COVID-19 OUTBREAK IN TAIWAN
PU-ZHAO KOW AND JENN-NAN WANG
Abstract. The world has been suering from the impact of COVID-19 pandemic since the virus was rst detected in the late December of 2019. The economic and social disruption caused by the pandemic is devastating. Until now, even though several vaccines have been approved by WHO and a lot of countries have accelerated vaccine rollout in an eort to prevent further outbreak of the pandemic, the world is still under the threat of new variants of COVID-19 virus.
Unlike most countries, for the past one and a half years, Taiwan was virtually virus- free and people were doing business as usual. However, new cases of infection emerged dramatically in mid-May this year. To curb the spread of the virus, Taiwanese government imposedLevel 3 epidemic alerton May 20, 2021. Level 3 epidemic alert isstill in eectat the time of writing this paper. Out of curiosity, we are interested in predicting the current trend of Taiwan's pandemic combining a simple mathematical model and deep learning. Our model is a modi cation of multi-chain Fudan-CCDC models introduced in [PSY+20a,PSY+20b].
In [PSY+20a,PSY+20b], the spread of virus depends entirely on two parameters infec- tion rate and isolation rate. These two parameters are assumed to be piecewise constants in [PSY+20a, PSY+20b]. However, we were not able to make a satisfactory prediction of the COVID-19 pandemic outbreak in Taiwan using their models. Therefore, we consider more general parameters, not necessarily piecewise constants, in our model.
There are two major di culties in predicting the epidemic with a mathematical model.
One di culty is that the only available data is the number of daily infected cases provided by the government. All other parameters and variables in the model are unknown. We are able to approximate the parameters and the unknown variables based on the mathematical model using the available data of daily infected cases. Another di culty is that all parameters are unknown when we perform the prediction. Hence, it becomes impractical to use the mathematical model to do the forecasting. To overcome this di culty, we make use of long short-term memory neural network (LSTM), a machine learning algorithm.
Acknowledgments
The authors would like to thank Professor Jin Cheng1for some conversations on the mathe- matical model. They would also like to express their gratitude to Pu-Yun Kow2for numerous suggestions on implementing machine-learning algorithm. This work is partially supported by the Ministry of Science and Technology (MOST) Taiwan.
1. Introduction
The COVID-19 pandemic has been the news headline for almost one year and a half since the virus was rst reported in December, 2019. The new coronavirus is contagious and
2020 Mathematics Subject Classi cation. 92D30.
Key words and phrases. Taiwan COVID-19 outbreak, mathematical model, multi-chain Fudan-CCDC model, long short-term memory neural network (LSTM). .
1Vice Dean of School of Mathematical Sciences, Fudan University, China.
2Department of Bioenvironmental System Engineering, National Taiwan University, Taiwan.
1
such a long period. Until now, even though several vaccines have been approved by WHO and a lot of countries have accelerated vaccine rollout in an eort to prevent further outbreak of the pandemic, the world is still under the threat of new variants of COVID-19 virus. The path to normality may be further delayed.
Unlike most countries, Taiwan was virtually virus-free for the past one and a half year thanks to prompt responses at the onset of global pandemic. However, the number of new infected cases suddenly arose in mid-May, 2021 (largely due to theαvariantvirus). It caught the government o guard. To prevent further spread of the virus, Taiwanese government imposed a Level 3 epidemic alert on May 20, 2021. Key restrictions in the alert include that all people must weak masks at all times outdoors, and indoor gatherings are limited to ve people, while outdoor gatherings are restricted to ten. A wide range of business and public venues are to be closed, with the exception of essential services such as police departments, hospitals, and government buildings. All school classes are conducted online.
In order to better understand the outbreak of COVID-19 pandemic in Taiwan and to make a reasonable prediction of the trend, we propose a simple epidemic mathematical model. We are especially interested in the prediction when the Level 3 alert is likely to be lifted. Since last year, there are a large pool of mathematical literature discussing the COVID-19 pandemic from dierent aspects. It is not possible review all of them here. We only mention some of them which are closely related to our interests, [CCJL20a, CCJL20b, LJYC20, LSCC20, SCC20,SCCC20,SZCC20,SZY+20,XCY+21,YCL+20]. Our model is mainly inspired by the multi-chain Fudan-CCDC model [PSY+20a,PSY+20b], where the infection and the isolation rates play the important role in predicting the trend of the pandemic. We remark that a multi- chain Fudan-CCDC model (with constant infection and isolation rates) is indeed equivalent to a single-chain Fudan-CCDC model with piecewise constant parameters. However, we were not able to make a satisfactory prediction of the recent COVID-19 pandemic outbreak in Taiwan using their models with piecewise constant infection and isolation rates. Therefore, we want to propose a modi ed mathematical model and consider more general parameters, not necessarily piecewise constants, in the model.
