Coronavirus Critical Curves: Crucial Insights from Mathematical Modelling

Analysis of trends, perspectives and lessons for Kenya and Africa

Grim Coronavirus Statistics: Summary of Key Metrics

We are starting the weekend in Kenya with more than a hundred confirmed cases of the novel coronavirus infections. The confirmed global cases have exceeded one million. Taking the reported cases of infections, the average global death rate of 5% and the average recovery rate of 21% paint grim statistics. These global figures, so far, translate into 7 deaths for every one million people on Earth! From research into the latest data from 21st March 2020, the business-as-usual scenario gives the global equation:

, which projects the worldwide coronavirus cases to hit 1.5 – 1.7 million by 7th April 2020. This insight introduces a new week likely to record 2 million coronavirus cases worldwide by12th April 2020.

This study report has covered 14 countries which have posted enough data for trend analysis, including China, with data from 10th January to 31st March 2020. Over the study period, the observed daily growth rates in the exponential phases varied from 11% in Hong Kong to 26% in the USA, staggering rates by any measure. Based on this scope, it has been taking on average about a month (33 days) for the critical exponential phase of growth in coronavirus infection cases to commence. Nearly three weeks (17 days) has been the average time of stay of the critical exponential phase of growth in coronavirus infection cases, followed by a marked drop off the exponential trajectory. The lesson for Kenya and the rest of Africa is that, depending on the efficiency and effectiveness of responses, it may take a wide range of two weeks to two months for different countries to start recording an exponential rise in total COVID-19 cases.

Kenya started recording a sharp rise in confirmed coronavirus cases on 27th March 2020 (with a total of 31 cases). From the data range covering 27th March to 3rd April 2020, Kenya’s simulation curve becomes

. Though this may not be the critical exponential phase yet, it is already instructive enough for simulating the scenario of possibilities to inform government preparedness and coordinated action. Taking lessons from the observed rates in the 14 countries, let’s assume an optimistic scenario where the observed exponential growth phase lasts only three weeks. Assuming a business-as-usual scenario of 20% daily growth rate in confirmed cases and a pessimistic scenario of 25% daily growth rate over this phase (considering also that the ongoing curfew and growing measures of caution may prevent shooting to high rates of 30% and above), this study projects in the next few weeks of April the following numbers for Kenya:

  • Business-as-usual scenario – 4th April (149); 9th April (404); 14th April (1,098); 19th April (2,985); 24th April (8,113); 30th April (29,936).
  • Pessimistic scenario – 4th April (221); 9th April (774); 14th April (2,700); 19th April (9,426); 24th April (32,899); 30th April (147,443).

As detailed in the projection table, by 24th April 2020, Kenya could be staring at total confirmed COVID-19 cases of above 8,000 in the business-as-usual (BAU) case and more than 30,000 in the pessimistic scenario. The seemingly small difference of 5% in rate could be responsible for a wide gap by 30th April 2020: 30,000 in the BAU scenario or a stunning figure of almost 150,000 cases in the pessimistic scenario. This is why citizens and the government must take extra caution to contain the exponential spread of COVID-19, which thrives more on mass transmission by group contacts.

Warning Signs from the Battered West

As at 3rd April 2020, in terms of total coronavirus cases per one million population, Spain had a high index of 2500, Switzerland 2200, Italy 1900, and Germany 1000 (to the nearest hundreds). The USA was still leading at over 245,000 cases for a single country. Since her first confirmed case on 13th March 2020, Kenya has this weekend recorded a death rate of 3% of the total coronavirus cases and a recovery rate of almost 4%. If the emerging global trends with key lessons from the hardest-hit countries are anything to go by, then Kenya as a model case of African experience must up her game in the face of the explosive COVID-19 realities.  

The Science-Policy Interface: Quest for Projections in aid of Decision Making

The COVID-19 pandemic commands the attention of all policymakers at these critical moments to the projections scientists are making to inform plans to contain its explosion. In Kenya and other African countries, where COVID-19 has been a latecomer, there is room to learn from the experiences of the countries visited earlier by COVID-19 and are now bearing the brunt of the pandemic. Decision support for policy and strategy needs reliable models for simulation of scenarios and gaining insights into the most effective intervention points, known as high-leverage points. Health ministries need a compact bandwidth of the most likely scenarios to guide the effective planning and management of the pandemic. This is where science comes in through applied mathematical modelling.

Basics of Modelling for People in a Hurry

Models are important decision-support tools for policy and planning. Models cannot replicate reality. At best, models are only selective abstractions of the real world within margins of error befitting the purposes for which they are built. Models help us to reduce uncertainties into estimable risks amidst the divergent scenarios which accompany non-linear occurrences, such as the pervasive spread of pandemics. In an era of data-driven decisions, the value of data as the raw material for quality modelling is gaining currency.

Models should be as simple as possible and as complex as necessary. The rule of thumb in modelling is to generate a range of possibilities to guide decision making, not to engage in the wild goose chase of trying to accurately predict the (uncertain) future. The axiom attributed to the British statistician, George E.P. Box, applies, “All models are wrong, but some are useful.”

Models can be deterministic or stochastic depending on the law governing the level of uncertainties they handle. In terms of knowledge and data, we have two main categories: mechanistic models and statistical models. The former type is useful where we have good knowledge or hypothesis of the mechanisms driving the system despite limited data availability, with just a bit of some historical data. The latter type is applicable where we have sufficient data on system behaviour over time and can use algorithms to create a model of dependencies.

