text:
\section{Introduction \label{intro}}

Clinical data on the number of cases of Covid-19 (SARS-CoV-2) are subject to severe temporal confounding, as the rate of testing and criteria for testing have been changing rapidly on the same time scale as the infections, particularly in the early weeks and months of the epidemic. Because these are samples of convenience where the ascertainment fraction is changing and unknown, the data can clearly not be used to infer the actual number of infections. Neither, under normal circumstances, would statisticians recommend attempting to estimate the effective reproduction number of the pathogen from such data, since given the data problems the estimates must necessarily be driven strongly by the modelling assumptions \citep[see e.g.][\S 1.6 for general discussion of the problems with inference from non-random samples]{levine2001}. Indeed generically it is often very difficult to infer epidemiological parameters from clinical data, without the results being informed as much by the prior beliefs encoded in the model as by the data \citep[e.g.][]{lancet-ifr}. Much less problematic are estimates based on randomized surveillance testing, as now conducted in the UK by the office for national statistics (see Supporting Information for discussion of inferring incidence from testing data).

However some clinical data directly measure the quantity of epidemiological interest. This is the case for deaths with Covid-19 and for fatal disease duration. While not perfect, these data are less compromised than the data on cases. Deaths are reliably recorded and clinical grounds for suspecting Covid-19 are relatively clear for fatal cases, although accurately attributing death to a single cause is clearly not always possible. Good records are also often kept for such cases, with the result that there are several published studies on fatal disease duration \citep[][see section \ref{ddist}]{verity2020ifr,linton2020incubation,wu2020covid}. Although only possible with a delay of some weeks, it is of interest to establish what these relatively high quality data imply about the time course of infections, without strong modelling assumptions.

Two types of daily death data are available. Daily reported deaths \citep[e.g.][]{worldometer} typically show marked weekly fluctuations as a result of weekly patterns in reporting delays, and may exclude deaths in some locations (such as nursing homes). Registered death data, such as the ONS data in the UK \citep{ons-deaths}, contain deaths in all locations and record exact date of death. \cite{nhs-covid-daily} publishes equivalent data for hospital deaths in England. The weekly cycle is less pronounced in these data, but their release is necessarily delayed relative to the daily reported deaths, although recent work partially overcomes this delay problem, by modelling the delays to enable `now-casting' of deaths by actual death date: see \cite{stoner2020deathcasting}. The right column of Figure \ref{infections} shows ONS, NHS and Swedish daily deaths by date of death (without now-casting).

The purpose of this paper is to show how a relatively straightforward statistical approach can be used to infer the fatal infection trajectory in the UK in a data driven way that makes the minimum of strong modelling assumptions. The approach is also applied to data from Sweden, the western European country offering the greatest policy contrast to the UK. Sweden never implemented full lockdown, sticking to less restrictive NPIs \citep[broadly aimed at `optimal mitigation' rather than `suppression' in the terms used by][who projected around 40 thousand deaths for this policy]{report12}. Meaningful quantification of the aggregate strength of restrictions that are intrinsically multivariate is difficult, but in terms of their aggregate economic impact, Swedish GDP dropped by about 2.9\% in 2020 compared to about 9.9\% for the UK.  Particular questions of interest are when the decline in fatal infections started in the UK and Sweden, whether UK infections were in substantial decline before full lockdown, whether the pathogen reproduction number was below 1 before lockdown, and how the timing of fatal incidence decline relates to the timing of the easing of lockdown.

Answers to these questions may contribute to judging the proportionality of lockdown measures in the UK context, where there is strong statistical evidence for very large preventable life loss being associated with economic deprivation, and of economic deprivation being increased by economic shocks. This evidence is reviewed in detail in \cite{marmot-review-10}. For example the deprivation related life loss that the current UK population was due to suffer before the Covid crisis was 140-240 million life years \citep[or 2-3.5 years per capita, see][figure 2.3, for example]{marmot-review-10}. The range depends on whether the life expectancy of the lower decile or the lower half of the deprivation distribution is used as the reference for achievable life-expectancy. In examining the effects of the 2008 financial crisis and its aftermath, Marmot documents sharp reductions in life expectancy growth in the UK, which would imply a life loss burden in the 10s of millions of years. However attribution of such reduction-relative-to-trend is obviously very difficult. Less problematic is the 9 million life year loss implied by the increase in life expectancy gap between the more and less deprived halves of the UK population since 2008 \citep[7 weeks per capita, see][figure 2.5, for example]{marmot-review-10}: given the evidence presented in the review, this is more difficult to attribute to causes unrelated to the 2008 economic shock. The Bank of England estimates the shock to the UK economy caused by the response to Covid-19 to have been the largest for over 300 years, so there is a clear danger of substantial life loss being caused, given the historical data for the UK. For example, a feature of the 2008 crisis already repeated in 2020, is the reliance on a large programme of quantitative easing. Quantitative easing is credibly argued to directly increase economic inequality via mechanisms related to asset price inflation \citep[e.g.][]{fontan2016QE,domanski2016}.  There is some literature attributing some short-term life saving to recessions \citep[see e.g.][]{covid-life-loss-govt}, but the effects are modest relative to the long term effects reviewed by Marmot.
For comparison with the above figures, the life loss that might have occurred from a minimally mitigated Covid-19 epidemic appears to be in the region of 3 million life years (2.5 weeks per capita). This is based on \cite{ons-lifetables} lifetables, ONS Covid-19 fatality by age data, a mid range infection fatality rate estimate of 0.006, a somewhat high herd immunity threshold of 0.7 and a 1 year lower bound life expectancy adjustment for co-morbidities based on \cite{hanlon2020}. It is broadly in line with the UK government estimates \citep{covid-life-loss-govt}. Given that 9 million life years, associated with the substantially smaller economic shock of 2008, is not negligible relative to 3 million life years potentially losable to Covid-19, there is obviously a delicate balance to be struck in the UK context, and evidence based on assumption light inference should probably play a role in shaping that balance. Another indicator of the difficulty of achieving the right balance is that the usual UK threshold for approving a pharmaceutical intervention is \pounds 30,000 per life year saved. On the basis of economic costs detailed in \cite{obrjuly20}, and the preceding life loss figures, the non-pharmaceutical interventions used in the UK appear to have a cost per life year saved that is an order of magnitude higher than this (excess government borrowing is projected to peak at \pounds 660 Billion in the OBR central scenario, for example). This discrepancy in willingness to pay  may lead to a problem of opportunity cost, as the same money can not be spent on preventing other life loss, such as that associated with economic hardship.

The remainder of the paper is structured as follows. Section \ref{ddist} discusses the available information on the distribution of fatal disease durations, and how to combine it while adequately characterizing the associated uncertainty. Section \ref{sec.models} introduces a simple generalized additive model for direct modelling of the daily deaths trajectories, and shows how it can be extended to infer the trajectory of fatal infections, either directly or by inferring the trajectory of the pathogen effective reproduction number, $R$, in a simple epidemic model. Since the extensions are not standard models, and are relatively expensive to compute with using standard Bayesian software, section \ref{sec.methods} outlines methods allowing computationally efficient inference with the models. Section \ref{sec:results} presents the main results on infection trajectories, and also the estimation of $R$. Section \ref{sec:check} discusses possible problems with the approach, in particular examining whether smoothness assumptions could be leading to substantial bias in inferred timings. Replication code and data are provided in the Supporting Information.

\section{Fatal disease duration \label{ddist}}

Data on the incubation period from infection to onset of symptoms are analysed in many papers, for example \cite{lauer2020covid} found that the period is 2 to 11 days for 95\% of people, with a median of 5.2 days. A meta-analysis by \cite{McAloon20} suggests a log-normal distribution with log scale mean and standard deviation of 1.63 and 0.50. The uncertainty in this distribution is negligible in comparison to the uncertainty in the distribution of times from onset of symptoms to death discussed next.

Several studies estimate the distribution of time from onset of symptoms to death, while properly controlling for the right truncation in the fatal duration data. \cite{verity2020ifr} found that the distribution of time from onset of symptoms to death for fatal cases can be modelled by a gamma density with mean 17.8 and standard deviation 8.44, based on 24 patients from Wuhan. \cite{wu2020covid} suggested a gamma density model with mean 20 and standard deviation 10 based on 41 patients from Wuhan. \cite{linton2020incubation} found that a log normal model offers a slightly better fit, and estimated a mean of 20.2 days and standard deviation of 11.6 days from 34 patients internationally. These distributions are shown in the left panel of Figure \ref{o2d}. A simple meta-analysis approach was used to combine the models. Samples of the correct size were simulated from each model and a log normal model was estimated by maximum likelihood for the combined resulting sample ($n=99$). A further log normal was also fitted (minimizing Kullback Leibler divergence) to the infection to death distribution implied by the fitted onset to death distribution and \cite{McAloon20} infection to  onset distribution (treated as independent). This process was repeated to generate replicate distributions. These replicate distributions were treated as draws from the distribution of infection to death distributions in subsequent analysis. 100 such draws are shown in fig \ref{o2d}.  The log normal was chosen because the careful analysis of \cite{linton2020incubation} found it to be a better model than the gamma.

In addition, under strict conditions, I was able to access data on fatal disease durations for deaths occurring in English hospitals. Access to data with hospital acquired infections filtered out was not possible, so is was necessary to treat these data as a mixture of hospital and community acquired infections, as detailed in the Supporting Information. The resulting inferred fatal disease duration distribution for community acquired infection is plotted in blue in Figure \ref{o2d} and is consistent with the published studies.

