
Citation: Fulin Dang, Chunxue Wu, Yan Wu, Rui Li, Sheng Zhang, Huang Jiaying, Zhigang Liu. Cost-based multi-parameter logistics routing path optimization algorithm[J]. Mathematical Biosciences and Engineering, 2019, 16(6): 6975-6989. doi: 10.3934/mbe.2019350
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Since the onset of the Covid-19 pandemic in early 2020 an enormous research effort has been underway on the modelling and prediction of various aspects of the pandemic. For accessible reviews concentrating on general features of this modelling, see, for example, Vespignani et al. (2020), Poletto et al. (2020) and Gnanvi et al. (2021), while for discussion of the growth models widely used for predicting Covid-19 infections and deaths, see Tovissodé et al. (2020) and Shen (2020). Central to this modelling, the analysis of the mass of time series data accumulating daily and weekly from the coronavirus pandemic has become ever more important as the pandemic has progressed through its numerous phases. Spiegelhalter and Masters (2021) provide an accessible introduction to such data issues, paying particular attention to the evidence emerging from the U.K.
It is becoming increasingly apparent that econometric techniques are particularly suited to analysing this data: see, for example, Li and Linton (2020), Manski and Molinari (2020) and the review by Dolton (2021). Much of the research using these techniques has focused on short-term forecasting of cases, hospital admissions and deaths, with notable examples being Doornik et al. (2020), Doornik et al. (2021) and Harvey et al. (2021). It has also been directed at generalising, to stochastic settings, compartmental epidemiological models, such as the well-known "susceptible (S), infected (I) and recovered or deceased (R)", or SIR, model, as in Korolev (2020) and Pesaran and Yang (2021).
The focus of the present paper is rather different, however, in that we investigate the changing dynamic relationship between infections, hospital admissions and deaths using daily data from England. Section 2 thus considers the relationship between hospital admissions and subsequent deaths and introduces two models that might be useful for this task: the recently proposed balanced growth model of Harvey (2020) and the more familiar autoregressive distributed lag/error correction model used widely to analyse economic time series (see, for example, Banerjee et al., 1993, for detailed development and Mills, 2019, chapters 12 and 14, for a more introductory treatment). Section 3 extends the analysis to examining the prior relationship between infections and hospital admissions, while Section 4 links the two sets of models together before discussing the advantages and disadvantages of the two modelling procedures.
Figure 1 shows daily hospital admissions and deaths in England between 19th March 2020 and 31st October 2021.1 Both admissions and deaths show pronounced multiple wave patterns with admissions obviously leading deaths, but the shifting nature of the relationship between the two series is clearly discernible. How might this evolving and dynamic relationship be modelled? Attention is focused in this paper on two approaches: balanced growth modelling and the use of autoregressive distributed lags.
1The focus here is on data from England as U.K.—wide hospital admissions rely on different definitions across the home nations.
Let daily deaths due to Covid-19 in England be denoted yt, t=1,2,⋯,T, with their cumulation being Yt=∑tj=1yj, so that the growth rate of daily deaths is gy,t=yt/Yt−1. Similarly, denote daily hospital admissions due to Covid-19 by xt, their cumulation by Xt=∑tj=1xj, and their growth rate by gx,t=xt/Xt−1.
Following Harvey and Kattuman (2020), we initially assume that there is balanced growth between daily deaths and hospital admissions lagged k days, which implies the regression model.
log(gy,t)=δ+log(gx,t−k)+εtt=k+1,⋯,T | (1) |
where εt is an error term assumed to independently and identically distributed through time with zero mean and variance σ2ε, which is denoted εt∼ IID(0,σ2ε). The "equilibrium" relationship between the two growth rates is given by
gy,t=exp(δ)gx,t−k | (2) |
Between daily deaths and admissions, yt and xt, the equilibrium is then
yt=exp(δ)(Yt−1/Xt−k−1)xt−k | (3) |
Allowing for a lag structure in the leading admissions series in (1) gives
log(gy,t)=δ+∑kj=hβjlog(gx,t−j)+εt | (4) |
where h<k and ∑βj=1, a restriction that may be imposed by rewriting (4) as
log(gy,t)−log(gx,t−k)=δ+∑k−1j=hβj(log(gx,t−j)−log(gx,t−k))+εt | (5) |
so that βk=1−∑k−1j=hβj, a restriction that ensures that there is indeed balanced growth. The corresponding equilibrium relationship is then
gy,t=exp(δ)∏kj=hgβjx,t−j=exp(δ)−gx,t−k | (6) |
where ˉgx,t−k is the weighted geometric mean of gx,t−h,⋯,gx,t−k. The levels equilibrium is thus
yt=exp(δ)(Yt−1/ˉXt−k−1)ˉxt−k=Δˉxt−k | (7) |
where ˉxt−k and ˉXt−k−1 are the corresponding weighted geometric means of xt−h,⋯,xt−k and Xt−1−h,⋯,Xt−1−k, respectively. Thus Δ measures the long-run response of deaths to an increase in hospital admissions: if daily admissions increase by 100 then deaths will increase by 100Δ after k days.
