MODELS, ESTIMATION, SIMULATION, AND PREDICTION

Let ${Y_t}$ denotes the time series of the number of confirmed cases by the time t or cumulative confirmed cases at t (Figure 1A). Our objective is to model ${X_t} = {{{Y_{t \,+ \,1}}} \over {{Y_t}}} - 1$ , say the daily relative increment (RI) (Figure 1B). The model that we are going to apply to represent the time series of daily RIs has 5 positive parameters $\left( {b{\hbox,\, }IR{\hbox,\, }K{\hbox,\, }\theta {\hbox,\, }a} \right)$ :

$$\eqalign{ {\rm{For}}\,t = 1,...,b\,\,\,\,\,\,\,\,\,\,{X_t}\~Normal\left( {IR,{\raise0.7ex\hbox{${2*I{R^2}}$} \!\mathord{\left/ {\vphantom {{2*I{R^2}} a}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$a$}}} \right), \cr {\rm{and}},\,{\rm{for }}t = b + 1,b + 2,...\,\,\,\,\,\,\,\,\,{X_t}\~Normal\left( {{\raise0.7ex\hbox{$K$} \!\mathord{\left/ {\vphantom {K {{t^\theta }}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${{t^\theta }}$}},{\raise0.7ex\hbox{${{K^2}}$} \!\mathord{\left/ {\vphantom {{{K^2}} {a{t^{2\theta }}}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${a{t^{2\theta }}}$}}} \right), \cr} $$

where b is the number of days (at the beginning of the spread of the disease) that the daily RI is extreme.^{Reference Jamshidi, Rezaei, Bekrizadeh and Zargaran9} The rationale behind these 2 different patterns is that, after a while, the results of the measurements taken by governments and people to fight against the diseases appear. Figure 1B shows that the time series model is suitable to represent the considered data. IR represents the average of the RI in the first period of the spread of the disease when the time series is stationary. After b days, the daily RI starts falling regarding the model ${X_t} \propto {t^{ - \theta }}for \ t = b + 1{\hbox,\, }b + 2{\hbox,\, } \ldots $ . Therefore, $\theta $ determines the acceleration of the falling of the RIs after the first days of spreading. Also, a indicates the fixed ratio of the square of the mean to the variance $\bigg(\sqrt a = {{Expectation} \over {\vskip 1ptStandard \ deviation}}\bigg)$ , and, finally, K is the adjusting coefficient for the curve ${t^{ - \theta }}$ to fit the time series of the RI after the first period of the spread.^{Reference Jamshidi, Rezaei, Bekrizadeh and Zargaran9}

In all plots, the x-axis represents the days to April 14, 2020 (daily).

FIGURE 1 Information about the propagation of COVID-19 in the UK by April 14, 2020. A, Cumulative number of confirmed cases infected by COVID-19 from February 25 to April 14, 2020. B, Time series of RIs and the time of passing from the first stationary period to the new decreasing period (the red horizontal line represents the geometric mean of the RIs that is approximately equal to 0.3272, and the perpendicular line shows the border of the periods (b)). C, Number of deaths from COVID-19 in the UK from 03.03.2020 to 14.04.2020. D, CFR of COVID-19 in the UK from March 11, 2020 to April 14, 2020.

Notably, in the first formula, multiplying by 2 is done due to the higher fluctuation in the first period in comparison with the next days when the time series is smoother. There is another reason for justifying multiplying by 2; in the first days, the behavior of the time series of RI is more chaotic because the numbers are relatively small, so the changes in the ratios are more observable.

To estimate the parameters of the model, we

1. 1. Take b as the first point that the geometric mean of the RIs in the previous points exceeds $3\,{\hbox /}\,2$ times the geometric mean of the next three points.

   $${\eqalign{{\widehat b} = \, {\rm{min}}\,\{ n|Geo{\hbox. }mean\left( {{X_{n \,+ \,1}}{\hbox,\, }{X_{n \,+ \,2}}{\hbox,\, }{X_{n \,+\, 3}}} \right) \cr \lt {2 \over 3}\,Geo{\hbox. }mean\, \left( {{X_1}{\hbox,\, }{X_2}{\hbox,\, } \ldots {\hbox,\, }{X_n}} \right)\} }}$$
   or,
   $${\eqalign{{\widehat b} = \, {\rm{min}\,}\{ n|\root 3 \of {\left( {1 + {X_{n \,+\, 1}})\left( {1 + {X_{n \,+ \,2}}} \right)(1 + {X_{n \,+ \,3}}} \right)} - 1 \cr \lt {2 \over 3}\root n \of {\left( {1 + {X_1})\left( {1 + {X_2}} \right) \ldots (1 + {X_n}} \right)} - 1\} .}}$$

2. Graphically, this turning time usually can be identified as the time when the plot of RIs falls irreversibly (Figure 1B). According to Figure 1B, $\widehat b = 13$ , and the perpendicular red line depicts this turning time.