Finally, after the peer reviewed version of this paper was published, I realized that Figure 12 of  \cite{isaric.oct20} gives the cumulative distribution function of the time from hospitalization to death for 24421 patients in the ISARIC study who had died by the time of the report. Furthermore this is at a point in time at which recruitment to the study was at a low rate, so that right truncation problems should be relatively minor. The study also reports the mean and standard deviation of time from onset to hospitalization. Combining this information as described in the Supporting Information, gives the densities shown as dashed blue curves in  Figure \ref{o2d}. These data offer strong support for the distributions used in this paper, and in fact suggest that the results may be somewhat more certain than the plotted uncertainty bands imply.

\begin{figure}

\eps{-90}{.67}{ddist-I.eps}
\vspace*{-.5cm}
\caption{Fatal disease duration distributions. Left: onset to death. Dotted Verity {\em et al.} (2020) ; dashed Linton {\em et al.} (2020); dash-dot Wu {\em et al.} (2020); continuous blue is the log-normal mixture component for community acquired infection from the English hospital data; blue dashed is from the ISARIC study. Middle: combined Linton-Verity-Wu onset to death model, thick red is mean model, grey are 100 draws from the distribution of the combined model, thin blue are as left. Right: as middle, but combined infection to death model.}
\label{o2d}
\end{figure}

\section{Models \label{sec.models}}

This section first introduces a simple generalized additive model for daily death trajectories, and then shows how this can be extended to directly infer the trajectory of fatal infections (fatal incidence), without having to assume any particular dynamic model for the epidemic. The resulting model is no longer a generalized additive model and is the model that this paper advocates using. Its structure is such that any method for inference with the model can also be used for inference with the dynamic model of \cite{flaxman2020lockdown}, with appropriate restriction of the incidence trajectory to one representable with that model. The Flaxman model is presented to allow comparison of the results from the infection trajectory model with the apparently contradictory results of  Flaxman et al., but not to advocate its use.

{\bf Basic deaths series model}. Let $y_i$ denote the deaths or reported deaths on day $i$, assumed to follow a negative binomial distribution with mean $\mu_i$ and variance $\mu_i + \mu_i^2/\theta$. Let
\begin{equation}
\log(\mu_i) = f(i) + f_w({\cal{D}}_i) \label{basic}
\end{equation}
where $f $ is a smooth function of time measured in days, and $f_w$ is a zero mean cyclic smooth function of day of the week, ${\cal D}_i \in \{1,2,\ldots, 7\}$, set up so that $f^{[k]}_w(0) = f^{[k]}_w(7)$, where $k=0,1$ or 2  denotes order of derivative. $f(i)$ represents the underlying log death rate, while $f_w$ describes the weekly variation about that rate. The functions $f$ and $f_w$ can be represented using splines with associated smoothing penalties $\lambda \int f^{\prime\prime}(t)^2 dt$ and $\lambda_w \int f_w^{\prime\prime}({\cal D})^2 d{\cal D}$. Hyper-parameters $\lambda$ and $\lambda_w$ control the smoothness of the functions. The model is a straightforward generalized additive model and $(\lambda, \lambda_w)$ can be estimated as part of model fitting using a standard empirical Bayes approach as described in \cite{wood2017igam}. The model provides a good fit to both the reported deaths and ONS data. As expected $f_w$ is greatly attenuated for the ONS data (it vanishes for Swedish exact death date data).

{\bf Infection trajectory model}. To estimate the daily infection trajectory the model is extended by expressing $f(i)$ in terms of the time course of earlier infections. Let $f_c(i)$ be the function describing the variation in the number of eventually fatal infections over time. Let $\bf B$ be the square matrix such that $B_{ij} = \pi(i-j+1;\mu,\sigma^2)$ if $i\ge j$ and 0 otherwise . $\pi$ denotes an infection-to-death log normal density as discussed above. For the moment its parameters, $\mu$ and $\sigma^2$, are treated as fixed but this will be relaxed in section \ref{sec:dist-unc}. Given the continuity of the log normal, the given form for $B_{ij}$ can be viewed as approximating an integral of $\pi$ over each day, using the midpoint of the integrand -- it is straightforward to approximate the integral more accurately, but given that $\pi$ is  originally estimated from durations discretized to whole days, any precision gain is illusory.  If ${\bf f}_c = [f_c(0),f_c(1), \ldots ]\ts$ and $ {\bm \delta} = [\delta(1), \delta(2),\ldots]\ts$ then ${\bm \delta} = {\bf Bf}_c$, where $\delta(i)$ is the expected number of deaths on day $i$. $\log f_c(i)$ can be represented using a spline basis, again with a cubic spline penalty. Working on the log scale ensures that $f_c$ is positive, but is also appealing because it means that a cubic spline penalty on $\log f_c(i)$ can be interpreted as a first derivative penalty $\int r^\prime(t)^2 dt$, acting on the epidemiologists `intrinsic growth rate', $r$. The final infection trajectory model is then obtained by simply substituting $f(i) = \log \delta(i)$ into (\ref{basic}). $\bf B$ is rank deficient, so inferring $f_c$ can be viewed as an inverse problem: without regularization multiple solutions that oscillate from day-to-day are possible. This ambiguity is removed by the smoothing penalty on $\log f_c$.

{\bf Relaxed Flaxman model}. Since this work was originally undertaken in late April 2020, the work of \cite{flaxman2020lockdown} has appeared. Flaxman et al. make inferences about the reproduction number, $R$, and hence incidence rates, based on death trajectories and the fatal infection duration distribution of \cite{verity2020ifr}, but do so by modelling the pathogen effective reproduction number $R_t$ within a simple epidemic `renewal model'. \cite{flaxman2020lockdown} represent $R_t$ as a step function with steps allowed each time the government announced new containment interventions, and a sparsity prior promoting a small number of steps. In the notation of Flaxman {\em et al.} the expected number of infections each day (now total, rather than fatal) are denoted $c_t$. Given an initial $c_1$ the model is iterated from $t=2$ as follows
\begin{equation}
c_t = \left (1 - \sum_{i=1}^{t-1} c_i/N\right) R_t \sum_{\tau = 1}^{t-1} c_\tau g_{t-\tau} \label{epimod}
\end{equation}
where $N$ is the total initially susceptible population, $g_1 = \int_0^{1.5} \gamma(x) dx$ and $g_j = \int_{j-.5}^{j+.5} \gamma(x) dx$ for $j>1$. $\gamma$ is the p.d.f. of a Gamma distribution with shape parameter $6.5 \times 0.62^2$ and scale parameter $0.62^{-2}$. The $c_t$ values multiplied by the assumed infection fatality rate give ${\bf f}_c$. The level of the IFR only matters for the damping term in the first bracket of the expression for $c_t$ --- this has almost no effect in practice, a mid range value of 0.006 was used. The original assumptions about $R_t$ can be relaxed by representing $\log R_t$ using a spline basis, with associated penalty as for the other models, while $\log c_1$ is also treated as a free parameter. Hence $f_c$ in the infection trajectory model can simply be replaced by the Flaxman model with $\log R_t$ represented as a spline function. The model is otherwise unchanged. This model is presented only to allow comparison of this paper's results with those of \cite{flaxman2020lockdown}: its simple single compartment structure clearly does not meet the aim of inferring incidence with minimal assumptions.

\section{Methods \label{sec.methods}}

The infection trajectory and Flaxman renewal models are not standard models estimable with standard software. They can  be implemented in Bayesian software, such as JAGS or STAN, but inference typically takes several hours if this is done. Dealing adequately with the uncertainty in the disease duration distribution multiplies this cost by 1-2 orders of magnitude. To avoid these problems an empirical Bayes approach can be employed.

\subsection{Basic inferential framework}

Direct inference about (\ref{basic}) uses the empirical Bayes approach of \cite{wood2015plig} in which the smooth functions are estimated by penalized likelihood maximisation \citep[e.g.][]{green.silverman}, with the smoothing parameters and $\theta$ estimated by Laplace approximate marginal likelihood maximization. Writing ${\bm \beta}$ for the combined vector of  basis coefficients for $f$ and $f_w$, the penalized version of the log likelihood, $l({\bm \beta}),$ can be written
$$
l({\bm \beta}) - \frac{\lambda}{2} \int f^{[2]}(t)^2 dt - \frac{\lambda_w}{2} \int f^{[2]}_w({\cal D})^2 d{\cal D} =
l({\bm \beta}) - \frac{1}{2} {\bm \beta} \ts {\bf S}_\lambda {\bm \beta}
$$
where ${\bf S}_\lambda = \lambda {\bf S}_f + \lambda_w {\bf S}_w$: ${\bf S}_{f}$ and ${\bf S}_{w}$ are known constant positive semi-definite matrices. Smoothing parameters, $\lambda$ and $\lambda_w$, control the smoothness of $f $ and $f_w$. Let $\widehat {\bm \beta}$ be the maximizer of the penalized log likelihood, and $\bf H$ its negative Hessian at $\widehat {\bm \beta}$. Viewing the penalty as being induced by an improper Gaussian prior, ${\bm \beta} \sim N({\bf 0}, {\bf S}_\lambda^-)$, $\widehat {\bm \beta}$ is also the MAP estimate of ${\bm \beta}$. Furthermore in the large sample limit
\begin{equation}
{\bm \beta} |{\bf y} \sim N(\widehat {\bm \beta},({\bf H} + {\bf S}_\lambda)^{-1}). \label{gg}
\end{equation}
Writing the density in (\ref{gg}) as $\pi_g$, and the joint density of ${\bf y} $ and ${\bm \beta} $ as $\pi({\bf y}, {\bm \beta})$, the Laplace approximation to the marginal likelihood for the smoothing parameters $\bm \lambda$ and $\theta$ is $\pi({\bm \lambda},\theta) =\pi({\bf y}, {\bm \beta})/\pi_g({\bm \beta}|{\bf y})$. Nested Newton iterations are used to find the values of $\log({\bm \lambda}),\theta$ maximizing $\pi({\bm \lambda},\theta)$ and the corresponding $\widehat {\bm \beta}$ \citep[for details see][]{wood2015plig}.

Given (\ref{gg}) credible intervals for $f$ are readily computed, but it is also straightforward to make inferences about when the peak in $f$ occurs. Simply simulate replicate coefficient vectors from (\ref{gg}) and find the day of occurrence of the peak for each corresponding underlying death rate function, $f$.

\subsection{Extension for the infection and Flaxman models}

While inference about (\ref{basic}) using the preceding framework requires little more than a call to the {\tt gam} function in R package {\tt mgcv}, its application to the other models, which are not generalized additive models, requires more work.
For the  model formulated in terms of $f_c$ this requires expressions for the negative binomial deviance (or log likelihood) and its derivative vector and Hessian matrix w.r.t. the model coefficients.