When the two series are not on the same growth path, the model can be extended by replacing the intercept δ with a stochastic trend:
log(gy,t)=δt+∑kj=hβjlog(gx,t−j)+εt | (8) |
or
log(gy,t)−log(gx,t−k)=δt+∑k−1j=hβj(log(gx,t−j)−log(gx,t−k))+εt | (9) |
where δt is defined as
δt=δt−1−γt−1+ηtηt∼IID(0,σ2η) | (10) |
γt=γt−1+ζtζt∼IID(0,σ2ζ) | (11) |
i.e., δt is a random walk with a drift that is itself potentially a random walk. If σ2ζ=0 then γt=γt−1 and the drift is constant. On the other hand, if σ2η=0 then δt=2δt−1−δt−2−ζt−1 and δt will tend to evolve very smoothly, being known as an integrated random walk, or IRW. The equilibrium relationship in (6) is
gy,t=exp(δt)ˉgx,t−k | (12) |
so that the dynamic relationship between the two growth rates is given by exp(δt). In terms of daily deaths and admissions, yt and xt, we have
yt=exp(δt)(Yt−1/ˉXt−k−1)ˉxt−k=Δtˉxt−k | (13) |
Thus, an increase of 100 in hospital admissions will lead to an increase of 100Δt deaths in the following k days and this long run response will shift through time. As Harvey (2020) shows, this model may be arrived at by assuming that deaths and admissions follow Gompertz processes separated by k days, but such an assumption is not necessary.
Equation (9) may be fitted by casting it into state space form and employing the Kalman filter. Estimation is carried out by maximum likelihood using the predictive error decomposition and the estimates of δt used to compute the Δt series shown in Figures 3 and 7 below are the "smoothed" estimates, obtained by running the Kalman filter first forwards from t=k+1 to t=T and then backwards from t=T to t=k+1. Mills (2019, chapter 17) provides an introductory discussion to state space modelling and Harvey (1989) is the classic exposition. A value for the lag k must be obtained before estimation can be undertaken. This lag could be selected by considering the basic model (9) with h=0, setting k to an initial value kmax and then running the regression (9) for k=kmax,kmax−1,kmax−2,⋯, the sequence stopping when ˆβk is significant at some pre-chosen level of significance. Alternatively, clinical considerations may suggest an appropriate value for k and indeed h, and this simpler approach is also investigated. A further refinement may be to select only a subset of the lagged regressors log(gx,t),⋯,log(gx,t−k), thus determining h and including only significant lags. This may be done, for example, by using a stepwise least squares algorithm or other sequential testing procedure. The balanced growth assumption may be checked by including log(gx,t−k) (or, indeed, any other lag) in (9) and testing for its significance.
An autoregressive distributed lag (ARDL) model linking deaths and admissions may be specified in general as
yt=ϕ0+∑mi=1ϕiyt−i+∑ni=0θixt−i+ut | (14) |
where ut∼ IID(0,σ2u) is an error term. Typically, the lag lengths m and n will be unknown and must be determined from the data. An algebraically equivalent but often more convenient form of this ARDL(m,n) model, particularly for model specification and inference, is the error correction model (ECM)2
2Banerjee et al (1993) sets out and analyses in detail the algebraic equivalencies existing between the ARDL and ECM formulations. It should be emphasised that the recasting of (14) as (15) is a purely algebraic transformation and is not predicated on any particular statistical properties of the data. With integrated data, cointegration leads from an ARDL to an ECM via Granger's representation theorem (see Engle and Granger, 1987). The series here are not integrated, as may be demonstrated from standard unit root tests, and so we are in a stationary world in which the ECM (15) is simply a more convenient representation for our purposes than the ARDL (14).
∇yt=a0+∑m−1i=1ai∇yt−i+∑n−1i=0bi∇xt−i−c(yt−1−dxt−1)+ut | (15) |
where ∇ is the difference operator defined such that ∇yt=yt−yt−1 and where the coefficients of (14) and (15) are linked by the set of relationships
ϕ0=a0ϕ1=1+a1−cϕi=a−ai−1i=2,⋯,m−1ϕm=−am−1θ0=b0θ1=b1−b0+cdθi=bi−bi−1i=2,⋯,n−1θn=−bn−1 | (16) |
The error correction term is yt−1−dxt−1, which embodies the long-run, or equilibrium, relationship y=dx between deaths and admissions, with d being termed the long-run or total multiplier. An increase in hospital admissions of 100, say, will eventually increase deaths by 100d.
The speed at which the increase in deaths approaches the total multiplier depends on c, the speed of adjustment parameter. The smaller is c, the faster the speed of adjustment and the quicker the total multiplier is arrived at. The actual time path of adjustment depends upon the lag coefficients ψi in the distributed lag
yt=ψ0+∑∞i=0ψixt−i | (17) |
where
ψi=∑min(i,m)j=1ϕjψi−j+θi0≤i≤n | (18) |
ψi=∑min(i,m)j=1ϕjψi−ji>n |
This result is most easily obtained by utilising the lag operator B, defined such that Bjzt≡zt−j (note that the difference operator introduced in (15) may then be written as ∇=1−B). This allows (14), on ignoring the error term as we are only interested in the systematic dynamics here, to be written as
ϕ(B)yt=ϕ0+θ(B)xt | (19) |
where the lag polynomials ϕ(B) and θ(B) are defined as
ϕ(B)=1−ϕ1B−⋯−ϕmBm | (20) |
θ(B)=θ0+θ1B+⋯+θnBn | (21) |
Equation (19) can then be expressed as
yt=ϕ−1(B)ϕ0+ϕ−1(B)θ(B)xt=ψ0+ψ(B)xt | (22) |
where
ψ0=ϕ0/(1−ϕ1−⋯−ϕm)ψ(B)=ϕ−1(B)θ(B) | (23) |
Thus, the lag coefficients in (18) are obtained by equating coefficients of powers of B in ψ(B)ϕ(B)=θ(B). The total multiplier is then given by the sum of these lag coefficients, i.e., d=∑∞i=1ψi and the increase in deaths after l days is given by the lth interim multiplier dl=∑li=1ψi l=1,2,⋯ which will converge to d as l increases, i.e., dl→d as l→∞.