The chance of successfully forecasting the spread of the virus depends essentially on how good the estimates of the parameters using the public available data. Nonetheless, the only available data is the number of daily infected cases released by the government. All other parameters and variables in the model are unknown. We can overcome this obstacle by
tting the parameters and the unknown variables based on the mathematical model using the available data of daily infected cases. However, in the prediction step, all parameters are still unknown. Hence, it becomes impractical to use the mathematical model to do the forecasting. To overcome this di culty, we make use of long short-term memory neural network (LSTM), a machine learning algorithm.
In our approach, we divide the data obtained from the public data of the COVID-19 pandemic in two groups:
(i) Training set: these data are given to the network to train the system, i.e., adjusting the parameters in the model to minimize the tting error;
(ii) Validation set: these data are used to measure the accuracy of the system and to halt the training when the error is minimized.
After training our system, we then perform the prediction based on the trained model with the parameters determined by the network.
It is worth to mention that one can perform of the forecasting of the pandemic by feeding only the number of daily infected cases into the LSTM algorithm without referring to the mathematical model. This is a type of unsupervised learning. In our approach, we will make use of the mathematical model to produce the training dataset and train the system based on this dataset. In other words, our method can be seen as a supervised learning. In later section, we will provide an evidence to convince the reader that our method outperforms the unsupervised learning.
2. Mathematical model
We are now ready to describe the mathematical model. Let p be the transition probability from infection-to-illness onset. According to the early transmission dynamics in Wuhan of the COVID-19 disease, which studied in [LGW+20] by Chinese Center for Disease Control and Prevention (CCDC), p can be approximated by a log-normal distribution Lognormal(µ, σ2) with µ = 1.417 and σ2 = 0.4525, that is,
(2.1) p(t) = 0.5977
t e−1.105(ln t−1.417)2
. Here we remark that
(2.2) E[p] ≈ 5.172 and Z
t≤14
p(t) dt ≈ 0.965.
Therefore, in our model, it is reasonable to approximate (2.1) by p(t) ≈1t≤14
0.5977
t e−1.105(ln t−1.417)2
.
Remark 2.1. According to the introduction of COVID-19 by Taiwan Centers for Disease Control (Taiwan CDC), the incubation period from infection with the COVID-19 to the onset of disease is 1 to 14 days (mostly 5 to 6 days), which agrees to (2.2).
The quantities used in the dynamics of the COVID-19 infection in our model are listed below:
It: the cumulative infected people at time t, St: the number of symptomatic cases at time t, Ct: the cumulative con rmed cases at time t,
Qt: the number of infected people who are in quarantine but yet to be con rmed at time t,
Ht: the number of infected people who are neither in quarantine nor hospitalized at time t,
see Figure 2.1 for the relations between dierent quantities. We then denote
∆It:= It− It−1,
∆Ct:= Ct− Ct−1,
∆Qt:= Qt− Qt−1.
Infected ( It )
Infected and confirmed ( Ct )
Infected but not confirmed
Quarantined ( Qt )
Not quarantined ( Ht )
Figure 2.1. Relations of the quantities Based on the observation in [LGW+20], the following relation is proposed:
(2.3) St =
14
X
i=1
pi∆It−i, where pi = p(i).
Our model contains several key parameters:
βt: the infection rate,
γt : the con rmation rate for symptomatic cases, δt: the con rmation rate for asymptomatic cases, ℓt : the isolation rate.
We expect that δt≪ γtsince, for asymptomatic cases, their Ct values (cycle threshold values) in RT-PCR tests are still in the range of being categorized as negative cases (see Figure 3.5.) We assume that the infection chain begins at time t = 16. In this paper, we propose the following nonlocal linear COVID-19 epidemic model:
(2.4)
∆It = βtHt−1 ,for all integers t ≥ 16,
∆Ct= γtSt+ δtQt−1 ,for all integers t ≥ 16,
∆Qt= ℓt∆It− ∆Ct ,for all integers t ≥ 16, Ht= It− Ct− Qt ,for all integers t ≥ 16.
