Modelling COVID Cases and Fatalities using Engineering Concepts

I have been asked several times if my chemical engineering study was useful at all in economies. As I always used to answer, the concepts are omnipresent. Basic learning mechanisms are the same, complex models just add more sophistication to it. To show an example here, I utilized the concepts learnt in chemical kinetics, especially when I had learnt to the model reaction in series and parallel. When I observed the data of daily COVID cases and cumulative deaths, I realized a resemblance between it and what I had learnt in engineering models. So, I decided to explore.

Upon exploration, I found that indeed such models like SIR (basic one's) are being used by academicians to stimulate the spread of the novel corona-virus. They then build upon it to make it more realistic but at its core, the concepts are similar to those used in chemical kinetics.

The simplest SIR models simulate how the three states susceptible (S), infectious (I) and recovered (R) interact. This is similar to the series reaction studied in kinetics, i.e., S -> I -> R, with the rate of reactions being proxies for the rate of transmission from one state to another. Thus, we can mathematically derive and the number of cases etc and obtain the parameters which maximize the fit. These models assume homogeneity of the population and that the entire population exists in the mentioned states and that recovered patients get immune and so on. However, one can extract further information and incorporate it into our models to make them more realistic. For the time being I would take a simple model but with few modifications over the simplest SIR.

My model takes the structure as mentioned below,

Here, E is the population exposed to infection,

I is the population infected,

Q is asymptomatic population. For simplicity, I haven't tracked the infections and recoveries from this track. Such models under the name SAIRD are available in the literature but for the sake of simplicity I have not used it,

R is the population recovered,

and D is the population fatalities.

k1, k2, gamma and delta are constant parameters which determine the rate of transmission from one state to the other. k1 and k2 could be further broken down to bring more sophistication but here I have taken them as exogenous parameters.

Having set up the basic framework, I now would shift to pen and paper to derive mathematical formulations for [I] and [D] as a function of t by solving the set of differential equations analytically. This has been done because [I] would give the number of infections at each time (or day), for which we also have the data. Thus, by fitting theoretical results to observed per day cases we can obtain the parameter values.

The math is given below,

The downside of this model is that it doesn't capture the slow growth in the beginning. The rise of cases starts slowly but the model rockets early thereby had deviations. To take this into account, the early some-days (different for different countries) for [I] are modelled as linearly growing, which matched observations. The deaths during this period have been calculated accordingly. The math used for this is shown below,

a and b were taken to match the data for each country. 

Now, having obtained [I] and [D] for the entire period, the theoretical data and observational data is made to fit. To make a comparison feasible, for each country data was plotted as (number of cases observed on that day/total number of cases in the taken period) and similarly (number of deaths observed on that day/total number of cases in the taken period) on Y-axis vs days on the X-axis. The theoretical data was transformed accordingly as well.

The modelling is done for the United Kingdom, Italy, Germany and Spain.

• United Kingdom

• Italy

• Spain

• Germany

It can be seen that with the given parameter values, the model fits the data quite nicely.

The parameter values obtained, which maximize the fit, for each of the countries are tabularized below,

As per the model, the rate at which deaths rise in increasing order is Italy < Germany < Spain < the UK. It should be kept in mind that the rate of increase doesn't imply the total deaths follow that order. This is because total deaths are the product of rate and total infectious population.

Similar is the case for the infectious population which increases at the rate of k1 but reduces at the rate of k2 and delta*k2.