The logarithms of the growth rates of daily hospital admissions and subsequent deaths between 21st March 2020 and 31st October 2021 are shown in Figure 2. The series are evidently not on the same growth path so that a model of the form (9) rather than (5) is clearly required. Both clinical considerations and exploratory sequential testing along the lines suggested in section 2.1 above suggest setting h=0, thus allowing for the death of some patients on the day of their admission to hospital, and k=7, so that there is a one week delay between hospital admission and death.
Column (1) of Table 1 reports estimates of (9) with θlog(gx,t−7) included as an additional term. If θ is non-zero then balanced growth does not hold: with this term included,
β7=1−∑6j=0βj+θ | (24) |
(1) | (2) | |
β0 | 0.057(0.069)[0.408] | 0 |
β1 | −0.071(0.072)[0.323] | 0 |
β2 | 0.327(0.068)[0.000] | 0.364(0.048)[0.000] |
β3 | 0.039(0.072)[0.593] | 0 |
β4 | 0.148(0.068)[0.030] | 0.168(0.052)[0.001] |
β5 | −0.063(0.076)[0.406] | 0 |
β6 | 0.185(0.072)[0.010] | 0.201(0.061)[0.001] |
θ | −0.129(0.083)[0.118] | 0 |
ση | 0.000(3.497)[0.999] | 0 |
σζ | 0.002(0.001)[0.000] | 0.002(0.000)[0.000] |
σε | 0.212(0.004)[0.000] | 0.213(0.004)[0.000] |
Several of the coefficients are estimated to be insignificantly different from zero, including θ, so that balanced growth is confirmed. Column (2) of Table 1 reports the estimates of the model with β0,β1,β3,β5 and θ all restricted to be zero on using a 10% significance level; the remaining coefficients are all significantly positive (β7 is calculated to be 0.267(0.057)). It is also found that σ2η=0, so that δt will be a smoothly evolving IRW. Figure 3 shows the resulting estimate of Δt, the response of deaths to an increase in hospital admissions, calculated using the geometric mean
ˉXt−8=X0.364t−3X0.168t−5X0.201t−7X0.267t−8 | (25) |
and it is indeed seen to evolve smoothly. 95% confidence interval upper and lower bounds for Δt are also shown, and these indicate that Δt is indeed estimated precisely.
Δt reaches a maximum at 0.38 on 27th April 2020, just under three weeks after the peak in daily deaths. There are further local maxima of 0.28 on 5th December 2020 and 0.31 on 26th December 2020, both occurring during the second wave of deaths in late 2020/early 2021. Δt reaches a minimum of 0.06 on 9th June 2021, the previous minimum being 0.13 on 10th August 2020, and at the end of October 2021 it stood at 0.15. This suggests that, while Δt is positively related to the number of daily deaths, its magnitude has been declining relative to the number of deaths as clinical practice has improved over the course of the pandemic and the vaccination program has been rolled out. The long run response of deaths to an increase of 100 in hospital admissions has fallen from a maximum of 37 deaths (95% confidence interval (34,40)) during the first wave of the pandemic in April 2020, to around 30 (27,33) during the second wave at the turn of the year and down to just 6 (5,7) by the summer of 2021, although this had increased to 15 (13,17) by the end of October.
ARDL/ECM models were developed to characterise the relationship between daily deaths and admissions for the complete sample period to 31st October 2021 and for two sub-periods, the first ending on 31st January 2021, the second beginning on 1st February 2021, the "break point" being chosen to reflect the increasing vaccination uptake at this point. The models, estimated by nonlinear least squares, were initially selected using the AIC information criteria with insignificant coefficients (at the 10% level) then being sequentially removed to obtain a parsimonious ECM specification. Details of the chosen specifications are given in Table 2. HAC standard errors are shown in parentheses to accommodate any remaining autocorrelation and heteroskedasticity in the residuals. The most notable features of the models are the much lower estimate of the equilibrium parameter d in the second sub-period than in the first, which is accompanied by a much larger estimate of c. Note that the estimates of these parameters are all highly significant and, in terms of goodness of fit, the model for the second sub-period has much the superior performance, with a higher R2 and a lower equation standard error than the model for the first sub-period.
