19 pandemic are so different, that we do not expect the insights from these studies to transfer directly to the COVID-19 pandemic.

The COVID-19 pandemic has shown that many authorities are prepared to implement strict and dramatic emergency policies at short notice in order to slow down the spread of a serious pandemic. However, our understanding of the economic and human consequences of such measures is still incipient. At the same time, implementing such policies is delicate, and, if done improperly, authorities could damage the economy whilst still failing to lower the transmission rate of the disease. There is therefore an urgent need for research that develops guidelines for the use of these emergency measures, and to help policymakers understand their impacts and consequences over time.

We contribute to the study of the efficiency, impacts, and consequences of temporary emergency measures during a pandemic by incorporating a policy parameter into a model that integrates disease transmission dynamics and economic growth. Our model provides a theoretical framework for understanding the impact of emergency policies on the trajectory of the pandemic as well as the main economic variables, in light of their mutual interactions. To gain a deeper understanding of the emergency policies, we select parameter values that are consistent with the global economy under the impact of COVID-19, and numerically simulate a large number of scenarios for possible emergency policies. Using this simulation-based approach, we investigate the impact of altering the starting date of policy intervention, the intensity of the policy intervention, and the duration of the policy intervention. Altering the simulated policies along these three dimensions provides insights into the impact of the emergency measures on the trajectory of the pandemic and the development of the main economic variables. These insights could help policymakers design effective emergency responses to pandemics.

Our work differs from earlier studies in some important aspects. Principally, the period of interest in our study extends beyond the duration of the pandemic. Therefore, our model includes the dynamics of capital accumulation, population growth and pandemic deaths, which have not been jointly considered in previous studies. These components are important to study the impact of the pandemic on economic growth beyond the short- and medium term. Other novel aspects of our model include a relationship between the reduction in economic output and the reduction in the infection rate in the short run, a mortality rate that depends on the infection rate, and explicitly modelled excess costs of hospital admissions due to the pandemic. Although some of these features are included in previous models, they have not yet been combined into a single comprehensive framework. In addition, our study contains some early estimates of the economic impacts of COVID-19 on Europe's five largest economies, constructed by analysing changes in real-time and high-frequency data on electricity demand. We also make our data and custom computer code freely available, which will hopefully be useful to the research community and contribute to future developments in the area.

In addition to this introduction, this paper consists of three sections. The following section presents a mathematical framework that incorporates a model of disease transmission into a model of economic growth, shows how model parameter values were chosen to fit the model to the global economy during the COVID-19 pandemic, and details the numerical experiments that were performed. In the third section, we present, interpret, and discuss the results of the numerical experiments and their implications. In the final section, we summarise the main findings of the study.

\section{Pandemic in an Economic Growth Model}
Here we detail the integration of an epidemiological model into a neoclassical model of economic growth -- known as one of the ``workhorses'' of modern macroeconomics \citep{acemoglu_introduction_2011}. We first modify the SIR (Susceptible-Infected-Recovered) model of the spread of an infection, pioneered by \cite{kermack_contribution_1927}, then incorporate it into the a model setup similar to the classical Ramsey-Cass-Koopmans model in discrete time. We then explain how we select functional forms and parameters to represent the global economy and the global spread of COVID-19, before outlining a set of numerical experiments designed to give insight into the economic and epidemiological impacts of varying the starting time, intensity and duration of the policy interventions.

\subsection{Incorporating Pandemic Dynamics into a Neoclassical Model of Economic Growth}
Here we combine an epidemiological model with a model of economic growth, concentrating on three main bridges between the models. First, we assume that the spread of a pandemic reduces the labour force, since infected or deceased individuals will not work, and this reduces economic output. Second, we assume that society incurs additional direct costs, for instance due to the hospitalisation of infected individuals, and these costs must be covered with output that would otherwise have been consumed or invested. Third, we assume that governments may, through policy, simultaneously impact both the spread of the pandemic and the efficiency of economic production. These interactions between the spread of the pandemic and economic growth are the main focus of our model, which jointly represents the dynamics of the spread of the pandemic and the dynamics of economic growth.

The SIR model \citep{kermack_contribution_1927, brauer_mathematical_2012} is a simple Markov model of how an infection spreads in a population over time. This model divides a population ($N$) into three categories: Susceptible ($S$), Infected ($I$), and Recovered ($R$). In each period, the number of susceptible individuals who become infected is a product of the susceptible population, the number of individuals who are already infected, and an infection rate $b$. A given proportion of the infected individuals ($r$) also recover in each period.

To incorporate the SIR model into a model of economic growth, we make two adaptations to the basic SIR model. First, we introduce a distinction between recovered individuals ($R$) and deceased individuals ($D$), since recovered individuals will re-enter the labour force whereas deceased individuals will not: each period, infected individuals will recover at a rate $r$ and pass away at a rate $m$. Second, instead of considering the population to be of a fixed size, we allow the population to grow over time. Population growth is usually negligible at the timescale of interest for models of epidemics or pandemics, but it is significant in the timescales of economic growth. Therefore, we introduce a logistic model for population growth, and new individuals will be added to the number of susceptible individuals each period. Using two parameters, $a_1$ and $a_2$, to describe the population growth, we can describe the spread of the pandemic in the population as follows:
\begin{align}
    \label{eq:popgrowth}
    N_{t+1} &= a_1 N_t + a_2 N_t^2 - m I_t \\
    \label{eq:susceptible}
    S_{t+1} &= S_t + (a_1-1)N_t + a_2 N_t^2 - b S_t I_t \\
    \label{eq:infected}
    I_{t+1} &= I_t + b S_t I_t - r I_t - m I_t \\
    \label{eq:recovere}
    R_{t+1} &= R_t + r I_t \\
    \label{eq:deceased}
    D_{t+1} &= D_t + m I_t.
\end{align}
Many variations of the basic SIR model already exist, and it would be possible to incorporate more complex dynamics into the epidemiological model. However, this model will be sufficient for our current purposes.