First consider the negative binomial deviance for observation $i$,
$$
D_i = 2y_i\log\{\max(1,y_i)/\mu_i\} - (y_i + \theta) \log\{(y_i + \theta)/(\mu_i +\theta) \},
$$
$$
\dif{D_i}{\mu_i} = 2\left (\frac{y_i + \theta}{\mu_i + \theta} - \frac{y_i}{\mu_i} \right)
~~~\text{and}~~~ \dif{^2 D_i}{\mu_i^2} = 2 \left (
\frac{y_i}{\mu_i^2} - \frac{y_i+\theta}{(\mu_i + \theta)^2}
\right ).
$$
These need to be transformed into derivatives w.r.t. $\bm \beta$, as follows:
$$
\pdif{D_i}{\beta_j} = \dif{D_i}{\mu_i}\pdif{\mu_i}{\beta_j} ~\text{and}~
\pddif{D_i}{\beta_j}{\beta_k} = \dif{^2 D_i}{\mu_i^2} \pdif{\mu_i}{\beta_j}\pdif{\mu_i}{\beta_k} +
\dif{D_i}{\mu_i} \pddif{\mu_i}{\beta_j}{\beta_k}.
$$
Writing ${\bf X}^f$ and ${\bf X}^w$ for the model matrices for the smooth terms $\log f_c$ and $f_w$, we have ${\bm \delta} =  {\bf Bf}_c $ where ${\bf f}_c = \exp({\bf X}^f{\bm \beta}^f)$ (here $\exp(\cdot)$, division and multiplication are applied element-wise to vectors), and ${\bf f}_w = {\bf X}^w{\bm \beta}^w$. Then ${\bm \mu} = \exp(\log {\bm \delta} + {\bf f}_w )$, while
$$
\pdif{\bm \mu^{~}}{{\bm \beta}^f} = \text{diag}({\bm \mu}/{\bm \delta}){\bf B} \pdif{{\bf f}_c}{{\bm \beta}^f}, ~~~
\pdif{\bm \mu^{~}}{{\bm \beta}^w} = \text{diag}({\bm \mu}) {\bf X}^w,
$$
$$
\pddif{{\bm \mu}}{\beta_j^w}{\beta_k^w} = {\bm \mu} {\bf X}^w_{\cdot,j} {\bf X}^w_{\cdot,k}, ~~~
\pddif{{\bm \mu}}{\beta_j^f}{\beta_k^f} = \text{diag}({\bm \mu}/{\bm \delta}){\bf B}
\pddif{{\bf f}_c}{\beta_j^f}{\beta_k^f}
$$
$$
\text{~and~}
\pddif{{\bm \mu}}{\beta_j^f}{\beta_k^w} = \text{diag}({\bf X}^w_{\cdot,k}{\bm \mu}/{\bm \delta})
{\bf B} \pdif{{\bf f}_c}{{\bm \beta}^f} .
$$
For the given representation of ${\bf f}_c$
$$
\pdif{{\bf f}_c}{{\bm \beta}^f} = \text{diag}({\bf f}_c) {\bf X}^f ~~ \text{and} ~~
\pddif{{\bf f}_c}{\beta_j^f}{\beta_k^f} = \text{diag}({\bf f }_c) {\bf X}^f_{\cdot,j} {\bf X}^f_{\cdot,k}.
$$

When using the relaxed Flaxman model, the preceding derivatives of ${\bf f}_c$ have to be replaced with derivatives of ${\bf f}_c$ w.r.t. the coefficients of the spline representing $\log R_t$. Routine application of the chain rule to (\ref{epimod}) gives corresponding iterations for the derivatives of $c_t$, and hence ${\bf f}_c$, w.r.t these spline coefficients and $\log c_1$.

Given these expressions and the penalties, $\widehat {\bm \beta} $ can be obtained by Newton iteration, given smoothing parameters. To estimate smoothing parameters, the simplest approach is to fix the negative binomial $\theta$ at its estimate from model (\ref{basic}), and use \cite{wood2016gfs}, alternating generalized Fellner Schall updates of the smoothing parameters  with updates of $\widehat {\bm \beta} $ given those smoothing parameters.  This finds the smoothing parameters to approximately maximise the model marginal likelihood. The non-linearity of the renewal equation model means that some effort is required to get non-absurd starting values. I got these by a few minutes of experimentation with simple step functions for the initial $\log R_t$ to get death trajectories of roughly the shape and amplitude of the true trajectories (a close initial fit is not required: initial deviances 2 orders of magnitude greater than for the final fit were unproblematic).

Given $\theta$ and the smoothing parameters, the approximate posterior (\ref{gg}) could be used directly, or as the basis for the proposal distribution in a simple Metropolis Hastings sampler. A fairly efficient sampler results from alternating fixed proposals based on (\ref{gg}) with random walk proposals based on zero mean Gaussian steps with a shrunken version of the posterior covariance matrix. By this method, effective sample sizes $>5000$ for each coefficient took about 40 seconds computing on a low specification laptop. This was the approach used for the infection trajectory model. The results were indistinguishable from those produced at the cost of several hours of computing using JAGS \citep{plummer2003jags,coda} to simulate from the model posterior.

\subsection{Disease duration distribution uncertainty \label{sec:dist-unc}}

The methods so far perform inference conditional on fixed values for the parameters $\mu$ and $\sigma^2$ of the log normal density describing the infection to death duration distribution. In reality there is uncertainty about these parameters. To incorporate this uncertainty into the infection trajectory model, inference was run for each of the 100 sample distributions shown in grey in the right hand panel of Figure \ref{o2d}, and the resulting posterior samples were pooled, to give a sample from the unconditional posterior distribution of the model.

\section{Results \label{sec:results}}

\begin{figure}

\eps{-90}{.58}{infections.eps}
\caption{ In all plots black curves show  the posterior median while light grey and dark grey regions show respectively 95\% and 68\% confidence regions, including uncertainty in the fatal disease duration distribution. Day 0 is 13th March 2020, and the vertical red line marks the first day of UK lockdown. Top left: Inferred daily fatal infection rate, $f_c$, for the UK.  The scaled barchart shows the posterior distribution for day of peak infection with the peak day labelled. NPI start dates are marked by labelled vertical lines. Top right: Consistency check. In grey are 100 sets of death data simulated forward from the inferred median fatal infection profile. Symbols are the ONS daily death data for the UK on which inference is based. The dashed curves are 95\% confidence intervals for underlying death rate estimated by direct fitting of (\ref{basic}).
Middle row: As top row, but using the NHS England daily hospital death data. Note that the inferred infection trajectories are substantially different to time lagged versions of the deaths trajectories.
Bottom row: as the previous rows, but for Sweden.
 \label{infections}}
\end{figure}

Figure \ref{infections} shows the results of applying the model to the Office for National Statistics daily Covid-19 death data for the UK, to the NHS England hospital data and to the daily death data for Sweden from \cite{sweden-covid-daily}. The results include the uncertainty about the disease duration distribution shape.   ONS and NHS data are up to 27th June -- including later data simply narrows the uncertainty, while making negligible difference to the overall conclusions. The most notable feature of the results is that fatal infections are inferred to be in substantial decline before full lockdown (the same result was obtained by this method in early May 2020, based on the first 50 days of reported daily death data). Sweden appears most likely to have peaked only one or two days later (barring some systematic difference in fatal disease durations for Sweden), having introduced NPIs well short of full lockdown. The results also emphasise the fact that the infection trajectory is not simply a time shifted version of the death trajectory (assuming it was might lead to unwarranted delay in easing lockdown, for example). The difference in timing and shape of the inferred profile between the ONS and NHS data reflects the fact that the latter contain care home data. There is an argument for preferring hospital data for inferring community fatal infections, in that the care home epidemic is now known to have special features with at least some of the infection not coming from normal community transmission. See in particular \cite{comas2020care-home} for a discussion of care home deaths internationally, including the UK. In addition, in the UK, care home deaths were often attributed to Covid-19 without a test, especially after death certification guidelines were changed to encourage reporting of suspected, rather than confirmed Covid-19 deaths. The care home data therefore have some under-reporting of Covid deaths, followed by over-reporting (the signal of this is visible in ONS data in the change in non-Covid pneumonia deaths being reported, relative to normal, for example).

\begin{figure}

\eps{-90}{.55}{RCI.eps}

\caption{ Left: Estimates and confidence bands for the effective reproduction number, $R$, from a simple SEIR model given the inferred infection profile (incidence), $f_c$.  The assumed mean time to infectivity was $1/\gamma = 3$ days and the mean infectivity duration was $1/\delta = 5$ days. The labelled vertical bars show policy change dates in March 2020.
Given the rapidity of policy change relative to the epidemic's dynamic time scale, and government policy sometimes lagging behaviour, casual over interpretation of these timings should be avoided. Right: sensitivity analysis. Dashed blue -- time to infectivity was varied from 1 to 5 days. Grey -- duration of infectivity was varied from 2 to 10 days. Logs are natural. $R$ appears to be below 1 before full lockdown, but fell further after it.
 \label{Rfig}}
\end{figure}

Taken together the results for the UK and Sweden raise the questions of firstly whether full lockdown was necessary to bring infections under control, or whether more limited measures might have been effective, and secondly whether the several month duration of full lockdown was appropriate. These emphasise the desirability of statistically well founded direct measurement of epidemic size through randomized testing. Had such testing being carried out leading up to lockdown it would have been clearer if the measures preceding lockdown (see Figures \ref{infections} and \ref{Rfig
)  were working, or whether stronger restrictions were needed. Similarly such testing might have given earlier indication of when lockdown could be eased. Instead management was reliant on a complex modelling synthesis of expert judgement and problematic clinical case data. Less statistically problematic reconstructions, like the one presented here, are clearly only possible weeks after the fact. Note that while it is natural to interpret these fatal infection trajectories as proportional to the overall infection trajectories, that will only be the case if the infection fatality rate is constant over time. There is evidence for improvements in hospital care from late March onwards that suggest that this is might not be the case \citep[see][]{dennis2021covid-improve}. The Supporting Information includes a sensitivity analysis of this issue: it has the potential to right shift the peak incidence by up to a day and to lead to somewhat less rapid decay of the incidence trajectory.

\subsection{Inferring R}

Much public debate has focused on the effective reproduction number, $R$, and in theory it is possible for a decline in the rate of infections to be only temporary as a result of $R$ dropping but remaining above one. Could it be that the declines in $f_c$ seen before lockdown were of this short term type, and that renewed increase would therefore have occurred without full lockdown? The answer appears to be no. $R$ is all but impossible to measure directly, so inference about it requires assumption of an epidemic model. However, given an epidemic model it can be directly inferred from the reconstructed infection profile. For example consider a simple SEIR model: $\dot{S} = -\beta S I$, $\dot{E} = \beta S I - \gamma E$, $\dot{I} = \gamma E - \delta I$ (here $\delta I $ is the rate of recovery {\em or} progression to serious disease). $\widehat f_c$ is a direct estimate of $\beta S I$ (to within a constant of proportionality), so by solving  $$\dot{E} = \widehat f_c - \gamma E, ~~~~~~\dot{I} = \gamma E - \delta I $$ (from 0 initial conditions) the direct estimate $R = f_c/(I\delta)$ is readily computed (any constant of proportionality cancels in $R$). A different epidemic model could be used here of course: see \cite{diekmann1990R} for calculation of $R$ in general from a model. Figure \ref{Rfig} shows the results using $\widehat f_c$ for the English hospital data for plausible values of average time to infectivity of $1/\gamma = 3$ days and mean duration of infectiousness of $1/\delta = 5$ days, along with sensitivity analysis for these values. The credible intervals shown include the uncertainty about the fatal disease duration distribution. $R $ appears to be below 1 before full lockdown.