In terms of purpose, we use prediction models for forecasting with some measure of prediction error; narrative models are aimed at transforming mental models through the persuasive power of sharing a convincing storyline of likely outcomes; and inferential models are used to test hypotheses using a measure of p value for statistical significance. In terms of the finiteness of intervals in data or events, we classify models into discrete models and continuous models. Discrete variables can be classified into a distinct countable number of numeric values but continuous variables, like an infinite continuum, cannot fit into such a definite range.

In a complex chain of causality like the case of coronavirus across different countries and cultures, it is important to note that the assumed relationships between independent variables (x) and dependent variables (y) get modified significantly by (i) moderating variables inherent in culture and attitudes and (ii) intervening variables such as health systems and environmental variables. For practical simplicity, reliable model outputs can still be obtained using sufficient infection data as inputs – in an exercise where the modeller doesn’t have to be overly concerned with intricate mechanisms within the imaginary “black box”.

Against this background and given the growing data on COVID-19 cases, it should be clear why statistical models are a good choice for decision support. In Africa, where coronavirus has been a latecomer, advice for policy and planning can benefit from statistical modelling to gain a good understanding of the scenario of possibilities in aid of practical interventions.

Modelling the Critical Curves of Coronavirus Cases

This release has analysed the growth trend of the novel coronavirus infections in 14 countries from January to March 31, 2020. They include China and the others which have been bearing the brunt of COVID-19 such as the USA, Italy, Spain, and Germany. In the table are the key parameters analysed from the data on total reported cases, death rates, recovery rates, and the mathematical equations derived to simulate the critical growth phases.

The following curves of the growth trends have been plotted for the critical growth phases of coronavirus infections. As shown, the cases have generally been tracing an exponential trajectory before dropping in rate towards the desired “flattening of the curve”. It is well known that for an exponential equation

 given time interval (t), the growth rate (r) is the critical component of high leverage. With rising numbers in X, even a change as small as 0.1% in r can cause an immense surge in growth.

Insights from the Models and Implications for Interventions in Africa

Since the data has been drawn from experiences outside Africa, the uniqueness of African countries is a strong factor in weighing the scenario of possibilities. Key considerations must be given to the status of health infrastructure and the glaring structural differences in governance, technological, environmental, sociocultural, demographic, financial and economic settings. The following key metrics and lessons come out of the 14 countries studied and offer key points for policy interventions in Kenya/Africa.

  • Nearly three weeks (17 days) has been the average time of stay of the critical exponential phase of growth in coronavirus infection cases, before a marked drop off this growth trajectory. Note: Consider that the 14 countries have superior social and technology infrastructure to the one in Kenya/Africa.
  • Generally, the longer the time it has taken the countries since their first reported cases to reach the critical exponential phase, the shorter this phase has stayed on before tipping towards a flatter curve. Switzerland has been a clear exception to this general trend. Note: Measures that help reduce community transmission like social distancing are key to flattening the curve. A faster rise in coronavirus cases in early stages may portend a massive challenge ahead for a country in trying to flatten the curve.
  • Germany has presented an interesting combination of a low death rate and a relatively high recovery rate within the hard-hit European region. Over the review period of January – March 2020, the leading death rates among the 14 countries have been recorded in Italy (12%), Indonesia (9%), Spain (9%), and the Netherlands (8%). Ranging barely 1% to under 4%, the observed recovery rates have been particularly low in Sweden, the UK, the Netherlands, and the USA. Note: Containing COVID-19 summons a keener look into the most infected demographic. Indonesia and the USA boast younger median ages (30’s) than the European countries’ (40’s), yet their single-digit record of recovery rates has been much lower than the double digits in Italy and most of the other European countries. This is counterintuitive, if not suggestive of a skewed trend of infections across the demographic divide.
  • For the aggregated global statistics, a delay of 22 days since 10th January when China reported the first cases accompanied the start of the critical exponential phase. The daily growth rate was barely 5% from 1st February and by 21st March had doubled to a daily rate of 10% with the estimation equation

This brings the projected global estimate, assuming the business-as-usual scenario, into the range of 1.5 – 1.7 million cases by 7th April 2020 – introducing a new week in April likely to end with a record of 2 million cases worldwide. Note: Factoring in the observed global rates, the business-as-usual scenario for Kenya/Africa would be to work with a daily growth rate of 20% in the exponential phase.

  • The highest daily growth rates over the critical exponential phase for the 14 countries have been recorded in the USA (26%), Israel (21%), Spain (20%) and the UK (20%). The other rates have been ranging from 11% in Hong Kong, 13% in Italy, and 17-18% in the rest of the group. Note: Factoring in the observed country rates, the pessimistic scenario for Kenya/Africa would be to prepare to confront a daily growth rate which could even reach 30% in the critical exponential phase.
  • There were a few exceptions to the dominant exponential trend in this group of countries during their critical growth phases. The reported daily growth in the total cases in China traced more of a linear trend with a slope of about 3000. The growth in the total reported cases in Switzerland and Sweden also defied the exponential trend, rather tracing quadratic curves. Italy’s cases have also recently tended more towards a quadratic curve. Note: Containing coronavirus cases following linear and quadratic curves may be easier than the stealthy but eventually explosive exponential curves. This implies that these countries are likely to see a significant lowering in the rates of infections. Latecomers that are not managing the pandemic well can, therefore, overtake the present record holders.

Nashon J. Adero

Nashon, a geospatial expert, lecturer and trained policy analyst applies dynamic models to complex adaptive systems. He is a youth mentor on career development and the founder of Impact Borderless Digital.