The model of economic growth assumes that a representative household chooses what quantity of economic output ($Y$) to consume ($C$) or save (invest) each period in order to maximise an infinite sum of discounted utility, represented by a logarithmic utility function. Output is produced by combining labour and capital ($K$) using a technology represented by a Cobb-Douglas production function with constant returns to scale and total factor productivity $A_t$. However, we allow pandemic policy, represented by $p$, to reduce the total output, and furthermore assume that only susceptible and recovered individuals are included in the labour force:
\begin{gather}
    Y_t = (1-p) A_t K_t^\alpha (S_t+R_t)^{1-\alpha},
\end{gather}
in which $\alpha$ represents the output elasticity of capital, and total factor productivity $A_t$ grows at a constant rate $g$:
\begin{gather}
    A_{t+1} = (1+g)A_t.
\end{gather}
Assuming that physical capital ($K_t$) depreciates at a rate of $\delta$ from one period to the next and that the pandemic causes direct costs $H(\cdot)$ to society, the capital stock accumulates according to the following transition equation:
\begin{gather}
    \label{eq:capitalaccumulation}
    K_{t+1} = (1-\delta) K_t - C_t - H(\cdot).
\end{gather}

Given a utility discount factor $\beta$, we assume that a benevolent social planner chooses an infinite stream of consumption $\{ C_t \}_{t=0}^\infty$ to optimise the discounted sum of logarithmic utility, solving the following maximisation problem:
\begin{gather*}
    \max_{\{ C_t \}_{t=0}^\infty } \sum_{t=0}^\infty \beta^t N_t \ln{\left( \frac{C_t}{N_t} \right) },
\end{gather*}
while respecting the restrictions represented by equations \eqref{eq:popgrowth} through \eqref{eq:capitalaccumulation}. Since our period of interest is actually finite, this maximisation problem can be solved numerically by backwards induction, provided that the terminal period is chosen so far in the future that it will not interfere with the period of interest\footnote{The implementation of the model, \emph{Macroeconomic-Epidemiological Emergency Policy Intervention Simulator}, is available at \url{https://github.com/iantrotter/ME3PI-SIM}.}.

\subsection{Representing the Global Economy and COVID-19}
The model presented in the previous subsection relies on parameters for population dynamics, the spread of a pandemic, production of economic output, and the accumulation of physical capital. In order to perform computational experiments, we need to determine realistic numerical values for these, as well as initial values for the state variables.

One fundamental issue that we must address is that the study of epidemics and economic growth usually consider different timescales: whereas the spread of an epidemic is usually analysed at a daily or weekly timescale, economic growth is usually studied at an annual, or even a decennial, timescale. To reconcile these differences, we choose a daily timescale for our model. A daily timescale is suitable for studying the spread of a pandemic, since pandemics spread rapidly and their health effects pass almost entirely within a short timeframe. However, daily resolution is an unusual choice for a model of economic growth, as capital accumulation, technological progress, and population growth are almost negligible from one day to the next. During a pandemic, however, daily movements of individuals in and out of the labour force could have a large impact on economic production, and indirectly on the accumulation of capital -- and in order to adequately capture these effects, we choose a daily resolution for the model. Parameter values for both the parameters belonging to the economic components, as well as the epidemiological components, are therefore chosen to represent a daily timescale.

\paragraph{Population Growth} The parameters for the logistic population growth model, $a_1$ and $a_2$, were selected by first estimating a linear regression model on annual global population data from the World Bank between 1960 and 2018\footnote{Available at \url{https://data.worldbank.org/indicator/SP.POP.TOTL}, accessed on 2020-05-04}. The estimation results are shown in Table \ref{tab:pop}, and the regression coefficients $a_1^y$ and $a_2^y$ -- representing the parameters of an annual model -- were converted into their corresponding daily values by calculating:
\begin{gather*}
    a_1 = 1 + \frac{a^y_1-1}{365} \quad\quad a_2 = \frac{a^y_2}{365}.
\end{gather*}
The fitted values of the population model is shown in Panel (a) of Figure \ref{fig:calibration}, and appear to match the historical data for global population closely.
\input{population_model_parameters.tex}

\paragraph{Capital Stock} We imputed the global physical capital stock by combining annual data on global gross physical capital formation, available for the period between from the World Bank\footnote{Available at \url{http://api.worldbank.org/v2/en/indicator/NE.GDI.TOTL.KD}, accessed on 2020-05-04.}, with an assumed physical capital depreciation rate of $\delta = 4.46\%$. This depreciation rate corresponds to the median value of the national depreciation rates for 2017 listed in the Penn World Tables 9.1\footnote{Available for download at \url{www.ggdc.net/pwt}, accessed on 2020-05-04.}, whose distribution is shown in Panel (b) of Figure \ref{fig:calibration}. The resulting estimated level of global physical capital stock from 1990 to 2019 is shown in Panel (c) of Figure \ref{fig:calibration}.