A useful feature of the $R$ estimates is to emphasise that the analysis in this paper in no way suggests that lockdown did not have an effect on transmission. Even if $R$ was below one before lockdown, full lockdown can only have reduced it further, and the estimates in Figure \ref{Rfig} are obviously consistent with this. Note, however that the recovery in $R$ after the post lockdown dip is to be expected, given the simple fact that $R$ is the number of new infections created per infection, {\em averaged over the population of infections}, not the population of people. Broadly speaking, at lockdown the population of people, and infections, was split into the locked down population, where infections could create few new infections, and the `unlocked' population where the reproductive rate of the pathogen was higher (assuming lockdown had an effect). An initial average over all infections is then dominated by those infections in the locked down population, giving a low $R$ (especially once the possibilities for infecting locked down household members have been exhausted). As the infections in the locked down population die out, the proportion of all infections that are in the unlocked population must increase -- so that an average over all infections must yield a higher $R$ again.

\subsection{The Flaxman model}

\begin{figure}

\eps{-90}{.55}{renewal.eps}

\caption{Results from the epidemic model of Flaxman {\em et al.} (2020), with the assumptions on $R$ relaxed: $\log R$ is assumed smooth and continuous. Left: the inferred $R$ from fitting the NHS hospital data. The inferred $R$ trajectory is similar to the one shown in Figure \ref{Rfig}, despite the different model structure. Intervals do not include disease duration distribution uncertainty here. Middle: the corresponding fatal infection profile. Right: the simple sanity check as in Figure \ref{infections}.
 \label{renewal}}
\end{figure}

As noted above in section \ref{sec.models}, \cite{flaxman2020lockdown} also analysed death trajectories, using a simple epidemic model, but came to conclusions apparently contradicting Figure \ref{Rfig}. They  concluded that only after full lockdown did $R$ drop below 1, and that fatal infections continued to increase up until the eve of full lockdown. \cite{flaxman2020lockdown} used the \cite{verity2020ifr} fatal disease duration distribution, so the difference in results does not lie there. To describe the epidemic dynamics Flaxman {\em et al.} use the simple single compartment discrete renewal model (\ref{epimod}). Within that model they assume that $R$ is constant between the imposition of interventions, but can undergo a step change at each intervention: the steps are free model parameters. This model for $R$ is quite restrictive.  In particular it does not  allow $R$ to change after lockdown, despite the fact that at lockdown the population has been stratified in a way that the renewal model does not represent, so that some compensating flexibility in $R$ is likely to be required to avoid modelling artefacts. At the same time the model is rather underdetermined preceding lockdown, because of the frequent intervention changes. This indeterminacy in the model is addressed by using a sparsity promoting prior on the step changes in $R$, which favours few larger changes, rather than several smaller changes (see the supplementary material for Flaxman {\em et al.} for a description of this prior). When using the model to simultaneously model multiple European countries there is a further assumption that the intervention effects are the same for all countries (despite the different order of their implementation) and that only the lockdown effect varies between countries. It seems likely to be difficult to pick up effects of the interventions preceding lockdown from such a model structure.

A relaxed version of the Flaxman model in which $\log R_t$ is a continuous function is described in section \ref{sec.models}. The results from using this model for inference using the NHS hospital data are shown in Figure \ref{renewal}. The relaxation of the assumptions on $R$ brings the results (for the UK) into alignment with those in the rest of this paper, and into broad  consistency with developments later in the year, which are otherwise difficult to square with \cite{flaxman2020lockdown}.

\subsection{Later infection waves}

While the initial motivation for this work was to provide reasonably timely analysis for the first wave, based on the limited data available by May 2020, the methods scale readily to the much longer data runs available by early 2021. The only change is that it makes sense to use an adaptive smoother \citep[see e.g.][\S 5.3.5]{wood2017igam} for $f(t)$, in which the degree of smoothness is allowed to vary with time. The longer data runs make it feasible to estimate the multiple smoothing parameters that this entails. Using an adaptive smooth guards against artefacts driven by the smoothness that is appropriate on average, for all the data, not being appropriate at times of rapid change.

 \begin{figure}

\eps{-90}{.55}{epidemic.eps}

\caption{Inference for the English hospital deaths data up to mid February 2021, including disease duration uncertainty. Top: Inferred fatal incidence. Grey symbols are the hospital deaths from which incidence is inferred. Red vertical lines mark the start of each of the three English lockdowns. Note that improvements in medical treatment mean that the IFR is very likely not to be constant between the first and later waves, so comparing their relative sizes is difficult. Bottom: the inferred $R$ using the simple SEIR approach. NPI impositions short of lockdown are marked by dotted vertical lines, relaxations are marked by dashed lines.
 \label{epidemic}}
\end{figure}

The results of this application are shown in Figure \ref{epidemic}. Note that likely changes in infection fatality rate as a result of improved hospital treatment mean that the relative sizes of the fatal infection incidence curves in the first and subsequent waves can not be interpreted as reflecting the relative sizes of total incidence (the later incidence curves would need to be scaled up somewhat). Causal over-interpretation
of the $R$ curves should be avoided, not least because there is no reason to expect Covid-19 not to display the seasonality in transmission common to other respiratory illnesses. However, the results are obviously inconsistent with full lockdowns having caused $R<1$, since cause should not happen after effect. Further, the drop in $R$ seen after the initial NPIs were introduced, but before full lockdown, does seem consistent with the levels of $R$ later achieved while measures short of lockdown were in place. The interesting feature of $R$ apparently increasing from quite early in the second lockdown, might relate to the spread of the new variant, but of course also occurs at a time when respiratory infections generally start to increase. Likewise the further increase until mid December, may well be due to the new variant, but increased activity in the run up to Christmas is also likely to be a factor -- incidence appears to peak over the Christmas to New Year period. Vaccine rollout seems virtually certain to be a major factor in pushing down $R$ and fatal incidence from December. The vaccine has been given to those most at risk first, so the constant IFR assumption required to interpret fatal incidence as proportional to total incidence obviously no longer holds. This further implies  that the inferred $R$ is in some sense only the $R$ relevant to the `at serious risk' population. Of course, it could be argued that for epidemic management purposes, the fatal incidence and the corresponding $R$ are of primary interest.

Interestingly the pattern observed at the second lockdown and in the preceding months is consistent with the results reported by \cite{rep41} who analysed regionally stratified death, hospital occupancy and testing data for 2020 up until December, using a highly detailed age structured SEIR with added health service compartments. The entire trajectory up until December is also consistent with the results of \cite{wood2021rep41}, who re-implemented the Knock et al. model, but removed some of its very strong modelling assumptions around the first lockdown.

\section{Model checking \label{sec:check}}

While standard residual checks indicate no problem with the model from the point of view of statistical fit, there are three issues which could potentially undermine the results, and a further issue relating to interpretation.

The first relates to the infection to death interval distribution and the fact that the death data contain an unknown proportion of patients whose infection was hospital acquired. These patients are likely to have had shorter disease durations, since they were already sufficiently unwell or frail to be in hospital. This paper has inferred when the fatal infections would have occurred if they were all community generated, since it is the community infections that are of interest with respect to the effects of lockdown, social distancing etc. Without knowing even the proportion of deaths from hospital acquired infection it is anyway not possible to do otherwise.

The presence of hospital infections in the death data will bias inference about the dynamics of community fatal infections if it substantially changes the shape of the deaths profile, relative to what would have occurred without hospital infection. Broadly, if the trajectory of hospital acquired infection deaths peaked earlier than the overall trajectory, then the community infection peak will be estimated to be earlier than it should be (since the true community infection death peak is then later). Conversely, if the hospital acquired infection deaths peaked later, then the community infection peak will be estimated as being later than it should be. The degree of bias will depend on the proportion of hospital acquired infections and the degree of mismatch in timings. It is difficult to judge which alternative is more likely: standard epidemiological modelling assumptions would imply that the more community acquired cases are hospitalised the more hospital infections would occur and that hospital infections will lag community cases. But against this, hospital acquired fatal disease durations are likely to contain a higher proportion of shorter durations. In any case the proportion of hospital acquired infections in the death series would have to be quite high for the issue to substantially modify the conclusions.

The second issue is that age dependency in the duration distribution coupled with shifts in the age structure of deaths over time could also be problematic. However, as documented in the Supporting Information, the data for England and Wales show remarkably little variation in the age structure of Covid-19 fatalities over the course of 2020, while analysis of English hospital data apparently shows little evidence for age dependence in the disease duration distribution.

\begin{figure}

\vspace*{-.6cm }

\eps{-90}{.55}{dilation.eps}

\vspace*{-.5cm}

\caption{Model checking plots in which the smoothness assumptions are relaxed around lockdown by a time dilation, in order to allow accurate capture of any extremely discontinuous infection profile in this region. The top row shows the method reconstructing an extreme simulation scenario in which there was no reduction in transmission rate up until lockdown, and then an instantaneous drop. Left: the reconstruction (plot meaning as Figure \ref{infections}) with the true simulated daily infections shown dashed. Right: forward simulation from the median profile as in Figure \ref{infections}. The blue symbols are the simulated death data used for inference. The bottom row is for the NHS England hospital data under the time dilated model. Even this model deliberately modified to promote a very abrupt change at lockdown suggests that the infection rate was probably declining before lockdown.