We would like to give further explanations of the model (2.4). The rst equation indicates that the increase of newly infected cases at t is equal to the infection rate βt times the hidden cases Ht−1on the previous day t−1. The parameter βtand variables It, Htare unknown. The infection rate βtdepends on several factors such as face masks requirement, social distancing, and personal hygiene, etc. The second equation explains the increase of con rmed cases at t. It comes from two parts. The rst part is the number of symptomatic cases that have positive tests (RT-PCR test). This part is determined by the con rmation rate γt and the number of symptomatic cases St. The second part is the number of con rmed positive cases that come from the asymptomatic people who are in quarantine at day t − 1.
The third equation of (2.4) indicates the change of people in quarantine at time t. This quantity is equal to the isolation rate ℓttimes the increase of infected cases at time t subtracts the con rmed cases at time t. The isolation rate ℓt depends on the government's policy in response to the severity of the pandemic, for example, the level of epidemic alert. Finally, the last equation de nes the number of hidden cases.
3. Interpolation and model fitting
Sine all parameters and initial conditions of variables in (2.4) are not known, it is impossible to solve the equation in practice. The only data we know is the cumulative con rmed cases (or the daily con rmed cases). We arrange the time series data ˆC = { ˆCt}15+dt=16 of con rmed cases, where d > 1 is arbitrary depending on dierent scenario. Here we choose d = 85. To
t the model, we use the data (number of daily con rmed cases) from April 22, 2021 to July 15, 2021, which correspond to the indices t = 16 and t = 15 + 85 = 100, respectively. We remark that the choice of the starting index t = 16 is due to (2.3). In other words, the rst signi cant data is the number of con rmed cases on April 22, 2021, i.e., ˆC16. Therefore, it is reasonable to assume that
Ct= Qt= 0, for all integers 1 ≤ t ≤ 15,
before the outbreak of the pandemic. Note that the time series of I, Q, H, β, γ, δ, ℓ are all unknown. Our strategy is to interpolate β, γ, δ, ℓ by minimizing the following objective quantity:
(3.1) ∥C − ˆC∥2ℓ2 =
15+d
X
t=16
|Ct− ˆCt|2, d = 85.
We will minimize (3.1) by implementing the Matlab® function fminsearch, which locates the minimum of a unconstrained multi-variable function using derivative-free method. Since the parameters β, γ, δ, ℓ are non-negative, we have to use the function wisely3. In the work, we use the ow chart as described in Figure 3.1. The interpolation results are shown in Figure 3.2, Figure 3.3, Figure 3.4, and Figure 3.5.
3https://puzhaokow1993.github.io/homepage/Publications/upload/COVID19_Taiwan_Program/
main_program.html
Initial
guesses Optimization
block Parameters (input)
Optimization
block Parameters (output)
Large quantity
Small quantity Replacing Parameters (input)
by Parameters (output)
Optimization block
END
MATLAB®
fminsearch
Replacing all negative values by 0 output
Computing objective
quantity
Figure 3.1. Flow chart of minimizing the objective quantity (3.1).
To obtain better interpolation results, it is recommended to repeat the optimization block in Figure3.1. On the other hand, from our experience, it is better to choose the default values for fminsearch, while not to set larger values on MaxFunEvals orMaxIter inoptimset.
May 18 Jun 01 Jun 15 Jun 29 Jul 13 2021 0
5000 10000 15000
number of people
Fitting curve 250 Fitting curve 150 Fitting curve 50 Fitting curve data
Figure 3.2. The tting curve of cumulative con rmed cases based on the available data from April 22, 2021 to July 15, 2021. Here we only plot the results from May 18 to July 15 which will be used in the prediction later. We want to point out that the tting results before May 18 is rather noisy.
May 18 Jun 01 Jun 15 Jun 29 Jul 13
2021 0
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
absolute error (number of people)
Figure 3.3. This gure shows the approximation error |Ct − ˆCt| at time t,
where Ct is a minimizer of (3.1) obtained from solving the mathematical model (2.4).