\paragraph{Production} Following \cite{nordhaus_optimal_1992}, we set the output elasticity of capital to $\alpha = 0.3$. We then combine annual data of global output from the World Bank between 1990 and 2018\footnote{Available at \url{http://api.worldbank.org/v2/en/indicator/NY.GDP.MKTP.PP.KD}, accessed on 2020-05-04.} with the global population and the imputed global stock of physical capital, in order to estimate the total factor productivity, $A_t$, and its growth rate, $g$. This gives an annual total factor productivity growth rate of around $1.3\%$, with a corresponding daily growth rate of around $g=3.55 \times 10^{-5}$ over the period. The modelled global production is shown in Panel (d) of Figure \ref{fig:calibration}, and fits the observed data relatively well, although the modelled production level slightly overestimates global production at some points.

\paragraph{Utility} We select an annual utility discount rate that corresponds to an annual rate of $\rho=8\%$. This discount rate allows the simulated investment from the model to match the observed gross physical capital formation in the period between 1990 and 2010, as shown in Panel (d) of Figure \ref{fig:backtest}. Although this discount rate appears somewhat high, it is not unreasonable if we take into consideration that the model represents the global economy, and that large parts of the global population consists of low-income households with high discount rates.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{Fig_Calibration.pdf}
    \caption{Calibration of the economic parameters. Panel (a): Global population and one-step-ahead predicted population from the fitted Gordon-Schaefer population growth model. Panel (b): National capital depreciation rates from the Penn World Tables 9.1 for 2017, with the dashed black line marking the median value $\delta=4.46\%$. Panel (c): Imputed values for daily global physical capital stock. Panel (d): Daily modelled gross world product, using a Cobb-Douglas production function with imputed daily capital stock, interpolated values for daily global population, output elasticity of capital set to 0.3, a growth rate for total factor productivity correspodning to 0.014 annually, and initial total factor productivity set to match actual production in 1990.}
    \label{fig:calibration}
\end{figure}

\paragraph{Excess Direct Pandemic Costs} A pandemic directly causes additional costs to society, which is captured by the function $H(\cdot)$ in our mathematical model. To model this cost, we look to the literature which has estimated the excess hospital admission costs for a recent similar pandemic, the H1N1 pandemic in 2009: for Spain \citep{galante_health_2012}, Greece \citep{zarogoulidis_health_2012}, Australia and New Zealand \citep{higgins_critical_2011}, New Zealand \citep{wilson_national_2012}, and the United Kingdom \citep{lau_excess_2019}. Figure \ref{fig:costs} shows the direct hospitalisation costs attributed to the H1N1 pandemic in various countries, along with the number of hospital admissions. Based on these previous cost estimates for the H1N1 pandemic, we use a flat cost of $u = 5,722$ USD per hospital admission (see Table \ref{tab:direct_costs}), corresponding to the solid red line in Figure \ref{fig:costs}. Although we assume a flat cost per admission, it may be more reasonable in other contexts -- for instance when applying the model to specific regions -- to consider a cost function with an increasing marginal cost: as hospital capacity becomes constrained in the short-run during a surge of admissions, one could expect the unit cost to increase. However, in the context of our global model, we do not distinguish between the regions in which the cases occur, and therefore cannot accurately capture such a saturation effect. Therefore, we choose a direct cost function that is simply linear in the number of hospital admissions.

We assume that $h=14.7\%$ of the confirmed infected cases will be admitted to hospital\footnote{This hospitalisation rate corresponds to the median of USA state level hospitalisation rates reported in the daily COVID-19 reports from the Center for System Science and Engineering at John Hopkins University, on the 10\textsuperscript{th} of May 2020, available at \url{https://github.com/CSSEGISandData/COVID-19}.}, and our direct cost function is given by:
\begin{gather*}
    H_t = u h b S_t I_t.
\end{gather*}
\input{direct_cost_model_parameters}
\begin{figure}
    \centering
    \includegraphics[width=0.5\textwidth]{Fig_Direct_Costs.pdf}
    \caption{Direct costs of the H1N1 pandemic, based on data compiled by \cite{lau_excess_2019}.}
    \label{fig:costs}
\end{figure}

\paragraph{Infection, Recovery and Mortality Rates} To estimate the mortality and recovery rates, $r$ and $m$, we solve the transition equations for the number of recovered $R_t$ (equation \eqref{eq:recovere}) and the number of deceased $D_t$ (equation \eqref{eq:deceased}) for their respective parameters:
\begin{gather*}
    r = \frac{R_{t+1}-R_t}{I_t}, \quad m = \frac{D_{t+1} - D_t}{I_t}.
\end{gather*}
Using daily data for the number of confirmed, recovered and deceased cases, made available by John Hopkins University\footnote{Available at \url{https://github.com/CSSEGISandData/COVID-19}, accessed on 2020-05-06.}, we can calculate the recovery and mortality rates for each day, as shown in the bottom two rows of Figure \ref{fig:epidemicparameters}.