\label{dilation}}
\end{figure}

The third issue is whether the smoothing penalty on $\log f_c$ would lead to systematic mis-timing of the estimated peak under the scenario of a very asymmetric peak in the true infection profile around lockdown. To investigate this, data were simulated from a model in which the underlying infection rate increased geometrically, doubling every 3 days until lockdown, when the rate dropped immediately to 0.2 of its peak value, shrinking thereafter by 5\% per day. Fatal infections were simulated as Poisson deviates with the given underlying rate. This model is an extreme scenario, in which measures prior to full lockdown had no effect, and the effect of lockdown was instant, as if the locked down population (i.e. those not in essential work) had isolated alone, rather than increasing their contact with members of their household while drastically reducing it with everyone else. However it is the scenario implicit in much public discussion in the UK, at least at the time that this work was originally conducted. Under this scenario, the method does indeed tend to incorrectly estimate the infection peak as 2 to 3 days before lockdown, rather than the day before, as it struggles to accommodate the drop.

The naive approach to this issue is to introduce a parameter at lockdown representing an instantaneous drop in infections. However doing so introduces a very strong structural assumption into the model, undermining the aim of avoiding strong assumptions. This approach also has the serious side effect of introducing non-parametric smoothing boundary effects on both sides of the break. These boundary effects severely compromise inference in the most interesting region of the infection profile, while simultaneously increasing the importance of the structural assumption at the expense of the data. Indeed when such a model is built it estimates a large drop even from data simulated from a smooth infection profile. It also estimates such a drop if the drop's location is moved (for simulated or real data).

A better approach is to use a smooth time-dilation to relax, but not eliminate, the model smoothness assumptions in the vicinity of lockdown. The dilation is made sufficient that the model can accurately  capture the extreme scenario in the simulation, but without imposing a break and boundary effects. In particular $f_c$ and its smoothing penalty are computed with respect to a version of time which makes the day before, of and after lockdown count as 3.5, 6 and 3.5 days, respectively. Obviously regular un-dilated time is used for mapping infections to deaths. For the extreme simulation, the model then correctly gives most posterior probability to the day before lockdown as the peak. In contrast the same model for the real data has very low probability of the peak being the day before lockdown rather than earlier.

Figure \ref{dilation} shows the results from fitting the time dilated model to the extreme simulation scenario and to the NHS England hospital data. Even this model, deliberately modified to favour a very abrupt change at lockdown, suggests that infections started to decline before lockdown, with the most likely day for the peak only 1 day later than with the un-dilated model. The Supporting Information includes similar checks for the \cite{flaxman2020lockdown} model, with similar conclusions.

 Finally, interpretation of the fatal incidence trajectories as proportional to the overall incidence trajectories rests on an assumption that the infection fatality rate is constant over time. There is evidence that the hospitalized case fatality rate declined in the two months or so after the peak of the first wave of infections \citep{dennis2021covid-improve}, with this effect not explicable by any detectable change in patient characteristics. However, on the ground changes in the severity threshold for admission would be very difficult to detect, seem likely at times when some hospital's were at or near capacity, and could also contribute to such a pattern. The Supporting Information includes a check of the impact that the reported improvements would have on the shape of inferred overall incidence. The peak incidence could be shifted by as much as a day later, and there would be a somewhat slower decline in incidence relative to the results plotted in Figure \ref{infections}.

\section{Discussion \label{disc}}

The analysis in this paper does not absolutely prove that substantial decline in fatal infections in the UK preceded the first full lockdown by several days, but it very strongly suggests that this is what happened, and the ISARIC fatal disease duration distribution data, not available in the peer reviewed version of the paper, only strengthens this point further. Since the peer reviewed publication, information has also become available from the UK's two main Covid surveys based on proper randomized statistical sampling. The Office for National Statistics PCR prevalence survey has now published its estimates of incidence. For the second and third lockdown these support the analysis in this paper. See Figure \ref{ons}. Furthermore the REACT-2 survey \citep{ward2021react} has reconstructed numbers of newly symptomatic cases per day from survey participants testing positive for Covid-19 antibodies. Lagging these by the published mean time from infection to first symptoms gives the left plot of figure \ref{ons}. This agrees with the results of this paper for all three lockdowns. Some differences in relative peak sizes between the series are to be expected if there are changes in infection fatality rate, and as a result of the vaccination program. Taken together these results, from the 3 highest quality data sources available for the UK epidemic, leave little room for reasonable doubt that incidence was in decline before each lockdown. Apparent demonstrations of the contrary result appear to rely on building it into the modelling assumptions, or on highly informal reasoning about timings or correlations, or on a view that scientific opinion, in sufficient quantity, outweighs data and measurement.

 \begin{figure}

\vspace*{-.6cm }

\eps{-90}{.55}{react2-ons.eps}

\vspace*{-.5cm}

\caption{Left, REACT-2 onset of new symptoms per day digitized from \cite{ward2021react}, lagged by the mean 5.8 days from infection to first symptoms. Day 1 is Jan 1 2020. Red lines are the lockdowns. Rather than simple lagging, this paper's method could be applied to the onset dates to infer infection dates given the published  infection to onset distribution, but because none of the onset peaks are after lockdown this can not change the qualitative conclusion (although it would sharpen the peaks a little). Right ONS published estimated incidence and 95\% confidence limits. Red lines are at lockdowns 2 and 3 (the survey was not running at lockdown 1).
\label{ons}}
\end{figure}

The timing result in turn suggests that the measures, and possibly the spontaneous reactions to rising deaths and cases, preceding lockdowns were probably sufficient to bring the epidemic under control, and that community infections, unlike deaths, were probably at a low level well before the first lockdown was eased. Such a scenario would be consistent with the infection profile in Sweden, which began its decline in fatal infections shortly after the UK, but did so on the basis of measures well short of full lockdown.

The analysis does not in itself say what would have happened without full lockdowns, and must obviously be weighed against other evidence. No currently available analysis will conclusively determine what would have happened without full lockdowns, and the state of the art in causal inference is obviously a very long way from being able to answer this question. Models based on approximations to the mechanisms of epidemic transmission do not allow reliable answers to these causal questions either. This is particularly so given the paucity of data with which to validate their component assumptions - a paucity that only grows more acute as more detail is included in the models. These are not weather or climate models, based on the bulk properties of enormous numbers of physically well understood interactions of simple molecules, tested and refined against huge quantities of carefully measured calibration data collected worldwide over decades. Rather they are best working approximations constructed by experts given the limited information that could be rapidly assembled in a matter of months, and subject to all the uncertainty this implies. A model does not become a valid basis for casual inference merely by being described as mechanistic.  As the above reanalysis using the Flaxman model serves to emphasise: the inclusion of model structure aiming to represent mechanism is no guarantee of improved statistical inference, and certainly not a justification for treating inference with mechanism based models as causal.

Since this work was first undertaken other low assumption analyses have appeared, in particular looking for the coincidence of NPI introductions and changepoints in incidence, for example in Germany and Spain. The results of this paper are in some alignment with such analyses for Germany \citep{Wieland2020, kuchenhoff2020}, which also suggest that a decline in incidence preceded the first full lockdown. Both are based on case data, which are problematic even in Germany which had mass (but not randomized) testing in place from the start of the epidemic. However it seems likely that the biases in case data would lead to the start of decline in incidence being estimated as later than it really was, rather than earlier, so the qualitative conclusion is likely to be robust. In Spain, \cite{santamaria2020} also identify substantial changes in rate of change of incidence before Spanish lockdown based on death series, but not sufficient to suggest a decline in incidence before lockdown. Based on pre-print versions of the current paper a number of researchers have also attempted to employ the basic idea of dynamic model free inference about incidence profiles, but via a simplified method. This method tries to impute date of infection by subtracting a random draw from the fatal duration distribution from each deceased patient's  death date. This process is replicated to obtain an expected incidence profile. The method is invalid, as duration of disease is not independent of time of death, and it will tend to incorrectly show much less steep, or no, decline before lockdown. See the Supporting Information for a full discussion.

The results of applying the method to data up to mid February 2021, as well as being consistent with the independent measurements in Figure \ref{ons}, also  provide a picture consistent with the results for the first lockdown. In particular the results preceding the first lockdown appear consistent with how the epidemic progressed under later restrictions short of lockdown. This is not the case for the published analyses suggesting high $R$ and surging incidence on the eve of the first lockdown. The fact that school re-opening does not appear to be followed by an increase in $R$ is interesting: whether it relates to people deciding to keep school children apart from the vulnerable, which is anecdotally plausible, or to other factors, is unclear.  While tempting, it is difficult to interpret the later patterns in terms of the new, apparently more infectious, variant that emerged in late 2020: there is confounding with seasonality of transmission, behavioural changes around the end of year holidays and with the roll out of effective vaccines from late December onwards. Greater clarity on these issues may emerge in future, particularly if the UK ONS Covid surveillance data eventually becomes public in raw form.