31st March 2020–31st October 2021 | 31st March 2020–31st January 2021 | 1st February 2021–31st October 2021 | |
c | −0.052(0.014)[0.000] | −0.056(0.020)[0.005] | −0.181(0.020)[0.000] |
d | 0.308(0.025)[0.000] | 0.328(0.035)[0.000] | 0.164(0.010)[0.000] |
a0 | −3.080(1.052)[0.004] | −2.925(1.369)[0.033] | −3.679(0.846)[0.000] |
a1 | −0.438(0.056)[0.000] | −0.374(0.055)[0.000] | −0.549(0.078)[0.000] |
a2 | −0.344(0.055)[0.000] | −0.400(0.068)[0.000] | −0.346(0.068)[0.000] |
a3 | – | – | −0.263(0.093)[0.005] |
a4 | – | – | −0.339(0.092)[0.000] |
a5 | 0.119(0.054)[0.028] | – | – |
a6 | 0.108(0.064)[0.095] | – | – |
a7 | 0.091(0.053)[0.089] | – | 0.107(0.058)[0.067] |
a8 | – | – | 0.141(0.062)[0.024] |
a9 | – | – | 0.193(0.079)[0.015] |
a10 | – | – | – |
a11 | – | – | −0.110(0.061)[0.075] |
b0 | 0.055(0.012)[0.000] | 0.056(0.013)[0.000] | 0.043(0.018)[0.015] |
b1 | 0.041(0.013)[0.002] | 0.034(0.016)[0.029] | 0.036(0.019)[0.068] |
b2 | – | – | – |
b3 | – | – | – |
b4 | – | – | 0.036(0.017)[0.041] |
b5 | 0.027(0.015)[0.069] | 0.041(0.015)[0.008] | – |
b6 | 0.030(0.014)[0.030] | 0.041(0.010)[0.000] | – |
b7 | 0.027(0.015)[0.067] | 0.042(0.013)[0.002] | – |
b8 | 0.031(0.016)[0.058] | 0.037(0.015)[0.011] | – |
b9 | 0.060(0.015)[0.000] | 0.076(0.016)[0.000] | −0.047(0.022)[0.034] |
b10 | 0.058(0.012)[0.000] | 0.062(0.013)[0.000] | – |
b11 | 0.050(0.013)[0.000] | 0.065(0.015)[0.000] | – |
R2 | 0.469 | 0.512 | 0.597 |
σu | 17.91 | 20.80 | 11.13 |
T | 580 | 307 | 273 |
m | 8 | 3 | 12 |
n | 12 | 12 | 10 |
The time paths of the interim multipliers dl for the two sub-periods are shown in Figure 4. The paths have been smoothed to ensure that they are monotonically non-declining, since the number of deaths that follow a given increase in hospital admissions cannot fall with time! With ˆd=0.328(0.035) in the first sub-period ending on 31st January 2021, an increase of 100 in hospital admissions will therefore eventually lead to a further 33 deaths (95% confidence interval (26,40)). From Figure 4 it is seen that approximately 95% of these deaths (31) occur within 14 days of admission to hospital.
In contrast, for the second sub-period beginning on 1st February 2021, ˆd=0.164(0.010), so that now an increase of 100 in hospital admissions eventually leads to just 16 further deaths (95% confidence interval (14,18)). Moreover, only 69% of these deaths (11) occur within 14 days of admission to hospital. It would thus appear that, as the vaccination programme was rolled out along with other improvements in clinical practice, so the hospital death rate was more than halved with patients being kept alive for longer.
As well as the linkage from hospital admissions to deaths, there is also the prior link from testing positive (i.e., becoming infected) for Covid-19 to admission into hospital. This has become particularly important to analyse since the roll-out of the vaccination program in the U.K: is there evidence that vaccination has "broken the link between cases and admissions", as has been stated several times by the government? To investigate whether this is indeed the case, we analyse English data on daily positive cases and hospital admissions from 1st September 2020 to 31st October 2021, as shown in Figure 5. Earlier data have been excluded both because of the limited extent of testing during the early months of the pandemic and the hiatus in cases and admissions during the summer months of 2020.
We first fit balanced growth models to the logarithms of hospital growth, gx,t, and case growth, gw,t=wt/Wt−1, where wt and Wt are daily and cumulative cases, respectively. These logarithms are shown in Figure 6. From both clinical considerations and exploratory sequential testing a lag of k=14 was chosen, along with h=1.
Estimates of the balanced growth Equation (9) are shown in Table 3 after the elimination of insignificant coefficients. Here ση>0 so that, with both σζ and γt estimated to be very small, ∇δt≈ηt, a driftless random walk. The consequent Δt series, estimated under the assumption of balanced growth (which is questionable here since θ appears to be significantly negative: ˆθ=−0.321(0.064)), is shown in Figure 7, along with 95% confidence interval upper and lower bounds. These show that Δt is estimated extremely precisely. The trend in Δt values is generally upwards until the middle of February 2021, reaching a maximum of 0.12 (95% confidence interval (11,13)) on 12th December 2020 (i.e., an additional 120 hospital admissions result from an increase of 1000 positive test cases), after which it turns down, reaching a minimum of 0.02 (0.019,0.021) on 21st July 2021 (20 additional admissions from an increase of 1000 cases), and was still around this value at the end of October. This suggests that there is indeed evidence that the rollout of the vaccination program has had an impact on the relationship between cases and subsequent hospital admissions.