3. 2. Calculate the geometric mean of the RIs from $t = 1$ to $t = b$ as the estimation of the parameter IR:

   $${\widehat {IR}} = \root {\hskip 2pt\vskip -2pt\scale 50%\widehat}} {\hskip -1pt b}} \of {\left( {1 + {X_1})\left( {1 + {X_2}} \right) \ldots (1 + {X_{{\hskip 2pt\vskip -3pt\scale 60%\widehat}} {\hskip -1pt b}}} \right)} - 1{\hbox.\, }$$

   The horizontal red line of Figure 1B represents the estimation of the parameter IR. To some extent, the shift from stage 1 to stage 2 can be seen as a shock because it affects both the mean and the variance.

4. 3. Obtain the estimation of the parameters $\theta $ and K by using the following linear relation:

   $$\eqalign{{X_t} \cong {\raise0.7ex\hbox{$K$} \!\mathord{\left/ {\vphantom {K {{t^\theta }\,}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{t^{\theta} }\,}$}} \Rightarrow {\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {{\chi _t}}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{X_t}}$}} \cong {\raise0.7ex\hbox{${{t^{\theta} }}$} \!\mathord{\left/ {\vphantom {{{t^\theta }} K}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$K$}} \Rightarrow \ln \left( {{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 {{\chi _t}}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{X _t}}$}}} \right) \cong \theta \ln (t) \minus \ln (K) \cr \Rightarrow \ln \left( {{X_t}} \right) \cong - \theta \,{\rm{In}}\left( t \right) + {\rm{ln}}\left( K \right) \cr \Rightarrow \ln \left( {{X_t}} \right) \cong \theta \ln \left( {{1 \over t}} \right) + {\rm{ln}}\left( K \right).{\mkern 1mu} \cr} $$

   Due to the classification of the periods during the calculation of the estimation of b, it is concluded that ${\raise0.7ex\hbox{$K$} \!\mathord{\left/ {\vphantom {K {{t^\theta }}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{t^{\theta} }}$}} < IR$ for all $t = b + 1{\hbox,\, }b + 2{\hbox,\, } \ldots $ .

5. Multiply all the observations after $t = b$ by ${{{t^{\theta} }} \over K}$ to have an identical mean and variance for all of the newly obtained data ( ${W_t}$ ):

   $${X_t}\sim Normal\left( {{\raise0.7ex\hbox{$K$} \!\mathord{\left/ {\vphantom {K {{t^\theta }}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{t^{\theta} }}$}}{\hbox,\, }{\raise0.7ex\hbox{${{K^2}}$} \!\mathord{\left/ {\vphantom {{{K^2}} {a{t^{2\theta }}}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${a{t^{2\theta }}}$}}} \right) \Rightarrow {W_t} = {{{t^{\theta} }} \over K}{X_t}\sim Normal\left( {1{\hbox,\, }{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 a}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$a$}}} \right)$$

   Therefore, the variance of the newly obtained data is a good candidate for estimating ${\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 a}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{$a$}}$ . So $\widehat a = {1 \over {{S_W}^2}}$ [9].

Accordingly, we get the following estimations to simulate the time series of daily RIs from February 25 to April 14, 2020, by the model:

$$\left( {\widehat b{\hbox,\ }\widehat {IR}{\hbox,\ }\widehat K{\hbox,\ }\widehat {\theta} {\hbox,\ }\widehat a} \right) = \left( {13{\hbox,\, }0{\hbox.}3272{\hbox,\, }88{\hbox.}2050{\hbox,\, }1{\hbox.}9283{\hbox,\, }0{\hbox.}3382} \right)$$

Now, we are going to model the ratio of the deaths (Figure 1C) to confirmed cases (Figure 1A), which is called the case fatality rate or case fatality ratio (CFR):

$$\eqalign{ CF{R_t} = {{The\;cumulative\;number\;of\;death\;in\;the\;day\;i} \over {The\;cumulative\;number\;of\;confirmed\;cases\;in\;day\;i - 5}} \cr {\rm{ (Figure 1D)}} \cr} $$