\section*{Acknowledgments}

Thanks to Nicole Augustin, John Coggon, Dan Cookson, Peter Green, Dirk Husmeier, Anna Jewell, Jason Matthiopoulos,  Jonty Rougier and Ernst Wit for various suggestions and discussions of aspects of the paper.
I am grateful to Robert Verity for his unsolicited offer to help get access to the duration data for England, for making the effort to get the permissions and follow through on this, and for being willing to repeat the original data extraction to remove right truncation problems. I am also very grateful to a non-attributable source who provided key information on the necessity of filtering out patients with onset after hospitalization from such data. I thank two fact checkers who, although they declined to correct an article at {\tt fullfact.org} to remove misrepresentations of this paper, \cite{linton2020incubation} and \cite{flaxman2020lockdown} and some other related statistical flaws, alerted me to the ISARIC study.

The peer review process for this paper was unusual, with 5 referees and 3 editors/associate editors providing comments, which influenced the paper as follows. The paper was first rejected by {\em JRSSC (Applied Statistics)} in early May 2020 for insufficient methodological novelty, and then by {\em Nature} and {\em Nature Communications}, all without refereeing. In late May it was sent to {\em PLOS ONE} and reviewed by an epidemiological referee: they argued that the paper should not be published as, although the substance of the paper appeared correct, a reader might interpret it as implying that lockdowns were unnecessary. They also argued that the drop in incidence could have occurred despite $R>1$ before lockdown: this latter point inspired the simple $R $ calculation presented in the paper. Editor Abdallah M. Samy rejected the paper (at the end of July) on the basis of this first report, a decision formally overturned on appeal in early August. A second referee then strongly recommended publication in a detailed review (dated late August), and provided useful references on the care home epidemic. At the same round a third referee commented briefly on the paper, stating that recessions save lives and consequently suggesting removal of one paragraph of discussion from the paper. On 20 November Abdallah M. Samy formally invited a major revision, but added his own report stating that the paper was unacceptable, arguing that the discussion of possible human costs of lockdown was inappropriate, that incidence {\em can} be meaningfully inferred from case data (how was not stated), that statistical analysis could suggest nothing whatsoever about what might have happened without lockdown (a dynamic epidemic model is apparently required), that the study needed an ethics committee statement, and that the paper was remiss in not citing Flaxman et al (the update including the Flaxman model having been supplied 2 months earlier). These points motivated the transfer of the paper to the statistical journal {\em Biometrics} and the discussion of causal inference in the discussion. One referee and the AE at {\em Biometrics} made the very useful suggestion to analyse the data up to mid February 2021, and provided helpful extra references, substantially improving the paper. The editor's comments were also helpful. A further (theoretical epidemiologist) referee understood the paper's aim to be implementation of the Flaxman et al. model as a GAM, motivating much clearer signposting of several technical aspects. Their comments on age structure effects and the feasibility of PCR testing based incidence estimation led to the sections in the Supplementary Material on these points, and they also suggested giving calendar dates on plots making them much more readable.

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Mirza ( or ; ) is a name of Persian origin. It is used as a surname or prefix to identify patriarchal lineage.

It is  a historical royal and noble title, denoting the rank of a royal prince, high nobleman, distinguished military commander, or a scholar. Specifically, it was used as a title by (and today signifies patriarchal lineage to) the various Persian Empires, the Nogai Horde, Shirvanshahs and Circassians of the European Caucasus, as well as the Muslim Rajputs and mainly the Mughals / Moguls, both of the Indian Subcontinent. It was also a title bestowed upon members of the highest aristocracies in Tatar states, such as the Khanates of Kazan and Astrakhan.

Etymology
The original title Mīrzā or Merzāh is derived from the Persian term Amīrzādeh which literally means child of the Amīr or child of the ruler. Amīrzādeh in turn consists of the Arabic title Amīr (English: Emir), meaning "commander" and the Persian suffix zādeh, meaning "son of" or "lineage of". Due to vowel harmony in Turkic languages, the alternative pronunciation Morza (plural morzalar; derived from Persian) is also used.

History
Mirza first emerged during the 15th century as an appellative term for members of the Timurid dynasty, adopted in deference to their progenitor, the Central Asian conqueror Timur, who used Amir as his principal title. During the early Timurid period, Mirza preceded a prince's given name, therefore adhering to the Persian fashion, though subsequently the Turkish style was adopted, with the title instead being placed after. This was continued by later rulers such as the Aq Qoyunlus, Safavids and Mughals.

Originally restricted to only kings and princes, the title eventually spread among other social groups, though only the former could have it placed after their given name. During the 16th century, the Safavids conferred it upon high-ranking viziers such as Mirza Shah Hossein and Mirza Ata-Allah Isfahani. By the Qajar period, the title simply marked a person as a clerk or a literate man of consequence. Writing in 1828, Frederic Shoberl records that "as a prefix to the name, it may be assumed by, or conferred on any person. It is right, however, to observe, that none but well-educated men, or such as follow respectable professions, or hold honourable posts, take the title of mirza."

Persian empires

Safavid dynasty

Padeshah Ali Mirza
Prince Alqas Mirza
Prince Mohammad Baqer Mirza
Vizier Mirza Shokrollah Isfahani
Vizier Mirza Salman Jaberi
Vizier Mirza Shah Hossein

Afsharid dynasty

Prince Reza Qoli Mirza
Prince Shahrokh Mirza
Prince Ebrahim Mirza
Statesman Mirza Mehdi Khan
Prince Nader Mirza

Qajar dynasty

Prince Iraj Mirza
Prince Malek Mansur Mirza Shao es-Saltaneh
Prince Bahram Mirza Sardar Mass'oud
Prince Abbas Mirza
Prince Abdol Majid Mirza
Prince Ali-Mohammad Mirza
Prince Bahram Mirza
Prince Djahangir Mirza
Prince Eskandar Mirza
Prince Hamid Mirza
Prince Khanlar Mirza
Prince Khosrow Mirza
Prince Muhammad Mirza
Prince Mahmoud Mirza
Prince Mohammad Hassan Mirza
Prince Mohammad Hassan Mirza II
Prince Ahmad Shah Qajar
Prince Nosrat-od-Dowleh Firouz Mirza
Prince Firouz Mirza Nosrat-ed-Dowleh Farman Farmaian III
Prince Ali Mirza Qajar
Prince Nosrat al-Din Mirza Salar es-Saltaneh
Prince Abdol-samad Mirza Ezz ed-Dowleh Saloor
Prince Mass'oud Mirza Zell-e Soltan
Prince Mohammad Ali Mirza Dowlatshah
Prince Mohammad Taqi Mirza
Prince Kamran Mirza Nayeb es-Saltaneh

Shirvanshahs
Three consecutive titular kings of Shirvan, of the Shirvanshah Dynasty (present-day Azerbaijan), adopted the title as well following the death of Gurban Ali.
 Gasim Mirza
 Kavus Mirza
 Abu Bakr Mirza

Circassians

Circassian dynasty

The hereditary title of Mirza was adopted by the nobility class of the Circassians. Idar of Kabardia, also known as "Mirza Haydar Temruk Bey", was the great-grandson of Prince Inal – Sultan of Egypt the founder of the "Temruk dynasty" of the Kabardian princes, known in Russia as the "Cherkassky" a Circassian princely family.

Circassian nobility with the name Mirza include:
 Mirza Haydar Temruk Bey (ca. 1470 – 1571)
Princely Issues:
 Temruk Mirza (ca. 1501 – 1571)
 Kambulat Mirza (ca. 1510 – 1589)
 Zhelegot Mirza (ca. 1520- ?)

Russian empire
Under Catherine the Great, empress of Russia, the Mirzas gained equal rights with the Russian nobility due to their extreme wealth. Abdul Mirza was given the title Prince Yusupov, and his descendant Prince Felix Yusupov married Princess Irina Alexandrovna of Russia, the only niece of Tsar Nicholas II.

Mughal empire

In the Indian subcontinent, the title Mirza was borne by an imperial prince. It was adopted as part of ones name, implying relationship to the Mongol dynasties like the Mughal dynasty (the Imperial House of Timur). In the traditional naming sequence of the Indian royal families, the title can be placed both before the name and after it, such as Prince Mirza Mughal or Prince Kamran Mirza. Prince Khusrau Mirza was the grandson of Emperor Babur (Babur Mirza), son of Emperor Jahangir and a brother of Emperor Shah Jahan. Emperor Akbar Shah II was known as Prince Mirza Akbar before his coronation. Emperor Babur took the imperial title of Padishah on 6 March 1508, before which he used the title Mirza.

Mughal dynasty

 Mirza Zahiruddin 1494–1530, first Mughal Emperor.
 Mirza Nasiruddin 1530–1539 & 1554–1555, second Mughal Emperor.
 Mirza Jalaluddin 1556–1605, third Mughal Emperor.
 Mirza Nuruddin Salim 1605–1627, fourth Mughal Emperor.
 Mirza Salafuddin Shahryar 1627–1628, de facto Ruler
 Mirza Shahabuddin Khurram 1628–1658, fifth Mughal Emperor.
 Mirza Muhiuddin Aurangzeb 1658–1707, sixth Mughal Emperor.
 Mirza Qutbuddin Azam 1707, seventh Mughal Emperor.
 Mirza Mu'azzam 1707–1712, eighth Mughal Emperor.
 Mirza Muizuddin Muhammad 1712–1713, ninth Mughal Emperor.
 Mirza Muinuddin Muhammad 1712–1719, tenth Mughal Emperor.
 Mirza Shamsuddin 1719, eleventh Mughal Emperor.
 Mirza Rafiuddin 1719, twelfth Mughal Emperor.
 Mirza Muhammad Nekusiyar 1719, thirteenth Mughal Emperor.
 Mirza Nasiruddin Roshan Akhtar, fourteenth Mughal Emperor.
 Ahmad Shah Mirza 1720–1748, fifteenth Mughal Emperor.
 Azizuddin Beg Mirza 1754–1759, sixteenth Mughal Emperor.
 Jalaluddin Mirza 1759–1760, seventeenth Mughal Emperor.
 Muhi-ul-Millat Mirza 1788–1806, eighteenth Mughal Emperor.
 Mirza Akbar 1806–1837, nineteenth Mughal Emperor.
 Sirajuddin Mirza 1837–1857, Mughal Emperor.
 Mirza Abdullah, son of Bahadur Shah II 1850–1897, Mughal Emperor.
 Mirza Aziz Koka (1542–1624), foster-brother of Akbar, son of Ataga Khan

Imperial families of Central India and Bengal

The archaic Bengali form of Mirza was Mridha in Bengal and Bihar.
 Mirza Shuja ud-din Muhammad Khan, second Nawab of Bengal, Bihar and Orissa.
 Mirza Asadullah, third Nawab of Bengal, Bihar and Orissa.
 Mirza Muhammad Ali, fourth Nawab of Bengal, Bihar and Orissa.
 Mirza Mohammad Siraj, fifth Nawab of Bengal, Bihar and Orissa.
 Mirza Hassan Ali Khan Bahadur, 18th Nawab of Bengal, Bihar and Orissa.
 Mirza Wasif Ali Khan, 19th Nawab of Bengal, Bihar and Orissa.
 Mirza Waris Ali Khan, 20th Nawab of Bengal, Bihar and Orissa.

Royal family of Awadh

 Mirza Muhammad Muqim Ali Khan, second Nawab Subahdar of Awadh (Oudh)
 Mirza Amani Asif-ud-Dawlah, fourth Nawab Subahdar of Awadh
 Mirza Asif Jah Wazir Ali Khan, fifth Nawab Subahdar of Awadh
 Mirza Wajid Ali Shah, eleventh Nawab Subahdar (fifth King) of Awadh

Rajput dynasty

Rajputs of Northern India

Originally being adversaries and opponents to the Mughal Emperors, the title Mirza was also adopted by the Muslim Rajputs of Northern India. The Rajput imperial families were descendants of ancient Indo-Aryan warriors who strategically formed blood alliances with Mughal aristocracy. The Rajputs were rulers of princely states comprising vast territories of Northern India, including the Punjab Region, Kashmir and Rajasthan. Inter-marriage between Mughal aristocracy and Rajput aristocracy became very common and various factions of Rajput kingdoms embraced the Islamic faith, giving rise to the term "Muslim Rajputs". Rajput rulers were also granted the title Mirza on account of being high-ranked commanders in the Mughal military. The meaning of Mirza (Persian origin) is identical to the meaning of Rajput (Sanskrit Origin).

