key: cord-216208-kn0njkqg
authors: Botha, Andr'e E.; Dednam, Wynand
title: A simple iterative map forecast of the COVID-19 pandemic
date: 2020-03-23
journal: nan
DOI: nan
sha: 
doc_id: 216208
cord_uid: kn0njkqg

We develop a simple 3-dimensional iterative map model to forecast the global spread of the coronavirus disease. Our model contains at most two fitting parameters, which we determine from the data supplied by the world health organisation for the total number of cases and new cases each day. We find that our model provides a surprisingly good fit to the currently-available data, which exhibits a cross-over from exponential to power-law growth, as lock-down measures begin to take effect. Before these measures, our model predicts exponential growth from day 30 to 69, starting from the date on which the world health organisation provided the first `Situation report' (21 January 2020 $-$ day 1). Based on this initial data the disease may be expected to infect approximately 23% of the global population, i.e. about 1.76 billion people, taking approximately 83 million lives. Under this scenario, the global number of new cases is predicted to peak on day 133 (about the middle of May 2020), with an estimated 60 million new cases per day. If current lock-down measures can be maintained, our model predicts power law growth from day 69 onward. Such growth is comparatively slow and would have to continue for several decades before a sufficient number of people (at least 23% of the global population) have developed immunity to the disease through being infected. Lock-down measures appear to be very effective in postponing the unimaginably large peak in the daily number of new cases that would occur in the absence of any interventions. However, should these measure be relaxed, the spread of the disease will most likely revert back to its original exponential growth pattern. As such, the duration and severity of the lock-down measures should be carefully timed against their potentially devastating impact on the world economy.

March 2020, the world health organisation (WHO) characterised the 2019 outbreak of coronavirus disease (COVID-19) as a pandemic, referring to its prevalence throughout the whole world 1 . The outbreak started as a pneumonia of an unknown cause, which was first detected in the city of Wuhan, China. It was reported as such to the WHO on the 31st December 2019, and has since reached epidemic proportions within China, where it has infected more than 80 000 citizens, to date. During the first six weeks of 2020 the disease spread to more than 140 other countries, creating wide-spread political and economic turmoil, due to unprecedented levels of spread and severity.

The rapid spread of COVID-19 is fuelled by the fact that the majority of infected people do not experience severe symptoms, thus making it more likely for them to remain mobile, and hence to infect others 2 . At the same time the disease can be lethal to some members of the population, having a globally averaged fatality ratio of 4.7%, so far. It is most likely this particular combination of traits that has made the COVID-19 outbreak one of the largest in recorded history.

In late 2002 and early 2003 a similar outbreak took place with the occurrence of severe acute respiratory syndrome (SARS). Although the etiological agent of SARS is also a coronavirus, the virus was not able to spread as widely as in the current case. One possibility why the SARS outbreak was less devastating than the current outbreak is, paradoxically, due to its much higher fatality ratio (almost 10% globally), making it too severe to spread easily.

While there are a number of models available for the global spread of infectious diseases 3 , some even containing very sophisticated traffic layers 4 , relatively few researchers are making use of simpler models that can provide the big picture without difficult to interpret unambiguously. In the latter category of relatively simple models we could find only a discrete epidemic model for SARS 5 , and more recently, a comparison of the logistic growth and susceptible-infected-recovered (SIR) models for COVID-19 6 .

In our present work we develop a simple discrete 3-dimensional iterative map model, which shares some similarities with the classic SIR model. We show that our model fits the currently-available global data for COVID-19. The fact that the available data for the pandemic can be fitted well by a simple model such as ours suggests that past and current interventions to curb the spread of the disease, globally, may not be very effective.

As a model for the global data we use a 3-dimensional iterative map, given by

where x i is the total number of confirmed cases, y i is the number of new cases and z i is the global population, on any given day i. We denote the only fitting parameter by α, while c is a fixed parameter equal to the fraction of people who have died from the disease. According to the latest available data from the WHO (see Table 2 in Methods ), c = 0.04719. By using Levenberg-Marquardt (least squares) optimisation 7 we find α = 1.14594, for the initial condition x 30 = 75152 y 30 = 359.63 and z 30 = z 0 = 7.7000 × 10 9 .

We briefly describe the physical content of Eqs. (1). The first equation simply updates the total number of cases by setting it equal to the previous total number of cases, plus the number of new cases. Here the factor of 1/z 0 has been introduced for convenience, to ensure that the proportionality constant α remains close to unity. In the second equation the number of new cases is assumed to be proportional to the previous number of new cases multiplied by the previous number of susceptible people. The third equation keeps track of the global population by subtracting the estimated number of people who have died each day, based on the fraction c. Figure 1 shows a comparison of the data with the model, as well as a forecast made up to the 200th day. As we see in Table 1 ). The forecast made in Figure 1 (b) (corresponding to the last row of Table 1 ), predicts that approximately a quarter of the worlds population, i.e. ≈ 1.83/7.7 = 0.24, would have had COVID-19 by the 200th day. The peak of the pandemic is expected to occur on day 129, when about 65 million daily new cases can be expected. We also predict that by the beginning of August 2020, hardly any new cases should occur; however, the total number of lives lost by then could be as high as 86 million.