value | |
β1 | 0.086(0.024)[0.000] |
β2 | 0 |
β3 | 0.108(0.018)[0.000] |
β4 | 0 |
β5 | 0.118(0.025)[0.000] |
β6 | 0.150(0.029)[0.000] |
β7 | 0.094(0.029)[0.001] |
β8 | 0.053(0.023)[0.020] |
β9 | 0.096(0.020)[0.000] |
β10 | 0 |
β11 | 0 |
β12 | 0.045(0.025)[0.073] |
β13 | 0.067(0.031)[0.031] |
β14 | 0.183(0.028)[0.000] |
ση | 0.046(0.004)[0.000] |
σζ | 0.001(0.0006)[0.061] |
σε | 0.054(0.003)[0.000] |
θ | −0.321(0.064)[0.000] |
ARDL/ECM models were fitted to the sub-periods 1st September 2020–31st January 2021 and 1st February–31st October 2021, with the resulting estimates and statistics shown in Table 4. The estimate of the long-run parameter d is 0.087 (0.006) in the early period but just 0.027 (0.001) in the later period, i.e., up to the end of January 2021 an increase of 1000 positive test cases ultimately led to 87 (95% confidence interval (75,99)) additional hospital admissions, whereas from February such an increase led to only an additional 27 (95% confidence interval (25,29)) admissions. The estimate of the speed of adjustment parameter, c, on the other hand, has remained almost the same across the two periods, just increasing from 0.103 to 0.118. Again, the fit of the second sub-period model is vastly superior to that of the first.
9th September 2020–31st October 2021 | 9th September 2020–31st January 2021 | 1st February 2021–31st October 2021 | |
c | −0.010(0.005)[0.071] | −0.103(0.020)[0.000] | −0.118(0.016)[0.000] |
d | 0.038(0.014)[0.006] | 0.087(0.006)[0.000] | 0.027(0.001)[0.000] |
a0 | − | − | − |
a1 | −0.241(0.035)[0.000] | −0.198(0.095)[0.038] | −0.488(0.049)[0.000] |
a2 | − | − | −0.236(0.064)[0.000] |
a3 | −0.178(0.066)[0.007] | –0.287(0.088)[0.001] | −0.351(0.086)[0.000] |
a4 | – | −0.121(0.068)[0.075] | −0.117(0.059)[0.050] |
a5 | − | – | – |
a6 | 0.131(0.031)[0.000] | – | − |
a7 | 0.304(0.046)[0.000] | 0.294(0.121)[0.017] | 0.213(0.061)[0.001] |
a8 | 0.265(0.024)[0.000] | 0.254(0.088)[0.005] | 0.110(0.057)[0.055] |
a9 | − | −0.230(0.081)[0.005] | − |
a10 | 0.128(0.051)[0.013] | – | – |
a11 | 0.115(0.049)[0.020] | – | −0.129(0.065)[0.050] |
a12 | −0.134(0.037)[0.000] | −0.234(0.094)[0.014] | 0.152(0.049)[0.002] |
b0 | 0.003(0.000)[0.000] | 0.005(0.001)[0.001] | 0.002(0.001)[0.001] |
b1 | − | −0.005(0.002)[0.009] | − |
b2 | – | – | – |
b3 | 0.003(0.001)[0.005] | – | –0.002(0.001)[0.077] |
b4 | – | – | −0.002(0.001)[0.003] |
b5 | 0.004(0.001)[0.000] | − | −0.002(0.001)[0.020] |
b6 | 0.005(0.001)[0.008] | − | – |
b7 | 0.004(0.001)[0.003] | − | 0.002(0.001)[0.016] |
b8 | 0.005(0.002)[0.004] | 0.003(0.001)[0.044] | – |
b9 | 0.002±0.001 | − | – |
b10 | − | − | 0.002(0.001)[0.089] |
R2 | 0.445 | 0.442 | 0.628 |
σu | 72.71 | 112.37 | 33.18 |
T | 419 | 146 | 273 |
m | 13 | 13 | 13 |
n | 10 | 9 | 11 |
The interim multipliers for admissions are plotted in Figure 8. In the period up to the end of January 2021, approximately 85% of hospital admissions occur within 28 days of a positive test. In contrast, for the second sub-period beginning on 1st February 2021, only 79% of these admissions occur within 28 days of a positive test. It would thus appear to be clear that, as the vaccination programme was rolled out, along with other improvements in medical practice, so fewer people were admitted to hospital.
Given the "causal structure" inherent in the relationship between cases, hospital admissions and deaths, the models in Sections 2 and 3 may be linked together. If the equilibrium responses from the balanced growth models are now denoted Δy,t and Δx,t, to use an obvious extension of notation, then the response of deaths to an increase in cases is given by the product Δy,t×Δx,t−7 to ensure an appropriate timing match-up, and this series is shown in Figure 9.3 This product reached a maximum of 33 deaths per thousand cases on 18th December 2020 (95% confidence interval (28,38)) and by the end of October 2021 had declined to just 3 (2,4).
3Using (7) we have yt=Δy,txt−7 and xt=Δx,twt−14, so that yt=Δy,tΔx,t−7wt−21.