Other notable people named Mirza

Sport
 Sania Mirza, Indian tennis player, former WTA number 1
 Fadi Merza, Syrian-Austrian middleweight kickboxer, former world champion kickboxing and Muay Thai
 Mirza Ali Baig, Pakistani mountaineer
 Mirza Bašić, Bosnian tennis player, winner 2018 Sofia Open
 Mirza Begić, Bosnian-Slovenian Olympic basketball player
 Mirza Delibašić, Bosnian basketball player
 Mirza Teletović, Bosnian basketball player
 Mirza Varešanović, Bosnian football manager
 Rasul Mirzaev, Russian MMA fighter

Academics and literature
 Heidi Safia Mirza (born 1958), British academic
 Iraj Mirza, Persian folk poet, also known as Jalaal-al-mamalek.
 Mirza Abu Taleb Khan, tax-collector and administrator from northern India, writer of an early travel guide to Europe.
 Mirza Athar Baig is a Pakistani novelist, playwright and short story writer.
 Mirza Gʻafur Gʻulom, Uzbek poet, writer, and literary translator, considered one of the most influential Uzbek writers of the 20th century.
 Mirza Ghalib (born Mirza Asadullah Baig Khan), an Urdu and Persian poet from South Asia who adorned the Mughal court.
 Mirza Kalich Beg, Sindhi writer.
 Mirza Khan of "Mirza & Sahiba", a tragic romance story, based on true events, which is enshrined in Punjabi literature and commonly told in the Punjab region. Mirza of "Mirza & Sahiba" was of Kharal Muslim Rajput / Muslim Jat tribe of Puar Rajput descent.
 Muhammad Munawwar Mirza, a prominent scholar, historian, writer and intellectual from Pakistan.
 Nawab Mirza Khan "Daagh", Urdu poet.

Arts and entertainment
 Bashir Mirza also known as BM, a Pakistani painter.
 Haroon Mirza, Anglo-Pakistani artist born in London.
 Mirza Kadym Irevani, Azerbaijani artist.
 Shazia Mirza, English comedian.
 Aziz Mirza (born 1947), Indian film director, producer and writer.
 Dia Mirza, Indian actress and former "Miss Asia Pacific" titleholder.
 Mastan Haider Mirza, Indian Mafia boss, mobster and filmmaker; popularly known as the first "celebrity gangster" of Bombay.
 Mirza Babayev, Azerbaijani movie actor and singer. Honored Artist of the Azerbaijan SSR and People's Artist of Azerbaijan.
Mirza Nadeem Baig Mirza Nazeer Baig Mughal better known by his stage name Nadeem Baig, a Pakistani actor, singer and producer.
 Mohib Mirza is a Pakistani actor and television host.
 Saeed Mirza, Indian film director and screenwriter, considered one of the most influential parallel cinema movie makers in India.

Government
 Mirza Fakhrul Islam Alamgir, Bangladeshi politician, the incumbent secretary general of the Bangladesh Nationalist Party (BNP) since 2016
 Mirza Abbas, Bangladesh Nationalist Party politician. He was the Mayor of Dhaka City Corporation from 1991 to 1993
 Fahmida Mirza, first female Speaker of the National Assembly of Pakistan
Hasnain Mirza, lawyer, barrister and Member of Parliament who was a Member of the Provincial Assembly (MPA)
 Iskandar Ali Mirza, the first President and 4th Governor General of the Islamic Republic of Pakistan in 1956
Mirza Ali Asghar Khan Amin al-Soltan, the last prime minister of Iran under Naser al-Din Shah Qajar
 Mirza Ismail, Prime Minister, Jaipur (Diwan of Jaipur) (1942–1946)
Mirza Kuchik Khan, early 20th-century revolutionary, Gilani Nationalist and the president of the Republic Of Gilan
 Mirza Muzaffar Ahmad, a Federal Finance Minister and Finance Secretary, Chairman of Planning Commission of Pakistan Executive director of the World bank
Mirza Nasrullah Khan, the first Iranian Prime Minister
Munira Mirza, Deputy Mayor of London and the Director of the Number 10 Policy Unit under UK Prime Minister Boris Johnson
Zulfiqar Mirza, Pakistani politician affiliated with the Pakistan Peoples Party (PPP)

Judges and advocates
 Mirza Aziz Akbar Baig, former Vice Chairman of the Pakistan Bar Council.
 Zafar Hussain Mirza, Pakistani judge and the father of former Home Minister of Sindh Zulfiqar Mirza.

Journalists
 Janbaz Mirza was a writer, poet, and journalist from Pakistan.
 Tahir Mirza was a senior Pakistani journalist and former editor of Dawn, Pakistan's oldest and most widely circulated English-language newspaper.

Military
Mirza Ahmed Bey, one of the original Punjabi soldiers of the famed "Hodson's Horse" regiment of the British Indian Army, pictured in the historical 1858 photograph. He was a descendant of Mirza Hakim Bey, after whom the Indian village Hakimpur, Gurdaspur District is named.
 Mirza Aslam Baig a Former Chief of Army Staff of Pakistan.
 Mirza Baqi, 17th-century Mughal general
 Mirza Ghulam Murtaza was the chieftain of Qadian and a commander with the Sikh Army.
 Mirza Kuchak Khan was a Persian revolutionary who led the Jungle Movement in the northern jungles of Gilan Province.

Nobility and royalty
 Khan Muhammad Mirza, architect during the Mughal Era.
 Mirza Ali Behrouze Ispahani, Bangladeshi businessman from the Ispahani family and the chairman of M. M. Ispahani Limited.
 Mirza Hadi Baig, Mughal nobleman from Samarkand who was granted 80 villages by Babur.
 Mirza Najaf Khan, Indian courtier and Commander-in-Chief of the Mughal Imperial Army.
 Sikandar Mirza, Armenian courtier within Akbar's Mughal Empire
 Umer Sheikh Mirza, ruler, father of Mughal Prince Mirza Babar

Other people
 Mirza Ghulam Ahmad, Indian religious leader

See also

 Rana (title)
Baig, Bey
Beg Khan
Emir
Khanzada
Begzada
Morza
Mughal (tribe)
Muslim Rajput
Nawab
Shahzada
Sultanzada
Mirza Ghassemi
Sayyid

Further reading

"Life of a Mirza" Chapter 7 (pp. 225–227) The Empire of the Great Mughals: History, Art and Culture (2004), by Annemarie Schimmel, .
Mirzah in The Wordsworth Dictionary of Phrase and Fable by Ebenezer Cobham Brewer, .
MI'RZA in Chambers's Encyclopaedia: A Dictionary of Universal Knowledge For the People. .
A. Jaimoukha The Circassians: A Handbook Routledge, Palgrave, 2001, pp. 157–60, .

Footnotes

References
Specific

Sources
Brewer's Dictionary of Phrases and Fable, 16th edition Revised by Adrian Room, 1999
WorldStatesmen
Etymology Online

Noble titles
Surnames
Court titles
Royal titles
Titles in Iran
Titles in Azerbaijan
Titles in Russia
Titles in Bangladesh
Turkish titles
Turkic culture
Titles in Pakistan
Persian words and phrases
Pakistani families
Muslim families
Mughal nobility
Ottoman titles

redpajama_set_name: RedPajamaWikipedia


text:
Marky Mark, Andrea Boccelli and Aretha Franklin's free live concert
The Festival of Families - a giant free concert on Saturday on the Ben Franklin Parkway. It's emceed by Mark Wahlberg, features Andrea Boccelli, Juanes, Aretha Franklin and more. Pope Francis is expected to make a cameo.
Where: The Ben Franklin Parkway
When: September 26, 2015 from 7:30 p.m. to 10:00 p.m.
Chris Krewson
chris@lionpublishers.com @ckrewson
Chris Krewson is the executive director of LION Publishers, a national nonprofit association that serves local journalism entrepreneurs build sustainable news organizations, and the founding editor of Billy Penn. He lives in Havertown.


text:
The conservation and heritage charity for the
Explore Nature
Enjoy The Park
Sir Walter Scott and the discovery of the Trossachs and Loch Lomond.
Around 1790, Sir Walter Scott, as an apprentice in his father's law office, had to deliver legal documents to some 'difficult' tenants in Balquhidder Glen. Trouble was expected, and Scott picked up an escort of six soldiers and a sergeant from Stirling Castle. From this sergeant – who knew the ground well - he heard more tales of Rob Roy, to add to those he had already learned from an old Highland client of his father who had known Rob personally.
He further added to his growing familiarity with the Rob Roy country in the 1793 when, as a newly qualified advocate, he visited several of the homes of other legal associates and friends who came from the area around Callander.
For some years afterwards he pursued his career and also became known as the author of 'The Lay of the Last Minstrel' and 'Marmion'. He decided to put his store of knowledge of the geography and folklore of The Trossachs and Loch Lomond area into another creative project. By the summer of 1809 he was at work on 'The Lady of the Lake'. He rode from Loch Venachar to Stirling to check the time it would take his hero Fitz-James who makes the journey in the story. He then systematically studied the terrain around Loch Lomond. He even read out the first part of the poem while staying with the Duke of Montrose at Buchanan Castle.
'The Lady of the Lake' was published in 1810 and was a sensation in its day. Huge numbers of visitors then descended on the Trossachs - far more than the then primitive 'tourism infrastructure' could cope with! However, the visitors found the scenery very much to their taste.
This 'cult of the picturesque' was reinforced by artists who painted suitable Highland scenes sold as prints to meet demand. Much artistic licence can be noted in the works of this era. Alexander Nasmyth, for example, who painted Loch Lomond as well, thought that his Loch Katrine view could only be enhanced by inserting a classical temple into the composition, in the manner of the classical landscape allusions of the earlier painter Claude.
Park Clans
Literary Park
Tourist Era
Park Place Names
Park People
John Groome Diaries
Park Stories
Loch Lomond Song
Jacobite Uprising
Glen Finglas Scandal
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The Remarkable Rev Robert Kirk
Tourists in the Trossachs
Tom Weir
Park Heritage Group Links
Sponsor Tree Planting
From as little as £12 help plant native trees with protective tubing and stakes to restore some of the Park's special landscapes. Click here to donate online.
Designed and Developed by Mucky Puddle | Friends of Loch Lomond & The Trossachs is a registered Scottish charity (SC015389)


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text:
News > Companies
Midway Air calls it quits
September 12, 2001: 3:38 p.m. ET
Bankrupt airline sees drop in travel, shuts down; 1,700 workers lose jobs
NEW YORK (CNNfn) - Bankrupt Midway Airlines said Wednesday it is ceasing operations immediately in order to preserve the company's value, in anticipation of a sharp drop in air travel after Tuesday's terrorist attacks.
Midway said 1,700 employees will be terminated effective immediately and will be paid at their next scheduled payday.
"We are deeply troubled by the impact this action will have on our customers, employees, creditors and community," a Midway spokesman said. "Unfortunately, we simply do not have the resources necessary to permit us to reorganize in this environment."
The company said it will begin returning aircraft and look for purchasers for assets immediately. All ticket holders for future travel will be given refunds or tickets on another carrier.
Instructions for refunds are available on the company's Web site.
Midway filed for Chapter 11 bankruptcy protection on Aug. 14, letting 700 workers go.
At the time of the announcement, all airline flights in the United States still were grounded by the Federal Aviation Administration one day after hijacked airliners destroyed the World Trade Center's twin towers in New York and damaged the Pentagon near Washington, D.C.
Industry experts have predicted demand for air travel will fall because of consumers' safety concerns after the attacks.
Midway said it anticipates paying all obligations incurred following its Chapter 11 bankruptcy filing on Aug. 14 and will propose a reorganization plan to deal with all obligations that arose prior to the filing.
-- from staff and wire reports
Skies safe from layoffs - Aug. 24, 2001
Midway Airlines