In Table 1 we see that the fitting parameter α, and hence the predictions made by the model, do change somewhat as more of the available data is used in the fitting procedure. To see the variation in α more clearly we have plotted the first and second columns of Table 1 in Figure 2 . It shows that, as more data is used, there seems to be a general upward trend in α, until day 64. At the same time the increase in α is not monotonic, since α appears to oscillate. While we are not sure, at this stage, whether α is converging, or whether it will continue to increase (or decrease) generally in the future, we note that the variations in α and the predictions made by the model, are relatively small over the last two weeks. Thus it seems like, as more data is used to calculate α, the variations will become smaller -assuming that there is no systematic errors in the current or future data (see Discussion).

Last day α

x 200 × 10 9 max{y} × 10 6 Day of max{y} In Figure 2 we also plot (blue solid line) the mean valueᾱ = 1.156 over the last ten days. The oscillations of the calculated data points about this line give an indication of the uncertainty inᾱ over the last ten days. As a rough estimate of the Figure 2 . Variation of the fitting parameter α as more and more of the available data is used in the fitting procedure. We see that the value of α seems to have stabilised over the last ten days, as discussed in the main text.

uncertainties in the predictions made by the model, we also calculate the means and standard deviations of the other quantities that are given the last ten rows given in Table 1 . This results in x 200 = (1.94 ± 0.06) billion, max{y} = (74 ± 5) million, peak day = 126 ± 2, 'Deaths' = (86 ± 3) million. While we realise that this method may not result in rigorous estimates for the uncertainties involved, we provide it here merely as rough estimate of the sensitivity of our simple model to the new data 3/7 coming in, as the pandemic continues.

From the trend that can be seen in Table 1 , it seems that our current model actually provides a best case scenario prediction since, as more data becomes available, the resulting predictions become less and less optimistic, i.e. in terms of the total number of lives lost, etc. Furthermore, as the disease spreads there will probably be many more unreported cases, either due to asymptomatic responses, or simple because the numbers are now becoming too large to manage (see, for example Ref. 8 ). In developing countries such as South Africa there is also a relatively large percentage of people with compromised immunity, due to the high prevalence of human immunodeficiency virus (HIV), and this also could result in the coronavirus having a much larger impact than our model of the current global data shows. Another factor to consider is the reliability of the WHO data itself. At present, this data is probably the most accurate we will ever have. However, as things progress, there will be a much greater chance of unreported cases, since people are now being instructed to contact the hospitals only if they experience severe symptoms. This means that all other cases are unlikely to be tested/confirmed. Our present model does not take into account such details.

One can of course try to answer more specific questions with a more sophisticated model, like the discrete model we mentioned for SARS 5 ; however, here we have been more interested in developing a very simple model that brushes over the details and only captures the essential, large scale behaviour.

As we have already alluded to, our model may not be suitable for individual countries, because it does not include many factors that may be necessary to predict the spread of the disease in specific situations. Additionally, one must realise that the population of one country is much smaller that the world's, and the initial interventions taken could range from minimal to very severe, as in Italy and China, for example. In contrast with this, on a global scale, the population is essentially limitless, and it is nearly impossible to impose restrictions on everybody. Hence, it is our contention that the virus will spread more naturally on a global scale, almost as if it were left completely unchecked.

The direct human cost of such an unchecked spread could be truly devastating 9 . On the one hand it could result in a catastrophic loss of tens of millions of lives, as our model predicts, but on the other hand, all the (possibly ineffective) measures being taken by individual countries to contain the virus, could also have fatal consequences. So far these measures have included enforced quarantine, which has led to a severe slowdown in economic activity and manufacturing production, principally due to declining consumption and disrupted global supply chains 10 . (As an example of the severity of the slowdown in production, several major car manufacturers are gradually halting production in major manufacturing hubs throughout the developed world 11 .) This decline, coupled with the associated economic uncertainty, has had knock on effects in the form of historically unprecedented stock market falls 12 . Although the stock market is more of an indicator of the future value of the profits of listed corporations, their collapsed share prices could trigger severe financial crises because of a spike in bankruptcies. (The debt of US corporations is the highest it has ever been 13 .) The inevitable loss of jobs will also lead to an inability to pay bills and mortgages, increased levels of crime, etc. In principle, such a major decline in economic conditions could also result in a large-scale loss of life, which should be weighed carefully against the direct effects that the unimpeded global spread COVID-19 could have.

We have fitted our model to the data shown in Table 2 .

For the reader's convenience, the complete python script for the optimisation is provided on the following page. In this script, the function leastsq(), imported from the module scipy.optimize 15 , uses Levenberg-Marquardt optimization to minimize the residual vector returned by the function ef(). The function leastsq() is called from within main(), which reads in the data and sets up the initial parameter and the other two quantities (the initial values x[0] and y[0]) for optimisation. These three quantities are then passed to leastsq(), via the vector v0.

For the data in Table 2 , the output from the script should be: 

who-director-general-s-opening-remarks-at-the-media-briefing-on-covid

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A. E. B. would like to acknowledge M. Kolahchi and V. Hajnová for helpful discussions about this work. Both authors wish to thank A. Thomas for uncovering some of the related literature.

 New Deaths  30  75204  1872 2009  31  75748  548  2129  32  76769  1021 2247  33  77794  599  2359  34  78811  1017 2462  35  79331  715  2618  36  80239  908  2700  37  81109  871  2762  38  82294  1185 

A. E. B devised the research project and performed all the numerical simulations. Both authors analysed the results and wrote the paper.

The authors declare no competing interests.