May 18 Jun 01 Jun 15 Jun 29 Jul 13 2021 0
1 2 3 4 5 6
Figure 3.4. The interpolation results of parameters βt and ℓt. The forms of the parameters are not assumed a priori. They are determined from the optimization algorithm.
May 18 Jun 01 Jun 15 Jun 29 Jul 13
2021 0
0.2 0.4 0.6 0.8 1 1.2
Figure 3.5. The interpolation results of two screen rates since, for asymp- tomatic cases, their Ct values (cycle threshold values) in RT-PCR tests are still in the range of being categorized as negative cases.
4. Prediction of pandemic
In this section, we will discuss the main theme of this work forecasting of the pandemic.
We want to use the data in npast past days to predict the con rmed cases in the nfuturefuture
days. Intuitively, we assume that the following mapping is satis ed
(4.1) Ytrain = f (Xtrain),
where f is a nonlinear function, and
Ytrain=
Ct+1
...
Ct+nfuture
,
Xtrain=
Ct
...
Ct−npast+1
,
βt
...
βt−npast+1
,
γt
...
γt−npast+1
,
δt
...
δt−npast+1
,
ℓt
...
ℓt−npast+1
. We usually refer the array Xtrain as the input time series, and Ytrainas the target time series.
It is impossible to determine the nonlinear function f. Therefore, in this work, we aim to approximate f using a machine-learning algorithm, called the long short-term neural network (LSTM). Here we use theTensorFlow Keras API module[AAB+15] inPython 3language, via the Scienti c Python Development Environment (Spyder), to execute the machine learning algorithm4. We will sketch the idea of LSTM in next section. It is interesting to compare our model (4.1) with the model:
(4.2) Ytrain= g( ˜Xtrain),
where g is a nonlinear function and
X˜train=
Ct
...
Ct−npast+1
,
In other words, (4.2) predicts the trend of the pandemic using only the data of con rmed cases in previous npast days without referring to the model (2.4). It can be seen as an unsupervised learning, while our model (4.1) is a type of supervised learning. The comparison is shown in Figure 4.1 below.
Figure 4.1. Comparison of models (4.1) and (4.2) using LSTM algorithm.
The result in Figure 4.1 suggests that the prediction model (4.1) works better than the model (4.2). It is noted that the LSTM algorithm is probabilistic. Any prediction curve is only one realization of the process. Therefore, we need to re-execute the program for several
4https://puzhaokow1993.github.io/homepage/Publications/upload/COVID19_Taiwan_Program/
prediction_COVID19_export.pdf
Figure 4.1 shows the superiority of our model (4.1). Finally, we perform the prediction of the COVID-19 epidemic future trend in Taiwan from July 16 to July 27 based on our model (4.1). The prediction result is shown in Figure 4.2 and Figure 4.3. The relative error in Figure 4.3 is de ned by
|Ct− ˆCt| Cˆt
× 100%,
where Ct and ˆCt represent the predicted cumulative con rmed cases (prediction) and the actual cumulative con rmed cases (validation), respectively.
From our prediction, we are quite con dent of claiming that the recent outbreak of the COVID-19 pandemic in Taiwan has been brought under control and the lifting of the epidemic alert level (from Level 3 to Level 2) on July 27, 2021, is very likely. Another promising development is that people are eagerly getting vaccinated. Our lives have been disrupted recently, but we can see the light at the end of the tunnel.
Figure 4.2. Prediction of the daily COVID-19 epidemic trend in Taiwan from July 16 to July 27, 2021.
Figure 4.3. This gure shows the relative prediction error at time t.
5. Description of the long short-term neural network (LSTM)
In this section, we want to brie y describe the LSTM. Let {xt}t∈N, where xt = (x(1)t , · · · , x(n)t ) ∈ Rn, be a time series. We assign the cell of state (ct, ht)at each time t, where ct ∈ Rn repre- sents the memory in t-cell, and ht ∈ Rn represents the hidden state of t-cell. In LSTM, t-cell learns from the following two sources of information:
• xt (the data at time t),
• ht−1 (hidden state of (t − 1)-cell),
and we denote ˜ct = ˜ct(xt, ht−1) the input vector. Moreover, the memory is preserved from the previous time step. Therefore, we write
(5.1) ct= F(t)(ct−1) + I(t)(˜ct) for all time t.