To estimate the infection rate, $b$, we solve equation \eqref{eq:infected} for the parameter $b$:
\begin{gather*}
    b = \frac{I_{t+1} - (1-r-m)I_t}{S_t I_t}.
\end{gather*}
Taking into account that $I_t$ in the model refers to the number of active cases, whereas the data reports the accumulated number of cases, and using the population growth model to help estimate the number of susceptible individuals, we calculate the daily infection rates, shown in the top row of Figure \ref{fig:epidemicparameters}. As the infection rate $b$ varies over time, we choose a relatively high infection rate to represent the infection rate in the absence of policy intervention, $b_0 = 2.041\times 10
^{-11}$, which equals the upper quartile (75\%) of the observed infection rates. As we simulate different intervention policies, this base infection rate $b_0$ will be modified.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{Fig_Parameters_TimeHist.pdf}
    \caption{Calibration of the parameters for the SIR model. Panel (a) and (b): the development and distribution of the SIR infection rate, $b$. Panel (c) and (d): the development and distribution of the SIR recovery rate, $r$. Panel (e) and (f): the development and distribution of the SIR mortality rate, $m$.}
    \label{fig:epidemicparameters}
\end{figure}

We notice from Figure \ref{fig:epidemicparameters} that the mortality rate appears to rise and fall together with the infection rate\footnote{\cite{alvarez_simple_2020} have also made this observation, and included the effect in their model.}. This might reflect that lack of capacity in a health system increases mortality rate, and this is therefore a feature that we would like to capture. We therefore estimate $m$ as a function of $b$:
\begin{gather*}
    m = k_1 b^{k_2},
\end{gather*}
in which $k_1$ and $k_2$ are constants. Table \ref{tab:mortality} shows the regression for estimating the parameters $k_1$ and $k_2$, and the fitted function is shown in Figure \ref{fig:mortalityrate}, together with the observed values of daily infection and mortality rates.
\input{mortality_rate_model_parameters}
\begin{figure}
    \centering
    \includegraphics[width=0.5\textwidth]{Fig_Mortality_x_Infection.pdf}
    \caption{Global mortality rate and infection rate.}
    \label{fig:mortalityrate}
\end{figure}

For the recovery rate in the simulations, we select the median of the daily recovery rates calculated from the data, $r=0.02099$, which implies that slightly over 2\% of infected individuals recover from one day to the next. The infection rate, $b$, however, will be determined individually for each scenario, and will reflect the pandemic policy simulated in each of the scenarios. The mortality rate $m$ will be determined by the infection rate $b$, according to the relationship between them we estimated earlier.

\paragraph{The Production-Infection Trade-Off} Our model assumes that pandemic policy mainly impacts the spread of the pandemic through manipulating the infection rate $b$, and mainly infects economic growth by affecting production of economic output, $Y_t$. Our model contains a single parameter, $p$, to represent pandemic policy, which directly represents the shortfall in global production. In order to analyse the trade-off between production and the infection rate, however, we must establish how the infection rate, $b$, is impacted by the policy parameter $p$ -- that is, we must quantify how the infection rate responds to foregone production.

We expect the relationship between the infection rate, $b$, and the GDP shortfall, $p$, to exhibit two specific characteristics. Firstly, we expect a reduction in the infection rate as the GDP shortfall increases, because we assume pandemic policies are designed to reduce the infection rate, which result in a shortfall in the economic production as a side-effect. Secondly, we expect the reduction in the infection rate to be greater at first, because we expect measures to be enacted in order from more to less effective, and from less to more disruptive. That is, the infection rate reductions exhibit a form of decreasing returns in the GDP shortfall. We suggest that the percentage reduction in the infection rate, $\Delta b(\%)$, responds to the percentage reduction in GDP, $\Delta \text{GDP}(\%)$, as follows:
\begin{gather*}
    \Delta b(\%) = q_1 \Delta \text{GDP}(\%)^{q_2},
\end{gather*}
in which $q_1$ and $q_2$ are constants, and $q_2 \in (0,1)$.

The shortfall in production associated with each infection rate is, however, not straight-forward to estimate. Firstly, GDP data is published several months late, which means that a whole pandemic might have long since passed by the time the GDP data is published. Secondly, GDP data are generally aggregated into months, quarters or years, which would make it difficult to associate with a particular infection rate, which can vary greatly over these timeframes. Therefore, we infer the daily reduction in economic production from the shortfall in electricity consumption: electricity consumption data for many countries is available near real-time -- often at sub-hourly resolution -- and electricity consumption is known to correlate well with economic activity (see, for instance, \cite{trotter_climate_2016} and \cite{rodriguez_climate_2019}).

Data on electricity demand (load) is available for many European countries from the ENTSO-E Transparency Platform\footnote{Available at \url{https://transparency.entsoe.eu/}, accessed on 2020-05-13.}, and we utilise data from Europe's five biggest economies, which have all been significantly impacted by COVID-19: France, Germany, Italy, Spain, and the United Kingdom. To measure the daily \textit{shortfall} in electricity consumption, however, we must compare the observed electricity consumption to what it would have been under normal conditions. Therefore, we first need to create a counterfactual representing the electricity consumption under normal conditions. We use the automated forecasting procedure by \cite{taylor_forecasting_2017} to calibrate models on the daily national electricity consumption (load) data from 2015 until March 1, 2020. This period does not include the main impacts of the pandemic, such that forecasts from the models for the period from March 1, 2020 to May 10, 2020 can act as counterfactuals -- how electricity consumption would have been expected to develop under normal conditions. These counterfactuals may then be compared to the observed values in the same period, and allows us to calculate the daily electricity consumption shortfall in terms of percentages.

