Fight the COVID-19
WITH the rapid march of the coronavirus, COVID-19, across the planet, and WHO sounding out the alarm to the world to brace for a pandemic, it is timely to consider how mathematicians could quickly model the growth pattern of the number of positive cases arising from an infectious virus and obtain fairly accurate short-term estimates which can be extremely valuable for decision makers.
It is well-known in the mathematics literature that growth curve modelling, which does not include hypothetical kinetics of various aspects of the virus transmission, but which rests solely on the basis of reported number of positive cases, provides the simplest, yet rigorous, means of studying the dynamics of the infected population. Scientists have argued that the basic epidemiological parameters associated with the spread of a virus were not fully understood.
Thus, mathematical models, which were constructed on the basis of the hypothetical kinetics of various aspects of the virus transmission were partially pedagogical and may not include many of the factors, which could and should be included in more realistic models. These pitfalls could be avoided by studying the growth patterns which typically show an "S" shape. There are suitable mathematical models that can be used to study it. In general, these models can be grouped into four types.
Type I models are useful tools of analysis in situations where an initial slow growth rate, followed by a rapid growth rate, could again exhibit slow growth rate over a time interval due to some inhibiting factors, which include the impact of a country's response to slowing down the spread of the virus.
Type I models have well-known shortcomings, including;
(a) the data may not reflect the true numbers; and
(b) the inability to predict abrupt changes because epidemiological data are not incorporated.
However, with respect to point (a) if the data reported were true reflections of the size of the infected population, then the data would be sufficient to study the dynamics of the population, given that a reported case is independent of the mode of transmission and/ or the behaviour or type of the infected individual.
It is best to stress here that Type I models simply attempt to fit a suitable pre-defined function to observed positive cases, new or accumulative. Assumptions about incubation period, mode of transmission, and epidemiological and biological assumptions, are not incorporated in Type I models. These assumptions are incorporated in more advanced models designated as Type II, Type III and Type IV.
Just for the sake of illustration, suppose that one of the small Pacific islands is at the initial stages of the growth of COVID-19 positive population and data have been captured in the first 12 weeks (three months) showing the following number of new confirmed cases (note that the data shown are actual data associated with a virus reported in the paper Nonlinear Dynamics of the Observed HIV-Positive Population Size written by the author in 2003 and published in the medical journal Public Health Dialog):
We want to have estimations of new cases for each week in the next three months. It is ideal to attempt Type I modelling for small Pacific Islands. Some advantages of Type I models are simplicity and elegance because they depend only on initial observed data.
A well-known Type 1 model is the Logistic Curve, which can capture the beginning of the "S" growth. An infected population typically shows such an initial growth pattern, as shown above by the observed data: Incorporating the initial observed data, the Logistic Curve shows that after an initial rapid growth between the 6th week and the 15th week, the estimated number of new cases stabilises toward the end of the 24th week (6th month) on the assumption that the preventive measures are working.
While what has been presented is hypothetical, the modelling provides a scientifically accepted pattern of growth that mimics the "S" shape typically induced by an infectious virus – an initial steady trend followed by a steep rise followed by a steady trend. If the trend stabilises, it means that there are still infected people, but preventive measures suppress any growth. If the trend drops, it means that either a cure has been found, or preventive measures are working effectively, or the virus has killed all its hosts in quarantine.
A silver lining to this doom-and-gloom scenario is that we have the analytical tool to help our country fight off COVID-19 if, if not when, it arrives to our shore.
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