In standard LSTM, we assume that
F(t)(ct−1) = ft◦ ct−1, where ft ∈ Rn, (5.2a)
I(t)(˜ct) = it◦ ˜ct, where it ∈ Rn, (5.2b)
c˜t= σc(y)|y=Wcxt+Ucht+bc, (5.2c)
where ◦ denotes the Hadamard product, that is, the elementwise product. Here, σc: R → R is an activation function, and usually the hyperbolic tangent function is chosen. Hence, (5.1) becomes
(5.3) ct= ft◦ ct−1+ it◦ ˜ct for all time t,
where ˜ct is computed in (5.2c). In the machine-learning terminology, the sequence {ft}t∈N are called the forget gates, and {it}t∈N are called the input gates.
The hidden state ht at time t depends only on ct. Since ht will serve as an input in (t + 1)-cell, hence it can be regard as the output of the t-cell. In LSTM, it is usually to de ne (5.4) ht= ot◦ σh(ct), where ot∈ Rn,
where σh : R → R is an activation function, and normally the hyperbolic tangent function is used. The sequence {ot}t∈N are called the output gates. We now provide further explanations of the gates. Assume that the vectors ft, it, otall depend on xt and ht−1 only. In LSTM, we usually choose
ft= σg(y)|y=Wfxt+Ufht+bf, it= σg(y)|y=Wixt+Uiht+bi, ot= σg(y)|y=Woxt+Uoht+bo,
where σg : R → R is another activation function. In most applications, one takes σg a Sigmoid function. A owchart description of LSTM is given in Figure 5.1 below.
ct − 1 (memory) ct (memory) ct + 1 (memory)
Input ̃ct
[depending on training data xt as well as the hidden state ht − 1 in the memory cell #(t − 1)]
Input ̃ct + 1
[depending on training data xt + 1 as well as the hidden state ht in the memory cell #t ]
Input gate Input gate
Forget gate Forget gate
ht (hidden state)
ht − 1 (hidden state) ht + 1 (hidden state)
Output gate Output gate Output gate
Input ̃ct − 1
[depending on training data xt as well as the hidden state ht − 2 in the memory cell #(t − 2)]
Input gate
Figure 5.1. Long short-term memory neural network
Data availability
In this paper, we use the data of COVID-19 daily con rmed cases available from Taiwan CDC via Commonwealth Magazine.
Conflict of Interest The authors declare that they have no con icts of interest.
References
[AAB+15] Martín Abadi, Ashish Agarwal, Paul Barham, Eugene Brevdo, Zhifeng Chen, Craig Citro, Greg S.
Corrado, Andy Davis, Jerey Dean, Matthieu Devin, Sanjay Ghemawat, Ian Goodfellow, Andrew Harp, Georey Irving, Michael Isard, Yangqing Jia, Rafal Jozefowicz, Lukasz Kaiser, Manjunath Kudlur, Josh Levenberg, Dandelion Mané, Rajat Monga, Sherry Moore, Derek Murray, Chris Olah, Mike Schuster, Jonathon Shlens, Benoit Steiner, Ilya Sutskever, Kunal Talwar, Paul Tucker, Vincent Vanhoucke, Vijay Vasudevan, Fernanda Viégas, Oriol Vinyals, Pete Warden, Martin Wattenberg, Martin Wicke, Yuan Yu, and Xiaoqiang Zheng, TensorFlow: Large-scale machine learning on heterogeneous systems, 2015, Software available fromtensor ow.org.
[CCJL20a] Y. Chen, J. Cheng, Y. Jiang, and K. Liu, A time delay dynamic system with ex- ternal source for the local outbreak of 2019-ncov, Applicable Analysis (2020), 12 pp., doi:10.1080/00036811.2020.1732357,arXiv:2002.02590.
[CCJL20b] , A time delay dynamical model for outbreak of 2019-nCoV and the parameter identi ca- tion, Journal of Inverse and Ill-posed Problems 28 (2020), no. 2, 243250,doi:10.1515/jiip-2020- 0010,arXiv:2002.00418.