The relationship between electricity consumption and production has been the subject of many studies (often as variants of ``income elasticity of electricity consumption''), and, synthesising the studies into a useful heuristic, we assume that a 1\% decrease in electricity consumption is associated with a 1.5\% reduction in GDP. The estimated GDP shortfall for France, Germany, Italy, Spain and the United Kingdom are illustrated in Panel (a) of Figure \ref{fig:prodinfecttime}, which shows a clear increase in the GDP shortfall throughout March 2020, and a stable shortfall of around 10\%-20\% throughout April and the start of May 2020, appearing to correspond closely to the lockdown periods of these countries. Panel (b) of Figure \ref{fig:prodinfecttime} shows the reduction in the infection rate for the five countries over the same time period, with the base period for the infection rate considered to be the first seven days in March, and it is clear that the infection rate has decreased as the GDP shortfall has dropped, which is consistent with our expectations. The scatter plot of the estimated GDP shortfall and the reduction in infection rates, shown in Figure \ref{fig:prodinfectscatter}, shows a clear relationship between infection rate reductions and GDP shortfall. The red line in Figure \ref{fig:prodinfectscatter} shows the model, with constants estimated as in Table \ref{tab:infxgdp}. The model fits the observations well, and the estimated values for the constants, $q_1$ and $q_2$, conform to our expectations.
\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{Fig_Production_and_Infection_Timeseries.pdf}
    \caption{Weekly estimated GDP shortfall for France, Germany, Italy, Spain, and the United Kingdom. The GDP shortfall is based on the difference between actual and expected electricity consumption. The average weekly infection rate is based on the number of confirmed cases.}
    \label{fig:prodinfecttime}
\end{figure}
\begin{figure}
    \centering
    \includegraphics[width=0.5\textwidth]{Fig_Production_x_Infection_Scatter.pdf}
    \caption{Weekly estimated GDP shortfalls for France, Germany, Italy, Spain, and the United Kingdom, based on the difference between actual and expected electricity consumption. The average weekly infection rate is based on the number of confirmed cases. The red line represents the model estimated on the data.}
    \label{fig:prodinfectscatter}
\end{figure}
\input{infection_gdp_regression}

\paragraph{} Table \ref{tab:parameters} summarises the chosen values for the parameters in the model. Having defined parameter values such that the model represents the global economy under the impact of COVID-19, we can now define scenarios that can be simulated numerically, and provide insight into the impact of policy on both economic growth and the spread of the pandemic.

\begin{table}
    \centering
    \caption{Parameter values.}
    \resizebox{\columnwidth}{!}{%
    \begin{tabular}{llrl}
		\toprule
         \textbf{Parameter} & \textbf{Description} & \textbf{Value} \\
		\midrule
		 $a_1$, $a_2$ & Logistic population growth (annual) & 1.028, -2.282$\times$10$^{-12}$ \\
		 $\delta$ & Capital depreciation rate (annual) & 4.46\% \\
         $\alpha$ & Output elasticity of capital & 0.3 \\
         $g$ & Growth rate of total factor productivity (annual) & 1.3\% \\
         $\rho$ & Utility discount rate (annual) & 8\% \\
         $u$ & Cost per hospital admission & 5,722 USD \\
         $h$ & Hospital admissions per confirmed case & 14.7\% \\
         $r$ & Daily recovery rate per active infection & 2.1\% \\
         $b_0$ & Base infection rate (no intervention) & $2.041 \times 10^{-11}$ \\
         $k_1, k_2$ & Mortality rate parameters, $m = k_1 b^{k_2}$ & 12.561, 0.717 \\
         $q_1, q_2$ & Infection rate parameters, $\Delta b(\%) = q_1 \Delta\text{GDP}(\%)^{q_2}$ & 3.677, 0.238 \\
		\bottomrule
    \end{tabular}
    }
    \label{tab:parameters}
\end{table}

\subsection{Policy Experiments}
To have a basis for comparison, we first simulate two baseline scenarios: the \textit{No Pandemic} scenario, in which no pandemic occurs, and the \textit{No Intervention} scenario, in which the pandemic occurs with no direct intervention ($p=0$). Comparing the remaining scenarios to the first baseline scenario (\textit{No Pandemic}) will help us understand the impact of the pandemic. In addition, the baseline scenario will provide the initial conditions for population and capital stock at the start of the pandemic, as observations are not yet available. Comparing the remaining scenarios to the second baseline scenario (\textit{No Intervention}) will help us understand the impact of the simulated policy intervention. The initial values for these simulations are shown in Table \ref{tab:baselines}.

Having established the baseline scenarios, we run a series of simulations to investigate three fundamental aspects of the policy intervention. First, we alter the timing of the start of the intervention, to explore the advantages and disadvantages of starting the intervention early or late. Second, we alter the intensity of the intervention, to investigate the differences in the impacts between light and severe interventions. And, finally, we alter the duration of the intervention. That is, by running numerical experiments that vary the policy interventions in commencement, intensity and duration, we answer three fundamental policy questions: ``When?'', ``How much?'', and ``For how long?''. When taken together, these experiments will provide insight into the economic and health impacts of varying policies along these three dimensions, and highlight the trade-offs that policymakers must consider:
\begin{description}
    \item[When to intervene?] Holding the intervention intensity and duration fixed at 10\% and 26 weeks, we run simulations altering the start of the policy intervention between April 09, 2020, and June 02, 2020.
    \item[How much?] Holding the starting date of the intervention fixed at March 12, 2020 -- the date when the WHO declared COVID-19 to be a pandemic -- and the duration fixed at 26 weeks, we alter the intensity of the intervention from 5\% to 25\%, in steps of 10 percentage points.
    \item[For how long?] Keeping the starting date of the intervention fixed at March 12, 2020, and the intervention intensity fixed at 10\%, we alter the duration of the intervention between 4 weeks and 76 weeks.
\end{description}
The initial values used in all these simulations are the same as in the \textit{No Intervention} scenario, specified in Table \ref{tab:baselines}. Taken together, these three sequences of simulations will provide important and actionable insights into the impacts of policy intervention on both economic growth and on the spread of the pandemic that will help policymakers understand the relevant trade-offs.