Similarly, denoting dx and dy as the total multipliers from the ARDL/ECM models, with dx,l and dy,l being the accompanying interim multipliers, then the products dx×dy and dx,l×dy,l provide the total multiplier and set of interim multipliers for the response of deaths to an increase in cases, these being shown for the two sub-periods in Figure 10. The total multipliers are 28 deaths per thousand cases (95% confidence interval (18,38)) for the first sub-period and 4 (3,5) for the second, both consistent with the balanced growth estimates.
Both modelling approaches provide a consistent story: as clinical practice has evolved and the vaccination program rolled out, so there has been a substantial decline in hospital admissions and subsequent deaths for a given level of infection. Are there grounds for preferring one approach over the other? Each have their advantages and disadvantages.
Balanced growth models have the advantage of providing a time varying equilibrium response which can therefore be tracked through time. They have the disadvantages of requiring balanced growth to hold, which might not be the case, and a fixed adjustment period dependent on the setting of the lag k. Their specification is also very precise with little flexibility in the setup and, moreover, estimation requires specialised software. ARDL/ECM models, on the other hand, have greater flexibility in their specification and may be estimated by routine regression software. The adjustment to the long-rum response is freely estimated rather than fixed but the multiplier is not time varying, with evolving relationships having to be investigated through fitting the models over sub-periods of the data, which have to be selected by the investigator or by using break-point procedures and tests, such as those developed by Bai and Perron (1998). Given the limited time period available and the clear waves in the data, we have chosen not to follow this latter route here. However, as more data becomes available, and particularly given the more recent wave of the pandemic associated with the omicron variant, formal testing for break-points would be a useful extension of the ECM modelling approach. Given these competing benefits and drawbacks, it would thus seem sensible to keep an open mind and to use both models to track the behaviour of key Covid variables through time.
While this paper has focused on the relationship between infections, hospital admissions and deaths in England for the period of the pandemic up to the end of October 2021, it is clear that the models may be used for similar data from other countries and time periods. They may also be used for data on infections, admissions and deaths disaggregated into age groups and regions, if such data are available, as it is for England, where such research is ongoing.
The author declares that there are no conflicts of interest in this paper.
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1. | Laura Grassini, Statistical features and economic impact of Covid-19, 2023, 5, 2689-3010, 38, 10.3934/NAR.2023003 | |
2. | Gunnar Bårdsen, Ragnar Nymoen, Dynamic time series modelling and forecasting of COVID-19 in Norway, 2024, 01692070, 10.1016/j.ijforecast.2024.05.004 |
(1) | (2) | |
β0 | 0.057(0.069)[0.408] | 0 |
β1 | −0.071(0.072)[0.323] | 0 |
β2 | 0.327(0.068)[0.000] | 0.364(0.048)[0.000] |
β3 | 0.039(0.072)[0.593] | 0 |
β4 | 0.148(0.068)[0.030] | 0.168(0.052)[0.001] |
β5 | −0.063(0.076)[0.406] | 0 |
β6 | 0.185(0.072)[0.010] | 0.201(0.061)[0.001] |
θ | −0.129(0.083)[0.118] | 0 |
ση | 0.000(3.497)[0.999] | 0 |
σζ | 0.002(0.001)[0.000] | 0.002(0.000)[0.000] |
σε | 0.212(0.004)[0.000] | 0.213(0.004)[0.000] |