[LGW+20] Q. Li, X. Guan, P. Wu, Wang X., L. Zhou, Y. Tong, R. Ren, Kathy S.M. Leung, Eric H.Y. Lau, Jessica Y. Wong, X. Xing, N. Xiang, Y. Wu, C. Li, Q. Chen, D. Li, T. Liu, J. Zhao, M. Liu, W. Tu, C. Chen, L. Jin, R. Yang, Q. Wang, S. Zhou, R. Wang, H. Liu, Y. Luo, Y. Liu, G. Shao, H. Li, Z. Tao, Y. Yang, Z. Deng, B. Liu, Z. Ma, Y. Zhang, G. Shi, Tommy T.Y. Lam, Joseph T.
Wu, George F. Gao, Benjamin J. Cowling, B. Yang, Gabriel M. Leung, and Z. Feng, Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus-Infected Pneumonia, N Engl J Med 382 (2020), 11991207,doi:10.1056/nejmoa2001316.
[LJYC20] K. Liu, Y. Jiang, Y. Yan, and W. Chen, A time delay dynamic model with external source and the estimation of reproductive number for the outbreak of novel coronavirus pneumonia (in Chinese)
「局部新冠肺炎时滞模型及再生数的计算」, Control Theory and Applications 37 (2020), no. 3, 453460,doi:10.7641/CTA.2020.00068.
[LSCC20] X. Luo, N. Shao, J. Cheng, and W. Chen, Modeling the trend of outbreak of COVID-19 in the diamond princcess cruise ship based on a time-delay dynamic system (in Chinese) 「基于时滞动 力学模型对钻石公主号邮轮疫情的分析」,Chinese Math. Modeling Appl. 「数学建模及其应用 杂志社」1 (2020), 1522,link.
[PSY+20a] H. Pan, N. Shao, Y. Yan, X. Luo, A. Ahmadi, Y. Fadaei, J. Cheng, and W. Chen, Multi-chain Fudan-CCDC model for COVID-19 in Iran,medRxiv:10.1101/2020.04.22.20075630.
[PSY+20b] H. Pan, N. Shao, Y. Yan, X. Luo, S. Wang, L. Ye, J. Cheng, and W. Chen, Multi-chain Fudan- CCDC model for COVID-19 - a revisit to Singapore's case, Quant. Biol. 8 (2020), no. 4, 325335, doi:10.1007/s40484-020-0224-3,medRxiv:10.1101/2020.04.13.20063792.
[SCC20] N. Shao, J. Cheng, and W. Chen, The reproductive number r0 of COVID-19 based on estimate of a statistical time delay dynamical system,medRxiv:10.1101/2020.02.17.20023747.
[SCCC20] N. Shao, Y. Chen, J. Cheng, and W. Chen, Some novel statistical time delay dynamic model by statistics data from CCDC on novel coronavirus pneumonia (in Chinese) 「关于新型冠状病毒 肺炎一类基于CCDC统统计数据的随机时滞动力学模型」, Control Theory and Applications 37 (2020), no. 4, 697704,doi:10.7641/CTA.2020.00069,link.
[SZCC20] N. Shao, M. Zhong, J. Cheng, and W. Chen, Modeling for COVID-19 and the prediction of the number of the infected based on Fudan-CCDC (in Chinese) 「基于FUDAN-CCDC模型对新冠肺 炎的建模和确诊人数的预测」, Chinese Math. Modeling Appl. 「数学建模及其应用杂志社」1 (2020), 2932,link.
doi:10.1002/mma.6345.
[XCY+21] S. Xiao, G. Cheng, R. Yang, Y. Zhang, Y. Lin, and Y. Ding, Prediction on the number of con rmed Covid-19 with the FUDAN-CCDC mathematical model and its epidemiology, clinical manifestations, and prevetion and tretment eects, Results in Physics 20 (2021), 103618, 8 pp., doi:10.1016/j.rinp.2020.103618.
[YCL+20] Y. Yan, Y. Chen, K. Liu, X. Luo, B. Xu, Y. Jiang, and J. Cheng, Modeling and prediction for the trend of outbreak of NCP based on a time-delay dynamic system (in Chinese) 「基于一类时 滞动力学系统对新型冠状病毒肺炎疫情的建模和预测」, Sci. Sin. Mathematica 50 (2020), no. 3, 385392,doi:10.1360/SSM-2020-0026.
Department of Mathematics, National Taiwan University, Taipei 106, Taiwan.
Email address: [email protected]
Institute of Applied Mathematical Sciences, National Taiwan University, Taipei 106, Tai- wan.Email address: [email protected]