\begin{table}
    \centering
    \caption{Parameters used for the baseline scenarios.}
    \resizebox{\columnwidth}{!}{%
    \begin{tabular}{llrrrrrrrrrr}
		\toprule
		\midrule
        \textbf{Scenario} & \textbf{Start date} & $N_0$ & $I_0$ & $R_0$ & $D_0$ & $b_0$ & $A_0$ & $K_0$ \\
		\midrule
        \textbf{No Pandemic} & 2019-01-01 & 7.634$\times 10^{9}$ & 0 & 0 & 0& 0 & 1.880 & 2.775$\times 10^{14}$ \\
        \textbf{No Intervention} & 2020-01-22 & 7.718$\times 10^9$ & 510 & 28 & 17 & 2.041$\times 10^{-11}$ & 1.906 & 2.827$\times 10^{14}$ \\
		\bottomrule
    \end{tabular}}
    \label{tab:baselines}
\end{table}

\section{Results and Discussion}
\subsection{Backtest 1990-2010}
Before presenting the main simulation results, we first present the results of a backtest. This is shown in Figure \ref{fig:backtest}, and shows that the model captures the main features of the observed historical data. Although the backtest in this case is not an out-of-sample test, due to lack of data, the backtest provides strong support for the economic components of the model.

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{Fig_Backtest.pdf}
    \caption{Model backtest results. Panel (a): The simulated daily development of the global physical capital stock and the daily imputed global physical capital stock. Panel (b): Global daily simulated and observed population. Panel (c): Simulated and observed daily gross world production. Panel (d): Simulated and observed daily global gross physical capital formation.}
    \label{fig:backtest}
\end{figure}

\subsection{Baseline Scenarios}
The results of simulating the baseline scenarios -- \textit{No Pandemic} and \textit{No Intervention} -- are shown in Figure \ref{fig:baselines}. As expected, the \textit{No Pandemic} scenario is characterised by steady economic growth, and no infected or deceased individuals. The \textit{No Intervention} scenario, however, shows a large and abrupt drop of around 45\% in production during the first half of 2020, as the pandemic spreads through the population. The number of active infections peaks in mid-June, 2020. As the pandemic subsides, a large proportion of the labour force never returns as the mortalities reach 1.75 billion people, and production recovers only to 85\% of its pre-pandemic value before 2021. Although growth in economic production resumes after the pandemic, production remains 20\%-25\% below the production in the \textit{No Pandemic} scenario until the end of the simulation in 2030. In summary, the \textit{No Intervention} scenario shows substantial loss of human life, as well as a lasting and significant negative impact on production, and we expect that shrewd policy intervention could partially mitigate these impacts.
\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{Fig_Baselines.pdf}
    \caption{Simulation results for the baseline scenarios.}
    \label{fig:baselines}
\end{figure}

In the following, we run model simulations to gain insight into when to start the policy intervention, to what degree to intervene, and for how long the intervention should last.

\subsection{When to Intervene?}
We first run a series of simulations to examine the question of when a possible policy intervention should start. In this series of simulations, the intervention intensity is held fixed at 10\% (that is, the intervention causes a 10\% decline in production), and the duration of the intervention is held fixed at 26 weeks. Multiple simulations are run with differing starting dates for the policy intervention. This series of simulations is shown in Figure \ref{fig:startdates}, with three possible starting dates for the policy intervention: April 9, May 21, and July 2.
\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{Fig_StartDates.pdf}
    \caption{Simulation results when varying the starting date of the policy intervention.}
    \label{fig:startdates}
\end{figure}

Examining Panel (f) of Figure \ref{fig:startdates}, we note that intervening on July 2 allows the pandemic to spread almost identically to the \textit{No Intervention} scenario -- that is, July 2 is too late for effective intervention because the peak in active infections has passed, most of the damage is already done, and the pandemic is decelerating by itself. However, by intervening on July 2 when the number of active infections is near its highest, many mortalities are avoided, and the human and economic damage is somewhat lower than in the \textit{No Intervention} scenario. Further, we note that intervening on April 2 does not appear to significantly alter the course of the pandemic or mitigate its effects -- intervening so early in the pandemic only serves to delay the main wave of infections.

The intervention starting on May 21 -- about one month before the peak of the \textit{No Intervention} scenario -- appears to be the most effective of our simulations, both considering the economic impacts and the final mortality rate. May 21 appears to be just before the inflection point of the \textit{No Intervention} scenario, and the number of infections is growing at its highest rate. Between the three simulated scenarios, this is by far the preferred option.