31st March 2020–31st October 2021 | 31st March 2020–31st January 2021 | 1st February 2021–31st October 2021 | |
c | −0.052(0.014)[0.000] | −0.056(0.020)[0.005] | −0.181(0.020)[0.000] |
d | 0.308(0.025)[0.000] | 0.328(0.035)[0.000] | 0.164(0.010)[0.000] |
a0 | −3.080(1.052)[0.004] | −2.925(1.369)[0.033] | −3.679(0.846)[0.000] |
a1 | −0.438(0.056)[0.000] | −0.374(0.055)[0.000] | −0.549(0.078)[0.000] |
a2 | −0.344(0.055)[0.000] | −0.400(0.068)[0.000] | −0.346(0.068)[0.000] |
a3 | – | – | −0.263(0.093)[0.005] |
a4 | – | – | −0.339(0.092)[0.000] |
a5 | 0.119(0.054)[0.028] | – | – |
a6 | 0.108(0.064)[0.095] | – | – |
a7 | 0.091(0.053)[0.089] | – | 0.107(0.058)[0.067] |
a8 | – | – | 0.141(0.062)[0.024] |
a9 | – | – | 0.193(0.079)[0.015] |
a10 | – | – | – |
a11 | – | – | −0.110(0.061)[0.075] |
b0 | 0.055(0.012)[0.000] | 0.056(0.013)[0.000] | 0.043(0.018)[0.015] |
b1 | 0.041(0.013)[0.002] | 0.034(0.016)[0.029] | 0.036(0.019)[0.068] |
b2 | – | – | – |
b3 | – | – | – |
b4 | – | – | 0.036(0.017)[0.041] |
b5 | 0.027(0.015)[0.069] | 0.041(0.015)[0.008] | – |
b6 | 0.030(0.014)[0.030] | 0.041(0.010)[0.000] | – |
b7 | 0.027(0.015)[0.067] | 0.042(0.013)[0.002] | – |
b8 | 0.031(0.016)[0.058] | 0.037(0.015)[0.011] | – |
b9 | 0.060(0.015)[0.000] | 0.076(0.016)[0.000] | −0.047(0.022)[0.034] |
b10 | 0.058(0.012)[0.000] | 0.062(0.013)[0.000] | – |
b11 | 0.050(0.013)[0.000] | 0.065(0.015)[0.000] | – |
R2 | 0.469 | 0.512 | 0.597 |
σu | 17.91 | 20.80 | 11.13 |
T | 580 | 307 | 273 |
m | 8 | 3 | 12 |
n | 12 | 12 | 10 |
value | |
β1 | 0.086(0.024)[0.000] |
β2 | 0 |
β3 | 0.108(0.018)[0.000] |
β4 | 0 |
β5 | 0.118(0.025)[0.000] |
β6 | 0.150(0.029)[0.000] |
β7 | 0.094(0.029)[0.001] |
β8 | 0.053(0.023)[0.020] |
β9 | 0.096(0.020)[0.000] |
β10 | 0 |
β11 | 0 |
β12 | 0.045(0.025)[0.073] |
β13 | 0.067(0.031)[0.031] |
β14 | 0.183(0.028)[0.000] |
ση | 0.046(0.004)[0.000] |
σζ | 0.001(0.0006)[0.061] |
σε | 0.054(0.003)[0.000] |
θ | −0.321(0.064)[0.000] |
9th September 2020–31st October 2021 | 9th September 2020–31st January 2021 | 1st February 2021–31st October 2021 | |
c | −0.010(0.005)[0.071] | −0.103(0.020)[0.000] | −0.118(0.016)[0.000] |
d | 0.038(0.014)[0.006] | 0.087(0.006)[0.000] | 0.027(0.001)[0.000] |
a0 | − | − | − |
a1 | −0.241(0.035)[0.000] | −0.198(0.095)[0.038] | −0.488(0.049)[0.000] |
a2 | − | − | −0.236(0.064)[0.000] |
a3 | −0.178(0.066)[0.007] | –0.287(0.088)[0.001] | −0.351(0.086)[0.000] |
a4 | – | −0.121(0.068)[0.075] | −0.117(0.059)[0.050] |
a5 | − | – | – |
a6 | 0.131(0.031)[0.000] | – | − |
a7 | 0.304(0.046)[0.000] | 0.294(0.121)[0.017] | 0.213(0.061)[0.001] |
a8 | 0.265(0.024)[0.000] | 0.254(0.088)[0.005] | 0.110(0.057)[0.055] |
a9 | − | −0.230(0.081)[0.005] | − |
a10 | 0.128(0.051)[0.013] | – | – |
a11 | 0.115(0.049)[0.020] | – | −0.129(0.065)[0.050] |
a12 | −0.134(0.037)[0.000] | −0.234(0.094)[0.014] | 0.152(0.049)[0.002] |
b0 | 0.003(0.000)[0.000] | 0.005(0.001)[0.001] | 0.002(0.001)[0.001] |
b1 | − | −0.005(0.002)[0.009] | − |
b2 | – | – | – |
b3 | 0.003(0.001)[0.005] | – | –0.002(0.001)[0.077] |
b4 | – | – | −0.002(0.001)[0.003] |
b5 | 0.004(0.001)[0.000] | − | −0.002(0.001)[0.020] |
b6 | 0.005(0.001)[0.008] | − | – |
b7 | 0.004(0.001)[0.003] | − | 0.002(0.001)[0.016] |
b8 | 0.005(0.002)[0.004] | 0.003(0.001)[0.044] | – |
b9 | 0.002±0.001 | − | – |
b10 | − | − | 0.002(0.001)[0.089] |
R2 | 0.445 | 0.442 | 0.628 |
σu | 72.71 | 112.37 | 33.18 |
T | 419 | 146 | 273 |
m | 13 | 13 | 13 |
n | 10 | 9 | 11 |
(1) | (2) | |
β0 | 0.057(0.069)[0.408] | 0 |
β1 | −0.071(0.072)[0.323] | 0 |
β2 | 0.327(0.068)[0.000] | 0.364(0.048)[0.000] |
β3 | 0.039(0.072)[0.593] | 0 |
β4 | 0.148(0.068)[0.030] | 0.168(0.052)[0.001] |