Modeling the time series of CFRs is conducted by the adjusted form of the second part of the above model. As the CFR follows a falling trend, and the data

$$\left( {\widehat K{\hbox,\ }\widehat {\theta} {\hbox,\ }\widehat a} \right) = \left( {0{\hbox. }5430{\hbox,\, }0{\hbox. }2427{\hbox,\, }0{\hbox. }0869} \right)$$

regarding COVID-19 worldwide has shown a short-term stabilization point around 7% for the mean of CFR, we use the average of short-term average and the decreasing model:

$$\matrix{ {{\rm{for = 1{\hbox,\, }2{\hbox,\, }}} \ldots } {{Z_t}\sim mean\left( {0{\hbox.}07{\hbox,\ }Normal\left( {{\raise0.7ex\hbox{$K$} \!\mathord{\left/ {\vphantom {K {{t^\theta }}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${{t^{\theta} }}$}}{\hbox,\, }{\raise0.7ex\hbox{${{K^2}}$} \!\mathord{\left/ {\vphantom {{{K^2}} {a{t^{2\theta }}}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${a{t^{2\theta }}}$}}} \right)} \right)} \cr } \!{\hbox.\, }$$

Calculating the distribution, mean, and variance of ${Z_t}$ , it is concluded that

$${Z_t}\sim Normal\left( {{\raise0.7ex\hbox{$K$} \!\mathord{\left/ {\vphantom {K {2{t^\theta }}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${2{t^{\theta} }}$}} + 0{\hbox.}035{\hbox,\, }{\raise0.7ex\hbox{${{K^2}}$} \!\mathord{\left/ {\vphantom {{{K^2}} {4a{t^{2\theta }}}}}\right.\kern-\nulldelimiterspace}\!\lower0.7ex\hbox{${4a{t^{2\theta }}}$}}} \right){\hbox.\, }$$

Therefore, we apply a similar method to find the estimate of the parameters. Accordingly, we obtain the following estimates for the model to represent the CFRs from March 13 to April 14.

All the simulations and calculations in the present work have been done by means of Matlab R2015b, and the average curves and confidence intervals are calculated based on 1000 simulations of the fitted models.

According to Figure 1B, the time series of RIs had experienced its extreme values during 13 days after February 25, while it had been fluctuating around the line introducing the geometric mean of the RIs. Thereafter, the time series had a slight downward trend and almost experienced fewer number than 0.33 in the second period: from the 13th day until April 14.

Figure 2A illustrates how the obtained model fit the real daily RIs of confirmed cases infected by COVID-19 from March 9 to April 14. It is observable that the trend of the real data is similar to the pattern of the bounds and average curves. The model predicts that RI is decreasing from around 4.5% to about 1.5%, and its 80% confidence interval on 30.05.2020 is 0.48%–2.34% (Figure 2B).

FIGURE 2 Fitting RIs by the model. A, The real RIs of the number of confirmed cases, the 80% confidence interval, and the average curve, 09.03.2020–14.04.2020. B, The predicted 80% confidence interval, the average curve, and 2 realizations of the RIs based on the model from April 15, 2020 to May 30, 2020.

Despite the increasing scheme of the cumulative number of confirmed cases, the trend of daily RI is downward, and the count of new confirmed cases follows a falling pattern. In the first half of April, the count had been fluctuating around 4500, while based on the model, it is predicted that the number of new cases decreases to around 3100 as the point estimation and 1330–6450 with the probability of 80% on May 31, 2020 (Figure 3A). Based on the obtained results, the cumulative number of cases infected by COVID-19 starts rising from 93 873 on April 14 to 282 000 cases on May 30, 2020, with 80% confidence interval equal to 242 000–316 000 (Figure 3B).

FIGURE 3 The 80% confidence interval, the average curve, and 2 realizations of the number of A, new daily and B, total confirmed cases based on the model from April 15, 2020 to May 30, 2020.

According to the model representing the data concerning the CFR, the trend of the CFR in the next 46 days will be decreasing by a mild acceleration. Our forecast establishes that the CFR is falling around 11% by May 30. The 80% confidence interval for this index was obtained at 8–15% (Figure 4A). Therefore, we expect that almost 11% of 282 000 confirmed cases will die from the pandemic in the UK by the end of May 2020. This means that the UK will see around 31 000 deaths up to May 30. Figure 4B promotes this raw forecast. According to Figure 4B, the UK will lose around 23 000 people dying from COVID-19 in May and the second half of April 2020. Accordingly, the interval of 28 000 to 50 000 deaths is obtained for the total death up to the end of May 2020.

FIGURE 4 A, The 80% confidence interval, the average curve, and 2 realizations of the CFR; B, cumulative number of deaths based on the model from April 15, 2020, to May 30, 2020.