It seems that timing the policy intervention is of great importance to mitigate both the human and the economic impacts. Although we do not believe that the exact dates hold for the COVID-19 pandemic in particular, these simulations lead us to interesting insights: policy intervention appears to be most effective when the number of active infections is approaching its inflection point, and is growing at its highest rate. An intervention that is too early will only serve to delay the critical phase of the pandemic, and an intervention after the peak has occurred will obviously do nothing to lower the peak. Although it may be difficult to know beforehand when a pandemic will enter its critical phase, the timing of the policy intervention is of paramount importance, and our results suggest that authorities should implement emergency policies only when the disease is sufficiently spread. However, we expect that the exact specification of \textit{sufficiently spread} to vary significantly from place to place, depending on local conditions.

This finding contradicts the claims of \cite{guan_global_2020}, who argue that interventions should be ``earlier, stricter and shorter'': instead, our results show that starting the intervention before the disease is sufficiently spread will either simply delay the critical phase of the pandemic (if the intervention indeed is kept ``short''), or prolong the intervention (if the intervention is extended). The study by \cite{eichenbaum_macroeconomics_2020} focuses on ``starting too late'' and ``ending too early'', yet our results suggest that policymakers also need to avoid starting \emph{too} early.

\subsection{How Much Intervention?}
In this series of simulations, we keep the starting date of the policy intervention fixed at March 12 and the duration of the policy intervention fixed at 26 weeks, whilst varying the intervention inensity. We simulate policies that reduce production by 5\%, 15\%, and 25\%, and the simulation results are shown in Figure \ref{fig:degrees}.
\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{Fig_Degrees.pdf}
    \caption{Simulation results when varying the intensity of the policy intervention.}
    \label{fig:degrees}
\end{figure}

The three simulations with different intervention intensity, shown in Figure \ref{fig:degrees}, suggest that varying the intensity of the intervention mainly alters the timing of the pandemic, but does little to mitigate the economic and human impacts: apart from a delay in the main phase of the pandemic, most variables behave similar to the \textit{No Intervention} scenario.

This indifference between the intensities of the interventions is likely related to the relationship we identified between the GDP shortfall and the infection rate reduction, as shown in Figure \ref{fig:prodinfectscatter}. There are strong diminishing returns, such that even an intervention of a low intensity (5\%) already reduces the infection rate substantially (60\%), and that additional measures have a lower effect on the infection rate.

Essentially, the intensity of the intervention -- above a certain minimum level -- appears to be less important than the timing of the intervention. Again, this finding contradicts the study of \cite{guan_global_2020}, our results indicating that intervention should perhaps be \emph{less} strict, as the intervention intensity faces strong diminishing returns.

\subsection{For How Long to Intervene?}
To analyse the impact of the duration of the policy intervention, we vary the duration of the policy intervention, whilst maintaining the intervention intensity fixed at 10\% and the starting date fixed at March 12. Figure \ref{fig:durations} shows the results of simulating intervention durations of 4 weeks, 28 weeks, 52 weeks and 76 weeks. It is clear from the figure that the duration of the policy intervention can have a large impact on the trajectory of the pandemic, and its human and economic aftermath.
\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{Fig_Durations.pdf}
    \caption{Simulation results when varying the duration of the policy intervention.}
    \label{fig:durations}
\end{figure}

From Figure \ref{fig:durations}, it seems that policies with longer durations clearly lead to lower human and economic impacts, with a 76-week duration -- the longest of our simulations -- showing dramatically lower number of total mortalities, as well as a much quicker post-pandemic recovery of production. The key appears to be that 76 weeks is sufficient to include the peak in active infections of the trajectory determined by the reduced infection rate. This observation suggests that policies with a lower intensity would require a shorter duration, whereas policies with a higher intensity would require a longer duration -- something which may, at first, appear counter-intuitive.

Our results partially support the findings of \cite{eichenbaum_macroeconomics_2020}, who warn of ending the intervention ``too early'', but contradict the claims of \cite{guan_global_2020} that intervention should be ``short'' -- although the duration of the intervention should naturally be as short as possible. Our results suggest that stricter policies will require longer durations.

\subsection{Considerations, Limitations, and Concerns}
We have tried to make sensible modelling choices in this study, but, like all models, our model is a simplification that focuses only on certain aspects and ignores others. The simulation results should not be understood \emph{literally}: the intention of our model has never been to provide numerically accurate predictions, but to generate insights into the impacts of policy interventions by analysing the dynamics of the system as a whole. Although the insights from the numerical simulations can contribute to improving policy interventions during pandemics, it is important to appreciate the limitations of our model and results.

We are concerned about the quality of the parameters used for the epidemiological part of the model: there is a great deal of doubt and uncertainty about the quality of the official datasets on the spread of the pandemic -- that is, the number of confirmed cases, recovered cases and mortalities. There is a general sense that the number of confirmed cases is not representative for the number of infections, as testing is severely lacking in many regions. The lack of testing also affects the number of mortalities due to the pandemic, and number of recovered cases. Although we have used the data that is available without much discrimination, we share the concerns of many other researchers as to the quality of this data.

It is also unusual for a model of economic growth to operate at a daily resolution. We do not think this directly invalidates our resulting insights, although it means that the parameter values may appear unusual to researchers and practicioners, and that special care must be taken in the interpretation of the results. An alternative would be to develop the model in continuous time, which might be more familiar to some. However, in that case it would be necessary to discretise the model later for performing numerical experiments -- the model would, in the end, be the same, so presenting the model directly in discrete time appears to be a simpler alternative.