β5 | −0.063(0.076)[0.406] | 0 |
β6 | 0.185(0.072)[0.010] | 0.201(0.061)[0.001] |
θ | −0.129(0.083)[0.118] | 0 |
ση | 0.000(3.497)[0.999] | 0 |
σζ | 0.002(0.001)[0.000] | 0.002(0.000)[0.000] |
σε | 0.212(0.004)[0.000] | 0.213(0.004)[0.000] |
31st March 2020–31st October 2021 | 31st March 2020–31st January 2021 | 1st February 2021–31st October 2021 | |
c | −0.052(0.014)[0.000] | −0.056(0.020)[0.005] | −0.181(0.020)[0.000] |
d | 0.308(0.025)[0.000] | 0.328(0.035)[0.000] | 0.164(0.010)[0.000] |
a0 | −3.080(1.052)[0.004] | −2.925(1.369)[0.033] | −3.679(0.846)[0.000] |
a1 | −0.438(0.056)[0.000] | −0.374(0.055)[0.000] | −0.549(0.078)[0.000] |
a2 | −0.344(0.055)[0.000] | −0.400(0.068)[0.000] | −0.346(0.068)[0.000] |
a3 | – | – | −0.263(0.093)[0.005] |
a4 | – | – | −0.339(0.092)[0.000] |
a5 | 0.119(0.054)[0.028] | – | – |
a6 | 0.108(0.064)[0.095] | – | – |
a7 | 0.091(0.053)[0.089] | – | 0.107(0.058)[0.067] |
a8 | – | – | 0.141(0.062)[0.024] |
a9 | – | – | 0.193(0.079)[0.015] |
a10 | – | – | – |
a11 | – | – | −0.110(0.061)[0.075] |
b0 | 0.055(0.012)[0.000] | 0.056(0.013)[0.000] | 0.043(0.018)[0.015] |
b1 | 0.041(0.013)[0.002] | 0.034(0.016)[0.029] | 0.036(0.019)[0.068] |
b2 | – | – | – |
b3 | – | – | – |
b4 | – | – | 0.036(0.017)[0.041] |
b5 | 0.027(0.015)[0.069] | 0.041(0.015)[0.008] | – |
b6 | 0.030(0.014)[0.030] | 0.041(0.010)[0.000] | – |
b7 | 0.027(0.015)[0.067] | 0.042(0.013)[0.002] | – |
b8 | 0.031(0.016)[0.058] | 0.037(0.015)[0.011] | – |
b9 | 0.060(0.015)[0.000] | 0.076(0.016)[0.000] | −0.047(0.022)[0.034] |
b10 | 0.058(0.012)[0.000] | 0.062(0.013)[0.000] | – |
b11 | 0.050(0.013)[0.000] | 0.065(0.015)[0.000] | – |
R2 | 0.469 | 0.512 | 0.597 |
σu | 17.91 | 20.80 | 11.13 |
T | 580 | 307 | 273 |
m | 8 | 3 | 12 |
n | 12 | 12 | 10 |
value | |
β1 | 0.086(0.024)[0.000] |
β2 | 0 |
β3 | 0.108(0.018)[0.000] |
β4 | 0 |
β5 | 0.118(0.025)[0.000] |
β6 | 0.150(0.029)[0.000] |
β7 | 0.094(0.029)[0.001] |
β8 | 0.053(0.023)[0.020] |
β9 | 0.096(0.020)[0.000] |
β10 | 0 |
β11 | 0 |
β12 | 0.045(0.025)[0.073] |
β13 | 0.067(0.031)[0.031] |
β14 | 0.183(0.028)[0.000] |
ση | 0.046(0.004)[0.000] |
σζ | 0.001(0.0006)[0.061] |
σε | 0.054(0.003)[0.000] |
θ | −0.321(0.064)[0.000] |
9th September 2020–31st October 2021 | 9th September 2020–31st January 2021 | 1st February 2021–31st October 2021 | |
c | −0.010(0.005)[0.071] | −0.103(0.020)[0.000] | −0.118(0.016)[0.000] |
d | 0.038(0.014)[0.006] | 0.087(0.006)[0.000] | 0.027(0.001)[0.000] |
a0 | − | − | − |
a1 | −0.241(0.035)[0.000] | −0.198(0.095)[0.038] | −0.488(0.049)[0.000] |
a2 | − | − | −0.236(0.064)[0.000] |
a3 | −0.178(0.066)[0.007] | –0.287(0.088)[0.001] | −0.351(0.086)[0.000] |
a4 | – | −0.121(0.068)[0.075] | −0.117(0.059)[0.050] |
a5 | − | – | – |
a6 | 0.131(0.031)[0.000] | – | − |
a7 | 0.304(0.046)[0.000] | 0.294(0.121)[0.017] | 0.213(0.061)[0.001] |
a8 | 0.265(0.024)[0.000] | 0.254(0.088)[0.005] | 0.110(0.057)[0.055] |
a9 | − | −0.230(0.081)[0.005] | − |
a10 | 0.128(0.051)[0.013] | – | – |
a11 | 0.115(0.049)[0.020] | – | −0.129(0.065)[0.050] |
a12 | −0.134(0.037)[0.000] | −0.234(0.094)[0.014] | 0.152(0.049)[0.002] |
b0 | 0.003(0.000)[0.000] | 0.005(0.001)[0.001] | 0.002(0.001)[0.001] |
b1 | − | −0.005(0.002)[0.009] | − |
b2 | – | – | – |
b3 | 0.003(0.001)[0.005] | – | –0.002(0.001)[0.077] |
b4 | – | – | −0.002(0.001)[0.003] |
b5 | 0.004(0.001)[0.000] | − | −0.002(0.001)[0.020] |
b6 | 0.005(0.001)[0.008] | − | – |
b7 | 0.004(0.001)[0.003] | − | 0.002(0.001)[0.016] |
b8 | 0.005(0.002)[0.004] | 0.003(0.001)[0.044] | – |
b9 | 0.002±0.001 | − | – |
b10 | − | − | 0.002(0.001)[0.089] |
R2 | 0.445 | 0.442 | 0.628 |
σu | 72.71 | 112.37 | 33.18 |
T | 419 | 146 | 273 |
m | 13 | 13 | 13 |
n | 10 | 9 | 11 |