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A silent debate seems to be raging as to which is better being a 'stay at home' or a 'working mother'. A few people consider it their utmost duty to inform you that if you choose to not work you are probably wasting your education and talent. On the contrary some assert that they can be a better mother if they stay at home. Indeed being a 'sahm' in recent times to me seems no less than a luxury . It is the easiest way to follow your child's round the clock developments unless they are school going. Not to forget a house wafting in the smell of elaborately cooked meals, at times done fresh everyday. Those frequent cake or cookie making sessions , doing various craft , watch your kids do a jiggle at the tune on the T.V but rarely get to watch anything remotely adult or a simple trip to the park, library or the mall. The list is endless.
And then there is the other kind of mother who seeks to acquire the best of both worlds. It was my short work-experience which changed my perspectives and expanded my vision about the working mother. Closer home my mother in law had been a working mother all her life and I was well aware of her home management techniques which were a world apart from my mothers' who always stayed at home cooking warm meals and being there for us when we returned from school. Not to miss on saying all the smart tips and tricks that I learnt from my mother in law.
It is no mean feat to be a working mother. The basic foundation of her life lie upon her organizing skills.
It also depends on your ability to take stress and that is going to be loads. Whoever said that sahm are stress free . Add to that some extra stress loads and put a final cherry on top of meeting deadlines at home and outside. That's the life of a working mother. If you are lucky you might just get the time to breathe.
When you are returning from work in the evening the food is not going to put itself on the table and the dishes are not going to clean themselves. Somebody got to do what needs to be done. Besides unless you have managed to learn a few tips that makes it easier for you to be food-ready every time you are home you must be ready to avail the service from elsewhere. Their is only so much you can buy from your neighbourhood restaurant especially when you have kids unless you have thrown any concerns of your children's good health to the winds and McDonald fries and burgers is your staple diet. On the contrary, If you are someone who is probably spending most of your time in the kitchen, this life is not for you.
As a working mommy surely you have also tried loads of kitchen and cooking hacks so that you spend the least time cooking.
Cooking in bulk becomes a part of one's life.
In addition while I was working I would always wake up 2 hours earlier than normal so it entailed waking up when the sun was not even up in the sky. Early to bed or earlier to rise or something like that!
Mainly life is all about planning and execution, their is little time to think or relax. You have the time to put your feet up only when you are not a mommy. Imagine mommy's never get a leave even if their is the weekend she happens to be on duty albeit a 24*7 one. Another feature which plays a huge role in your life is the number of children and most importantly their age and if they are really young ,their temperament. Not all children take to the day-care life really well. Some kids make it turn out real ugly.
How easy it is for you to manage your children is also an important factor. Being a mother of two and providing for a baby was a time consuming affair for me. Also it really matters who you leave your children with, it also helps if its family. However not every person in the world today can afford that luxury in this globalized world.
Not to forget availing of childcare services is an expensive affair at times. If you are making very little profit then it makes no point to slog at work.
Working outside can drain your energy as well and by the time you are back all you can do is sit on your couch, grab some food and stare at the T.V listlessly. Well that was only a fantasy that you can have. Remember ? You are a mother!
Remember your kids will grow up!
Hoever sahm are more at the risk of developing mental health issues which is why it is also important for the sahm to have something that she enjoys doing and which is a source of self esteem and self satisfaction for her. This is why a lot of mom are turning towards alternative careers working from the proximity of her home such as blogging , owning a franchisee or a small business, whatever seems to work.
The problem is not the way a mother decides to rear a child -by staying at home or being a career woman. It arises when society starts deciding which is better. The choice should remain with the mother alone to decide what is best for her child .It takes a lot of extra hard work at times in most cases, for the working mom. However if every woman stayed at home we would not have that awesome teacher or doctor or scientist or whatever it is that you do. At the same time the mother's who choose to stay home for their child deserve all the respect for choosing to do whatever makes them happy. As for me I hope to do whatever little I can do well enough. As for now I am enjoying writing this. So for me the debate ends when I realize its all about enjoying what you are doing and whatever it is that makes you happy as a woman , chances are that probably you will do it well. I would best conclude that all mothers are the best, no matter how they do it!
It would be better if a parent stay home when the kids are young. It is a guaranteed that the kids won't get abused or neglected if they were to go to day care.
I know some working mothers who would give their right arm to stay home with their children…..
I know these days a lot of people are distrustful of leaving their kids behind with strangers , I agree with you!
The parents have every reason to be distrustful…..
There are recipes called "dump dinners" where you go shopping on your day off and pre-prep dinners to be tossed into the crock-pot, they are a working mom's savior as well as a stay at home moms lifeline, I've found. You can make them as deliciously nutritious as you want, with veggies and meats and such. They're easy to prepare and if your children are middle school or high school aged, they can "dump dinner" in the crock pot when they get home so it's already ready come dinner time. Most of them take little to no time to prep and cook within 3-4 hours, or you can do them on low heat and put them in yourself before going to work or on your lunch break if you live close enough! Being a stay at home mom, I've taken to home schooling my girls due to the lack luster educational material and $$ requirements for sending them to public school since they require you to buy oodles of stuff that they usually don't even use all of.
Hi dear, I actually wrote this blog months back and I so agree with you over the goodness of the crock pot. I am lately discovering its goodness and wonder why I did not know earlier. Like you said it is such a life saver. Life can be boring when so much time is taken up with cooking.
I am also intrigued by homeschooling. My daughter is usually crying every morning when I pack her off to school. I agree with what you said about school these days.
Homeschooling seems to be something that requires both a lot of time and dedication as well as a willingness to take a moment when you become overwhelmed. I'm still adjusting to that last part! I've picked up homeschooling mainly for the fact that we can do it at any given time of the day, we can study what we want. Right now we are working on building respect for others, proper table manners, when to say please and thank you, while also doing alphabet tracing, learning to read, colors and shapes, with a touch of potty training thrown in. My girls may only be 2 & 3 years old but they have much to learn!


Changing an organization's culture is one of the most complex challenges that a business leader will ever face.
But there are times when a cultural shift is essential for a business to survive and grow. That's precisely the challenge facing HumanSystems® (HSI®), a Guelph-based consulting and applied research company focused on improving the performance and well-being of people in complex systems.
Founded in 1982 by Bob Webb, a former British Army engineer officer and University of Guelph professor, the company has a well-established reputation for developing innovative solutions to complex problems involving people, performance and technology, particularly in the defence sector. However, over time the company's concentration on defence work made it vulnerable.
To help figure out how to make that happen, HSI® turned to Innovation Guelph. Starting with workshops at IG on topics such as storytelling and marketing communications, HSI® also enrolled in the Fast Lane program that provides diagnostic and tune-up services to help established companies identify problems and overcome barriers to growth.
With 15 to 18 staff and a network of research associates, HSI® handles 30 to 50 projects a year: analyzing, defining and understanding complex issues; collaborating with other technical specialists to solve organizational and technical challenges with human users in mind; testing and evaluating equipment from the user's perspective; developing and evaluating custom e-learning and training programs. The guiding principle in all their work is that people are the most important part of any system.
Working with IG mentors Steve Barrett and Niel Palmer, the company has begun the process of shifting from a highly technical mindset, well-attuned to navigating the complexities of government contracting, to a proactive, profit-oriented approach needed to serve a more diversified client base.
But change doesn't come easy, and Palmer praises the HSI® team for being open to the possibilities.
To help HSI® plot a new course, Barrett and Palmer recommended a 90-day "strategic sprint" designed to break up a long, complex change process into focused, achievable goals. The first sprint involved taking a critical look at areas such as: defining the ideal client; value propositions; communications and marketing practices; sales and billing processes, human resources and more.
"The 90-day sprint allows you to focus on the items that you need to focus on now," says Palmer. "The sprints help keep everybody on the same page and minimize distractions caused by other longer-term projects or priorities," he adds.
Key issues are tackled at weekly "Tiger Team" meetings — an approach that emphasizes frank discussions, intense focus, out-of-the-box thinking, and short timelines.
"With this company, it's not just about identifying particular problems — it's about a cultural shift. The cultural shift needs to gain traction internally in order for the company to become efficient at identifying and exploiting new opportunities. That's why these 90-day sprints are so important. You've got to keep your eye on the ball, focus on the target, get some things done, and move on," Palmer explains.
The Tiger Team group has made considerable progress and recently began their third 90-day sprint. The team is adapting a LEAN management approach to streamline their business processes, improve productivity and focus on client needs. They are also working to better align the goals and expectations of the technical/field staff with the business/administration side of the organization and establishing a dedicated professional services marketing team.
"The value or return on investment (ROI) of the company's expertise in promoting performance, safety and efficiency in defence systems is transferable to other systems. While this transfer of knowledge is most readily recognized by other high risk sectors such as utilities, and emergency services where human performance is critical to avoid catastrophic failure, it is also relevant anywhere people work, live or play. Our great challenge in expanding to broader markets is to explain this potential ROI in terms that resonate with managers in other sectors," says Ste-Croix.