The parameter values were chosen for the model to represent the global economy and the global spread of COVID-19. There are, however, large differences between regions in the world. For instance, the five countries used for estimating the economic impact of policy measures -- France, Germany, Italy, Spain, and the United Kingdom, which were chosen for their data availability -- are probably not entirely representative for the rest of the world. There is also no global central government that implements global policy, and our insights are therefore not directly applicable by any specific authority. The purpose of the study, however, was not to generate recommendations for specific actions, but to generate insights into the impacts and trade-offs that policy interventions must consider. Regional, national and local policy can differ from ``global'' policy -- and probably should, as policy can be optimised to local conditions -- but the insights on intervention timing, intensity and duration may nevertheless be useful at these levels also. It would be possible to adapt the model for use at regional, national or local scales. In this case, we would recommend considering replacing the linear admission costs with a specification that allows for increasing marginal costs of admissions, which might better reflect increasing costs in the short run due to capacity saturation.

For calibrating the model, we estimated the economic impacts using an estimate of the shortfall in electricity consumption. Although we believe this approach is valid, it is difficult to assess the accuracy of the economic impact estimates. The approach based on electricity consumption could also be complemented by other near-real-time, high-resolution data sources that are believed to correspond well to economic activity, such as satellite observations, mobile phone movement and activity data, and urban traffic data. However, these data are not usually as widely available as electricity consumption data.

The model does not incorporate any demographic heterogeneity. Since some pandemics appear to affect people with certain demographic characteristics differently, this may bias the results. For instance, the mortality rate of COVID-19 appears to differ greatly between old and young people: if the disease has a greater impact on groups that were not originally included in the work force anyway, the model could exaggerate the economic impact of the pandemic by disconsidering demographic heterogeneity. We do not believe this to affect the main insights derived from our model simulations, since we do not think it substantially alters the dynamics of the system. However, it would certainly be an issue for the ``predictive accuracy'' of the simulations.

Since the model is deterministic, agents in the model have perfect foresight from the very start of the simulation. This is, naturally, not true in the real world, in which there are large uncertainties about future developments. This gives the model agents an unrealistic ability to plan for the future, and the economic portion of the model should therefore be considered an optimistic path. Another detail that also may positively bias the outcomes, is that the model does not include structural damages -- such as bankruptcies, institutional change, changing habits, and so forth -- and affords the model agents much more flexibility than economic agents may have in reality, where they may be facing additional restrictions.

Finally, we only simulated very simple policies for the purpose of understanding the impact of altering the policy in a very specific way. For instance, superior policies can be made relatively easily by allowing the intensity of the intervention to vary during the pandemic. Our examples, in which starting dates, intensity and duration are fixed, only served for illutration and to understand some of the dimensions of policy intervention.

We reiterate that our purpose has not been to provide numerically accurate predictions -- nor the means to generate accurate predictions -- of the evolution of the COVID-19 pandemic or the global economy. We have only explored particular aspects of effective policy responses to a pandemic, using a very high-level and theoretical approach, and it is with this in mind that our results are most appropriately appreciated. Our research does not aim to offer specific guidance for world authorities on the handling of the COVID-19 pandemic, but to analyse how a pandemic interacts with the global economy and thus help establish a set of of general guidelines.

\section{Conclusion}
We have presented a mathematical model for the joint evolution of the economy and a pandemic, based on incorporating the dynamics of the SIR model that describes the spread of epidemics into a neoclassical economic growth model framework. This model is subsequently adapted to represent the global economy under the impact of the COVID-19 pandemic by selecting appropriate functional forms and parameter values. The model includes a parameter that represents policy, by which economic production can be lowered in exchange for a reduced infection rate.

Using the calibrated model, we simulate the joint evolution of the economy and the pandemic for a series of policy assumptions in order to discover what is the most effective timing of a policy intervention, what intensity of policy intervention is most effective, and how long policy intervention should last.

Our experiments suggest that it is most effective to start the policy intervention slightly before the number of confirmed cases grows at its highest rate -- that is, to wait until the disease is sufficiently spread. Not only does this help lower the peak in active infections, it also reduces the economic impact and the number of mortalities. Starting too early can delay the pandemic, but does not otherwise significantly alter its course, whereas starting after the peak in active infections can obviously not impact the peak.

Furthermore, altering the intensity of the intervention does not appear to greatly influence the evolution of the pandemic nor the economy, other than cause minor delays. We ascribe the lack of effect to the concave relationship that we estimated between intervention intensity and infection rate reduction, as a large reduction in infection rate can be achieved by sacrificing a modest proportion of economic production, and appears to show strong decreasing returns thereafter. Our estimates suggest that a 60\% reduction in the infection rate can be achieved by sacrificing only 5\% of production, whereas a 70\% reduction in infection rate could be achieved for a 10\% reduction in production.

Altering the duration of the intervention showed that interventions with a longer duration lead to significantly lower mortalities and a quicker post-pandemic recovery in economic production. The key observation is that the policy must include the peak of the \textit{new} path set out by the reduced infection rate: in short, policy intervention should last until the peak has passed. Therefore -- somewhat counter-intuitively -- stricter policies should last longer, and less strict policies should last shorter.

Although the scenarios we present are not necessarily numerically accurate as predictions -- mostly due to generalisations made for modelling purposes, large regional variations, and large uncertainties in the parameters -- our conclusions are based mainly on the dynamics revealed by the policy experiments, and not specifically on their numerical values. As such, we hope that our model can serve as a tool for enhancing our understanding of the design of effective policies against the spread of pandemics, and that our insights can contribute to this discussion and provide general guidelines for policymakers.

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